University of Ljubljana Faculty of Mathematics and Physics Seminar - 1st year, 2nd cycle degree Quantum chaos Author: Ana Flack Mentor: izred. prof. dr. Marko šnidari Ljubljana, May 2018 Abstract The basic ideas of quantum chaos are presented in this seminar. It begins with a short overview of the properties of classical chaos and continues with the introduction of the quantum version of chaos. The random matrix theory and its role in quantum chaos is discussed. At the end dynamical localisation in the quantum kicked rotator is explained.
Contents 1 Introduction 1 2 Classical chaos 1 3 Quantum chaos 2 3.1 Quantum chaos and the Correspondence principle.................. 2 3.2 Characteristic time scales................................ 3 4 Random matrix theory 4 4.1 Time reversal symmetry................................. 4 4.2 Gaussian ensembles for Hermitian matrices...................... 5 4.3 Level repulsion...................................... 5 4.4 Periodically driven systems............................... 6 4.4.1 Floquet operator................................. 6 4.4.2 Circular ensembles................................ 7 5 Dynamical localisation and kicked rotator 7 5.1 The classical model.................................... 7 5.2 Quatum kicked rotator.................................. 7 6 Conclusion 9 1 Introduction Chaotic behaviour of classical dynamical systems has been discovered trough the study of the celestial dynamics. In the nineteenth century Poincaré proved that the three body system can not be analytically solved. This resulted in the emergence of a new theory of dynamical systems. After the discovery of quantum mechanics the question about chaos in quantum systems arose naturally. However, it has turned out that the answer is not as simple as it may seem. The roots of quantum chaos can be traced back to the 1950 when the spectra of complex nuclei were analysed and the topic is still active today. In this seminar some basic concepts of quantum chaos will be discussed. First a short review of classical chaos is presented then quantum chaos is introduced. Some results from random matrix theory are given and their role in quantum chaos is discussed. At the end dynamical localisation is presented on the model of the quantum kicked rotator. 2 Classical chaos If quantum chaos is to be discussed we must rst understand some basic concepts of classical chaos. In this chapter the main ideas will be presented. Classical dynamical chaos is a property of classical dynamical systems, that is usually characterised by exponential sensitivity of trajectories to the small variations of initial conditions and unpredictability. This notion is quite interesting because it shatters the idea of predictability in classical mechanics [1]. Exponential sensitivity of motion is mathematically described by Lyapunov exponents. If x is an arbitrary reference orbit and x some neighbour orbit we denote the dierence between them at time t as δ t = x t x t. Maximal Lyapunov exponent can be therefore dened as [2] Λ = lim lim t δ 0 0 1 t log( δ t δ 0 ). (1) If it is positive, one may conclude that system in question in chaotic. Not every system is fully chaotic, many systems display mixed dynamics. In gure 1 the Poincaré sections for standard 1
map (that will be discussed later) are shown, each for a dierent value of parameter K. Far-most left picture shows regular motion, on the middle we can see mixed dynamics, and on the far right we see fully developed chaos [3]. Figure 1: Poincare section of the standard map for dierent parameters K, reproduced from [3]. It is important to notice that exponential instability implies decay of correlation functions [2]. Correlation function is dened as C ab (t) = dµ(x)a(x)b(x(t)) dµ(x)a(x) dµ(x)b(x), (2) where a and b are observables and dµ is invariant measure, usually just dx. From the denition it is clear that when C ab is zero, the observables a and b are independent. So C ab 0 when t implies that the system has a property of mixing [4]. In other words if system has mixing we can conclude that two subsets of phase space are statistically independent if they are suciently separated in temporal domain. On the other hand mixing is needed if we want to statistically describe dynamical system. In this case we can introduce probability theory and instead of trajectories we can observe distribution functions. The time evolution of distribution functions is described by the Louiville equation [2]. Other main characteristic of chaos is the unpredictability. It can be mathematically understood by