Lyapunov decay in quantum irreversibility

Lyapunov decay in quantum irreversibility

Ignacio García-Mata Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), CONICET-UNMdP, Mar del Plata, Argentina Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET), Argentina    Augusto J. Roncaglia Departamento de Física “J. J. Giambiagi” and IFIBA, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina    Diego Wisniacki Departamento de Física “J. J. Giambiagi” and IFIBA, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina

The Loschmidt echo – also known as fidelity – is a very useful tool to study irreversibility in quantum mechanics due to perturbations or imperfections. Many different regimes, as a function of time and strength of the perturbation, have been identified. For chaotic systems, there is a range of perturbation strengths where the decay of the Loschmidt echo is perturbation independent, and given by the classical Lyapunov exponent. But observation of the Lyapunov decay depends strongly on the type of initial state upon which an average is done. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as it was shown for quantum maps. In this work we introduce an analogous quantity for systems with infinite dimensional Hilbert space, in particular the quantum stadium billiard, and we show clearly the universality of the Lyapunov regime.

Keywords: Quantum Chaos, Irreversibility, Foundations of Quantum Mechanics

03.65.Yz, 03.65.Ta, 05.45.Mt

I Introduction

Understanding the emergence of irreversibility from the basic laws of physics has been a longstanding problem. Although, since the eighteenth century it is known that the second law describes the arrow of time, its microscopic foundation has been matter of debate until these days Fermi (); Maxweedemon (). The main problem is that classical mechanics is time-symmetric and cannot explain the emergence of the thermodynamic arrow of time. This contradiction has been apparently resolved with the understanding of chaos. The sensitivity to initial conditions of chaotic systems, along with the notions of mixing and coarse graining, has been the main argument to explain irreversibility in classical systems Gaspard-book ().

In quantum mechanics, the situation is more involved. Due to the linearity of the Schrödinger equation there is no sensitivity to initial conditions, and therefore the origin of irreversibility in quantum mechanics lies elsewhere. For this reason, an alternative idea was proposed by Peres Peres1984 (). He suggested that quantum mechanics is sensitive to perturbations in the evolution rather than to the initial conditions. A suitable dynamical quantity to study such a behaviour was coined fidelity or Loschmidt echo (LE), which is defined as,


where is an evolution operator and, is a corresponding perturbed one, an initial state, and the parameter characterises the strength of the perturbation. Thus, Eq. (1) can be interpreted in two different ways. On the one hand, it is the overlap between an initial state evolved forward up to time with the evolution operator , and the same state evolved backward in time with a perturbed evolution operator . On the other hand, it can also be interpreted as the overlap at time of the same state evolved forward in time with slightly different Hamiltonians. While the first interpretation gives the idea of irreversibility, the second is related to the sensitivity to perturbations of quantum evolutions.

The LE has been intensively studied in the first decade of this century Gorin2006 (); Jacquod2009 (); DiegoScholar (), and several time and perturbation regimes were clearly identified using different techniques like random matrix theory, semiclassical, and numerical simulations DiegoScholar (). The progress in experimental techniques has permitted to study the LE in various different settings like NMR PastawskiRMN (), microwave billiardsexpbilliards (), elastic waves lobkis () and cold atoms coldatoms (). The most relevant result in connection with chaos and irreversibility is that the LE has a regime where the decay becomes independent of the perturbation strength and it is given by the Lyapunov exponent Jalabert2001 (), a classical measure of the divergence of neighbouring trajectories Gaspard-book (). The Lyapunov regime has been observed in several systems Jacquod2001 (); PhysRevE.65.055206 (); PhysRevE.68.056208 (); Cucc2004 (); PhysRevE.70.026206 (); PhysRevE.67.056217 (); Wang2004 (); Pozzo2007 (). In these works a crucial feature is that the initial states need to be coherent (Gaussian) wavefunctions Diego2002 (). Besides, an average on initial condition or perturbations is required.

