University of Ljubljana Faculty of mathematics and physics Department of Physics Gregor Vidmar The influence of Tunneling on Eigenvalue Statistics and Dynamics in the Mixed-Type Systems Doctoral thesis ADVISER: Prof. Dr. Marko Robnik Ljubljana, 2008 Univerza v Ljubljani Fakulteta za matematiko in fiziko Oddelek za fiziko Gregor Vidmar Vpliv tuneliranja na statistiko in dinamiko energijskih stanj sistema meˇsanega tipa Doktorska disertacija MENTOR: prof. dr. Marko Robnik Ljubljana, 2008 CAMTE This PhD thesis has been performed at the CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor CAMTE To doktorsko delo je bilo izdelano na CAMTP - Centru za uporabno matematiko in teoretiˇcno fiziko Univerze v Mariboru i Abstract In this thesis we have studied dynamical tunneling and random matrix theory in the field of quantum chaos. First, we have considered the spectra of quantum Hamilton systems characterised by the mixed-type classical dynamics. In the semiclassical limit the Berry-Robnik (BR) statistics applies whereas at larger values of heS one can notice deviations from BR due to localisation and tunneling effects. We have derived a 2-level random matrix model which can be treated analytically and the iV-level random matrix model which has been treated numerically. Both models describe tunneling effects. The coupling between the regular and the chaotic levels due to tunneling is assumed to be Gaussian distributed. The results are predicted to apply in mixed-type systems at low energies. The two-level model describes many features of large matrices. The proposed 2 and the N level spacing distributions have two parameters, the BR parameter p, characterising the classical phase space, and the coupling parameter a. The same procedure has been followed for the all-to-all couplings as well. Second, we have studied the mushroom billiard introduced by Bunimovich, especially in terms of its level spacing distribution, avoided crossing distribution and dynamical tunneling rates. The mushroom billiard has a nice property of sharply divided phase space into precisely one regular and one chaotic region. By changing the appropriate parameter one can easily identify the regular or the chaotic states without considering the Wigner functions. First we have studied the level spacing distribution to test the random matrix model and, then, we have focused on the avoided crossings appearing between the regular and the chaotic states. The splittings, appearing at such avoided crossings, indicate the strength of dynamical tunneling. Larger splitting presuppose stronger tunneling effect. With the help of Fermi’s golden rule we have calculated the dynamical tunneling rates for each regular state of the mushroom billiard. We have compared the results from the microwave experiment and those from the expanded boundary integral numerical method with the already-existing analytic prediction derived by Ketzmerick and his co-workers. We have observed excellent agreement in the accessible regime of low energies. Third, we have discussed various ensembles of real symmetric matrices with the dimensions N = 2 to N = oo for a variety of distributions of matrix elements. For N = 2 there exist the exact analytic results obtained by Grossmann and Robnik whereas for N = oo one can rely on the theoretical findings by Hackenbroich and Weidenmu¨ller (HW). According to these findings the local spectral fluctuations are exactly described by the GOE if the limiting distribution of the eigenvalues is smooth and restricted to a finite interval. We have numerically shown, according to HW, that such a transition to the universal behaviour is pretty fast and that it does really not occur if one or both of the conditions from the theory mentioned are not fulfilled. We have tested this findings for the box, the exponential, the Cauchy-Lorentz and the singular (power law times exponential) distribution function of matrix elements. ii PACS numbers: 02.50.Cw Probability theory 02.70.Pt Boundary-integral methods 03.65.Sq Semiclassical theories and applications in quantum mechanics 03.65.Xp Tunneling, traversal time, quantum Zeno dynamics 05.45.-a Nonlinear dynamics and nonlinear dynamical systems 05.45.Mt Quantum chaos; semiclassical methods 05.45.Pq Numerical simulations of chaotic systems Keywords: dynamical tunneling, regular, chaotic, mixed-type system, energy levels, level spacing distribution, level dynamics, mushroom billiard, avoided crossings, tunneling rates, random matrix theory, microwave experiments iii Povzetek V tej disertaciji študiramo dinamično tuneliranje in teorijo naključnih matrik na področju kvantnega kaosa. Najprej obravnavamo energijske spektre kvantnega Hamiltonskega sistema z mešano klasično dinamiko. V semiklasični limiti, ko je efektivna Planckova konstanta heS dovolj majhna, velja statistika Berryja in Robnika (BR), medtem ko pri večjih vrednostih freff (manjših energijah) vidimo odstopanja od BR zaradi lokalizacije in tunelskih efektov. Izpeljemo 2-nivojski model naključnih matrik, ki ga lahko obravnavamo analitično in iV-nivojski model naključnih matrik, ki ga obravnavamo numerično. Oba modela opisujeta tunelske efekte. Predpostavljamo Gaussovo, eksponentno ali škatlasto porazdelitev tunelske sklopitve med regularnimi in kaotičnimi nivoji. Rezultati se predvidoma uporabljajo v mešanih sistemim pri nizkih energijah. Dvonivojski model opisuje večino lastnosti večjih matrik. Predlagani 2- in iV-nivojski porazdelitvi po razmikih med sosednjimi nivoji imata dva parametra, BR parameter p, ki opisuje klasični fazni prostor, in sklopitveni parameter a. Isti postopek naredimo tudi v primeru sklopitve vseh nivojev z vsemi. Nadalje eksperimentalno in numerično študiramo gobasti biljard, ki ga je vpeljal Bunimovich, posebej porazdelitev po razmikih med sosednjimi nivoji, porazdelitev izognjenih križanj in koeficiente dinamičnega tuneliranja. Gobasti biljard ima to lepo lastnost, da je fazni prostor ostro ločen v natanko eno regularno in eno kaotično komponento. S spreminjanjem ustreznega parametra razvrstimo stanja v regularna in kaotična brez računanja Wignerjevih funkcij. Najprej se osredotočimo na porazdelitev po razmikih med sesednjimi nivoji z namenom testiranja modela naključnih matrik, potem pa na izognjena križanja med regularnimi in kaotičnimi stanji. Razcep v izognjenem križanju nam pove o velikosti dinamičnega tuneliranja in posledično o stabilnosti določenega regularnega stanja. Večji razcep pomeni večje tuneliranje. Z upoštevanjem Fermijevega zlatega pravila smo dobili koeficiente dinamičnega tuneliranja za vsako regularno stanje gobastega biljarda. Primerjamo rezultate mikrovalovnega ekperimenta in numeričega računa z razširjeno metodo integriranja po robu z že obstoječo analitično teorijo Ketzmericka in sodelavcev. Našli smo zelo dobro ujemanje v dosegljivem območju nizkih energij. Nazadnje obravnavamo ansamble realnih simetričnih matrik dimenzij N = 2 do N = oo za različne porazdelitve matričnih elementov. Za N = 2 imamo eksaktne analitične rezultate Grossmanna in Robnika, medtem ko imamo zaiV=co teorijo Hackenbroicha in Weidenmüllerja (HW). Ta pravi, da so lokalne fluktuacije spektrov natančno popisane z GOE, če je limitna porazdelitev lastnih vrednosti gladka in omejena na končen interval. Numerično pokažemo, daje tak prehod k univerzalnemu obnašanju, ki ga napoveduje HW, zelo hiter in da se dejansko ne zgodi, če ena ali druga predpostavka v HW ni izpoljena. To testiramo za škatlasto, eksponentno, Cauchy-Lorentzovo in singularno (potenčna pomnožimo z eksponentno) porazdelitev matričnih elementov. iv PACS števila: 02.50.Cw Verjetnostna teorija 02.70.Pt Metoda integriranja po robu 03.65.Sq Semiklasične teorije in aplikacije v kvantni mehaniki 03.65.Xp Tuneliranje, čas prečenja, kvantna Ženo dinamika 05.45.-a Nelinearna dinamika in nelinearni dinamični sistemi 05.45.Mt Kvantni kaos; semiklasične metode 05.45.Pq Numerične simulacije kaotičnih sistemov Ključne besede: dinamično tuneliranje, regularno, kaotično, sistem mešanega tipa, energijski nivoji, porazdelitev po razmikih med sosednjimi nivoji, dinamika stanj, gobasti biljard, izognjena križanja, koeficienti tuneliranja, teorija naključnih matrik, mikrovalovni eksperimenti v Acknowledgement I would like to thank my supervisor Marko Robnik for all his effort, organisation and support during all the years of my research presented in this thesis. I would also like to thank Hans-Ju¨rgen St¨ockmann for his cooperation, great ideas and remarks. I am also grateful to Ulrich Kuhl for his great and invaluable help in the experiments and for many answers to my numerous questions. I would also like to thank Marko Vraniˇcar, Igor Mozetiˇc and Valery Romanovski for their unselfish help at CAMTP. I am also grateful to Roland Ketzmerich and his team from Dresden, Helena M. David and Ruven H¨ohmann for their excellent cooperation. I would like to express my gratitude to Tomaˇz Prosen and, especially, to Gregor Veble for all their advice and software assistance. In the end, I would like to thank Brigita Retar and Petra Puˇcnik for proofreading the English and the Slovenian part of the present thesis. vi vii Zahvala Rad bi se zahvalil mentorju Marku Robniku za ves trud, organizacijo in podporo vsa leta raziskovanja, ki je sedaj predstavljeno v tej disertaciji. Potem bi se rad zahvalil Hans-Ju¨rgenu St¨ockmannu za zelo dobro sodelovanje, ideje in komentarje. Rad bi se zahvalil Ulrichu Kuhlu za veliko in prijazno pomoˇc v ˇcasu bivanj v Marburgu, za vso podporo pri eksperimentalnem delu in izvrednotenju podatkov ter za odgovore na veliko mojih vpraˇsanj. Rad bi se zahvalil tudi Marku Vraniˇcarju, Igorju Mozetiˇcu in Valeriju Romanovskemu za nesebiˇcno pomoˇc na CAMTP-ju ter skupini Rolanda Ketzmericka iz Dresdena, Heleni M. David in Ruvenu H¨ohmannu za dobro sodelovanje. Zahvala gre tudi Tomaˇzu Prosenu in posebej Gregorju Vebletu za koristne diskusije in programsko pomoˇc. Na koncu bi se rad zahvalil Brigiti Retar in Petri Puˇcnik za lektoriranje angleˇskega in slovenskega dela te disertacije. viii ix Dedication I dedicate this thesis to my family. X xi Posvetilo Disertacijo posvečam svoji družini. xii Contents 1 Introduction 1 1.1 Billiards ..................................... 1 1.2 Quantum chaos ................................. 2 1.3 Dynamical tunneling .............................. 3 1.4 Random matrix theory and level statistics .................. 5 1.5 Non-Gaussian RMT ............................... 7 2 The distorted Berry-Robnik level spacing distribution 9 2.1 The Berry-Robnik level spacing distribution ................. 9 2.2 General distortion of the general level spacing distribution ......... 10 2.3 Gaussian distortion of the BR distribution .................. 12 2.4 Analytical studies of the distorted Berry-Robnik distribution ........ 22 2.4.1 Small S behaviour of PDABR(S) ..................... 22 2.4.2 Large S behaviour of PDABR(S) and PDTBR(S) ............. 23 2.5 Simulations with random matrices ....................... 24 2.5.1 The antenna distorted BR distribution (all-to-all couplings) . . . . 24 2.5.2 The tunneling distorted BR distribution ............... 31 2.6 Improvements on the tunneling-distorted BR distribution .......... 34 2.7 Dependence of the model on the matrix element statistics .......... 38 3 Dynamical tunnelinginmushroom billiards 41 3.1 Mushroom billiard - classical .......................... 41 3.2 Mushroom billiard - quantum ......................... 43 3.3 Microwave experiments ............................. 45 3.3.1 The similarity between QM and ED .................. 45 3.3.2 Resonances ............................... 46 3.3.3 Harmonic inversion ........................... 48 3.4 Level Statistics ................................. 50 3.4.1 Level dynamics ............................. 51 3.4.2 The level spacing distribution - comparison with the RMT prediction 53 3.4.3 Absorber in the foot .......................... 53 3.5 Expanded boundary integral method ..................... 53 3.5.1 Corners and the accuracy of EBIM .................. 57 3.5.2 Level dynamics - increasing the depth l ................ 61 3.5.3 The level spacing distribution - comparison with the RMT prediction 64 3.6 Avoided crossings ................................ 64 3.7 Tunneling rates ................................. 72 xiv 3.7.1 Fermi’s golden rule ........................... 73 3.7.2 Tunneling rates - the experiment ................... 