Univerza v Ljubljani Fakulteta za matematiko in fiziko Oddelek za fiziko Marko ˇ nidariˇc STABILNOST KVANTNE DINAMIKE Disertacija Mentor: izr. prof. Tomaˇz Prosen Zunanji mentor: prof. Thomas H. Seligman Ljubljana, 2004 University of Ljubljana Faculty of Mathematics and Physics Physics Department Marko ˇ nidariˇc STABILITY OF QUANTUM DYNAMICS Dissertation advisor: Prof. Tomaˇz Prosen external advisor: Prof. Thomas H. Seligman Ljubljana, 2004 Abstract The stability of quantum systems to perturbations of the Hamiltonian is studied. This stability is quantified by the fidelity, an overlap of an ideal state obtained by the unperturbed evolution, and a perturbed state obtained by the perturbed evolution, both starting from the same initial state. Dependence of fidelity on the initial state as well as on the dynamical properties of the system is considered. In particular, systems having a chaotic or regular classical limit are analysed. The fidelity decay rate is given by an integral of the correlation function of the perturbation and is thus smaller the faster correlation function decays. Quantum systems with a chaotic classical limit can therefore be more stable than regular ones. If the perturbation can be written as a time derivative of another operator, meaning that the time averaged perturbation vanishes, fidelity freezes at a constant value and starts to decay only after a long time inversely proportional to the perturbation strength. In composite systems stability of entanglement to perturbations of the Hamiltonian is analysed in terms of purity. For regular systems purity decay is shown to be independent of Planck’s constant for coherent initial states in the semiclassical limit. The accelerated decoherence of macroscopic superpositions is also explained. The theory of fidelity decay is applied to the stability of quantum compu-tation and an improved quantum Fourier transform algorithm is designed and shown to be more stable against random perturbations. Keywords: quantum stability, fidelity, purity, quantum chaos, de-coherence, entanglement, quantum computation, quantum Fourier transformation, the kicked top, Jaynes-Cummings model. PACS numbers: 03.65.Sq Semiclassical theories and applications 03.65.Yz Decoherence; open systems; quantum statistical methods 03.67.Lx Quantum computation 03.67.Mn Entanglement production, characterization, and manipulation 03.67.Pp Quantum error correction and other methods for protection against decoherence 05.45.-a Nonlinear dynamics and nonlinear dynamical systems 05.45.Mt Quantum chaos; semiclassical methods 05.45.Pq Numerical simulations of chaotic systems 42.50.Ct Quantum description of interaction of light and matter; related experiments i Povzetek V tem delu smo ˇstudirali stabilnost kvantnih sistemov na motnje v hamiltonovi funkciji. Mera za stabilnost je kvantna zvestoba, ki je kvadrat skalarnega produkta med moteno valovno funkcijo, dobljeno z evolucijo generirano z moteno hamiltonovo funkcijo, in nemoteno valovno funkcijo, dobljeno z nemoteno evolucijo, obakrat pa zaˇcnemo iz istega zaˇcetnega stanja. Raziskali smo odvisnost pojemanja kvantne zvestobe od zaˇcetnega stanja in od dinamike, pri ˇcemer smo si podrobno ogledali primera sistemov s kaotiˇcno in regularno klasiˇcno limito. Hitrost pojemanja kvantne zvestobe je sorazmerna z integralom kvantne korelacijske funkcije motnje in je torej manjˇsa, tem hitreje pojema korelacijska funkcija. Navidez presenetljiva posledica tega je, da so lahko kvantni sistemi s kaotiˇcno klasiˇcno limito bolj stabilni kot pa regularni sistemi. Zanimiv poseben primer so motnje, ki jih lahko zapiˇsemo kot odvod nekega drugega operatorja in katerih ˇcasovno povpreˇcje je niˇc. Za takˇsne motnje so kvantni sistemi bistveno bolj stabilni, saj kvantna zvestoba dolgo ˇcasa ostane konstantna in zaˇcne pojemati ˇsele po zelo dolgem ˇcasu. Raziskali smo tudi stabilnost kvantne prepletenosti v sklopljenih sistemih. Za regularne sisteme in lokalizirana stanja smo pokazali, da je dekoherenca bistveno hitrejˇsa, ˇce je zaˇcetno stanje superpozicija makroskopsko loˇcenih stanj, t.i. stanja Schrodignerjeve maˇcke. Teoretiˇcna spoznanja o pojemanju kvantne zvestobe smo uporabili za izboljˇsanje stabilnosti kvantnega raˇcunanja. Sestavili smo izboljˇsan algoritem za kvantno Fourierovo transformacijo, ki je bolj stabilen ob prisotnosti nakljuˇcnih motenj. Kljuˇcne besede: kvantna stabilnost, kvantna zvestoba, kvantna ˇcistost, kvantni kaos, dekoherenca, prepletenost, kvantno raˇcunanje, kvantna Fourierova transformacija, brcana vrtavka, model Jaynes-Cummings iii Acknowledgement I would like to thank my advisor Prof. Tomaˇz Prosen for giving me the opportunity to work in his group, guiding me and showing me the way. It has been a great pleasure to work with him, sharing the excitement of new discoveries. Initially small project gradually grew, eventually surpassing all our expectations and resulting in the present thesis. Thanks also go to Prof. Thomas H. Seligman for coopera-tion on parts of this thesis and for warm hospitality during my stay in Cuernavaca. In particular, he has been a very careful reader of the final manuscript, pointing out english mistakes and confusing statements. v Contents I Razˇsirjeni povzetek ix 1.1 Zgodovina ......................................... ix 1.2 Sploˇsna motnja ...................................... xii 1.3 Motnja s ˇcasovnim povpreˇcjem niˇc ........................... xx 1.4 Sklopitev z okolico .................................... xxvi 1.5 Uporaba: kvantno raˇcunanje .............................. xxxiii 1 Introduction 1 1.1 Historical Overview .................................... 3 1.1.1 Quantum Fidelity ................................ 3 1.1.2 Classical Fidelity ................................. 7 1.1.3 Entanglement ................................... 9 1.2 Outline .......................................... 10 2 Fidelity 11 2.1 Models for numerics ................................... 15 2.1.1 The Kicked Top ................................. 15 2.1.2 The Jaynes-Cummings model .......................... 17 2.2 Average Fidelity ..................................... 18 2.2.1 Time Averaged Fidelity ............................. 18 2.2.2 State Averaged Fidelity ............................. 22 3 General Perturbation 23 3.1 Mixing Dynamics ..................................... 23 3.1.1 Long Time Behaviour .............................. 26 3.2 Regular Dynamics .................................... 30 3.2.1 Action-angle Operators ............................. 31 3.2.2 Coherent Initial States .............................. 33 3.2.3 Random Initial State ............................... 37 3.3 Time Scales ........................................ 39 3.3.1 Illustration with Wigner Functions ....................... 42 3.3.2 Coupled Kicked Tops .............................. 43 4 Special Case: Zero Time Averaged Perturbation 49 4.1 Mixing Dynamics ..................................... 51 4.1.1 The Plateau .................................... 51 4.1.2 Beyond the Plateau ............................... 54 vii viii CONTENTS 4.2 Regular Dynamics .................................... 57 4.2.1 The Numerical Model .............................. 60 4.2.2 Coherent Initial State .............................. 62 4.2.3 Echo Resonances for Coherent Initial States .................. 68 4.2.4 Random Initial States .............................. 72 5 Coupling with the Environment 77 5.1 Reduced Fidelity, Purity Fidelity ............................ 77 5.1.1 Inequality Between Fidelity, Reduced Fidelity and Purity Fidelity ..... 78 5.1.2 Uncoupled Unperturbed Dynamics ....................... 79 5.1.3 Linear Response Expansion ........................... 80 5.1.4 Numerical Illustration .............................. 82 5.2 Mixing Dynamics ..................................... 85 5.3 Regular Dynamics .................................... 86 5.3.1 Beyond Linear Response ............................. 86 5.3.2 The Jaynes-Cummings Model .......................... 90 5.4 Separation of Time Scales ................................ 93 5.4.1 Fast Mixing Environment ............................ 94 5.4.2 Fast Regular Environment ............................ 97 5.5 Freeze in a Harmonic Oscillator ............................. 99 5.5.1 The Plateau .................................... 101 5.5.2 Beyond the Plateau ............................... 103 5.6 Decoherence for Cat States ............................... 105 6 Application: Quantum Computation 111 6.1 General Framework .................................... 112 6.1.1 Time Dependent Perturbations ......................... 113 6.2 Quantum Fourier Transformation ............................ 113 7 Conclusion Appendices A Spin Wigner functions 119 120 121 B Coherent State Expectation Values 123 I Razˇsirjeni povzetek I.1 Zgodovina Navidezni paradoks ireverzibilnosti je zaposloval fizike vse od rojstva termodinamike. Osnovo drugega zakona termodinamike, ki pravi, da lahko entropija sistema s ˇcasom le naraˇsˇca, je postavil Boltzmann (1872). To naraˇsˇcanje entropije je v navideznem nasprotju z mikroskopskimi zakoni, ki so reverzibilni. Ce je nek potek dogodkov reˇsitev enaˇcb gibanja, potem je tudi obraten potek, ko ˇcas “teˇce” nazaj, tudi moˇzna reˇsitev enaˇcb gibanja. Torej, ˇce pade kozarec na tla in se razbije, potem je tudi obraten potek moˇzen, ˇcrepinje se na tleh sestavijo in “skoˇcijo” nazaj na mizo. Takega dogodka seveda ˇse nikoli ni nihˇce opazil, kar nam ravno pripoveduje drugi zakon termodinamike, razbit kozarec na tleh ima veˇcjo entropijo kot pa cel kozarec na mizi. Ta tako imenovani paradoks reverzibilnosti ponavadi pripisujejo Josefu Loschmidtu, ki ga je beˇzno omenil na koncu ˇclanka (Loschmidt, 1876) v katerem je poskuˇsal ovreˇci Maxwellsko porazdelitev hitrosti molekul v termiˇcnem ravnovesju. Lotil se je tudi Boltzmannovega zakona o naraˇsˇcanju entropije in v zaljuˇcku ˇclanka zapisal: “...Das ber¨uhmte Problem, Geschehenes ungeschehen zu machen, hat damit zwar keine Losung...”. Boltzmann (1877) je ˇze naslednje leto odgovoril na oˇcitke Loschmidta in poudaril kljuˇcno vlogo verjetnosti pri interpretaciji drugega zakona ter pomen zaˇcetnega pogoja, ki je dejansko odgovoren za “zlom simetrije” na obrat ˇcasa. Velika veˇcina zaˇcetnih pogojev je takih, pri katerih se kozarec ne bo sam od sebe sestavil, tako da ˇce izberemo neko nakljuˇcno zaˇcetno stanje, se bo sistem z veliko verjetnostjo obnaˇsal tako kot veleva entropijski zakon. Prva zapisana omemba paradoksa ireverzibilnosti pa ni bila Loschmidtova, temveˇc jo je objavil William Thompson (pozneje Lord Kelvin), dasiravno je moˇzno, da je Loschmidt omenil paradoks v privatnih diskusijah z Boltzmannom. Boltzmann in Loschmidt sta bila namreˇc dobra prijatelja in sta delala na Fizikalnem Inˇstitutu na Dunaju v ˇcasu, ko je bil direktor Joˇzef Stefan. Leta 1869 se je Boltzman sicer preselil v Graz, a je v obdobju 1873-1876 ponovno delal na Dunaju. Za podrobnejˇso Boltzmannovo biografijo glej Cercignani (1998). V ˇclanku iz leta 1874 je Thompson (Thompson, 1874) podal presenetljivo moderen pogled na ireverzibilnost in drugi zakon termodinamike. Ob opisovanju prevajanja toplote in izenaˇcevanja temperatur je zapisal: “...If we allowed this equalization to proceed for a certain time, and then reversed the motions of all the molecules, we would observe a disequalization. However, if the number of molecules is very large, as it is in a gas, any slight deviation from absolute precision in the reversal will greatly shorten the time during which disequalization occurs... Furthermore, if we take account of the fact that no physical system can be completely isolated from its surroundings but is in principle interacting with all other molecules in the universe, and if we believe that the number of these latter molecules is infinite, then we may conclude that it is impossible for temperature-differences to arise spontaneously...”. Zanimivo ix x I. Razˇsirjeni povzetek vpraˇsanje, ki se zastavlja je, kakˇsen je ta “kratki ˇcas” v katerem bo obrnjena evolucija ˇse sledila prvotni? Odgovor na to nam da zvestoba (fidelity). Zvestobo ali Loschmidtov odmev dobimo kot razdaljo med dvema stanjema dobljenima z malenkost razliˇcnima evolucijama, pri ˇcemer obakrat zaˇcnemo iz istega zaˇcetnega stanja. Zanimivi sta tako kvantna kot klasiˇcna zvestoba. Pri kvantni zvestobi je razdalja med stanjema doloˇcena s skalarnim produktom med obema kvantnima stanjema, pri klasiˇcni pa s prekrivalnim integralom med ustreznima gostotama v faznem prostoru. Predmet tega dela bo izkljuˇcno kvantna zvestoba. Kljub temu, da je zvestoba pomembna za razumevanje osnov termodinamike, pa je originalna motivacija za njen ˇstudij priˇsla iz kvantne mehanike. Kvantna mehanika je gotovo najuspeˇsnejˇsa fizikalna teorija 20. stoletja. ˇ e samo industrija polprevodnikov je izrednega pomena. Kljub temu pa vse te naprave, ki sicer temeljijo na principih kvantne fizike, ne izkoriˇsˇcajo posameznih kvantov, temveˇc so mezoskopske narave. ˇ ele v osemdesetih letih 20. stoletja je eksperimentalna fizika toliko napredovala, da so postali mogoˇci zares kvantni poskusi s posameznimi kvanti. Relativno mlado podroˇcje fizike, ki se ukvarja s tem kako takˇsne kvantne pojave izkoristiti za procesiranje in prenaˇsanje informacij je kvantna informacijska teorija (Nielsen & Chuang, 2001). Kvantna informacijska teorija je zdruˇzila kvantno mehaniko z informacijsko teorijo in raˇcunalniˇstvom. Ob uporabi kvantnih sistemov je mogoˇca npr. teleportacija kvantnih stanj, varna kominikacija preko javnih povezav ali pa kvantno raˇcunanje. Nobeno izmed teh stvari ni mogoˇce doseˇci samo s klasiˇcnimi sredstvi. Kvantno raˇcunalniˇstvo je zaenkrat ˇse v povojih in omejeno na laboratorijske eksperimente z le nekaj qubiti. Glavno oviro predstavljajo nezaˇzelene motnje v evoluciji, ki naredijo proces raˇcunanja izredno nestabilen. ˇ eleli bi si torej karseda stabilen kvantni raˇcunalnik. Mera za stabilnost je kvantna zvestoba, torej moramo najprej razumeti kako se obnaˇsa kvantna zvestoba, ˇce jo hoˇcemo izboljˇsati. Prve ˇstudije kvantne zvestobe so bile narejene v okviru kvantnega kaosa. Kaos v klasiˇcnih sistemih ponavadi opredelimo kot eksponentno obˇcutljivost na variacijo zaˇcetnih pogojev, oddaljenost dveh bliˇznjih orbit s ˇcasom eksponentno naraˇsˇca. Ker so naravni zakoni v osnovi kvantni, se postavi vpraˇsanje, kako je s to karakterizacijo kaosa v kvantni mehaniki. S tem se ukvarja podroˇcje kvantnega kaosa. Kljub dinamiˇcni definiciji klasiˇcnega kaosa pa so bile karakteristike klasiˇcnega kaosa najprej raziskane v stacionarnih lastnostih kvantnih sistemov kot so lastne vrednosti in lastne funkcije (Haake, 1991; Stockmann, 1999). Prvi je kvantno zvestobo F (t) ˇstudiral Peres (1984), F(t) := |(?(t)|??(t))|2 = |(?(0)|U0†(t)U?(t)|?(0))|2, (I-i) kjer je |?(t)) = U0(t)i)(0)) stanje ob ˇcasu t dobljeno z nemoteno evolucijo U0(t) = e-iHot/h in stanje |??(t)} = U?(t)|?(0) | dobljeno z moteno evolucijo U?(t) = e-iHst/—h, kjer je motena hamiltonova funkcija H? = H0 + ?-V. Kljuˇcen korak je bil, da je opazoval obˇcutljivost na spremembo dinamike, to je na variacijo hamiltonove funkcije, in ne na spremembo zaˇcetnega pogoja, kot je to v navadi v klasiˇcni mehaniki. Zaradi nakoliko ponesreˇceno izbranih zaˇcetnih pogojev pa ni priˇsel do najbolj sploˇsnih zakljuˇckov. Po Peresovem ˇclanku je veˇc kot deset let vladalo zatiˇsje, ponovno pa je zadeva postala zanimiva ob koncu ’90 (Ballentine & Zibin, 1996). Najprej je skupina Pastawskega izvedla serijo NMR eksperimentov, v katerih so merili kvantno zvestobo oziroma polarizacijski odmev (Levstein et al. , 1998; Usaj et al. , 1998; Pastawski et al. , 2000). Na teoretiˇcnem podroˇcju sta Jalabert & Pastawski (2001) semiklasiˇcno izpeljala t.i. Lyapunovo pojemanje kvantne zvestobe v kaotiˇcnih sistemih, F (t) ~ e-?t, ki je neodvisno od jakosti motnje ?. Ta reˇzim je prisoten za dovolj velike motnje ? in je popolnoma klasiˇcne narave (Veble & Prosen, 2004). Lyapunov reˇzim pojemanja kvantne zvestobe so ˇstudirali tudi Cucchietti et al. (2002a). Kot zelo uspeˇsen pristop k ˇstudiju kvantne zvestobe se je izkazal razvoj po korelacijskih funkcijah motnje V (Prosen, 2002), glej tudi Prosen & ˇ nidariˇc (2001). Pojemanje kvantne zvestobe v kaotiˇcnih sistemih so ˇstudirali I.1. Zgodovina xi tudi Jacquod et al. (2001) in Cerruti & Tomsovic (2002). Glede na jakost motnje ? lahko kvantna zvestoba pojema kot eksponentna funkcija ali pa kot gaussova za zelo majhne motnje. Vsi ti reˇzimi, skupaj z obnaˇsanjem v primeru regularne dinamike, so bili podrobno obdelani glede na pojemanje korelacijskih funkcij (Prosen & ˇ nidariˇc, 2002). Jedro priˇcujoˇcega dela se napaja predvsem iz tega ˇclanka, v katerem je bila prviˇc uporabljena tudi klasiˇcna zvestoba. Po teh ˇclankih je bilo v manj kot 3 letih objavljenih veˇc kot 50 del na temo kvantne in klasiˇcne zvestobe. Za podrobnejˇsi pregled literature glej zgodovinski pregled v angleˇski verziji (Section 1.1). Na tem mestu bom na kratko podal le pregled klasiˇcne zvestobe, ki sicer ni tema tega dela, je pa zelo pouˇcna primerjava med klasiˇcno in kvantno zvestobo, saj je, kot bomo videli, obnaˇsanje obeh lahko zelo razliˇcno. Klasiˇcna zvestoba Klasiˇcna zvestoba je definirana kot prekrivalni integral med nemoteno in moteno gostoto v faznem prostoru (Prosen & ˇ nidariˇc, 2002), Fklas(t) := dx?0(x,t)?s(x,t), (I-ii) kjer je ?s(x, t) gostota dobljena z moteno evolucijo, ?0(x, t) pa z nemoteno, x pa je toˇcka v faznem prostoru. V kaotiˇcnih sistemih ostane klasiˇcna zvestoba blizu ena vse do ˇcasa tv ? ln(?/?)/? (Benenti & Casati, 2002; Benenti et al. , 2003b), kjer je ? velikost zaˇcetnega stanja, ? pa eksponent Lyapunova. Za tem se zaˇcne eksponentno pojemanje Fklas(t) = e-A(f-^) (Veble & Prosen, 2004). V sistemih z difuzijo nastopi po koncu reˇzima Lyapunova difuzijsko pojemanje. Asimptotsko pojemanje klasiˇcne zvestobe za velike ˇcase je doloˇceno z najveˇcjo lastno vrednostjo Perron-Frobeniusovega operatorja in je lahko eksponentno ali pa potenˇcno (Benenti et al. , 2003b). Hitrost pojemanja klasiˇcne zvestobe v kaotiˇcnih sistemih ni odvisna od jakosti motnje ? v nobenem od naˇstetih reˇzimov, so pa od ? odvisni ˇcasi ob katerih le ti nastopijo. Kvantna zvestoba na drugi strani je odvisna od ?, tako da se ujemanje med klasiˇcno in kvantno zvestobo konˇca ˇze po kratkem Ehrenfestovem ˇcasu tE, ki naraˇsˇca le logaritemsko s Planckovo konstanto, tE ? -ln—h. V regularnih sistemih (npr. integrabilnih) in za lokalizirana zaˇcetna stanja je pojemanje klasiˇcne zvestobe lahko dveh tipov (Benenti et al. , 2003a). Ce motnja povzroˇci predvsem spremembo frekvence torusana katerem je zaˇcetno stanje, potem je pojemanje enostavno posledica balistiˇcnega oddaljevanja nemotenega in motenega paketa. Casovna skala na kateri zvestoba pojema skalira kot ? 1/?, sama funkcijska oblika pa je odvisna od oblike zaˇcetnega stanja, npr. za koherentna stanja je ta gaussova. Na drugi strani, ˇce motnja spremeni predvsem obliko torusa (frekvenca pa ostane v povpreˇcju enaka), je pojemanje klasiˇcne zvestobe potenˇcno, in sicer Fklas(t) ? 1/(?t)d za sistem z d prostostnimi stopnjami. Katerega od obeh tipov pojemanja bomo imeli, je odvisno od motnje ter od lege zaˇcetnega stanja, ne pa od jakosti motnje ?. V grobem lahko reˇcemo, da bo pojemanje potenˇcno za motnje katerih ˇcasovno povpreˇcje je niˇc, za take z neniˇcelnim povpreˇcjem pa bo pojemanje balistiˇcno. Kot bomo videli, bomo tudi za kvantno zvestobo imeli dva razliˇcna reˇzima pojemanja glede na ˇcasovno povpreˇcje motnje, vendar z eno veliko razliko. Medtem ko se pri klasiˇcni zvestobi spremeni le funkcijska odvisnost, glede na to ali je povpreˇcna motnja niˇc ali ne, se pri kvantni zvestobi zgodijo veliko bolj dramatiˇcne stvari. Ce je povpreˇcna motnja niˇc, kvantna zvestoba “zamrzne” na konstantni vrednosti-platoju, in ponovne zaˇcne padati ˇsele po zelo dolgem ˇcasu. Od sedaj naprej bo beseda tekla le o kvantni zvestobi, klasiˇcna bo na nekaj mestih sluˇzila le za primerjavo. xii I. Razˇsirjeni povzetek I.2 Sploˇsna motnja Pojemanje kvantne zvestobe je odvisno od tega ali je ˇcasovno povpreˇcje motnje niˇc ali pa je neniˇcelno. V sploˇsnem bo ˇcasovno povpreˇcje motnje neniˇcelno in to bo prvi primer, ki si ga bomo ogledali. Zaradi enostavnosti se bomo v celotnem delu omejili na propagatorje U(t), ki se dajo zapisati kot potenca nekega osnovnega propagatorja za “en korak”. Tak osnovni propagator bomo zapisali kot U0 za nemoteno evolucijo in U? za moteno, propagator za t korakov pa bo enostavno U0t in U?t. Casovni indeks t bo torej tekel po celih ˇstevilih, ki bodo merila ˇstevilo osnovnih enot ˇcasa. S tem formalizmom zajamemo dve pomembni skupini sistemov, take s ˇcasovno periodiˇcno hamiltonovo funkcijo, v tem primeru je U0 kar propagator ˇcez eno periodo, in pa ˇcasovno neodvisne hamiltonske sisteme, pri katerih je ˇcas t sicer zvezen indeks. Z naˇsim formalizmom dobimo zvezen primer z enostavno zamenjavo vsot ˇcez diskreten indeks t z integralom po ˇcasu. Poljubno moteno evolucijo U? (za en korak) lahko zapiˇsemo kot U? = U0exp(-iV?/h), (I-iii) kjer je ? brezdimenzijska jakost motnje, V pa generator motnje. Kvantna zvestoba - sploˇsna teorija Kvantno zvestobo (I-i) lahko zapiˇsemo kot priˇcakovano vrednost operatorja odmeva M?(t), F(t) = |?(0)|M?(t)|?(0))|2, M?(t) := U0- tU?t. (I-iv) Na nekaj mestih bo ugodno zapisati kvantno zvestobo kot sled produkta zaˇcetne gostotne matrike ?(0) in gostotne matrike po odmevu ?M(t) ?(0) = |?(0))(?(0)|, ?M(t) = M?(t)?(0)MJ(t), (I-v) kot F (t) = tr?(0)?M(t)] = |f(t)|2, f(t) = tr[?(0)M?(t)]. (I-vi) Tukaj smo vpeljali tudi kompleksno amplitudo zvestobe f(t). Operator odmeva je ugodno izraziti z operatorji motnje v interakcijski sliki. Ce imamo nek operator A, potem je njegova interakcijska slika A (t) enaka * A(t) := U0- tAU0t. (I-vii) Cˇe to naredimo za V, lahko operator odmeva M?(t) zapiˇsemo kot M?(t) = exp (-iV(t - 1)?/h) · · · exp (-iV(0)?/h) = Texp (-i?(t)?/h), (I-viii) kjer je T operator ˇcasovne ureditve, ?(t) pa je t-1 ?(t):= ^ V(j). (I-ix) j=0 V zveznem primeru imamo preprosto ?(t) := t 0 dt'V(t'). Ce uporabimo Baker-Campbell-Hausdoffovo (BCH) formulo eAeB = exp (A + B + [A, B]/2 + · · ·), lahko operator odmeva zapiˇsemo z eno ek- sponentno funkcijo, M?(t) = exp- i ?(t)? + -?(t)? 2 + ···, (I-x) ?A(t) lahko smatramo tudi kot Heisenbergovo sliko dobljeno z nemoteno evolucijo. I.2. Sploˇsna motnja xiii s hermitskim operatorjem ?(t) i t-1t-1 ?(t) := —h ^^ [V(j),V(k)]. (I-xi) j=0 k=j To zadnjo obliko operatorja odmeva (I-x) bomo uporabili za praktiˇcno vsa teoretiˇcna razglabljanja. Ce ima operator motnje V dobro klasiˇcno limito, potem imata tudi operatorja ?(t) in ?(t) dobri klasiˇcni limiti, ki ju lahko preprosto dobimo, ˇce zamenjamo kvantni komutator s Poissonovim oklepajem, (-i/—h)[•,•] › {•,•}. Praktiˇcno glavni rezultat, ki vsebuje veˇcino fizike, se da dobiti ˇze s preprostim razvojem operatorja odmeva po potencah ?. Ce tak razvoj vstavimo v izraz za kvantno zvestobo, dobimo v najniˇzjem redu F(t) = 1 - —h L C(j, k) + O(?4) = 1 - —h { - (?(t))2} + O(?4), (I-xii) kjer smo priˇcakovano vrednost v zaˇcetnem stanju oznaˇcili kratko (A) := {?|A|?), C (j, k) pa je kvantna korelacijska funkcija motnje, C(j, k) := (V(j)V(k)) - (V(j)) {V(k)). (I-xiii) V zveznem primeru ˇcasovno neodvisnih hamiltonskih sistemov je ta preprost izraz enak 1-F(t) = 2Jtft C(t',t")dt'dt". (I-xiv) —h 00 Zadeva ni niˇc drugega kot izraz za kvantno zvestobo v pribliˇzku linearnega odziva, tudi sama oblika zelo spominja na razne Green-Kubove formule. Padec kvantne zvestobe 1-F(t), oziroma “disipacija kvantne informacije”, je dana z integralom korelacijske funkcije motnje. Do sem vse lepo in prav, preseneˇcenje pa nastopi, ko pomislimo kako se te korelacijske funkcije obnaˇsajo za razliˇcne sisteme. V kaotiˇcnih sistemih (oz. takih z lastnostjo meˇsanja) korelacijske funkcije pojemajo proti niˇc in to v sploˇsnem hitreje tem bolj kaotiˇcen je sistem. To pomeni, da bo dvojni integral korelacijske funkcije tem manjˇsi ˇcim bolj kaotiˇcen je sistem in bo torej kvantna zvestoba pojemala poˇcasneje tem bolj kaotiˇcna je dinamika! To se zdi nekako v nasprotju s klasiˇcnimi priˇcakovanji. Nenazadnje smo videli, da klasiˇcna zvestoba v kaotiˇcnih sistemih pojema z Lyapunovim koeficientom in torej pojema hitreje ˇcim bolj kaotiˇcen je sistem. Na tem mestu naj povemo le, da s kvantno-klasiˇcno korespondenco seveda ni niˇc narobe, vso to “kontradiktorno” obnaˇsanje se namreˇc zgodi ˇsele po koncu ujemanja med kvantno in klasiˇcno zvestobo. Poglejmo si sedaj podrobneje kvantno zvestobo v kaotiˇcnih sistemih. Kaotiˇcni sistemi Za kaotiˇcno dinamiko z lastnostjo meˇsanja bodo korelacijske funkcije v sploˇsnem pojemale, tako da lahko definiramo transportni koeficient ? 2 t^oo 2t j=1 t?2(t)) ? := -C(0) + ^C(j) = lim---- , (I-xv) kjer smo predpostavili, da je korelacijska funkcija homogena v ˇcasu, C (j) = C(k, k + j). V semi-klasiˇcni limiti se bo kvantna korelacijska funkcije ujemala s klasiˇcno in ? bo skonvergiral za ˇcase xiv I. Razˇsirjeni povzetek veˇcje od nekega klasiˇcnega ˇcasa meˇsanje tmix. Dvojni integral korelacijske funckije bo za t > tmix naraˇsˇcal linearno s ˇcasom, kvantna zvestoba pa bo v pribliˇzku linearnega odziva linearno pojemala, F(t) = 1-2(?/h)2?t. - (I-xvi) Ob nekoliko stroˇzji zahtevi, da imamo t.i. n-toˇckovno meˇsanje, to pomeni daje korelacijska funkcija {V(j1) ¦ ¦ ¦ V(j2m)) v limiti j2m-j1 ->¦ oo znatna le kadar so indeksi sparjeni, z razlikami j2k-j2k-1 < lahko izraˇcunamo kvantno zvestobo do vseh redov v ? in dobimo eksponentno pojemanje F (t) = exp(-t/?m), ?m = 2 —h2 (I-xvii) kjer lahko v semiklasiˇcni limiti uporabimo kar klasiˇcni ?cl dobljen iz klasiˇcne korelacijske funkcije. Hitrost pojemanja kvantne zvestobe ?m je v tem reˇzimu odvisna od jakosti motnje ?, torej se kvantna in klasiˇcna zvestoba ne bosta ujemali. Enak reˇzim eksponentnega pojemanja je bil izpeljan tudi na precej drugaˇcen naˇcin ob uporabi Fermijevega zlatega pravila (Jacquod et al. , 2001; Cerruti & Tomsovic, 2002), zato vˇcasih to eksponentno pojemanje zvestobe imenujejo tudi reˇzim Fermijevega zlatega pravila. 10 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 0 500 1000 1500 2000 2500 3000 x=(8S) t Slika I.1: Kvantna zvestoba za kaotiˇcno dinamiko v brcani vrtavki in tri razliˇcne jakosti motnje, ? = 5x 10-4, 1 x 10-3 in 3 x 10-3 (polna, ˇcrtkana in pikˇcasta krivulja). Ravna ˇcrta je teoretiˇcna napoved (I-xvii). Da bi tudi numeriˇcno preverili napovedano eksponentno pojemanje, smo z numeriˇcno simulacijo izraˇcunali kvantno zvestobo v sistemu brcane vrtavke. Propagator za osnovni korak je U0 = U(?) = exp(-i?Sy/2)exp (-i?^S j, (I-xviii) kjer so Sx,y,z standardni spinski operatorji, [Sk,Sl] = i?klmSm. Velikost spina S doloˇca efektivno Planckovo konstanto h = 1/S. Moten sistem je dobljen tako, da za ? spremenimo parameter ?, U? := U(? + ?), kar pomeni, da je klasiˇcna limita operatorja motnje V kar Vcl = 2 z, (I-xix) I.2. Sploˇsna motnja xv ˇce je z koordinata na sferi, torej normalizirana klasiˇcna vrtilna koliˇcina. Na sliki I.1 so numeriˇcni rezultati za S = 4000 in ? = 30, ko je ustrezna klasiˇcna dinamika popolnoma kaotiˇcna. Vidimo, da se teoretiˇcna napoved (I-xvii) ujema z numeriko. Za izraˇcun teoretiˇcnega ?m smo uporabili klasiˇcno izraˇcunan ?cl = 0.00385. Poudariti je tudi potrebno, da v tem reˇzimu kvantna zvestoba ni odvisna od zaˇcetnega stanja. Obnaˇsanje za dolge case Eksponentno pojemanje kvantne zvestobe smo izpeljali ob predpostavki kvantnega meˇsanja, to je da korelacijska funkcija pojema proti 0. Ta predpostavka pa je upraviˇcena le v limiti neskonˇcno dimenzionalnega Hilbertovega prostora, M ->¦ oo. Za konˇcne dimenzije Hilbertovega prostora M bomo po dovolj dolgem ˇcasu imeli efekte zaradi konˇcnega prostora. Integral korelacijske funkcije ne bo skonvergiral h ?, temveˇc bo naraˇsˇcal linearno s ˇcasom. Korelacijska funkcija C(j, k) bo imela namreˇc plato zaradi konˇcnega M. V takem primeru lahko definiramo povpreˇcno vrednost korelacijske funkcije C, 1 t~1 C= lim 2 Y. C(j,k). (I-xx) t^oo t2 j,k=0 Za kvantne sisteme s kaotiˇcno klasiˇcno limito lahko C izrazimo s klasiˇcnim ?cl, in sicer C = 4?jr. (I-xxi) Dvojna korelacijska vsota (I-xiv) bo torej naraˇsˇcala kot 2?clt le do ˇcasa tH, ko bo postal del Ct 2 dominanten. Cˇe izenaˇcimo oba prispevka, dobimo za tH tH = -JV. (I-xxii) To je t.i. Heisenbergov ˇcas, kije dan kar z dimenzijo Hilbertovega prostora, ki skalira kot M ~ —h~d. Za ˇcase veˇcje od tH je torej priˇcakovati kvadratiˇcno naraˇsˇcanje korelacijske vsote s ˇcasom. To smo preverili z numeriˇcnim eksperimentom, katerega rezulati so na sliki I.2. Vsi podatki za brcano vrtavko so enaki kot prej, le S = 1500. Vidimo, da do ˇcasov okoli tH dvojna korelacijska vsota sledi ?cl, nato pa se zaradi konˇcnega M in s tem neniˇcelnega C— zaˇcne naraˇsˇcanje C—t2. Za dovolj dolge ˇcase bo torej kvantna zvestoba pojemala kvadratiˇcno in ne linearno kot pri eksponentnem pojemanju. Ob dodatni predpostavki, ki se izkaˇze za upraviˇceno v semiklasiˇcni limiti kaotiˇcnih sistemov, da so matriˇcni elementi motnje porazdeljeni po gaussovi porazdelitvi, lahko izpeljemo pojemanje kvantne zvestobe do vseh redov v ?. Rezultat je gaussovo pojemanje F (t) = exp (-(t/?p) 2), ?p = J4?c-?. (I-xxiii) Takˇsno pojemanje nastopi ˇsele po Heisenbergovem ˇcasu tH, do tam pa je pojemanje eksponentno (I-xvii). Podoben efekt zaradi konˇcne dimenzije M kot smo imeli pri korelacijski funkciji, bomo imeli tudi pri sami zvestobi. Za velike ˇcase kvantna zvestoba ne bo padla popolnoma k vrednosti 0, temveˇc bo fluktuirala okoli neke asimptotske vrednosti doloˇcene s konˇcno velikostjo prostora. To asimptotsko vrednost zvestobe lahko izraˇcunamo kot ˇcasovno povpreˇceno vrednost kvantne zvestobe F. Raˇcun je enostaven, z rezultatom F = 1 , (I-xxiv) xvi I. Razširjeni povzetek o 0.1 0.01 0.001 10000 100000 Slika 1.2: Korekcijska vsota a(t) = J2jk=oC(j,k)/2t (polna krivulja) in ustrezna klasična vsota Cel (t) = J2j k=oCci(j, k)/2t (črtkana krivulja) za kaotično brcano vrtavko. Linearno naraščajoča premica daje asimptotsko naraščanje zaradi Č / 0. pri čemer je /x med 1/2 in 1, odvisno od začetnega stanja. Zgornji rezultat velja, če je začetno stanje sestavljeno iz veliko lastnih stanj nemotenega propagatorja. 0.1 0.01 0.001 0.0001 1000 S(dt) Slika 1.3: Kvantna zvestoba za 5 < 5P in kaotično brcano vrtavko. Črtkana in pikčasta krivulja sta za 5 = 1 • 10~6 in 5 • 10~6, polna krivulja pa je teoretično gaussovo pojemanje (I-xxiii). Kvantna zvestoba bo torej pojemala le do asimptotskega platoja F. Če hočemo opaziti gaussovo pojemanje zvestobe (I-xxiii), mora le ta biti še vedno blizu 1 ob Heisenbergovem času. Od tod vidimo, da bomo imeli gaussovo pojemanje le za dovolj majhne jakosti motnje, ko je eksponentno pojemanje dovolj počasno. Mejni čp je I.2. Sploˇsna motnja xvii Spet smo teoretiˇcno napoved o gaussovem pojemanju za ? < ?p preverili z numeriˇcno simulacijo taiste brcane vrtavke pri S = 1500. Rezultati so na sliki I.3. Reˇzim gaussovega pojemanja kvantne zvestobe za kaotiˇcne sisteme imenujemo vˇcasih tudi perturbacijski reˇzim. Regularni sistemi Z regularnimi sistemi imamo v mislih sisteme, pri katerih bo dvojna vsota korelacijske funkcije naraˇsˇcala kvadratiˇcno s ˇcasom, torej kadar ima korelacijska funkcija neniˇcelni plato C. Pri tem imamo v mislih sisteme, kjer je C neniˇceln zaradi dinamike in ne zaradi konˇcne dimenzije M, kot je to pri kaotiˇcnih sistemih. Neniˇcelni C bomo imeli tipiˇcno v integrabilnih sistemih ali pa v sistemih blizu integrabilnosti. Ce oznaˇcimo s tave (klasiˇcni) ˇcas po katerem povpreˇcna korelacijska funkcija (I-xx) skonvergira k C, potem lahko za t > tave definiramo povpreˇcno motnjo V, V = lim - V V (k) = lim ?(t) . (I-xxvi) Oznaˇcimo z |?k) lastna stanja nemotenega propagatorja U0. Neniˇcelni matriˇcni elementi V so enaki kar diagonalnim elementom operatorja V, = Y,Vkk|?k)(?k|, (I-xxvii) V k Vkk|?kX?k|, kjer so Vkk = {?k|V|?k). Povpreˇcna korelacijska funkcija ni niˇc drugega kot C = {V 2) - (V)2 . Za (?kV tave operator odmeva aproksimiramo z Ms(t) = exp (-iV?t/h), t > tave, (I-xxviii) pri ˇcemer smo zanemarili viˇsji red ?2?(t), ki postane pomemben za velike ˇcase t ~ 1/?2. Kot bomo videli, se bo zvestoba spreminjala na krajˇsi ˇcasovni skali ~ 1/?, tako da je ta pribliˇzek upraviˇcen. Izraˇcun kvantne zvestobe lahko v regularnih sistemih precej poenostavimo, ˇce uvedemo akcijsko kotne spremenljivke j in ?. Lastne vektorje ustreznih kvantnih akcijskih operatorjev lahko oznaˇcimo z vektorjem kvantnih ˇstevil n, torej |n), z lastnimi vrednostmi —hn. Ker je operator povpreˇcne motnje V diagonalen, to pomeni da komutira z nemoteno evolucijo, ga lahko izrazimo samo z akcijskimi operatorji. Priˇcakovano vrednost operatorja odmeva, ki je enaka amplitudi zvestobe f(t) (I-vi), lahko tedaj zapiˇsemo f(t) = 53exp (-i?tV(hn)/—h)Dp(hn), Dp(hn) = (n|?|n), (I-xxix) kjer je ? gostotna matrika zaˇcetnega stanja. Ce sedaj v vodilnem semiklasiˇcnem redu nadomestimo kvantni V z ustrezno klasiˇcno limito v ter zamenjamo vsoto po kvantnih ˇstevilih n z integralom po akcijah, dobimo semiklasiˇcni izraz f(t) =—h ~d fddj exp f- ? tv(j)\ dp(j), (I-xxx) kjer je dp(j) klasiˇcna limita Dp(hn). Pribliˇzek z integralom ˇcez akcije velja do ˇcasov ~ 1/?. Za razliko od kaotiˇcnih sistemov je kvantna zvestoba za regularno dinamiko odvisna od zaˇcetnega stanja. Ogledali si bomo dva primera, koherentna in nakljuˇcna zaˇcetna stanja. Pri nakljuˇcnih zaˇcetnih stanjih je vsak koeficient v razvoju nakljuˇcno gaussovo ˇstevilo. xviii I. Razširjeni povzetek Koherentna stanja Za koherentna stanja ima klasična gostota v prostoru akcij dp(j) gaussovo obliko, dp(j) = (n/vr)d/2|detA|1/2exp(-(j -j*).A(j-j*)/h). (I-xxxi) Matrika A dimenzije d x d nam poda obliko začetnega koherentnega paketa v sistemu z d pro-stostnimi stopnjami, j* pa lego središča začetnega stanja. Integral v izrazu za amplitudo zvestobe (I-xxx) lahko izračunamo z metodo stacionarne faze. Rezultat za kvantno zvestobo je F(t)=exp{-(t/rr)2}, 1 2h Tr = r d^v'-A^v'7 kjer smo z v' označili vektor odvodov po akcijah, izvrednoten v začetni točki j* j = dv(j*) dj (I-xxxii) (I-xxxiii) Kvantna zvestoba za koherentna stanja v regularnih sistemih pojema gaussovo. Za numerično 10u 10"' 10"' 10"' 10"' 10 10 •tic----------1----------------1------------- **\ \ \ + + + -fr- +¦ _j- + + % Z + i: +4 X + + + + + + : \ / ++ +++ ++ tb " +¦ ++ _ T ++ T ++ ++ ¦ *t ++ __ 0 50 100 150 200 250 300 350 400 t Slika 1.4: Kvantna zvestoba za regularno brcano vrtavko in koherentno začetno stanje, 5 = 0.0025. Polna črta je teoretična napoved (I-xxxv), plusi pa rezultat simulacije. Navpična črta podaja teoretično vrednost za periodo utripanja tb (I-xxxvi). demonstracijo smo ponovno vzeli brcano vrtavko, vendar tokrat z a = 0.1, tako da smo v regularnem režimu. Dinamika za a = 0 je enostavna rotacija za 7r/2 okoli osi y, tako da povprečno motnjo in njen odvod v tem približku zlahka izračunamo, v = \(l-f), (I-xxxiv) Akcijska spremenljivka j je kar koordinata y. Ob upoštevanju A = 1/(1 — y2) takoj dobimo teoretično napoved (I-xxxii) F(t)=exp(-52St2y2(l-y2)/8). (I-XXXV ) 1.2. Splošna motnja xix 0.1 0.01 0.001 1000 x=Sdt Slika 1.5: Kvantna zvestoba za regularno brcano vrtavko in naključno začetno stanje, 5 = 0.01. Ravna pikčasta črta je asimptotsko potenčno pojemanje ~ t~l, črtkana nihajoča krivulja je numerika za eno naključno stanje, polna črta je numerika za povprečje čez cel Hilbertov prostor, pikice pa so razlika s teoretičnim rezultatom (I-xxxviii). Na sliki 1.4 so numerični rezultati za S = 100 in koherentno stanje pri y = 0.77. Ujemanje s teoretično napovedjo je odlično. Opazimo pa, da se po določenem času kvantna zvestoba ponovno povrne do približno 1. Ta efekt je opazen le v sistemih z eno prostostno stopnjo (d = 1), kjer imamo lahko utripanje zaradi dveh bližnjih frekvenc nemotenega in motenega torusa. Teoretična napoved za ta čas utripanja % je za naš primer 27T tb = — > (I-xxxvi) 2vr 6y' in se lepo ujema z eksperimentom. V več kot eni dimenziji, zaradi v splošnem inkomenzurabilnih frekvenc, utripanja ne bo. Naključna stanja Za naključno začetno stanje lahko v vodilnem semiklasičnem redu kvantno zvestobo povprečimo kar čez cel akcijski prostor, torej vzamemo konstanten dp(j) = (2i:h)d/V, s prostornino klasičnega faznega prostora V. Integral po akcijah lahko ponovno izvrednotimo z metodo stacionarne faze. Asimptotsko je pojemanje zvestobe za velike čase potenčno F(t) (27T) 3d V2 5t, (I-xxxvii) Za naš enostavni numerični primer regularne brcane vrtavke (enak primer kot za koherentna stanja) lahko za a = 0 eksplicitno izračunamo akcijski integral (I-xxx) in dobimo F(i) = m er fti^e^VŠŠi) (I-xxxviii) kjer je erfi(z) = -^j= JqZ e * dt kompleksna erf funkcija z limito lim^^oo lerfi^e171"/4^)! = 1, h kateri se oscilirajoče približuje. Vidimo, da imamo v primerjavi s semiklasičnim rezultatom (I-xxxvii) xx I. Razˇsirjeni povzetek nihajoˇce popravke v obliki funkcije erfi. Takˇsni nihajoˇci popravki so sploˇsna lastnost stacionarne faze na konˇcnih prostorih. I.3 Motnja s ˇcasovnim povpreˇcjem niˇc Za nedegeneriran spekter nemotenega propagatorja U0 smo videli (I-xxvii), da ima povpreˇcna motnja V le diagonalne matriˇcne elemente. To pa pomeni, da lahko tako motnjo V definiramo s ˇcasovnim odvodom novega operatorja W kot V = W(1) - W(0), W(t) := U0- tWU0t. (I-xxxix) V primeru zveznega ˇcasa je ta definicija V := (d/dt)W = —i[H0, W]. Hitro vidimo, da so matriˇcni elementi operatorja W v bazi nemotenega propagatorja preprosto W* = exp(i(?j- ?))-1 , (I-xl) kjer so e-i^ lastne vrednosti nemotenega propagatorja U0. V tem razdelku se bomo ukvarjali z motnjami katerih ˇcasovno povpreˇcje je niˇc, torej imajo niˇcelne diagonalne matriˇcne elemente, in jih lahko izrazimo z operatorjem W. Takˇsna motnja je lahko npr. posledica simetrije, vendar pa morajo matriˇcni elementi V naraˇsˇcati gladko vstran od diagonale, ˇce hoˇcemo, da W nima singularnosti. Ce izrazimo koliˇcine z W namesto z V, se marsikateri izraz poenostavi. Vsota ?(t) je preprosto ?(t) = W(t)-W(0). (I-xli) Za motnje z neniˇcelnim ˇcasovnim povpreˇcjem smo potrebovali le operator ?(t), da smo izraˇcunali pojemanje kvantne zvestobe. Sedaj temu ni tako. Za dolge ˇcase bo postal pomemben tudi ˇclen ?(t) (I-xi) v izrazu za operator odmeva. Ta operator lahko zapiˇsemo kot kjer je ?(t) = ?R(t)--[W(0),W(t)], R:=-[W(0),W(1)], (I-xlii) t-1 ?R(t) := Y, R(t), R(t) := U0- tRU0t. (I-xliii) t'=0 Ker sedaj norma operatorja ?(t) (I-xli) ne naraˇsˇca s ˇcasom, bo kvantna zvestoba do nekega ˇcasa t2 ? 1/?, ko postane pomemben tudi ?(t), “zamrznila” na neki ne-padajoˇci vrednosti, dani z ? Fplat(t) = exp - ? {W(t)-W(0)} (I-xliv) To konstantno vrednost, za katero bomo videli da ni odvisna od ˇcasa, bomo poimenovali plato zvestobe. Vidimo tudi, da bomo imeli plato ne glede na to kakˇsna je dinamika ali zaˇcetno stanje. Mi si bomo ogledali primera kaotiˇcne in regularne dinamike, ko lahko eksplicitno izraˇcunamo vse koliˇcine. Za ˇcase veˇcje od t2 bo postal dominanten ˇclen ?2?(t)/2, ki zaradi svoje enostavne oblike (I-xlii) nudi zanimivo interpretacijo. Ce zanemarimo ˇclen [W(0), W(t)], ki ne bo pomemben za velike ˇcase, potem je ?