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arXiv:2407.08780 · Physical review. E

Effects of stickiness on the quantum states of strongly chaotic open systems

Miguel A. Prado Reynoso, Edson M. Signor, Sandra D. Prado, Léa Ferreira dos Santos

Published Dec 27, 2024• 3 citations
arXivHTMLPDF

Abstract

Focusing on the standard map of the kicked rotor, we investigate the effects of classical stickiness (orbits temporarily confined to a region of a chaotic phase space) on an open quantum system. The regions of stickiness that we identify emerge in the strongly chaotic map. By scanning the system's phase space with a leak, we show that stickiness reduces the degree of delocalization of the quantum states and the decay rate of generic initial states. We find an excellent correspondence between the classical dwell time and finite-time Lyapunov exponents with the quantum dwell time and Wehrl entropies of the quantum states. Knowledge of the structure of the classically chaotic trajectories can thus be used to determine the best placement of leaks to enhance or decrease the degree of delocalization of quantum states and to modify the quantum dynamics of open systems.

DOI
10.1103/physreve.110.l062201

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Focusing on the standard map of the kicked rotor, we investigate the effects of classical stickiness (orbits temporarily confined to a region of a chaotic phase space) on an open quantum system. The regions of stickiness that we identify emerge in the strongly chaotic map. By scanning the system's phase space with a leak, we show that stickiness reduces the degree of delocalization of the quantum states and the decay rate of generic initial states. We find an excellent correspondence between the classical dwell time and finite-time Lyapunov exponents with the quantum dwell time and Wehrl entropies of the quantum states. Knowledge of the structure of the classically chaotic trajectories can thus be used to determine the best placement of leaks to enhance or decrease the degree of delocalization of quantum states and to modify the quantum dynamics of open systems.
In this work, we study quantum chaos by focusing on the evolution of initially close states in the dynamics of the Quantum Kicked Rotor (QKR). We propose a novel measure, the Quantum Lyapunov Exponent (QLE), to quantify the degree of chaos in this quantum system, analogous to its classical counterpart. We begin by modeling the momentum space and then the QLE is computed through analyzing the fidelity between evolving states, offering insights into the quantum chaotic behavior. Furthermore, we extend our investigations to various initial states: localized, uniform, spreading, contracting and oscillating in momentum space. Our results unveil a diverse range of dynamical behaviors, highlighting the complex nature of quantum chaos. Finally, we propose an innovative optimization framework to represent a complex state as a superposition of the aforementioned states, which has potential implications for visualizing and understanding the dynamics of multifaceted quantum systems.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
In this work, we study a family of fully chaotic billiards that exhibits only rotational symmetries, whose geometry is based on the C3C_3C3​ symmetry system proposed by Leyvraz, Schmit, and Seligman~(LSS) in 1996. Quantum spectral analyses are performed on billiards with symmetry CnC_nCn​~(the billiard repeats itself under rotations of 2π/n2π/n2π/n), where nnn is the symmetry parameter. In these systems, there are subspaces of singlets~(invariant under time reversal) and doublets~(not invariant under time reversal). For the LSS billiard, it has been established both numerically and experimentally that the corresponding subspectra follow the Gaussian Orthogonal Ensemble~(GOE) statistics for singlets and the Gaussian Unitary Ensemble~(GUE) statistics for doublets. From a classical perspective, the shapes of these billiards allow certain subregions of phase space to be visited more frequently by chaotic trajectories, a phenomenon known as stickiness. We investigate the relationship between the fraction of sticky regions in classical phase space and the deviations of the energy subspectra from GOE and GUE statistics. Our results suggest the existence of correlations between the energy distributions associated with different symmetry subspaces. In addition, we discuss aspects related to the superposition of the different energy subspectra.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
The de Broglie-Bohm (BB) approach to quantum mechanics gives trajectories similar to classical trajectories except that they are also determined by a quantum potential. The quantum potential is a "fictitious potential" in the sense that it is part of the quantum kinetic energy. We use quantum trajectories to treat quantum chaos in a manner similar to classical chaos. For the kicked rotor, which is a bounded system, we use the Benettin et al. method to calculate both classical and quantum Lyapunov exponents as a function of control parameter K and find chaos in both cases. Within the chaotic sea we find in both cases nonchaotic stability regions for K equal to multiples of π. For even multiples of π the stability regions are associated with classical accelerator mode islands and for odd multiples of π they are associated with new oscillator modes. We examine the structure of these regions. Momentum diffusion of the quantum kicked rotor is studied with both BB and standard quantum mechanics (SQM). A general analytical expression is given for the momentum diffusion at quantum resonance of both BB and SQM. We obtain agreement between the two approaches in numerical experiments. For the case of nonresonance the quantum potential is not zero and must be included as part of the quantum kinetic energy for agreement. The numerical data for momentum diffusion of classical kicked rotor is well fit by a power law DNβ in the number of kicks N. In the anomalous momentum diffusion regions due to accelerator modes the exponent β(K) is slightly less than quadratic, except for a slight dip, in agreement with an upper bound (K2/2)N2. The corresponding coefficient D(K) in these regions has three distinct sections, most likely due to accelerator modes with period greater than one. We also show that the local Lyapunov exponent of the classical kicked rotor has a plateau for a duration that depends on the initial separation and then decreases asymptotically as O(t-1lnt), where t is the time. This behavior is consistent with an upper bound that is determined analytically.