75 3.7.3 Tunneling rates - the numerics ..................... 80 4 Numerical studies of non-Gaussian real symmetric random matrices 85 4.1 2D real symmetric random matrices ...................... 85 4.1.1 Box (uniform) distribution ....................... 85 4.1.2 Exponential distribution ........................ 86 4.1.3 Cauchy-Lorentz distribution ...................... 86 4.1.4 Singular times exponential distribution ................ 87 4.1.5 Comparison with numerics ....................... 87 4.2 Numerical calculations with higher dimensional non-Gaussian random matrices ....................................... 88 4.2.1 Box (uniform) distribution ....................... 89 4.2.2 Exponential distribution ........................ 89 4.2.3 Cauchy-Lorentz distribution ...................... 89 4.2.4 Singular times exponential distribution ................ 92 5 Summary and conclusions 95 Bibliography 98 Appendices 105 A ?1 calculation for the mushroom billiard 107 B Theoretical analysis of the tunneling rates in the mushroom billiard 111 C Numerics with particular solutions for the tunneling rates in the mushroom billiard 115 Daljˇsi slovenski povzetek 117 i Deformirana porazdelitev Berryja in Robnika po razmikih med sosednjimi nivoji ........................ 118 ii Simulacije z nakljuˇcnimi matrikami ...................... 120 iii Gobasti biljard ................................. 123 iv Mikrovalovni eksperimenti in numeriˇcna metoda EBIM ........... 125 v Dinamika lastnih energij gobastega biljarda .................. 126 vi Izognjena kriˇzanja in tuneliranje ........................ 127 vii Numeriˇcna ˇstudija negausovskih realno simetriˇcnih matrik ......... 131 1 Chapter 1 Introduction In classical mechanics there exist two types of motion - regular and chaotic. In an integrable system regular motion is stable quasiperiodic or periodic and takes place on the N-dimensional invariant tori in the classical phase space of 2N dimensions where N is the number of degrees of freedom. Since chaotic orbits are characterised by the exponential divergence of nearby orbits, they have positive Lyapunov exponents. The K-system is characterised by the chaotic region in the classical phase space with positive Lyapunov exponents and with a positive Liouville measure. If the system is ergodic as well, i.e. if almost any orbit can visit an arbitrarily small neighbourhood of any other point on the (2N - 1)-dimensional energy surface, we say that the system is fully chaotic. However, in nature the generic (or typical) systems are neither integrable (regular) nor fully chaotic but they are of mixed type. Their motion is regular on invariant tori for certain initial conditions and chaotic for complementary ones. 1.1 Billiards The difference between the two types of motion can easily be demonstrated in billiards. The 2D closed billiard system is characterised by a free-point particle elastically bouncing off the boundary of a given Euclidean domain. The 2D billiard is integrable, if apart from the energy there exists another constant of motion. The invariant tori on the surface of section appear as one dimensional. All the orbits are stable. The only billiards, proven to be integrable, are the circular, the rectangular and the elliptic one. In the circular billiard there appears the angular momentum as the additional constant of motion. Thus, the invariant tori in this case can be either rational (resonant) or not, depending on the reflection angle selected. If the reflection angle is a rational multiple of ?, the torus is resonant and all the orbits on this torus are periodic. If, on the other hand, the reflection angle is an irrational multiple of ?, the orbits on the torus are quasiperiodic whereas the motion is ergodic on this torus. In the case of the semicircular billiard the second constant of motion is the absolute value of the angular momentum. In the rectangular billiard the constants of motion are the absolute values of momenta parallel to the walls of the billiard in question. The phase portrait of the elliptic billiard reminds one of the mathematical pendulum. Apart from the total energy the constant of motion is the product of the angular momenta about both foci of the ellipse. On the other hand the Sinai billiard was the first proven fully chaotic billiard (Sinai, 1970), whose disk (a circular obstacle) is located in the rectangle. At that time it was 2 Chapter 1. Introduction believed that the exponential divergence of orbits can only be achieved with the help of a non-convex boundary in a billiard. A few years later L. A. Bunimovich invented the stadium billiard (Bunimovich, 1974) and proved, that it is ergodic, that it has a mixing property and that it is a K system, which means that it is fully chaotic. The next fully chaotic billiard is the cardioid billiard whose chaoticity is proven in (Markarian, 1993). So far these are the only three rigourously proven chaotic billiards. At this point one should mention the family of Robnik billiards as an example of ’practically’ chaotic billiards (Robnik, 1983) for A > 1. This billiard family is described by the complex conformal map of the unit circle, w = z + \z2 which determines the boundary (for A = 12 one gets the cardioid billiard). 1 Another interesting example of chaotic billiards comes from the family of Africa billiards (Berry and Robnik, 1986) for A > 0.2, defined as a conformal map of the circle as well: w = z + Xz2 + AzVf. The most frequent mixed-type systems are KAM systems in which fractal structures of the islands of stability coexist with the chaotic sea. The Robnik and the Africa billiards are KAM systems which are characterised by a continuous transition from an integrable to a fully chaotic billiard with the continuously changing (shape) parameter A. There are other examples of KAM billiards, from cosine billiard to annular billiard (Bohigas et al., 1993a), etc. However there exists a mixed-type billiard which does not share the complex structures of many regular islands surrounded by the chaotic sea. This billiard, which is characterised by a sharply divided phase space, is called the mushroom billiard. It was invented by L. A. Bunimovich and it was classically treated in (Bunimovich, 2001; Altmann et al., 2005, 2006; Dietz et al., 2006; Tanaka and Shudo, 2006). This thesis discusses the characteristics of the mushroom billiard and its quantum mechanical counterpart. 1.2 Quantum chaos In quantum mechanics classical behaviour is related to various quantum objects, from wave functions to energy spectra of the corresponding counterparts, etc. The field in physics, which deals with chaos in quantum mechanics, is termed quantum chaos. One should emphasize the fact that in quantum mechanics chaos does not exist in the time domain since the Schr¨odinger equation is a linear equation. Therefore, quantum chaos was originally meant to be a study of quantum mechanics of classically chaotic systems. Some recent developments have proved that certain quantum integrable or chaotic systems, such as the spin chains which are presented in (Prosen, 1998b; Pineda and Prosen, 2007), are unique and do not have a classical limit. There arises a question of criteria distinguishing between regular and chaotic quantum systems. Apart from the Wigner or the Husimi functions of eigenstates lying in the quantum phase space, the most significant aspects are the statistical properties of the spectra of bound (Hamiltonian or Floquet) systems. These statistical properties are universal. The energy levels of integrable quantum systems are uncorrelated and possess the Poissonian statistics whereas in chaotic systems the neighbouring levels repel each other, which results in energy level repulsion. On the basis of the so-called Principle of Uniform Semiclassical Condensation (PUSC) of Wigner functions in the true semiclassical limit one can identify two types of eigen-states in mixed-type quantum systems, i.e. those of the regular and the chaotic type 1The so-called Pascal snail curve (i.e. Pascal lima¸con) only resembles the Robnik billiard, but it is not the same as this billiard. 1.3. Dynamical tunneling 3 (Percival, 1973), depending on where they ’live’ in the quantum phase space. But in order to achieve such a clear classification and uniform extendedness (no localisation) of the Wigner functions, one should really be in the semiclassical limit: otherwise, there can emerge the so-called hierarchial states (Ketzmerick et al, 2000), ’living’ in the chaotic sea in the vicinity of the hierarchy of the regular islands. This can happen if the value of the effective Planck constant is not small enough. Due to the formal similarity of the Schr¨odinger equation to other wave equations of physics the concept of quantum chaos can be generalized to the field of wave chaos. In these systems one may observe many similarities between wave mechanics and a suitably defined ’classical’ mechanics - ray dynamics. One should also study the morphology of the eigenfunctions and eigenmodes of the underlying wave equation. In Napoleon times there existed certain pictures named after E. F. Chladni (1756-1827). Chladni pictures (St¨ockmann, 1999) show the nodal lines of the first randomly-distributed dust on vibrating glass or metal plates; this phenomenon is shown qualitatively. Regular nodal patterns are typical for the integrable systems whereas the random ones appear in the chaotic systems of the underlying ray dynamics. To obtain more precise results people have used water surface waves, vibrating blocks, ultrasonic fields in water-filled cavities and microwave cavities. The Schr¨odinger equation of the 2D billiard system and the wave equation, which describes the electrical field of the low frequency modes in a thin microwave resonator of the same planar shape, are equivalent from the point of view of mathematical physics. The great advantage of microwave cavities is in their size which is in the range of about one meter, compared to the real implementations of quantum billiards (i.e. quantum dots, quantum wells, quantum corrals) with the range of /im or even nm (for a review see (St¨ockmann, 1999)). In the last fifteen years microwave cavities have become the main experimental research tool in the field of quantum chaos. In this research the cavity is mushroom billiard-shaped. 1.3 Dynamical tunneling Classically, individual regions of regular and chaotic motion are separated from one other. However, from the perpective of quantum mechanics they are coupled by tunneling. This process has been termed ’dynamical tunneling’ (Davis and Heller, 1981) since it occurs across a dynamically generated barrier in phase space. The dynamical tunneling effect differs greatly from the majority of the familiar cases of quantum mechanical tunneling which involve tunneling through a classically forbidden region between two separate regions of coordinate space. The EBK (Einstein-Brillouin-Keller) quantized tori do not intersect in phase space, but their projections onto the coordinate space usually overlap, so there is no spatial separation of the two wavefunctions participating in the dynamical tunneling process. One can remark that if one treats the problem strictly mathematically, one can notice that such dynamically forbidden regions are connected by complex classical trajectories. Of course, complex algebra is needed in this case. Although the tunneling phenomena have been systematically studied in the context of quantum chaos for more than twenty years, we still have not managed to explain them completely. Dynamical tunneling was first discussed in (Davis and Heller, 1981) where the authors treated the 2D anharmonic model potential and its eigenstates and they also tried to explain certain splittings in the vibrational states of molecules. They also showed that 4 Chapter 1. Introduction symmetry breaking increases the splittings of the doublets. A few years later some important general considerations of the tunneling between the tori in phase space (Wilkinson, 1986) and the narrowly avoided crossings (Wilkinson, 1987) have been presented. Dynamical tunneling between symmetry related regular regions has been studied in many systems: in a driven anharmonic oscillator and in a driven double well oscillator (Lin and Ballentine, 1990, 1992), in coupled quartic oscillators (Bohigas et al., 1990, 1993b; Tomsovic and Ullmo, 1994; Leyvraz and Ullmo, 1996), in