(t) = ?R(t) tak, kakor je bil ?(t) v primeru sploˇsnje motnje (I-ix) z neniˇcelnim V, ˇce le V zamenjamo z R. Pojemanje kvantne zvestobe po platoju bo torej enako kot “renormalizirana” zvestoba, ki je posledica motnje R z jakostjo ?R := ?2/2. Uporabimo lahko kar vso teorijo iz prejˇsnjega razdelka. 1.3. Motnja s časovnim povprečjem nič xxi Kaotična dinamika Plato zvestobe 0.01 0.001 Slika 1.6: Plato kvantne zvestobe za kaotično brcano vrtavko in jakost motnje 5 = 10~2. Zgornja krivulja je za koherentno začetno stanje, spodnja pa za naključno. Vodoravni črti sta teorija (I-li), navpična črta pa podaja lego teoretičnega L2 (I-lvi). S črnimi pikami je označen numerični izračun klasične zvestobe za koherentno začetno stanje. V primeru kaotične dinamike (Prosen & Znidarič, 2004) nam računanje poenostavi pojemanje korelacij za čase večje od tmiX. Za vrednost platoja (I-xliv) lahko tako zapišemo Fplat^\G(5/h)(exp(iW5/h))\2, kjer je G (z) generatrisa definirana s klasičnim povprečjem G(z) := (exp(-izw))cl = - J ddqddp exp{-izw(q,p)}, (I-xlv) (I-xlvi) w je klasična limita"'" operatorja W. Za koherentna začetna stanja (CIS) nam drugi del v Fp\at samo vrti fazo, tako da imamo F§H = \G(S/h)\2, (I-xlvii) neodvisno od lege začetnega koherentnega stanja. Na drugi strani dobimo za naključna začetna stanja F§H = \G{6/h)\*- (I-xlviii) Med obema velja zanimiva univerzalna relacija Fi^t — (-^piat) • V približku linearnega odz 2 tmix faktoriziramo prispevka ?(t) in ?R(t), tako da imamo F(t) = Fplat ?2 /exp f-i2—h?R(t) 2 t>t2,tmix. (I-lii) Postopek izraˇcuna priˇcakovane vrednosti drugega ˇclena je popolnoma enak kot za sploˇsno motnjo, le renormalizirano motnjo R in jakost ?R = ?2/2 moramo vzeti. Prvi faktor, Fplat, je odvisen od zaˇcetnega stanja, kot smo izraˇcunali v prejˇsnjem razdelku, drugi pa ne. Za ˇcase manjˇse od Heisenbergovega ˇcasa je pojemanje eksponentno - ? 2?Rt\, t tH. (I-lv) Ce primerjamo obe moˇzni obliki asimptotskega pojemanja zvestobe z vrednostjo platoja (I-xlix), lahko ocenimo ˇcas t2, ko se plato zvestobe konˇca, "min ?R ?,?R?2 t2wmin Hcl,cl2 . (I-lvi) Ker lahko imamo eksponentno pojemanje le ˇce je t2 < tH, nam to ob zahtevi Fplat ~ 1 (torej ?/h < 1), ter da je eksponentna zvestoba ob tH majhna, da pogoj za ˇstevilo prostostnih stopenj sistema d > 2, ˇce hoˇcemo v semiklasiˇcni limiti imeti eksponentno pojemanje. V sistemih z eno prostostno stopnjo eksponentnega pojemanja torej ni mogoˇce videti. Zaradi tega smo tokrat za numeriˇcni model izbrali sklopljen sistem dveh brcanih vrtavk z unitarnim propagatorjem U = exp (-i?Sz1Sz2) exp (-i?Sy1/2) exp (-i?Sy2/2), (I-lvii) 1.3. Motnja s časovnim povprečjem nič xxiii za motnjo pa smo vzeli W = A®1 + 1®A, A = SI/2S2. (I-lviii) Velikost spina je S = 100 za obe vrtavki, sklopitev pa je e = 20, tako da smo v kaotičnem režimu (klasično). Na sliki 1.7 je primerjava med tremi izračuni kvantne zvestobe: teoretično napovedjo za eksponentno (I-liii) oz. gaussovo (I-lv) pojemanje, "renormalizirano" dinamiko, pri kateri smo vzeli za operator motnje R z jakostjo 52/2, ter neposrednim numeričnim izračunom. Pri teoretičnih napovedih smo uporabili klasično izračunan ctr = 9.2 • 10~3 in pa teoretično izračunano vrednost platoja Fpiat. 0 500 10001500 2000 2500 3000 3500 4000 4500 5000 t Slika 1.7: Kvantna zvestoba po platoju za sistem dveh brcanih vrtavk. Za močno motnjo 5 = 7.5 • 10~2, slika (a), dobimo eksponentno pojemanje (I-liii), za majhno motnjo 5 = 2 • 10~2, slika (b), pa gaussovo (I-lv). Polna črta je numerika, krogci so numerika za "renormalizirano" dinamiko z motnjo R in jakostjo 52 /2, prekinjena krivulja pa je teorija. Regularna dinamika V primeru regularne dinamike bomo uporabili akcijsko-kotne spremenljivke (Prosen & Znidarič, 2003b), kot smo to storili že za splošno motnjo. Vrednost platoja zvestobe bo dana z operatorjem S(t). V vodilnem redu lahko uporabimo kar klasično limito operatorja W, torej w(j,6), tako da imamo X(t)^w(j,0 + wt)-w(j,0). (I-lix) Podrobnejši račun pokaže, da je za t > ti vrednost platoja neodvisna od časa, tako da lahko časovno odvisen S (t) nadomestimo z njegovim časovnim povprečjem, kar nam da amplitudo kvantne zvestobe /plat = /exp (i^w(j, 0)J j j^exp [~^w(3,* (I-lx) kjer je povprečje vzeto čez klasični fazni prostor, časovno povprečje pa smo nadomestili s povprečjem po kotih. Kot smo povedali, je ta izraz upravičen za t > ti, ki je za naključna stanja ti ~ 5°h°, za koherentna stanja pa nekoliko daljši, ti = — min (m ¦ LIA lfl,Tm 4 m^o -1/2 oc ft-1/2, (I-lxi) xxiv I. Razˇsirjeni povzetek kjer je minimum vzet ˇcez vse nihajne naˇcine m spremenljivke w, matrika ? pa je matrika odvodov frekvenc po akcijah, ?jk := ??j(j*)/?jk. Za koherentna stanja ima ˇcas t1 nazorno razlago. To je t.i. regularni Ehrenfestov ˇcas, po katerem se zaˇcetni paket velikosti —h, zaradi razliˇcnih frekvenc torusov na katerih se razteza, razleze na klasiˇcno velikost ~ —h0. To je tudi ˇcas, do katerega se ujemata klasiˇcna in kvantna zvestoba. Vidimo tudi, da t1 divergira, torej uporaba ˇcasovnega povpreˇcja za fplat ni upraviˇcena, za sisteme z ? = 0, to je za harmonski oscilator. Ta primer bomo obravnavali posebej. Pojemanje zvestobe po platoju bo na drugi strani doloˇceno z ?(t), ki ga bomo za dolge ˇcase nadomestili z Rt oziroma klasiˇcnim rt. Uporabili bomo teorijo za sploˇsne motnje na tej “renor-malizirani” dinamiki. Poglejmo si kar konkreten primer za brcano vrtavko. Nemoten propagator vzamemo kar U0 = exp (-i?S2z/2S), (I-lxii) tako da je akcijska spremenljivka koordinata z, j = z, za motnjo pa vzamemo V = Sx/S, v = y1 - j2 cos ?. (I-lxiii) Klasiˇcna limita operatorja W je sin (? - ?/2) w(j,?) = -/1- j2 i /2, (I-lxiv) s frekvenco ? = ?j. Plato zvestobe Za koherentna zaˇcetna stanja (CIS) nam da klasiˇcno povpreˇcje v izrazu (I-lx) kar vrednost argumenta na mestu paketa, tako da takoj dobimo Fcis ^ 1 plat = (2?)2d d ?exp-—hw(j*,?) . (I-lxv) Izraz je podoben tistemu za kaotiˇcno dinamiko (I-xlvii), le da smo tam imeli v formuli za generatriso povpreˇcje ˇcez cel fazni prostor, medtem ko imamo sedaj le povpreˇcje ˇcez kote, akcija pa je doloˇcena z lego zaˇcetnega stanja j*. Plato je sedaj odvisen od lege zaˇcetnega koherentnega stanja. Za naˇso motnjo w (I-lxiv) je integral elementaren, z rezultatom ?S V j , (I-lxvi) kjer je J0 niˇcta Besselova funkcija z razvojem za majhne argumente J02(x) = 1 - x2/2 + • • •. Naj omenimo, da je rezultat integrala Besselova funkcija za vse motnje z le enim Fourierovim nihajnim naˇcinom. Za nakljuˇcna zaˇcetna stanja (RIS) je klasiˇcno povpreˇcje v izrazu za fplat kar fazno povpreˇcje, kar nam da RIS FR plat (2?) d 2 (2 V diCIS (j) (I-lxvii) Torej je plato zvestobe za RIS povpreˇcje platoja za CIS. Tukaj pa se lahko pojavi teˇzava, ˇce imamo v akcijskem prostoru toˇcke, kjer je frekvenca enaka niˇc, ?(j) = 0 (oz. m ¦ ? = 0), saj bomo imeli 1.3. Motnja s časovnim povprečjem nič XXV tedaj v izrazu za w (I-hriv) divergentne člene. Za naš numerični primer se to zgodi v j = 0. Rešitev je enostavna, le spomniti se je treba, da je integral dejansko zamenjava za vsoto čez hn in da so frekvence semiklasični izrazi za razlike lastnih vrednosti k (oz. u> = 0) dejansko nič. Torej moramo namesto integrala zapisati vsoto, divergentne člene pa preprosto izpustiti, 77.RIS r^ -fplat — 1 m-u>(hn)^2Trk E ^it(^) (I-lxviii) Na sliki 1.8 so numerični rezultati za S = 1600, a = 1.1 in 5S = 0.32, ter koherentno in naključno začetno stanje. Zaradi bližnje divergence pri j = 0 je plato veliko nižji za naključno začetno stanje kot pa za koherentno. Za koherentno začetno stanje opazimo tudi resonance, ob katerih je zvestoba blizu 1. Te resonance odmeva so posebnost sistemov z eno prostostno stopnjo in lokaliziranih začetnih stanj. Pojavijo se ob večkratnikih osnovnega časa tr, 2vr huj'(j*) (I-lxix) S=1600,dS=0.32 10 100 1000 t/S' 10000 Slika 1.8: Plato kvantne zvestobe za regularno vrtavko. Na levi je plato za koherentno začetno stanje, na desni pa za naključno začetno stanje. Vodoravna črta je teorija za koherentno (I-lxvi) in naključno stanje (I-lxviii). Krogci na sliki za koherentno stanje podajajo ustrezno klasično zvestobo, ki sledi kvantni le do ti oc l/\/fi. Po platoju Po koncu platoja je pojemanje kvantne zvestobe dano z "renormalizirano" dinamiko. Če je plato znaten, je potrebno upoštevati tudi vrednost platoja, ki pa ga ne moremo faktorizirati tako kot za kaotične sisteme. Tukaj se bomo omejili na primer, ko je plato majhen in ga lahko zanemarimo. Za koherentna začetna stanja je pojemanje po platoju gaussovo, F(t)^exp{-(t/rrr)2}, 7~ — 52 8h ¦A-lf> (I-lxx) xxv i I. Razširjeni povzetek z odvodom klasične limite r časovnega povprečja operatorja R. Čas L2, ko se plato neha, lahko za koherentna stanja ocenimo kot L2 = min {const K1'2 8 2, const U 1'25 1}. Za naključna začetna stanja bo pojemanje po L2 ~ l/# potenčno, ftran\d h F (t) X ( — j , tran = COnst X —^. (I-lxxi) (I-lxxii) Rezultati numerične simulacije za S = 1600, a = 1.1 in 5S = 0.064, skupaj s teoretičnimi napovedmi, so na sliki 1.9. 0.1 r 0.01 0.001 r 100 1000 Slika 1.9: Kvantna zvestoba po platoju za regularno vrtavko. Levo je gaussovo pojemanje za koherentno začetno stanje, desno pa potenčno za naključno stanje. V obeh primerih je prekinjena črta teoretična napoved. 1.4 Sklopitev z okolico Kvantna zvestoba ni vedno najbolj primeren kriterij stabilnosti. Ponavadi je sklopitev sistema z okolico neizogibna, četudi je majhna. V takem primeru je evolucija unitarna le, če obravnavamo naš centralni sistem skupaj z okolico. Pri tem bomo z okolico imenovali del celotnega sistema, ki nas ne zanima, centralni sistem pa je tisti, ki nas zanima. Hilbertov prostor celotnega sistema je tenzorski produkt H = Hc He, kjer indeks "c" pomeni centralni podsistem, "e" pa okolico. Kvantna zvestoba, ki je skalami produkt valovnih funkcij na celotnem sistemu, prav gotovo ne bo prava mera za stabilnost, če nas zanima le stabilnost centralnega sistema. Zvestoba je namreč lahko majhna tudi, če se samo del valovne funkcije, ki se nanaša na okolico razlikuje, centralni del pa je enak. Zaradi teh pomanjkljivosti bomo vpeljali dve novi meri za stabilnost, reducirano zvestobo (reduced fidelity) (Znidarič & Prosen, 2003) in čistost odmeva (purity fidelity) (Prosen & Seligman, 2002). Najprej definirajmo reducirane gostotne operatorje sistema, ki vsebujejo vso informacijo o stanju centralnega sistema, pc(t) := tre[p(t)], pf(t) := tre[pM(t)], (I-lxxiii) I.4. Sklopitev z okolico xxvii kjer je tre[»] sled ˇcez okolico. Zaˇcetno stanje sistema bo vedno produktno stanje |?(0)> = |?c(0)) |?e(0)) =: |?c(0); ?e(0)>. (I-lxxiv) Ce smo kvantno zvestobo definirali kot sled na celotnem prostoru (I-vi), F (t) = tr[?(0)? M (t)], potem je smiselni kriterij za stabilnost centralnega podsistema sled produkta reduciranih matrik -reducirana zvestoba, FR(t) := trc[?c(0)?Mc(t)]. (I-lxxv) Reducirana zvestoba FR(t) meri “bliˇzino” med prvotno reducirano matriko in reducirano matriko po odmevu. Ena najbolj ˇcudnih lastnosti kvantne mehanike so prepletena stanja. Zanimivo vpraˇsanje je, ali je neko stanje prepleteno ali ne. Za ˇcista stanje je kriterij za prepletenost preprost. Ce je ˇcistost (purity) I enaka 1, potem je stanje produktno, sicer pa je prepleteno. Cistost je definirana kot I(t) := trc[?2c(t)], ?c(t) := tre[|?(t)>(?(t)|]. (I-lxxvi) Zanimiva je tudi ˇcistost odmeva FP, ki nam pove ali je stanje po odmevu ?M(t) prepleteno, FP(t) := trc[{?Mc(t)}2]. (I-lxxvii) Vse tri koliˇcine, zvestoba, reducirana zvestoba in ˇcistost odmeva merijo stabilnost na motnje. Ce je motena evolucija enaka nemoteni, so vse enake 1, sicer pa manjˇse. Zvestoba meri stabilnost celotnega sistema, reducirana zvestoba le centralnega podsistema, ˇcistost odmeva pa le faktorizacijo stanja (prepletenost). Zdi se, da je zvestoba najstroˇzji kriterij stabilnosti. To domnevo potrdi neenakost, ki jo lahko dokaˇzemo za te tri koliˇcine ( ˇ nidariˇc & Prosen, 2003; Prosen et al. , 2003a). Velja naslednja neenakost F2 (t) < FR2 (t) < FP (t). (I-lxxviii) Zanimiva netrivialna posledica te neenakosti je, da bomo v primerih ko zvestoba zamrzne, torej za motnje z niˇcelnim ˇcasovnim povpreˇcjem, imeli enak efekt tudi za reducirano zvestobo in za ˇcistost odmeva. Pogosto je zaˇzeljena nemotena evolucija taka, v kateri sta centralni podsistem in okolica nesklo-pljena, torej je U0 faktoriziran U0 = Uc®Ue. (I-lxxix) V tem primeru lahko reducirano zvestobo in ˇcistost odmeva ˇse poenostavimo. Reducirano zvestobo lahko, namesto kot primerjavo med zaˇcetnim stanjem in stanjem po odmevu, smatramo tudi kot primerjavo med nemotenim in motenim reduciranim stanjem, FR(t) = trc[?c(0)?Mc(t)] = trc[?c(t)?f(t)]. (I-lxxx) Podobno lahko vidimo, da je ˇcistost odmeva v tem primeru nesklopljene nemotene dinamike enaka kar ˇcistosti motene dinamike I(t), FP(t) = trc[{?Mc(t)}2] = trc[{?cf(t)}2] = I (t). (I-lxxxi) Ker je ta poseben primer fizikalno ˇse posebej zanimiv, bomo v numeriˇcnih simulacijah vedno vzeli nesklopljeno nemoteno dinamiko. xxviii I. Razširjeni povzetek Linearni odziv Velikokrat nam za analizo stabilnosti zadostuje že linearni odziv, saj so zanimive vrednosti ravno v okolici 1. Podobno kot zvestobo lahko do najnižjega reda v d razvijemo tudi -Fr(L) in -Fp(t), l-F(t) = (^) l-FR(t) = (-) l-Fp(i) = 2^0 (E(i)(l-pc)®(l-pe)E(i)), (I-lxxxii) kjer sta pc,e reducirani matriki začetnega produktnega stanja p(0) = pc<%> Pe- Kaotični sistemi 10u 10"' -icr 10"' 10"' 10 10 V i N^fR2 - V. F2 - 0 500 10001500 2000 2500 3000 3500 4000 4500 5000 t Slika 1.10: Pojemanje F2(t),F^(t) in I (t) za kaotični sistem dveh brcanih vrtavk. Polna črta je teoretična napoved (I-lxxxiii), kjer smo uporabili klasično izračunan ac\ = 0.056. Vodoravni črti podajata asimptotsko vrednost reducirane zvestobe in čistosti, 1/4002 in 1/200. Za kaotične sisteme bodo vse tri količine pojemale na začetku linearno s časom. Za dovolj dolge čase bo matrika S (i) blizu naključni matriki. V tem primeru lahko ocenimo razliko -Fr(L) — F (t) ~ 1/We in Fp(t) — F2(t) ~ 1/Wc + 1/We. Torej lahko za dovolj velike dimenzije obeh podsistemov A/"c,e pričakujemo, da bodo vse tri količine pojemale enako. Za kvantno zvestobo smo izpeljali, da ta pojema eksponento (I-xvii), torej bo Fp(t) « Fl(t) « F2(t) = exp (-2i/rm), (I-lxxxiii) s rm = h2/252e = 30, za kota pa ?c)e = ?/2.1. Velikost spina je S = 200, motnja pa V = 1 Sz®Sz, (I-lxxxvi) S2 ® z jakostjo ? = 8- 10-4. Na sliki I.10 vidimo, da je ujemanje s teorijo odliˇcno. Regularni sistemi Za regularne sisteme se bomo omejili na produktna koherentna zaˇcetna stanja. Oblika koherentnega paketa za centralni podsistem je doloˇcena z matriko ?c dimenzije dc, za okolico pa z ?e dimenzije de, v sistemu z dc + de prostostnimi stopnjami. Postopek za izraˇcun FR(t) in FP(t) je podoben kot za F(t). Za dovolj dolge ˇcase lahko operator odmeva aproksimiramo z M5(t) ->¦ exp(-i?V—t/h), V nadomestimo s klasiˇcnim v(j) in zadevo razvijemo okoli lege zaˇcetnega paketa. Za reducirano zvestobo je rezultat gaussova funkcija, podobno kot za zvestobo, FR(t) = exp -CRt2, CR = -—h v—c ?-1v—c , (I-lxxxvii) ( -—h2CRt2 J, CR = -—h v—c ?-1v—c) , kjer je matrika v'c dana z v—, := ?v(j*) , (I-lxxxviii) ˇce razdelimo vektor akcij na del ki pripada centralnemu podsistemu in del za okolico, j =: (jc, je). S ˇcistostjo odmeva je nekoliko veˇc dela, rezultat pa je FP(t) = vdet +(?t)2 u,, u:= ?-1v—c e?-s1v—e c, (I-lxxxix) kjer je matrika drugih odvodov v—ce dimenzije dc x de dana z v—c c v—c e ), v—(j):=w(jh. (I-xc) Matrika u, ki doloˇca pojemanje ˇcistosti odmeva, je dimenzije dc x dc. Determinanta pod korenom (I-lxxxix) je polinom v (?t)2, in sicer reda med 1 in dc, odvisno od matrike u. V sistemih z 1 + de prostostnimi stopnjami je u kar ˇstevilo, ˇcistost odmeva pa je enostavno FP(t) = ^ 1 2, dc = 1. (I-xci) Za numeriˇcno demonstracijo bomo vzeli isti model kot za kaotiˇcni reˇzim, a s parametri ?c>e = 0, ?c = ?/2.1 ter ?e = ?/^7. Velikost spina je S = 100, motnja jakosti ? = 0.01 pa S4Sz®2 V = —S2 S2. (I-xcii) Rezultati numeriˇcne simulacije so na sliki I.11. XXX I. Razširjeni povzetek 10000 Slika 1.11: Pojemanje F(L),Fr(L) in I(t) za sistem brcanih vrtavk v regularnem režimu in za produktno koherentno začetno stanje. Teoretični napovedi (polni krivulji) za -Fr(L) (I-lxxxvii) in F(t) (I-xxxii) se popolnoma prekrivata z numeriko (simboli). Teorija za čistost (I-xci) je narisana s črtkano krivuljo. Zamrznitev za harmonski oscilator Za motnje, katerih časovno povprečje je enako nič, smo že na osnovi neenakosti napovedali, da bomo imeli podobno zamrznitev kot za zvestobo tudi za reducirano zvestobo in čistost. To si bomo ogledali v tem razdelku. Da pa se ne bi preveč ponavljali, bomo obravnavali sistem dveh harmonskih oscilatorjev za katerega je O, = 0 (I-hri), tako da teorija za regularne sisteme predstavljena v poglavju 1.3 ne deluje. Za konkretni numerični primer bomo vzeli model Jaynes-Cummings, ki se uporablja v kvantni optiki. Predstavlja sklopljen sistem kvantiziranega elektromagnetnega polja (harmonski oscilator) in spina atomov. Dodaten razlog, da smo izbrali ta sistem je tudi, da se ga da eksperimentalno realizirati. Zanimali se bomo le za koherentna začetna stanja. Naš nemoten sistem bo tako podan s hamiltonovo funkcijo Ho = fiuja+a + heSz. Vektor frekvenc je konstanten (neodvisen od akcij) u> = (e,uj) spin, harmonski oscilator pa okolico. Moten sistem H§ = Ho h V = (a+S+ + aS-). (I-xciii) Centralni podsistem bo predstavljal - 5V bomo dobili z motnjo (I-xciv) Takšna motnja predstavlja del dipolne sklopitve med poljem in dipolnim momentom atoma in je tako realistični primer motnje. Ima to dodatno lastnost, da je njeno časovno povprečje nič, V = 0, kar pomeni da kvantna zvestoba zamrzne na platoju. Zanima nas torej, kakšna je vrednost tega platoja, in kakšen je ta za reducirano zvestobo in čistost. Plato Najprej se lotimo izračuna zvestobe. Plato bo, podobno kot za motnjo z O, / 0, dan z i^piat = | (exp (-i6{W(t) - W(0)}/h)) |2. (I-xcv) 1.4. Sklopitev z okolico xxxi Za razliko od splošne motnje pa sedaj ne moremo uporabiti časovnega povprečja, saj za O, = 0 čas ti (I-bri), ko postane časovno povprečje upravičeno, divergira. Označimo z s(j,6,t) fazo, s{3,e,t):=w{j,e + ut)-w{j,e). Za lokalizirana stanja lahko s razvijemo okoli lege začetnega paketa, s(j,e,t) = s(f,e*,t) + s'-(j-j*,e-e*)^ z vektorjem odvodov , fds(j*,0*,t) ds(j*,0*,t). s : = ¦ dj dO (I-xcvi) (I-xcvii) (I-xcviii) Ker je s(j, 6, t) sedaj odvisen tudi od kotov, ne morem uporabiti enostavnega integrala čez akcijski prostor, temveč moramo uporabiti povprečje čez cel fazni prostor. Oblika koherentnega stanja v faznem prostoru bo določena z 2d x 2d matriko D. Povprečje eksponentne funkcije s fazo s(j,6,t) nam tedaj da *S = exP ( -^^har J , ^ar := ^ ¦ D V. 52 (I-xcix) V območju linearnega odziva imamo FS*f = 1 — ^-^har- Glavna razlika v primerjavi z rezultatom za sisteme z O, / 0 (I-lxvi) je, da je sedaj velikost platoja reda S2/h, medtem ko je bila prej S2/h2. Za harmonski oscilator je torej plato za l/h višji. Vendar pa za velike d pojema gaussovo, torej hitreje kot za O, / 0, kjer je bilo to pojemanje potenčno. 10" _ 10"' 10"' 10"' 10"' ¦ft 10" i i l l v 'i , i '' ' h t " - l J i v V J -'I! II11| i fl if « L L ji H \ 20 40 60 80 100 120 140 160 180 200 t Slika 1.12: Plato 1 — F (t) (zgornji simboli) in 1 — I (t) (spodnji) za model Jaynes-Cummings in koherentno začetno stanje. Prekinjeni črti sta teoretični napovedi za zvestobo in čistost (I-xcix,I-cii). Za naš model Jaynes-Cummings dobimo za koeficient z/har po nekaj računanja ^har (cu + e)2 sin 2 (ijJ + e i){(l-ic)+2je ic2 + (i-ic2)cos2 [ec + ee + t ,uj + e (I-c) xxxii I. Razˇsirjeni povzetek kjer so akcijsko-kotne spremenljivke dane z —hSt^y/1-jc2e*0*, a^-iv2jee^. (I-ci) Za reducirano zvestobo in ˇcistost smo izraz za linearni odziv izraˇcunali kar s kvantimi priˇckovanimi vrednostmi, z rezulatom 1-*R = 52 S 2 2(+ t2je\j 2 + (1-j 2 )cos 2(ec + ee + t +X 1-1 = 522 2(+ t\(1-jc). (I-cii) Pri vseh treh vrednost platoja niha, in ni neodvisna od ˇcasa kot je za sisteme z ? = 0. Za ˇcistost je plato neodvisen od h. Na sliki I.12 so numeriˇcni rezultati za S = 50, S = 0.01, parametri pa so uj = e = 0.3. Po platoju Za pojemanje po platoju lahko spet uporabimo “renormalizirano” dinamiko z operatorjem R. Podrobnosti ne bomo navajali, edina razlika od prejˇsnjih primerov je, da se sedaj klasiˇcna zvestoba ujema s kvantno, prav tako se klasiˇcna ˇcistost ujema s kvantno. To je potrebno primerjati s sistemi z ? = 0, ko korespondenca traja le do ˇcasa t1. Dekoherenca Kvantna evolucija bo povzroˇcila, da bodo kvantna stanja v sploˇsnem prepletena. Kljub temu pa prepletenosti ne opazimo za makroskopska stanja. Znan je t.i. paradoks Schrodingerjeve maˇcke, ki je v superpoziciji mrtvega in ˇzivega stanja. Razlog za odsotnost prepletenosti za makroskopska stanja je dekoherenca. V tem razdelku bomo pokazali, kako lahko teorijo pojemanja reducirane zvestobe uporabimo, da razloˇzimo dekoherenco za makroskopske superpozicije. Zaˇcetno stanje sistema bo produkt koherentnega stanja okolice in vsote dveh koherentnih stanj za centralni podsistem, |V(0)> = -7 (h) + |t2» ® |a), (I-ciii) ˇce sta |n)2) dve koherentni stanji centralnega podsistema. Radi bi torej pokazali, da bo dekoherenca tem hitrejˇsa, ˇcim bolj oddaljeni bosta stanji |n) in |r2), torej da je dekoherenca hitrejˇsa za makroskopske superpozicije. Na ravni reducirane matrike je dekoherenca naslednji proces, Pc ~ - (|r1)(n| + |r2)(r2| + |n)(r2| + |r2)(r1|) t - (|r1)(n| + |r2)(r2|) . (I-civ) Po nekem ˇcasu dekoherence tdec ˇcistost pade iz zaˇcetne /(0) = 1 za koherentno superpozicijo, na i"(tdec) = 1/2 za nastalo meˇsano stanje po dekoherenci. Cistost za posamezna koherentna stanja pojema na skali td neodvisni od h, td ~ 1/5, ˇce je S jakost sklopitve med podsistemoma. Za ˇcase t < td bo torej evolucija U(t) posameznih pro-duktnih stanj ˇse vedno produktno stanje, tako da bomo imeli |X1> ® |A) ~ U(t)|n) |a) in |X2>® |&) ~ U(t)Rc|n}^ |a), kjer smo zapisali |r2) =: Rc|n), z neko rotacijsko matriko Rc. Drugo stanje |X2) |f32) lahko torej smatramo kot stanje dobljeno iz istega zaˇcetnega stanja |n) |a) kot prvo, a z nekoliko moteno evolucijo Us(t) := U(t)Rc. Ker so koherence reducirane matrike I.5. Uporaba: kvantno raˇcunanje xxxiii (izvendiagonalni elementi) sorazmerni z (ß1Iß2), kvadrat tega skalarnega produkta pa je reducirana zvestoba za okolico, takoj vidimo povezavo z reducirano zvestobo. Pojemanje izvendiagonalnih matriˇcnih elementov bo torej dano z reducirano zvestobo, ki za koherentna stanja pojema kot exp(-l2CRt2/h2), z CR = 12—h(v'e ¦ ?-1v'e). Jakost motnje smo tukaj oznaˇcili z lin je doloˇcena z matriko Rc in propagatorjem U(t), in je ponavadi kar sorazmerna z “razdaljo” med stanjema |?1) in |?2). Ker je CR oc d 2 bo, ˇce ima okolica veliko prostostnih stopenj (de > dc), reducirana zvestoba za okolico pojemala hitreje kot reducirana zvestoba za centralni podsistem in bomo imeli po deko- herenci, t > tdec, ko bodo (ß1|ß2) majhni, ˇse vedno pribliˇzno |?1,2> ~ \?1,2), s ˇcimer smo potrdili predviden scenarij dekoherence (I-civ). Za superpozicijo makroskopskih stanj bo torej dekoherenca dana s pojemanjem FR(t), ki pada na skali odvisni od h, tdec ~ —h/l, in ne z I(t) za posamezna koherentna stanja, ki pojema na daljˇsi skali td ~ 1/?. Vse skupaj bomo demonstrirali na modelu Jaynes-Cummings (I-xciii), pri ˇcemer bomo za motnjo vzeli —h 2= katere ˇcasovno povpreˇcje ni niˇc. Hamiltonova funkcija sklopljenega sistema bo tako H? = —h?a+a + —h?Sz+?V. Za superpozicijo dveh koherentih stanj spina (I-ciii) lahko v semiklasiˇcni limiti izraˇcunamo korelacijsko funkcijo, ki nastopa v linearnem odzivu za ˇcistost 1 - I(t) = 2(?/h)2C P t2 (I-lxxxii), in dobimo l2 CP = - + 0(h2), l2 :=sin2?1+sin2?2-2sin?1sin?2cos(?1-?2). (I-cvi) V = -—h=(a + S- + aS+), (I-cv) 8S -2sin?1sin ?2cos(?1- Izraz za l ima nazoren pomen, je namreˇc kar razdalja med projekcijama obeh spinskih paketov na ravnino x-y. Spomnimo se, da je koeficient C P za eno koherentno stanje sorazmeren s h2 = 1/S2. Za naˇs primer makroskopskih superpozicij je C P oc h, saj je dan s pojemanjem reducirane zvestobe. Casovna skala pojemanja ˇcistosti za makroskopsko superpozicijo je torej tdec = 2y—h/(?l), medtem ko da raˇcun za posamezno koherentno stanje pojemanje na skali td = 1/(?cos2 (?/2)>/2). Za l > —h bomo torej imeli pospeˇseno dekoherenco. I.5 Uporaba: kvantno raˇcunanje Kvantna informacijska teorija je relativno mlada veda, za pregled glej (Nielsen & Chuang, 2001; Steane, 1998; Ekert & Josza, 1996). Kljub temu pa so nekateri rezultati, kot je na primer varna kvantna komunikacija, ˇze komercialno uporabni. Kvantno raˇcunanje je na drugi strani ˇse vedno omejeno na majhne laboratorijske eksperimente. Glavna ovira pri delovanju kvantnega raˇcunalnika so teˇzave pri natanˇcnem kontroliranju kvantne evolucije. Ce hoˇcemo zgraditi kvantni raˇcunalnik z veliko qubiti, je potrebno razumeti, kako te napake pokvarijo evolucijo in kako jih je moˇzno ˇcim bolj zmanjˇsati. Z drugimi besedami, radi bi imeli visoko kvantno zvestobo. Vhodni podatki za kvantni raˇcunalnik so zakodirani v kvantnem stanju |r). Pri samem procesu raˇcunanja nato apliciramo kvantna vrata Ut na tem zaˇcetnem stanju. Kvantna vrata so neke enostavne unitarne transformacije na naˇsem prostoru. Ponavadi so to transformacije na enem ali dveh qubitih, torej na enem ali dveh delcih, ˇce so qubiti kar delci. Ce s T oznaˇcimo ˇstevilo vseh vrat, imamo na koncu raˇcuna stanje UT ¦ ¦ ¦ U2U1\r). Produkt vseh kvantnih vrat U:=UT--- U2U1 imenujemo kvantni algoritem. Rezultat raˇcuna sedaj dobimo z meritvami na tem konˇcnem stanju. Ce je kvantni raˇcunalnik sestavljen is n osnovnih enot - qubitov, potem je velikost Hilbertovega prostora M = 2n in v zaˇcetno stanje \r) lahko zapiˇsemo eksponento (v n) mnogo podatkov. Trik, zakaj so lahko kvantni raˇcunalniki za doloˇcene stvari hitrejˇsi kot klasiˇcni leˇzi v tem, da lahko nekatere xxxiv I. Razˇsirjeni povzetek algoritme zapiˇsemo s polinomsko mnogo vrati T. Tako lahko v polinomsko korakih naredimo neko transformacijo na eksponentno velikem prostoru. Takih kvantnih algoritmov pa ni veliko, eden med najbolj uporabnimi je kvantna Fourierova transformacija (QFT). Cˇe se omejimo na unitarne motnje*, lahko poljubna motena kvantna vrata zapiˇsemo kot Ut?:=exp(-i?Vt)Ut. (I-cvii) Celoten moten kvantni algoritem je U?(T) = UT? • • • U1?. Mera za stabilnost algoritma je kvantna zvestoba na koncu algoritma F(T) = | (M?(t)) |2, M? := U\T)U?(T), (I-cviii) kjer je U(t) produkt vseh vrat do ˇcasa t, U(t) := Uf-U1, in podobno za U?(t). Sedaj lahko kvantno zvestobo razvijemo do najniˇzjega reda v ?, ter vse skupaj povpreˇcimo ˇcez cel Hilbertov prostor T F(T) = 1-?2 ]T C (t, t'), (I-cix) t,t'=1 s povpreˇcno korelacijsko funkcijo C(t,t'):= 1 tr[Vt(t)Vt/(t)], (I-cx) in Vt(t) := U(t)VtU(t). Glede na dosedanjo teorijo je razlika ta, da je sedaj motnja V eksplicitno odvisna od ˇcasa (vrat), kar oznaˇcuje spodnji indeks t, Vt. Cˇe so motnje med posameznimi vrati ˇcasovno nekorelirane (npr. ˇsum), potem je korelacijska funkcija diagonalna, C (t, t') oc ?tt. Sˇe veˇc, v tem primeru lahko tudi operator odmeva faktoriziramo in za primer, ˇce so tudi matriˇcni elementi med seboj nekorelirani, 1 kar velja npr. za nakljuˇcne matrike, lahko izraˇcunamo kvantno zvestobo do vseh redov v ?, {VjkVlm) = i?jm?kl, (I-cxi) gue (I-cxiii) pri n = 10 za originalni algoritem (desno) in za izboljˇsan (levo). Originalni QFT algoritem ima T = 60 vrat, medtem ko jih ima izboljˇsan T = 105. sliki I.13, kjer je narisana velikost korelacijske funkcije za oba algoritma pri n = 10 qubitih. Izkaˇze se, da je korelacijska vsota za izboljˇsan algoritem bistveno manjˇsa kot za originalni QFT, ˇceprav ima veˇc vrat in torej apliciramo pokvarjena vrata veˇckrat. Originalni algoritem ima T ~ n2/2 vrat, medtem ko jih ima izboljˇsan T ~ n2. Dvojna korelacijska vsota ? naraˇsˇca za nepopravljen algoritem kot ? oc n3, kar je posledica platojev v strukturi korelacijske funkcije (vidno na sliki), za izboljˇsan algoritem pa le ? oc n2, saj so znatni le diagonalni korelacijski elementi. Asimptotsko je dobra aproksimacija za pojemanje kvantne zvestobe eksponentna funkcija, F (T) ^exp(-??2). (I-cxiv) To smo preverili z numeriˇcno simulacijo, rezultati katere so na sliki I.14. Kot vidimo, se asimptotska odvisnost ujema z napovedano, kar pomeni, da je izboljˇsan algoritem, kjub veˇcjemu ˇstevilu vrat, bistveno bolj stabilen za motnje, ki se dajo modelirati z nakljuˇcno matriko. xxxvi I. Razširjeni povzetek Slika 1.14: Odvisnost F (T) od števila qubitov za d = 0.04 in nepopravljen QFT (plusi) in izboljšan algoritem QFT (križci). Dve polni črti sta exp (—52{0A7 n3 — 0.76n2 + 2.90n}) (spodnja) in exp (—52{1.22n2 + 1.78n}) (zgornja). Za konec naj dodamo, da izboljšan algoritem ni tako dober, če deluje motnja samo na nekaj qubitih. Zdi se, da je za motnje ki delujejo na omejen podprostor celotnega Hilbertovega prostora, konstrukcija boljšega algoritma bistveno težja. Chapter 1 Introduction In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite. —Paul Dirac Irreversibility of macroscopic behaviour has attracted attention ever since Boltzmann introduced his tf-theorem in a seminal paper in 1872 (Boltzmann, 1872), together with the equation bearing his name. The tf-theorem is a precursor of what is today known as the second law of thermodynamics, stating that entropy can not decrease in the course of time. The problem was how to reconcile this “arrow of time” with the underlying reversible microscopic laws. How come that macroscopic systems always seem to develop in one direction whether the underlying dynamics is symmetric in time? This so called reversibility paradox is usually attributed to Josef Loschmidt. He mentioned it briefly at the end of a paper published in 1876 (Loschmidt, 1876) discussing the thermal equilibrium of a gas subjected to a gravitational field, in an attempt to refute Maxwell’s distribution of velocities for a gas at constant temperature. He also questioned Boltzmann’s monotonic approach towards the equilibrium. Discussing that, if one would reverse all the velocities, one would go from equilibrium towards the initial non equilibrium state, he concludes with : “...Das ber¨uhmte Problem, Geschehenes ungeschehen zu machen, hat damit zwar keine Losung...”. Bolzmann was quick to answer Loschmidt’s objections in a paper from 1877 (Boltzmann, 1877), pointing out the crucial importance of the initial conditions and of the probabilistic interpretation of the second law. The fact that for macroscopic systems we always observe a definite “arrow of time” is a consequence of a vast majority of the initial conditions representing an equilibrium state. If we choose the initial state at random it will almost certainly evolve according to the second law of thermodynamics. Of course this probabilistic interpretation still does not solve the problem. Resolution lies in the initial condition, chosen to be a macrostate representing a very small part of the phase space, i.e. having a low entropy. So, provided we start with a low entropy state, the above probabilistic arguments can be used (Lebowitz, 1999) to explain the second law. The question of the initial low entropy state of say the whole universe still remains. For a popular account of this “cosmological” subject see Penrose (1989). The first written account of the reversibility paradox is actually not due to Loschmidt but due to William Thompson (later Lord Kelvin), although it is possible that Loschmidt mentioned the paradox privately to Boltzmann before. Boltzmann and Loschmidt became good friends while working at the Institute of Physics in Vienna around 1867 (directed at the time by the Slovenian 1 2 Chapter 1. Introduction physicist Joˇzef Stefan). In 1869 Boltzmann moved to Graz, but returned to Vienna in the period 1873-1876. For a detailed biography of Boltzmann see Cercignani (1998). In a vivid paper (Thompson, 1874) from 1874 Thompson gave a very modern account of irreversibility. When discussing the heat conduction and the equalization of temperature he says: “...If we allowed this equaliza-tion to proceed for a certain time, and then reversed the motions of all the molecules, we would observe a disequalization. However, if the number of molecules is very large, as it is in a gas, any slight deviation from absolute precision in the reversal will greatly shorten the time during which disequalization occurs... Furthermore, if we take account of the fact that no physical system can be completely isolated from its surroundings but is in principle interacting with all other molecules in the universe, and if we believe that the number of these latter molecules is infinite, then we may conclude that it is impossible for temperature-differences to arise spontaneously...”. The interesting question is then, how short is this short disequalization time? The quantity in question is nothing but the fidelity or the Loschmidt echo as it is sometimes called. We evolve the system forward in time with the unperturbed evolution, then backward in time with the perturbed evolution, and look at the overlap with the initial state. It can be considered in classical mechanics as well as in quantum mechanics. The decay time of the fidelity will then be the time of disequalization in question. Despite its importance for thermodynamics the fidelity was not considered until some years ago, with motivation coming from quantum rather than classical theory. Quantum theory is arguably the greatest achievement in 20th century physics. There are estimates (Tegmark & Wheeler, 2001) that up to 30% of the gross national product of the US relies on quantum devices. Alone the semiconductor industry is of enormous importance. Still, all these quantum devices do not manipulate individual quanta but rather exploit macroscopic phenomena involving many particles. Experiments involving individual quantum systems became possible only in the ’80 with the progress made in e.g. manipulation of cold atoms in traps, single electron devices, entangled photons etc.. This was so to say the experimental birth of what is now called quantum information theory (Nielsen & Chuang, 2001). Quantum information theory married quantum mechanics with information theory and with computer science. It deals with means of processing and transmitting information, and by using quantum systems can achieve things not feasible in any classical way. For instance, one can teleport a quantum state, or perform secure communication over a public channel or do a quantum computation. By using quantum resources to do a computation one is able for instance to factorize a number in a polynomial time, which is presently not known to be possible by classical computer. Also, quantum computers are very efficient in simulating other quantum systems, answering the problem posed by Feynman (1982). Namely, he asked whether it is possible to build a computing machine whose size will grow only linearly with the size of a quantum system simulated on it. With classical computers this is not possible since the size of the Hilbert space needed merely to describe the system grows exponentially with the number of particles. We do not know yet if it is possible to build a quantum computer that will achieve that goal, but on paper quantum computer will be able to do the trick. We say on paper because presently one is able to perform laboratory computations only on a few qubit (less than 10) quantum computer. The main obstacle are errors in the evolution, either due to unwanted coupling with the environment or due to internal errors. Therefore, the main goal is to build a stable quantum computer, resistant to such perturbations. The usual benchmark for stability is fidelity and therefore one ought to understand the behaviour of fidelity in different situations to know hot to maximise it. Yet again, the original push to study fidelity came neither from quantum information theory nor from thermodynamics, but from the field of quantum chaos. The exponential instability of classical systems is a well known and much studied subject. As the underlying laws of nature are quantum mechanical the obvious question arises how this 1.1. Historical Overview 3 “chaoticity” manifests itself in quantum systems whose classical limit is chaotic. The field of quantum chaos mainly dealt with stationary properties of classically chaotic systems, like spectral and eigenvector statistics. Despite classical chaos being defined in a dynamical way it was easier to pinpoint the “signatures” of classical chaos in stationary properties (Haake, 1991; Stockmann, 1999). There were not as many studies of the dynamical aspects of quantum evolution in chaotic systems, some examples being studies of the reversibility of quantum evolution (Shepelyansky, 1983; Casati et al. , 1986), the dynamical localization (Fishman et al. , 1982; Grempel et al. , 1984), energy spreading (Cohen, 2000), wave-packet evolution (Heller, 1991). Classical instability is usually defined as an exponential separation of two nearby trajectories in time. In quantum mechanics the state of a system is completely described by a wave function and so one could be tempted to look at the sensitivity of quantum mechanics to variations of the initial wave function. But quantum evolution is unitary and therefore preserves the dot product (i.e. the distance) between two states and so there is no exponential sensitivity with respect to the variation of the initial state. This, at first sight perplexing, conclusion has been reached because we compared two different things. Classical mechanics can also be stated in terms of a Liouiville propagation of phase space densities and this is also unitary. If we want to compare quantum and classical mechanics we have to compare them on the same footing. Quantum mechanics is a probabilistic theory, the wave function just gives the probabilities of measurement outcomes. Therefore we should also formulate classical mechanics in a probabilistic way as a propagation of probability densities in phase space. The idea, first proposed by Peres (1984), was to study not the sensitivity to the variation of the initial condition but with respect to the variation of the evolution. He compared two slightly different evolutions starting from the same initial state - the quantum fidelity. For classical systems fidelity gives the same exponential sensitivity to perturbations of the evolution as to perturbations of initial conditions; the two things are equivalent. The quantum fidelity though can behave in a very different way, as we will see in the present work. The fidelity lies at the crossroad of three very basic areas of physics: thermodynamics, quantum information theory and quantum chaos. The features of the fidelity turned out to be very interesting, as one would expect for such a crossroad. 1.1 Historical Overview 1.1.1 Quantum Fidelity The quantum fidelity F(t), being the square of the overlap of the state |?(t)) = U0(t)|?(0)) obtained by the unperturbed evolution U0(t) and the state |?S(t)) = U5(t)|?(0)} obtained by the perturbed evolution Us(t), F(t) := |?(t)|?(t))|2 = |{?(0)|Ukt)U(t)|?