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Physical Review

Quantum Mechanical Liouville Equation for a System in Contact with a Reservoir

C. R. Willis, Peter G. Bergmann

Published Oct 1, 1962• 22 citations
Publisher

Abstract

In this paper we derive the equations of motion satisfied by the density matrix of an arbitrary physical system in contact with an infinite thermal reservoir of quantum mechanical free particles. We show that for weak but sustained interactions between system and reservoir, the reservoir is represented by a linear time-independent kernel in the system density matrix space. We obtain the kernel for the three possible models of quantum mechanical free particles which are boson and fermion reservoirs in which the system-reservoir interaction conserves the number of particles of the reservoir and a boson reservoir in which reservoir "particles" may be created or destroyed. In order to derive the equations of motion without using the inconsistent "rerandomization of phases" after each interaction, we generalize the Wigner-Weisskopf theory of line broadening of spectral lines to the equations of motion for the density matrix. We next show that the kernel preserves the normalization, positive-definiteness, and Hermiticity of the system density matrix and causes the system to approach equilibrium monotonically. In the limit of a classical reservoir our results include a derivation, without the repeated use of random phasing, of Bloch's equations of motion for the density matrix of a spin system imbedded in a thermal reservoir.

DOI
10.1103/physrev.128.391

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In this paper we derive the equations of motion satisfied by the density matrix of an arbitrary physical system in contact with an infinite thermal reservoir of quantum mechanical free particles. We show that for weak but sustained interactions between system and reservoir, the reservoir is represented by a linear time-independent kernel in the system density matrix space. We obtain the kernel for the three possible models of quantum mechanical free particles which are boson and fermion reservoirs in which the system-reservoir interaction conserves the number of particles of the reservoir and a boson reservoir in which reservoir "particles" may be created or destroyed. In order to derive the equations of motion without using the inconsistent "rerandomization of phases" after each interaction, we generalize the Wigner-Weisskopf theory of line broadening of spectral lines to the equations of motion for the density matrix. We next show that the kernel preserves the normalization, positive-definiteness, and Hermiticity of the system density matrix and causes the system to approach equilibrium monotonically. In the limit of a classical reservoir our results include a derivation, without the repeated use of random phasing, of Bloch's equations of motion for the density matrix of a spin system imbedded in a thermal reservoir.
In this paper we study the approach to equilibrium of an arbitrary physical system in contact with a thermal reservoir when the internal dynamics of the system is described by its Hamiltonian and the interaction with the reservoir can be described by a linear stochastic kernel which in general can be time dependent. We find a generalization of the Markov process represented by the master equation to the space of density matrices and we prove that the necessary and sufficient condition that any solution of an equation of our general structure approach equilibrium monotonically is that equilibrium is a stationary solution. We prove that the Helmholtz free energy is a minimum as a functional of the density matrix and thus is a suitable measure of deviation from canonicity. We also find another measure of deviation which is equally good and easier to use. We show, as an example, that Bloch's microscopic density matrices fulfill our criteria and thus approach equilibrium monotonically. Finally, we extend our results for the thermal reservoir to more general reservoirs.
We consider a finite quantum system under slow driving and weakly coupled to thermal reservoirs at different temperatures. We present a systematic derivation of the quantum master equation for the density matrix and the out-of-time-order correlators. We start from the microscopic Hamiltonian and we formulate the equations ruling the dynamics of these quantities by recourse to the Schwinger-Keldysh nonequilibrium Green's function formalism, performing a perturbative expansion in the coupling between the system and the reservoirs. We focus on the adiabatic dynamics, which corresponds to considering the linear response in the ratio between the relaxation time due to the system-reservoir coupling and the time scale associated to the driving. We calculate the particle and energy fluxes. We illustrate the formalism in the case of a qutrit coupled to bosonic reservoirs and of a pair of interacting quantum dots attached to fermionic reservoirs, also discussing the relevance of coherent effects.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
We analyse the discrete-time dynamics of a model of non-interacting fermions coupled to an infinite reservoir formed by a bosonic quantum walk on Z{\mathbb Z}Z. This dynamics consists of consecutive applications of free evolutions of the fermions and bosons followed by a local coupling between them. The unitary operator implementing this coupling accounts for energy exchanges between the system and reservoir while it preserves the number of fermions. The free fermion evolution is given by a second-quantized single-particle unitary operator satisfying some genericity assumptions. The free boson evolution is given by the second-quantized shift operator on Z{\mathbb Z}Z. We derive explicitly the Heisenberg dynamics of fermionic observables and obtain a systematic expansion in the large-coupling regime, which we control by using spectral methods. We also prove that the reduced state of the fermions converges in the large-time limit to a mixture of infinite-temperature Gibbs states in each particle sector.
The density operator of the arbitrary physical system must be positive definite. Employing the general master equation technique which preserves this property, we derive equations of motion for the density operator of an active atom which interacts collisionally with the reservoir of perturber atoms. The obtained general relations applied to the two-level atom yield Bloch-Boltzmann equations (BBE) which, as it is the case with master equation approach, are linear in the matrix elements of the active-atom density operator. The obtained BBE guarantee that positivity is preserved, which needs not to be the case with the results known from literature. We argue that our results are the correct ones and as such should be used in practical applications. Moreover, the structure and the terms which appear in our set of BBE seem to allow simpler and more straightforward physical interpretation.