annular billiard (Doron and Frischat, 1995; Frischat and Doron, 1998; Dembowski et al., 2000), in certain experiments with cold atoms (Steck et al., 2001; Hensinger et al., 2001; Mouchet et al., 2001), in the time periodic kicked Harper system (Brodier et al., 2001, 2002; Eltschka and Schlagheck, 2005), etc. In such systems there exist the so-called quasi-modes (Arnold, 1989) lying on the symmetric tori in phase space. Quasi-modes are wavefunctions which are constructed on a single torus. These functions fulfill the Schr¨odinger equation and would degenerate if tunneling was not present. But they are not eigenstates because they do not fulfill symmetry claims, ’real’ eigenstates are constructed via the linear combination of the quasi-modes lying on symmetry related tori. Therefore, if a quasi-mode, which is constructed on one of the tori, develops for a long period of time, it eventually evolves into its symmetric partner whereas its classical trajectories forever remain trapped on a single torus. While quasi-modes also emerge within the familiar 1D tunneling across the barrier, they are often surrounded by the chaotic sea in the context of dynamical tunneling. Thus, the presence of chaotic states increases the tunneling rates and causes an extreme sensitivity to external parameter variation. This process is called chaos-assisted tunneling (CAT). The tunneling, thus, annihilates the degeneracies and creates the splittings. Energy splitting is semiclassically given by ?E = A exp(-S/h) = A exp -- pdq , (1.1) " B where A is a constant associated with the energy of a quasi-mode, which weakly and mostly algebraically depends on h, whereas S is the classical action for the trajectories from the ’forbidden’ region along the path B. A special role in this semiclassical treatment is performed by the tori, fulfilling the EBK quantisation conditions, where the action integrals calculated for the N independent closed paths Cj on the torus are given by Jj = f Pdq= 2vrfr n, + -4 . (1.2) Here nj is the quantum number, j = 1...N where iV is the number of degrees of freedom of the system whereas v, is the Maslov index which counts the number of caustics encountered on the path Cj. Thus, in the far semiclassical limit, where the actions are very large, the level splittings decrease, which reduces the tunneling between the two symmetric EBK levels (1.2) according to (1.1). For the sake of clarity one should emphasise that without any chaos involved, there always exists a direct tunneling between symmetry related quasi-modes. But if chaos is present, tunneling is usually much stronger and this is the reason why CAT is so important. We can see this clearly if the rate of chaoticity is treated as a perturbation, as it can be observed in the case of the model of two coupled quartic oscillators in (Tomsovic 1.4. Random matrix theory and level statistics 5 and Ullmo, 1994). Direct tunneling is preferred for smaller couplings whereas CAT is preferred in the case of larger couplings. In this process, which can be best described as a 3-level mechanism, two regular states (the doublet) and one chaotic state (which is not part of the doublet) function as the media in the neigbourhood of the two quasi-degeneracies. The reason why CAT is often preferred to direct tunneling lies in the fact that chaotic orbits from the chaotic region between two tori connect these tori classically and, thus, this type of classical transport supports quantum tunneling. Tunneling rates also increase in the case of resonance-assisted tunneling. This type of tunneling occurs if in regular islands the complex KAM structure of resonant tori becomes important (Brodier et al., 2001, 2002; Eltschka and Schlagheck, 2005). Dynamical tunneling has also been studied from a single regular region to the chaotic sea: in the kicked rotor (Hanson et al., 1984), in the kicked rotors with complex classical dynamics (Shudo and Ikeda, 1995) and via localisation suppression of the tunneling effects (Ishikawa et al., 2007), in the ionisation of hydrogen atoms in polarized microwaves (Za-krzewski et al., 1998), during the decay of quantum accelerator modes (Sheinman et al., 2006), in electronic quantum transport through nanowires (Feist et al., 2006), etc. Tunneling is essential whenever two eigenvalues, one from the chaotic and one from the regular region, are degenerate. Due to tunneling there emerges a quantum-mechanical mixture of the respective eigenfunctions ?C and ?R, so the degeneracy vanishes. This type of dynamical tunneling is widely discussed in the present thesis, but also CAT emerges in a slightly different way from the one presented above. Unlike the 1D tunneling through a barrier, it is extremely difficult to quantitatively predict dynamical tunneling. Results have only been found for specific systems or system classes so far, most recently for 2D quantum maps. The latter has been studied with the help of a fictitious integrable system (B¨acker et al., 2008b). However, a precise knowledge of tunneling rates is of great importance. Apart from our spectral statistics model in systems characterised by the mixed phase space recently studied examples are the eigen-states affected by the flooding of regular islands (B¨acker et al., 2005, 2007) and emission properties of optical micro-cavities (Wiersig and Hentschel, 2008). So far the dynamical tunneling rates have not been fully quantitatively predicted since they require fitting by the factor of about 6 in the case of the annular billiard (Frischat and Doron, 1998) and by the factor of about 100 in the case of the mushroom billiard (Barnett and Betcke, 2007). In Chap. 3 we have presented a combined experimental, theoretical, and numerical investigation into the tunneling rates in mushroom billiards. The tunneling rates were calculated with the help of Fermi’s golden rule of time perturbation theory applying to the avoided crossings between the regular and the chaotic eigenstates. The most important results have been published in (B¨acker et al., 2008a). Although the theoretical and one of the numerical parts of the research have not been provided by the author of the present thesis they are also briefly presented and discussed for the sake of completeness. Another significant topic is avoided-crossing distribution for the mushroom billiard (which is presented in Chap. 3). 1.4 Random matrix theory and level statistics Random matrix theory (RMT) (Mehta, 1991; Guhr et al., 1998) has important applications in many branches of physics such as elementary particle physics, nuclear physics, 6 Chapter 1. Introduction atomic physics, molecular physics, solid state physics, and, especially, in quantum wells (Narimanov and Stone, 1999). In quantum chaos RMT has proved to be an excellent model for the statistical properties of energy spectra of chaotic Hamiltonian systems. It goes back to the original paper (Casati et al., 1980) and to the classic paper (Bohigas et al., 1984) where the conjecture about the universality of RMT for classically fully chaotic systems was formulated. The conjecture has been theoretically supported by the dynamical and the semiclassical theory of spectral rigidity (Berry, 1985). The next important step beyond this approximation has been achieved in (Sieber and Richter, 2001), followed by the development of an expanded semiclassical theory (Mu¨ller et al., 2004; Heusler et al., 2004; Mu¨ller et al., 2004). Classically integrable quantum systems show the Poissonian statistics, which is well known and corroborated by numerous analytical and numerical works (Berry and Tabor, 1977; Robnik and Veble, 1998). In chaotic systems spectral fluctuations obey the statistics of the Gaussian orthogonal (GOE), the unitary (GUE) or the simplectic (GSE) ensembles (depending on the existence of antiunitary symmetry (Robnik, 1986) and on an internal degree of freedom - spin). Usually the well known Wigner surmise, Eq. (2.4), well describes the level spacing distribution for the infinite dimensional GOE level spacing distribution. For higher-order level spacing distributions of chaotic systems useful approximate closed form formulae have been derived in (Abul-Magd and Simbel, 2000). The spectral statistics in mixed-type systems rests upon the PUSC of the Wigner functions, upon the theoretical foundations in (Berry, 1977), and upon the theory of superposition of the so-called E(k,L) statistics (Berry and Robnik, 1984). E(k,L) statistics is the probability of having exactly k levels in an interval of the length L after unfolding, where the level spacing distribution P(S) is the second derivative of the gap probability E(0,S). The theory states that the E(k,L) statistics factorise in the strict semiclassical limit of a sufficiently small effective Planck constant heS, where the factorisation is a direct consequence of the statistical independence of the sequences of the regular and the irregular levels. The Berry-Robnik (BR) theory and its resulting formulae have been verified in many different systems in the asymptotic semiclassical regime: in the ’far’ semiclassical limit of quantised standard map on the torus (Prosen and Robnik, 1994a,b), in the periodically pulsed spin system (Jacquod and Amiet, 1995), in the 2D semiseparable oscillator (Prosen, 1995), in the quartic generic KAM billiard with the border described by r(0) = 1 + a cos(40) with the simple phase space structures reflected in the lower transition point (Prosen, 1998a), in the E(k, L) statistics for the standard map on the torus and the quartic billiard (Prosen and Robnik, 1999), in the E(k,L) statistics with a generalised Wigner surmise tested on the Henon-Hiles potential (Abul-Magd and Simbel, 2000), in the system with a sharply divided phase space of kicked-rotor-type map in the ’near’ semiclassical regime (Malovrh and Prosen, 2002), in the Andreev billiard (Kaufmann et al., 2006), etc. There have been a number of attempts in literature to describe correctly the level statistics in the mixed-type systems and all these approaches interpolate between the RMT and the Poissonian statistics, but, unlike the BR, they are not based on sound physical grounds, and they have not been confirmed in the semiclassical limit. For a review see Chapter 3.2.2 of (St¨ockmann, 1999). If the semiclassical limit is not reached, one observes deviations from the BR behaviour which emerge due to both the localisation effects and the tunneling between the regular and the chaotic regions in the quantum phase space of the Wigner functions of the eigen- 1.5. Non-Gaussian RMT 7 states. This results in the linear behaviour of P(S) at small S, as predicted qualitatively in (Berry and Robnik, 1984), and in the fractional power law level repulsion (which was first presented in (Prosen and Robnik, 1994b)) for S between the linear level repulsion regime and the BR tail. Presumably, the main reason for the fractional power law level repulsion lies in the localisation effects (for more details see (Izrailev, 1989, 1990)). Here, we would like to offer a new random matrix model in order to generalise the BR level spacing distribution by including the tunneling effects between eigenstates. We will not consider localisation effects, except for the special case where p can be replaced by some effective pe//; for example in the case of flooding (B¨acker et al., 2005, 2007) or in the case of hierarchical states (Ketzmerick et al., 2000). So, we will assume that the PUSC is fulfilled and the level splittings are affected by tunneling mechanisms which couple two regular and two chaotic levels and also regular-chaotic levels through the chaotic levels functioning as the intermediary. Thus, our purpose is to mimic tunneling by means of a two-level random matrix model where the non-diagonal elements correspond to the tunneling matrix elements. Following some recent advances in the non-Gaussian and the non-normal random matrices in (Grossmann and Robnik, 2007a,b) we will try to verify to what extent the properties of the matrix ensemble remain structurally stable, i.e. robust, against certain variations of the model properties such as the statistics of the matrix elements. Finally, we will use the theoretical results to describe the level spacing distribution of the eigenvalues of a various configurations of the mushroom billiard, both on the basis of the experimental data for the microwave cavities as well as on the basis of the numerical data obtained by the expanded boundary integral method. This work, published in (Vidmar et al., 2007), is presented in Chap. 2 and, partially, in Chap. 3 of the present thesis. 