Kolmogorov-Sinai (K-S) entropy which carries the information about the amount of information produced by the system in one step [4]. Positive K-S entropy is necessary and sucient condition for chaos [1]. The nature of unpredictability is explained by Alekseev-Brudno theorem stating that information needed for prediction of new segment of trajectory is proportional to its length and independent of the full previous length of trajectory. In other words, the needed additional information can not be gathered from the previous motion [2]. Both sides of dynamical chaos, the exponential sensitivity and unpredictability, can be connected. For example the Kolmogorov-Sinai entropy (h KS ) is less or equal than the sum of positive Lyapunov exponents. If the system is Hamiltonian and some other conditions hold, the Pesin's theorem states that [4] h KS = λ j. (3) λ j>0 The question that arises after studying classical chaos is: How does chaotic behaviour manifests in quantum systems? Next chapters will try to shed some light on this topic. 3 Quantum chaos 3.1 Quantum chaos and the Correspondence principle Niels Bohr's Corespondence principle states: For those quantum systems with a classical analogue, as Planck's constant becomes vanishingly small the expectation values of observables behave like their classical counterparts [9]. 2
Therefore it is only natural that one tries to nd chaos in quantum world. However, when we try to dene chaotic behaviour in quantum realm, we are faced with a number of diculties. The rst hurdle that one has to overcome is that the trajectory in quantum mechanics is not a well dened concept. According to the fundamental uncertainty principle, p x, the position and momentum can not be determined simultaneously. Therefore the exponential sensitivity of trajectories (in the ordinary sense) is not useful denition of chaos. The next problem we must face is the linearity of Schrödinger's equation and the unitary time evolution. In nite classical system chaos emerges from non-linearity of equations of motion. For this reason one would not expect chaos to emerge from linear Schrödinger's equation [10]. Nevertheless if we consider classical chaos from the viewpoint of the probability densities, the Liouville's equation dρ = {ρ, H(x, p)}, (4) dt is the one that describes the dynamics. In the equation ρ stands for phase space distribution function and {} denotes Poisson's brackets. This equation is also linear just as the Schrödinger's equation, therefore the mere linearity of Schrödinger's equation is not sucient to exclude the possibility of quantum chaos [1]. Similarly the unitary time evolution, Ψ(t) = U ψ(0), that governs the quantum dynamics implies that the distance or the overlap of two states in phase space (Hilbert space) is constant in time. It follows that the small dierences of initial states will not grow in time. However the classical Liouville's equation has the same property. The overlap of two Liouville's densities is constant in time, and still this does not prevent chaotic behaviour [11]. It is still not agreed upon what the exact denition of quantum chaos is, dierent authors have taken dierent approaches to resolve the mystery of quantum chaos. Popular technique is to study quantum systems which have chaotic classical limit, although chaos can also emerge in quantum systems without classical limit, for example spin chains. However it is clear that quantum chaotic systems obey some characteristic time scales that will be discussed in the next section. 3.2 Characteristic time scales Figure 2: Visual representation of time scales. Line 1 corresponds to random time scale and line 2 to relaxation time scale, adapted from [2]. One way of tackling the seemingly severe problem of quantum chaos was proposed by Casati and Chirikov. On the basis of numerical simulations they proposed idea of dierent characteristic time-scales where quantum systems display all the properties of classical chaos. This stems from the notion that the distinction between discrete and continuous spectra is unambiguous only in the limit t [12]. The known Ehrenfest's theorem states that motion of quantum mechanical wave packet follow classical motion as long as the packet stays narrow enough [10]. Following this reasoning we can conclude that the motion during this time is as chaotic as in classical limit. However, due to the uncertainty principle the wave packet spreads with time. The timescale where this happens is described by the shortest timescale, namely the Eherenfest time given by the equation t E = ln q Λ, (5) where q stands for the semi-classical parameter q = A 0 /, where A 0 represents the volume of phase space explored by classical trajectory and Λ is the maximal Lyapunov exponent [12][10]. From the equation 5 it can be seen that t E is logarithmically short in. 3
