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Journal of Physics A Mathematical and General

Quantum dynamics of a two-level system coupled to a bosonic degree of freedom in terms of Wigner phase space distributions

N Klenner, M Doucha, Jasmin Weis

Published Dec 21, 1986• 4 citations
Publisher

Abstract

The quantum dynamics of the Hamiltonian of a two-level system coupled to a boson mode is formulated in terms of the Wigner matrix. The trace of the Wigner matrix gives complete information for the bosonic degree of freedom. The time evolution and the long time average of the trace of the Wigner matrix is calculated numerically for small perturbations and in resonance. Some aspects of 'quantum chaos' of the two-level system are discussed.

DOI
10.1088/0305-4470/19/18/028

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The quantum dynamics of the Hamiltonian of a two-level system coupled to a boson mode is formulated in terms of the Wigner matrix. The trace of the Wigner matrix gives complete information for the bosonic degree of freedom. The time evolution and the long time average of the trace of the Wigner matrix is calculated numerically for small perturbations and in resonance. Some aspects of 'quantum chaos' of the two-level system are discussed.
The Hamiltonian of a two-level system coupled to a boson mode is transformed to Bargmann's Hilbert space of analytical functions. The quantum dynamics is calculated with a suitable choice of a complete set of orthonormal functions in Bargmann space. A semiclassical approximation which is formulated as a dynamical Hamiltonian system is compared with the quantum dynamics. For the parameters under consideration, semiclassical dynamics is not at all appropriate to approximate the quantum dynamics. These parameters are suited to view the system as a two-site model of small-polaron dynamics.
Abstract In the framework of a novel dissipative scheme, we have investigated the quantum dynamics of an oscillating system interacting with two reservoirs with different temperatures trough different time-dependent coupling functions. The reduced density matrix, quantum optical characteristic functions, and (quasi) distribution functions like Husimi, Glauber–Sudarshan and Wigner functions on the phase space of the oscillator are obtained. The problem has been generalized to the case where the oscillator is interacting with n distinctive reservoirs, and a quantum current and an effective reservoir is introduced. Finally, the quantum dynamics of a two-level system interacting with two reservoirs has been investigated, and the exact reduced density matrix is obtained.
We study the dynamics of a three-mode bosonic system with mode-changing interactions. For large mode occupations the short-time dynamics is well described by classical mean-field equations allowing us to study chaotic dynamics in the classical system and its signatures in the corresponding quantum dynamics. By introducing a symmetry-breaking term we tune the classical dynamics from integrable to strongly chaotic, which we demonstrate by calculating Poincar\'e sections and Lyapunov exponents. The corresponding quantum system features level statistics that change from Poissonian in the integrable to Wigner-Dyson in the chaotic case. We investigate the behavior of out-of-time-ordered correlators (OTOCs), specifically the squared commutator, for initial states located in regular and chaotic regions of the classical mixed phase space and find marked differences between the two cases. The short-time behavior is well captured by semi-classical truncated Wigner simulations directly relating these features to properties of the underlying classical mean-field dynamics. We discuss a possible experimental realization of this model system in a Bose-Einstein condensate of rubidium atoms, which allows reversing the sign of the Hamiltonian required for measuring OTOCs.
The classical limit n>>0 is investigated for a two-level system coupled to a boson mode. The usual formulations are found to neglect a shift lambda 2 of the spectrum where lambda is the strength of the coupling. A scaling law for the highly excited energy levels is derived and tested numerically. An insight is obtained into the characteristic properties of the energy level distribution which is of interest in connection with possible 'quantum chaotic' behaviour. The interpretation of the asymptotically effective Hamiltonian as a one-dimensional tight-binding model for the two-band Stark ladder is discussed.
Based on the experimental evidence that impurities contribute to the dissipation properties of solid-state open quantum systems, we provide here a description in terms of nonlinear quantum Langevin equations of the role played by two-level systems in the dynamics of a bosonic degree of freedom. Our starting point is represented by the description of the system-environment coupling in terms of coupling to two separate reservoirs, modeling the interaction with external bosonic modes and two-level systems, respectively. Furthermore, we show how this model represents a specific example of a class of open quantum systems that can be described by nonlinear quantum Langevin equations. Our analysis offers a potential explanation of the parametric effects recently observed in circuit-QED cavity optomechanics experiments.
Nonequilibrium quantum dynamics of a system comprising a (pseudo) spin-\textonehalf{} object coupled to a boson degree of freedom is studied. It is shown that the time evolution of the system is described by a set of elementary scattering processes. Through these processes the system approaches an equilibrium in which the spin is in a mixed state that on average maximizes its entropy. However, the irregular behavior of the system is unrelated to "quantum chaos" as the Hamiltonian is integrable.