1.5 Non-Gaussian RMT The last part (Chap. 4) of the thesis, also presented in (Robnik et al., 2007), contains the eigenvalue calculations for different random matrix ensembles where the distribution of the matrix elements is non-Gaussian. The importance of the Gaussian ensembles of the RMT has been corroborated by theory in (Hackenbroich and Weidenmu¨ller, 1995), where the authors have considered other random matrix ensembles than the Gaussian ones (GOE, GUE and GSE). They have proved (using the supersymmetric techniques) that the local spectral fluctuations (after spectral unfolding) obey universal statistical laws described by the Gaussian ensembles in the limit of the infinite dimension N › ?, independent of the distribution of the matrix elements, provided that two conditions are fulfilled: (i) the energy level distribution function in this limit should be smooth, and (ii) it should be confined to a finite interval. Thus, this important finding is some kind of a central limit theorem for all the spectral fluctuations of random matrices. Usually the renowned Wigner formula (Mehta, 1991) well describes the infinite dimensional GOE (or also GUE and GSE) level spacing distribution. Recently in (Grossmann and Robnik, 2007a) the authors have also studied 2D non-normal Gaussian matrices (with asymmetric variance) and a number of non-Gaussian ensembles of 2D real symmetric random matrices in (Grossmann and Robnik, 2007b) where explicit analytic results for level spacing distribution have been obtained. It has been shown there that the level repulsion 8 Chapter 1. Introduction is a very robust phenomenon, which only depends on the behaviour of the distribution of the matrix elements, i.e. on its regularity at zero value. If the distribution functions of the diagonal and the off-diagonal matrix elements are regular and nonzero at zero value, there always appears the linear level repulsion. If the distribution functions in question are regular and at least one of them is zero at zero value, then the level repulsion is quadratic. If the distribution of the matrix elements is singular at zero value (e.g. an integrable power law times exponential), there appears the fractional power law level repulsion (Prosen and Robnik, 1994b; Prosen, 1995). The Cauchy-Lorentz distribution is also interesting, since it does not contain a finite first moment and produces P(S) which is not normalisable to the unit first moment (mean level spacings). It is straightforward, interesting and important to generalise these analytic 2D results to the higher dimensional case N > 2 where analytical results are practically impossible. Our extensive numerical calculations have shown that for regular matrix ensembles there applies the HW theorem. In the last two singular cases (mentioned earlier) all the assumptions from HW theory are not fulfilled, i.e. condition (ii) is not realised and, due to this, the systems strongly violate the GOE behaviour permitted by the HW theorem which is presented in Chap. 4. 9 Chapter 2 The distorted Berry-Robnik level spacing distribution 2.1 The Berry-Robnik level spacing distribution Gap probability E(0,S) is the probability that there are no levels on the interval of the length S after spectral unfolding. The level spacing distribution P(S) for an unfolded spectrum is equal to the second derivative of gap probability: P(S) = d2E(0, S)/dS2. Gap probability factorises upon a statistically independent superposition of independent level sequences. From here onwards gap probability is denoted by E(S). The basic finding in the Berry-Robnik (BR) picture (Berry and Robnik, 1984) is that the E(S) for the total spectrum of a mixed-type system is the product of all the gap probabilities whose arguments must be weighted by the classical parameters Pi measuring the relative volume of the phase space of the regular component (i = 1), of the largest chaotic component i = 2, of the next largest chaotic component (i = 3), etc. We have focused on the simple (and usually sufficient) approximation of only two components, i.e. the regular component i = 1, with P1 = p, and the chaotic component one i = 2, with p2 = 1 - p, so that p1+p2 = 1. Thus, gap probability for the entire spectrum equals E(S) = E1(p1S)E2(p2S). The BR level spacing distribution can be presented by Pbr(S) = ^2 E1(P1S) E2(p2S). (2.1) Here we have introduced two quantities related to E(S). The first one, F(S), indicates the probability that the level spacing is larger than S. The second one is the cumulative level spacing distribution which equals: W(S) = 0 P(x)dx. The relations between two quantities can be represented as: F(S) = -dE/dS = 1 - W(S). Now the BR level spacing distribution (2.1) can be rewritten thus Pbr(S) = p21P1E2 + 2p1p2F1F2 + p22E1P2 , (2.2) where the argument of each quantity with the index % equals Xi = PiS, % = 1, 2. For the chaotic case we have to apply the GOE (GUE, GSE) results for infinite matrices, for which there exist no closed form expressions. For the analytical case the Wigner distribution, which is an exact GOE result for the 2 x 2 matrices (while the GUE and the GSE are not treated here), is often used as an excellent approximation. The discrepancy between the 10 Chapter 2. The distorted Berry-Robnik level spacing distribution Wigner distribution and the exact result for the infinite matrices is only up to 2%, so the Wigner approximation is suitable for many practical and, especially, for many analytical purposes. In the regular case, i = 1, we will use the Poissonian statistics (Berry and Tabor, 1977) E1(x1) = F1(x1) = P1(x1) = e-x\ (2.3) whereas in the chaotic case, % = 2, we will employ the Wigner (2D GOE) approximation; see (Mehta, 1991). P2(x2) = — exp i-^4) , ^(r2) = 1 - W2(x2) = exp (~^4) , (2.4) and L2(2:2) = 1-erf y^A =erfcy^A (2.5) where erf (x) = -^ /0X e~u2 du is the error integral whereas erfc(x) = 1 - erf (x) represents its complement. With (2.3) - (2.5) the BR level spacing distribution function (2.2) can be written as (Berry and Robnik, 1984) Pbr(S) = e~plS ie- 22 (2Plp2 + ^L^\ + p?erfc (^f^] X . (2.6) FLF 2 ,l 2 The PBR and its first moment are normalised. The second moment (S2) = 2 j™ E(S) dS can sometimes be useful as well. For the BR level spacing distribution the second moment can be expressed in a closed form {2 \ 1 - e 2 erfc ( -^ ] \ (S 2)br = - 1 - e2erfc-^ . (2.7) Pi In case of the Poissonian statistics (Pl = 1, p2 = 0) (S2) equals 2, while in the case of the Wigner distribution (Pl = 0, p2 = 1) {S2} equals 4/vr. 2.2 General distortion of the general level spacing distribution Let us consider 2x2 real symmetric matrices A = (Ay) where i,j = 1 or 2. Only the difference between the eigenvalues is of relevance in the present context. Without any loss of generality (Grossmann and Robnik, 2007a) one may assume that the trace of A vanishes A= ( ? M , (2.8) o —a where a and b are real. Thus, for the eigenvalues of A one obtains Ai,2 = ±Va2 + b2 , (2.9) 2.2. General distortion of the general level spacing distribution 11 where the difference A1 - A2 between the both eigenvalues equals 2y/a2 + b2. Thus, the level spacing distribution is given by +00 +(» / / da db 5 LS - 2V —00 —00 P(S)= dad&čS-2a2 + &2U„(a)#fc(&), (2.10) —00 —00 where 5(a;) is the Dirac delta function whereas #0(a) and #,(&) are the normalised probability densities of the diagonal and the off-diagonal matrix elements a and b respectively. In the above construction P(S) is defined for S > 0. P(S) is automatically normalised, +00 +OO oo 00 +oo +oo oo _________ / P(S) dS = / / / da db dS 5 LS - 2Va2 + b2\ ^a(a) ^6(6) 0 —00 —00 0 ^a(a) da / ^6(6) d6 / 5 (^ - 2Va2 + b2) dS 0 = 1 , (2.11) but this is not true for its first moment; in general /0°° S P(S) dS=1. If we now introduce the polar coordinates a = rcos^, 6 = rsin^ (2.12) where r G [0,oo) and 0 <
\ p = 0.50 (j = 0.04 - 0.6 — 0.4 N - 0.2 - - o n .........i ........i..... . . . TTTTTt..... Figure 2.6: The same as Figs. 2.1 - 2.5 presenting the regular fraction p = 0.5 and various values of the level coupling parameter a = 0.01,0.02,0.03,0.04. This is, in fact, the weighted mean of the undistorted and the antenna-distorted (distorted by the all-to-all couplings) BR distributions. For both limiting cases p = 0 and p = 1 the original GOE and PE (the Poissonian ensemble) level spacing distributions are recovered as expected. One should note that the tunneling-distorted BR distribution (2.22) is normalised to unity whereas its first moment (i.e. the mean level spacing) is not. In order to compare the theoretical level spacing distribution with the level spacing distribution from the real data (experimental or numerical ones) the rescaling, or the normalisation of the first moment to one, has to be performed exactly as suggested in Eq. (2.20). In the end we obtain the final theoretical tunneling-distorted BR distribution function by using the rescaling factor BT pdbr(s) = bt PDbr(btS)} with BT o X PlBR(x) dx. (2.23) The BTs for the set of parameters (p, a) are shown in the forth column of Tab. 2.1. They appear in the range [1,1.1] for the parameters selected. Again, for each set of parameters (p, a) the integrals in Eqs. (2.23) and (2.17) are evaluated numerically The resulting P%%R(S)s are plotted in Figs. 2.1 - 2.6 with the full red curve. One can clearly see that there are no examples of repulsion. Why does this happen? For mathematical reasons PE'BR(S) is a superposition of PßBR(S) and PBr(S) where the latter has this property This feature is discussed in more detail in the Subsec. 2.5.2. At larger values of S of the distribution PlBR(S) one also notices the overshooting and, _ 22 Chapter 2. The distorted Berry-Robnik level spacing distribution asymptotically, Pdbr(S) also behaves like the stretched BR distribution. 2.4 Analytical studies of the distorted Berry-Robnik distribution We will calculate analytical approximations for the antenna-distorted BR level spacing distribution PßBR given in Eq. (2.17), i.e. for small S and for large S, by using (2.6) for Per in the general expression (2.17). 2.4.1 Small S behaviour of PßBR(S) Let us expand PBR(x) from (2.6) into the Taylor series at small x, oo Pbr(x) = ^akxk. (2.24) fc=0 Then PLBR(S) is given by the series, from Eqs. (2.17) and (2.24), Pdbr(S) = ^^Y,akS k r/2 coSk
/ X K 7T/2 /xTf + 1 -. 7 cosfc (/? exp (-asm2
1. The main contribution to the integral comes from the interval close to
14 all the results are practically the same. We have also tested matrices with N = 50 (and smaller) and observed a very good agreement as well. Therefore, we believe that the results of the model do not depend on the dimension of the matrices, so, for instance, the option N = 100 and unf = 20 would be equally or (even) slightly more appropriate, as indicated in Tab. 2.2.
We have also tested the phenomenological rule with unf (for N = 1000) by using the GOE random matrices (p = 0, a = 0) where the optimal agreement with the exact unfolding (using the Wigner semicircle rule) has been sought. The results confirm the ones obtained from the X2 test, which means that unf = 30 is the best possible choice in this case where the unf does not influence the results to such a large extent.
If we consider the results of PDABR(S) and PDA BnRN(S) for sufficiently small values of a, we notice one interesting feature: smaller spacings are affected to much a larger extent
2.5. Simulations with random matrices
31
than bigger ones (for the non-normalised distributions the effect would be even greater). For the 2D model this can be explained in the following way: the spacing S = En+l - En of the coupled system, which is expressed in terms of the spacing of the uncoupled system So = En+i,o - En,o = 2a > 0 as well as in terms of the coupling b (for the definition of matrix elements a and b see Eq. (2.8)), is given by
S=Sl + 4b2 (2.41)
If Enfl and Lra+i)0 are equal (i.e. a = 0), then S = 2b, otherwise S = S01 + ^L. If a is
small, which we have assumed here, the coupling b is small as well, i.e. b < SQ. Thus, we can only take the first order in the Taylor expansion and get
S =So + 2^. (2.42)
The So from the denominator explains why smaller spacings change to much a larger extent than bigger ones. It seems that the two-level process dominates in the iV-dimensional model as well.
