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arXiv:0809.4585 · Physical Review E

Nonperturbative approach to quantum Brownian motion

Subhasis Sinha, P. A. Sreeram

Published May 13, 2009• 5 citations
arXivHTMLPDF

Abstract

Starting from the Caldeira-Leggett model, we derive the equation describing the quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms. This nonperturbative approach yields explicit analytical expressions for the temperature dependence of the diffusion constants. At high temperatures, additional momentum diffusion terms are suppressed and classical Langevin equation can be recovered and at the same time positivity of the density matrix is satisfied. At low temperatures, the diffusion constants have a finite positive value. However, below a certain critical temperature, the master equation does not satisfy the positivity condition as proposed by Dekker.

DOI
10.1103/physreve.79.051111

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Starting from the Caldeira-Leggett model, we derive the equation describing the quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms. This nonperturbative approach yields explicit analytical expressions for the temperature dependence of the diffusion constants. At high temperatures, additional momentum diffusion terms are suppressed and classical Langevin equation can be recovered and at the same time positivity of the density matrix is satisfied. At low temperatures, the diffusion constants have a finite positive value. However, below a certain critical temperature, the master equation does not satisfy the positivity condition as proposed by Dekker.
Caldeira and Leggett (CL), in a seminal paper, derived a master equation describing Markovian quantum Brownian motion. Such an equation suffered from not being completely positive, and many efforts have been made to solve this issue. We show that when a careful mathematical analysis is performed, the model considered by CL leads to a nondissipative master equation. We argue that the correct way to understand the master equation derived in the CL regime is to consider it non-Markovian. Moreover, we show that if one wants to provide a microscopic description of quantum Brownian motion with the CL model, one always needs to consider a non-Markovian dynamics. We conclude that dissipation is a genuinely non-Markovian feature.
We generalize the classical theory of Brownian motion so as to reckon with non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a free particle and a harmonic oscillator, it is shown that such non-Markovian effects account for the differentiability of the Brownian trajectories as well as the breakdown of the energy equipartition of statistical mechanics at short times in some physical situations. This non-Markovian approach is also extended to look at anomalous diffusion. Next, we bring in the dynamical-quantization method for investigating open quantum systems, which does consist in quantizing the classical Brownian motion starting directly from our non-Markovian Klein-Kramers and Smoluchowski equations, without alluding to any model Hamiltonian. Accordingly, quantizing our non-Markovian Klein-Kramers in phase space gives rise to a non-Markovian quantum master equation in configuration space, whereas quantizing our non-Markovian Smoluchowski equation in configuration space leads to a non-Markovian quantum Smoluchowski equation in phase space. In addition, it is worth noticing that non-Markovian quantum Brownian motion takes place in presence of a generic environment (e.g. a non-thermal quantum fluid). As far as the special case of a heat bath comprising of quantum harmonic oscillators is concerned, a non-Markovian Caldeira-Leggett master equation and a thermal quantum Smoluchowski equation are derived and extended to bosonic and fermionic heat baths valid for all temperatures.
Caldeira and Leggett (CL) in a seminal paper derived a master equation describing Markovian Quantum Brownian motion. Such an equation suffered of not being completely positive, and many efforts have been made to solve this issue. We show that, when a careful mathematical analysis is performed, the model considered by CL leads to a non dissipative master equation. We argue that the correct way to understand the master equation derived in the CL regime is to consider it non-Markovian. Moreover, we show that if one wants to provide a microscopic description of Quantum Brownian motion one always needs to consider a non-Markovian dynamics. We conclude that dissipation is a genuinely non-Markovian feature.
The standard Brownian motion master equation, used to describe thermal damping, is not completely positive, and does not admit a Monte Carlo method, important in numerical simulations. To eliminate both these problems one must add a term that generates additional position diffusion. He we show that one can obtain a completely positive simple quantum Brownian motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic Schroedinger equation (SSE), closely analogous to Langevin's equation, that has no equivalent Markovian master equation. Considering a specific example, we show that this SSE is sensitive to nonlinearities in situations in which the master equation is not, and may therefore be a better model of damping for nonlinear systems.
The "standard" Brownian motion master equation, used to describe thermal damping, is not completely positive, and does not admit a Monte Carlo method, important in numerical simulations. To eliminate both these problems one must add a term that generates additional position diffusion. He we show that one can obtain a completely positive simple quantum Brownian motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic Schrödinger equation (SSE), closely analogous to Langevin's equation, that has no equivalent Markovian master equation. Considering a specific example, we show that this SSE is sensitive to nonlinearities in situations in which the master equation is not, and may therefore be a better model of damping for nonlinear systems.
The dynamics of an open quantum system in the high-temperature regime of the Caldeira-Leggett model using a Bohmian language is investigated. By expressing the density matrix in polar form, generalized continuity and Hamilton-Jacobi equations are obtained. Thermal effects appear explicitly in the former, while they influence the latter only indirectly through an effective potential. Remarkably, the effective potential does not vanish in the classical limit and cannot, in general, be decomposed into simple additive quantum and thermal contributions. Instead, it retains a nontrivial structure leading to a residual temperature dependence in the resulting dynamics. As a consequence, the resulting temperature-dependent-classical trajectories remain deterministic even in the presence of a thermal environment. This behavior contrasts with the stochastic dynamics observed in the Langevin description and reflects the ensemble-based nature of the present approach. To further explore the quantum-to-classical transition, a scaled version of the Caldeira-Leggett equation is introduced by scaling both the density matrix and Planck constant through a quantumness parameter. This parameter takes the value one in the fully quantum regime and smoothly approaches zero in the classical limit. Applications to Gaussian states demonstrate a gradual suppression of coherence and interference, governed jointly by thermal effects and the transition parameter.