(0))|2, (1.1) has been first used as a measure of quantum stability by Peres (1984), see also his book (Peres, 1995). The fidelity can also be interpreted as an overlap of the initial state and the echo state obtained after forward unperturbed evolution followed by a backward perturbed evolution, i.e. after evolution with Uj5(t)U0(t). Quite generally we can imagine unperturbed evolution being governed by an unperturbed Hamiltonian H0 and the perturbed evolution by slightly perturbed Hamiltonian H5 = H0 + ?V, with ? a dimensionless perturbation strength and V the perturbation operator. Peres reached non-general conclusion that the decay of fidelity is faster and has a lower asymptotic value for chaotic than for regular classical dynamic. As we will see the general situation can be exactly the opposite. Non decay of the fidelity for regular dynamics in Peres’s work was due to a very special choice of the initial condition, namely that a coherent wave packet was placed 4 Chapter 1. Introduction in the centre of a stable island. Such a choice is special in two ways, first the centre of an island is a stationary point and second the number of constituent eigenstates of the initial state is very small. After Peres’s work the subject lay untouched for about a decade. In 1996 Ballentine & Zibin (1996) numerically studied a quantity similar to fidelity. Instead of perturbing the backward evolution (i.e. the Hamiltonian), they took the same backward evolution but instead perturbed the state after forward evolution by some instantaneous perturbation, like shifting the whole state by some ?x. They also looked at the corresponding classical quantity. The conclusion they reached was that for chaotic dynamics quantum stability was much higher than the classical one, while for regular dynamics the two agreed. All these results were left mainly unexplained. Gardiner et al. (1997, 1998) proposed an experimental sheme for measuring the fidelity in an ion trap. Somehow related to the studies of stability was also a work by Schack & Caves (1992, 1993, 1996), where they studied how much information about the environment is needed to prevent the entropy of the system to increase. Fidelity studies received new impetus by a series of NMR experiments carried out by the group of Pastawski. In NMR echo experiments are a standard tool. The so called spin echo experiment of Hahn (1950) refocuses free induction decay in liquids due to dephasing of the individual spins caused by slightly different Larmour frequencies experienced due to magnetic field inhomogeneities. By an appropriate electromagnetic pulse the Zeeman term is reversed and thus the dynamics of non-interacting spins is reversed. The first real interacting many-body echo experiment was done in solids by Rhim et al. (1970). Time reversal, i.e. changing the sign of the interaction, is achieved for a dipolar interaction whose angular dependence can change sign for a certain “magic” angle, that causes the method to be called magic echo. Still, the magic echo showed strong irreversibility. Much later, Zhang et al. (1992) devised as sequence of pulses enabling a local detection of polarisation (i.e. magnetic moment). They used a molecular crystal, ferrocene Fe(C5H5)2, in which the naturally abundant isotope 13C is used as an “injection” point and a probe, while a ring of protons 1H constitutes a many-body spin system interacting by dipole forces. The experiment proceeds in several steps: first the 13C is magnetised, then this magnetisation is transfered to the neighbouring 1H. We thus have a single polarised spin, while others are in “equilibrium”. The system of spins then evolves freely, i.e. spin diffusion takes place, until at time t the dipolar interaction is reversed and by this also spin diffusion. After time 2t the echo is formed and we transfer the magnetisation back to our probe 13C enabling the detection of the polarisation echo. Note that in the polarisation echo experiments the total polarisation is conserved as the dipole interaction has only “flip-flop” terms like S+S3_+1, which conserve the total spin. To detect the spin diffusion one therefore needs a local probe. With the increase of the reversal time t the polarisation echo - the fidelity - decreases and Zhang et al. obtained approximately exponential decay. The nature of this decay has been furthermore elaborated in Pastawski et al. (1995). The group of Pastawski then performed a series of NMR experiments where they studied in more detail the dependence of the polarisation echo on various parameters (Levstein et al. , 1998; Usaj et al. , 1998; Pastawski et al. , 2000). They were able to control the size of the residual part of the Hamiltonian, which was not reversed in the experiment and is assumed to be responsible for the polarisation echo decay. For small residual interactions they obtained a Gaussian decay while for a larger ones the decay rate saturated and was independent of the perturbation strength, i.e. of the size of the residual interaction. While there is still no complete consensus on the interpretation of these experimental results they triggered a number of theoretical and even more numerical investigations. We will briefly list them in chronological order. Using the semiclassical expansion of the quantum propagator Jalabert & Pastawski (2001) derived a perturbation independent fidelity decay for localised initial states and chaotic dynamics, F(t) ~ e"Af, also called a Lyapunov decay due to its dependence on the Lyapunov exponent ?. 1.1. Historical Overview 5 Perturbation independent fidelity decay obtained for strong perturbations has also been studied numerically in a Lorentz gas (Cucchietti et al. , 2002a), the perturbation being in the mass of the particle. In the same paper the authors also studied the asymptotic fidelity saturation value which, for the strong perturbations considered, is independent of the perturbation strength, but becomes perturbation dependent for smaller perturbations (see Section 2.2.1). Studying fidelity turned out to be particularly fruitful in terms of the correlation function (Prosen, 2002), see also (Prosen & ˇ Znidariˇc, 2001). For coherent initial states, the fidelity of regular systems decays as a Gaussian while for chaotic systems we have different regimes, the most prominent being the perturbation dependent exponential decay. The decay of quantum fidelity in chaotic systems has also been studied by Jacquod et al. (2001) and by Cerruti & Tomsovic (2002). For sufficiently small per-turbations one gets a Gaussian decay (also called a perturbative regime), for intermediate ones the so-called Fermi golden rule regime of exponential, perturbation dependent decay, while for still stronger perturbations we get the Lyapunov decay. All these regimes, including the fidelity decay for regular dynamics, and for different initial states, were carefully discussed using the correlation ˇ function approach in Prosen & Znidariˇc (2002). Several interesting results were obtained, perhaps the most surprising one being that in a certain range of parameters we can, by increasing chaoticity of the corresponding classical system, increase quantum fidelity, i.e. improve the stability of quan-tum dynamics. Different time and perturbation scales were discussed as well as their dependence on the number of degrees of freedom. It is well known that the quantization of classical system is not unique, i.e. there are different quantizations leading to the same semiclassics. Kaplan (2002) compared the quantization ambiguity in chaotic and regular systems, reaching a conclusion that in chaotic systems the quantization ambiguity is supressed as compared with regular ones. Karkuszewski et al. (2002) connected the fidelity decay with the decay of the off-diagonal ma-trix elements of the reduced density matrix, and therefore with decoherence. They claimed that quantum systems, whose classical limit is chaotic, are particularly sensitive to perturbations due to small scale structures in their Wigner functions. Actually, what their results show, was just the dependence of the fidelity decay on the size of the initial state. Fidelity decay is faster for random initial states than for coherent ones and there are no quantum effects in the regime they studied. For related comments see also (Jordan & Srednicki, 2001; Jacquod et al. , 2002). The transition between Fermi golden rule and the Lyapunov decay has been further considered in a Bunimovich stadium (Wisniacki et al. , 2002), see also (Cucchietti et al. , 2002b) for a study of Lyapunov decay. We should stress that the Lyapunov decay of quantum fidelity is purely a consequence of the quantum-classical correspondence. There is nothing “quantum” in it and can be explained in terms of the classical fidelity displaying the same perturbation independent decay as the quantum fidelity (Veble & Prosen, 2004). The fidelity decay can also be connected with the local density of states, although not in a straightforward manner (Wisniacki & Cohen, 2002). Fidelity decay in mixed systems, having a coexisting regular and chaotic components, is not as well studied as in purely chaotic or regular situation. In (Weinstein et al. , 2002a) they studied the fidelity decay for initial states placed at the border between regular and chaotic regions and observed a power-law fidelity decay although over less than an order of magnitude. Jacquod et al. (2003) studied averaged fidelity decay in regular systems for perturbations with a zero time average for which they predict a universal power-law decay with a power 3/2. This does not agree with our findings, see discussion at the end of Section 4.2.2. If the packet after an echo in the chaotic system drifts exponentially away from its position at the beginning, the fidelity can decay also in a super-exponential way (Silvestrov et al. , 2003). In the same paper the authors also consid-ered the influence of different averaging procedures on the decay of short-time fidelity. From the correlation function approach one easily sees that the decay of the fidelity does not only depend 6 Chapter 1. Introduction on the unperturbed dynamics, say being regular or chaotic, but also on the perturbation. If the correlation function decays to zero sufficiently fast, we can get exponential decay also for regular unperturbed dynamics. Such is the case for instance if the perturbation is modelled by a random matrix (Emerson et al. , 2002), but note that such perturbation has no direct classical limit. Instead of the fidelity one can also study the Fourier transformation of the fidelity, see (Wang & Li, 2002). In (Wisniacki, 2003) it was studied how different perturbations of the billiard border influence the short-time fidelity decay. For an attempt to give a uniform approximation of the fidelity decay in the crossover regime between the Gaussian perturbative and the exponential Fermi golden rule decays using random matrix theory see (Cerruti & Tomsovic, 2003b,a) and also (Gorin et al. , 2004). The asymptotic saturation level of the fidelity has been studied in (Weinstein et al. , 2002c), ˇ confirming results of (Prosen & Znidariˇc, 2002). Perturbative calculation of the fidelity decay in disordered systems by field theoretical method, i.e. diagrammatic expansion of Green’s function, has been done by Adamov et al. (2003). Recently Van´iˇcek & Heller (2003) devised an efficient numerical scheme for a semiclassical evalu-ation of the quantum fidelity. The method consists of transforming intractable (due to exponentially many contributing orbits) semiclassical expressions in coordinate space into an initial momentum space representation. A surprising quantum phenomenon of a prolonged stability, called freeze of ˇ fidelity, has been described in Prosen & Znidariˇc (2003b) for regular systems. Later, it has been ˇ generalised to arbitrary dynamics, in particular to chaotic systems (Prosen & Znidariˇc, 2004). The decay of fidelity in regular one dimensional systems has been studied in (Sankaranarayanan & Lakshminarayan, 2003) and they also numerically observed a very short correspondence between the classical and the quantum fidelity for initial states placed in a rotational part of phase space, where the average perturbation is zero. This is nothing but the freeze of fidelity, not present in classical fidelity. The Lyapunov regime, being of purely classical origin, and its borders of validity ˇ have been furthermore elaborated in (Cucchietti et al. , 2003b), see also (Prosen & Znidariˇc, 2002) for a detailed discussion of borders within which different regimes are valid. They also stressed the importance of noncommutativity of the limits —h › 0 and ? › 0 in recovering the classical ˇ behaviour, as already explained in (Prosen & Znidariˇc, 2002). A nontrivial question, addressed by Hiller, Kottos, Cohen and Geisel (Hiller et al. , 2004) concerns the optimal time of the unper-turbed evolution, i.e. we fix the duration of the unperturbed evolution and seek the duration of the perturbed evolution for which the fidelity will be maximal. The quantum fidelity decay in weakly chaotic systems when one might not get the Lyapunov or the Fermi golden rule decay has been explored in (Wang et al. , 2004). The fidelity for various random matrix models has been analysed by Gorin et al. (2004), see also (Cerruti & Tomsovic, 2003b,a). For further random matrix results and their relevance for quantum computation see also (Frahm et al. , 2004). In (Iomin, 2003) the fidelity of a nonlinear time-dependent chaotic oscillator with respect to the time-dependent per-turbation of its frequency has been studied analytically. Most of the studies so far focused on a few degrees of freedom systems, the exception being (Prosen, 2002). For additional results on the stability of many-body systems see (Izrailev & Castan~eda-Mendoza, 2004). There has also been a large number of papers dealing with the stability of quantum computation, i.e. studies of the fidelity for specific quantum algorithms and perturbations, mainly relying on numerical simulations. Some of these include (Miquel et al. , 1996, 1997; Gea-Banacloche, 1998, 1999, 2000). The group of Berman studied in detail the stability of the Ising quantum computer, see for instance (Berman et al. , 2001, 2002a,b) and references therein and also (Celardo et al. , 2003). The group of Shepelyansky studied the stability of many different quantum algorithms (Georgeot & Shepelyansky, 2000; Song & Shepelyansky, 2001; Benenti et al. , 2002; Terraneo & Shepelyansky, 2003), see also (Benenti et al. , 2003c) and (Bettelli, 2004). The scaling of errors with various 1.1. Historical Overview 7 parameters is easily explained by our correlation function approach. With the advances in experiments manipulating individual quantum systems it has become possible to actually measure quantum fidelity. Apart from NMR echo experiments already mentioned, one is able to measure the quantum fidelity on a few qubit NMR quantum computer (Weinstein et al. , 2002b). Particularly promising candidates are ion traps Gardiner et al. (1997, 1998), ultra cold atoms in optical traps (Andersen et al. , 2003; Schlunk et al. , 2003). Experiments with microwave resonators in billiards are also under way (Schafer et al. , 2003). 1.1.2 Classical Fidelity Classical fidelity can be defined in an analogous way as quantum fidelity and has been first used in (Prosen & ˇ nidariˇc, 2002). It is an overlap integral of two classical densities in phase space, obtained by unperturbed and perturbed evolutions, Fclas(t) := fdx?0(x,t)?s(x,t), (1.2) where ?s(x,t) is the density in phase space obtained by a perturbed evolution ?s(x), where ?s(x) is a volume preserving flow in phase space. The density at time t can be obtained by backward propagating phase space point x, ?s(x,t) = ?(?j1 (x),0). Note that in order for fidelity to be normalized to 1, the classical density has to be square normalized, jdx?2(x,0) = 1. Classical fidelity behaves markedly different from quantum fidelity. For chaotic systems and localized initial states the classical fidelity agrees with quantum only up to the short Ehrenfest time, logarithmic in Planck’s constant, when the quantum-classical correspondence breaks down. For regular systems and if the time average perturbation is nonzero though, classical fidelity follows quantum fidelity. On the other hand, if the time averaged perturbation is zero, quantum fidelity exhibits the so called freezing (Chapter 4), while classical fidelity does not, except in the non-generic case of a harmonic oscillator (Section 5.5). The classical fidelity will depend on the stability of orbits in phase space. Linear response will therefore depend on the stability of the flow and this involves the derivatives of the flow, a derivative being an unbounded operator. For chaotic systems for instance, the derivatives grow exponentially in time due to orbit separation, a simple consequence of the famous Lyapunov instability. In a phase space picture, the classical dynamics can produce structures on an arbitrary small scales, even if we start from a smooth density. Resolution of the quantum mechanics on the other hand is limited by a finite Planck constant. We will give a brief overview of the known results about classical fidelity. This will help us to understand the differences with quantum fidelity which will be the object of study in the present work. The literature on classical fidelity is not nearly as extensive as on quantum fidelity. Linear response calculation has been done in (Prosen & ˇ nidariˇc, 2002). Numerical results on the classical fidelity and its correspondence with the quantum fidelity in chaotic systems and in systems exhibiting diffusion have been presented in (Benenti & Casati, 2002). Classical fidelity in regular and chaotic systems has also been theoretically discussed in (Eckhardt, 2003). A detailed explanation of the asymptotic decay in chaotic systems has been given in (Benenti et al. , 2003b) and a theoretical explanation of the Lyapunov decay for short times in (Veble & Prosen, 2004). Classically regular systems on the other hand have been worked out in (Benenti et al. , 2003a). Chaotic Systems The classical fidelity in chaotic systems will go trough different decay regimes as time increases. We will consider a localized initial state of size ?. Starting from t = 0 fidelity stays close to 1 until 8 Chapter 1. Introduction time tv (Benenti & Casati, 2002; Benenti et al. , 2003b), t~?ki? , (1.3) where ? is the Lyapunov exponent. The time tv can be thought of as the time in which the initial perturbation is “amplified” to the size of the initial packet and ?5 starts to differ from ?0. For a rigorous derivation of t„ as well as for a discussion of multi-degree of freedom systems, where there is a cascade of times tv, see (Veble & Prosen, 2004). After tv the so called Lyapunov decay F(t) = exp {-?(t - t)} sets in and lasts until time t5 determined by the spreading of the packet over the whole phase space. The Lyapunov decay has been explained by Veble & Prosen (2004) and for systems with more than one degree of freedom there is a whole cascade of Lyapunov decays determined by the Lyapunov spectrum*. If we have a classical system with diffusion, such that the phase space is much larger in one direction, say q,pe [0,2?] x [0, L], with L > 2?, time t5 will be given by (Benenti et al. , 2003b) After ts fidelity will decay diffusively F(t) ~ 1/y/Dt with the diffusion constant D, until the diffusive process reaches the phase space boundary also in p-direction, i.e. at tD ~ L2/D. Note that this diffusive regime is present only if L > 2?. After tD the asymptotic decay of classical fidelity begins. This asymptotic decay is determined by the largest Ruelle-Pollicott resonance (Benenti et al. , 2003b), i.e. the eigenvalue of the Perron-Frobenius operator, and is thus the same as the asymptotic decay of classical correlations. If there is a gap in the spectrum of the Peron-Frobenius operator, this decay will be exponential, otherwise it can be power law. Note that this asymptotic decay rate does not depend on the perturbation strength ? but only on the phase space size L. For large times fidelity will decay towards the asymptotic value F = F (t ->¦ oo) determined by the ratio of the initial packet size and the phase space size. In order to see the asymptotic regime of fidelity decay one has to look at F(t) - F. Furthermore, for the fidelity at the end of the Lyapunov decay ts to be larger than F we must have ? > F. The decay rate for classical fidelity in chaotic systems, apart from the non-decaying “shoulder” until t„, does not depend on the perturbation strength ? (decay time borders do depend though) and is therefore independent of the perturbation itself. This must be contrasted with quantum fidelity, which as we will see does depend strongly on perturbation strength and type. Regular Systems The classical fidelity decay for regular systems has been explained (Benenti et al. , 2003a) by studying changes in the action-angle variables caused by the perturbation. There is a competition between two contributions to the fidelity decay. It can decay as a consequence of the ballistic separation of the perturbed and unperturbed packets caused by different frequencies of the perturbed and unperturbed tori on which the initial packet is placed, or it can decay due to different shapes of unperturbed and perturbed tori. In the latter case the decay is caused by transitions of the perturbed packet between unperturbed tori. Therefore, if the perturbation predominantly changes the frequency of tori, the classical fidelity will exhibit a ballistic decay. The shape of this ballistic decay is determined by the shape of the initial packet. If the packet is a coherent state having a Gaussian shape, the decay will be Gaussian. On the other hand, if the perturbation predominantly ?In case of drift of packets one can get a super-exponential instead of exponential decay of fidelity. For sufficiently small times quantum fidelity will show the same phenomena, see (Silvestrov et al. , 2003). 1.1. Historical Overview 9 changes the shape of tori the decay will be algebraic F(t) ~ 1/(?t)d, in a system with d degrees of freedom. Which type of decay one gets, depends on the shape of the perturbation but not on its strength ?, provided it is small enough. The type of the decay can also depend on the position of the initial packet. Note that the above results were derived under the assumption that the angles across the packet are spread over 2?, i.e. t > 2?/(?j??/?j), where ?j is the width of the initial packet in the action direction. Incidentally, this time is equal to the time t1 (4.50) after which quantum fidelity freezes at a constant value if the time averaged perturbation is zero (see Section 4.2). Furthermore, the change of the actions across the packet caused by the perturbation must be small, i.e. ?j > ?. This last condition translates for coherent packets into ? < h1/2, also being the upper limit of validity of approximations used in the calculation of the quantum fidelity plateau (4.61). Comparing to quantum fidelity, the case of ballistic decay of classical fidelity corresponds to perturbations having a nonzero time average, discussed in Chapter 3. For such perturbations quantum fidelity agrees with the classical one under certain conditions. The case of the algebraic decay corresponds to the perturbation with a vanishing time average. In this case though, the quantum and the classical fidelity do not agree. What is more, quantum fidelity displays an intriguing new feature called freezing (see Section 4.2) and decays only on a much longer time scale ~ 1/?2. Whereas only the functional dependence of the classical fidelity decay changes depending on the perturbation type, always decaying on a time scale ~ 1/?, the decay of quantum fidelity drastically changes if one has a perturbation with a zero time average. 1.1.3 Entanglement The literature on decoherence is exhaustive and we will here list only those more or less directly related to our work. Time independent perturbative expansion has been used by Ku¨bler & Zeh (1973) to study the eigenvalues of the reduced density matrix. A similar expansion was used much later for the purity (Kim et al. , 1996). Note that these perturbative approaches are not equivalent to our linear response expressions presented in Chapter 5 as they correspond only to the short time regime in which the correlation function is constant. Among the early studies of entropy growth and its relation to the chaoticity of the underlying system is the one by Alicki et al. (1996). Miller & Sarkar (1999) observed a linear entropy growth with the slope given by the Lyapunov exponent. Gorin & Seligman (2002, 2003) studied the purity decay for random matrix models. Random matrix asymptotic value of the purity (or of linear entropy) was later re-derived in (Bandyopadhyay & Lakshminarayan, 2002b) and numerically observed in chaotic systems. Prosen & Seligman (2002) first defined the purity fidelity, generalising the purity to echo dynamics. The purity fidelity and its relation with the fidelity in chaotic and regular systems was studied also in (Prosen et al. , 2003b) using a correlation function approach which was used also by Tanaka et al. (2002) for studying chaotic systems. The connection between the purity and the fidelity was put on a firm ˇ ground by a rigorous inequality between the two (Znidariˇc & Prosen, 2003; Prosen et al. , 2003a). The classical analog of decoherence has been studied in (Gong & Brumer, 2003c,b,a). Using a perturbative approach the influence of the type of the perturbation and of the dynamics on the quantum-classical correspondence were explored, see also (Angelo et al. , 2004) for a study of quantum-classical correspondence of entanglement. The entanglement in weakly coupled composite ˇ systems was studied in (Znidariˇc & Prosen, 2003) as well as (Fujisaki et al. , 2003), reaching the same conclusion, namely that the increase of chaos can inhibit the production of entanglement. Subsequently the inequality between fidelity and purity was used in (Cucchietti et al. , 2003a). 10 Chapter 1. Introduction Entanglement production in a coupled baker’s map has been studied in (Scott & Caves, 2003) and in coupled kicked tops in (Bandyopadhyay & Lakshminarayan, 2002a). Similar semiclassical methods as for the evaluation of the fidelity have been used also for the purity (Jacquod, 2004), ˇ predicting an exponential decay in chaotic systems, confirming prediction in (Znidariˇc & Prosen, 2003), and algebraic decay in regular systems. The predicted power of the algebraic decay though does not agree with our theory and numerical results presented in Section 5.3.1. The entanglement under echo situation during a quantum computation has been studied in (Rossini et al. , 2003). Entanglement in weakly coupled kicked tops has been studied also in (Demkowicz-Dobrzan´ski & Ku´s, 2004), among other things also for random initial states. 1.2 Outline In Chapter 2 we introduce the quantum fidelity and briefly review numerical models used for the illustration of the theory. The material in sections dealing with the average fidelity has been mainly ˇ published in (Prosen & Znidariˇc, 2002) and (Prosen et al. , 2003c). The core of Chapter 3 has been ˇ published in (Prosen & Znidariˇc, 2002), which is the main paper presenting the correlation function approach to fidelity. New, unpublished material consists of numerical demonstration of a Gaussian distribution of diagonal matrix elements in chaotic systems, relevant for a Gaussian perturbative decay of fidelity. New is also the discussion about the average fidelity in regular systems as well as the last part of Section 3.3 with the illustration of the fidelity decay in terms of Wigner functions and some additional figures hopefully clarifying the relations between different decay regimes. The ˇˇ material of Chapter 4 has been published in (Prosen & Znidariˇc, 2003b) and (Prosen & Znidariˇc, 2004). We unified the approach in mixing and regular situation making the exposition of a regular case more concise. New material is the explanation of the average fidelity decay in regular systems with zero time average perturbation. About half of the contents of Chapter 5 has been published ˇ in (Prosen et al. , 2003b; Znidariˇc & Prosen, 2003; Prosen et al. , 2003a). The new material consists of the purity fidelity and the reduced fidelity calculation in regular systems beyond linear response, most of the discussion about the Jaynes-Cummings model, the section describing the freeze in a harmonic oscillator and the last section explaining the accelerated decoherence of cat states. The application of the fidelity theory to the improvement of the quantum Fourier transformation has ˇ been published in (Prosen & Znidariˇc, 2001) and an extension to the Ising model of quantum computer, not presented in the present work, in (Celardo et al. , 2003). Chapter 2 Fidelity Basic research is what I am doing when I don’t know what I am doing. —Werner von Braun Quantum fidelity F between two general density matrices p and a, representing either a pure or a mixed state, has been used by Uhlmann (Uhlmann, 1976), F(p,a) = (trJp1/2ap1/2) . (2.1) tr\/n1/2ap1/2 He called it a transition probability and the name fidelity was introduced by Jozsa (Jozsa, 1994). It is symmetric with respect to the exchange of p and a and in the case of one density matrix being a pure one, the general expression for fidelity simplifies into F = (^|a|^} and if both are pure it is F(^,(p) = |(^|(p}|2, (2.2) where obviously p := |^)(^| and a := |(p)((p|. We will always deal with pure states and the latter definition (2.2) will be sufficient for us. Note that sometimes the name fidelity is used for a quantity without a square, i.e. for a fidelity amplitude. We will study stability of quantum dynamics with respect to the perturbation of evolution and the quantity studied will be the fidelity between states obtained by unperturbed and perturbed evolutions, starting from the same initial state. Let us denote the initial state by |^(0)), and the states at time r as |V(r)> = U0(r)|V(0)), |Mt)) = U?(t)|4)(0)), (2.3) where U0(t) is a unitary propagator from time 0 to time r and U?(t) is a perturbed propagator. The overlap between the perturbed and unperturbed states, denoted by F(t), will serve us as a criterion of stability, F(t) = |(^?(r)|^(r))|2 = |/(r)|2, /(r) = (^(0)|M?(r)|^(0)), (2.4) where we introduced a complex fidelity amplitude /(r) and the unitary echo operator M?(t) = uI(t)U?(t). (2.5) 11 12 Chapter 2. Fidelity The fidelity as defined in (2.4) is a real quantity between 0 for orthogonal states and 1 iff the two states are equal (up to a phase) and is a standard measure of stability. Up to now we have not specified the propagators U0(?) and Us(?) and to simplify the matter we will limit ourselves to the case where a propagator for time ? can be written as a power of some basic single-step propagator U0. That is, we will use time index t measuring the number of basic units of duration ?0, ? = t?0. We therefore write a single step propagator U0 = U0(?0) and similarly for the perturbed evolution Us = Us(?0). The propagator for t steps is now simply the t-th power of a basic propagator U0(?0t) = Uf0. These discrete time formalism allows us to treat two interesting cases at once: the case of time-independent Hamiltonian and the case when the Hamiltonian is time periodic function H(?) = H(? + ?0) with a period ?0. The latter case includes the so-called kicked systems which will be used for the numerical demonstration of our theory. Note that the theory can be easily generalised also to time-dependent Hamiltonians, only notation becomes more cumbersome. For an example see Section 6.2 describing the application of fidelity theory to the quantum Fourier transformation algorithm. The general perturbed evolution can be written in terms of the perturbation generator V as Us = U0exp(-iV??0/h), (2.6) where ? is a dimensionless perturbation strength. The above equation (2.6) can be considered as a definition of a hermitian operator V, given the unperturbed and perturbed one time step propagators U0 and Us, respectively. In the case of Hamiltonian dynamics with H0 generating unperturbed evolution as Uf0 = exp (-iH0t?0/h) and with the perturbed Hamiltonian of the form Hs = H0 + H'? we have V = H' + O(?0?). The difference between H' and V therefore goes to zero in the limit of either small perturbation strength ? or in the limit of small time step ?0. The latter limit, namely ?0 ->¦ 0, corresponds to the case of time-independent Hamiltonians, where the time step ?0 can be chosen arbitrarily small. From now on we will drop the irrelevant parameter ?0 (i.e. take a unit of time to be ?0) in all equations, so that the discrete time index t has units of time. We will use the same letter t for a discrete time index as well as for a continuous time (limit ?0 ->¦ 0) on few occasions. Whether t is a discrete time index or a continuous will be clear from the context. Our definition of fidelity (2.4) is in discrete time formulation F(t) = |?(0)|MS(t)|?(0))|2, Ms(t) := U^U. (2.7) The fidelity is just the expectation value of the echo operator. It can be equivalently expressed in terms of the initial pure density matrix ?(0) and the echo density matrix ?M(t) (sometimes refered to as the Loschmidt echo) obtained from ?(0) by evolving it with the echo operator Ms(t) ?(0) = |?(0))(?(0)|, ?M(t) := M5(t)?(0)Ml(t), (2.8) as F(t) = tr?(0)?M(t)], f(t) = tr[?(0)Mff(t)]. (2.9) On several occasions we will be interested in the fidelity averaged over some ensemble of initial states. In such a case the density matrix ?(0) in the above definitions of fidelity and fidelity amplitude f(t) (2.9) must be replaced by appropriate mixed density matrix. We will frequently use a uniform average over whole Hilbert space of dimension M, ?(0) = 1/M, see Section 2.2.2 for details. Using our definition (2.6) of perturbed dynamics in terms of the perturbation generator V, the echo operator can be rewritten in a more useful way by writing V in the interaction picture*, i.e. *It can also be considered as the Heisenberg picture of the unperturbed dynamics. 13 for an arbitrary operator A its interaction picture A(t) is A(t) = U0- tAU0t, (2.10) and so the echo operator is M?(t) = exp (-iV(t - 1)?/h) · · · exp (-iV(0)?/h) = Texp (-i?(t)?/h), (2.11) where T is a time-ordering operator and ?(t) is the sum of operators V(j), t-1 ?(t):= ^ V(j). (2.12) j=0 In the case of the time-independent Hamiltonian one has ?(t) = t0 V(t')dt'. The form of the echo operator (2.11) is nothing but the interaction picture of the perturbed propagator, familiar in quantum field theory. Methods of quantum field theory have been actually used by Adamov, Gornyi and Mirlin (Adamov et al. , 2003) to calculate the fidelity in a disordered system. Remember that the time dependence of V(t) comes from the interaction picture (2.10) and so V(t) is time dependent even if original perturbation V (2.6) is not, which is the case throughout our derivations. The same definition of fidelity, just by replacing density matrices with the densities in phase space, can be used for classical fidelity, see Veble & Prosen (2004). The echo operator M?(t) can we written as an exponential function of a single operator by using the Baker-Campbell-Hausdorff (BCH) formula e A e B =exp(A + B + -[A, B] + —[A, [A, B]] + —[B, [B, A]] + · ··. (2.13) Applying the BCH formula on the product form of M?(t) (2.11) gives to the order ?2 in the argument of the exponential function with the hermitian operator ?(t) being M?(t) = exp - i ?(t)? + -?(t)? 2 + ···, (2.14) i t-1t-1 ?(t) = —h^2^2[V(j),V(k)]. (2.15) j=0 k=j Both operators ?(t) and ?(t) standing in the expression for the echo operator have a well defined classical limit, provided V has a classical limit. The classical limit of ?(t) can be obtained by replacing commutator with the Poisson bracket {•, •}, (-i/h)[•, •] › {•, •}. The BCH form (2.14) of the echo operator will be particularly useful in the case of regular dynamics. For now let us list three different possible fidelity decays depending on the behaviour of operator ?(t). For mixing dynamics the fluctuations of ?(t) give the dominant contribution, i.e. terms like (?2(t)) grow linearly with time as ? t. For regular dynamics second moment (?2(t)) ? t2 grows quadratically with time, corresponding to the existence of a nontrivial time-averaged perturbation V(t). In certain cases we can have even (?2(t)) ? t0 which results in the so-called freeze of fidelity. Only in this last case is the fidelity decay caused by the operator ?(t). 14 Chapter 2. Fidelity If we want the fidelity expressed as a power series to all orders in ? the BCH formula becomes too cumbersome. An easier approach is to just expand the product form of the echo operator M?(t) (2.11), giving the fidelity amplitude f(t) = 1 + Y (-'mm y T(V(j1)V(j2)---V(jm)), (2.16) m m!hm ^ "1=1 jl,...,jm=0 where we introduced the notation (•) = (?(0)| •|?(0)). One can see, that the fidelity is expressed in terms of m-point quantum correlation functions of the perturbation generator V. In the semiclas-sical limit one is able to replace quantum correlation functions with the classical ones and therefore the quantum fidelity is expressed in terms of classical quantities. One should keep in mind that the classical fidelity behaves distinctively different from quantum fidelity (see Chapter 1.1.2) and therefore it is not obvious that quantum fidelity can be expressed in terms of classical quantities. Power series expansion of the classical fidelity for example does not yield a fruitful result (for chaotic systems) due to unboundedness of the classical operators and the classical fidelity can not be expressed in a similar way as the quantum fidelity (2.16). Fidelity F(t) is now obtained by taking absolute value square of f(t) (2.16). Only even orders in ? survive as odd orders just rotate the phase of the fidelity amplitude f(t), the lowest order being quadratic in ? F(t) = 1 - ? ]T C(j, k) + 0(?4) = 1 - ?-2 { - (?(t))2} + 0(?4), (2.17) with C(j, k) being the quantum correlation function C(j, k) = (V(j)V(k)) - (V(j)) (V(k)). (2.18) The second order expansion of the fidelity (2.17) is one of the central theoretical results. Although it is very simple and is just the lowest order expansion it contains most of the essential physics of fidelity decay. Furthermore, there is one very pragmatic reason why it is sufficient and higher orders are not needed. In all practical and experimental applications where the fidelity is the relevant quantity, one is mainly interested in a regime of high fidelity, i.e. of high stability. If the interesting range is say F(t) > 0.9, higher orders will give only corrections of order 0.01. To see this, let us denote the second order term with x = (?/h)2J2jkC(j,k). From the expansion (2.16) one can see that the term with ?2m will be at most of the order (?t/h)2m ~ x2m. So if 1 - F(t) « x2 < 1 one can safely neglect higher orders in ?, i.e. terms of order x2m with m > 1, irrespective of the values of individual parameters like ?, h or time t. The range of validity of the lowest order expansion in (2.17) is limited only by the value of 1 - F(t). Furthermore, in certain cases such as mixing dynamics (Section 3.1), regular dynamics (Section 3.2) or in the so-called freeze of fidelity (Chapter 4) the series (2.16) can be resummed to all orders in ? and one gets an expression for the fidelity valid in the whole range from 1 to its asymptotic value. Let us now discuss in more detail the lowest order term in the expansion of fidelity which can be in the case of continuous time-independent Hamiltonian written as 1-F(t)= ? 