2.5.2 The tunneling distorted BR distribution
In order to describe the tunneling between the regular and the chaotic phase space regions in physical systems we will assume that the non-zero off-diagonal matrix elements only occur between the regular and the chaotic states but not within the regular or the chaotic block itself. In the cross coupling case the two-level approximation expresses the tunneling-distorted Berry-Robnik formula P^R(S) which is a weighted mean of the antenna-distorted and the undistorted Berry-Robnik formula which is normalised to (S) = 1, as presented in Eqs. (2.22) and (2.23). Fig. 2.11 shows the results from the simulations of PE'brn(s) and from the two-level expression for PlBR(S), for a = 0.05 (as also applied for the all-to-all couplings) and for a number of p values. Fig. 2.12 represents PE'brn(s) and the PEbr(S) for a = 0.1 and for a number of p values. Fig. 2.13 displays the results for p = 0.5 and a = 0.01,0.02,0.03,0.04. In all the cases the agreement between the iV-level simulation and the two-level analytical formula (2.22) is rather encouraging. The agreement is better for larger values of p where the regular part dominates.
Although the general agreement for a = 0.05 is still good, there appear significant deviations between the two-level approximation and the results from the N-dimensional simulations for small values of S. In the iV-dimensional simulation the linear level repulsion is always observed in cases of small values of S while the two-level approximation starts with a non-zero value at S = 0. For a steep linear level repulsion the region in S, which should appear within an exponentially small interval oc exp(-const./heff) typical of tunneling phenomena, is indeed very small (see the insets in Figs. 2.11 - 2.13). This reflects that the accidental degeneracies, which occur generically in the regular part of the spectrum, are not lifted in the case of the two-level approximation. In the N-dimensional simulations, on the other hand, all the degeneracies are lifted in the regular block although there are no direct tunneling matrix elements; there only appears the second order tunneling which couples two regular states indirectly, i.e. via one or more chaotic ones. Here the two-level approximation is certainly not appropriate. 3
3Here the term ’degeneracies’ also includes nearby lying states, i.e. the so-called quasi-degeneracies.
32 Chapter 2. The distorted Berry-Robnik level spacing distribution
Figure 2.11: The numerical histogram for the N-level (N = 1000) tunneling distorted BR level spacing distribution P]j%RN(S) is compared with the pEbr(s) from the 2-level model (2.23) for the fixed coupling parameter a = 0.05 and for various sizes of the regular regions p = 0.10,0.25,0.35,0.50,0.75,0.90. For the sake of comparison the blue dashed line represents the BR distribution with the same p. Only the couplings between the regular and the chaotic eigenvalues are allowed in this case; couplings cannot occur between two regular states or two chaotic ones. The insets show the small S behaviour of all the three curves plotted. The tunneling formula (2.23) is in good agreement with the numerical simulations. For the lower left plot with p =0.75 there are 10 times more objects in the statistics.
We should also mention that neither localisation (Leyvraz and Ullmo, 1996; Ishikawa et al., 2007) nor flooding effects (B¨acker et al., 2005, 2007) have been treated so far. Typically, localisation of chaotic states (as a deviation from the PUCS-uniformity) would
2.5. Simulations with random matrices
33
1.0
1.0
1.0
0 8 P ö = 0.25 = 0.10
0.6 "f\ " 1 -
0.4 K V °' -j -
0.2 -
o n ....................
Figure 2.12: The same as in Fig. 2.11 but for the doubled cross-coupling (tunneling) strength a = 0.10 and for various values of the relative size of the regular regions p = 0.10,0.25,0.35,0.50,0.75,0.90. Again only cross couplings of the integrable with the chaotic states are allowed here; couplings cannot occur between two regular states or two chaotic ones. All the plots have the same number of objects in the statistics.
be modelled by the suppression of tunneling between certain basis-states. Localisation near regular (stability) islands (classically stickiness) would be modelled by replacing ? with some larger effective ?eff, whereas (partial) flooding of regular (stability) islands would be modelled by some smaller effective ?eff.
34 Chapter 2. The distorted Berry-Robnik level spacing distribution
0.0
1.U ......... p = 0.50 a = 0.02
n R
\ I - f J^aV^Ir^M,_
0.6 1 \l If f -
0.4 >t I - -
0.2 n n .................... , , TTTtt......
0.0
Figure 2.13: The same comparison of level spacing distributions as presented in Figs. 2.11 and 2.12, but now for the fixed ? = 0.5 (equal for both the regular and the chaotic parts) and for other (smaller and larger) coupling ? = 0.01, 0.02, 0.03, 0.04, 0.20, 0.30 affected by tunneling. Again only cross couplings between the integrable and the chaotic eigenvalues are allowed. In the upper left plot with ? = 0.01 there appear 10 times more objects in the statistics.
2.6 Improvements on the tunneling-distorted BR distribution
Sec. 2.5.2 proves that the 2D model has certain disadvantages. This section represents two possible improvements to this model.
The formulation from Eq. (2.21), which presents the distribution of tunneling matrix elements, could be replaced by
2.6. Improvements on the tunneling-distorted BR distribution
35
gb(b) = 2p(1-p)—1=exp—b— +[1-2p(1-p)]—1=exp—^ , (2.43)
where oo (Hackenbroich and Weidenmu¨ller, 1995), provided that the level density in this limit is smooth and confined to a finite interval. For 2D ensembles an explicit analytic theory has recently been performed (Grossmann and Robnik, 2007a,b) for a variety of matrix element distribution functions, including the exponential and the box (the uniform) distributions. Extensive numerical studies of such ensembles at high dimensions of the matrices (see Chap. 4) confirm the Hackenbroich-Weidenmu¨ller prediction and show that the transition to the universal behaviour is pretty fast. If one or both of the Hackenbroich-Weidenmu¨ller conditions are not fulfilled, e.g. for the Cauchy-Lorentz distribution of the matrix elements, there appear deviations from the GOE level spacing distribution.
Bearing in mind all this we would like to explore the sensitivity of the distorted BR level spacing distribution as regards the choice of the matrix element statistics. Thus, in addition to the Gaussian distribution treated earlier, as defined in (2.16), we will consider the box (the uniform) distribution
9b(b)
-y/3
if | 6 |< aV3 and zero, otherwise,
(2.59)
and the exponential distribution
9b(b)
2^
exp
( \b\2
(2.60)
both with the variance a2.
It is obvious that the deviations from a Gaussian distribution are not as large as one might expect, which is clearly demonstrated in Figs. 2.15, 2.16, representing the results for the normalized antenna-distorted BR distribution P^RN(S), for the Gaussian, for
_
_
40 Chapter 2. The distorted Berry-Robnik level spacing distribution
Figure 2.16: The same as Fig. 2.15 for the box (instead of the exponential) distribution of the off-diagonal matrix elements: the histogram from the numerical simulations with the matrices N = 1000 well agrees with the 2D formula (solid curve), but both of them only slightly deviate from the Gaussian model (dotted curve).
the exponential and for the box (the uniform) distribution with the same dispersion a2. These are various theoretical curves with the same a2. One should emphasise that no best curve fitting has been applied. One notices that the 2D theory for P$%R(S) from Sec. 2.3 is a suitable approximation for the random matrix ensembles of the dimension N = 1000. Nevertheless, the curves slightly differ from one another, especially in the central region of 0.3 < S < 0.6 whereas the small S behaviour (the level repulsion) and the large S behavior (the tail) are in good agreement with the model where the gb(b) is the Gaussian distribution function. In fact, the tail for the exponential distribution is, analytically, the same as one for the Gaussian model (2.37). The ratio of the slopes at S = 0 for the exponential and the Gaussian case equals yfii whereas this ration equals V^/6 in the case of the box and the Gaussian distribution. This ration remains the same within 2% after one has rescaled the first moment to unity by using the stretch factor BA. The agreement in the case of the Gaussian and the other ensembles is, therefore, quite satisfactory.
One intuitively expects that the statistical spectral properties of random matrix ensembles mostly depend on the variance of the matrix elements and not on other details of the ensemble unless these become singular in one way or another.
Some basic results from this chapter are tested in the following one.
41
Chapter 3
Dynamical tunneling in mushroom billiards
3.1 Mushroom billiard - classical
The mushroom billiard (shown in Fig. 3.1) was introduced in (Bunimovich, 2001). It is composed of a semicircle - the head, and a rectangle - the foot. We have used the rectangle for the foot although some other shapes, such as a triangle, etc., may be used instead. The mushroom billiard is a 2D autonomous system with the Hamiltonian H(p,q) = p2/2M + V (q), where (q,p) are 2D coordinates and momenta of the particle whereas M is its mass and V is its potential. The potential is zero inside and infinite outside the billiard domain D.
d A
b
2a 1
Figure 3.1: The geometry of the mushroom billiard: R = radius of the semicircle, 2a = width of the foot, l = height of the foot, b = position of the foot (the larger distance from the edge).
The energy is the only global constant of this billiard. There exists one additional local constant of motion for the orbits which do not enter the foot, i.e. the absolute value of the angular momentum. The orbits with the same angular momentum form a semicircular caustic. There is one critical caustic which rigorously separates the orbits into the regular and the chaotic ones. The orbits inside the head with a larger or equal caustic forever remain in the head whereas the other orbits eventually enter the foot, so they are chaotic (top two of Fig. 3.2). Since the billiard system has precisely one chaotic and one integrable component in the phase space, which are well separated from one
42
Chapter 3. Dynamical tunneling in mushroom billiards
another (bottom of Fig. 3.2), this system is particularly attractive for analysis. Such a clear separation of the phase space first appeared in certain classical maps (Lee, 1989; Malovrh and Prosen, 2002).
1.0
0.5
Q- 0.0 i'
0.5 -:
1.0
o
3
5
6
Figure 3.2: Top-left: Example of a regular (70 reflections) orbit in the mushroom billiard. Top-right: Example of a chaotic (100 reflections) orbit. Bottom: Phase space portrait. The abscissa is the border coordinate s (of the reflection point) beginning at the right ?2 corner of the semicircle whereas the ordinate p is the sinus of the reflection angle.
The regular part of the phase space of the mushroom billiard is simple since it consists of a single regular island. Due to this the volume of the regular (or the chaotic) part of the phase space can be calculated much easier than in the case of a KAM system. The group from Dresden, which was lead by R. Ketzmerick, derived an analytic expression for the effective chaotic area of the mushroom billiard with the central (b = R - a) foot
3.2. Mushroom billiard - quantum
43
Ach = 2la +
R2 arcsin (|) + av^R2^2
(3.1)
a
where Ac7l is the effective chaotic area of the configuration space of the mushroom billiard. If one wants to obtain the fraction p2 of the chaotic component, the Ach has to be divided by the entire area of the billiard, i.e. A = 2 la + ^. If the foot is not in the central position (b=R-a), then instead of a in [... ] from Eq. (3.1) one employs a' defined as: a' = 2a + b-R. This calculation is represented and confirmed in Appendix A.
While the regular component of the phase space is very simple, the chaotic one consists of a complex distribution of families of marginally unstable periodic orbits (MUPOs). The first class of MUPOs corresponds to the orbits bouncing between the parallel walls in the foot of the mushroom billiard. Similar MUPOs can be found in many other billiards with parallel walls. The other, more interesting class of MUPOs, corresponds to periodic orbits in the chaotic region which remain trapped in the head of the mushroom billiard (Altmann et al., 2005, 2006). Since these sticky chaotic orbits, characterised by an almost regular motion and by the zero Lyapunov exponent, have measure zero, they do not affect the ergodicity of the system. Nevertheless, due to the MUPOs with long periods, one can understand certain dynamical properties of the system, such as the transport and the decay of correlations. As presented in (Altmann et al., 2005) these MUPOs result in an exponent 7 = 2 for the asymptotic power-law decay of the commutative recurrence time distribution Q(T) oc T~\ If P(t)dt indicates the probability that the first return to the chosen region will happen at the time between t and t + dt], NT is the number of recurrences in that region with the recurrence time t > T whereas N represents the total number of recurrences, so that then Q(T) = L~T P(t) = lim^«, ^. In fully chaotic hyperbolic systems Q(T) decays exponentially. This definitely proves that the system discussed has the property of stickiness.