The next time scale is given by the relaxation time which is larger then the Eherenfest time and it bears great importance for the statistical description. In this region quantum motion is similar to the classical and subsequently diusion and statistical relaxation take place [12]. After this time quantum uctuations start to govern the motion. In the gure 2 the general structure of quantum mechanics on the (q, t) plane is presented. Localisation will be discussed later. In between both time scales is the so called quantum pseudo-chaos where diusion and relaxation happen. True chaos manifests only for the limited time [2]. In the region of true chaos semiclassical approximations can be used, however if we are out of the domain of the true chaos the random matrix theory (RMT) turns out to be quite useful. This theory and its application to the quantum chaos will be presented in the next chapter. 4 Random matrix theory The main idea of the theory is the use of Hermitian random matrices as model of complicated Hamiltonians. Random matrix theory (RMT) was introduced by Eugene Wigner, Dyson, Mehta and others in nineteen fties. Wigner used random matrices for description of the spectra of complex nuclei. His idea was that such systems and their Hamiltonians are so complicated that random matrices would be a good approximation. [5]. At rst it was believed that RMT can be used only for describing systems with high number of degrees of freedom. However, a few decades later Bohigas, Giannoni and Schmit conjectured that the theory can also be applied to simpler systems with as little as two degrees of freedom, provided that their classical analogue is chaotic [3]. It should be mentioned that RMT can be further justied due to the fact that it can be derived from the so called Gutzwiller semiclassical trace formula, which formulates the density of states only with classical parameters such as orbit stability matrix and classical action [5], however this area of chaos theory will not be further discussed in this seminar. In the following section some important results from RMT and their connection to the quantum chaos will be presented. 4.1 Time reversal symmetry Symmetries, especially time reversal symmetry, play crucial role in the RMT. According to symmetries Hamiltonians can be divided into three universality classes. The Schrödinger's equation i ψ(x, t) = Hψ(x, t) is time reversal invariant if for any solution ψ(x, t) also ψ (x, t ) where t = t is solution and there exists a unique connection between ψ and ψ. Any time reversal operator T must be antiunitary [13]. If the system has this symmetry, T commutes with H [H, T ] = 0. (6) First systems with broken time reversal symmetry will be considered. For the RMT it is important to know which group of transformations preserves the symmetries and eigenvalues of the Hamiltonian. Systems that are not time reversal invariant are in general not real, however as all physical Hamiltonians they are Hermitian. Therefore the group of transformations that preserve Hermicity and the eigenvalues is the group of unitary transformations [13]. If the time reversal symmetry is not broken we have to consider two cases. In the rst case there are no interactions of particles with spin 1 2, thus the Hamiltonian can always be represented as a real matrix. The canonical transformations conserving this property are orthogonal matrices [5]. Lastly the systems with spin 1 2 interactions are to be considered. If the system is time reversal invariant and has no additional symmetries, the group of canonical transformations are symplectic matrices. However if such system has some other geometric symmetries the canonical transformations can also be unitary or orthogonal. [13]. Which transformations are canonical for a Hamiltonian will play a important role in determining the elements of random matrices as it is explained in the next section. 4