2 ffC(t',t)dt'dt". (2.19) —h 00 This linear response expression is reminiscent of Green-Kubo like formulas. It says that the “dissipation of quantum information” 1 - F(t) equals the double integral of the correlation function of 2.1. Models for numerics 15 the perturbation V. An interesting and somehow counterintuitive conclusion can be drawn from it, namely, the smaller the integral of time correlation function the higher fidelity will be. In the semi-classical limit the quantum correlation function approaches the classical one, provided the initial state |?(0)> also has a well defined classical limit, and one will see a fast decay of the correlation function for chaotic systems while for regular systems the correlation function typically will not decay. The double integral of the correlation function will therefore grow as oc t for chaotic systems and like oc t2 for regular systems. The fidelity will in turn decay more slowly for chaotic systems than for regular ones. Or in other words, the more chaotic the systems is, the slower decay of quantum fidelity it will have, i.e. the more stable it is to perturbations. This must be contrasted with the behaviour of the classical fidelity (see Section 1.1.2) which is just the opposite. The more chaotic the system is, the faster classical fidelity will decay. 2.1 Models for numerics We will always use numerical simulations to compare with the theoretical derivations. Two classes of systems will be used. One group will be the so-called kicked top models, describing dynamics of a spin “kicked” by some external field. In the case of a kicked top the Hamiltonian is time periodic and the propagator U0 or Us represent a Floquet map over the period of one kick. The second example will be a Jaynes-Cummings system familiar in quantum optics. The Jaynes-Cummings model is a time-independent model with two degrees of freedom representing a harmonic oscillator (boson) coupled with a spin (fermion). In the following two subsections we will present both models and briefly describe their properties, while the actual numerical results will be presented in each chapter as needed. 2.1.1 The Kicked Top The kicked top has been introduced by Haake, Ku´s and Scharf (Haake et al. , 1987) and has served as a numerical model in numerous studies since then (Haake, 1991; Shack et al. , 1994; Fox & Elston, 1994; Alicki et al. , 1996; Miller & Sarkar, 1999; Breslin & Milburn, 1999). In addition the kicked top might also be experimentally realizable (Haake, 2000). In Chapter 3, discussing the fidelity decay for general perturbations, we will use a unitary one step propagator U0 = U(?, ?) = exp (-i?Sy) exp ( -i?2S J, (2.20) where Sx)y)z are standard spin operators [Sk, Si] = i?km Sm and ?, ? are two parameters determining dynamical'properties. The propagator for t steps is Uf0. Half-integer (integer) spin S determines the size of the Hilbert space M = 2S + 1 and the value of the effective Planck constant h = 1/S. The perturbed propagator Us is obtained by perturbing the parameter ? U5 = U(?,? + ?), (2.21) so that the perturbation generator V is 2 z . Sz V = -S . (2.22) In the classical limit S ->¦ 00 the area preserving map corresponding to U0 can be obtained from the Heisenberg equations for spin operators, S(1) = Uj0SU0. The classical map is most easily written 16 Chapter 2. Fidelity in terms of a unit vector on a sphere r = (x, y, z) = S/S as x' = cos ? (x cos (?z)-y sin (?z)) +z sin? y' = y cos (?z)+x sin (?z) (2.23) z' = z cos? - sin? (x cos (?z)-y sin (?z)). The classical perturbation generator (2.22) is simply ^ V › v = z2/2. The angle ? in the propagator is usually set to ?/2 whereas we will use two different values, ? = ?/2 and ? = ?/6. For these two values of ? the classical correlation function displays two different decays towards zero, a monotonic decay for ? = ?/6 and an oscillatory decay for ? = ?/2. The phase space of the classical map (2.24) is regular for small values of ?, at ? ? 3 (see e.g. (Peres, 1995)) most of tori disappear and for still larger ? the system is fully chaotic. For ? = ?/2 the unperturbed propagator U0 (as well as the perturbed one) commutes with the operator Ry of a ? rotation around y-axis, Ry = exp (-i?Sy). In addition there is an antiunitary symmetry so that the Hilbert space is reducible to three invariant subspaces. Following notation in Peres’s book (Peres, 1995) we denote them with EE, OO and OE with the basis states (here we assume S to be even) EE: |0),{|2m) + |-2m)}/ ? 2 NEE = S/2 + 1 OO: {|2m-1)-|-(2m-1))}/ ? 2 NOO = S/2 OE: {|2m)-|-2m)}/ ? 2,{|2m-1) + |-(2m-1))}/ ? 2 MOE = S, where m runs from 1 to S/2 and |m) are standard eigenstates of Sz. For ? = ?/2 the subspaces EE and OO coalesce as Ry is the only symmetry left and we have just two invariant subspaces. Except if stated otherwise the initial state will always be chosen from subspace OE (i.e. initial coherent state will be projected onto OE subspace) so that the size of the relevant Hilbert space will be N = S. In Chapter 4, dealing with the so-called freeze of fidelity, we will take a slightly different form of the propagator which will be presented in the mentioned chapter. Apart from the one dimensional kicked top we will also use a system of two coupled kicked tops to demonstrate the dependence of the fidelity decay on the number of degrees of freedom. The one step propagator for two coupled kicked tops is chosen to be Us = U1(?, ?)U2(?, ?) exp (-i(? + ?)V/—h). (2.25) Propagators U1 and U2 are the usual single kicked top propagators (2.20) acting on the first and the second top, respectively, and the last term with operator V is responsible for the coupling of strength ? for unperturbed evolution and ? + ? for perturbed one. The operator V will be left unspecified for now as we will use different Vs. The initial state |?(0)) used for fidelity evaluation will be either a random state with the expansion coefficients cm = (m|?(0)> being independent Gaussian complex numbers or a coherent state. A random state might be the most relevant for quantum computations for instance as it contains the most information and the states used in quantum computation are expected to be “random”. Coherent state centred at the position r* = (sin?*cos?*,sin?*sin?*,cos?*) is given by expansion s / 2S \1/2 |?,?*>= E [S + m) coss+m(?/2)sins-m(?/2)e-im|m). (2.26) †We will use lowercase letters for classical quantities. 2.1. Models for numerics 17 Equivalently, it can be written in terms of a complex parameter ? as |?,?*)= exp(?S-)|S), ? = ei^tan(?/2), (2.27) with S± = Sx ± iSy. In the semiclassical limit of large spin S the expansion coefficients of coherent state cm go towards cm ~ exp(-S(m/S - z*)2/2(1 - z*2))e-im^. Coherent states have a well defined classical limit and this enables to compare the quantum fidelity for coherent initial states with the corresponding classical fidelity. The initial classical phase space density corresponding to a coherent state is (Fox & Elston, 1994) ?clas(?, ?) = J— exp {-S[(? - ? )2 + (?- ?*)2 sin2 ?]}. (2.28) The above density is normalised as J?2clasd? = 1. 2.1.2 The Jaynes-Cummings model The Jaynes-Cummings model (Jaynes & Cummings, 1963; Tavis & Cummings, 1968), see also (Meystre & Sargent III, 1990, p. 336), is a system of a coupled harmonic oscillator and a spin. It can be realized experimentally in the cavity electrodynamics experiments (QED) by sending a beam of atoms trough a cavity. The electromagnetic field in the cavity is quantized with the Hamiltonian —h?a+a, and the spin degree of freedom —h?Sz of atoms interacts with the electromagnetic field. The dominant interaction is a dipolar d · E, with the monochromatic field E = ?E0(aeikr - a+e-ikr). For two level atoms only y-component of the dipole moment d is nonzero and is proportional to Sy =-i(S+- S-). All this results in the Hamiltonian H = —h?a + a + —h?Sz + —h {G (aS+ + a + S-) + G' (a + S+ + aS-)\ , (2.29) ? 2S1 J with boson lowering/raising operators a, a+, [a, a+] = 1. Planck’s constant is chosen as h = 1/S so that the semiclassical limit is reached for S › ? . If we are close to a resonance ? = ? the rotating-wave approximation can be made in which the fast oscillating term G' can be neglected and only the G-term is retained (oscillating with small ? -?). In our discussion we will predominantly focus on this situation of G' = 0. In addition, frequencies ? and ? are usually in a GHz regime, while the coupling G is of the order of kHz and is therefore small. If either G = 0 or G' = 0 the model is integrable with an additional invariant being the difference or the sum of the spin and boson (harmonic oscillator) quanta. The initial state will be always chosen as a product state of spin and boson coherent states. The spin coherent state is the same as above (2.26), while the coherent state for a boson is |?) = eQO+-Q*°|0> = exp (-|?|2/2) V ??=|k), (2.30) with ? being a complex parameter determining the position of the coherent state and |k) is an eigenstate of operator a+a having k boson quanta. The classical density corresponding to the boson coherent initial state with a complex parameter ? = ?r + i?i is ?class(q,p) = \— exp I-S \(q - ^ 2/S?r) 2 + (p - y/2/S?i) 2]\. (2.31) 18 Chapter 2. Fidelity The normalisation is as usual J?2dassdq dp = 1. The classical Hamiltonian is obtained by taking the limit S ->¦ oo and defining new canonical quantum operators, q1 and p1 for a boson and q2 and p2 for a spin \-(q1 + ip1), \1- a = -(q1 + ip1), 1 -p22eiq2 = S+/S. (2.32) They satisfy commutation relations [q1,p1] = i—h and [q2,p2] = i—h. In the limit S ^ oo they commute and can be replaced by the classical variables resulting in the classical Hamiltonian Hclass = -(p21 + q12) +?p2 + G+^1-p22 q1 cosq2 - G.^1-p22 p1 sinq2 - 2?S, (2.33) with G± = G ± G'. 2.2 Average Fidelity Sometimes the average fidelity is of interest, i.e. the fidelity averaged over some ensemble of initial states. Such an average fidelity is also more amenable to theoretical treatment. Easier to calculate is the average fidelity amplitude f(t) which is of second order in the initial state |?) while the fidelity F(t) is of fourth order in |?). We will show that the difference between the average fidelity amplitude and the average fidelity is semiclassically small. In a finite Hilbert space the fidelity will not decay to zero but will instead fluctuate around some small plateau value. The value of this plateau equals to a time averaged fidelity. For ergodic systems this time averaged value equals to the phase space averaged one. 2.2.1 Time Averaged Fidelity We want to calculate the value of fidelity in the limit t ->¦ oo that is its asymptotic value for large time. For a finite Hilbert space size M the fidelity will start to fluctuate for long times due to a discreteness of the spectrum of the evolution operator. The size of this fluctuations can be calculated by evaluating a time average fidelity F F = mlim.-t=: F(t). (2.34) This is easily done if we expand the initial state in eigenbasis of the unperturbed propagator U0 and denote the matrix elements between unperturbed and perturbed eigenstates by Pkl = exp(-i?l)|?l), U?|??l)=exp(-i??l)|??l), Pkl = {?k|??l). (2.35) U0|?l)=exp(-i?l)|?l), U?|??l) =exp(-i??l)|??l), We denoted the eigenphases of unperturbed and perturbed one-step propagator with ?l and ??l, respectively. The matrix P is unitary and in the case when both eigenvectors can be chosen to be real it is orthogonal. This happens if U0 and U? commute with an antiunitary* operator T whose square is identity §. The fidelity amplitude can now be written f(t) = L (P?)lmPml exp (-i(??l - ?m)t), (2.36) lm »Antiunitary operator must satisfy being the matrix elements of the initial density matrix in the unperturbed eigenbasis. To calculate the average fidelity F we have to take the absolute value square of f(t). Averaging over time t we will assume that the phases are nondegenerate exp (i(??l - ??l +?m- ?m>)t) = ?mmi?ll. (2.37) This results in the average fidelity F— = Ys|(?P)ml|2|Pml|2. (2.38) ml The time averaged fidelity therefore understandably depends on the initial state ? as well as on the “overlap” matrix P. For small perturbation strengths, say ? smaller than some critical ?rm, the unitary matrix P will be close to identity. Using P ->¦ for ? < ?rm (2.38) gives us F—weak = J2?2l l. (2.39) l One should keep in mind that for the above result Fweak we needed eigenphases to be nondegenerate, ??l = ?m (2.37), and at the same time P ->¦ . This approximation is justified in the lowest order in ?, when off diagonal matrix elements are |Pml2 oc ?2. On the other hand, for sufficiently large ? | ?rm and complex perturbations V one might assume P to be close to a random matrix with independent real or complex matrix elements Pml. Then we can average expression (2.38) over a Gaussian distribution oc exp (-ßW|Pml|2/2) of matrix elements Pml, where we have ß = 1 for orthogonal P and ß = 2 for unitary P. This averaging gives (|Pml|4) = (4 - ß)/ M2 and (|Pml|2) = 1/M for the variance of Pml (brackets (•) denote here averaging over the distribution of matrix elements and not over the initial state). The average fidelity for strong perturbation can therefore be expressed as strong = 4-ßY.?2ll + JfY. |?lm|2. (2.40) l l^=m The point of crossover ?rm from weak (2.39) to strong (2.40) perturbation regime is system dependent and can not be discussed in general apart from expecting it to scale with h similarly as a mean level spacing ?rm ~ —h d . We will discuss the value of F for three different initial states: (i) First, let us consider the simplest case when the initial state is an eigenstate of U0 say, ? = |?1)(?1| with matrix elements ?lm = ?l,1?m,1. For weak perturbations this gives (2.39) Fweak = 1, therefore the fidelity does not decay at all. This result can be generalised to the case when ? is a superposition of a number of eigenstates, say K of them, all with approximately the same weight, so that one has diagonal density matrix elements of order ?ll ~ 1/K, resulting in Fweak ~ 1/K. On the other hand, for strong perturbation ? > ?rm we get Fstrong = (4 - ß)/AT for an initial eigenstate. Summarising, for an initial eigenstate we have time averaged values of fidelity Fweak = 1, Fstrong = (4 - ß)/M. (2.41) With this simple result we can easily explain the numerical result of Peres (Peres, 1995) where no-decay of fidelity was found for a coherent initial state sitting in the centre of an elliptic island, thus being a superposition of a very small number of eigenstates (it is almost an eigenstate). The behaviour in generic case may be drastically different as described in the present work. 20 Chapter 2. Fidelity (ii) Second, consider the case of a random pure initial state \tp) = J2m cm\m), giving pmi = cmc*. The coefficients cm are independent random complex Gaussian variables with variance l/M, resulting in averages (\pim\2) = l/M2 for m / I and (pf,) = 2/M2 (average is over Gaussian distribution of cm). Using this in expressions for average fidelity (2.39) and (2.40) we get Fweak = 2/M, strong = l/M. (2.42) For random initial state there is therefore only a factor of 2 difference between finite size fluctuating plateau for weak and for strong perturbation. The result for weak perturbation agrees with the case (i) where we had Fweak ~ I/K if there were K participating eigenvectors. (iii) Third, for a uniform average over the whole Hilbert space, i.e. taking a non-pure initial density matrix p = l/M, we have i^weak = l/M, Fstrong = (4 - P)/M2. (2.43) As expected, the fluctuating plateau is the smallest for an uniform average over the whole Hilbert space and strong perturbation. s=ioo; S=200 S = 400; s=8oo: J____l_l______I___ 1e-05 0.0001 0.001 0.01 0.1 1 10 d Figure 2.1: Dependence of time averaged fidelity (multiplied by the Hilbert space size M = S) F on 5 is shown for a chaotic kicked top system and Hilbert space average p = l/M, i.e. our case (iii). The transition from weak to strong perturbation regime is seen (2.43). Horizontal full lines are the theoretical predictions -Fstrong (2.43), while the theoretical result for the weak regime corresponds to 1. Observe that the average fidelity F (2.38) is of fourth order in matrix elements of P, the same as the inverse participation ratio (IPR) of the perturbed eigenstates. Actually, in the case of initial eigenstate, our case (i), the average fidelity (2.38) can be rewritten as F = J2m \Pim\4, exactly the IPR. The inverse of the IPR is a number between 1 and M which can be thought of as giving the > 0.1 CO 0.01 2.2. Average Fidelity 21 approximate number of unperturbed eigenstates represented in the expansion of a given perturbed eigenstate. For an average over the whole space, case (iii), we have instead F = J2im [Pim^/N2, i.e. the average IPR divided by TV. The time averaged fidelity is thus directly related to the localisation properties of eigenstates of U§ in terms of eigenstates of Uq. However, except for the pathological case of the initial state being a small combination of eigenstates of Uq with weak perturbation, the fidelity fluctuation is always between the limiting values 2/7V, and 3/7V2. Therefore, the fidelity will decay only until it reaches the value of finite size fluctuations and will fluctuate around F thereafter. The time too when this happens, F (too) = F, depends on the decay of fidelity and will be discussed in subsequent chapters. To illustrate the above theory we have calculated the average fidelity (2.38) for a kicked top with a propagator (2.20). As an initial state we used p = 1/J\f, i.e. the case (iii), where the Hilbert space size is determined by the spin value, N = S (OE subspace). We calculated the dependence of F on 5 for two cases: a chaotic one for kicked top parameters a = 30, 7 = 7r/2 shown in Figure 2.1 and a regular one for a = 0.1, 7 = 7r/2 shown in Figure 2.2. In both cases one can see a transition from the weak perturbation regime Fweak = 1/A/" to the strong regime Fstr0ng = 3/A/"2 for large 5. In the chaotic case the critical 5rm can be seen to scale as 5rm ~ h = I/S. In the regular situation, the strong perturbation regime is reached only for a strong perturbation 5 ~ 4, where the propagator Us itself becomes chaotic. Namely, the transition from the regular to chaotic regime in the kicked top happens at around a = 3, see e.g. (Peres, 1995). Still, if one defines 5rm as the points where the deviation from the weak regime starts (point of departure from 1 in Figure 2.2) one has scaling 03 0.1 CO 0.01 22 Chapter 2. Fidelity 2.2.2 State Averaged Fidelity In previous section we used the initial density matrix ?(0) = /N in expression (2.9) for the fidelity amplitude to calculate the average fidelity over the whole Hilbert space. As the fidelity is of fourth order in the initial state |?(0)), whereas the fidelity amplitude is bilinear in |?(0)) the average fidelity is not equal to the quantity obtained by first averaging fidelity amplitude and subsequently squaring it. In general the difference between the two depends on the ensemble of states over which we average. Let us look only at the simplest case of averaging over random initial states, denoted by ((•)). In the asymptotic limit of large Hilbert space size N › ? the averaging is simplified by the fact that the expansion coefficients cm of a random initial state in an arbitrary basis become independent Gaussian variables with variance 1/N. Quantities bilinear in the initial state, like the fidelity amplitude or the correlation function, result in the following expression {{?|A|?)) =: ((A)) = ((c*m Aml cl)) = ^ trA, (2.44) 1 where A is an arbitrary operator. The averaging is done simply by means of a trace over the whole Hilbert space. For the fidelity F(t) which is of fourth order in |?) we get ((F(t))) = (cC[M?(t)]ml cl cp [M?(t)]*pr c*)) = |((f(t)))|2 + N . (2.45) The difference between the average fidelity and the average fidelity amplitude is therefore semiclas-sically small (Prosen et al. , 2003c). There are two reasons why averaging over random initial states is of interest. First, in the field of quantum information processing this are the most interesting states as they have the least structure, i.e. can accomodate the largest amount of information. Second, for ergodic dynamics and sufficiently long times one can replace expectation values in a specific generic state |?) by an ergodic average. If the system is ergodic on the whole space one can calculate the correlation function C(j, k) (2.18) by means of a simple trace (2.44), so that it does not depend on the initial state, or even calculate it classically in the leading semiclassical order, which greatly simplifies theoretical derivations 1 For regular systems, where the decay of fidelity depends on the initial state, the ergodic averaging differs from the average in a specific initial state, although one can still be interested in the behaviour of the average fidelity. Such an averaging will be discussed in the section describing the decay of fidelity in regular systems. We have seen that for sufficiently large Hilbert spaces there is no difference between averaging the fidelity amplitude or the fidelity or taking a single random initial state. For mixing dynamics the long time fidelity decay is independent of the initial state even if it is a non-random, wheras in the regular regime it is state dependent. For instance, the long time Gaussian decay (Section 3.2.2) depends on the position of the initial coherent state. The fidelity averaged over this position of the initial coherent state might be of interest and will not be equal to the fidelity averaged over random initial states. We will discuss averaging over coherent states at the end of Section 3.2.2 describing long time fidelity decay for coherent initial states. ¶In the case of autonomous systems the canonical or micro-canonical averaging should be used instead. Chapter 3 General Perturbation An expert is a person who has made all the mistakes that can be made in a very narrow field. —Niels Bohr 3.1 Mixing Dynamics Here we assume that the system is mixing such that the correlation function of the perturbation V decays sufficiently fast; this typically (but not necessarily) corresponds to globally chaotic classical motion. Due to ergodicity we will assume the initial density matrix to be ?(0) = /Af, so that all averages over a specific initial state can be replaced by a full Hilbert space average, (.) = tr(.)/JV (Section 2.2.2). For any other initial state (e.g. in the worst case for the minimal wave packet – coherent state) one obtains identical results on F(t) for sufficiently long times*, i.e. longer than the Ehrenfest time tE « ln(1/—h)/? (for a classically chaotic system with maximal Lyapunov exponent ?) needed for a minimal wave packet to spread effectively over the accessible phase space (Berman & Zaslavsky, 1978). The state averaged quantum correlation function is homogeneous in time, i.e. C(j, k) = C(k - j), so we simplify the second order linear response formula for the fidelity (2.17) ? 2 f t~1 1 F(t) = 1 - —h2 < tC(0) + 2 Y, (t - j)C(j) \ + 0(?4). (3.1) If the decay of the correlation function C(j) is sufficiently fast, namely if its integral converges on a certain characteristic mixing time scale tmix, then the above formula can be further simplified. For times t > tmix we can neglect the second term under the summation in (3.1) and obtain a linear fidelity decay in time t (in the linear response) F(t) = 1-2(?/h)2?t, (3.2) with the transport coefficient ? being ? = -C(0) + yC(j)= lim ?2(t))-^))2 . (3.3) 2 j=1 t^oo 2t ?The exception might be systems with localized states. 23 24 Chapter 3. General Perturbation In the continuous time case a is just the integral of the correlation function. Note that a has a well defined classical limit obtained from the classical correlation function and in the semiclassical limit this classical ac\ will agree with the quantum one. We can make a stronger statement in a non-linear-response regime if we make an additional assumption on the factorisation of higher order time-correlations, n—point mixing. This implies that 2m-point correlation (V(ji) ¦ ¦ ¦ V(J2m)} is appreciably different from zero for J2m — j\ —>¦ oo only if all (ordered) time indices {jk, k = 1... 2m} are paired with the time differences within each pair, J2k~J2k-i, being of the order or less than tmiX. Then we can make a further reduction, namely if t » mtmix the terms in the expansion of the fidelity amplitude f(t) (2.f6) are -0.015 10" 10" 10" 10" 10" 10" ^x- x„ -x- ~x 'X- x~ x- -x- -x -x- x- x—;k -x -x- x- _i____________i_ 10 t 15 20 #=X X X X >K X X X >K X X X X X X X X X X X X X X X X X X -Č; 10 15 20 25 30 35 40 t Figure 3.f: The classical correlation function (2.f8) of perturbation V (3.6) for chaotic kicked top and 7 = 7r/6 (top solid curve) and 7 = 7r/2 (bottom broken curve). The finite time integrated correlation function is shown in the inset. T E (V(ji)V(j2)---V(j2m))^ jl,---,J2m=0 t-l T E {V{h)V{J2)) ¦ ¦ ¦ (V(J2m-l)V(j2m)) jl,---,J2m=0 (2m)! m!2m (2at)r (3.4) The fidelity amplitude is therefore f(t) = exp (—52at/h2) and the fidelity is F(i)=exp(-i/rm), h2 7"rn — 252acl7 (3.5) with a mixing decay time-scale rm = 0{5~2) and a classical limit of the transport coefficient ac\. We should stress again that the above result (3.5) has been derived under the assumption of true quantum mixing which can be justified only in the limit M —> 00, e.g. either in the 3.1. Mixing Dynamics 25 semiclassical or the thermodynamic limit. Thus for the true quantum-mixing dynamics the fidelity will decay exponentially. The same result has been derived also by a quite different approach, using a Fermi golden rule (Jacquod et al. , 2001; Cerruti & Tomsovic, 2002). That is why this regime of exponential fidelity decay is sometimes called a Fermi golden rule regime. To numerically check the above exponential decay, we will use the kicked top (2.20) with parameter a = 30, giving a totally chaotic classical dynamics. As argued before, one can calculate the transport coefficient a (3.3) by using the classical correlation function of the perturbation (2.22), Vcx = v = ^z2. (3.6) We consider two different values of kicked top parameter 7, namely 7 = 7r/2 and 7 = tt/6. The classical correlation functions can be seen in Figure 3.1. The correlation function (obtained by averaging over 105 initial conditions on a sphere) is shown in the main frame. The correlation functions have qualitatively different decay towards zero for the two chosen 7's. In the inset the convergence of classical a (3.3) is shown, where one can see that the mixing time is tmiX ~ 5. The values of ac\ are ac\ = 0.00385 for 7 = 7r/2 and ac\ = 0.0515 for 7 = tt/6. These values are used 10~1 10~2 10~3 f 10-4 10~5 10"6 10" 0 500 1000 1500 2000 2500 3000 x=(dS)2 t Figure 3.2: Quantum fidelity decay in the chaotic regime for 7 = 7r/2 and three different perturbation strengths (5 = 5x 10~4, 1 x 10~3 and 3 x 10~3 (solid, dashed and dotted curves, respectively) is shown. The chain line gives theoretical decay (3.5) with the classically calculated a seen in Figure 3.1. to calculate the theoretical decay of fidelity F(t) = exp (—52S22ac\t) (3.5) which is compared with the numerical simulation in Figures 3.2 and 3.3. We used averaging over the whole Hilbert space, p(0) = l/S and checked that that due to ergodicity there was no difference for large S if we choose a fixed initial state, say a coherent state. As fidelity will decay only until it reaches its finite size fluctuating value F (2.43) we choose a large S = 4000 in order to be able to check exponential decay over as many orders of magnitude as possible. In Figure 3.2 the decay of quantum fidelity is shown 26 Chapter 3. General Perturbation for 7 = 7r/2. The agreement with the theory is excellent. Note that the largest 5 shown corresponds to rm ~ 1 so the condition for n—point mixing t » tmiX is no longer satisfied. The agreement with theory is still good due to the oscillatory nature of the correlation decay (see Figure 3.1) fuMling the factorisation assumption (3.4) on average. In Figure 3.3 for 7 = 7r/6 a similar decay can be 10L 10" 10" 10" 10" 10" 10" \ ' ' i r \cx g=p/6 ~ ~ \\ '''¦• ~. \\ \ \\\ \ S\'''-. - r V—A;\x\>-,.. / - r - 50 100 x=(dS)2 t 150 200 Figure 3.3: Similar figure as 3.2, only for 7 = 7r/6 and perturbation strengths 5 = 1 x 10 4, 2 x 10 4 and 3 x 10~4 (solid, dashed and dotted curves, respectively). seen. In both cases fidelity starts to fluctuate around F calculated in the section 2.2.1 for times larger than too- 3.1.1 Long Time Behaviour We assumed that the quantum correlation function C(j) decays to zero and its integral converges to a. For a system having a finite Hilbert space of size M, the correlation function asymptotically does not decay but has a non-vanishing plateau Č due to finite TV, similarly as we have a finite asymptotic value of the fidelity F. This will cause the double correlation sum to grow quadratically with time. Because this plateau Č is small, the quadratic growth will overtake linear growth 2ac\t only for large times. The time averaged correlation function C(j,k) (2.18) can be calculated assuming a nondegenerate unperturbed spectrum (/>& as 1 ¦00*2 t-i C = lim ^ ]T C(j,k) = Y^Pkk(Vkk)2 j,k=0 / ,PkkVkk \ k / (3.7) where pkk are diagonal matrix elements of the initial density matrix p(0) and Vkk are diagonal matrix elements of the perturbation V in the eigenbasis of the unperturbed propagator Uq. One can see that Č depends only on the diagonal matrix elements t, in fact it is equal to the variance The case when diagonal matrix elements are zero is the subject of Chapter 4. 3.1. Mixing Dynamics 27 10 O CO 0.1 0.01 0.001 ¦ tH /'¦ 0A02US/ : g=p/6 / ) ^ / /0.007 7t/S : \ / xf 9=p/2 ¦ : —"^^-——/ ^—-—-^^¦// ¦ 10 100 1000 10000 100000 Figure 3.4: The finite time quantum correlation sum ait) = Y?j k=o C(j, k)/2t (solid curves) together with the corresponding classical sum ac\(t) = Y?fk=oCc\{j,k)/2t (dashed curves saturating at ac\ and ending at t ~ 1000) is shown for the chaotic kicked top. Quantum data are for a full trace p = 1/7V with S = 1500. Upper curves are for 7 = 7r/6 while lower curves are for 7 = tt/2. Chain lines are best fits for asymptotic linear functions corresponding to Ct/2 = 0.0077t/S for 7 = 7r/2 and O.W3t/S for 7 = vr/6. of the diagonal matrix elements. Since the classical system is ergodic and mixing, we will use a version of the quantum chaos conjecture (Feingold & Peres, 1986; Wilkinson, 1987; Feingold et al. , 1989; Prosen & Robnik, 1993; Prosen, 1994) saying are independent Gaussian random variables with a variance given by the Fourier transformation S(u) (divided by M) of the corresponding classical correlation function Cc\(j) at frequency u = k) = S(0)t we can write C = 2S(0) 4acl N N ' (3.8) Because of ergodicity, for large TV, Č does not depend on the statistical operator p used in the definition of the correlation function, provided we do not take some non generic state like a single eigenstate \ tu, and provided 5 is sufficiently small, the correlation sum will grow quadratically and the linear response fidelity reads F(t) = 1 Č24(7C1 2 h2 N ¦ (3.11) To derive the decay of fidelity beyond the linear response regime one needs higher order moments of diagonal elements of perturbation V. If we use the BCH form of the echo operator (2.14) and discard the term involving * T(t), we have the fidelity amplitude f(t) = J2kexP(~^kk^t/h)/M, where we choose an ergodic average p = 1/7V. In the limit TV —> 00 we can replace the sum with an integral over the probability distribution of diagonal matrix elements p(Vkk) = p(V), f(t)= dVp(V)exp(-iV6t/h). (3.12) *in Chapter 4 we will see that the size of F(t) term grows at most linearly with time and so can be neglected because of S2 prefactor. 3.1. Mixing Dynamics 29 For long times the fidelity amplitude is therefore a Fourier transformation of the distribution of diagonal matrix elements. For classically mixing systems the distribution is conjectured to be Gaussian with the second moment equal to Č (3.8). This is confirmed by numerical data in Figure 3.5. The mean value of diagonal matrix elements is perturbation specific and is for our choice of the perturbation (2.22) Lfc Vkk/(2S + 1) = (2S + l)^ + 1)/12S2. From the figure we can see that the distribution is indeed Gaussian with the variance agreeing with the theoretically predicted Č = Aac\/S. The Fourier transformation of a Gaussian is readily calculated and we get a Gaussian fidelity decay F(t)=exp(-(t/rp)2), M h Aad5 (3.13) In order to see a Gaussian fidelity decay for mixing systems the perturbation strength 5 must be sufficiently small. If it is not, the fidelity will decay exponentially (3.5) to its fluctuating plateau F before time Lh when the Gaussian decay starts. Demanding that the mixing decay time rm is smaller than tu = J\f/2 gives the critical perturbation strength 5P, s* = 7br- (al4) For 5 < 5P we will have a Gaussian decay (3.13) otherwise the decay will be exponential (3.5), for details see Section 3.3. 0.1 0.01 0.001 0.0001 10 100 1000 s(dtr Figure 3.6: Quantum fidelity decay for 5 < 5P in the chaotic regime. For 7 = 7r/2 data for 5 = 1 • 10~6 (solid curve) and 5 • 10~6 (dotted curve) are shown. For 7 = tt/6, 5 = 3 • 10~7 (solid) and 1 • 1CT6 (dotted) are shown. Note that for both 7 the curves for both 5 practically overlap. The chain curves are theoretical predictions (3.13) with classically computed ac\. Again we numerically checked the predicted Gaussian decay for the chaotic kicked top with a = 30 and S = 1500 and a full trace average over Hilbert space. The results of numerical simulation, together with the theory are shown in Figure 3.6. 30 Chapter 3. General Perturbation The regime of Gaussian decay is sometimes referred to as the perturbative regime (Cerruti & Tomsovic, 2002; Jacquod et al. , 2001) because it can be derived using the lowest order perturbation theory. Writing first order corrections in phases ??k = ?k + Vkk?/h and for overlap matrix Pkl = ?kl + 0(?) (2.36) one gets the Fourier transform formula (3.12). 3.2 Regular Dynamics As opposed to mixing dynamics which was characterised by a linear growth of the double sum of the correlation function, the regular regime will typically exhibit quadratic growth. By a regular regime we again refer to the behaviour of the correlation function. Typically, its double sum will exhibit quadratic growth for integrable or mixed (KAM) systems. In such case, we can define a time average correlation function, denoted by C 1 t~1 C= lim i V C(j,k). (3.15) t^oo t2 ^ j,k=0 For nonzero C we will call this a “regular regime”. Of course, we have AA-dependent nonzero value of C also in mixing systems (3.8), as discussed in Section 3.1.1. Here we consider only systems where C exists in the limit of M ->¦ oo, i.e. nonzero C is a consequence of the dynamics and not of the finite Hilbert space size. Due to non-ergodicity all expectation values (like C) depend on the initial state, and we can not use ergodic average over the whole Hilbert space as for mixing dynamics. The time in which time averaging of the correlation function converges will be denoted by tave. For times t > tave the linear response fidelity decays quadratically in time - —h2 2 F(t) = 1 - —h2Ct 2 . (3.16) In contrast to the mixing regime one does not need any further assumptions in order to go beyond th V the linear response formula. Namely, for times t > tave we can define a time average perturbation V = lim - V V(k) = lim ?(t) . (3.17) Observe that V is by construction a constant of motion, [V, U0] = 0. In a mixing regime V is trivial (in the limit M ->¦ oo), i.e. proportional to the identity §, whereas for regular dynamics nontrivial V exists. The special case of V = 0 will be considered in Chapter 4. In the case of nondegenerate spectrum of the unperturbed propagator U0, the time average is simply the diagonal part of V in the eigenbasis \?k) of the unperturbed propagator, namely V— = Y,Vkk\?k)(?k\, (3.18) k where as usual Vkk = (?k\V\?k). Note that the average correlation function is C = (V 2) - (V)2 . (3.19) - §Actually it can be made zero by subtracting from V the identity operator which only rotates a phase of the fidelity amplitude and does not affect the fidelity itself. 3.2. Regular Dynamics 31 For times t > tave the operator ?(t) is dominated by the linearly growing term Vt and one can neglect contributions not growing with time. The BCH form of the echo operator can therefore be simply written as M?(t) = exp (-iV?t/h), t > tave, (3.20) where we neglected the term ~ ?(t)?2 as it becomes important only at large times ~ 1/?2, whereas the first term will cause the fidelity to decay already in time ~ 1/?. The above form of the echo operator will be the main ingredient of theoretical calculation of the fidelity decay in the regular regime. As the operator V commutes with the unperturbed propagator it is diagonal in the eigenbasis \?k). In integrable systems the basis states \?k) can be ordered in a very special way. Namely, there exist quantum numbers which are eigenvalues of canonical action operators having a very simple algebra. Using action-angle operators will make derivations easier and will furthermore enable us to use classical action-angle variables in the leading semiclassical order, thereby approximating quantum fidelity in classical terms. So before proceeding with the evaluation of f(t) for various initial states, let us have a look at action-angle operators. 3.2.1 Action-angle Operators Since we assume the classical system to be completely integrable (at least locally, by KAM theorem, in the phase space part of interest) we can employ action-angle variables, {jk, ?k, k = 1... d}, in d degrees of freedom system. In the present section, dealing with the regular regime as well as in Section 4.2.1 describing quantum freeze of fidelity in regular systems, we shall use lowercase letters to denote classical variables and capital letters to denote the corresponding quantum operators. For instance, the quantum Hamiltonian will be given as H(J, ?) whereas its classical limit will be written as h(j,?). As our unperturbed Hamiltonian is integrable, it is a function of actions only, i.e. h0 = h0(j). The solution of classical equations of motion is very simple, j(t) = j, ?(t) = ? + ?(j)t (mod2?) (3.21) with a dimensionless frequency vector ?(j) := ®Mj1. (3.22) The classical limit v (j, ?) of our perturbation generator V can be written as a Fourier series v(j,?)= J2 vm(j)eim?, (3.23) mezd where a multi-index m has d components. The classical limit of the time-averaged perturbation V isv = v0(j), i.e. just the zeroth Fourier mode of the perturbation. In quantum mechanics, one quantises the action-angle variables using the famous EBK procedure (see e.g. (Berry, 1977)) where one defines the action (momentum) operators J and angle operators exp(im • ?) satisfying the canonical commutation relations, [Jk,exp(im-?)] = —hmkexp(im-?), k = 1,...,d. (3.24) 32 Chapter 3. General Perturbation As the action operators are mutually commuting they have a common eigenbasis |n) labelled by d-tuple of quantum numbers n = (n1,..., nd), = —h(n + ?)