If one perturbs the system by employing the uniform transverse magnetic field (and inserts a charged particle) the situation changes. In this case there emerge the KAM tori whereas the phase space is not sharply divided anymore - now it exhibits the hierarchy of KAM-like islands and cantori which are surrounded by the chaotic sea.
3.2 Mushroom billiard - quantum
Quantum mechanically the billiard inside the domain V is described by the time-independent Schr¨odinger equation with the Hamiltonian #(p,q), the eigenenergies Ej and the eigenstates ipj
- ^ipi(q) = Eiipi(q), (3.2)
where h is the Planck constant divided by 2vr whereas V2 is the Laplace operator; we use 2M = h = 1 in the following. In the case of hard walls represented in mushroom billiard the Dirichlet boundary conditions have to be applied: Vn(q)U = 0.
The eigenstates are either mainly regular or mainly chaotic, depending on the phase space region they concentrate on. One clearly sees (Fig. 3.4) the two types of wavefunc-tions which are already classified in (Percival, 1973). In this thesis we have used the desymmetrised mushroom billiard when the foot has been located in the central position, so that we would not have to deal with two symmetry classes of eigenstates.
44
Chapter 3. Dynamical tunneling in mushroom billiards
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r
Figure 3.3: The radial part NmnJm (jmnr) of the quarter circle eigenfunctions from Eq. 3.3. The upper two plots show these functions for the fixed radial quantum numbers: n = 1 and m = 2,4,. . . ,28 on the upper left plot (black), n = 2 and m = 2,4,.. . ,24 on the upper left plot (red), n = 5 and m = 2,4,. . .,14 on the upper right plot (orange), n = 6 and m = 2, 4,. . .,12 on the upper right plot (violet). The lower two plots represent the fixed azimuthal quantum numbers: m = 2 and n = 1, 2, ... , 8 the lower left and m = 20 and n = 1, 2, . .., 8 the lower right plot.
The regular wavefunctions resemble those of the Hamiltonian H1/4 for the quarter circle. If R = 1, its eigenenergies and eigenfunctions, written in polar coordinates (r,tp) read
L1/4 = i and *P™(r < 1, Y>) = NmnJm (jmnr) sin(mip), ip^(r >1,ip) = 0 (3.3)
These are characterized by the radial (n = 1, 2,...) and the azimuthal (m = 2, 4,...) quantum numbers. Here Jm is the m-th Bessel function whereas jmn indicates its n-th root and Nmn = -t/8/tv/Jm-1(jmn) is the normalisation constant. The radial part NmnJm (jmnr) of the quarter circle eigenfunctions is shown in Fig. 3.3.
In Fig. 3.4 (the right picture) we have resolved the quantum numbers (m, n) for a mushroom regular eigenstate and written down the energies of this regular and the neig-bouring chaotic state; these energies are very close together. As evident from the following dynamical tunneling can, in fact, explain this small splitting.
3.3. Microwave experiments
45
m=26 n=2
1/4
1344.816
Figure 3.4: A neighbouring ’chaotic’ and ’regular’ state in the desymmetrised mushroom billiard. The probability density |?(q)|2 is plotted.
3.3 Microwave experiments
3.3.1 The similarity between QM and ED
By first employing the Maxwell equations one obtains the Helmholtz equation for the propagation of electromagnetic waves (Jackson, 1982)
V2 B + fc2 B
0,
(3.4)
with a wavenumber k = ^, where z/ is the frequency and c is the speed of light. Now one applies the cartesian coordinates and the cylindrical waveguide with walls from any ideal conductor, with any cross section oriented along the z direction. Using the ansatz
E(x,y,z,t) B(x,y,z,t)
Eq. (3.4) can be simplified to
1 f E(x,y)ei(±kzZ~27TVt) = B(x,y)ei(±kzZ~27TUt)
[v
x,y
k
nfEl B
0.
(3.5)
(3.6)
The boundary conditions for the EM field at the ideal conducting boundary have to be taken into account: Etang = 0 and Bnorm = 0 where neither the electric field at the surface along the axis z nor the surface currents are allowed. If one encloses the infinite long waveguide at z = 0 and z = d, one gets the resonator of thickness d.
At this point we will select the components of the field parallel to the axis z (all the other components can be derived from them). The boundary conditions are written as
Ez(x,y,z)\Bcrandary = 0,
dBz(x,y,z) dn
I Boundary
0.
(3.7)
_
_
_
46
Chapter 3. Dynamical tunneling in mushroom billiards
In the z direction the condition kz = *f with n = 0,1,2,... has to be fulfilled as well. There are two classes of solutions: the transversal-electric (TE) and the transversal-magnetic (TM) modes. For both, respectively, one finds
Bz(x,y) = 0 results in 2vr, so that, at small values of S, P(S) « S/(2na0b0), which well agrees with Eq. (2.14). The integral in (4.11) can be calculated analytically, resulting in
P(S) = A____<*y^ + PW+*
2ira0b0 (a2 + ß2 + a2 ß^VTTa2\JY^ß2
where a2 = S2/(4a2) and ß2 = S2/(46g). The asymptotic behaviour of P(S) at large 5 is characterised as an inverse quadratic power law,
P(S) « 4(a; + M (4.13)
implying that, due to divergence, P(S) does not have the first moment.
4.1.4 Singular times exponential distribution
The normalized singular distributions are defined as
9a(a) = Ca\a\-^e-XM, gh{b) = Cb\b\-^e-XbW, (4.14)
with the normalization constants
Q = A|"w/(2r(l -fa)). (4.15)
Here % = a, b and the exponents m < 1 whereas Y{x) is the gamma function. These distribution functions are singular but integrable power laws for a, b -> 0 and they decay nearly exponentially in the tails. For the level spacing distribution it follows
ns) = ca s (s)^+H) f '^H^+AtSiM), (4,6)
but this formulae still have no analytical solutions. One can evaluate it for small argument S where the exponential can be approximated by 1. This results in the following level repulsion law, i.e. the fractional exponent power law
/ q -(ßa+ßb) TVl _
2~ 2T(1
P{S) = CaCb S- 2 %/2 2\ (4.17)
The power law distribution of matrix elements leads to the power law repulsion in the level spacing distribution.
4.1.5 Comparison with numerics
Fig. 4.1 shows the numerically evaluated P(S) for 2D random matrices defined by the four ensembles mentioned above and compared to the theoretical curves. An excellent agreement is observed for certain typical parameter values.
Chapter 4. Numerical studies of non-Gaussian real symmetric random 88 matrices
Figure 4.1: The numerical histograms are compared with the theoretical curves for the four ensembles from Sec. 4.1 (not unfolded, except for the singular times exponential case). M is the number of matrices taken from the ensemble. Top-left: Box distribution, a0 = \/2/2, b0 = 1/2, M = 108. Top-right: Exponential distribution, Aa = y/2, Xb = 1, M = 108. Bottom-left: Cauchy-Lorentz distribution a0 = 1/2, b0 = 1/2, M = 108. Bottom-right: Singular times exponential distribution: Aa = y/2, Xb = 1, /x„ = 0, /x6 = 1/2, M = 107; this distribution is unfolded.
4.2 Numerical calculations with higher dimensional non-Gaussian random matrices
We have generalized the random matrix ensembles from Sec. 4.1 from N = 2 to the higher dimension N by applying the distribution ga(a) for all the diagonal elements of the N-dimensional real symmetric matrix and the distribution gb(b) for all the off-diagonal matrix elements.
This study primarily intends to determine whether there exists the transition (in the cases 4.1.1 and 4.1.2), or maybe there is not any (in the cases 4.1.3 and 4.1.4), from the nonuniversal behaviour at small N values to the Hackenbroich-Weidenmu¨ller (HW) universal GOE behaviour at N = ?. In the numerical studies of the spectra we have used two different types of spectral unfolding which is necessarily approximate due to the finite dimension of the spectra. For small matrices we have calculated the level spacings for the M representatives from the given ensemble (M is the number of matrices drawn from the ensemble). The spacings obtained have then been divided by the mean spacing calculated for the entire set. In the case of large matrices this rule does not make sense since the density of eigenvalues is strongly non-uniform. Therefore, in the case of larger matrices, we have applied the phenomenological rule (see Subsec. 2.5.1), which means that we have selected a certain number of the nearest neighbours (typically unf = 20 for
4.2. Numerical calculations with higher dimensional non-Gaussian random matrices 89
N = 120), so that the local mean level spacing has been calculated by averaging the entire set. One should emphasise that N = 120 and unf = 20 in the phenomenological unfolding procedure means 120 - 20 = 100 effective energy levels (for comparison: in Sec. 2.5 only the effective points have been treated). The two unfolding procedures differ in the mean used; this mean can be either global or local, so it is sometimes hard to decide which type of mean is to be applied in the calculation. The results (unavoidably) depend on the unfolding method selected although, in most cases, its influence is not as large as one might expect at first sight.
4.2.1 Box (uniform) distribution
As observed in Fig. 4.2 the transition from N = 2 to GOE is quite fast since for N = 3, 4 one still notices certain deviations from the GOE (approximated by the Wigner distribution) whereas at N = 7 the agreement with the Wigner distribution is already perfect despite the fact that we have used the unfolding procedure for small matrices in this case (the procedure is described in the previous section). At the dimension N = 120 one observes perfect agreement with the GOE distribution for both the unfolding procedures, i.e. for small matrices as well as for the 20 neighbours-rule. This demonstrates that, at least in this case, the final results relatively weakly depend on the unfolding procedure. The last plot in Fig. 4.2 shows the eigenvalue distribution for the dimension N = 120, so one clearly sees that the limiting distribution is smooth and confined to a finite interval. Since the assumptions from the HW theorem are fulfilled in this case, the transition to the GOE behaviour is expected (see the plots from Fig. 4.2).
4.2.2 Exponential distribution
In comparison with the previous subsection the transformation of the level spacing distribution as a function of N is relatively slower here, but it completes at N = 120, which confirms the theoretical prediction (as observed in Fig. 4.3). The level spacing distribution approaches the Wigner distribution already at N = 4. Then the value of N increases from 7 to 10 and 20 so that the agreement is almost perfect. The last plot in Fig. 4.3 re-confirms this for the dimension N = 120. Like in the case of the box distribution the distribution of the eigenvalues for N = 120 is smooth and confined to a finite interval (but this is not shown in the figure).
