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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Microscopic foundation of a finite-temperature stochastic Schrödinger equation

Peter Goetsch, Robert Graham, Fritz Haake

Published Feb 1, 1996• 13 citations
Publisher

Abstract

We present a microscopic derivation of a stochastic Schrödinger equation for an oscillator valid at finite temperatures. Damping and noise arise from interaction with a heat bath. Vacuum and thermal noise drive the wavevector in rather different ways which we illuminate by treating the fate of an initial Schrödinger cat state. While the decoherence of the two macroscopically distinct components of such a state is thermally enhanced, the rise of a classically interpretable signal indicating `life' or `death' is not. Suitable averages over the noises could be obtained from a well known master equation and demand interpretation in terms of ensembles, but results contingent on single realizations of the noises may be related to single runs of experiments.

DOI
10.1088/1355-5111/8/1/012

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We present a microscopic derivation of a stochastic Schrödinger equation for an oscillator valid at finite temperatures. Damping and noise arise from interaction with a heat bath. Vacuum and thermal noise drive the wavevector in rather different ways which we illuminate by treating the fate of an initial Schrödinger cat state. While the decoherence of the two macroscopically distinct components of such a state is thermally enhanced, the rise of a classically interpretable signal indicating life' or death' is not. Suitable averages over the noises could be obtained from a well known master equation and demand interpretation in terms of ensembles, but results contingent on single realizations of the noises may be related to single runs of experiments.
The interrelationship between the non-Markovian stochastic Schr\"odinger equations and the corresponding non-Markovian master equations is investigated in the finite-temperature regimes. We show that the general finite-temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite-temperature reservoir in the rotating-wave approximation. The exact convolutionless master equation for the damped harmonic oscillator is obtained by averaging the quantum trajectories, relying upon no assumption of coupling strength or time scale. The master equation derived in this way automatically preserves the positivity, Hermiticity, and unity.
A form of the Schr\"odinger equation is given in which the time variable is replaced by a complex time-temperature variable and the interaction with the heat bath is represented by a stochastic interaction operator with Gaussian white noise. It is shown that at large times the ensemble averaged solution yields the equilibrium spatial probability density of quantum statistical mechanics. As equilibrium is approached, we also show how the master equation for a canonical ensemble can be derived from this form of the Schr\"odinger equation.
We study the behavior of two weakly coupled quantum systems, described by a separable density operator. One of them is a single oscillator, representing a microscopic system, while the other is a set of oscillators, which perform the role of a reservoir in thermal equilibrium. From the Liouville-Von Neumann equation for the reduced density operator, we devise the master equation that governs the evolution of the microscopic system, incorporating the effects of temperature via Thermofield Dynamics formalism and suitably redefining the vacuum of the macroscopic system. As applications, we study an oscillator interacting with a set of boson oscillators at finite temperature, usually employed in quantum mechanics to describe a Brownian particle; and investigate the behavior of an oscillator in the presence of a heat bath consisting of a set of fermion oscillators.
We propose physical interpretations, also valid for temperatures different from zero, for stochastic methods which have been developed recently to describe the evolution of a quantum system interacting with a reservoir. As opposed to the usual reduced density operator approach, which refers to ensemble averages, these methods deal with the dynamics of single realizations, and involve the solution of stochastic Schrödinger equations. These procedures have been shown to be completely equivalent to the master equation approach when ensemble averages are taken over many realizations. We show that these techniques are not only convenient mathematical tools for dissipative systems, but may actually correspond to concrete physical processes, for any temperature of the reservoir. We consider a mode of the electromagnetic field in a cavity interacting with a beam of two- or three-level atoms, the field mode playing the role of a small system and the atomic beam standing for a reservoir at finite temperature, the interaction between them being given by the Jaynes-Cummings model. We show that the evolution of the field states, under continuous monitoring of the state of the atoms which leave the cavity, can be described in terms of either the Monte Carlo wavefunction (quantum jump) method or a stochastic Schrödinger equation, depending on the system configuration. We also show that the Monte Carlo wavefunction approach leads, for finite temperatures, to localization into jumping Fock states, while the diffusion equation method leads to localization into states with a diffusing average photon number, which for sufficiently small temperatures are close approximations to mildly squeezed states. We prove analytically that, in the quantum jump situation, the system evolves in the mean towards a Fock state, even if an infinite number of photon-number amplitudes is present in the initial state.
A master equation for a quantum system coupled to a heat bath with stochastically fluctuating energy levels is derived by making use of the ensemble averaging and the averaging with respect to a stochastic process in the bath. Relaxation terms are determined in the Born approximation with respect to the system-bath interaction and the damping parameters related to a relaxation kernel are specified. In parallel with the spectral strength of the bath, the damping parameters determine the transient times for the Markovian description creating the physical origin of the slippage [A. Suarez, R. Sibey, and I. Oppenheim, J. Chem. Phys. 97, 5101 (1992)]. The influence of energy fluctuations of the bath is analyzed for a two-level system including the solution of the corresponding non-Markovian equation for the level population difference. The conditions for the formation of Boltzmann's thermal ratio between steady-state populations are evaluated as well.
The methodology of stochastic description for dissipation, a generic scheme to decouple the interaction between two subsystems, is applied to the study of dissipative dynamics in quantum optics. It is shown that the influence of the coupled thermal or vacuum field on the quantum mode can be exactly represented by the induced stochastic fields. The quantum mode thereby satisfies a stochastic differential equation and dissipation effect due to the coupling with the environment is obtained through statistical averaging. Within the framework of stochastic description, it is demonstrated how to derive the master equation for a single optical mode interacting with the bosonic bath. A numerical algorithm for solving the master equation in which the coefficients are determined by a set of integral equations is discussed and a comparison with the known results is displayed. The derivation of the master equation for the spontaneous decay of two-state atoms in the vacuum is also presented.