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IntechOpen eBooks

Dissipative Quantum System and Energy Balance

Jishad Kumar

Published Oct 6, 2022• 1 citation
Publisher

Abstract

We discuss how various parts of a quantum many-body system exchange energies at thermal equilibrium. To show this, we assume a quantum system is coupled to a many-body environment (at thermal equilibrium with a bigger environment) consisting of a large number of independent and non-interacting quantum harmonic oscillators above a stable ground state. Once the coupling to a large environment is switched on, the system dissipates its energy continuously to the environment until it reaches equilibrium with the latter. We use the Quantum Langevin equation to show such energy exchange at equilibrium. We conclude that different parts of a physical system can exchange energies even at absolute zero temperature.

DOI
10.5772/intechopen.106474

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We discuss how various parts of a quantum many-body system exchange energies at thermal equilibrium. To show this, we assume a quantum system is coupled to a many-body environment (at thermal equilibrium with a bigger environment) consisting of a large number of independent and non-interacting quantum harmonic oscillators above a stable ground state. Once the coupling to a large environment is switched on, the system dissipates its energy continuously to the environment until it reaches equilibrium with the latter. We use the Quantum Langevin equation to show such energy exchange at equilibrium. We conclude that different parts of a physical system can exchange energies even at absolute zero temperature.
The role of random force in maintaining equilibrium in a dissipative quantum system is studied here. We compute the instantaneous power supplied by the fluctuating (random) force, which provides information about the work done by the random force on the quantum subsystem of interest. The quantum Langevin equation formalism is used here to verify that, at equilibrium, the work done by the fluctuating force balances the energy lost by the quantum subsystem to the heat bath. The quantum subsystem we choose to couple to the heat bath is the charged oscillator in a magnetic field. We perform the calculations using the Drude regularized spectral density of bath oscillators instead of using a strict ohmic spectral density that gives memoryless damping. We also discuss the energy balance for our dissipative quantum system and in this regard it is to be understood that the physical system is the charged magneto-oscillator coupled to the heat bath, not the uncoupled charged magneto-oscillator.
In a recent paper [Phys. Rev. E 48, 1547 (1993)] Li, Ford, and O'Connel argue that the power lost through dissipation by a system in equilibrium with a thermal reservoir is compensated by the action of a fluctuation force that the reservoir exerts on the system. Their analysis is quantum mechanical and their expression for the power supplied by the reservoir remains positive definite even at absolute zero. This Comment outlines an analysis of a harmonic oscillator coupled to a loss mechanism and shows how the quantum-mechanical formalism should be interpreted to avoid unphysical conclusions. An explicit distinction is made between thermal and zero-point fluctuations, and the physical significance of the latter is discussed.
Abstract This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.
Experimentalists have come to temperatures very close to absolute zero at which physics that was once ordinary becomes extraordinary. In such a regime quantum effects and fluctuations start to play a dominant role. In this context we study the simplest open quantum system, namely, a free quantum Brownian particle coupled to thermal vacuum, i.e. thermostat in the limiting case of absolute zero temperature. We analyze the average energy [Formula: see text] of the particle from a weak to strong interaction strength c between the particle and thermal vacuum. The impact of various dissipation mechanisms is considered. In the weak coupling regime the energy tends to zero as [Formula: see text] while in the strong coupling regime it diverges to infinity as [Formula: see text]. We demonstrate it for selected examples of the dissipation mechanisms defined by the memory kernel [Formula: see text] of the Generalized Langevin Equation. We reveal how at a fixed value of c the energy E(c) depends on the dissipation model: one has to compare values of the derivative [Formula: see text] of the dissipation function [Formula: see text] at time [Formula: see text] or at the memory time [Formula: see text] which characterizes the degree of non-Markovianity of the Brownian particle dynamics. The impact of low temperature is also presented.
The process of diffusion constitutes the dissipative motion of a classical particle or group of particles subjected to many-body interactions with their surroundings. When the surroundings consist of an infinitely large system that can rapidly recover from any deviation from thermal equilibrium such that its temperature remains roughly uniform, the system qualifies as a heat bath. Further, when the degrees of freedom of the heat bath are integrated out because of the implied timescale separation, the dynamics of the subsystem appear dissipative, characterized by parameters such as friction coefficient, diffusion constant, and others.
The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.