4.2.3 Cauchy-Lorentz distribution
The Cauchy-Lorentz distribution is interesting, since its first moment diverges. One also knows that according to Eq. (4.13) the tail of P(S) forN = 2 is proportional to 1/S2, so it does not have the first moment either. In the limit of very large Ns, such as N = 120, one clearly sees (Fig. 4.4) that the density of the eigenvalues is not confined to a finite interval. Thus, one of the assumptions from the HW theory is not fulfilled in this case, so in the limit N -> oo one might expect certain deviations from the universal GOE behaviour. This is indeed what one notices in Fig. 4.4, which shows the level spacing distribution for N = 120 for the unfolding procedure with unf = 20 neighbours. At smaller Ns, such as N = 3, 4, 7 and, also, at N = 120 if one uses the unfolding procedure typical of small matrices, we have observed an unusual behaviour of P(S) which differs
Chapter 4. Numerical studies of non-Gaussian real symmetric random 90 matrices
Figure 4.2: The numerical histograms compared with the Wigner curve for the ensemble with the box distribution for a0 = ?2/2, b0 = 1/2. M is the number of matrices from the ensemble. Top-left: N = 3, M = 106, unfolding for small matrices. Top-right: N = 4, M = 106, unfolding for small matrices. Middle-left: N = 7, M = 106, unfolding for small matrices. Middle-right: N = 120, M = 104, unfolding for small matrices. Bottom-left: N = 120, M = 104, unfolding with unf = 20 neighbours. Here the curve represents the exact GOE result. Bottom-right: the eigenvalue distribution for N = 120, M = 104, showing that the distribution is smooth and confined to a finite interval.
in each case; clearly, this behaviour is influenced by the fact that the mean energy level spacing approaches infinity when N › ?. If certain singularities of the Cauchy-Lorentz distribution of the matrix elements are eliminated (by cutting off the tails at a large but finite a, b), one observes the immediate transition to the GOE behaviour, which well agrees with the HW theory (see Fig. 4.4). The cutting is realised via the transformation from the uniform distributed random numbers abox, bbox to the Cauchy-Lorentz distributed random numbers a, b. This transformation is represented by
4.2. Numerical calculations with higher dimensional non-Gaussian random matrices 91
Figure 4.3: The numerical histograms compared with the Wigner curve for the ensemble with the exponential distribution for Aa = y/2, Xb = 1. M is the number of matrices from the ensemble. Top-left: N = 3, M = 106, unfolding for small matrices. Top-right: N = 4, M = 106, unfolding for small matrices. Middle-left: N = 7, M = 106, unfolding for small matrices. Middle-right: N = 10, M = 105, unfolding for small matrices. Bottom-left: N = 20, M = 105, unfolding with unf = 6 neighbours. Bottom-right: N = 120, M = 103, unfolding with unf = 20 neighbours. Here the curve represents the exact GOE result.
a = a0 tan
k abox 2 cl0
b = b0 tan
2 b0
(4.18)
e),a0(l - e)] and
where abox, bbox are uniformly distributed on the intervals [-a0(l - t;,a0v^ -[-60(l-e), 60(l-e)] respectively. Thus, a, b are Cauchy-Lorentz distributed with ga(a), gb(b)
from Subsec. 4.1.3 up to the border acut = a0 tan f (1 - e) and bcut = b0 tan f (1 - e) where \a\ < acut and |6| < bcut. The value e = 0.01 has been chosen for the last plot from Fig. 4.4.
Chapter 4. Numerical studies of non-Gaussian real symmetric random 92 matrices
Figure 4.4: The numerical data for the ensemble with the Cauchy-Lorentz distribution for a0 = V2/2, 60 = 1/2. M indicates the number of matrices selected from the ensemble. For the sake of comparison the thin curve represents the Wigner distribution. Top-left: the density of the eigenvalues for N = 120, M = 103 which is obviously not confined to a finite interval. Top-right: N = 120, M = 103, unfolding with unf = 20 neighbours, the smooth curve is the Wigner distribution for the sake of comparison. Middle-left: N = 3, M = 106, unfolding for small matrices. Middle-right: N = 4, M = 106, unfolding for small matrices. Bottom-left: N = 7, M = 106, unfolding for small matrices. Bottom-right: N = 120, M = 103, unfolding with unf = 20 neighbours. Here the tails of the matrix element distribution function have been cut off at the specific value of e = 0.01, due to which the level spacing distribution immediately approaches the GOE behaviour.
4.2.4 Singular times exponential distribution
The singular (times exponential) distribution from Sec. 4.1.4 (4.14) at the dimension N = 2 might trigger off a completely new phenomenon, i.e. the fractional power law level repulsion (see Fig. 4.1). Surprisingly, if this ensemble is generalised to the higher
4.2. Numerical calculations with higher dimensional non-Gaussian random matrices 93
Figure 4.5: The numerical histograms compared with the Wigner curve (except for N = 120 where the exact GOE appears) for the ensemble with the singular times exponential distribution for A„ = y/2, \b = 1, ßa = 0, and ßb = 1/2, except in the last plot (bottom-right) where [ib = 0.9. M indicates the number of matrices chosen from the ensemble. Top-left: N = 3, M = 105, unfolding for small matrices. Top-right: N = 4, M = 105, unfolding for small matrices. Middle-left: N = 7, M = 105, unfolding for small matrices. Middle-right: N = 120, M = 5 x 103, unfolding with unf = 20 neighbours. Here the curve represents the exact GOE result. Bottom-left: N = 120, M = 5 x 103, the eigenvalue density. Bottom-right: N = 3, M = 105, unfolding for small matrices.
dimensions of N > 2, for pa = 0 and pb = 1/2, one observes a transition to the linear level repulsion already at the dimension N = 3. Fig. 4.5 shows a relatively fast transition to the GOE distribution appearing with the increasing values of N. One can verify that the level density for N = 120 is confined to a finite interval, so the HW prediction applies in this case. If one increases the value of /ib to /ib = 0.9, i.e. closer to the nonintegrable singularity /ib = 1, this influences the P(S) behaviour so strongly that the fractional power law level repulsion can be observed in the end.
Chapter 4. Numerical studies of non-Gaussian real symmetric random 94 matrices
Figure 4.6: The numerical histograms compared with the Wigner curve for the ensemble with the singular times exponential distribution for Aa = y/2, Xb = 1, ßa = 0, and ßb varying from 0.9 to 0.9999. In all the cases the dimension of the matrices was N = 120 and the unfolding was calculated via unf = 20 neighbours. M indicates the number of matrices chosen from the ensemble. Top-left: ßb = 0.9, N = 120, M = 2 x 103. Top-right: ßb = 0.95, N = 120, M = 2 x 103. Middle-left: ßb = 0.97, N = 120, M = 2 x 103. Middle-right: fib = 0.99, N = 120, M = 2 x 103. Bottom-left: ßb = 0.999, N = 120, M = 103, one can also observe the Poissonian curve P(S) = exp(-S). Bottom-right: fib = 0.9999, N = 120, M = 2 x 104 and the Poissonian curve P(S) = exp(-S) as well.
However, as the exponent /ib increases and approaches the nonintegrable singularity ßb = 1, in the case of large matrices with N = 120 there appears the transition from the GOE (at smaller values of ßh) to the Poissonian (exponential) if ßb is sufficiently close to the value 1 (see Fig. 4.6). At the intermediate values of ßb, such as ßb « 0.99, we have conjectured the fractional power law level repulsion, althoug additional analytical as well as numerical analysis is needed to provide more quantitative predictions and descriptions.
95
Chapter 5
Summary and conclusions
This thesis mainly discusses dynamical tunneling (a concept introduced in (Davis and Heller, 1981)) in the systems with the mixed (i.e. regular-chaotic) type dynamics. We have considered two different aspects: (i) we have studied the level spacing distribution for a general mixed-type system and tested the theory for the mushroom billiard levels and (ii) we have analysed the tunneling rates as well as the avoided-crossing distribution for the mushroom billiard. Additionally, we have calculated the level spacing distribution for the random matrices of non-Gaussian ensembles in order to model the fractional power-law level repulsion.
In Chap. 2 we have devised a new random matrix model which describes the distorted Berry-Robnik level spacing distribution affected by the tunneling between the regular and the chaotic states, denoted by PEbrn(s). We have derived a two-level analytic formula for PEbr(S) which agrees with the results for higher dimensional random matrices at not too large values of the coupling parameter. We have also developed an analytic two-level model creating the level spacing distribution P$BR(S) for the all-to-all level couplings, which well applies in the cases of general perturbation such as the presence of an antenna in a microwave resonator. Since this all-to-all two dimensional model, where all the levels are coupled, perfectly agrees with the iV-dimensional simulations denoted by Pdbrn(s), this is, certainly, a very successful model. Its exact integral representation PßBR(S) can be analytically expressed in a closed form for small as well as for large values of S. The same applies for the tunneling-distorted Berry-Robnik distribution PlBR(S) where only the regular and the chaotic levels are coupled. However, an overall good, simple analytic approximation has not been devised yet. We have also shown that the non-Gaussian models for the all-to-all couplings (i.e. the exponential and the box distributions for the off-diagonal matrix elements) well agree with the Gaussian model although they deviate from it significantly a the intermediate values of S. However, they almost perfectly agree with the corresponding two-dimensional model.
The experimental and the numerical data for the mushroom billiard (see Chap. 3) can be well described by the theoretical models employed. If the energy increases towards infinity (the semiclassical limit), a should approach zero (and there appears the BR behaviour), which does not necessarily happen quickly and monotonously. We have noticed that a somehow decreased (but not monotonously), but this cannot be confirmed by the results obtained, so one cannot make any definite conclusions at this point. The reason, presumably, lies mainly in the theory since two important things were not taken into account, i.e (i) the splittings of regular states due to the presence of chaotic states and
96
Chapter 5. Summary and conclusions
(ii) localisation effects. There may be other reasons as well, i.e. the experimental and the numerical inaccuracy of the mushroom levels or, even, the specific properties of the very mushroom billiard, such as scarred (Dietz et al., 2007) or bouncing-ball eigenfunctions. In the future we intend to improve the theory in the way discussed in Sec. 2.6, so that it can be tested in other systems, billiards or maps, or even in the recalculated mushroom billiard levels with the new method of particular solutions. In the case of the mushroom billiard the semiclassical limit is reached very quickly, i.e. already after a few thousands of levels. For the numerical level dynamics in this high lying regime one should definitely apply the method of particular solutions since the EBIM would be too slow in this case. Experimentally, without using the superconducting cavity - in (Dietz et al., 2007) the authors resolved the first 938 resonances and some additional ones as well, but they did not reach more resonances since they only measured one billiard configuration at different antenna positions whereas we reached the 600th branch in the level dynamics measurement - obviously, this semiclassical limit is unreachable.
According to the overall picture of the level spacings in the mixed-type systems (Robnik and Prosen, 1997), there exists the regime of linear level repulsion at small S whereas the regime of the fractional power law level repulsion appears at larger S followed by the Berry-Robnik tail. Thus, the theoretical approach presented seems to be very promising. The linear level repulsion, which always exists due to tunneling, is a very robust phenomenon indeed (Grossmann and Robnik, 2007b).
Recently, V.A. Podolskiy and E.E. Narimanov have tried to correct the Berry-Robnik level spacing distribution by modelling the tunneling effects in another way, which has resulted in the formula presented in (Podolskiy and Narimanov, 2003b) and based on (Podolskiy and Narimanov, 2003a). However, this derivation is incomplete and inconsistent since it does not incorporate the normalisation and since the physical grounds and the results are questionable, so this approach is not discussed in any further details.
Localisation effects should cause the fractional power law level repulsion, as captured by the Brody-like distribution (Prosen and Robnik, 1994a,b). We have tried to model localisation in Chap. 4 of the thesis where we have numerically studied four non-Gaussian ensembles of real symmetric random matrices for various dimensions N, from N = 2 to N = 120, defined by (a) the box, (b) the exponential, (c) the Cauchy-Lorentz and (d) the singular times exponential distribution. We have studied the transition to the universality regime at N -> oo, precisely described by the GOE from the random matrix theory, as predicted in (Hackenbroich and Weidenmu¨ller, 1995), provided that two conditions are fulfilled, i.e. (i) the limiting eigenvalue distribution is smooth, and (ii) it is confined to a finite interval.