The dependence of the LE with the type of initial state can be removed by considering an average over initial states according to the Haar measure for finite dimensional systems zanardi2004 ()


where is the Hilbert-Schmidt product between the operators, and is the dimension of the Hilbert space. Thus, the average fidelity amplitude


which is directly related to the LE, is a state-independent quantity. This quantity was studied in detail for quantum maps garciamaNJP (). Analytical results were obtained using a semiclassical theory known as dephasing representation (DR) vanicek2003 (); vanicek2004 (); vanicek2006 (). It is shown that has two clear decay regimes. For short times, the decay rate depends on the perturbation and it is predicted considering random dynamics. This corresponds to the limit of infinite Lyapunov exponent. If the strength of the perturbation is small enough, this regime lasts up to the saturation point. The other regime, was obtained considering that the perturbation is completely random. That is, after each step of the map, the perturbation contributes with a random phase to each trajectory. In that case, using the DR and transfer matrix theory it is shown that the asymptotic decay rate of is controlled by the largest classical Lyapunov exponent . Numerical tests of the analytical predictions were given for the quantum baker and a family of perturbed cat maps (see Sect. II below).

In this work we go one step further by studying in a realistic system. We consider a particle inside a stadium billiard that is perturbed by a smooth potential consisting of a number of Gaussians randomly distributed inside the cavity. A 2-D billiard has an infinite-dimensional Hilbert space. For this reason, instead of computing Eq. (3) for a complete set, we consider an initial state defined as an incoherent sum of all the energy projectors from the ground state up to a given high energy level. We have numerically computed this quantity that we call for the quantum stadium billiard and show that it has similar behaviour to that observed for quantum maps. For short times, has a decay that depends on the perturbation strength. But, after a crossover and for sufficiently large perturbation strength we can clearly see that decays exponentially with a decay rate given by the Lyapunov exponent of the classical billiard. In order to confirm these results, we have also computed using the DR. We also show that the DR describes very well the quantum behaviour and that the Lyapunov regime is also clearly observed in this approximation.

The rest of the paper is organised as follows. In Sec. II we summarise the results obtained in Ref. garciamaNJP () in quantum maps. We show that the DR works very nicely to describe the quantum behaviour and we show the different decay regimes of . In Sec. III we show the for the stadium billiard. Final remarks and outlook are given in Sec. IV.

Ii quantum maps

For this work to be self contained, in this section we briefly review the results previously obtained in garciamaNJP () for quantum maps on a two dimensional phase space with periodic boundary conditions (2-torus). These maps have the essential ingredients of chaotic systems and are simple to treat numerically and sometimes even analytically. The torus geometry implies that, upon quantisation, position and momentum are discrete and related by the discrete Fourier transform (DFT). The Hilbert space then has finite dimension , and the semiclassical limit is given by . An efficient Planck constant can be thus defined as .

As a tool we use the DR vanicek2003 (); vanicek2004 (); vanicek2006 () which avoids some of the drawbacks of other semiclassical methods. The fidelity in the DR can be written as


where is the Wigner function of the initial state , and


is the action difference evaluated along the unperturbed classical trajectory. If is a maximally mixed state then Eq. 4 is an average over a complete set (and becomes basis independent),


this quantity is the semiclassical expression for the fidelity given in Eq. (3). Here for simplicity we write phase space variables and , and their differentials, as one dimensional. But Eq. (6) holds for arbitrary dimensions. For maps time is discrete so, for the reminder of this section, we define , with integer.

We use the DR to study the decay of the fidelity in the chaotic regime as follows. First we suppose that the system is very strongly chaotic, . This is essentially equivalent to assuming that the evolution is random without any correlation. In order to compute we partition the phase space into cells and consider that the probability of jumping from one cell to the other is uniform. It can then be shown that


where is the action difference on the cell at time . Taking the limit , we get


The absolute value of can then be written as




Then, if the dynamics is completely random, which is approximately the case for strongly chaotic systems, then the fidelity decays exponentially with a rate . As we shall see, this decay also explains the short time behavior regardless of because for short times the dynamics can always be supposed to be uncorrelated.

