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Physica Scripta

Energy balance for a dissipative quantum system

Jishad Kumar

Published Jul 17, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0031-8949/89/9/095701

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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 this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modeled as a collection of independent quantum harmonic oscillators. We derive closed form expressions for the mean kinetic and potential energies of the charged dissipative magneto-oscillator in the forms ${E}_{k}=\ensuremath{\langle}{\mathcal{E}}_{k}\ensuremath{\rangle}$ and ${E}_{p}=\ensuremath{\langle}{\mathcal{E}}_{p}\ensuremath{\rangle}$, respectively, where Ek{\mathcal{E}}_{k}Ek​ and Ep{\mathcal{E}}_{p}Ep​ denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is twofold; the first one is over the Gibbs canonical state for the thermostat, giving Ek{\mathcal{E}}_{k}Ek​ and Ep{\mathcal{E}}_{p}Ep​, and the second one, denoted by $\ensuremath{\langle}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\rangle}$, is over the frequencies $\ensuremath{\omega}$ of the bath oscillators which contribute to Ek{E}_{k}Ek​ and Ep{E}_{p}Ep​ according to probability distributions ${\mathcal{P}}_{k}(\ensuremath{\omega})$ and ${\mathcal{P}}_{p}(\ensuremath{\omega})$, respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay and Drude and radiation bath spectral density functions on the typical properties of ${\mathcal{P}}_{k}(\ensuremath{\omega})$ and ${\mathcal{P}}_{p}(\ensuremath{\omega})$. Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.
We investigate the quantum dissipative dynamics of a charged magneto-oscillator(CMO) coupled to a heat bath of harmonic oscillators in the presence of an additional field-bath interaction. The problem is formulated within the framework of the generalized quantum Langevin equation, where the external confining potential modifies the random force correlations and the memory kernel of the environment, and the effective spring constant. Using the corresponding non-Markovian master equation, we examine the influence of the field-bath coupling on the temporal decay of the reduced density matrix of the system. We show that the additional interaction leads to an enhancement of the decoherence rate by altering the dissipative response of the bath. Thus our results shed light on the role of field-bath coupling in the quantum to classical transition of a CMO. The resulting fluctuation-dissipation relation is analyzed, and the position autocorrelation, position-velocity correlation and velocity autocorrelation are obtained explicitly. The dependence of these experimentally accessible quantities on the field-bath coupling parameter is discussed. We outline an experimental proposal for testing our theoretical predictions. The study sheds light on reservoir engineering which is central to quantum technologies.
We obtain the quantum Langevin equation (QLE) of a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a quantum heat bath through momentum variables. The bath is modeled as a collection of independent quantum harmonic oscillators. The QLE involves a random force which does not depend on the magnetic field, and a quantum-generalized classical Lorentz force. These features are also present in the QLE for the case of particle-bath coupling through coordinate variables. However, significant differences are also observed. For example, the mean force in the QLE is characterized by a memory function that depends explicitly on the magnetic field. The random force has a modified form with correlation and commutator different from those in the case of coordinate-coordinate coupling. Moreover, the coupling constants, in addition to appearing in the random force and in the mean force, also renormalize the inertial term and the harmonic potential term in the QLE.
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.
In Chapter I of this dissertation, we present a pedagogical introduction to the basic concepts of stochastic theory and review the progress made as well as the outstanding unsolved problems in the field. For the rest of this dissertation, we use a generalized quantum Langevin equation (GLE) approach to investigate various properties of quantum dissipative systems. In Chapter II, calculations for the displacement and random force correlation functions for Brownian motion are generalized to the case of an arbitrary heat bath for a damped harmonic quantum oscillator. The mean square displacement of such an oscillator is then evaluated for both Ohmic and blackbody radiation heat baths, to determine the effects of many parameters on the localization of the oscillator. In Chapter III, the formalism is extended to the Brownian motion of a charged particle in an external magnetic field as well as in a potential. The influence of the magnetic field on the memory function and random force is determined, with the blackbody radiation heat bath analyzed as a special case. For a charged harmonic oscillator, the generalized susceptibility is obtained, which enables us to derive the symmetrized position correlation functions using the fluctuation-dissipation theorem. In addition, we obtain the free energy of the system by generalizing the "remarkable formula" of Ford, Lewis, and O'Connell. Explicit calculations are performed for Ohmic and blackbody radiation heat baths. Furthermore, the effect of dissipation on the localization of the oscillator in an Ohmic heat bath at zero temperature is shown to differ qualitatively from that without the magnetic field. Finally, we formulate retarded Green's functions and symmetrized position correlation functions for the oscillator, reach some general conclusions, and make explicit calculations for the Ohmic heat bath. For the special case of Brownian motion at both zero and nonzero temperatures, we prove two general asymptotic relations between the retarded Green's functions and the displacement correlation functions, which we use to evaluate the long-time behaviors of the latter from those of the former, for both the Ohmic heat bath and a rather general class of heat baths discussed extensively in the literature.