# Statistical properties of a modified standard map in quantum and classical regimes

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## Abstract

We present a model—a modified standard map. This model has interesting properties that allow quantum–classical correspondences to be studied. For some range of parameters in the classical phase space of this model, there exist large accelerator modes. We can create a family of maps that have large accelerator modes.

## Keywords

Low-dimensional chaos Accelerator modes Quantum chaos Husimi function## 1 Introduction

*P*is the momentum,

*I*is moment of inertia,

*t*is time,

*T*is the period of the force, and

*V*(

*Q*) is a periodic function-external potential. We can write the Hamilton equations for the dynamics of our model as

*f*(

*Q*) and

*V*(

*Q*) are related by the standard formula:

*Q*according to the formula \(V(Q)=\cos {Q}\).

For the models considered in this article, the so-called accelerating modes (AMs) emerge [4, 6, 7] in phase space for a wide range of parameters. There are areas of the phase space where momentum increases (or decreases) approximately linearly in time. In the case of SM, there are two kinds of AM. However, areas of AM in SM are quite small. Using our modifications, we can create AMs that are several times larger and parametrically control their areas. We can also propose modification when only one kind of AMs exist with quite large areas. Simultaneously, trajectories of the area outside the accelerator modes can “stick” to their boundaries and follow a linear evolution for a very long time. This phenomenon leads to statistically interesting results in the form of power-like tails in the momentum distribution function [8].

## 2 Modified standard map

*n*th step, \(P_{n}\) represents momentum in the

*n*th step, and

*f*(

*Q*) is a piecewise linear continuous periodic function:

With this modification, it is much easier to investigate the quantum counterpart of the classical map, especially effects like tunneling from the AM or the stickiness of its boundary (see Fig. 1). Using a proper choice of parameters, the area of AM in the phase space is relatively large (see Fig. 2). It should be noted that MSM as well as SMs is obtained from Hamilton equations in accordance with the assumption that the time dependence of the acting force is a periodic Dirac \(\delta \)-function; thus, we are considering the so-called kicked model. This also allows the exact form of the quantum map to be obtained. However, in the case of continuous and smooth time dependence in the quantum description, one can apply the well-known Floquet Hamiltonian.

## 3 Quantum map

*T*. Due to the form of the Hamiltonian, the evolution operator from time \(t=0\) to \( t=T\) may be written as a product of two simple operators:

*Q*; therefore, we have discrete momentum representation with \(P_m=m\hbar , \; m=\ldots ,-2,-1,0,1,2,\ldots \). In numerical computations, we take into account only a finite number of

*P*-values say \(P_m=-N\hbar ,\ldots ,(N-1)\hbar \) and consequently a finite number of discrete

*Q*-values: \(Q_j=2\pi j/(2N),\; j=0,1,2,\ldots ,2N-1\). The transitions between both representations are given by the discrete Fourier transform and its inversion:

## 4 Classical and quantum correspondence

*D*with a complex parameter. We can say that the function \(\psi _{\varphi }(Q)\) is a product of Gaussian and linear chirp. The formula above can also be used in the cylinder case if the Gaussian function is sufficiently narrow, i.e., if its width is much smaller than \(2\pi \).

In Fig. 4, we present the Husimi function for the initial state (rotated Gaussian) localized in AM in the cell described by the Cartesian product: \([0, 2\pi ) \times [-\pi , \pi )\) and five of its iterations for \(n=1,2,10,100,1000\). We can observe that the Husimi functions of these iterations are localized in AMs in cells of phase space described by \([0, 2\pi ]\times [(2n-1)\pi , (2n+1)\pi ]\), where *n* is the iteration number. Moreover, the form of the Husimi function changes slightly. However, there exists tunneling from AM to the remaining phase space. In order to investigate the properties of this tunneling, we calculate the probability \(W(n) = \sum _{i} |\psi (P_{i})|^{2}\), where \(P_{i} \in [(2n - 1)\pi , (2n + 1)\pi ]\) at each iteration of the initial state. When proceeding in this way, however, we are calculating rather the escape rate from the cell with AM than from AM itself.

The results of numerical calculations for the chosen value of *D* and \(\hbar \) are presented in Figs. 5 and 6. In the first one, we present Wigner functions of the initial state (\(\hbar =2\pi /24, D=0.35803\)) (left side of the figure) and of the same state after 4000 iterations in the cell with a number \(n = 4000\) (right side of the figure). We can see that the state is still well localized in AM as \(W(4000)\approx 0.996\).

*D*, we obtain a good fit to the straight line for results with \(n>5000\). A value of \(\beta =9.44 \times 10^{-7}\) was estimated for iterations with \(n>30000\). Such a calculated value of the \(\beta \) parameter is practically the same as in the previous figure (\(D=0.358\)).

*P*-values with constant velocity. At the same time, it possesses a long tail for positive

*P*-values with the front which moves with constant velocity as well (the front jumps by \(2\pi \) at every iteration). In order to verify the nature of this tail, we consider a state after a large number of iterations as is presented in Fig. 9 on the left side. The value of the wave function amplitude is drawn in two scales, which can be seen on the right side of this figure. The upper figure presents a linear fit to the results in log-log scale that has a high coefficient of determination. It is a strong numerical argument that the asymptotics of the fragment of \(|\varPsi (P)|\) under consideration has an inverse power character as in the classical case (see Fig. 2 in [5]).

Taking into account the results of our numerical calculations, we can state that basic properties of quantum distribution retain the properties of the classical model (despite quite a large Planck constant). The AMs still have an influence on quantum evolution as in the classical model. Both in the classical picture and the quantum one, densities have a pronounced maximum for some values of momentum while for larger values of momentum they have inverse power asymptotics.

## 5 Further modifications of the standard map

*f*(

*Q*) with a four-part piecewise linear continuous periodic function for which \(\langle P_{n+1}-P_n\rangle =0\). For certain parameter values, we can observe the emergence of two kinds of AM in phase space. In one type of AM, the P value grows approximately linearly during successive iterations. In the other type of AM, the movement goes in the opposite direction. Both types of AM are presented in Fig. 10.

We can also modify function (5) requiring that it be differentiable, i.e., that it be a function of class \(C^n\). It can be realized by joining its adjacent linear parts by a polynomial of order \(n+1\) on some small intervals of length *D*. All coefficients of the linear and polynomial parts can be uniquely determined by periodicity, continuity of functions, and their derivatives. The proper choice of model parameters allows us to obtain AM in the phase space (see Fig. 11). In the limit \(D \rightarrow 0\), this function becomes *f*(*Q*) defined by formula (5).

## 6 Conclusions

We have presented a model that is a modification of the SM. The kicking function is a smooth piecewise linear function parametrized by one parameter (besides position of three intervals). For properly chosen parameters, there exist large AMs. This feature simplifies the numerical analysis of the quantum mapping. Our analysis shows that AM and its stickiness property are retained in the quantum picture. The quantum evolution of states localized in AM follows classical evolution, i.e., the quantum map translates the localized state in AM to the next AM, practically without changing its Husimi function, with the exception of very small tunneling to the rest of phase space. Quantum probability density in momentum representation preserves many essential features of classical momentum density. Numerical results show that states which are localized in the chaotic part of phase space delocalize under the influence of quantum mapping in a manner similar to the classical case. For a sufficiently long time (a number of a map iterations), density has an inverse power asymptotic for large momentum with a characteristic front moving with constant velocity. This model is relatively easy to analyze when instead of kick, i.e., \(\delta \)-time dependence, we use a continuous periodic function. In such a case, one has to apply Floquet theory.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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