To unveil the intermediate time regime we consider the limit of random perturbation. In garciamaNJP (), using the DR it is shown that, for the baker map with a random perturbation, the fidelity can be written as a sum of products of transfer matrices




the digits define position and momentum


in symbolic dynamics (see e.g. Saraceno1994 ()). A point in phase space is then , and one step of the map consists in shifting the point to the right. The letter in the previous equations indicates a truncation size of the symbolic dynamics expansion. Defining the unit norm vector , Eq. (11) can be written in compact form as


The properties of the fidelity are then determined by the spectrum of the finite matrix . In particular the asymptotic decay is ruled by the largest eigenvalue (in modulus) of . Considering the special structure of the transfer matrices for the bakers map, it was shown that


where is the largest Lyapunov exponent of the baker map. This analytical result was further extended to more general types of maps garciamaNJP ().

We show numerically these two regimes for a family of perturbed cat maps dematos ()


For simplicity let . For integer, these maps are uniformly hyperbolic and for small enough the Lyapunov exponent is approximately given by

Figure 1: Fidelity as a function of discrete time using straight quantum calculation (points) and DR (black solid line) for the perturbed cat map, with (circles), corresponding to and (squares) corresponding to . Here and and . The dashed (gray) lines correspond to the Lyapunov regime, . The dotted (red) line marks the short time decay, , with obtained from Eq. (10).

For simplicity from now on we take . These maps can be written in the general form


and can be simply quantized as a product of two operators


Many well known quantisations of classical maps can be expressed in this way, e.g. the kicked Harper map Leboeuf1990 (), and the Chirikov standard map DimaScholar (). For the numerical examples we consider


as the perturbing “forces” of Eq. (18)

In Fig. 1 we show two things. On the one hand, the almost prefect agreement of the DR calculation of against the straightforward quantum result. On the other hand, it is shown that the two different exponential regimes can be distinguished. For the sake of clarity we show results for and which correspond to and respectively.

Figure 2: Quantum calculation of the fidelity for the perturbed cat map with and different values of the perturbation, . The dashed red lines show the small time behavior . The values of obtained from Eq. (10) are shown in the inset.

In Fig. 2 we show a detailed example illustrating how different the two regimes can be. There, five examples of for different values of are displayed. It can be clearly observed that the initial decay rate is given by . In the inset we show as a function of the perturbation and the points mark the decay rate value indicated by the dashed (red lines) in the main panel. After this short time decay there is a revival and then the fidelity again decays exponentially with a rate given by , except in the case where .

From the evidence of Figs. 1 and 2 a behaviour like


can be hinted. The decay given by the rate is explained by an initial lack of correlations. If the dynamics is strongly chaotic, then this is the decay that dominates throughout the evolution. This can be simulated by random evolution. In other cases there is a crossover from to the perturbation independent Lyapunov regime. In garciamaNJP () a random perturbation model was used to demonstrate this crossover, and also the crossover time could be inferred.

Iii Stadium billiard

In the previous sections we show that is a suitable quantity to characterise quantum irreversibility in d-dimensional systems. It does not depend on the initial conditions because it is the trace of the echo operator . Moreover, in the case of abstract maps, the DR can be used to show analytically that there is a Lyapunov regime that does not depend on the type of initial states, contrary to what happens in the case of the LE. Now we will study the behaviour of a similar quantity in a realistic system, a particle inside a billiard. In this system, the Hilbert space is infinite-dimensional and it is not possible to compute the trace of the echo operator. For this reason, we consider


where the initial density function , a microcanonical state located in an energy window that start in the ground state up to the -th excited state. We note that is related to a well known quantity in non-equilibrium statistical mechanics: the probability of doing work . This can be seen by considering a system with Hamiltonian that is in an initial equilibrium state . At the energy is measured and a quench is done. Then, the system evolves a time and another energy measurement is done. If and are the results of the measurements, the work done on the system is . Then, it is easy to show that the probability of work is the Fourier transform of


here we consider the absolute value of this quantity Eq. 25.