4.2 Gaussian ensembles for Hermitian matrices As was mentioned at the beginning of this section the idea of RMT is to replace the Hamiltonian of a system with the ensemble of random matrices. In this way the spectrum and its properties can be obtained without the diagonalization of the original supposedly complex Hamiltonian. However we are not completely free when choosing the elements of the random matrices. Two restrictions should be taken under the consideration. The probability density P (H) for the elements of random Hamiltonian can be obtained just by these two conditions. First is that the matrix elements are to be uncorrelated, for the case of 2 2 matrix this implies P (H) = P 11 (H 11 )P 22 (H 22 )P 12 (H 12 ). (7) Secondly the P (H) must be invariant under any canonical transformation, that is orthogonal, unitary or symplectic depending on the type of the system. For the orthogonal class this means P (H) = P (H ), H = OHO T, (8) where O is a orthogonal matrix. The same holds for unitary and symplectic class if we use unitary or symplectic matrix instead of orthogonal. Probability should also be normalised as dh 11 dh 22 dh 12 P (H) = 1. (9) With this restrictions in mind we can obtain the probability density P (H), which has Gaussian form and is [13] P (H) = Ce AT rh2, (10) where C and A are constants, C is determined by normalization and A xes the unit of energy. There are three ensembles Gaussian ortogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble, (GSE) one for each of the canonical transformations presented above [13]. 4.3 Level repulsion From the above probability density we can calculate many properties of interest, for example the correlated eigenenergy and eigenvector distribution, averaged density of states, level spacing, nearest neighbour distribution and many more. I will focus solely on the level spacing distribution P (S), that measures the probability of two successive energy levels being separated by the distance S [3]. This distribution can be derived from the eigenvalue distribution. For the 2 2 matrices the distributions are known as Wigner surmises and have the following form [13] π 2 s exp ( π 4 s2 ), GOE, 32 P (S) = π s 2 exp ( 4 2 π s2 ), GUE, (11) 2 18 3 6 π s 4 exp ( 64 3 9π s2 ), GSE. Although the equations are for the 2 2 case they are still able to suciently precisely describe the spectra of random matrices with arbitrary rank [5]. The expression for higher rank matrices are very complex and hard to derive therefore the Wigner surmises are commonly used. The above formulae bear great importance due to the fact that they show dierent degrees of level repulsion for dierent ensembles. Level repulsion is one of the main predictions of the RMT. If one considers the limit of small level spacings S 0, it is clear that GOE exhibits linear level repulsion, GUE quadratic and GSE quartic level repulsion. When analysing level repulsion it is necessary to divide levels of H into multiples, each with xed values for all observables except H [13]. The reason for this is that levels corresponding to dierent quantum numbers do not have a reason not to cross. 5
The importance of level repulsion is evident only after we know what kind of distribution corresponds to the case of integrable models. In the case of completely integrable model it is presumed that eigenvalues are completely uncorrelated and therefore P (S)dS is the conditional probability to nd a eigenvalue in the interval between S and S+dS from a given eigenvalue [5]. Subdividing the interval of length S into N parts and writing the probability that no other eigenvalue is in the interval, we obtain P (S)dS = lim (1 s )ds. (12) N N After taking the limit we get the exponential or Poisson level spacing distribution P (S) = e S, (13) where is assumed that the mean level spacing is set to one. From this equation it is evident that in the completely integrable case there is Figure 3: Level spacing distribution P (S) for Poissonian (p), orthogonal (o), unitary (u) and symplectic (s) ensembles reproduced from [13]. no level repulsion and therefore the level distribution can be used as a indicator of chaotic systems. In the gure 3 the distributions from equation 11 and the Poissonian level spacing distribution are presented. 4.4 Periodically driven systems Aforementioned ensembles and distributions are relevant for autonomous systems (time independent Hamiltonian). However for the better understanding of the kicked rotator model presented in the last chapter the use of RMT for periodically driven systems shall also be explained. 4.4.1 Floquet operator In the case of periodically driven system we must write the Schrödinger's equation with the explicitly time dependent Hamiltonian i ψ(t) = H(t)ψ(t). (14) The solution for the t > 0 can be written as a time ordered exponential [13] U(t) = T {exp[ i If the system is periodically driven the Hamiltonian is of the following form t 0 dt H(t )]}. (15) H(t + nτ) = H(t), n = 0, ±1, ±2,... (16) The time evolution operator for one period is called the Floquet operator. With this operator we can describe the dynamics in quite a simple form [13] ψ(nτ) = F n ψ(0), (17) where F stands for the Floquet operator. Due to the unitarity of the Floquet operator its eigenvalues (eigenphases or quasi-energies) are on the unit circle F Φ v = e iφv Φ v. (18) 6