|n) (3.25) J|n) = —h(n + ?)|n) where 0 ? ?k ? 1 are the Maslov indices which are irrelevant for the leading order semiclassical approximation we will use. It follows from (3.24) that the angle operators act as shifts exp(im·?)|n) = |n + m). (3.26) ·?)|n) = |n + m). The Heisenberg equations of motion can be trivially solved in the leading semiclassical order by simply disregarding the operator ordering, J(t) = e iH0t/—h Je -iH0t/—h = J, eim·e(t) = eiH0t/heim·®e-iH0t/—h ?= eim·?(J)teim·e (3 27) in terms of the frequency operator ?(J). Throughout this paper we use the symbol ?= for ’semiclas-sically equal’, i.e. asymptotically equal in the leading order in h. Similarly, time evolution of the perturbation observable is obtained in the leading order by substitution of classical with quantal action-angle variables in the expression (3.23) V(t) = e iH0t/—h Ve -iH0t/—h ?=ST^ (J)e im·?(J)t e im·@ (3 28) m Operator V(J) is diagonal in the eigenbasis |n) and therefore the expectation value of the echo operator (3.20) in the initial density matrix ? is f(t) = 53exp (-i?tV(h{n + ?})/—h)D?(hn), D?(hn) = (n|?|n). (3.29) (n|?|n). For pure initial states D? is just D? = |(?|n)|2. This is still the exact quantum mechanical expression of the fidelity amplitude. Now we make a leading order semiclassical approximation by replacing quantum V with its classical limit v and replacing the sum over quantum numbers n with the integral over classical actions j. The replacement of the sum with the action space integral (ASI) is valid up to classically long times ta, such that the variation of the argument in the exponential across one Planck cell is small, ta =1 ? 0 /?. (3.30) Subsequently we will see that the fidelity decays on shorter times and so the approximation is justifiable. By denoting with d?(j) the classical limit of D?(hn) we arrive at the fidelity amplitude f(t) ? h~d fddj exp f- ? tv(j)\ d ? (j). (3.31) This expression will be used to evaluate the fidelity for different initial states. 3.2. Regular Dynamics 33 3.2.2 Coherent Initial States We proceed to evaluate fidelity amplitude (3.31) for coherent initial state centred at (j*,?*) in action-angle phase space. The expansion coefficients of the initial density matrix ? = |j*, ?*)(j*, ?*| for a general coherent state in d degrees of freedom system can be written as '—h\d/4 1 (n|j*,?') = —h |det?|1/4exp - (Jn- j*)- ?(Jn -j*)-in-?* \ , (3.32) where ? is a positive symmetric d x d matrix of squeezing parameters and Jn = —h(n + ?) is an eigenvalue of operator J in eigenstate |n). The classical limit of D? (3.29) is therefore d?(j) = (—h/?)d/2|det?|1/2exp(-(j - j*) • ?(j - j* ')/—h). (3.33) The ASI for the fidelity amplitude can now be evaluated by the stationary phase method. To lowest order in ?t the stationary point js, i.e. the zero of [-i?tv - (j - j*) ¦ ?(j - j*)], is at js=j* + 2 ?-1v' + 0(?2), where the derivative of the average perturbation is For ?t < 1 the stationary point is simply at j* and the fidelity is F(t) = exp -(t/?r 2 , ?r = 1«/?L. (3.34) (3.35) -(t/?r)2, The fidelity amplitude on the other hand has an additional phase, f(t) = exp { -(t/?r)2/2 } exp(-iv(j*)?t/h). (3.36) (3.37) Comparing the fidelity (3.36) with the linear response formula (3.16) we see that the average correlation function for a coherent initial state is C = 1h(v' ¦ ?-1v—). We have already briefly explained the behaviour of classical fidelity in Section 1.1.2, for details see (Benenti et al. , 2003a). For regular dynamics one gets a ballistic decay of classical fidelity for perturbations predominantly changing frequencies of tori or power-law decay for perturbations predominantly changing the shape of tori. In the ballistic case, provided one is allowed to replace the perturbation v with its time average v(j), one gets the same Gaussian decay of the classical fidelity as for the quantum one (3.36). Therefore, in the regime where v determines the decay of classical fidelity, classical and quantum fidelity agree. From the expression for the echo operator (3.20) one can in this case interpret the fidelity as the overlap between the initial coherent state and the state obtained after the evolution with the Hamiltonian ?v. Because v is a function of actions only, the classical equations of motion are very simple, namely only the frequency of classical motion is changed by the amount ?? = v' ?. This change in frequency causes the “echo” packet M?(t)|?(0)> to move ballistically away from its initial position and as a consequence fidelity decays. For coherent state the shape of the packet is Gaussian and therefore fidelity decay is also Gaussian. For other forms of localized initial packets the functional form of the fidelity will be different but 34 Chapter 3. General Perturbation with the same dependence on the ballistic separation "speed" v . If V = 0 we obviously have v' = 0, this situation will be discussed in Chapter 4 describing a freeze of fidelity. In one dimensional systems (d = 1) another phenomena should be observable. After long time the echo packet will make a whole revolution around the torus causing the fidelity to be large again. This will happen after the so-called beating time tb determined by 5v'{j*)% = 2-7T, 2vr v'o (3.38) This beating phenomena is particular to one dimensional systems as in general the incomensurability of frequencies will suppress the revivals of fidelity. 400 Figure 3.7: Quantum fidelity for a coherent state of the regular kicked top, a = 0.1, S = 100, 7 = 7r/2 and d = 0.0025. The full curve is the theoretical decay (3.41) and the vertical line is indicates the theoretical beating time (3.38). Let us now illustrate the above theory with numerical experiment. We again take the kicked top (2.20), but this time with 7 = 7r/2 and a = 0.1 resulting in regular classical dynamics. The perturbation is the same as for mixing situation, i.e. a perturbation in a resulting in a classical perturbation v = z2/2 (2.22). Let us denote by #, ¦ 00, observed in a class of kicked spin chains by Prosen (2002). Agreement beyond linear response with the Gaussian decay is frequently observed also for a finite d, e.g. in a spin model of quantum computation (Prosen & ˇ nidariˇc, 2001) or even in a one dimensional kicked top seen in Figure 3.10. 38 Chapter 3. General Perturbation 0.1 0.01 0.001 10 100 1000 x=Sdt Figure 3.10: Fidelity decay for the regular kicked top, a = 0.1, 7 = tt/2, 5 = 0.01, S = 100. The solid curve gives the results of a numerical simulation for p = 1/7V on the OE subspace (2.24). Isolated dots denote the difference between numerical calculation and analytic formula (3.48) for a —>¦ 0, |Fnum.(i) — Fanaii(t)\. The dotted line gives a predicted asymptotic decay oc i_1, and the dashed/chain curves are the predicted fidelity decays at small times, namely the second order expansion F(t) = 1 — (St5)2/180, and 'improved' one by an exponential. The wiggling dashed curve represents the numerics for a single random state. For numerical demonstration we take the same kicked top system as for the case of coherent initial states, a = 0.1, 7 = 7r/2 and S = 100 and perturbation in parameter a, giving v = (1—j2)/4. Rather than comparing the numerical results with the asymptotic formula (3.46) we will try to directly calculate the fidelity decay in order to demonstrate oscillatory corrections due to a finite Hilbert space. We start with the quantum expression for the fidelity amplitude (3.29), using the eigenvalues (2m — l)2 of operator S2 on OE subspace, F(t) = S/2 S Y, exp (-idt(2m - lf/iS) m=\ (3.47) For large S we can replace the sum with an integral and get F(i) = 1st erfi(^/4V^Št) (3.48) where erfi(z) = ^A= JqZ e * dt is a complex error function with the limit lim-n^oo |erfi(ie17r/4y/š)| = 1 to which it approaches by oscillating around 1. We therefore have an analytic expression for the fidelity (3.48) in the case of an uniform average over the whole Hilbert space or, equivalently, over one random initial state. Its asymptotic decay is proportional to t~l which agrees with the general semiclassical asymptotic (3.46) but in addition we have an oscillatory erfi correction due to a finite 3.3. Time Scales 39 space. For small times we can get initial quadratic linear response decay by expanding the full theoretical formula (3.48) or more easily by simply calculating C according to formula (3.19). One can even use the classical calculation. For ? = 0 we have v = (1 - y2)/4, so the classical average correlation functions is — = —(y4- 2 ) = —, (3.49) 1 (y4-y22) = 1 , where the averages are performed over a uniform distribution on a sphere. In Figure 3.10 we compare theoretical decay (3.48) with the numerical as well as with the linear response formula F(t) = 1 - (?S)2Cclt2. Also, beyond the linear response there is a significant agreement with the Gaussian approximation F(t) = exp (-?2S2Cclt2) obtained by exponentiating the linear response expression. 3.3 Time Scales Let us close the present chapter with an overview of time scales of different fidelity decays depending on parameters ? and h and on the initial state. For regular dynamics we have only three relevant time scales, the classical averaging time tave in which the average perturbation operator converges to V, the quantum fidelity decay time ?r and the time too when the fidelity reaches a fluctuating plateau due to a finite dimension of Hilbert space. For times smaller than tave decay is system and state specific and can not be discussed in general. After tave the fidelity decay is first quadratic in time as dictated by the linear response formula (3.16). The decay time ?r scales as ~ y—h/? (3.36) for coherent initial states and as ~ h/? (3.46) for random initial states. Beyond linear response the functional dependence of the decay is Gaussian for coherent initial states and power law for random initial states with the before mentioned decay time ?r. This decay persists until the finite size plateau F (2.38) is reached at time t^. The time too again depends on the initial state as well as on the Hilbert space dimension M. For random initial states the power law decay gets faster with increasing dimensionality d of the system, and is conjectured to approach a Gaussian decay in the thermodynamic limit. For mixing dynamics we have a more complicated situation. There are five relevant time scales (even six for coherent initial states): The classical mixing time tmix on which correlation functions decay; the quantum decay time of the fidelity ?m; the Heisenberg time tH after which the system starts for “feel” finiteness of Hilbert space; the decay time ?p of perturbative Gaussian decay present after tH; the time too when the fidelity reaches finite size plateau; for coherent initial states we have in addition the Ehrenfest time tE up to which we have quantum-classical correspondence. Depending on the interrelation of these time scales, i.e. depending on the perturbation strength ?, Planck’s constant h and the dimensionality d, we will also have different decays of fidelity. All different regimes can be reached by fixing h and increasing ?. Let us follow different decay regimes as we increase ? (shown in Figure 3.11): (a) For ? ¦ 0 and in the thermodynamic d ->¦ oo limit. For coherent initial states the quantum correlation function relaxes on a slightly longer time scale, namely on the Ehrenfest time tE------ln h/?. Until tE quantum packet follows the classical trajectory and afterwards interferences start to build leading to the breakdown of quantum-classical correspondence. Equating ?m = tE gives the upper border for coherent states ?e = --l= J? = ?mix J-ln—h . (3.54) (d) For ? > ?mix the perturbation is so strong that the quantum fidelity decays before tmix, i.e. perturbed and unperturbed dynamics are essentially unrelated and fidelity decays almost instantly. For coherent initial states the upper border of regime (c) is at ?E which is smaller than the lower border of regime (d) ?mix which opens up the possibility of another regime between (c) and (d), namely for ?E < ? < ?mix the fidelity will decay within the Ehrenfest time. In this regime the decay of quantum fidelity is the same as the decay of classical fidelity and can be explained in terms of classical Lyapunov exponents (Veble & Prosen, 2004). Note that the relative width of this regime ?mix/?E = ln (1/—h)/?tmix grows only logarithmically in 1/h, i.e. much slower than the width of regime (c). vl 3.3. Time Scales 41 In Figure 3.11: Schematic view of different fidelity decay regimes for mixing dynamics. For details see text. The fidelity decay time scales as ~ 1/? for regular systems, while it is ~ 1/?2 for mixing dynamics. This opens up an interesting possibility: is it possible that the fidelity would decay faster for regular systems than for chaotic ones? The answer is yes. Demanding ?r < ?m we find that for sufficiently small ? one will indeed have faster fidelity decay in regular systems. This will happen for {?r = hC1/2/2?cl (xh random init.state ?c = h3/2Jv' ¦ ?~1v'/8?c2 l oc —h3/2 coherent init.state We explicitly wrote the result for regular initial states ?r and coherent initial states ?c as the two have different scaling with —h. We can see that for random initial states ?r scales in the same way as ?mix and so one has faster decay of fidelity in regular systems just provided ? < ?r ~ ?mix. For a coherent initial state this can be satisfied above the perturbative border ? > ?p only in more than one dimension d > 1. In one dimensional systems ?p and ?c have the same scaling with —h and whether we can observe faster decay of fidelity in regular systems than in chaotic ones depends on the values of ?cl and v'. We stress that our result does not contradict any of the existing findings on quantum-classical correspondence. For example, a growth of quantum dynamical entropies (Alicki et al. , 1996; Miller & Sarkar, 1999) persists only up to logarithmically short Ehrenfest time tE, which is also the upper bound for the so-called “Lyapunov” decay of the fidelity (Jalabert & Pastawski, 2001; Cucchietti et al. , 2002a) and within which one would always find Freg(t) > Fmix(t) (for coherent states) above the perturbative border ? > ?p, whereas our theory reveals new nontrivial quantum phenomena with a semiclassical prediction (but not correspondence!) much beyond that time. If we let —h —> 0 first, and then ? —>¦ 0, i.e. we keep ? » ?r,c(h), then we recover the result supported by classical intuition, namely that the regular (non-ergodic) dynamics is more stable than the chaotic (ergodic and mixing) dynamics. On the other hand, if we let ? —> 0 first, and only after that —h —> 0, i.e. satisfying (3.55), we find somewhat counterintuitive result saying that chaotic (mixing) dynamics is more stable than the regular one. We can conclude the section by saying that we have three non-commuting limits, namely the semiclassical limit —h —> 0, the perturbation strength 42 Chapter 3. General Perturbation limit ? ->¦ 0, and the thermodynamic limit d ->¦ oo, such that no pair of these limits commutes. Similar regimes as for the fidelity decay were also obtained by Cohen (1999, 2000); Cohen & Kottos (2000) when studying parametrically driven systems and energy dissipation. 3.3.1 Illustration with Wigner Functions In order to demonstrate faster decay of fidelity for a regular than for a chaotic system we did a numerical simulation on the kicked top. For chaotic situation we choose ? = 30, while for regular one we take ? = 0.1 as before. Other parameters are the same for the chaotic and the regular case, ? = ?/2, S = 100 and a coherent initial state at (??,??) = ?(1/>/3, 1/>/2). Perturbation of strength ? = 1.5 • 10-2 is in parameter ?. In Figure 3.12 we show the fidelity decay for both cases and an illustration of states in terms of Wigner functions. Spin Wigner function is a distribution on a sphere with a nice property that if we have two states ?1 and ?2 and their corresponding Wigner functions W?1 and W?2 the following equality holds, tr(?1?2) = lW?1W?2d?. (3.56) For an exact definition of a spin Wigner function see Appendix A. Wigner functions enable us to represent fidelity, which is an overlap of the initial state ?(0) with the echo state ?M(t) (2.8), as a phase space overlap integral of two Wigner functions. Also, the classical quantity analogous to Wigner function is just the classical density in phase space. But note that Wigner function is not necessarily positive whereas the classical density is, so the positivity of a Wigner function is a necessary condition (but not sufficient) for the classicality of a state. In Figure 3.12 we show for the chaotic (triangles in the fidelity plot and pictures above) and the regular case (pluses in the fidelity plot and pictures below) two series of Wigner functions at different times (t = 0,60,120): the Wigner function after the unperturbed forward evolution (row labeled “forward”) and the Wigner function after the echo (row labeled “echo”). We also show the structure of the classical phase space. In the inset the same data for the fidelity decay is shown on a longer time scale and the vertical line shows the theoretical position of tb. The fidelity is the overlap between the echo Wigner function and the initial Wigner function. For our choice of coherent initial states, the initial Wigner function is a Gaussian in the semiclassical limit (2.28). For chaotic dynamics the forward Wigner function develops negative values around the Ehrenfest time after which the quantum-classical correspondence is lost. For the regular dynamics this correspondence persist much longer, namely until the “integrable” Ehrenfest time ~ —h-1/2 after which the initial wave packet of size ~ —h1/2 spreads over the phase space. For a detailed study of Wigner functions in chaotic systems see (Horvat & Prosen, 2003; Lombardi & Seligman, 1993) and references therein. The echo Wigner function for regular dynamics moves ballistically from the initial position, causing the Gaussian decay of fidelity. But note that for regular dynamics the echo Wigner function does not have negative values even if they occur in the forward Wigner function. In our case the quantum fidelity agrees with the classical one for regular dynamics. In a chaotic case on the other hand, the echo image stays at the initial position and diffusively decays in amplitude, causing the fidelity to decay slower than in regular case. Classical fidelity follows quantum fidelity in the chaotic regime only up to the chaotic Ehrenfest time. Finally, we would like to illustrate the dimensional dependence of the fidelity decay. We have seen that in d > 1 border ?c is above the perturbative border and so one can get faster regular decay in a wider range of ?. In addition, strong revivals of fidelity in one dimensional regular systems for coherent initial states should be absent for d > 1. 3.3. Time Scales 43 t=0 t=60 t=120 forward echo 0.01 0.001 0.0001 echo forward Figure 3.12: Fidelity decay for chaotic (top curve and pictures) and regular (bottom curve and pictures) kicked top. Initial conditions and the perturbation are the same in both cases (see text for details). Wigner functions after forward and echo evolution are shown. 3.3.2 Coupled Kicked Tops As already remarked, for a one dimensional system (d = 1), the 'surprising' behaviour of the regular decay time being smaller than the mixing one, rr < rm, is for coherent initial states possible only 44 Chapter 3. General Perturbation around the border (3.50) 5P (unless ac\ is very small) where the exponential decay in the mixing regime goes over to a Gaussian decay due to a finite TV. However, for more than one degree of freedom, such behaviour is generally possible well above the finite size perturbative border 5P. In order to illustrate this phenomenon we will now briefly consider a numerical example of a pair of coupled kicked tops where d = 2. 10C 10" 10" 10" 10" 10 10 0 500 10001500 2000 2500 3000 3500 4000 4500 5000 t Figure 3.13: Fidelity decay for two coupled kicked tops, 5 = 8-10-4 and S = 200. The upper curves are for e = 20 (mixing regime), the solid curve for a coherent initial state and the dashed curve for a random one. The lower dotted curve is for e = 1 (regular regime) and a coherent initial state. The exponential and Gaussian chain curves give, respectively, the expected theoretical decays (3.5) and (3.36), with the Gaussian decay time determined by the best fit. We take a simplified version of coupled kicked tops (2.25) with a unitary propagator --- „—lo-Sly -if "Sby p—ieSizS2z/S U(e) =e-^2^ye-12i'2ye-^i^2,/i>^ (3^57) where S± and S% are two independent quantum angular momentum vectors. The perturbed propagator is obtained by perturbing the parameter e, so that U§ = U{e + 5). The perturbation generator is therefore V = -^SlzS2z, (3.58) with h = 1/S. We have used the propagator (3.57) over the full (2S + l)2 dimensional Hilbert space, without taking into account the symmetry classes of the double kicked top. The classical limit is obtained by S —> oo and writing the classical angular momentum vectors in terms of two unit vectors on the sphere r\$ = Si^/S. In component notation we get the classical map = *i,2 (3.59) = 2/1,2 cos(ez2,i) + «1,2 sin(e22,i) z'l2 = -iri,2Cos(eir2,i) + ž/1,2 sin(ez2,i) #1,2 ž/1,2 3.3. Time Scales 45 We have chosen two regimes, non-ergodic (KAM) regime for e = 1 where a vast majority of classical orbits are stable, and mixing regime for e = 20 where no significant traces of stable classical orbits were found and where the correlation sum was to a very good accuracy given by the first term only °" frv = ¦&*«&& = hi1+i)'- <3-60) The value of a if the whole correlation sum is calculated is a = 0.058, only slightly larger than 1/18 given by C(0). Our motivation here was to compare the regular and mixing fidelity decays for the coherent initial state which is here the product of spin coherent states (2.26) for each top |0,Y>> = |02,Y>2>®|#i,Y>i>. (3.61) In Figure 3.13 we show the fidelity decay at S = 200 and 5 = 8 • 10~4 in regular and mixing Figure 3.14: Numerically calculated decay time of the quantum and classical fidelity for the double kicked top. The two full curves show quantum fidelity, the one with a higher slope for the chaotic regime and the other for the regular regime. Symbols are decay times of classical fidelity, diamonds for the regular and squares for the chaotic regime. The two dotted straight lines are theoretical decay time predictions for the regular (3.63) and chaotic (3.62) regimes. The dotted curve (overlapping with squares) is the theoretical classical decay time rcias = In (0.25/č)/A. Vertical lines show the theoretical position of perturbation borders (3.64). The shading and the letters (a),(c) and (d) correspond to the regimes described in Figure 3.11. The Zeno regime corresponds to very short times r < 1 (i.e. strong perturbations d > 0.4). cases starting from the coherent state (3.61) with (#i, Se towards a genuinely quantum regime, where in a chaotic regime we can increase the decay time by increasing chaoticity - shaded region for the three smallest 5. 3.3. Time Scales 47 These theoretical borders are denoted with vertical lines in Figure 3.14. We can see, that in regular regime the quantum and the classical fidelity agree in the whole range of ? and furthermore agree with the theoretical ?r. In the chaotic regime things are a bit more complicated. By decreasing perturbation from ? = 1 we are at first in regime of very strong perturbation where the fidelity decay happens faster than any dynamical scale and it does not depend on whether we look at chaotic or regular system or quantum or classical fidelity. This is the so-called Zeno regime. For smaller ? the regular and chaotic regimes start to differ but in the chaotic regime we still have quantum-classical correspondence. This correspondence breaks down around ?mix where the quantum fidelity starts to follow the theoretical ?m, while the classical fidelity decay is ?clas = log (0.25/?)/?, with 0.25 being a fitting parameter (depending on the width of the initial packet) and ? = 0.89 is the Lyapunov exponent, read from Figure 3.15. For a brief explanation of this classical decay see Section 1.1.2. Incidentally, in our system at ? = 5 the classical mixing time is very short", tmix ~ 1, and we see that the correspondence breaks already slightly before ?E. The quantum fidelity decay time ?m is valid until a perturbative border ?p is reached, when a finite Hilbert space dimension effects become important and the decay time becomes equal to ?p. Note that for ? < ?c we indeed have a faster fidelity decay for a chaotic dynamics than for a regular one. Another interesting aspect of our correlation function formalism is that the decay rate of the fidelity in a mixing situation is proportional to the integral of the correlation function ?. As stronger chaoticity will usually result in a faster decay of C(t) and therefore in smaller ?, this means that increasing chaoticity (of the classical system) will increase quantum fidelity, i.e. stabilise quantum dynamics. Of course, for this to be observable we have to be out of the regime of quantum-classical correspondence. All this is illustrated in Figure 3.15, where we show similar decay times as in Figure 3.14, i.e. the same system, initial condition and S = 100, but depending on the parameter ? for six different perturbation strengths ?. Parameter ? controls the chaoticity of the classical dynamics. At ? = 1 we are in the regular regime and for larger ? we get into the chaotic regime, also seen from the dependence of the Lyapunov exponent. We can see that in the regular regime (? < 2) the classical fidelity agrees with the quantum one regardless of ?. In the chaotic regime though, the agreement is present only for the two largest ? shown, where we have ? > ?mix (3.64). For ? < ?c and for chaotic dynamics (three smallest ?) we get into the non-intuitive regime (shaded region in Figure 3.15) where the quantum fidelity will increase if we increase chaoticity. Note that this growth of the decay time stops at around ? ~ 4 because the classical mixing time tmix gets so small that the transport coefficient is given by its time independent term ? = C(0)/2 alone. "This is the reason why ? (3.60) is almost entirely given by C(0). Chapter 4 Special Case: Zero Time Averaged Perturbation One should always keep an open mind, but not so open that one’s brains fall out. —Bertrand Russell (?2(t)) ?2(t)oct0, j,k = 1,...,Af. (4.1) In the previous chapter we considered general perturbations, for which the double correlation sum was growing with time either linearly for mixing dynamics or quadratically for regular systems. What about the third possibility, namely if it does not grow with time. This situation will be the subject of the present chapter. We will demand that (?2(t)) ~ t0 for any initial state, i.e. all matrix elements of ?2 must be O(t0), 2k(t)oct0, j,k = 1,...,Af. We will write the perturbation V as the sum of its time average V and the rest, called the residual part Vres V=:V + Vres. (4.2) For a nondegenerate? spectrum of the unperturbed propagator U0 we have seen that the time averaged perturbation equals to the diagonal part of V (3.18) therefore, the residual Vres is just the off-diagonal part of V and has zeros on the diagonal, (Vres)kk = 0. The operator ?(t) can then be written as ?(V, t) = Vt + O(t 0 ), where the second part depends just on Vres and does not grow with time. One can conclude that to satisfy condition (4.1) the time averaged perturbation must be zero, V = 0. The subject of the present chapter are perturbations with V = 0, also called residual perturbations because V = Vres. For non-residual perturbations all essential physics was contained in the operator ?(t). For residual perturbations on the other hand, the second term involving ?(t) in the BCH form (2.14) of the echo operator will also be important. Matrix elements of ?(t) in the eigenbasis of U0 are (?j|?(t)|?j) (?j|?(t)|?k) 1 h 1 e -ih(?j-?k) +e -i(?j-?k)(-t) = —hY,|Vjk|2cot[12(?k-?j)] + O(t-1), (4.3) = (Vjj-Vkk)Vjk---------1------------------+ O(t-1), j=k. t —h jj kk jk 2sin[ 1 (?k-?j)] O(t-1), *For degenerate spectra we have V = k l s(k - 4>l)Vkl\4>k){4>l\. 49 Ek 50 Chapter 4. Special Case: Zero Time Averaged Perturbation Note that the matrix elements of ?(t) can not grow faster than linearly with t, despite the double sum over time in the definition of ?. We see that, provided the perturbation is residual, the limit of time-averaged ?(t) defined as — = lim ?(t) = i lim 1 y y [V(t'), V(t")] (4.4) exists and is diagonal in the eigenbasis of U0: 1 h ?— = I^?—jj|?j)(?j|, ?—jj = -y|Vjk|2cot[12(?k-?j)]. (4.5) k=j The operator ?— is again a constant of motion, [U0, ?— ] = 0. Any residual perturbation can be defined in terms of another operator W by the following prescription V = W(1) - W(0), W(t) := U0- tWUt (4.6) In the continuous time case we would have definition V =: (d/dt)W = —hi[H0,W]. Indeed, given a residual perturbation one easily determines the matrix elements of W as Wjk := exp (i(?j-?k))-1 . (4.7) Note that instead of the unperturbed evolution operator U0 one could use any other unitary operator that has the same degeneracy and eigenvectors as U0 in the definition of V in terms of W (4.6). With the newly defined operator W, the expression for ?(t) is extremely simple, ?(t) = W(t)-W(0). (4.8) Similarly, the expression for ?(t) is also considerably simplified, and ?(t) = ?R(t)-—h[W(0),W(t)], R:=—h[W(0),W(1)], (4.9) t-1 ?R(t) := y R(t'), R(t) := U0- tRU0t. (4.10) t'=0 The operator ?(t) is, apart from the term —hi[W, W(1)], similarly as ?(t) the sum of R(t). In the continuous time case we have R := —hi[W,(d/dt)W] = —h-2[W,[W,H0]] and ?(t) = f0dt'R(t') -i[W, W(t)]. The fidelity decay will be given by the echo operator M?(t) = exp |- i C?(t)? + -?(t)? 2 + •••)}, (4.11) with ?(t) (4.8) and ?(t) (4.10) expressed in terms of the operator W. We see that for small times t t2 instead, the fidelity decay will be determined by the operator R in ?(t). 4.1 Mixing Dynamics 4.1.1 The Plateau We begin with the linear response evaluation of the fidelity plateau (4.14). For classically mixing dynamics we can assume that for times larger than some mixing time t1 the time correlations vanish in leading semiclassical order, C(t) ->¦ 0(h). Also, quantum expectation values become time independent and equal to their classical values in leading order. We will denote by (A)cl := j dµAcl the classical average value of observable A. Therefore, for times t1 < t < t2 the linear response expression for fidelity plateau, i.e. its time-independent value is Fplat = 1 - —h2(?02 + ?2c l), ?2t := (W2(t)) - {W(t))2 , (4.15) where ?2c l := {w2)cl-(w)2cl is time-independent classical limit of?2t with the classical limit w(q,p) of operator W. We will consider two kinds of initial states, namely coherent states and random states. 52 Chapter 4. Special Case: Zero Time Averaged Perturbation For coherent initial states (CIS) the mixing time t1 is equal to the Ehrenfest time, t1 ? - ln h/?. As the initial state variance of W is proportional to the spread of the CIS, ?20 ? h, it can be neglected with respect to ?2c l. The fidelity plateau for CIS is therefore FpClaIS = 1 - (?/h)2?2cl. For random initial states (RIS), the mixing time is h independent, t1 ? —h0 . Also initial state average for RIS is equal to ergodic average and the fidelity plateau is in this case FpRlaIS = 1 - 2(?/h)2?2cl. Summarising, the linear response value of the fidelity plateau is for CIS and RIS 1 - FpClaISt = ?4, 1 - FpRlaISt = 2 ?4, (4.16) plat ? —h 22< i.e. it is twice as large for RIS than for CIS. One can go beyond linear response in approximating (4.14) using a simple fact that in the leading order in h quantum observables commute, and as before, that for t > t1 the time correlations vanish, namely {exp(-i$r(W(t)-W))) ? (exp(-i|W(t)))(exp(i|W)). Fort > t expectation values become time independent and so Fplat = |(exp (-iw?/h))cl (exp (iW?/h)}|2 . (4.17) Note that the the right average is an average over initial state whereas the left average is the classical average over the invariant measure. The fidelity plateau can be compactly expressed in terms of a generating function G(?/h) G(z) := (exp(-izw))cl = V fddqddp exp{-izw(q,p)}. (4.18) For CIS one can neglect the initial state average over a localized packet, i.e. the second term, and gets FpClaISt = |G(?/h)|2. (4.19) For RIS on the other hand, initial state average is equal to the classical average in the leading order and we have F^? |G(?/h)|4. (4.20) We have a universal relation between the plateau for CIS and RIS, namely F$® ? (FCIS)2. If the argument z = ?/—h of the generating function is large, the analytic function G(z) can be calculated generally by the method of stationary phase. In the simplest case of a single isolated stationary point x* in N dimensions we obtain |G(z)| - 2z \det?Xj?Xkw(x*)\ 1/2 — 1 / 2 ?Xkw(x*)\ . (4.21) This expression gives an asymptotic power law decay of the plateau height independent of the perturbation details. Note that for a finite phase space we will have oscillatory diffraction corrections to Eq. (4.21) due to a finite range of integration Jdµ which in turn causes an interesting situation for specific values of z, namely that by increasing the perturbation strength ? we can actually increase the value of the plateau. We tested the above theory by numerical experiments. For the system we choose the kicked top, with slightly different order of factors as before (2.20). One step propagator is -i?^S exp(-i-Sy 1. (4.22) 4.1. Mixing Dynamics 53 1 0.99 0.98 0.97 0.96 0.95 0.94 1 0.1 0.01 0.001 K ¦-/-Wr/-"^, A^A^V ^^^^^y^. - (a) Figure 4.1: F(i) for the kicked top with perturbations 5 = 10~3 (a), and 5 = 10~2 (b). In each plot the upper curve is for CIS and the lower for RIS. Horizontal chain lines are theoretical plateau values, linear response (4.16) in (a) and full (4.20,4.19) in (b). Vertical chain lines are theoretical values of L2 (4.29). The full circles represent calculation of the corresponding classical fidelity for the CIS which follows quantum fidelity up to the Ehrenfest (log ti) barrier and exhibits no freezing. We take a = 30 ensuring fully chaotic classical dynamics whereas spin size is taken S = 1000 giving effective Planck's constant h = 1/S = 1 • 10~3. Again we choose two initial states, a random one and a coherent one centred at (#*, 2 2S2 (4.23) The perturbed propagator is as always U§ = Uoexp(—i5V/h). We choose two perturbation strengths. A weak d = 10~3 to check the linear response expressions (4.16). The results are seen in Figure 4.1a. The classical value of n\ used in theoretical formulas is easily calculated for w = z2/2 and one gets n\ = 1/45. In Figure 4.1b we show numerical results for larger 5 = 10~2 where the plateau is very low and full formulas must be used. Generating function (4.18) can be 54 Chapter 4. Special Case: Zero Time Averaged Perturbation calculated exactly without stationary phase approximation, resulting in G(?S) = j? _______ (4.24) We have an asymptotic power law decay as predicted by the general stationary phase formulat (4.21). In addition, due to a finite phase space we have a diffractive oscillatory erfi correction. Theoretical prediction for the plateau using G(?S) (4.24) agrees well with the numerical results shown in the figure. Small quantum fluctuations around the theoretical plateau values lie beyond the leading semiclassical approximation used in our theoretical derivations. Note also that the quantum fidelity and its plateau values have been expressed (in the leading order in h) in terms of classical quantities only. Yet, the freezing of fidelity is a purely quantum phenomenon as one can also see in Figure 4.1 where the classical fidelity does not exhibit freezing. 4.1.2 Beyond the Plateau Next we shall consider the regime of long times t > t2. Then the second term in the exponential of the echo operator (4.11) dominates the first one, however even the first term may not be negligible for large ?. Up to terms of order 0(t?3) we can factorize Eq. (4.11) as Ms(t) « exp(-if (W(t) W))exp(-ig?(t)). When computing the expectation value we again semiclassical order operator ordering is irrelevant and that, since t > t1, any « exp(-i— - time-correlation can be factorized thus the second term —i[W, W(t)] of ?(t) (4.9) vanishes and we have F(t) = Fplat exp (-i2—h?R(t) , t>t2. (4.25) This result is quite intriguing. It tells us that apart from a pre-factor Fplat, the decay of the fidelity due to a residual perturbation is formally the same (in the leading semiclassical order when time ordering is not important) as the fidelity decay with a generic non-residual perturbation, eq. (2.11), when one substitutes the operator V with R and the perturbation strength ? with ?R = ?2/2. Thus we can directly apply the semiclassical theory of fidelity decay for general perturbations explained in Section 3.1, using a renormalised perturbation R of renormalised strength ?R. Here we simply rewrite the key results in the ’non-Lyapunov’ perturbation-dependent regime, ?R < h. Using the classical transport rate ?r, ?R := lim(M2tM, (4.26) we have either an exponential decay F(t) = Fplat exp (-—?R t) , ttH. The crossover between the two decay regimes happens at the Heisenberg time tH = N/2 (4.28) (3.10). †Note that w = z2/2 has one stationary point in N = 1 dimension, despite the phase space being two-dimensional. 