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The Journal of Chemical Physics

Fokker–Planck quantum master equation for mixed quantum–semiclassical dynamics

Jinjin Ding, Yao Wang, Hou-Dao Zhang, Rui‐Xue Xu, Xiao Hong Zheng, YiJing Yan

Published Jan 10, 2017• 12 citations
Publisher

Abstract

We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.

DOI
10.1063/1.4973610

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We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.
The continued-fraction method of solving classical Fokker–Planck equations has been adapted to tackle quantum master equations of the Caldeira–Leggett type. This was done taking advantage of the phase-space (Wigner) representation of the quantum density matrix. The approach differs from those in which some continued-fraction expression is found for a certain quantity, in that the full solution of the master equation is obtained by continued-fraction methods. This allows us to study in detail the effects of the environment (fluctuations and dissipation) on several classes of nonlinear quantum systems. We apply the method to the canonical problem of quantum Brownian motion in periodic potentials both for cosine and ratchet potentials (lacking inversion symmetry).
Measurement and feedback control are essential features of quantum science, with applications ranging from quantum technology protocols to information-to-work conversion in quantum thermodynamics. Theoretical descriptions of feedback control are typically given in terms of stochastic equations requiring numerical solutions, or are limited to linear feedback protocols. Here we present a formalism for continuous quantum measurement and feedback, both linear and nonlinear. Our main result is a quantum Fokker-Planck master equation describing the joint dynamics of a quantum system and a detector with finite bandwidth. For fast measurements, we derive a Markovian master equation for the system alone, amenable to analytical treatment. We illustrate our formalism by investigating two basic information engines, one quantum and one classical.
In this work, we explore the range of validity of the semiclassical approximation of a quantum master equation designed to describe the c c ¯ dynamics in a quark-gluon plasma at various temperatures, in the quantum Brownian regime. We perform a comparative study of various properties, e.g., the charmonia yield, of the Wigner density obtained both with the Lindblad equation and with the associated semiclassical Fokker-Planck equation. The semiclassical description is found to reproduce with a remarkable accuracy the results obtained through the full quantum description. We show that, to a large extent, this can be attributed to the nonunitary components of the dynamics that result from the contact of the c c ¯ subsystem with the thermal quark-gluon bath, leading to a rapid classicalization of the subsystem.
A maximum entropy approach to time-dependent phenomena is employed to construct a quantum stochastic theory for adiabatic processes in condensed phases. This theory is illustrated by considering the relative motion of a two-particle system interacting with a heat bath. We present a fully quantum mechanical phase space master equation, which is new. In the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion, our master equation reduces to a simpler form, and the principle of detailed balance emerges. This limit is shown to be equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2. The simpler master equation is utilized in the development of a quantum mechanical nonlinear Fokker–Planck equation. For cases in which correlations involving spatial and momentum fluxes are short ranged, the nonlinear Fokker–Planck equation reduces to the usual linear form, with ’’collision’’ and ’’streaming’’ terms that contain quantum corrections to all orders in ℏ. The relationship between the classical limit of the present theory and Brownian motion theory based on the Chapman–Kolmogorov equation is established.
We model a quantum system coupled to an environment of damped harmonic oscillators by following the approach of Caldeira-Leggett and adopting the Caldirola-Kanai Lagrangian for the bath oscillators. In deriving the master equation of the quantum system of interest (a particle in a general potential), we show that the potential is modified non-trivially by a new inverted harmonic oscillator term, induced by the damping of the bath oscillators. We analyze numerically the case of a particle in a double-well potential, and find that this modification changes both the rate of decoherence at short times and the well-transfer probability at longer times. We also identify a simple rescaling condition that keeps the potential fixed despite changes in the environmental damping. Here, the increase of environmental damping leads to a slowing of decoherence.