Figure 3: Fidelity for the stadium billiard with initial state and a basis of states, as a function of the rescaled time . . In the inset we show the position of the four Gaussians and diameter of the circles is (blue indicate positive Gaussian and red negative). The dashed grey line is .

We have studied for a particle in the desymmetrised stadium billiard with radius and straight line of length (see inset of Fig. 3). The perturbation is a smooth potential consisting in a series of four Gaussians,


with the perturbation strength, their widths, , , and the positions of the centres and , . The eigenstates of the unperturbed stadium () are obtained using the scaling method vergini95 (). The states of the perturbed system are obtained by diagonalising the perturbed Hamiltonian in the basis of the stadium billiard. We have used several number of unperturbed states to check the convergence of the results. We point out that the perturbations considered in this work and in garciamaNJP () affect the whole phase space. There has been some theoretical Goussev2008 (); Ares2009 (); Hohmann (); Kober () and experimental efforts to study the effect of local perturbations. In theses works they either find a crossover from a Fermi-golden-rule regime to an exponential regime with rate given by the so-called escape rate (given by a representative size of the perturbation) Goussev2008 (); Kober (), or an algebraic decay regime Hohmann (). But, even though they also consider the average fidelity amplitude, they do not find a Lyapunov regime.Therefore in this work we only consider global perturbations.

Figure 4: (top) Decay rate for the short time decay in the perturbative regime, . As expected from the Fremi-Golden rule has approximately quadratic dependence with (the dashed red line is a fit where . (bottom) Characteristic time for the Gaussian decay , for small . The slope of the linear fit is , so until saturation. Initial state is and the basis used has 3134 states, the width of the Gaussian perturbation is .

In Fig. 3 we show for the stadium billiard. The unperturbed evolution is given by the free dynamics inside the cavity () and the perturbed one is with the potential of Eq. (27). Results for several perturbation strengths are shown. The initial microcanonical state corresponds to the first 500 eigenstates of the unperturbed system. The calculations were done using the first 3135 eigenstates of the stadium billiard. The convergence of the results were tested using a bigger basis of up to 5600 states. For a smaller basis of 1300 states the results are also well behaved.

Let us first analyse the small time behaviour. For small , decays exponentially . In Fig. 4 (top) we show as a function of . As expected, we can clearly see that . Such behaviour is referred to as the Fermi-golden-rule regime Jacquod2001 (). For , the short time decay is approximately a Gaussian function (see Fig. 3). In Fig. 4 (bottom) we show as a function of . We can see that after a transient, there is a region where does not depend on the perturbation strength .

The behaviour of the fidelity can be related to the spread of the initial state in the perturbed basis. The natural quantity to study this type of localisation properties is the inverse participation ratio (IPR) Fyodorov1995 (); Jacquod1995 (); Georgeot1997 (); Berkovits1998 (); Flambaum1994 (); Flambaum2001 (); PhysRevE.91.010902 (). The IPR of a perturbed eigenstate in the unperturbed basis is


(throughout this section ). The inverse of this quantity – also called the participation number– gives an estimation of the number of unperturbed states contributing to a given perturbed state. In Fig. 5 we show the inverse of the IPR averaged over the first 800 states, as a function of . We can see that it has an approximately quadratic growth up to . After that the inverse of the IPR grows linearly in the interval . Finally the growth rate tends to a saturation at value which is much smaller than the basis size. This shows that the perturbed states remain localised in energy. As an example, in the inset of Fig. 5 we show the for the state corresponding to the level for (marked by a red dot on the main panel). The exponential decay of the tails is a manifestation of its localisation. This localisation is responsible the plateau of shown in Fig. 4 (bottom). We remark that the three perturbation regimes of the short time decay of [see Figs.  3 and  4] are manifested in the IPR behaviour [Fig.  5].

Figure 5: Inverse of the average IPR for the first 800 perturbed states as a function of . Inset: for the state for .