4.1. Mixing Dynamics 55 0.1 0.01 0.001 Figure 4.2: Long time Gaussian decay (4.28) for the same CIS and parameters as in Figure 4.1a. The full curve is a direct numerical simulation, empty circles are numerical calculation using a renormalised perturbation R of strength 5r (4.30), while the chain curve gives the theoretical decay. Of course, the same consideration regarding the asymptotic saturation value of the fidelity F due to a finite TV applies here as well (see discussion of time scales in Section 3.3). The prefactor Fpiat can be calculated as described in the previous section, Eq. (4.17), and depends on the initial state. The exponential terms of (4.27,4.28) on the other hand do not depend on the initial state. From the two possible regimes of long-time fidelity decay we can also more precisely specify time Č2 when the plateau ends. Comparing the two factors in (4.27,4.28) with the plateau value (Eq. 4.16 for 5 < h), we obtain a semiclassical estimate of L2 *2 mm '*H«ii cr S' o-r52 (4.29) In Figure 4.1 we can see nice agreement of theoretical L2 with the results of the simulation. Interestingly though, as we have another time scale L2, the duration of the plateau, not present for a general case of non-residual perturbation, the exponential regime (4.27) can only take place if L2 < Lh> otherwise we immediatelly get a Gaussian decay after the plateau. If one wants to keep -Fpiat ~ 1 (i.e. high) and have an exponential decay in the full range until the asymptotic F ~ 1/W, the condition on dimensionality is imposed. Namely, demanding high plateau 5/h < 1 for t < L2 and low fidelity in the limit h —> 0 at the Heisenberg time, F(tn) = exp (—(5/h)4o-R/4hd~2), gives condition on dimensionality d > 2. In one dimensional systems and in the semiclassical limit the exponential decay (4.27) therefore can not be seen. We again compared theory with numerics. The system is the one dimensional kicked top already used in the previous section describing the plateau. The perturbation generator R for our choice of W is (4.30) R — „o3 WxSySz bzbybx 56 Chapter 4. Special Case: Zero Time Averaged Perturbation having a classical limit Rc\ = r = —xyz. We numerically calculated the integral of the classical correlation function (4.26) giving the transport coefficient ctr = 5.1 • 10~3. In Figure 4.2 we show three different sets of data. Direct numerical calculation, "a renormalised calculation" obtained by taking perturbed dynamics as U§ = Uoexp(—id-RR/h), i.e. perturbation generator R with the strength S-r = 52/2 and theoretical Gaussian decay (4.28) where all parameters have been calculated classically. Apart from the plateau prefactor Fpiat (which is close to 1) the decay does not depend on the initial state. Double Kicked Top 10l 10"1 10"' 10"' io-' 10"' 10c 10"1 0 10" 10" 10" 10" 1000 2000 3000 4000 5000 0 0.210s 0.410s 0.610s 0.810s 1.010s 1.2-KT t Figure 4.3: Long time fidelity decay for two coupled kicked tops. For strong perturbation d = 7.5 • 10~2 in (a) we obtain an exponential decay (4.27), and for smaller 5 = 2-10~2 in (b) we get a Gaussian decay (4.28). The three curves have the same meaning as in Figure 4.2. To demonstrate that for d > 1 we can also get an exponential decay of the fidelity, we also consider a two dimensional system. We look at two (d = 2) coupled tops Si and S2 described by a propagator Uq = exp (—ieSziSZ2) exp (—i7rSyi/2) exp (—i7rSy2/2), (4-31) with a perturbation generated by W = A1®t + t®A2, A = S2J2S2. (4.32) 4.2. Regular Dynamics 57 We set S = 1/h = 100, and ? = 20 in order to be in a fully chaotic regime. The initial state is always a direct product of spin coherent states centred at (?1,2, ?1,2) = (1,1) which is subsequently projected on an invariant subspace of dimension N = S(S+1) spanned by {Hoe ? Hr}sym, where Hr=H\ Hoe and {·}sym is a subspace symmetric with respect to the exchange of the two tops. The results of numerical simulation are shown in Figure 4.3. We show only the long-time decay, as the situation in the plateau is qualitatively the same as for d = 1. For sufficiently large perturbation one obtains an exponential decay shown in Figure 4.3a, while for smaller perturbation we have a Gaussian decay shown in Figure 4.3b. Numerical data have been successfully compared with the theory (4.27,4.28) using a classically calculated ?r = 9.2 · 10-3 together with theoretical Fplat, and with the “renormalised” numerics using the operator R, similarly as in Figure 4.2 for a one dimensional system. 4.2 Regular Dynamics The procedure of calculating the fidelity decay for the case of regular dynamics will be similar as for general perturbations in Section 3.2. We will use the classical action-angle variables (for definition see Section 3.2.1) and semiclassical methods to calculate the fidelity plateau as well as its long time decay. The only difference will be that in contrast to the case of a general perturbation, where only ?(t) was important in the BCH form of the echo operator, here the second term involving ?(t) will also be relevant for long times. The same form of the echo operator (4.11) has already been used for the case of a residual perturbation in mixing systems as described in the previous section. The existence of action-angle variables enables us to expand the classical limit v of the quantum perturbation generator V into Fourier series v(j,?):= Evm(j)eim·0. (4.33) The fact that the perturbation is residual is reflected in the zeroth Fourier coefficient which is zero, v0 ? 0, so we explicitly excluded this term from the summation. Classically this means that the perturbation only changes the shapes of tori as the average change of the frequency along the unperturbed tori is zero. For the explanation of the decay of classical fidelity in case of a residual perturbation see (Benenti et al. , 2003a). Similarly, we can expand the classical limit w(j,?) of a quantum observable W, w(j,?):= 5wm(j)eim·fl. (4.34) The zeroth Fourier mode w0 can be set to zero as it cancels in the definition of v in terms of w. Using these expansions we can easily calculate the leading semiclassical forms of ?(t) and ?(t). For ?(t) (4.8) we have ?(t) ? w(J,? + ?(J)t)-w(J,?) ? ]T wn(J)(eim·^J)t-1)eim·?. (4.35) Note that this is still an operator (capital J and ?) and is correct in the leading semiclassical order (the sign ?=). Coefficients wm can be also expressed in terms of vm as wm(j) = -ie-im·u,(i)t/2-------1m (4.36) 2sin ( In the continuous time case we have wm = -ivm/m · ?. vm(j) 2sin(1m·?(j)) 58 Chapter 4. Special Case: Zero Time Averaged Perturbation The operator ?(t) will determine the plateau. Long time decay on the other hand will be dictated by the operator ?(t). For long times when this decay will take place we can define an average — (4.4) and approximate ?(t) « — t. This approximation is justifiable as we will see, because the fidelity decay will happen on a long time scale ~ 1/?2, whereas the average of ?(t) converges in a much shorter classical averaging time tave. As one can see from the definition of ?(t) in Eq. (4.9), the average — is nothing but the time averaged operator R. For R the semiclassical limit r can be calculated using the Poisson brackets instead of commutators, r = -{w, w(1)}. When we average r over time, only the zeroth Fourier mode survives resulting in ?—(J)=r—(J), r(j) = - J2 m ¦ ?j {\wm(j)\2 sin[m ¦ ?(j)]} . (4.37) In the continuous time case we would have r— = - Em=0 m ¦ ? j {\wm\2 m • ?}. From the equation for wm (4.36) in terms of vm we see that the denominator sin (m • ?/2) could cause problems at points of vanishing m ¦ ?(j) = 0. This divergence carries over to all operators like — or ?(t). In classical perturbation theory this so-called “small denominator” problem is well known and cannot be avoided. For our quantum mechanical calculation though, there is an easy remedy. It is sufficient to remember that we are dealing with a residual perturbation, i.e. the one for which all matrix elements Vjk are zero if the corresponding eigenphases are equal, ?j = ?k. But the term m • ? is nothing else than the semiclassical approximation for ?j - ?k. Therefore, one can see that actually all the diverging terms have to be excluded when evaluating expectation values. This is furthermore confirmed if we compare the semiclassical expression for — (4.37) with the matrix elements of operator ?(t) (4.3). An example of such a calculation will be presented when describing the decay for random initial states. Using semiclassical expressions for ?(t) and — we can write the echo operator as Ms(t) = exp j- i (?(t)? + -?—t?2 j, (4.38) with ?(t) and — given in Eqs. (4.35,4.37). The third order term in the BCH form of the echo operator (4.38) can be shown to grow no faster than ?3t and can therefore be neglected. For times smaller than t2 (specified later) the term involving ?— can be neglected and the fidelity will exhibit freeze. The fidelity amplitude of the plateau is f(t)plat = (exp (-i??(t)/—h)> with semiclassical ?(t). Furthermore, for t > t1 the fidelity plateau is time independent and can be calculated by averaging the fidelity amplitude over time fplat =flim- ]Tf(t). (4.39) Time averaging the classical version of ?(t) = ?(j,?(t)) (4.35) is equivalent to averaging over the angle ?, resulting in the fidelity plateau fplat = /exp (i—hw( J, ?)) I dx2 exp f- ? w(J, x)\ \ , (4.40) where we assumed ergodicity of m -?t so we could replace a sum over t with an integral over angles. For long times t > t2 the second term in the echo operator becomes dominant. If the plateau is small (?¦ ?n,m/M. From this we immediately conclude that index mi in Eq. (4.45) must be zero, and as a consequence also m = 0 is zero. Therefore, for random initial states only terms with all ml = 0 survive the averaging over random ensemble (which is the same as if we would average over time instead of over random ensemble). The time scale t1 after which approximation with a time average is permissible is determined just by frequencies, t1 ~ 1/|m-?| and does not depend on h or perturbation strength ?, i.e. for random initial states we have t1 ~ ?0—h0. (4.46) 60 Chapter 4. Special Case: Zero Time Averaged Perturbation For localized initial states a little more work is needed to show that the expectation value equals the time average. For definiteness we will consider coherent initial states, for which the matrix elements have a Gaussian distribution (3.32). Furthermore, we will assume that the number of relevant Fourier modes is smaller than the width (in n) of the initial packet. As the width of the packet grows as —h-1/2 this is justifiable in the semiclassical limit provided w(j,?) is sufficiently smooth, so that its Fourier coefficients decrease sufficiently fast. In this approximation we have ?n,n+m>=d?(hn)eim'-?, (4.47) with d? a classical limit of the quantum structure function (3.33). The inner summation over quantum actions hn in Eq. (4.45) can now be replaced with an integral over the classical action j and the method of stationary phase can be used to calculate the resulting integral. Expanding frequencies around the centre of the packet j*, ?(j* + x) = ?(j*) + ?x + • • •, where ?jk := ??j(j*) (4.48) is a matrix of frequency derivatives, we can calculate the sum 5>(—hn)eim?(—hn)td?(—hn) = —h- d f ddjg(j)eim?(j)td?(j) n = g(j*)eim?(jt)t ?V/2 det ?\1/2 /ddx exp (- 1-x ¦ ?x + itm ¦ ?x) = g(j)eim?(jt)t exp (-m ¦ ??-1?Tm) . (4.49) We see that all terms with m = 0 decay to zero. The longest decay time scale of Gaussian envelopes is estimated as /—h \-1/2 t1= - min m ¦ ??-1?Tm) oc —h-1/2. (4.50) This means that for times longer than t1 only term with zero m will survive which is equivalent to time averaging the classical expression. Note that the Gaussian decay (4.49) is absent if ? = 0, e.g. in the case of a d-dimensional harmonic oscillator. Fidelity decay for a residual perturbation in a harmonic oscillator will be discussed for a Jaynes-Cummings model in Section 5.5. There may also be a general problem with the formal existence of the scale t1 (4.50) if the derivative matrix ? is singular, but this may not actually affect the fidelity for sufficiently fast converging or finite Fourier series (3.23). Time t1 can be interpreted as the integrable Ehrenfest time, up to which quantum-classical correspondence will hold. After t1 a quantum wave packet of size ~ \—h will spread over a classically large region of size ~ —h0 and interference phenomena will become important. This will also be reflected in the fidelity. As the perturbation wm will couple different tori, the echo packet will also start to exhibit interferences after t1 and therefore we can expect agreement between quantum and classical fidelity only up to time t1. This must be contrasted with the case of a general non-residual perturbation, where there were no upper limits on the correspondence between quantum and classical fidelity for localized initial packets under certain conditions. 4.2.1 The Numerical Model For the sake of numerical demonstration in the present section of residual perturbations in a regular dynamics we take an integrable kicked top model with a slightly different unperturbed one step 4.2. Regular Dynamics 61 propagator -i-SlS-ßi I, (4.51) where we introduced a second parameter ß. The classical map corresponding to U0 is simply a twist around z-axis xt+1 = xtcos(?(zt-ß))-ytsin(?(zt-ß)) yt+1 = yt cos (?(zt-ß))+xt sin (?(zt-ß)) (4.52) zt+1 = zt. Unperturbed evolution represents a continuous time system with the classical Hamiltonian h0 generating a frequency field ?(j) h0(j) = -?(j - ß) 2 , ?(j) = ?(j - ß). (4.53) Here we used a canonical transformation from a unit-sphere to an action-angle pair (j, ?) G [-1,1] x [0,2?), namely x = 1 - j2 cos ?, y = 1-j2sin?, z = j. (4.54) We perturb the Hamiltonian by periodic kicking with a transverse pulsed field in x direction, h?(j, ?, ?) = -?(j - ß)2 + ?y/1 - j2 cos ? JT ?(?-k). (4.55) k=—oo Perturbed quantum evolution is given by a product of two unitary propagators U? = U0exp(-i?Sx), (4.56) so the perturbation generator is V = Sx/S. (4.57) The classical perturbation has only one Fourier component, namely v(j,?) = 1-j2cos?, v±1(j) = -1-j2, (4.58) y1 - j2 cos ?, v±1 (j) = - y1- whereas zeroth Fourier component is zero, v0 = 0, indicating that we have a residual perturbation, v— = 0 and V = 0. The classical limit w of W is also readily calculated, giving -?/2) w(j,?) = 21-j22 - , (4.59) with ? = ?(j - ß). The reason we introduced parameter ß is to be able to control a possible singularity in w(j, ?) at points where ? = 0. 62 Chapter 4. Special Case: Zero Time Averaged Perturbation 4.2.2 Coherent Initial State The Plateau For times t < t1 quantum fidelity will follow the classical fidelity and will be system specific. For times t1 oo the agreement with semiclassical theory improves and that the size of the resonant spikes is of the same order as the drop in the linear response plateau. The insets show the data and the theory on the normal scale. for F(t) = exp (—(t/rr)2) (3.36) we only have to replace the perturbation d with 52/2 and v with 4.2. Regular Dynamics 65 r—, so that the fidelity decay is r 1 8—h F(t) ?=exp-(t/?rr)2, ?rr = -2?-1—7, (4.66) with the semiclassical r— given in Eq. (4.37) and its derivative —' := ?j r—. The decay time scales as ?rr ? h1/2?-2 and is thus smaller than the upper limit ta ? —h0?-2 of the validity of the stationary phase approximation used in deriving the Gaussian decay. Remember that the above formula is valid only if the plateau is small and the ?(t) term can be neglected, i.e. for ? < ?C-I1S/ 2h. For such small ? the crossover time t2 from the plateau to the long time decay can be estimated by comparing the linear response plateau (4.65) with the long time decay (4.66), (?/h)2?CIS ? (t2/?rr)2, resulting in t2 ? ?rr??CIS/—h ? —h- 1/2?-1. For stronger perturbations, namely up to ? ~ ? h time t2 can be simply estimated by ?rr. We therefore have ? t2 = min{1, %1C/I2S}?rr = min{const h1/2?-2, const —h-1/2?-1}. (4.67) Time scale t2 can be seen in Figure 4.4 as the point of departure of fidelity from the plateau value. Using our model with ? = 1.1 and ß = 0 and the position of the initial coherent state at (?*,?*) = (1,1) this can be calculated to be t2 = min{0.57 ?Š/?,0.57/(?2 ?Š)} (similarity of numerical prefactors is just a coincidence). These theoretical positions of t2 are shown with vertical chain lines in Figure 4.4 and are given by ?rr?C1/I2S?/h in Figures 4.5a,b while it is ?rr for a strong perturbation ?S = 3.2 in Figure 4.4c. Long time decay of fidelity is shown in figure 4.5. Theoretical Gaussian decay (4.66), shown with a chain curve, is again confirmed by numerical results. The average — given by classical r— is (4.37) = 2 1?(1 - j) + 2jsin(?j) . (4.68) 8sin2(?j/2) ?(1 - j2) + 2jsin(?j). The derivative r—' is r— = - 4cot (?j/2) - ? 4j + ?(1 - j2) cot (?j/2) /sin2 (?j/2) , which gives using ? = 1/(1 - j2) for spin coherent states the decay time (4.66) ?rr = 0.57?-2—h1/2. Note that we do not have any fitting parameters, except in the case of a strong perturbation (?S > 1, Figure 4.5c) where the prefactor and the exponent of a Gaussian had to be slightly adjusted due to the non-negligible effect of the first term in (4.38) [see caption for details]. Average fidelity We should remark that, although we obtain asymptotically Gaussian decay of fidelity for a single coherent initial state, one may be interested in an effective fidelity averaged with respect to phase-space positions of the initial coherent state, similarly as for the case of general perturbations (3.42). For times t < t2 when we have a plateau, the average fidelity (F(t))j will also have a plateau of the same height as the plateau for a random initial state calculated in Section 4.2.4, Equation (4.82). In the linear response regime this plateau is just twice as large as for a single coherent state. Long time decay of the average fidelity (F^j will be given by the phase space average of the regular decay time ?rr. The calculation is very similar to the one for a general perturbation described at the end of Section 3.2.2 so we wont repeat is here. Asymptotically we will have a power law decay with 66 Chapter 4. Special Case: Zero Time Averaged Perturbation the power determined by the order rj of zeros of f' ¦ A_1f' and dimensionality d (see Eq. 3.43) which is not universal as claimed by some authors (Jacquod et al. , 2003). To demonstrate asymptotic power law decay (3.43) we take the same regular kicked top model (4.51) as used throughout this section with parameter a = 1.1, /3 = 0 and spin size S = 1000. We take perturbation (4.57) of strength 5 = 5 • 10~4 and average fidelity over 200 coherent initial states placed randomly over sphere. The results are in Figure 4.6. The rr2r, i.e. f' ¦ A~lf' has for our f' (4.69) two zeros of the 0.01 Figure 4.6: Average fidelity decay (F(i)) • (4.71) for the regular kicked top with freezing (no fitting parameters!). The dotted line is the theoretical power law decay (4.71). The horizontal line gives the theoretical value of the plateau (4.84). first order, rj = 1, at j = ±1. The derivative (i.e. the first nonzero term in the expansion around zero) of the term f' ¦ A~lf' evaluated at j = —1 is c = 2a — sin a 2sin2 (a/2)' The asymptotic theoretical decay (3.43) should then be W)>, 8 S^SVc (4.70) (4.71) where for a = 1.1 the coefficient is c = 2.4. This theoretical decay agrees with numerics in Figure 4.6. The theoretical value of the plateau according to the Equation (4.84) for a random initial state should be 0.58 which also agrees with numerics. Illustration in Terms of Wigner Function All the phenomena described theoretically in the preceding subsections can be nicely illustrated in terms of the echoed Wigner function — the Wigner function Wpm(cos$, (p) of the echo-dynamics. 4.2. Regular Dynamics 67 Figure 4.7: [Movies online (Prosen & ˇ nidariˇc, 2003a)] Snapshots of the Wigner function of echo-dynamics for a quantized top a = 1.1, /3 = 0,7 = 0 with S = 200 and for 5 = 1.6 • 10 -3 (same as in Figures 4.4a,4.8a). The upper hemisphere is shown with j = costf G [0,1] on the vertical axis and 6 =

L2, even the frozen piece starts to drift ballisticly away from the position of the initial packet, thus explaining a fast Gaussian decay of fidelity. 4.2.3 Echo Resonances for Coherent Initial States 1 0.94 Ll_ 0.88 0.82 1 0.94 Ll_ 0.88 0.82 1 0.94 Ll_ 0.88 0.82 400 600 800 t 1000 1200 1400 Figure 4.8: Structure of echo resonances for coherent initial states of the modified quantum top Uo = exp(-iS[a(Sz/S)2/2 + j(Sz/S-f)3/6}), S = 200,5 = 1.6 • 10"3, for increasing values of ijj" = 7 = 0 (a), ijj" = 1 (b), and u" = 4 (c), which weakens and broadens the resonances. Note that in (a), u" = 0 we have the same data as in Figure 4.4a. Vertical chain lines show theoretical times ir/2 for ir, and ir for 2n resonances. Let us now discuss the behaviour of the fidelity for initial wave-packets in the regime of linear .(a) 1 ' 1 ' 1 ' w"=0 1 P 1 111. -nl M 1 1 |lr w"=1 _l__________1__________L_ 4.2. Regular Dynamics 69 response approximation in some more detail. We shall consider possible deviations from the time average plateau (4.40). Specifically we will explain the resonances observed e.g. in Figure 4.4. For such a resonance to occur the phases of terms (?k(t)} (4.44) have to build up in a constructive way and this is clearly impossible in a generic case, unless: (i) We have one dimension d = 1, so we sum up over a one-dimensional array of integers n in action space*. (ii) The wave-packet is localized over a classically small region of the action space/lattice such that a variation of the frequency derivative d?(j)/dj over this region is sufficiently small. In this subsection we thus consider a one-dimensional case, d = 1. For simplicity we will focus on a linear response regime, so we will study time-dependent terms (?2(t)) and (?(t)> with ?(t) given in Eq. (4.35). In the justification of the time average plateau, Section 4.2, we replaced a sum over n with an integral over action space. Here the time is not small enough and furthermore the very graininess of quantum actions is the source of resonances and we will have to explicitly consider sums over n. We seek a condition, such that consecutive phases in the exponential build up an interference pattern. 2?-resonance: Let us expand the frequency around the centre of the packet ?(j) =?* + (j - j*)? + -(j- j)2?" + ... (4.72) where ?* = ?(j), ? = d?(j)/dj and ?" = d2?(j*)/dj2. The phases in the sums of the form (4.49) with factors eimw^ra) come into resonance, for m = 1 and hence for any higher m > 1, when they change by 2? per quantum number, which happens at time tr, 2 /dj2. The phases in the sums —h ?'tr = 2?, tr = —h ? , (4.73) and its integer multiples. It is interesting to note that these resonant times correspond precisely to the condition for revivals of the wave-packet in the forward evolution (apart from a phase-space translation) studied in (Braun & Savichev, 1996) and (Leichtle et al. , 1996). In addition, we need that the coherence of linearly increasing phases is not lost along the size of the wave-packet, i.e. ? := m?'t?2 = —hm?t < 2?, (4.74) where we denoted by ?,- := ((j - j*)2)1/2 = 2^ the action-width of the initial wave-packet. Therefore, if ? < 2? we will observe echo resonances at integer multiples of 2?-resonance time tr. The shape of the echo resonance can also be derived. Let time t be close to ktr, k G , and write t = ktr + t where t' < tr, so that —h?'t < 2?. We can estimate the general time dependent sum over n in entirely analogous fashion to Eq. (4.49) by: (i) shifting the time variable to t', (ii) ‡In more than one dimension we would clearly need a strong condition on commensurability of frequency derivatives over the entire region of the action lattice where the initial state is supported. 70 Chapter 4. Special Case: Zero Time Averaged Perturbation incorporating the resonance condition, and (iii) approximating the resulting sum by an integral, due to the smallness of t', V g(—hn)e im?(hn)t d (—hn) = V g(—hn)e^{?+(—hn-j*)?+2(—hn-j*)2?'') (tI+t') d (—h) n n e im?*ty- g(hn)e im ( (hn-j)?'t'+l(hn-j)2?"t )d (—hn ) e im?*t (j ? ) ? d im ( (j-j*)?t/+I(j-j*)2?/t ) -A(j-j,)2 \j ? —h (4.75) We see that the resonance has a Gaussian envelope, modulated with an oscillation frequency « ??. At the resonance centre t' = 0, and assuming ?< 1, we can easily calculate the linear response terms in the fidelity, getting (?2(t)) = (?(t))2. Therefore, for small ? the fidelity at a 2? echo resonance is 1. We get a perfect revival of fidelity. For non-negligible ?, the resonance height dies as (1 + ?2)-1/2. As ? depends on time t, the resonances will decrease in height with the increasing order k of the resonance. That is, for the k-th resonance occurring at t = ktr we have ? = k ?m?-j, (4.76) and the magnitude of the resonances therefore falls as ~ 1/k. We will get strong and numerous resonances, i.e. small ?, provided that either the second derivative ?" is small, or the initial state is squeezed such that ? > 1. For example, if the second derivative vanishes everywhere, ?" = 0, then the resonances may appear even for extended states. This is the case for our numerical model, where resonances can be seen also for a random state in Figure 4.10. The Gaussian envelope of the resonance has an effective width ?r = ^^ 1 + ?2 ~ k/—h. (4.77) ?j In the semiclassical limit, the resonance positions scale as tr oc —h-1, while their widths grow only as ?r oc —h-1/2, so they are well separated. On the other hand, with increasing order k the resonance width grows, and they eventually start to overlap at k ~ 1/—h. This overlapping takes place at time ~ —h-3/2 and is smaller than t2 provided ? < ?C-I1S/ 2h. The structure of 2?-resonance is nicely illustrated in a numerical example in Figure 4.8, where we consider a slightly modified model with U0 = exp (-iS[?(Sz/S)2/2 + ?(Sz/S - j?)3/6]), (4.78) (-iS[?(Sz/S)2/2 + ?(Sz/S- corresponding to h0(j) = ?j2/2 + ?(j - j?)3/6 and a coherent initial state at (?, ?) = (1,1). For such a system ?" = ? may not be identically vanishing and ?' = ?. ?- and 2?/m-resonances: We note that one may obtain a resonance condition for a single time-dependent term (4.75) with fixed m even at a shorter time, namely for t = tr/m. This is trivially the case for perturbations with many, or at least more than one Fourier components with \m\ > 1. However, in such cases 4.2. Regular Dynamics 71 120 100 120 0 Figure 4.9: Two-time correlation function C(t',t") (2.18) for the quantized top with a = 1.1,7 = 0, S = 16, and coherent initial state at (#, - J2 Wm^ *\Am0* e for \t-{k + -%\ < Af, and( = ( E w^ m=2l+l *\ im9* *\2 1 - F(t) ^ 81 h2 m=2l+l Yl \wm(j*)\cos(mu)*t +ft Y, wm(j'")e 2 *\ im(6*+w*t) (4.79) ^m=2l+l where j3m are phases of complex numbers wm(j*)e . So we have learned that the fidelity at the peak of a 7r-resonance displays an oscillatory pattern, oscillating precisely around the plateau value -^piat (4-62) with an amplitude of oscillations equal to 1 — F^ so that the fidelity comes back to 1 close to the peak of the resonance. 72 Chapter 4. Special Case: Zero Time Averaged Perturbation Again, our numerical example illustrates such an oscillatory structure of ?-resonance in Figure 4.8. The resonances can also be nicely seen in ‘short-time’ Figure 4.4, and because ? = 0 also in the ’long-time’ Figure 4.5. In Figure 4.9 we depict the structure of ?- and 2?- resonance as reflected in the two-time correlation function C(t',t"). Note that the first intersection of the soliton-like-trains for t'-t" = const and t+t = const happens at tr/2 and produces a ?-resonance, while the second intersection at tr produces a 2?-resonance. In analogy to the emergence of a ?-resonance as a consequence of the contribution from the second moment of ?(t), even for the first Fourier mode m = 1, we shall eventually obtain also fractional 2?/p-resonances at times (k/p)tr in the non-linear-response regimes where higher moments (?p(t)> contribute to F(t) ~ {exp(-i?(t)?/h)). This is illustrated numerically in Figure 4.4c showing the case of strong perturbation ?S = 3.2 so that higher orders are important. One indeed obtains fractional resonances, some of which have been marked on the figure. 4.2.4 Random Initial States The second specific case of interest is that of random initial states. Here we shall assume that our Hilbert space has a finite dimension Af, like e.g. in the case of the kicked top or a general quantum map with a finite classical phase space, or it is determined by some large classically invariant region of phase space, e.g. we may consider all states \n) between two energy surfaces E1 < h0(hn) < E2 of an autonomous system. In any case we have the scaling M =(2 VTd, (4.80) where V is the volume of the relevant classical phase space. Throughout this section we will assume that the Hilbert space size M is sufficiently large, so that the difference between the expectation value in a single random state and an average over the whole space can be neglected. Also, as discussed in Section 2.2.2, the difference between the average fidelity amplitude squared and the average fidelity is semiclassically small. The Plateau To calculate the plateau we replace the quantum expectation value (•) with a classical phase space average, resulting in a fidelity amplitude fplat^dj V d ? expf- i? w(j,?) d j7( dd? (4.81) Interestingly, the plateau for a random initial state is just the average plateau for a coherent state squared, where averaging is done over the position of the initial coherent state. If we denote the plateau for a coherent initial state centred at j* by FpClaIS(j*) (Eq. 4.61), then the plateau for a random initial state FpRlaIS is simply F plat = (2?)d / (2 V d jF CIS ( (4.82) Similarly as in mixing dynamics there is a square relationship between the plateau for RIS and CIS. In the mixing case the plateau for CIS does not depend on the position of the initial state due to ergodicity, here though the plateau for CIS does depend on j? and one has to take a square of 4.2. Regular Dynamics 73 dS=0.32 1 10 102 103 104 105 t S=200 10 100 t 1000 10000 Figure 4.10: Short time fidelity for a quantized top with (3 = 1.4 (having no singularities) and a random initial state. In order to reduce statistical fluctuations, averaging over 20 random initial states is performed for S = 1600, and over 100 initial states for S = 200. The horizontal chain line shows the semiclassical theory (4.82). Echo resonances (two spikes for S = 200) are here present due to the special property u"(j) = 0 and will be absent for a more generic unperturbed system. The main figure shows the case of the weak perturbation 5S = 0.32, whereas the inset is for the strong perturbation 5S = 3.2. the average plateau for CIS. In the linear response approximation one has a simple formula for the plateau S2 J2vr)d 1 pRIS ¦ - i^ J^RIS = 2- V dd~" 3 2. wr »Cj)I' (4.83) In Section 4.2 we already discussed possible divergence problems in wm. If we express wm in terms of vm, Eq. (4.36), we have denominators sin (m • u>{j)/2). We therefore have divergences at points in phase space where m ¦ u>{j) = 2nk with k an integer. For a coherent initial state this was not a real problem as it would occur only if we placed initial packet at such a point. For random initial state though, there is an average over the whole action space in the plateau formula and if there is a single diverging point somewhere in the phase space it will cause divergence. The solution is very simple as explained in Section 4.2. The integral is actually an approximation for a sum over hn and so we have to retain the original sum over the eigenvalues of the action operator and exclude possible diverging terms. The formula for the plateau in the case of such divergences is pRIS r^ -fplat — 1 m-u>(hn)^2Trk E ^it(^) (4.84) where in the summation over n we exclude all terms for which for any constituent Fourier mode m we would have m ¦ u>{Tm) = 2nk. 74 Chapter 4. Special Case: Zero Time Averaged Perturbation 0.8 0.6 0.4 0.2 10 100 t 1000 10000 Figure 4.11: Short time fidelity decay for (3 = 0 (we have singularities) and a random initial state. The chain line shows the theoretical value of the plateau as computed from Eq. (4.84). Again we find an excellent confirmation of our theoretical predictions in the numerical experiment for the same system as for a coherent initial state (4.51). In the first calculation we choose the shift (3 = 1.4 so that we have no singular frequency in the action space. In Figure 4.10 we demonstrate the plateau, which in the case of random states starts earlier than for coherent states, namely at t\ oc h°5° (4.46). The value of the plateau can be immediately written in terms of the result for coherent states, 77.RIS r^ -fplat — djJ02 SS Vi^r 2sm{a(j-f3)/2} (4.85) The integral has to be calculated numerically. Horizontal chain lines in Figure 4.10 correspond to this theoretical values and agree with numerics, both for the weak perturbation 5S = 0.32 and the strong perturbation 5S = 3.2 (inset). Small echo resonances visible in the figures are due to the fact that the Hamiltonian is a quadratic function of the action and therefore u" = 0, so that the resonance condition (4.73) is satisfied also for extended states (4.74). For a more generic Hamiltonian these narrow resonant spikes will be absent. In Figure 4.11 we also demonstrate the plateau in the fidelity for the zero-shift case /3 = 0 with a singular-frequency, u(j = 0) = 0, where we again find an excellent agreement with the theoretical prediction (4.84). In this case the theoretical value has been obtained by replacing an integral in (4.85) with a sum as implied by Eq. (4.84) and summing over all quantum numbers except n = 0. Observe that the value of the plateau is much lower than in the case of a non-zero shift (3 = 1.4 in Figure 4.10 as quantum numbers around n = 0 will be close-to resonant. 4.2. Regular Dynamics 75 1 0.1 0.01 0.001 1 0.1 0.01 0.001 0.1 1 10 100 1000 d2 S t Figure 4.12: Long time power law fidelity decay for random states in the kicked top with /3 = 0 and a = 1.1, for S = 200 (a), and S = 1600 (b). Here averaging for S = 200 is performed over 1000 initial random states, otherwise all is the same as for Figure 4.10. The heavy chain line shows the theoretical asymptotic decay (4.87) with an analytically computed prefactor (no fitting parameters). The inset in the bottom figure shows the diffractive quotient between the numerical fidelity and the asymptotic formula (4.86) (chain line in the main figure). Long Time Decay After a sufficiently long time t > L2 fidelity will start to decay. To calculate this decay we have to evaluate the ASI formula (4.42). For a uniform average over Hilbert space we have Dp = 1/J\f with the classical limit dp(j) = (2i:h)d/V. The fidelity amplitude is therefore /(*) = ^ J ddj exp (-^r(J)) • (4.86) The stationary phase procedure is completely analogous to the situation for the case of non-residual perturbation, described in Section 3.2.3, Eq. (3.46). We will only write the asymptotic result, F(t)~(rf) ' W = const x^. (4.87) Here we should remember that the asymptotic formula (4.87) has been obtained as a stationary phase approximation of an integral in the limit of an infinite action space. If we have a finite region of the action space, as is the case for the kicked top, the stationary phase approximation gives an 76 Chapter 4. Special Case: Zero Time Averaged Perturbation additional oscillating prefactor, whose amplitude is damped as (h/t)1/2 for —h ->¦ 0 and/or t ->¦ oo, and which can be interpreted as a diffraction. This oscillating prefactor can be seen in numerical data for the fidelity in the inset of Figure 4.12. For random states the time t2 when the plateau ends is independent of h and is determined by the ratio of two competing terms in the BCH form or the echo operator, t2 ~ \??(t)/—h\/\?2?—/2—h\, therefore we have t2 ~ 1/?. (4.88) This agrees with the numerical results shown in Figures 4.10 and 4.11. Chapter 5 Coupling with the Environment When a man tells you that he got rich through hard work, ask him: ’Whose?’ —Don Marquis The fidelity might not always be the relevant measure of stability. Coupling with the environment is usually unavoidable so that the evolution of our system is no longer Hamiltonian. To preserve the Hamiltonian formulation we have to include the environment in our description. We therefore have a “central system”, denoted by a subscript “c”, and an environment, denoted by subscript “e”. The names central system and environment will be used just to denote two pieces of a composite system, without any connotation on their properties, dimensionality etc. The central system will be that part which is of interest and the environment the rest. The Hilbert space is a tensor product H = Hc®He and the evolution of a whole system is determined by a Hamiltonian or a propagator on the whole Hilbert space H of dimension M = McMe. The unperturbed state |?(t)> and the perturbed one |?S(t)) are obtained with propagators U(t) and Us(t) (2.3). Fidelity would in this case be the overlap of two wave functions on the whole space. But if we are not interested in the environment, this is clearly not the relevant quantity. Namely, the fidelity will be low even if the two wave functions are the same on the subspace of the central system and differ only on the environmental part. 5.1 Reduced Fidelity, Purity Fidelity We can define a quantity analogous to the fidelity, but which will measure the overlap just on the subspace of interest, i.e. on the subspace of the central system. Let us define a reduced density matrix of a central subsystem ?c(t) := tre[?(t)], ?Mc(t) := tre[?M(t)], (5.1) where tre[»] denotes a trace over the environment and ?M(t) = Mff(t)?(0)Mff(t)t is the so-called echo density matrix. Throughout this chapter we will assume that the initial state is a pure product state, i.e. a direct product, |?(0)> = |?c(0)) |?e(0)) =: |?c(0); ?e(0)>, (5.2) 77 78 Chapter 5. Coupling with the Environment where we also introduced a short notation |?c;?e> for pure product states. The resulting initial density matrix ?(0) = |?(0))(?(0)| is of course also pure. The fidelity was defined as F(t) = tr[?(0)? M (t)] (2.9) and in a similar fashion we will define a reduced fidelity ( ˇ nidariˇc & Prosen, 2003) denoted by FR(t), FR(t) := trc[?c(0)?Mc(t)]. (5.3) The reduced fidelity measures the distance between the initial reduced density matrix and the reduced density matrix after the echo. Note that our definition of the reduced fidelity agrees with the information-theoretic fidelity (2.1) on a central subspace Hc only if the initial state is a pure product state, so that ?c(0) is also a pure state. One of the weirdest features of quantum mechanics is entanglement. Of some interest is therefore also whether the coupling with the environment will produce entanglement and how fast. Due to the coupling between the central system and the environment the initial product state will evolve after an echo into the pure entangled state Ms(t)|?(0)) and therefore the reduced density matrix ?M(t) will be a mixed one. For a pure state |?(t)) the criterion for entanglement is very simple. It is quantified by a purity I(t), defined as I(t) := trc[?c2(t)], ?c(t) := tre[|?(t)){?(t)|]. (5.4) tre[|?(t))(?(t)|]. Purity, or equivalently von Neumann entropy tr(?ln?), is a standard quantity used in decoherence studies (Zurek, 1991). Iff the purity is less than one, I < 1, then the state |?) is entangled (between the environment and the central system), otherwise it is a product state. Similarly, one can define a purity after an echo, called purity fidelity (Prosen & Seligman, 2002) FP(t), FP(t) := trc[{?Mc(t)}2]. (5.5) All three quantities, the fidelity F(t), the reduced fidelity FR(t) and the purity fidelity FP(t) measure stability to perturbations. If the perturbed evolution is the same as the unperturbed one, they are all equal to one, otherwise they are less than one. The fidelity F(t) measures the stability of a whole state, the reduced fidelity gives the stability on Hc and the purity fidelity measures separability of ?M(t). One expects that fidelity is the most restrictive quantity of the three - ?(0) and ?M(t) must be similar for F(t) to be high. For FR(t) to be high, only the reduced density matrices ?c(0) and ?M(t) must be similar, and finally, the purity fidelity FP(t) is high if only ?M(t) factorizes. It looks though as fidelity is the strongest criterion for stability. 5.1.1 Inequality Between Fidelity, Reduced Fidelity and Purity Fidelity Actually, one can prove the following inequality for an arbitrary pure state |?) and an arbitrary pure product state |?c;?e) ( ˇ nidariˇc & Prosen, 2003; Prosen et al. , 2003a), |^c;?e|?)