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Studies of Classically Chaotic Quantum Systems within the Pseudo-Probablilty Formalism

Roberto Roncaglia

Published Aug 1, 1992• 0 citations
OpenAlex

Abstract

The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.

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The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.
Abstract Herein we establish a relation between quantum irreversibility and the chaotic semi-classical solutions for a spin-boson Hamiltonian system. We obtain quantum averages by numerically integrating the appropriate Liouville-Von Neumann equations of motion and find these averages to be less erratic than the corresponding chaotic semi-classical trajectories. However, the quantum averages are shown to be dissipative as measured by the entropy of the spin subsystem and to suppress the phenomenon of "revivals".
This dissertation sets out to examine some fundamental open questions in quantum physics regarding quantum-classical correspondence in regular versus chaotic systems. Specifically, we study these questions using approaches in quantum information science in an experimentally realized textbook model of quantum chaos - the quantum kicked top (QKT). \n \nThe effect of classical chaos on the generation of entanglement in spin systems has been a field of active research for a couple of decades. Whether high entanglement in these systems is a hallmark of chaos or not remains a widely debated topic. We explain the connection between entanglement and chaos in spin systems and resolve previous conflicting results. The previous studies have mostly drawn conclusions from numerical work on a few initial states in regular versus chaotic regions. We instead focus on stable and unstable periodic orbits because chaos emerges around unstable periodic orbits. We first propose a new set of criteria for determining whether quantum evolution will correspond to the classical trajectory in a localized manner at stable periodic orbits in periodically driven systems. These criteria can be used to calculate the quantum numbers that will lead to quantum-classical correspondence even in a deep quantum regime, and thus to quantify the well-known Bohr correspondence principle. Next, we analytically show a direct connection between entanglement generation and a measure of delocalization of a quantum state in spin systems. More concretely, we describe a method to calculate an upper bound on entanglement generation in any bipartition of spin systems, where the upper bound is a function of trace distance between the evolved state and the most localized classical-like separable states. This method along with our criteria for localized evolution enables us to explain the behaviour of entanglement in both deep quantum and semiclassical regimes for regular as well as chaotic regions. Hence, our analysis resolves the long-standing debates regarding the connection between classical chaos and quantum entanglement in deep quantum and semiclassical regimes. \n \nIn addition to the study of entanglement, we perform the first study of nonlocality, and the effect of chaos on its generation in the QKT. Since nonlocality and entanglement are inequivalent quantum resources, the effect of chaos on nonlocality merits an explicit study. Violations of Bell inequalities in the presence of spacelike separation among the subsystems imply nonlocality - meaning nonlocal correlations between subsystems of the total spin system. We show that the QKT evolution can lead to states that violate multiqubit Bell inequalities and hence provides a deterministic method to prepare nonlocal quantum states. Our numerical results suggest a correlation between delocalized evolution of a pure quantum state and generation of nonlocality in the quantum state. We further demonstrate that dynamical tunnelling - a classically forbidden phenomenon - in the QKT leads to the generation of Greenberger-Horne-Zeilinger (GHZ)-like states for even numbers of qubits. We analytically prove that these states are maximally nonlocal. On the other hand, we numerically show that any reduced state of the QKT obtained by tracing out a subsystem of the total spin system does not violate Bell inequalities. We provide an analytical explanation of the numerical results for 2−2-2−qubit reduced states by formulating and proving two general theorems regarding 2−2-2−qubit Bell inequalities. These theorems imply that any 2−2-2−qubit mixed state having a symmetric extension or symmetric purification cannot violate the Clauser-Horne-Shimony-Holt inequality. This highlights fundamental connections between two important and distinct concepts in quantum information science - Bell inequalities and symmetric extension of quantum states. \n \nApart from providing deeper insights into the fundamental questions of quantum-classical correspondence and new approaches to analyze quantum chaos, the methods developed in this thesis can be used to design quantum systems that can efficiently generate entanglement and nonlocality. Thus, our results could have interesting applications in quantum computing and quantum information science.
We describe a quantum many-body system undergoing multiple resonant tunneling which exhibits chaotic behavior in numerical simulations of a mean field approximation. This phenomenon, which has no counterpart in the classical limit, is due to effective nonlinearities in the tunneling process and can be observed in principle within a heterostructure. 03.65.-w, 05.45.+b, 73.40.Gk Typeset using REVT E X In current studies the problem of quantum chaos is usually posed in the following terms. We are given a system which exhibits chaotic behavior classically: does this property survive quantization? Typically one studies the spectrum of the quantum mechanical Hamiltonian operator. However spectral differences at the quantum level between classically integrable and chaotic systems do not seem to imply dramatic differences in the time evolution of the quantum mechanical observables. Relics of classical chaos exist in low dimensional systems in short wavelength or large quantum numbers limits ...
This paper discusses the problem of finding and defining chaos in quantum mechanics. While chaotic time evolution appears to be ubiquitous in classical mechanics, it is apparently absent in quantum mechanics in part because for a bound, isolated quantum system, the evolution of its state is multiply periodic. This has led a number of investigators to search for semiclassical signatures of chaos. Here I am concerned with the status of semiclassical mechanics as a distinct third theory of the asymptotic domain between classical and quantum mechanics. I discuss in some detail the meaning of such crucial locutions as the "classical counterpart to a quantum system" and a quantum system's "underlying classical motion". A proper elucidation of these concepts requires a semiclassical association between phase space surfaces and wave-functions. This significance of this association is discussed in some detail.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.