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Journal of Physics A Mathematical and General

Towards a satisfactory formulation of the quantum Langevin equation

Hiroshi Hasegawa, John R. Klauder, M. Lakshmanan

Published Feb 21, 1985• 7 citations
Publisher

Abstract

The quantum mechanical Langevin equation for a damped harmonic oscillator with operator-valued-noise, da t /dt=(-i omega - gamma )A t + lambda w(t), is reinvestigated. By assuming the canonical commutation relation at initial time, (A 0 , A 0 *)=1, and a Hermitian operator for the Gaussian quantum noise w(t), it is shown that Streater's procedure to satisfy (A t ,A t *)=1 for all later time yields an improved result: the noise operator w(t) can be determined universally and normalised such that, by choosing lambda 2 =2 gamma ( beta h(cross) omega ) -1 , the (symmetrised) power spectrum of w(t) is equal to the physically significant form 1 / 2 beta h(cross) omega coth( 1 / 2 beta h(cross) omega ), where beta is the inverse temperature of the heat bath, thus ensuring the beta -KMS correlation functions for any pair of operators.

DOI
10.1088/0305-4470/18/3/005

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The quantum mechanical Langevin equation for a damped harmonic oscillator with operator-valued-noise, da t /dt=(-i omega - gamma )A t + lambda w(t), is reinvestigated. By assuming the canonical commutation relation at initial time, (A 0 , A 0 *)=1, and a Hermitian operator for the Gaussian quantum noise w(t), it is shown that Streater's procedure to satisfy (A t ,A t *)=1 for all later time yields an improved result: the noise operator w(t) can be determined universally and normalised such that, by choosing lambda 2 =2 gamma ( beta h(cross) omega ) -1 , the (symmetrised) power spectrum of w(t) is equal to the physically significant form 1 / 2 beta h(cross) omega coth( 1 / 2 beta h(cross) omega ), where beta is the inverse temperature of the heat bath, thus ensuring the beta -KMS correlation functions for any pair of operators.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
The heat-bath frequency distribution, necessary to maintain the canonical commutation relations for all time for a quantum mechanical oscillator with non-Markovian features like memory-dependent damping, is shown to satisfy certain constraint relations. An algorithm is given to explicitly find the heat-bath frequency distribution in terms of a series expansion for all processes where the timescale for the non-Markovian memory kernel is much smaller than the inverse of the strength of damping term. In the non-Markovian case, the heat-bath distribution function exhibits dependence on the system characteristics. The KS periodicity conditions on the system Green functions as the system approaches equilibrium are established for the present case.
We develop a formulation of quantum damping theory in which the explicit nature of inputs from a heat bath, and of outputs into it, is taken into account. Quantum Langevin equations are developed, in which the Langevin forces are the field operators corresponding to the input modes. Time-reversed equations exist in which the Langevin forces are the output modes, and the sign of damping is reversed. Causality and boundary conditions relating inputs to system variables are developed. The concept of ``quantum white noise'' is formulated, and the formal relationship between quantum Langevin equations and quantum stochastic differential equations (SDE's) is established. In analogy to the classical formulation, there are two kinds of SDE's: the Ito and the Stratonovich forms. Rules are developed for converting from one to the other. These rules depend on the nature of the quantum white noise, which may be squeezed. The SDE's developed are shown to be exactly equivalent to quantum master equations, and rules are developed for computing multitime-ordered correlation functions with use of the appropriate master equation. With use of the causality and boundary conditions, the relationship between correlation functions of the output and those of the system and the input is developed. It is possible to calculate what kind of output statistics result, provided that one knows the input statistics and provided that one can compute the system correlation functions.
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a nonvanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact nonperturbative in the initial state.