|4?|^c|?c|?c)|2?trc[?c2], (5.6) where ?c:= tre[|?)(?|]. Proof. Uhlmann’s theorem (Uhlmann, 1976), i.e. noncontractivity of the fidelity, states that tracing over an arbitrary subsystem can not decrease the fidelity, tr[|?c;?e){?c;?e||?{?|] ? tr[|?c> {?c|?c]. (5.7) Then, squaring and applying the Cauchy-Schwartz inequality |tr[A^B]|2 ? tr[AA^] tr[BB^] we immediately obtain the wanted inequality (5.6). 5.1. Reduced Fidelity, Purity Fidelity 79 The rightmost quantity in the inequality I = tr[?2] is nothing but the purity of state |?) and so does not depend on |?c;?e). One can think of inequality (5.6) as giving us a lower bound on purity. An interesting question for instance is, which state c;?e) optimises this bound for a given |?>, i.e. what is the maximal attainable overlap |(?c;(| e|?)|4 (fidelity) for a given purity. The rightmost inequality is optimised if we choose |?c) to be the eigenstate of the reduced density matrix ?c corresponding to its largest eigenvalue ?1, ?c|?c) = ?1|?c). To optimise the left part of the inequality, we have to choose |?e) to be the eigenstate of ?e := trc[?] corresponding to the same largest eigenvalue ?1, ?e|?e) = ?1|?e>. The two reduced matrices ?e and ?c have the same eigenvalues (Araki & Lieb, 1970), ?1 > ?2 > ... > ?Nc. For such choice of |?c; ?e) the left inequality is actually an equality, |(?c;?e|?)|4 = ||2 =?21 and the right inequality is < No E ?12 < tr[?2] = ^ ?2, (5.8) with equality iff ?1 = 1. In the case when the largest eigenvalue is close to one, ?1 = 1 - e, the purity will be I = (1 - e)2 + 0(e2) ~ 1 - 2e and the difference between the purity and the overlap will be of the second order in e, I-|{?c; ?e|?)|4 ~ e2. Therefore, for high purity the optimal choice of |?c; ?s) gives a sharp lower bound, i.e. its deviation from I is of second order in the deviation of I from unity. For our purpose of studying stability to perturbations, a special case of the general inequality (5.6) is especially interesting. Namely, taking for |?) the state after the echo evolution M5(t)|?(0)} and for a product state |?c;?e) the initial state |?(0)) (5.2), we obtain F2(t) Ue. (5.10) This is a frequent situation if the coupling with the environment is “unwanted”, so that our ideal evolution U0 would be an uncoupled one. The reduced fidelity FR(t) and the purity fidelity FP(t) have especially nice forms in such case. The reduced fidelity (5.3) can be rewritten as FR(t) = trc[?c(0)?Mc(t)] = trc[?c(t)?f(t)], (5.11) where ?c(t) is the unperturbed state of the central system and ?sc(t) := tre[U5(t)?(0)U†(t)] the corresponding state obtained by perturbed evolution. Whereas for a general unperturbed evolution the reduced fidelity was an overlap of the initial state with an echo state, for a factorized unperturbed evolution it can also be interpreted as the overlap of the (reduced) unperturbed state at time t with a perturbed state at time t, similarly as the fidelity. 80 Chapter 5. Coupling with the Environment The purity fidelity can also be simplified for uncoupled unperturbed evolution. As the U0 is in factorized form, we can bring it out of the innermost trace in the definition of the purity fidelity and use the cyclic property of the trace, finally arriving at FP(t) = trc[{?Mc(t)}2] = trc[{?cf(t)}2] = I(t). (5.12) The purity fidelity is therefore equal to the purity of the forward evolution. The general inequality gives in this case F2(t) ? FR2(t) ? I(t), (5.13) and so the fidelity and the reduced fidelity give a lower bound on the decay of purity. Because the purity is frequently used in studies of decoherence this connection is especially appealing. In all our theoretical derivations regarding the purity fidelity we will assume a general unperturbed evolution, but one should keep in mind that the results immediately carry over to purity in the case of uncoupled unperturbed dynamics. Also a large part of our numerical demonstration in next two sections will be done on systems with an uncoupled unperturbed dynamics as this is usually the more interesting case. 5.1.3 Linear Response Expansion We proceed with the linear response expansion of the reduced fidelity (5.3) and the purity fidelity (5.5). We will use notation ?c := |?c(0))(?c(0)| for initial pure density matrix on a central system and ?e := |?e(0))(?e(0)| for the environment. The perturbed propagator is defined in exactly the same way as for the fidelity (2.6) in terms of the perturbation generator V. To order O(?4) we get, —hJ (?(t)(?-?c??e)?(t)> —hJ (?(t)(-?c)??(t)> 1-FP(t) = 2—h J (?(t)(-?c)?(-?e)?(t)>, (5.14) where (•) = tr[(?c??e)•] denotes the quantum expectation value in the initial product state and ?(t) is the sum of V(t) (2.12). If the expectation values are written explicitly in terms of expectations in the base states |j; ?), j = 1,...,Nc, ? = 1,...,Ne, with the convention that the first base state |1; 1) := |?c; ?e> is the initial state, we have 1-F(t) = 1-FR(t) = - 1-FP(t) = 2 (2 —hJ {(1;1|?2(t)|1;1)-(1;1|?(t)|1;1)2} (2 ( Me ^ —h?) (1;1|?2(t)|1;1)-5:|(1;?|?(t)|1;1)|2 (2 ( Me Mc —h| (1; 1|?2(t)|1; 1) - ]T |(1; ?|?(t)|1; 1)|2 - ]T |(j; 1|?(t)|1; 1)|2 . As one can see, the linear response expansion of course also satisfies the general inequality (5.9). The difference between FR(t) and F(t) as well as between FP(t) and F(t) is in off-diagonal matrix elements of operator ?(t). Somehow reminiscent perturbative expansion, although without time dependence, has been obtained in studying the eigenvalues of the reduced density matrix (Ku¨bler & Zeh, 1973). Depending on the growth of linear response terms with time we will again have two general categories, that of mixing dynamics and that of regular dynamics. 5.1. Reduced Fidelity, Purity Fidelity 81 Mixing Dynamics For mixing dynamics the correlations decay and the linear response term will grow linearly with time. For large times one can argue that ?(t) should look like a random matrix and the terms giving the difference between the fidelity and the purity fidelity and the reduced fidelity can be estimated as E^2|(j;1|?(t)|1;1)|2 Ej |[?(t)](j;1)(1;1)|2 1 (1;1|?2(t)|1;1) Ej,? |[?(t)](1;1 , ,(j,?)|2 ~ Ne, (5.16) because there are more terms in the sum for fidelity. Therefore we can estimate the difference FP(t) - F2(t) ~ 1/Nc + 1/Ne and FR(t) - F(t) ~ 1/Nc. Provided both dimensions Nc,e are large and for sufficiently long times, so the “memory” of the initial state is lost and the correlations decay, we can expect the decay of all three quantities to be the same. Regular Dynamics For regular dynamics on the other hand, ?(t) will not approach a random matrix but will grow with time with a well defined long time limit ?(t) ->¦ Vt. This will happen for times larger than the averaging time tave (3.17). Expectation value of ?2(t) will then grow quadratically with time. In a similar way as we defined the average correlation function C (3.15), we can also define a time average of the correlation functions occurring in the linear response expressions for FR(t) and FP(t), lim(?(t)(-?c)8?(t)) t^OO t2 — := li (?(t)( - ?c) ® ( - ?e)?(t)) = — V) -?c)®(- (-?e)V—). (5.17) In the linear response regime purity fidelity and reduced fidelity will therefore decay quadratically with time, if CR and CP are nonzero. For coherent initial states the average correlation function C is proportional to h (3.36) provided v—(j) is sufficiently smooth. In the semiclassical limit we can make an expansion around the centre of the wave packet v(j) w v—(j*) + v'(j*){j j*} + ---. pVp The second moment ( v—2)cl (v)2 proportional to the dispersion in j of the initial packet (3.33), i.e. to h. As we take initial coherent packet we have a dc + de dimensional squeezing matrix ? cl is then and we can write derivatives of classical v'=: (v'c,v'e), ? = as ?c 0 0 \ ?e , v'c:= ?v(j* ?jc , v—e:= ?v—(j), product (5.18) (5.19) where we split actions j into two components j =: (jc,je). The formula for C (3.36) can then be written as C = 1-—h(v—'c?-1v'c + v'e?e-1v—^, (5.20) By a similar method one can calculate also CR (5.17) and gets C—r = - —h(v—c?-1v—c). (5.21) 82 Chapter 5. Coupling with the Environment For CP on the other hand, the lowest order expansion of v' used for C— and CR gives zero. To get C—P we must expand v to the second order in (j - j?) v—(j) ~ v(j?)+v'(j?){j -j?} + -{j-j?} ¦ v—"(j?){j-j?} + - ¦ ¦, (5.22) where v"(j?) is a symmetric matrix of second derivatives evaluated at the position of initial packet j?, =ii v—il \ ?2v(j v—e v—c c v—e e, v—lk(j):=^dj ?) . (5.23) The only nonzero contribution to CP comes from off-diagonal terms v—ce and v—& = (v—c'e)T, with the final result being CP = -—h tr[u], u := ?-1v—c e?-^c. (5.24) —h)2 The result for CP is very interesting as it means that the purity fidelity in regular systems will decay as FP(t) = 2(?/h)2CPt2 on h-independent time scale because of CP oc h2. Note that to reach this conclusion of —h-independent decay, we need only the existence of a smooth classical limit of ?(t), therefore FP(t) will decay on —h-independent time scale even before V converges, i.e. for t < tavg. A simple special case of time averaged perturbation V is the tensor product form V = Vc V—e. Then we have v—& = v'c® v'—e /v (note that in order to comply with our previous notation we have v'c = (?cvc)ve). If in addition ?c,e are diagonal, then the average CP is CP = ——h v'c?-1v'c v'e?-1v'e . (5.25) (2 — —h) (v—c?c-1v—c) (v—e?-1v—e) . As we can see, for a tensor product forms of the time averaged perturbation there is a relation between C—, C—R and CP, v2CP =C—r(C—-C—r). (5.26) The purity fidelity will be high, i.e. decoherence will be slow, if we either make CR small or make C—r close to C—. Note that if CR is small, then the inequality (5.9) already tells that CP will be also small, but here we have a stronger result, that CP is of second order in h whereas CR is only of first order. 5.1.4 Numerical Illustration Here we would like to briefly demonstrate quadratic decay of the purity fidelity for regular systems and linear decay for chaotic systems. Furthermore, we will show that one can also have a situation where the purity fidelity FP(t) decays slower for chaotic systems than for regular. We take a Jaynes-Cummings model (Section 2.1.2) which is a two degrees of freedom model of a harmonic oscillator coupled to a spin. As initial state we always take a product state of two coherent states, for the spin the coherent state (2.26) is centred at (??, ??) = (1,1) and for the boson coherent state (2.30) we take ? = 1.15. We take two different sets of parameters. For the chaotic regime we choose ? = e = 0.3 and G = G' = 1, for which the energy of the initial state is E = 1.0 and the classical Poincare section shows a single ergodic component. The second set of parameters is ? = e = 0.3 and G = 1, while G' = 0, for which the systems is integrable and the energy is E = 0.63. The perturbation is chosen to be in the spin energy e, i.e. a detuning, corresponding to the perturbation generator V V = hSz. (5.27) 5.1. Reduced Fidelity, Purity Fidelity 83 Because the simulation is very time consuming for the chaotic case, we choose a small spin size S = A, resulting in an effective Planck's constant h = 1/4. Note that, as opposed to previous kicked top model, here the system is conservative, and the time index t is continuous. First we want to check the growth of linear response terms with time. In Figure 5.1 we show the correlation integral Sp(t) for fidelity and the difference Sp(t) — Sp(t) between the correlation integrals for fidelity Sp(t) and for purity fidelity Sp(t), SF(t) ¦¦= i(L(i)(l®l-pc®pe)L(i)> Sp(t) := i(E(i)(l-pc)®(l-pe)L(i)>. (5.28) Note that for chaotic systems Sp(t) should converge to 2a (3.3), where a is the transport coefficient, i.e. the integral of the correlation function. The difference Sp(t) — Sp(t) should on the other hand be of the order ~ 2a(l/Mc + 1/We) ~ o/2 (due to 1/WC + 1/We ~ 1/S). These two predictions are nicely confirmed by the lower two curves in Figure 5.1. We can see that the correction to the purity 1.6 1.4 1.2 Q. CO 1 CO 0.8 lL co 0.6 0.4 0.2 0 0 5 10 15 20 25 30 35 t Figure 5.1: The correlation integral Sp(t) and Sp(t) — Sp(t) (5.28) in the Jaynes-Cummings model for regular dynamics (upper curves) and chaotic dynamics (lower curves). The horizontal dashed lines (for the chaotic case) are at 2a = 0.2 and 0.2/4. The two linearly increasing dashed lines for regular dynamics have slopes 0 = 0.046 and 0(1 — 0.98/4). fidelity is really of order 1/S in the chaotic case and would therefore vanish in the semiclassical limit. For regular dynamics we expect Sp(t) to grow linearly with time, with the slope given by 0, Sp(t) —>¦ Ot, with 0 oc 1/S (5.20). The purity fidelity integral should grow as Sp(L) —> Opt with Op oc 1/S2 (5.24). The difference should therefore be SF(t) - SP(t) ~ Sp(l - const/S1). This is again confirmed in Figure 5.1. We checked that Sp oc 1/S2 also for larger S, up to S = 24. Next, we would like to demonstrate a faster decay of the purity fidelity in the regular regime than in the chaotic one. We set the perturbation strength to 5 = 0.005 for which we expect 84 Chapter 5. Coupling with the Environment Figure 5.2: Echo dynamics in the Jaynes-Cummings model. The square of the Wigner function for chaotic dynamics (top diagrams) and integrable dynamics (bottom diagrams) is shown as a function of time at times corresponding to the axis. The purity fidelity is shown in the main frame on the same time scale and for short times in the inset. Red curves give the integrable and blue curves the chaotic evolution. In the inset full curves show the complete numerics, symbols the evaluation starting from the numerical correlation integrals of Figure 5.1 and dashed curves the linear or quadratic approximation using a and Čp. the purity fidelity to become lower in the regular regime at the time determined by 1 — Fp (i) = 252S2CPt2 « 1 - F(t) = 52S22at, i.e. at t « 9, where we used a = 0.1 and CP = 0.011 as given in Figure 5.1. In Figure 5.2 we show the purity-fidelity for chaotic and regular regimes. We can see the crossover at the predicted time t p=s 9. We also illustrate the evolution of the purity fidelity with the square of the Wigner function corresponding to the reduced density matrix p^(t) on the central spin subspace. If Wpu (#, ¦ exp(—i5Vt/h). In the semiclassical limit we can use the classical limit v instead of V. Classical v (J) is a function of action operators only and so similarly to the evaluation of fidelity one can use the ASI (3.31) for the evaluation of purity fidelity. A partial trace over the environment gives an integral over je and due to a square in the definition of purity fidelity we end up with an integral over 2(dc + de) = 2d dimensions, if dc is the dimension of jc and de of je, FP(t) ^h~za /drexp V(jc,je) + V(j'c,j'e) ~ f (jc, Je)} dp(j)dJj'), (5.33) where dr = ddjddj', j = (jc,je), j' = (j'c,j'e) an(i dp(j) is the classical limit of (n\p\n) (3.31). Next we expand v(j) around the position of the initial packet (5.22). The constant term v(j*) and the linear terms cancel exactly as well as the diagonal quadratic terms, regardless of the position of the initial packet. The argument of the exponential function which remains is then .St v'L(f)Ue-j'e)] (5.34) 5.3. Regular Dynamics 87 with the matrix v" of second derivatives given in Equation (5.23). The resulting expansion can be used in FP(t) (5.33) to calculate the purity fidelity for initial states having well defined classical d?(j). For coherent initial states d?(j) are Gaussian (3.33) and so the whole integral is also a Gaussian, giving FP(t) = where L and V" are 2d dimensional matrices ^ det( 1 - i?tL-1V~") , (5.35) L:= /?c 0 0 0 \ 0 ?c 0 0 0 0 ?e 0 V 0 0 0 ?e/ , V" := 1 0 0 v" v— 0 0 v—c" -v" v'L ec 0 0 0 0 (5.36) with matrices of squeezing parameters ?c and ?e (5.18) of dimensions dc and de, respectively, and dcxde Vio dimensional matrix (5.23). Note that the determinant is real despite the imaginary unit. The determinant of the matrix matrices detA = det (A) det (D - CA-1B) = det (D) det (A - BD-1C), with m diagonal, we obtain -i?tL-1V" can be simplified using the following identity for block -----t (A) det (D - CA-1B) = det (D) det (A - BD-1C), (5.37) n dimensional matrices B and CT. Noting that the matrix - i?tL^~" has on the det ( - i?tL-1V") = det ( + (?t)2Z), (5.38) iC D) X with Z being the 2dc x 2dc dimensional matrix Z:= ?c--? 1v—c'e?e- -c 1v—c'e? 1v—e" -1v" -? ?- 1// v" ? v—ce e 1 v— // ec . To simplify the determinant of + (?t)2Z we use the following identity iC B) det A B = det (DA-BC), (5.39) (5.40) C D for square matrices A, B, C and D and commuting B and D, [B, D] = 0. Using this finally gives FP(t) = 1 det{ +(?t)2 u} , u:=?- t)" -1 v— v—ce (5.41) Note that u is the same dc x dc matrix we had in the expression for CP (5.24). This very simple, yet important result deserves a little discussion. It gives the purity fidelity (and thereby as a special case also the purity) decay for regular systems and coherent initial states to all orders in ?. Its validity is limited to times larger than the averaging time tave in which V converges and for sufficiently small h. Planck’s constant must be small to allow the replacement of V with its classical limit v and v must be smooth on the scale of the wave packet size —h. Furthermore, we replaced a sum over the quantum numbers with an integral over the actions, the validity of this being given by the condition ?t\v'^\—h2/—h < 1, i.e. ?tu < 1/h, if tu is the upper time limit of validity of the ASI. As we will see, the asymptotic decay of purity fidelity as given by Equation (5.41) is between 1/(?t) and 1/(?t)dc, depending on the matrix u. The FP(t) at the upper border tu is therefore between —h and hdc. The purity fidelity will for long times saturate at the plateau given by the finite Hilbert space size, FP w 1/Mc ~ —h dc . Comparing this, we see that the upper border tu coincides with w 1/A/c ~ , 88 Chapter 5. Coupling with the Environment the point where the asymptotic saturation FP is reached for one degree of freedom systems. For dc > 1 the ASI break time tu is only by a constant factor smaller than the time when we reach FP. Summarising, the purity fidelity decay (5.41) is valid from tave all the way to the asymptotic plateau FP provided h is sufficiently small, without any bound on ?. Let us now explore the asymptotic decay of FP(t). To simplify theoretical arguments, we assume that the squeezing parameters are all equal to one*, L = so that we have u = v'^v—cg. In 1 + de degrees of freedom systems (dc = 1), the matrix u is just a number, and the purity fidelity decays as FP(t) = ^ 1 2, dc = 1. (5.42) A single parameter u is already fixed by the linear response, i.e. by the value of CP (5.24). Asymptotically we get FP(t) x 1/(?t) decay regardless of the second dimension de. For general systems with dc > 1 one can see that the determinant det { + (?t)2 u} is a polynomial of order at most dc in (?t)2. Furthermore, as the matrix u is symmetric and positive definite, its eigenvalues are positive, meaning that all the coefficients of the polynomial are positive and the FP(t) is always less than one. In a special case, when the matrix u can be written as a tensor product of two vectors, like u = v'^v—cg =:x®y, the determinant is again simple regardless of the dimensions involved and we get FP(t) = T1+t1 u = x«y. (5.43) In such case we again get the asymptotic FP(t) ~ 1/(?t) decay, but here regardless of both dimensions. Matrix u has such a form for instance if all the matrix elements of v—#e are the same (e.g. for the perturbation v = LkJ (jc ® je)ki). Our result of course applies also to the purity decay in weakly coupled systems and'does not agree with the result of Jacquod (2004). Reduced Fidelity For the reduced fidelity a similar procedure based on the ASI can be used. In the expansion of v(j) around j* (5.22) the first non vanishing term is a linear one. The resulting Gaussian integrals are analogous to the ones in the calculation of the fidelity decay for coherent initial states (but there are twice as many), the final result being a Gaussian decay FR(t) = exp ( -—h2CRt 2 J, (5.44) with a single parameter CR = 12—h {v'c?c1v—c) given by the linear response alone (5.21). Numerical Illustration To numerically confirm the above formulas for the decay of purity fidelity and reduced fidelity we take a double kicked top model (2.25). To be in a regular regime we take ?c>e = 0 and ?c = ?/2.1 and ?e = ?/V7. Different unperturbed frequencies ?c = ?e are chosen in order to have a general situation, i.e. that the subsystems are not in resonance. Unperturbed dynamics is again uncoupled, ? = 0, so that the purity fidelity equals purity. The spin size is chosen S = 100 and the initial ?Usually this can be achieved by the right choice of actions. 5.3. Regular Dynamics 89 10000 Figure 5.4: Decay of F(t), i*R,(t) and I(t) in the regular regime of the double kicked top. For parameters see text. The theoretical Gaussian decay for the fidelity (3.36) and the reduced fidelity (5.44), with the theoretical value of Č (5.47) and Cr (5.48), overlap with the numerics (symbols) within the line width. Theoretical purity decay (5.42) (dotted curve) as determined by the theoretical u (5.49) also nicely agrees with the numerics (full curve) up to the plateau. product coherent state is placed at (#*, ip*)c = 7r(l/\/3, l/\/2) and (#*, cp*)e = 7r(l/\/3,3/\/7). The perturbation with strength 5 = 0.01 is of the form v = j-4s2z ® sl (5.45) Unperturbed classical evolution is very simple, namely rotation around y-axes for angles 7C and 7e for the central system and the environment, respectively. The classical limit of V is readily calculated and expressed in terms of the actions jc = yc and je = ye, V = \(1 "ic2)(l- ie2)- (5.46) The derivatives of v are v'c = — jc(l — j%)/2 and similarly for v'e. The squeezing parameter for the spin coherent states is A = 1/(1 — j2) and the average correlation function Č (5.20) is C = |(1 " ic2)(l " i2){ic2(l " Jl) + ie2(l " ic2)}, (5.47) which for our choice of the initial coherent packet gives C = 0.024/S1. For the reduced fidelity we get ^ (5.48) 1 CR = ^c2(l-jc2)(l-Je2), which evaluates to Cr = 0.021/S1. The above two parameters C and Cr completely determine the Gaussian fidelity (3.36) and reduced fidelity (5.44) decays. As we have 1 + 1 degrees of freedom 90 Chapter 5. Coupling with the Environment system, the purity is determined by a single parameter u (5.42). For our perturbation we get u = jc2je2(1 - jc2)(1 - je2), (5.49) giving u = 0.032 for our initial product coherent state. As our perturbation V is of the product form, we could as well use a universal relation between CP, CR and C (5.26) to calculate CP (or equivalently u). The numerical results are shown in Figure 5.4, together with the theory. Agreement is excellent. 5.3.2 The Jaynes-Cummings Model In this section we will consider in detail the stability of a Jaynes-Cummings model, described in Section 2.1.2, under various perturbations. The Jaynes-Cummings system can be experimentally realized and so it is a possible model on which one could experimentally study the quantum stability. In experiments one usually has only a co-rotating term, i.e. G' is zero, therefore we will focus on a situation when we have G' = 0 in the unperturbed dynamics and so the classical limit of the unperturbed system is integrable. For the initial state we will always choose a product of coherent states for a harmonic oscillator, given by a real parameter ? (2.30), and a spin, given by the initial position (?,?) (2.26). In studies of the reduced fidelity and purity fidelity we will consider the spin as a central system, and the harmonic oscillator as an environment. For regular dynamics with coherent initial states the decay of fidelity and reduced fidelity is Gaussian with the decay time determined by a single parameter C and CR, while the decay of the purity fidelity is FP(t) = 1/ 1 + u(?t)2 (5.42) and is again determined by a single parameter u. With this in view, it is sufficient to determine only the linear response parameters C (5.20), CR (5.21) and CP (5.24), to get the decay to all orders in ?. Therefore, we will focus only on the calculation of these coefficients. The calculation of the expectation values in the coherent initial state is described in Appendix B. Our unperturbed system will be For the perturbation we will look at four different situations: • variation of ?, corresponding to V = —ha+a =: [??] • variation of ?, corresponding to V = hSz =: [??] • variation of G, corresponding to V = ^(aS+ + a+S-) =: [?G] • variation of G', corresponding to V = ~j=(aS- + a+S+) =: [?G'], where we introduced short notation V = [?•] symbolically denoting the perturbation in the parameter •. In the last case of V = [?G'] and if G = 0 all correlations average to zero, corresponding to the residual perturbation (i.e. freezing), and this case will be considered separately in Section 5.5. As an additional simplifying assumption, we will assume that we have ? = ?, i.e. spin and an oscillator are in resonance. For G = 0 we have numerically checked that the average correlation functions are quite insensitive to the resonance condition. The only exception being the V = [?G] perturbation, for which C, CR and CP vanish if we are out of resonance and have G = 0. Because this freezing case will be studied separately, we do not loose generality by taking ? = ? but gain in the simpler theoretical calculations. Namely, in resonance the G term commutes with the unperturbed Hamiltonian, [aS+ + a+S-,H0] = 0. In addition, in experimental implementations we are usually close to the resonance condition. H0 = —h?a + a + —h?Sz + G=(aS+ + a + S-). (5.50) ? 2S 5.3. Regular Dynamics 91 V = [Se] = hSz V = [Su] = ha+a C 0.06 0.04 0.02 0 0 -, / 0.5 0.0 -0.5 ,,-, cos(J) 0.12 0.08 0.04 0 0 -, 0.5 0.0 -0.5 ,,-. cos(J) a R 0.04 0.03 0.02 0.01 0 0*1 2 3 J ~ bi 0.03 0.02 0.01 0 0 00 05 ,. cos(J .5 ) 0 0 -, 05 00 05 ,.. cos(J) 0.016 0.012 ~ 0.008 CP 0.004 0 ,rx\\M mmm?s^>: 0.5 0.0 -0.5 ,.-1 cos(J) 0.016 0.012 0.008 0.004 0 Lx\m. S32Ž27SC 0.5 0.0 0.5 ,.-1 cos(J) Figure 5.5: Numerically calculated dependence of C"s on the position of the spin coherent state ip and cos# for uj = e = 0.3, G = 1 (everything is independent of G provided it is nonzero), G' = 0, a = 1.15 and J = 4. Left column shows the results for e perturbation and the right for the perturbation in u. Perturbation of the frequency u We will first analyse the case when the perturbation consists of the variation of uj, so that the perturbation generator is V = ha+a. (5.51) Due to the resonance condition uj = e the unperturbed propagator Uo factorises into a part proportional to the G term and the rest. The perturbation in the Heisenberg picture is therefore y^ = eiG(o+S_+oS+)t/V2Š^a+a)e-iG(o+S_+oS+)t/V2Š (5.52) 92 Chapter 5. Coupling with the Environment To calculate the average correlation function one needs an integral of the perturbation ?(t) = J0 V(?)d? in the limit t ->¦ oo. As the constant G just re-scales time in the perturbation V(t) (5.52), the C’s which are an infinite time limit property of ?(t) will not depend on G, except possibly at G = 0 where the symmetry of H0 changes and we might have an effect due to degeneracies. Direct calculation in this simple case of G = 0 gives C— = 1 S , CR = CP = 0, (if G = 0), (5.53) for the initial product coherent state (2.26,2.30). Note that ?2/S has a well defined classical limit, namely the energy of the oscillator. The last two results are expected. The CP = 0 because the perturbed dynamics is also uncoupled and CR = 0 because the perturbation is only in the oscillator (“environment”) part of H0. For G > 0 we get an additional angle dependent term of order h2 = 1/S2 which can be seen in numerically calculated C’s in Figure 5.5. Therefore, there is a discontinuous jump in all three C’s at G = 0, the discontinuity being proportional to —h 2 and thus of higher order for C while it is the leading order term for CR and CP. Otherwise the values of the plateaus C are independent of the coupling G, but of course the time scale on which we get the convergence of C’s scales as 1/G. For small G the discontinuity will happen at large times. Perturbation of the spin energy ? In this case the perturbation generator is V = hSz. (5.54) Everything is analogous to the previous case and in the simple uncoupled case of G = 0 (and ? = ?) we get C = CR = S^? , CP = 0, (if G = 0). (5.55) The C now does not depend on the oscillator coherent state parameter ? but instead depends on the position of the spin coherent state. Also, the CR is now nonzero as we make the perturbation in our central system. At G = 0 there is again a discontinuity of order —h 2 in all three C’s, otherwise they are independent of G. For C and CR this discontinuity is of higher order in h and can be neglected in the semiclassical limit. Dependence of numerically calculated C’s on the initial position of the spin coherent packet is shown in Figure 5.5. One can see, that CP is equal for [??] and [??] perturbations. Therefore the decoherence (purity fidelity) is insensitive to which frequency we detune, whereas fidelity and reduced fidelity are not. Perturbation in the coupling G The perturbation generator is V = -—h=(a + S- + aS+). (5.56) We again take the resonant condition ? = ?. Now the perturbation is constant in time V(t) = V(0) regardless of the value of G and we can calculate all three C’s for an arbitrary G. These are written in Figure 5.6 next to each plot showing its dependence on the initial spin packet position. There is no dependence on G and also no discontinuity at G = 0, contrary to previous two examples. Also the C and CP are strictly larger than in the case of ? or ? perturbation whereas the CR is of order h and is smaller than for the ? perturbation. Now CP oc h2 is of course nonzero and gives us the decay time scale of purity fidelity (or purity in the case of G = 0). 5.4. Separation of Time Scales 93 V = [5G] = -t=(a+S. + aS+) C = ČR + ± sin2 ¦& 2S 0.15 0.1 0.05 0 0 -, / 0.5 0.0 -0.5 ,., _1 cos(J) ČR = Čp + §| (sin2 d sin2

, (5.58) giving the linear response decay of the fidelity F(t) = 1 - (?/h)2tSF(t), the reduced fidelity FR(t) = 1 - (?/h)2tSR(t) and of the purity (or generally purity fidelity) I(t) = 1 - 2(?/h)2tSP(t) can be significantly simplified in such situation. We will furthermore assume (Ve) = 0, with (A) = lim^oot"1 jt (A(?)) d? denoting a time average. This assumption corresponds to an equilibrium situation where the average “force” Ve vanishes. The integration over the fast variable Ve can be carried out and we get for t > tc > te SF(t) = 2?ejVc2) SR(t) = 2?e {(Vc2) - (Vc)2} SP(t) = 2?e {Vc2) - (Vcf} , (5.59) with ?e := lim (?2(t))/2t, with ?e(t) = / Ve(?)d?, (5.60) being the integral of the autocorrelation function for the environmental part of the perturbation Ve alone. The result does not depend on the initial state of the environment. Some interesting conclusions can be drawn from these linear response results (5.59). We can see that the decay time scale depends only on the time average diagonal correlations of the central system (Vc2(t)) and not on the full correlation function. This is a simple consequence of the separation of time scales and means that the decay of all three stability measures does not depend on the dynamics of the central system (e.g. being mixing or regular). Furthermore, reduced fidelity FR(t) and the purity I(t) will decay on the same time scale (5.59), meaning that the decay of the reduced fidelity is predominantly caused by the loss of coherence, i.e. entanglement between the two factor spaces. This in turn means that the reduced fidelity, which is a property of echo 5.4. Separation of Time Scales 95 dynamics, i.e. of comparison of two slightly different Hamiltonian evolutions, is equivalent to the decay of the purity. If the initial state of the central system pc(0) is a Gaussian wave packet (coherent state) then the dispersion (V,?) — (Vc) is by factor of order h smaller than (Vc2). Thus for coherent initial states of the central system, irrespective of the initial state of the environment, the -Fr(L) and I(t) are going to decay on a 1/h times longer time scale than F(t). We have therefore reached a general conclusion based on the assumption of chaotic fast environment (which is often the case), namely that the coherent states are most robust against decoherence (decay of purity), provided te Ve(t) and the average "force" tre[14(t)p] to vanish together with the exponential decay of environmental correlations of the form (5.61) for an arbitrary state, yields a master equation for the reduced density matrix p^(t) := tre[pM(L)], pf(t) = -^ae[Vc(t),[Vc(t),pf(t)}}. (5.62) Master equation is strictly valid in the Markovian limit of the correlation function being a delta function in time or if considered on time scales larger than the correlation time. 1 0.1 0.01 0.001 0.0001 1e-05 10 100 1000 10000 Figure 5.8: Decay of F2(t),F^(t) and I(t) for fast chaotic environment. The dashed line is an exponential function with the exponent given by the values of ae and (V2) (5.59) while the two dotted lines have slopes —2 and —1. For parameters see text. For numerical demonstration we use a double kicked top with: VCfi = Sz/S, S = 200, d = 1.5 • 10~3, coherent initial state at (#,ip)c,e = (tt/"\/3,tt/\/2) and parameters ac = 0, 7C = 7r/50 for the central system and ae = 30, 7e = tt/2.1 for the environment. Actually, we could take any value of ac and would get qualitatively similar results. The only advantage of using regular central dynamics ac = 0 is that it is then possible to explicitly calculate averages (Vc2) and (V,?) — (Vc) . Namely, if ac = 0 and 7C C 1 we get (K2) - (vc (K2) = ^(i-= Ž(1 vl) Vc, AS{ vl) (5.63) The values of these two quantities for our initial condition are (V,?) = 0.202 and (V,?) — (Vc) = 0.399/S1, and are shown in Figure 5.7 with two dotted lines (by pure coincidence we have ae = 0.20 ?« (V2)), together with numerically calculated time dependent (not yet averaged) (V^(t)) and (Vc(t)) — (Vc(t)) . This time dependent values oscillate on a time scale p=s 50, which is much longer 5.4. Separation of Time Scales 97 10000 1000 100 10 1 0.0001 0.001 0.01 0.1 1 d Figure 5.9: Times r at which F2(t), F^(t), I(t) fall to level 0.37 for different d and fast chaotic environment. Symbols give the numerics and lines give the theoretical dependence of r. All is for S = 100. For other parameters see text. than the time p=s 10 in which ae converges and so the assumption te oo If in addition the correlations of the system also do not decay then the correlation sum of the total system will grow as oc t2 which is just the regular regime already discussed before. Here, we will focus on a different situation where the integral of systems correlation function converges, i.e. the dynamics of the central system is mixing. We will additionally assume the average "position" Vc A O \ ^ 8 A + \+ ffi A e 98 Chapter 5. Coupling with the Environment Se >/t (- )/t : 1 10 100 1000 Figure 5.10: Correlation sums occurring in (5.66) (solid curves) for S = 200. The top chain line gives the best fit for ac and the two lower chain lines give theoretical time averaged correlation functions for the environment (5.67). All is for a fast regular environment. See text for details. to be zero (Vc) = 0. The transport coefficient of the central system ac is then ac := lim (Sc2(t))/2t, Ec(i) = /VC(LK. (5.65) The expressions for Sp(t), Sp(t) (5.28) and Sr(L) (5.58) are independent of time and can be simplified to Sf = 2acCe Sr = 2crcCe SP = 2<7c{ce-(W2}- (5-66) Note that now the reduced fidelity -Fr(L) decays on the same time scale as the fidelity F(t). This must be contrasted to the case of a fast mixing environment (5.59), where -Fr(L) decayed on the same time scale as the purity. If the initial state of the environment pe(0) is a coherent state, then purity will decay on a 1/fi times longer time scale than fidelity or reduced fidelity. On the other hand, for a random initial state of the environment, the average force (Ve) = 0 vanishes, and all three quantities decay on the same time scale. For the purpose of numerical experiment we chose Vc = Sz/S, and Ve = S^/S2 in order to have a less trivial situation of non-vanishing average force (Ve). The initial condition is again (#, /dj / 0. In the present section we will consider the case of a harmonic oscillator, for which O, = O and the theory explained in Section 4.2 can not be used. For the sake of numerical demonstration we will use a Jaynes-Cummings model (see Section 2.1.2) with the perturbation in the G' parameter, i.e. the only case not explained in Section 5.3.2. The initial state will be a product of coherent state for a spin and an oscillator. The perturbation will be V = -^L(a+S+ + aS-) = [5G'\. (5.68) 2S 100 Chapter 5. Coupling with the Environment 10000 1000 100 10 1 1e-05 I at 0.99 A A D 0.0001 0.001 0.01 d 0.1 Figure 5.12: Times r at which F2(t) (pluses), F^(t) (circles) and I(t) (triangles) fall to level 0.37, and times r when I(t) falls to 0.99 (squares) for varying 5 and fast regular environment. Symbols are the numerics and lines give theoretical dependences of r. Everything is for S = 100. Provided we have G' = O for the unperturbed system the correlation function has a zero time average regardless of other parameters. This is a simple consequence of the symmetries as the perturbation is a "counter-rotating" term, while the unperturbed Hamiltonian has only a "co-rotating" term. Due to the symmetry the perturbation is residual. The fact that the perturbation is residual has nothing to do with the perturbation G' breaking integrability. For instance, if we make a perturbation in G' and any other parameter at the same time, we will have a perturbation that breaks integrability but is not residual. Because we want to study the case of 0, = 0, we choose G = 0 so the unperturbed Hamiltonian, Ho = hua+a + heSz (5.69) is uncoupled. As a consequence purity fidelity equals purity. Residual perturbation can be written as a time derivative of another operator W (4.6), which is in our case dW „, m The echo operator can than be written as i W = 2S(u) + e) (a+S+ - aS-) . M5(t) = exp W)-^(o)}H-r(f)*2 (5.70) (5.71) with r(i) given by r(i) := j*R(T)dT-±[W(0),W(t)], 1 ru/ dW^ % R := - \W, —— \ =--------- hL dt J (u + e) *^7 *^Z ^^ (1^2 (5.72) 5.5. Freeze in a Harmonic Oscillator 101 For the semiclassical calculations we will also need the classical limits of W and R. Introducing canonical action angle variables for the spin (i.e. "central") system, hSz -> jc = cos 0, hS± -> yjl - jl e±i0c, 9C = -j, W = (e,(j), j = (jc,je)- The classical limits w of W and r of R are w ijj + e (5.73) (5.74) (5.75) (5.76) 5.5.1 The Plateau 10" 10" _ 10" 10" 10" 10" 20 40 60 80 100 120 140 160 180 200 t Figure 5.13: The plateau 1 — F(t) (upper symbols) and 1 — I(t) (lower symbols) for S = 50 and coherent initial state. Perturbation of strength 5 = 0.01 is in G', u = e = 0.3, G = 0, a = 1.15. Chain lines represent the theory (5.87,5.88) and circles full numerical simulations. The height of the plateau is given by the expectation value of the echo operator i^piat = | (exp (-i6{W(t) - W(0)}/h)) |2. (5.77) 102 Chapter 5. Coupling with the Environment For a harmonic oscillator one has to evaluate this expression explicitly and can not use time averaging as has been done for a general unperturbed dynamics in Section 4.2. We immediately see, that the leading semiclassical order will vanish for coherent initial states. We would get \exp(-i?s(j*,?