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Journal of Physics A Mathematical and General

Quantum dissipation of arbitrary strength from coupling to fermions

David Waxman

Published Apr 7, 1990• 1 citation
Publisher

Abstract

A single degree of freedom, analogous to the coordinate of a particle, is coupled to the density fluctuations of a filled sea of fermions. The fermions live in one spatial dimension and have a linear energy-momentum relation, as may be appropriate in the vicinity of the Fermi surface. By using the equivalence of fermions and bosons in one space dimension, the author shows that the classical limit of the above theory leads to a Langevin equation for the coordinate of the 'particle' in which there are no constraints on the strength of the friction constant. This is in distinction to existing models of fermionic heat baths which have upper limits on the value of the friction constant. The non-classical behaviour of the system is studied by determining the reduced equilibrium density matrix of the particle. The fermions, described by a functional integral over Grassmann fields, are uncoupled from the particle by a change of variables which is equivalent to a chiral gauge transformation. This generates a non-trivial Jacobian which is evaluated in an appendix and leads, in an appropriate limit, to identical results to those found by Caldeira and Leggett (1983) for an oscillator heat bath.

DOI
10.1088/0305-4470/23/7/019

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A single degree of freedom, analogous to the coordinate of a particle, is coupled to the density fluctuations of a filled sea of fermions. The fermions live in one spatial dimension and have a linear energy-momentum relation, as may be appropriate in the vicinity of the Fermi surface. By using the equivalence of fermions and bosons in one space dimension, the author shows that the classical limit of the above theory leads to a Langevin equation for the coordinate of the 'particle' in which there are no constraints on the strength of the friction constant. This is in distinction to existing models of fermionic heat baths which have upper limits on the value of the friction constant. The non-classical behaviour of the system is studied by determining the reduced equilibrium density matrix of the particle. The fermions, described by a functional integral over Grassmann fields, are uncoupled from the particle by a change of variables which is equivalent to a chiral gauge transformation. This generates a non-trivial Jacobian which is evaluated in an appendix and leads, in an appropriate limit, to identical results to those found by Caldeira and Leggett (1983) for an oscillator heat bath.
We analyze a model system of fermions in a harmonic oscillator potential under the influence of a dissipative environment: The fermions are subject to a fluctuating force deriving from a bath of harmonic oscillators. This represents an extension of the well-known Caldeira-Leggett model to the case of many fermions. Using the method of bosonization, we calculate one- and two-particle Green's functions of the fermions. We discuss the relaxation of a single extra particle added above the Fermi sea, considering also dephasing of a particle added in a coherent superposition of states. The consequences of the separation of center-of-mass and relative motion, the Pauli principle, and the bath-induced effective interaction are discussed. Finally, we extend our analysis to a more generic coupling between system and bath, which results in complete thermalization of the system.
The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.
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.
We consider a free particle coupled with finite strength to a bath and investigate the evaluation of its specific heat. A harmonic oscillator bath of Drude type with cutoff frequency ω D is employed to model an ohmic friction force with dissipation strength γ. Two scenarios for obtaining specific heat are presented. The first one uses the measurement of the kinetic energy of the free particle, while the second one is based on the reduced partition function. Both descriptions yield results which are consistent with the Third Law of thermodynamics. Nevertheless, the two methods produce different results that disagree even in their leading quantum corrections at high temperatures. We also consider the regime where the cutoff frequency is smaller than the friction strength, i.e. ω D < γ. There, we encounter puzzling results at low temperatures where the specific heat based on the thermodynamic prescription becomes negative. This anomaly is rooted in an ill-defined density of states of the damped free particle, which assumes unphysical negative values when γ/ω D > 1.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.
The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schrödinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.