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Journal of Physics A Mathematical and General

Quantum mechanical entanglements with chaotic dynamics

Atushi Tanaka

Published Sep 7, 1996• 18 citations
Publisher

Abstract

The effect of nonlinear dynamics on quantum correlations is studied by using the spin-kicked rotor, as a model system, which consists of a two-level system and a kicked rotor. Quantum-mechanical entangled states between the two-level system and the rotor form `tunnelling' doublets when the corresponding classical dynamics of the rotor is regular. In contrast to this, chaotic dynamics of the rotor makes chaotic entangled states after the destruction of these tunnelling doublets.

DOI
10.1088/0305-4470/29/17/020

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The effect of nonlinear dynamics on quantum correlations is studied by using the spin-kicked rotor, as a model system, which consists of a two-level system and a kicked rotor. Quantum-mechanical entangled states between the two-level system and the rotor form `tunnelling' doublets when the corresponding classical dynamics of the rotor is regular. In contrast to this, chaotic dynamics of the rotor makes chaotic entangled states after the destruction of these tunnelling doublets.
The Heisenberg spin-chain and the Quantum Kicked Rotor (QKR) are key models in condensed matter and quantum chaos respectively. The former, consisting of a line of qubits that interact with their neighbours through an exchange interaction, has recently received much attention in the context of quantum information and communication. In particular, it has been shown that a Heisenberg chain can be used as a channel for the transfer quantum states and the sharing of entanglement. The QKR, on the other hand, is a paradigm of quantum chaos. It is one of a valuable collection of quantum systems with a well de�ned classical limit that shows both regular and chaotic dynamics. This thesis investigates the dynamics of a Heisenberg chain subjected to periodic ±-kicks from a parabolic magnetic �eld. It will be shown that, when perturbed in this way, spin-chains can provide realizations of QKR systems. Using this connection, well-known features of the dynamics of kicked rotors can be used to manipulate the evolution of excitations and entanglement in the spin-chain. Conversely, results from the spin-chain analysis can inform our knowledge of the kicked rotor systems. The focus of the �rst part of this thesis is the dynamics of singly excited chains. In this case, the kicked Heisenberg chain is shown to be dynamically equivalent to a QKR. Similar correspondences can be found. Two examples, spin-chain realizations of the Double-Kicked Rotor and a double-well kicked rotor, are explored. Later sections are concerned with the dynamics of multiple excitations. These excitations interact - scattering or forming bound states of inverted spins. The kicked chain now corresponds to a system of coupled kicked rotors. It will be shown that by tuning the parameters it is possible to control the time-evolution of correlations in the chain.
Despite the periodic kicks, a linear kicked rotor (LKR) is an integrable and exactly solvable model in which the kinetic energy term is linear in momentum. It was recently shown that spatially interacting LKRs are also integrable, and results in dynamical localization in the corresponding quantum regime. Similar localized phases exist in other nonintegrable models such as the coupled relativistic kicked rotors. This work, using a two-body LKR, demonstrates two main results; first, it is shown that chaos can be induced in the integrable linear kicked rotor through interactions between the momenta of rotors. An analytical estimate of its Lyapunov exponent is obtained. Second, the quantum dynamics of this chaotic model, upon variation of kicking and interaction strengths, is shown to exhibit a variety of phases: classically induced localization, dynamical localization, subdiffusive, and diffusive phases. We point out the signatures of these phases from the perspective of entanglement production in this system. By defining an effective Hilbert space dimension, the entanglement growth rate can be understood using appropriate random matrix averages.
The kicked rotor provides a simple yet powerful model for introducing many of the central concepts of classical and quantum chaos. Despite its apparent simplicity, it exhibits rich dynamical behavior and has found applications across a wide range of fields, including atomic and optical physics, condensed matter physics, and emerging quantum technologies. This chapter begins by exploring foundational ideas using the kicked rotor as a unifying framework. We first discuss the transition from regular to chaotic motion in the classical system, and then introduce key quantum phenomena such as dynamical localization and quantum resonances. Special attention is devoted to the emergence of characteristic time scales and their role in the quantum-classical correspondence. To make these ideas more concrete, we also provide a brief overview of experimental realizations of the kicked rotor and its variants, illustrating how theoretical concepts are implemented in practice. In the second part of the chapter, we guide the reader toward more recent and advanced developments. Topics include near-resonant dynamics, topological features of kicked systems, the emergence of quantum dynamical phases inferred from classical transport properties, and extensions to non-Hermitian physics. We conclude with a discussion of open problems and future perspectives, outlining directions in which the kicked rotor continues to offer valuable insights.
The relation between the degree of entanglement and time scale of time-irreversible behavior is investigated for classically chaotic quantum coupled kicked rotors by comparing the entanglement entropy (EE) and the lifetime of correspondence with classical decay of correlation, which was recently introduced. Both increase on average drastically with a strong correlation when the strength of coupling between the kicked rotors exceeds a certain threshold. The EE shows an anomalously large fluctuation resembling a critical fluctuation around the threshold value of coupling strength where the entanglement sharply increases toward full entanglement. In this regime it can be shown that, although the correlation is hidden, EE and the lifetime of individual eigenfunctions also have a positive correlation that can be seen via another measure.
We analyze the interplay of chaos, entanglement, and decoherence in a system of qubits whose collective behavior is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement depends on the support of the initial state on regular versus chaotic Floquet eigenstates, whose phase-space distributions are concentrated on the corresponding regular or chaotic eigenstructures. We include the effect of decoherence via a realistic model and show that the signatures of chaos in the entanglement dynamics persist in the presence of decoherence. In addition, the classical chaos affects the decoherence rate itself.
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.