*,t)/h)\2 = 1, where we denoted s(j,?,t) :=w(j,? + ?t)-w(j,?). (5.78) - Recall that in case of ? = 0 this leading order already gave a nonzero contribution. Here though, we have to calculate the next order. For coherent initial states this is easily done using the stationary phase method, i.e. expanding the phase around the position of the packet to lowest order. The calculation is actually very similar to the calculation of the fidelity decay for coherent initial state and non-residual perturbations. Expansion of the phase gives s(j,?,t) = s(j*,?*,t) + s' ¦ (j - j*,? - ?*) + • • •, (5.79) s := (?s(j,?,t),?s(j,?,t)). (5.80) Note that now s(j, ?,t) depends also on the angles and therefore the ASI method cannot be used directly. Instead, we will use the classical averaging over the initial Gaussian distribution in a phase space. We will replace the quantum expectation value with the classical average. Using the compact notation x := (j,?) = (jc, je, ?c, ?e), we can write the classical density corresponding to the coherent initial state as ?(x;x*) = (2\ VdetDexp(-(x-x*)-D(x-x*)/h), (5.81) with x* being the position of the initial packet and a matrix D (of size 2d x 2d) determining the squeezing of the initial packet in d = dc + de degrees of freedom system. The above classical density is normalised as j?dx = 1. For our choice of the initial state being a product of a spin coherent state (2.28) and an oscillator coherent state (2.31) the matrix D is diagonal with elements The fidelity plateau is now calculated as D11 = 2 , D22 = —, D33 = 1 - jc, D44 = 2je. (5.82) Fplat = resulting in a Gaussian function jdx?(x;0)exp(-i?s>. x/h) (5.83) (-—h?har), FpClaISt = exp-—h?har, ?har := ~s'' D~1s'. (5.84) Note that the result is formally very similar to the expression for the fidelity decay for a non-residual perturbation (3.36). Linear response expansion of the plateau is of course FpClaIS = 1 - f ?har and so the plateau for a harmonic oscillator is by factor 1/—h higher than for general systems (4.62). Beyond linear response though, the plateau will decay faster (i.e. as a Gaussian) for a harmonic oscillator than for systems with ? = 0 where the asymptotic decay of the plateau was a power law (e.g. 1/(?S) for the numerical model used in Section 4.2). , 5.5. Freeze in a Harmonic Oscillator 103 For our Jaynes-Cummings model we have —t J- ? + ? c e 2 s(j, ?,t) = 2sin ?+?t ^ 2je(1 jc2) sin ?c + ?e + t?+?. (5.85) The coefficient ?har is then ?har = 2 2 sin 2 (?+? t) [ (1 - j 2 ) + 2je [j 2 + (1 - j 2 ) cos 2 ?c + ?e +t ?+?) 1 } . (5.86) The same ?har would be obtained from the quantum calculation of the second moment of ?(t). Results of the numerical calculation are shown in Figure 5.13. The initial packet was at (?, ?) = (1,1) and ? = 1.15, giving actions jc = 0.54 and je = 0.026 and angles ?c = 1 and ?e = ?/2. Because je < jc we can neglect the second term in ?har and get 1 - FpClaISt = 3.93?2Ssin2 (0.3t). (5.87) The agreement of this theoretical prediction with numerics can be seen in Figure 5.13. For the reduced fidelity and the purity one can go trough similar calculations. We will just list the linear response result obtained from the quantum expectation values, 1-FR = ? 2 S(?+?)2sin 2('?+? t)2jejc 2 + (1-jc 2 )cos 2^ c + ?e + t ? +?)| 1-I = ? 22 sin 2(? +? t\(1-jc) 2 . (5.88) Theory for the purity shown in Figure 5.13 also agrees well with numerics. The purity plateau for coherent initial states is —h-independent, just as the whole decay of purity. 5.5.2 Beyond the Plateau After a sufficiently long time, when the second term ?(t) becomes important in the echo operator (5.71), the decay will be determined by the operator R. As our R is non-residual, we can use the theory for general perturbations, just using a “renormalised” perturbation strength ?2/2. The fidelity and the reduced fidelity will decay as Gaussians, with the decay times given by C (5.20) and C—r (5.21), respectively, while the decay of the purity in our 1 + 1 degrees of freedom system will be I(t) = 1/y/1 + u(?t)2 (5.42), with u = 4C P /h2 (5.24). For our simple example, the average r— is equal to r and the derivatives occurring in the expressions for C’s are ?r— 2 ?r— 2 ?2r— 2 ?jc =-—(jc+je), ?je = -—jc, ?jc?je = -—. (5.89) Taking into account that the squeezing parameters for coherent initial states are ?c = 1/(1 - j2) and for an oscillator ?e = 1/(2je), we obtain C = —h 2 2 (jc+je)2(1-jc2) + 2j ------)2 (jc + je)2(1 - jc) + 2jc je Cr = —h 2 )2(jc + je) 2 (1-j (? + ?)2 Cp = —h2 (1 - j2)je. (5.90) 104 Chapter 5. Coupling with the Environment This then immediately gives the fidelity decay (4.66), the reduced fidelity (5.44) and the purity (5.42), F(t) = exp (-54S2Ct2/A), FR(t) = exp (-54S2CRt2/A), I(t) = ^j^^- (5'91) Note the ^-independent decay of purity. 102 103 104 105 106 t Figure 5.14: Long time fidelity and purity decay for the same data as in Figure 5.13. The theoretical decay for purity (chain curve)(5.91), agrees with the numerical I(t) until the asymptotic plateau is reached. Similarly, theory for the fidelity overlaps with the numerics. Symbols show the numerical result for classical fidelity which in this case agrees with the quantum fidelity. In Figure 5.14 we show numerical results for the Jaynes-Cummings model with spin size S = 1/h = 50 and a coherent initial state placed at (#, = 1=(|T1) + |T2>)®|a>. (5.94) We will assume that we are in a regular regime, for which the decay of purity of individual product coherent states has been derived before (5.41) and was seen do decay on a —t-independent time scale td ~ 1/5, where 5 is a coupling strength between the central system and the environment. For times smaller than the decoherence time of individual coherent states td, the initial cat state (5.94) will evolve into a superposition of product states, + |X2>®|A2>, (5.95) |V>(t)>«|x1>®|A> + |x2>®|A2>, where we used the notation |X1) ® |A> ~ U(t)|n) ® a) and |X2) ® | f2) ~ U(t)Rc|n) ® |a), where for the second state we have written |r2) =: Rc|n| with some unitary matrix Rc. Note that the propagator U(t) has a coupling of strength 5 between the two subsystems. The second state can therefore be thought of to be obtained from the same initial state |n) ® |a) under a slightly different, perturbed evolution Us(t) := U(t)Rc. The coherences in the reduced matrix (off diagonal elements in pc) of the evolved state (5.95) are proportional to the overlap (A|/32). The square of the overlap (A|/32) is in fact the overlap of two reduced density matrices, one obtained with the unperturbed evolution U(t) and the other with the perturbed one Us(t). Because the final states |A)2) as well as the perturbed evolution Us(t) depend on the initial states of the central system t Here the density has to be square normalised, fdxp2 = 1. 106 Chapter 5. Coupling with the Environment |?1,2>, the square of the overlap is nothing but the reduced fidelity * of the environment §. The decay of coherences and therefore of the purity will be given by the reduced fidelity of a product coherent initial state |?1) |?). As we have seen before (5.44), the CR for a coherent state in a regular regime is CR = 12—h(v'e ¦ ?-1v'e) giving the reduced fidelity decay exp (-l2CRt2/—h2), which in turn determines the decay of coherences and therefore also purity decay. Here the perturbation strength l depends on the unitary matrix Rc, i.e. on the “distance” between the initial states |?1,2) of the central system, as well as on U(t). This perturbation strength l is of course different than the coupling ? of the evolution U(t). For instance, in the simple case of |?2) being a space shifted packet |?1), |?2> = e-^P^|?1), the perturbation strength is proportional to the distance x between the two states. As the coupling ? causes decoherence of individual packets on a time scale td ~ 1/? while the decoherence of the cat state happens on the decay time scale of the reduced fidelity tdec ~ —h/l, it is sufficient to have l > ?—h in order to obtain faster decoherence for cat states. The separation l of the two packets constituing the cat state has to scale only as l ~ —h and therefore does not have to be macroscopically large in the semiclasical limit. For an environment with many degrees of freedom de, and if derivatives v'e are all nonzero, CR will be proportional to d2e. The reduced fidelity for the central system on the other hand will have CR proportional to d 2 c and so if de > dc, the reduced fidelity of the environment will decay faster than the reduced fidelity of the central system. Therefore, at the decoherence time tdec we will still have |?1,2> ~ |?1,2>, justifying the decoherence scenario (5.93). For cat states we therefore have CP oc h instead ofC—P oc h2 as for a single coherent state, resulting in an accelerated decoherence. Of course, the above argumentation is by no means a strict proof but is just a simple illustration of how an accelerated decoherence arises. An argumentation using the fidelity has been used by Karkuszewski et al. (2002), but without realizing the key role played by the time scales of purity and reduced fidelity decays. To repeat, several ingredients were needed: (i) the dynamics was assumed to be regular, (ii) for the reduced fidelity of product coherent states we have CR oc h, (iii) the purity of constituent product coherent states decays on much longer —h-independent time scale (i.e. CP oc —h 2 ) and approximation (5.95) was possible, (iv) furthermore, if de > dc the resulting decohered states |?) are still approximately equal to the initial states |?) of the central system as the reduced fidelity decay is faster for the environment than for the central system. All this together enabled us to “derive” a faster decoherence decay for macroscopic cat states. For chaotic dynamics and times longer than the Ehrenfest time ~ ln —h we have seen that fidelity, reduced fidelity and purity all decay on the same time scale irrespective of the initial state. A natural question arises: How does decoherence of macroscopic superposition in chaotic system behave? Actually, decoherence for macroscopic superpositions is so fast, that it happens on time scales shorter than any dynamical scale of the system (Braun et al. , 2001; Strunz et al. , 2003; Strunz & Haake, 2003). It therefore happens on an “instantaneous” time scale on which every system behaves as a regular one (i.e. no decay of correlation functions). The chaoticity therefore has no influence on decoherence of macroscopic superpositions. Numerical Example We take a Jaynes-Cummings model with the unperturbed Hamiltonian H0 = —h?a^a + ?hSz and the perturbation in the coupling G of strenght ?, V = [?G]. Purity fidelity is then equal to the purity which will be used as a measure of decoherence. The initial state will be a product state of a ‡Actually we used a slightly different definition of the reduced fidelity as an overlap of the initial and the echo state. §In general the states |/31,2) can not be obtained by a Hamiltonian evolution on the environmental subspace alone. 5.6. Decoherence for Cat States 107 Figure 5.15: Dependence of Čp (Eq. 5.99, exact expression) for the cat state (5.96) on the position angles of two constituent coherent packets. We have take (f\ = V8ficos2(^/2). For random initial state of a spin and a coherent packet for an oscillator we can calculate Čp using the expression (5.98) and get ČP = S+l random spin i.e.. (5.100) Again we have Čp oc h, just like for cat states, but with a smaller prefactor. If we had a random state on the whole Hilbert space, i.e. also for an oscillator, we would have Čp oc h°. Therefore, the decoherence rate for a cat state is between the decoherence rate of a single packet and the 5.6. Decoherence for Cat States 109 100 r-r 1 L—------------'-------'-----.......------------'-------'-----......' 1 10 100 s Figure 5.17: Dependence of numerically determined decay times of purity on S (= l/K) for a random spin initial state, a Schrodinger cat state and a coherent state. The perturbation strength is 5 = 0.01 while all other parameters are the same as for Figure 5.16. Theoretical decay times r = are shown with full lines. decoherence rate for a random state. The accelerated decoherence can be seen in Figure 5.16 where we show numerically calculated purity decays for all three different initial spin states. Revivals of purity on a classical time scale ~ 1/5 due to integrability of unperturbed Hamiltonian can also be seen. This revivals will be absent in a general non-integrable Hamiltonian. To further illustrate the scaling of decay times with S we show in Figure 5.17 numerically calculated decay times in which the purity decays to level 0.9 for all three different initial states, coherent, cat state and random state of a spin, while the oscillator is initially always in the same coherent state. Chapter 6 Application: Quantum Computation In theory, there is no difference between theory and practice. In practice, there is no relationship between theory and practice. —Anonymous Quantum information theory is relatively recent, for a review see (Nielsen & Chuang, 2001; Steane, 1998; Ekert & Josza, 1996). Its beginnings go back to ’80 and in recent years theoretical concepts have been demonstrated in experiments. While quantum cryptography, a method of provably secure communication, is already commercially available, quantum computation on the other hand is still limited to small laboratory experiments. In order to perform quantum computation you obviously need a quantum computer. A quantum computer can be considered as a many-body system of n elementary two-level quantum systems — called qubits. The union of all n qubits is called a quantum register |r). The size of the Hilbert space M and therefore the number of different states of a register grows with the number of qubits as M = 2n. Quantum computation then consists of the following steps: Load the data for the quantum computation in the initial state of the quantum register |r), resulting in a general superposition of M basis states. Then perform the actual computation, represented by a unitary transformation U. As U acts on an exponentially large space it is usually decomposed into simpler units U = UT--- U2U1. Such sequence of T elementary one-qubit and two-qubit quantum gates Ut, t = 1,2,..., T is called a quantum algorithm. • • • Finally, we read out the result of the computation by performing measurements on the qubits of the final register stateU | r). A quantum algorithm is called efficient if the number of needed elementary gates T grows with at most polynomial rate in n = log2AA, and only in this case it can generally be expected to outperform the best classical algorithm. At present only few efficient quantum algorithms are known, perhaps the most generally useful being the Quantum Fourier Transformation (QFT) (Shor, 1994; Coppersmith, 1994) which will also be the subject of our study. There are two major obstacles for performing practical quantum computation. First, there is the problem of decoherence (Chuang et al. , 1995) resulting from an unavoidable generally time- 111 112 Chapter 6. Application: Quantum Computation dependent perturbation due to the coupling between the qubits and the environment. If the perturbation couples only a small number of qubits at a time then such errors can be eliminated at the expense of extra qubits by quantum error correcting codes (Steane, 1996a; Calderbank & Shor, 1996; Steane, 1996b), for another approach see (Tian & Lloyd, 2000). Second, even if one knows an efficient error correcting code or assumes that quantum computer is ideally decoupled from the environment, there will typically exist small unknown or uncontrollable residual interaction among the qubits which one may describe by a general static perturbation. Therefore, understanding the stability of quantum algorithms with respect to various types of perturbations is an important problem, for some results on this topic see (Miquel et al. , 1996, 1997; Gea-Banacloche, 1998, 1999, 2000; Song & Shepelyansky, 2001; Georgeot & Shepelyansky, 2000; Berman et al. , 2001, 2002a,b; Celardo et al. , 2003). 6.1 General Framework Let us perturbe the t-th quantum gate by a perturbation* generated by Vt, Ut? :=exp(-i?Vt)Ut. (6.1) We set —h = 1 and use the superscript ? to denote a perturbed gate and the subscript t to denote a discrete time index, i.e. a gate number. We allow for different perturbations Vt at different gates. The perturbed algorithm is simply a product of perturbed gates, U? = UT? · · · U1?. Fidelity will again serve as a measure of stability and we have F(T) = | (M?(t)) |2, M? := U(-T)U?(T), (6.2) with U(t) :=Ut ···U1 and similarly for U?(t). For our generally time dependent perturbation the echo operator equals to M? = e-i?VT(T) · · · e-i?V2(2)e-i?V1(1), (6.3) where Vt(t) := U\t)VtU(t) is the perturbation of t-th gate Vt propagated with the unperturbed gates U(t) = Ut ···U1. Beware that Vt(t) is time dependent due to two reasons, one is due to the interaction picture (time in parentheses) and the second is that the perturbation itself is explicitly time dependent, i.e. different perturbation for different gate (time as a subscript). In quantum computation one is usually interested in initial states containing a maximum amount of information, thereby being close to random states. With this in view, we take the initial state average to be a trace over the whole Hilbert space, (•):= N tr(•). (6.4) By this prescription we study the average fidelity over the whole Hilbert space. Without sacrificing generality we furthermore assume the average perturbation to be traceless, LTt=1 trVt = 0. Trace of the average perturbation only changes the phase of the fidelity amplitude and has therefore no influence on the fidelity (i.e. probability). The linear response expansion of the fidelity then reads T F(T) = 1-?2 ]T C(t,t'), (6.5) t,t'=1 with the correlation function C(t,t'):= N tr[Vt(t)Vt (t)]. (6.6) ?It is expected that the computation is stable only when the evolution is close to unitary. 6.2. Quantum Fourier Transformation 113 Here we used the fact that the average trace vanishes so the term (?(t)> is zero. Decomposition of a given quantum algorithm U into quantum gates is by no means unique. An interesting question then is, given perturbations Vt, which form of the algorithm has the highest fidelity, i.e. is the most stable? We will see that the standard QFT algorithm can be rewritten in a non trivial way so that it becomes more stable against static perturbations. The guiding principle in the construction of this new algorithm will be to study the correlation function (6.5) and trying to minimise its sum. 6.1.1 Time Dependent Perturbations If our perturbation Vt is time dependent, i.e. we have different perturbations on different gates, then the decay of correlation function will not only depend on the unperturbed dynamics, but also on how strong these perturbations are correlated at different gates. Let us have a closer look at one extreme example. If the perturbation Vt is an uncorrelated noise, as it would be in the case of coupling to an ideal heath bath, then the matrix elements of Vt may be assumed to be Gaussian random variables which are uncorrelated in time, (Vjk(t)Vlm(t)) noise = 1 m?kl?tt . (6.7) Hence one finds (C(t,t'))noise = ?tt, where we have averaged over noise. In fact the average of the product in M? (6.3) equals to the product of the average and yields the noise-averaged fidelity = exp(-?2T), (6.8) ¦ oo (i.e. large number of qubits n) over the noise component, as soon as the quantum algorithm exhibits long time correlations of the operator Vstatic. If Vstatic = 0 is zero, the quantum computation can be stabilised by making “adiabatically” slow evolution of gates. In the following we will focus exclusively on the static perturbations being the worst ones. 6.2 Quantum Fourier Transformation We will consider Quantum Fourier Transformation algorithm (QFT) and will consider its stability against static random perturbations. The perturbation Vt = V will be a random hermitian matrix from a Gaussian unitary ensemble (GUE). Gaussian unitary ensamble is invariant under unitary transformations and the matrix elements in an arbitrary basis are independent random Gaussian variables (Mehta, 1991). Due to hermitian symmetry they are real on the diagonal and complex off-diagonal. The GUE matrices can be used to model quantum statistical properties of classically chaotic Hamiltonians and have been first applied to studies of nuclear resonances. Second moments of a GUE matrix V are normalised as (VjkVlm)GUE = ?jm?kl/M, (6.9) where the averaging is done over a GUE ensemble. 114 Chapter 6. Application: Quantum Computation Let us briefly describe the QFT algorithm. Basis qubit states in a Hilbert space of dimension M = 2n will be denoted by | k), k = 0,..., 2n - 1. The unitary matrix Uqft performs the following transformation on a state with expansion coefficients xk Af-1 Af-1 Uqft( Y, xk|k)) = Y, x~k|k>, (6.10) k=0 k=0 where xk = -^ Y$=-0 exp (2?ijk/M)xj. The resulting expansion coefficients xk are Fourier transformed input coefficients xk. The “dynamics” of the QFT is decomposed into three kinds of unitary gates: One-qubit gates Aj acting on j-th qubit where the basis is ordered as (|0>,|1», diagonal two-qubit gates Bjk = diag{1, 1, 1,exp(i?jk)}, with ?jk = ?/2k-j, and transposition gates Tjk which interchange the j-th and k-th qubit, Tjk|...j...k...} = |...k...j...). There are n A-gates, n(n - 1)/2 B-gates and [n/2] transposition gates, where [x] is the integer part of x. The total number of gates for the algorithm is therefore T = [n(n + 2)/2]. For instance, in the case of n = 4 we have a sequence of T = 12 gates (time runs from right to left) Uqft = T03T12A0B01B02B03A1B12B13A2B23A3. (6.12) For the GUE perturbation we can average the correlation function C(t, t') (6.6) over the GUE ensemble, resulting in 1 the gates Bjk are close to the identity and therefore cannot reduce the correlator. This slow correlation decay results in the correlation sum T ?:= Y, C(t,t'), (6.14) t,t'=1 being proportional to ? oc n3 (sum of the first n squares) as compared to the theoretical minimum ?ocTocn2. In view of this, we will now try to rewrite the QFT with a goal to achieve a smaller correlation sum, hopefully ? oc n2. From (6.13) we learn that the gates that are traceless (e.g. A-gates) reduce the correlator very efficiently. In the plain QFT algorithm (6.12) we have n - 1 blocks of B-gates, where in each block all B-gates act on the same first qubit, say j. In each such block, we propose 6.2. Quantum Fourier Transformation 115 0 20 40 60 gate t 80 100 0 10 20 30 40 50 60 4 6 -o o v 10 c 12 14 Figure 6.1: Correlation functions (C(t, L'))gue (6.13) for n = 10 qubits and static GUE perturbation. The right figure shows the standard QFT (6.12) with T = 60, while the left figure shows the IQFT with T = 105 gates. Colour represents the size of elements in a log-scale from red (e_0) to blue -14 and less). to replace Bjk with a new gate Gjk = RLB^, where a unitary gate Rjk will be chosen so as to commute with all diagonal gates B^ in the block, whereas at the end of the block we will insert Rjk in order to "annihilate" R-k so as to preserve the evolution matrix of a whole block. The unitarity condition RjkRjk = 1 and [Rjk, ~Bji] = 0 for all j,k,l leaves us with a 6 parametric set of matrices Rjk- By further enforcing trR^ = 0 in order to maximally reduce the correlator, we end up with 4 free real parameters in R^. One of the simplest choices, that turned out to be as suitable as any other, is the following Rjk — (o 0 -1 0 1 0 0 1 0 0 0 Vo 0 0 \ 1/ (6.15) with the basis states (\jk)) ordered as (00,10,01,11). The R gate can be compactly written as R ¦jk\ .cij ...bk = (-l \bk\ .. .a 3 ¦ {aj®bk) where © is an addition modulo 2, bar denotes a negation and a,j, bk are 0 or 1. Furthermore, we find that R-gates also commute among themselves, [RjTcRji] = 0, which enables us to write a sequence of R-gates whichever way we like, e.g. in the same order as a sequence of G's, so that pairs of gates Gjk, Rjk operating on the same pair of qubits (j, k), whose product is a bad gate Bjk, are never neighbouring. This is best illustrated by an example. For instance, the block B01B02B03 will be replaced by R01R02R03R1 01 B01R09B02R03B03 L03J R01R02R03G01G02G03. This is how we construct an improved quantum Fourier transform algorithm (IQFT). For the IQFT we need one additional type of gate, instead of diagonal B-gates, we use nondiagonal ones R and G. To illustrate the obvious general procedure we write out the whole IQFT algorithm for n = 4 qubits (compare with (6.12)) t^IQFT = T03T12A0R01R02R03G01G02G03A1R12R13G12G13A2R23G23A3. (6.16) Such IQFT algorithm consists in total of T = [n(2n + l)/2] gates (note that it does not pay of 116 Chapter 6. Application: Quantum Computation 1 0.95 0.9 0.85 0.8 t 0.75 Ll_ 0.7 0.65 0.6 0.55 0.5 3456789 10 n Figure 6.2: Dependence of the fidelity F(T) on the number of qubits n for the QFT (pluses) and the IQFT algorithms (crosses), for fixed 5 = 0.04. Numerical averaging over 50 GUE realizations is performed. The full curve is exp (—52{0A7n3 — 0.76n2 + 2.90n}) and the dashed one exp (—52{1.22n2 + 1.78n}). For n = 10 the trace is approximated by an average over 200 Gaussian random register states. to replace a block with a single B gate as we have done, so we could safely leave B23 = R23G23). The correlation function for the IQFT algorithm is shown on the left of Figure 6.1. Almost all off-diagonal correlations are greatly reduced (to the level oc 1/A/"2), leaving us only with a dominant diagonal. If we would have only diagonal elements, the fidelity would be F(T) = 1 — 52T, (as in the case of noisy perturbation or decoherence, however, with a different physical meaning of the strength scale 5) where the number of gates scales as T oc n2. From Figure 6.1 it is clear that we have a very fast correlation decay for the IQFT. Studying the scaling with n, the correlation sum v has decreased from v oc n3 to v oc n2. To further illustrate this, we have numerically calculated fidelity by simulating the quantum algorithm and applying the perturbation exp (—\5V) at each gate. The results are shown in Figure 6.2 where one can see much faster decay of the fidelity for the QFT than for the IQFT. Note that the IQFT has higher fidelity despite the perturbation being applied ~ n2 times for the IQFT and only ~ n2/2 times for the QFT. As we have argued before, the sum of 2-point correlator (6.5) gives us only the first nontrivial order in the ^-expansion. For dynamical systems, being either integrable or mixing and ergodic, we have shown in previous chapters that also higher orders of the fidelity can approximately be written as simple powers of the correlation sum z/, so that the fidelity has the simple form F(T) « exp(-z/L2). (6.17) Although quantum algorithm is quite inhomogeneous in time, we may still hope that exp (—vb2) is a reasonable approximation of the fidelity also to higher orders in 5. This is in fact the case as can be seen in Figure 6.3. Note also that the leading coefficient in the exponent for the IQFT 6.2. Quantum Fourier Transformation 117 fidelity, lim^^oo u/n2 = 1.22, is close to the theoretical minimum of 1. As the definition of what is 10° 10"1 ^ 10"2 Ll_ 10"3 10"4 10"5 0 0.05 0.1 0.15 0.2 0.25 0.3 d Figure 6.3: Dependence of the fidelity F(T) on d for the QFT (pluses) and the IQFT (crosses), for fixed n = 8. Solid curves are functions exp(—v52) (see text) with v calculated analytically (6.13) and equal to v = 216 for the QFT and 93.2 for the IQFT. a fundamental single gate is somehow arbitrary, the problem of minimising the sum v depends on a given technical realization of gates and the nature of the perturbation V for an experimental setup. It has been shown (Celardo et al. , 2003), that the IQFT algorithm improves stability against GUE perturbations also for a more realistic model of a quantum computer, namely for an Ising quantum computer. We should mention that the optimisation becomes harder if we consider few-body (e.g. two-body) random perturbation. This is connected with the fact that quantum gates are two-body operators and can perform only a very limited set of rotations on a full Hilbert space and consequently have a limited capability of reducing correlation function in a single step. For such errors the fidelity will typically decay with the square of the number of errors T (i.e. gates), like ~ exp(—52T2), that is the same as for regular systems, see also (Gea-Banacloche, 2000) for a similar result. This means that the very fact that the algorithm is efficient, having a polynomial number of gates, makes it very hard to reduce the correlation function and therefore causes a fast fidelity decay. However, our simple approach based on n-body random matrices seems reasonable, if the errors due to unwanted few-body qubit interactions can be eliminated by other methods. Chapter 7 Conclusion I hate quotations. Tell me what you know. —Ralph Waldo Emerson We have studied the decay of quantum fidelity, of reduced fidelity and of purity in quantum systems. We considered two extreme cases of system’s dynamics, mixing and regular. The dependence on the initial conditions, in particular for a random and coherent initial state, and the influence of the perturbation type on quantum stability has been analysed. For a general type of perturbation, having a nonzero diagonal matrix elements in the eigenbasis of the unperturbed system, i.e. having a nonzero time average, the fidelity decay depends on the mixing properties of system dynamics. For mixing systems in the Fermi golden rule regime fidelity decay is exponential, with the decay time given by the transport coefficient, which is in turn the integral of the correlation function of the perturbation. The decay time scales with Planck’s constant and with the perturbations strength as ?m ~ h2/?2. In this regime quantum fidelity decay is much slower than classical fidelity decay and moreover, it will in general be the slower the more chaotic the corresponding classical system is. This surprising result does not violate the quantum-classical correspondence though, as for large perturbation strengths, in the so called Lyapunov regime, quantum fidelity agrees with classical fidelity. Whether we observe the quantum or the classical behaviour depends on the order of two noncommuting limits, namely the semiclassical limit of vanishing Planck constant —h ->¦ 0 and the limit of vanishing perturbation strength ? ->¦ 0. For sufficiently long times fidelity decay in mixing systems does not depend on the initial state. In regular systems and for perturbations with a nonzero time average the fidelity decay for wave packets is governed by a ballistic separation of the packets. For coherent initial states the resulting decay is Gaussian with the decay time scaling as ?r ~ —h/?, and can be for sufficiently small perturbations smaller than for chaotic systems. For random initial states in a regular regime quantum fidelity decays according to a power law, F ~ (h/?t)d in d degrees of freedom system. In regular systems we also considered the decay averaged over random positions of the initial coherent state, resulting in an algebraic decay but with the power being system specific. By a semiclassical method we theoretically calculated all decay times in both cases of regular and mixing dynamics in terms of classical quantities only, despite the fact that the quantum and the classical fidelity do not agree for mixing systems. The quantum fidelity decay is markedly different for perturbations with a zero time average, which can be written as a time derivative of another operator. For such perturbations fidelity freezes at a constant plateau regardless of the dynamics and starts to again decay only after a much longer 119 120 Chapter 7. Conclusion time, scaling as ~ 1/?. This freezing is a pure quantum phenomenon as the correspondence with classical fidelity ends before the plateau starts and classical fidelity decays much faster. The only exception where the classical fidelity also exhibits freezing and agrees with quantum fidelity is for a harmonic oscillator, for which the plateau is higher than for other systems. We explicitly calculate the plateau value for mixing and regular dynamics and coherent and random initial states. For mixing dynamics a universal relation holds between the plateau FpRlaIS for random and FpClaIS for coherent states, (FpClaIS)2 = FpRlaIS. In the linear response regime the scaling of the plateau is 1 - Fplat ~ ?2/—h2 regardless of the dynamics. The asymptotic decay of fidelity after the plateau ends is also theoretically calculated in terms of a “renormalised” perturbation with strength ?2/2. For regular systems this long time Gaussian decay happens on a time scale ~ —h/?2 for coherent initial states whereas it is power law with the prefactor scaling as ~ h/?2 for random initial states. For mixing dynamics the long time decay does not depend on the initial state and is exponential with the decay time ~ h2/?4 or Gaussian with the decay time ~ —h1-d/2/?2. The crossover from the exponential to the Gaussian decay happens at the Heisenberg time. For one dimensional regular systems we also explain echo resonances, a sudden revivals of fidelity. We also study composite systems, composed of a central system and an environment, and connect the decay of purity with the decay of reduced fidelity. We prove a rigorous inequality between the fidelity (characterising the stability of the whole system), the reduced fidelity (characterising the stability of the central system) and the purity (characterising the entanglement). For mixing systems fidelity, reduced fidelity and purity all decay on the same time scale. For regular systems though, reduced fidelity has a Gaussian decay for coherent states whereas purity decays on an —h independent time scale. We explicitly calculate the purity decay and the power of the asymptotic algebraic decay depends on the perturbation and can range between 1 and dc, the dc being number of degrees of freedom of the central system. All decay constants are explicitly calculated. We also discuss an interesting case where the time scales of the central system and the environment are vastly different and one can use averaging over the faster system to simplify the theory. Decoher-ence for macroscopic superpositions of coherent states is derived and shown to be faster than for a single coherent state. Finally, we show an application of fidelity theory. By “randomising” the quantum Fourier transform algorithm we are able to make it more resistant against random perturbations from the environment. Appendix A Spin Wigner functions The Wigner function enables us to represent the quantum density matrices in a phase space and thereby compare it with the classical probability densities. If we have a one particle quantum system, described by a canonical pair [q,p] = ih, the Wigner function W? of a quantum state given by a density matrix ?^ is W?(q,p):= 1— fdx(q-x\?\q + x}exp(-i2px/h), (A.1) 2?—h dx(q-x\?\q if \x) is an eigenstate of operator q with an eigenvalue x. In this appendix we will use a hat for quantum operators. For spin state such a definition can not be used as the Hilbert space has different structure due to SU(2) commutation relations of spin operators. We would like to obtain a Weyl symbol of an arbitrary operator A acting on a Hilbert space of size 2S+1, i.e. on a state space of spin of size S. In the special case when the operator A is equal to a density matrix, the resulting Weyl symbol is called a Wigner function. Such functions have been first proposed by Agarwal (1981). The Weyl symbol WA(?, ?) will be a function of coordinates on a sphere. Furthermore, we would like the standard trace dot product for operators to carry over to Weyl symbols, i.e. we demand that the following equality should hold tr( (A^,B^f)= WAW%d?, (A.2) with d? = d? sin?d?. Spherical harmonic functions Ylm constitute an orthonormal basis on a sphere, [YlmYlm'*d? = ?ll/?mm/, (A.3) and so we can expand the Weyl symbol over Ym. On the other hand, an arbitrary operator A can be in turn expanded over multipole operators T^lm forming an orthogonal basis in the Hilbert space of operators, tr(Tmlff]) = ?ll ?mm. (A.4) Therefore we can write the operator A as 2S l A^ :=YsYl almT^lm, alm= tr(A^T^lmf). (A.5) l=0m=-l 121 122 Appendix A. Spin Wigner functions From this and the orthogonality property of Ylm we immediately see that the Weyl symbol defined as 2S l WA(?,?) :=Y1 Y, almYlm(?,?), (A.6) l=0m=-l will satisfy dot property (A.2) we demanded. The coefficients alm can be calculated using the explicit form of the multipole operators Tm:= S (-1)S-m-q?2l+1(-m- l S)|m + q)(q|, (A.7) with (- S - l S) being the Wigner 3 j symbol. The spin Wigner function for a state represented by a density matrix ?^ is obtained by taking A = ?^ in the above formulas. As the density matrix is a Hermitian operator, the resulting Wigner function is real. In the case of two pure states, ?1 = |?1)(?1|, ?2 = |?2X?2| the dot condition gives simply = W?1W?2d?. (A.8) For examples of Wigner functions of some simple states see e.g. (Dowling et al. , 1994). |?1|?2)|2 = Appendix B Coherent State Expectation Values Boson coherent states Expectation values of expressions involving creation and annihilation operators a+ and a for a harmonic oscillator coherent state |a) are frequently needed. Using the definition of |a) (2.30) it is easy to show the following equality (a|g(a+)f(a)|a)=g(a*)f(a), (B.1) with two polynomials g and f. Let p(a,a+) be some polynomial function and by : p(a,a+) : we will denote a polynomial in a normal order which can be obtained from p(a, a+) by using the commutation relation [a, a+] = 1 to bring all terms involving a+ to the left of terms with a. For instance, :aa+: = a+a + 1. Then we can write the expectation value of an arbitrary polynomial as = {a|:p(a,a+):|a)=:p(a,a*): ¦ (B.2) {a|p(a,a+)|a) = {a|:p(a,a+):|a) Spin coherent states For spin coherent states formulas are a bit more complicated due to different group structure. The easiest systematic method for the calculation of expectation values of polynomials in operators SX)y)z in coherent state |&*,ip*) (2.26) is using generating function formalism (Arecchi et al. , 1972). For brevity let us denote a spin coherent state with a complex parameter r := e1^* tan (#*/2), i.e. |t) := |#*,Y>*>. The following expression holds, y y y *«,*«}, S b zS c_|T2) =? L L x(^,ri,o , (B.3) where m| |2/= (1 + |ri|2)s(1 + |r2|2)s This formula, together with the commutation relations, can be used to calculate some of the lowest powers (r|Sz|r) = Sz 2 (r|Sz2|r) = S2z2 + S 1-z2) 123 124 Appendix B. 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