After the Gaussian decay shown in Fig. 3, we can see a second exponential decay with a rate given by the classical Lyapunov exponent of the stadium billiard. The Lyapunov exponent is , where corresponds to Benettin1978 (); Dellago1995 (). Here is the average velocity computed from the eigenenergies ( being the wavenumber of the eigenstate ) in the energy window considered. Evidently, the fidelity computed using an initial state and the fidelity obtained from the Haar measure for the quantum maps share the same decay behavior. A short time decay which depends on the characteristics and strength of the perturbation, followed by a Lyapunov regime depending on a classical feature.

Figure 6: for the stadium billiard with perturbation strength and width . The (red) circles correspond to the quantum evolution, with initial state and a basis of states. The solid line corresponds to DR calculation using initial conditions with energies chosen in the range corresponding to the first quantum energies. The dashed gray line corresponds to the Lyapunov decay.

We also compute using the semiclassical DR. This is a simple task due to the fact that unperturbed trajectories are geometrically obtained in the billiard and the perturbation only gives a phase as dictated by Eq. 4. To take into account the initial state, we compute the semiclassical using that the initial conditions are uniformly random inside the billiard, the same was assumed for the direction of the initial momentum. The modulus squared of the momenta are distributed as the eigenenergies of the unperturbed system. In Fig. 6 we show computed using the DR and the quantum results. We can see that the semiclassical approximation provides an accurate fitting of the quantum results. Moreover, the Lyapunov decay is clearly observed in the DR approximation.

Iv Conclusions

From the outset, the LE emerged as a viable quantity to characterise instability and irreversibility in quantum systems. A large amount of work was dedicated to describe the different regimes depending on the perturbation strength, but it received a real important boost when the Lyapunov regime was first described linking classical an quantum chaotic behaviour. However although it was shown to exist in many different systems, all the semiclassical and numerical calculations showed that the Lyapunov regime could only be observed if the initial states considered were “classically meaningful” Jacquod2009 () (typically coherent states). We found a solution to this problem considering the average fidelity amplitude, a basis independent quantity, which is closely related to the LE if one considers an average over the Haar measure (Eqs. (2) and (3)). Indeed, a recent work garciamaNJP (), briefly reviewed in Sect. II, shows analytically and numerically that for quantum maps on the torus the average fidelity amplitude decays as a double exponential, where the first decay rate is depends on the strength and type of perturbation, whereas the second decay rate is given by the classical Lyapunov regime.

But, although quantum maps have some generic properties of quantum chaos, they are not very generic systems themselves. So the challenge in this work was to take the analysis one step further and study the average fidelity amplitude in a more realistic system, paradigmatic of quantum chaos, like the stadium billiard. To overcome the problem of the infinite dimensional Hilbert space, where averaging over the Haar measure is unfeasible, we introduced an energy cutoff and considered the system to be initially in a state that has an equiprobable distribution over some energy window, in the same spirit of a microcanonical ensemble. In this way were able to recover the same behaviour of the fidelity amplitude as the one shown for quantum maps, in particular we could clearly observe the Lyapunov regime. Therefore we have made a step forward towards the settlement of this longstanding problem: we showed an example of a realistic system where the Lyapunov regime is observed, independently of the type of initial state, if the appropriate quantity – the fidelity amplitude – is considered.

Additionally, we have shown that these regimes were also manifested in the behaviour of the IPR, which is a very relevant quantity in the study of localisation and quantum chaos at the level of the structure of eigenstates. Finally, for completeness we studied the dynamics of the fidelity amplitude using the semiclassical DR approximation and showed a good agreement with the quantum results. This suggests that, since fidelity is also related to the Fourier transform of the work probability distribution after a quench silva08 (), further insight into this issue can be obtained by considering tools such as the semiclassical DR approximation.

The authors have received funding from CONICET (Grants No. PIP 114-20110100048 and No. PIP 11220080100728) ANPCyT (Grants No. PICT-2010-02483, No. PICT-2013-0621, No.PICT 2010-1556) and UBACyT.


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