In such systems the time invariance can be expressed as a property od the Floquet operator. It can be shown that the time reversed F is T F T 1 = F = F 1. (19) The reasoning presented for Hermitian matrices can be analogously applied to Floquet operators. If F possesses no time reversal invariance the group of canonical transformations is unitary, if there is time invariance the group is orthogonal and the same holds for symplectic ensemble [13]. 4.4.2 Circular ensembles Dyson introduced three new ensembles for the periodically driven systems. They are Circular orthogonal ensemble (COE), circular unitary ensemble (CUE) and circular symplectic ensemble (CSE). The name circular is used because the eigenvalues of the Floquet operator are unimodular, they are all on the unit circle. This dierence manifests in the constant probability of nding a eigenvalue on the unit circle [5]. This can also be seen by examining the joint distribution of eigenphases P ({Φ}) = P (Φ i Φ j ). The distribution is only a function of the dierences Φ i Φ j and therefore the circular ensembles are homogenous in the contrast to the Gaussian ensembles [13]. However in the limit of large systems N, level spacing distribution and also other quantities are the same as in the case of Gaussian ensembles [5]. 5 Dynamical localisation and kicked rotator In the last chapter the standard map or the kicked rotator will be briey presented. This model is immensely popular in the classical theory of dynamical systems as well as in the area of quantum chaos. Despite its simplicity the model exhibits many general features of quantum as well as classical chaos. Both classical and quantum versions have been thoroughly studied [14]. First let us look into the classical model. 5.1 The classical model Classical kicked rotor is model that describes free rotation of the pendulum which is periodically kicked by a gravitational potential. The Hamiltonian of such system is of the form H = p2 2 + k cos(φ) δ(t mτ), (20) m Z where p is the angular momentum, φ stands for angular displacement, k is kick strength and τ the kick period. The moment of inertia has been set to one. Motion can be divided to the free evolution and the kick. From the Hamiltonian the equation of the standard mapping can be derived φ n+1 = φ n + P n (mod 2π) (21) P n+1 = P n + K sin(φ n+1 ), (22) P n and φ n are the values right after the n-th kick. P n is rescaled momentum, P n = τp n, so that we are left with only one parameter K = τk. The system exhibits dierent types of motion for dierent values of K. For K = 0 the motion is regular, however it becomes more and more chaotic as the value of K is increased. Examples are shown in the picture 1 at the beginning. 5.2 Quatum kicked rotator 7
The above model can be quantised with the substitution p i φ and thus we obtain the Floquet operator [15] F = exp( i k cos(φ))exp(iτ 2 2 ). (23) φ2 In contrast with the classical model there are now two parameters, rst one is K = k and the second one is T = τ controlling the eective strength of quantum uctuations [13]. The classical parameter is connected to the new ones as K = K T. From the previous discussion about RMT and quantum chaos one would expect that in region where classical model exhibits complete chaos (K>7) the level spacing distribution should obey Wigner distribution. However this is not always the case. In the case of quantum kicked rotator now phenomenon called dynamical localisation emerges. In the generic case in the region where classical model is chaotic (K K crit 0.9176) and with K 1 the dynamical localisation can be observed [16]. We can see this by analysing the momentum diusion. If we compare classical and quantum rotators we obtain the results Figure 4: Classical (solid curve) and quantum (dotted curve) diusion for K = 5, K = 25, τ = 0.2, reproduced from [2]. presented on the gure 4. Quantum diusion follows the classical one until some characteristic time as discussed in the section about characteristic time-scales. After this time the quantum mechanical diusion is suppressed and eventually stopped. This behaviour can be attributed to the exponential localisation of eigenfunctions in the innite momentum space [16]. Exponential localisation means