In the first two cases (a) and (b) the two conditions are fulfilled, so the HW prediction is confirmed whereas in the case (c) the level spacing distribution clearly deviates from the GOE curve since the eigenvalue distribution is not confined to a finite interval. At the intermediate values of N the behaviour of P(S) is quite unusual indeed. In the case (d), for the singular times exponential distribution of the off-diagonal matrix elements, only a partial deviation from the HW theorem and the two conditions presented can be observed. This happens at those values of /ib which closely approach to the nonintegrable singularity /ib = 1. Thus, surprisingly, the fractional power law level repulsion, which is clearly manifested in the two-dimensional theory (Grossmann and Robnik, 2007b) and in its simulation, does not appear at the higher dimensions N > 2, except, possibly, for the case of a specific interval of /ib approaching the nonintegrable value 1, e.g. at /ib « 0.99.
97
However, only additional numerical and analytical studies could explain the behaviour of singular random matrix ensembles in more details. Indeed, the Hamilton operators of nearly integrable systems (slightly perturbed integrable systems in the KAM scenario) in the basis of the integrable part are quantally represented as sparsed matrices, which should exhibit singular distribution of matrix elements (Prosen and Robnik, 1993a).
Chap. 3 experimentally and numerically studies the tunneling rates 7mra in the mushroom billiards. Fermi’s golden rule is applied to all possible eigenstates V™ of the quarter circle which are also termed ’pure regular states of the mushroom billiard’ and which decay into the neighbouring chaotic states. The results have confirmed the theory especially devised for the mushroom billiard. We have observed a very good agreement without any free parameters including the error bars, which is unprecedented for billiards. On average an eigenstate of the quarter circle with quantum numbers (m, n) decays by about one order of magnitude slower than the one with quantum numbers (m,n+ 1). This result could be, potentially, applied in the field of microlasers, optical fibres, etc.
Surprisingly, Fermi’s golden rule which is derived for the continuum distribution of levels, well applies in closed systems, i.e. in systems with the discrete distribution of levels. Strictly speaking, this rule should be applied only in the case of the open mushroom billiard where the depth of the mushroom foot is infinite, but it is, obviously, suitable in this case as well.
In theory, the approach using a fictitious integrable system could, hopefully, be applied to other generic billiards where it is much more challenging and difficult to determine an appropriate Hreg. In the experiment presented the unavoidable coupling to the environment was preferred for small tunneling rates; however Fig. 3.27 shows that in the microwave experiment the coupling by the antenna is actually negligible beyond three orders of magnitude. This is a promising aspect for future experimental studies of more complex systems, especially in the cases where the numerical and theoretical results are not available yet.
Similarly, the tunneling rates could also be studied in the full chaotic systems, such as the Sinai or the stadium billiard. Here, one could study the decay rates of the bouncing-ball states into the (pure) eigenstates. In this way, one could learn a lot about the strength of interaction between the bouncing-ball modes and the eigenstates, one could, for example, determine how localisation influences the splittings at avoided crossings, to determine the relationship between tunneling and localisation. Similarly, this could also be studied in the mushroom billiard.
Chap. 3 also presents the results of the numerical study of the avoided-crossing distribution in the mushroom billiard. Both types of splittings, i.e. splittings at CC avoided crossings and splittings at RC avoided crossings, were studied separately. In the case of CC splittings we have found very good agrement with the 2 x 2 RMT prediction. However, one still cannot explain a ratio of the mean splitting and the mean level spacing which is smaller than predicted. Due to these new results, the RMT prediction may eventually be generalised to the CC splittings from the mixed-type systems.
98 Chapter5.Summary and conclusions
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107
Appendix A
pi calculation for the mushroom billiard
The group from Dresden has analytically derived the classical effective chaotic area of the desymmetrised mushroom billiard for the central position of the foot: b = R - a. The derivation is presented in this appendix together with our numerical results.
The fraction p2 of the chaotic part of the classical phase space volume of the 2D billiard system can be generally represented as
= ?PSC = JdqdpXch(q,p)6(E-H(q,p)) P2 ?ps Jdqdp5(E-H(q,p)) ' ( . )
where ?PS is the volume of the whole classical phase space and ?PSC is the volume of the chaotic part of the classical phase space; Xch(q, p) equals 1 if the trajectory with the coordinate q and the momentum p is chaotic, but if this trajectory is regular, Xch(q, p) equals 0. Due to the relation H(q, p) = 2j = E in billiards the ?PS is written as
f dqdpstE- 2-
?ps = dqdpS E-^—. (A.2)
The integration over the space coordinates provides the area A of the billiard. By replacing the double integral over the momenta with the integral over the energy one gets
?pS = 2nMA f dE5 (e - 2-\ . (A.3)
The remaining integral equals 1, which results in
?ps = 2ttMA. (A.4)
The phase space of the mushroom billiard (Fig. A.1) is given by {(r,^,i?) : (r, a is characterised by regular and chaotic dynamics. Its contribution to Ach, called / should be calculated as
where
Ach
Ia+ 4W
+ 1,
(A.6)
/
1
2^
/f R
I dip I rdr I
0 a
2lT
d&xch(r,(f,&).
(A.7)
Only the radial coordinate r determines whether there appear chaotic trajectory in the hat for r > a (see Fig. A.1). If one considers the trajectories crossing the point (r, reg> and |V>ch> are the eigenstates of different operators, Hieg and Hch, they are not necessarily orthogonal, (V'chlVw) = X with 0 < |x| < 1. In order to apply Fermi’s golden rule the orthonormalised states ^ were introduced |^reg) = |Vw), \M = (\ipch) ~ xlVw))/^1 - Ixl2, leading to
(V'chlVw) = 0.
For the coupling matrix element v = ftM Wreg) = $ch\H$mg) it follows
V = (Am\H - HreglÄeg) + (Äh |#reg|Vw)
1 (V'ch|Jff-Jffreg|V'reg)-^^(V'reg|Jff-Jffreg|V'reg)+0. (B.1)
v71 - Ixl2 v/1-lxl2
The leading order in X for billiards is represented by
v= [ 4>*ch(x,y)(H - Hreg)ijmg(x,y)dxdy + O(x). (B.2)
in
Eqs. (3.38) and (B.2) can be used if one wants to determine the dynamical tunneling
rates in billiards. Here one should emphasise that Hreg and |^g) have to be selected very
carefully, which is a difficult task in the case of a general billiard.
Appendix B. Theoretical analysis of the tunneling rates in the mushroom 112 billiard
This approach can now be applied to the desymmetrised mushroom billiard composed of a quarter circle and a rectangular foot (see Fig. B.1a) where we have chosen R = 1 in the following analysis. For the regular system HYeg the quarter-circle billiard H1/4 is a natural choice (with its eigenvalues E™ and the eigenstates V™ represented in Eq. (3.3)).
(a)
(b)
R
a
Figure B.1: The desymmetrised mushroom billiard: (a) Schematic picture of the coordinate systems used in the theoretical derivation. (b) The auxiliary billiard H1
w
/4
After evaluating Eq. (B.2) one obtains the undefined product of H - Hl/4 = -oo and ipff(x,y) = 0, for y < 0. Therefore, the auxiliary billiard H^4 is introduced (see Fig. B.1b) with a large but finite potential V(x,y < 0) = W > E, characterised by the eigenstates Vi/4W Eq. (B.2) is evaluated in the limit W -»• oo where H^4 approaches #i/4, which leads to
lim
W›?
pa pO
dx dyi)cb(x,y)(-W)i)^w(x,y) Jo-i
dx lim Wip7%w
JQ W^oo ' '
a dtbmn
dx lim l/4,W
-i
(x,y = 0) dyipch(x,y)exp(W
dy
(x,y
o
a dtbmn
dx^(x,y
a dtbmn
. d'r^f
0)-F
VF
V
T?mn ^1/4
y)
w-Lmra
0) dyVch(^,y) lim
Z W->oo
dyVch(x,y)exp( FFexp(FF
(x,y = 0)
(x,y
-i
0)4>ch(x,y
-Nmn
m
dx-Jm(jmnx)^ch(x,y o x
dyi)cb(x,y)2 5(y)
0) 0),
r
v
rprnn ^1/4
FF
y)
^1/4
v)
(B.3)
l
'f
__
a
__
__
__
o
__
__
__
__
__
a
__
113
where the y-integration was performed on ipT%w(x,y) = ip„Zw(x,y = 0)exp(W - E™y),
/a w ( i y) = r i /4 w ( i y = ) exp( yv — j-ja i a obtained from the Schr¨odinger equation for y < 0 and the continuity at y = 0. Further-
chbmn I---------------------
more^(x,y = 0) = JW - E™tl)™w(x,y = 0) has been used, obtained from the
continuity of the derivative at y = 0 and lim^^ V/Wexp( W - E™y) was replaced
by the Dirac delta function; Nmn is the normalisation defined in Sec. 3.2.
One should observe that the value of the chaotic eigenstates |V>ch> is only needed on the line y = 0. For these eigenstates a random wave decomposition (RWD) (Berry, 1977) was employed. Recently, the RWD has been generalised to the systems with the mixed phase space (B¨acker and Schubert, 2002). While this description accurately describes the behavior inside the billiard, it does not incorporate the effect of the boundary, e.g. near the corner, which, for m > 2, largely influences the (final) integral from Eq. (B.3). Therefore, the RWD model (Berry, 2002) was extended to the case of the corner with the angle 3vr/2 using the eigenstates with the Dirichlet boundary conditions (Lehman, 1959),
^(p, tf) ~ Jj J2 Cgj2i W^p) sin (—A' (B.4)
V ch s=l
where the polar coordinates (p, ) = NmnJm (jmnr) sm{rmp), ^{r >l, 0, izboljšanje teorije in seveda tudi izračun lastnih energij kakšnega drugega sistema mešanega tipa.
vi Izognjena kriˇzanja in tuneliranje
Veje lastnih stanj se lahko križajo ali pa tudi ne, v tem primeru dobimo t. i. izognjena križanja dveh vej z najbližjo razdaljo AE. V regularnem sistemu izognjenih križanj ni, v popolnoma kaotičnem sistemu pa so vsa križanja izognjena. Če ima tak sistem simetrijo na obrat časa, je v njem porazdelitev razcepov v izognjenih križanjih Gaussova, kar sledi iz 2 x 2 GOE modela naključnih matrik za izolirano izognjeno križanje (Zakrzewski et al., 1993). Normalizirana na enotski povprečni razcep se porazdelitev glasi:
128
Daljši slovenski povzetek
1.0
1/3
Co
0
1.0
0.8
0
2
S!
P
Or_ =
0.255 0.117 0.136 9898
2
Slika ix: Primerjava eksperimentalnih podatkov (histogram) z najbolj prilegajočo se teoretično krivuljo P^brn (zgornja) in numeričnih z najbolj prilegajočo se teoretično krivuljo P^brn (spodnja slika), kjer je rdeča krivulja za Gaussov, pikčasto črtkasta črna za eksponentni model, črtkasta modra za BR porazdelitev z istim p in pikčasta rjava za Wignerjevo porazdelitev. aG in (te sta vrednosti najboljših fltov za a za Gaussov in eksponentni model. N je število objektov v vsakem histogramu. Na zgornji sliki imamo nivoje na intervalu (jmin,jmax) = (100,300), na spodnji pa na intervalu (jmin,jmax) = (100,200).
Pgoe(AE)
7T
exp
AL2
n
(xv)
2
_
vi. Izognjena križanja in tuneliranje
129
70.0 70.5 71.0 71.5