that the probability amplitude in angular momentum space centred at p 0 decreases exponentially with the characteristic localisation length l as e p p 0 l. The consequences of dynamical localisation are also visible on the level spacing distribution. Instead of level repulsion predicted by RMT we get the exponential distribution [14]. This result can be explained by the fact that even when two eigenstates have very similar eigenphases they are far away from each other in momentum space due to the exponential localisation and have exponentially small overlap. Therefore there is no level repulsion and the Poisson level spacing distribution holds [14]. If the results of RMT are to be compared with this result we must rst transform the system so it has nite dimension. This must be done with great caution so that the unitarity of F is preserved [13]. Nevertheless it can be done and the results show that for the cases where the localisation length l is smaller than N, the eective dimension of the system the Poisson level spacing statistics hold. 8
Figure 5: Level spacing distribution for τ = 15π 2N+1, N = 25 a) for l N = 5, K = 20 and b) l N = 0.25, K = 5. Smooth curve is Wigner distribution. Reproduced from [17]. When the l becomes comparable or bigger than N we gradually transition to the statistics predicted for COE. In the gure 5 the results from numerical experiments are shown. On the left we see the case where localisation length is bigger then N. The results display level repulsion and are compatible with the Wigner distribution. On the right we see the case of intermediate statistics somewhere between exponential and Wigner distribution [17]. 6 Conclusion In this seminar only some basics concepts of quantum chaos have been presented. First the summary of classical theory has been presented. Then the quantum chaos and random matrix theory was introduced. At the end the dynamical localisation was presented on the model of kicked rotator. The concept of quantum chaos is essential for understanding properties of quantum systems, therefore it is relevant for many areas of research. For example the dynamical localisation is tightly connected to the solid state and Anderson model [13][5]. It is also immensely important for study of atomic nuclei and their spectra that are still being studied theoretically as well as experimentally. In recent years the research of many body quantum chaos has been analysed [18]. Furthermore the area is also connected to the development of quantum computers. Yet it seems that the question of quantum chaos is still not fully resolved and will surely be further investigated in the future. References [1] G. Belot and J. Earman, Chaos out of order, Stud. Hist. Philos. Sci. 28, 147 (1997). [2] G. Casati and B. Chirikov, The legacy of chaos in quantum mechanics in Quantum Chaos:Between order and disorder, edited by G. Casati and B. Chirikov (Cambridge University Press, New York, 1995) pp. 3-53. [3] D. Ullmo and S. Tomsovic, Introduction to quantum chaos, (2014). [4] T. Prosen, Dinami na analiza: Uvod v teorijo deterministi nega kaosa, Univerza v Ljubljani, (2018) [5] H. J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, New York, 1999). [6] F. Haake, Quantum Signatures of Chaos (Springer Verlag, Berlin, 2010). [7] J. M. McCaw, Quantum Chaos:Spectral Analysis of Floquet Operators, doctoral dissertation, School of Physics, University of Melbourne (2004). [8] M. A. Porter, An Introduction to Quantum Chaos (2001), arxiv:nlin/0107039v2. [9] E. Merzbacher, Quantum mechanics (John Wiley and Sons, New York, 1998) 9
[10] I. S. Gomez et al., About the concept of quantum chaos, Entropy 19, 205 (2017). [11] A. Peres, The Many Faces of Quantum Chaos, Chaos Solitons Fractals 5, 1069 (1995). [12] G. Casati, B. Chirikov, Quantum chaos: unexpected complexity, Physica D 86, 1069 (1995). [13] F. Haake, Quantum Signatures of Chaos (Springer Verlag, Berlin, 2010). [14] F. M. Izrailev, Simple models of quantum chaos: spectrum and eigenfunctions, Phys. Rep. 196, 299 (1990). [15] T. Manos and A. Robnik, Dynamical localization in chaotic systems: Spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems, Phys. Rev. E 87, 062905 (2013). [16] F. M. Izrailev, Intermediate statistics of the quasi-energy spectrum and quantum localisation of classical chaos, Phys. A:Math. Gen. 22, 865 (1989). [17] B. Chirikov and F. M. Izrailev, Quantum chaos: Localisation vs. ergodicity, Phys. D 33, 77 (1988). [18] P. Kos et al, Many-body quantum chaos: Analytic connection to random matrix theory Phys. Rev, arxiv:1712.02665 (2017). 10