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A Quantum Kac Model

Eric A. Carlen, Maria C. Carvalho, Michael Loss

Published Apr 26, 2018• 0 citations
OpenAlex

Abstract

We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the $N$ particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.

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We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the NNN particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.
A quantum kinetic master equation (QKME) for bosonic atoms is formulated. It is a quantum stochastic equation for the kinetics of a dilute quantum Bose gas, and describes the behavior and formation of Bose condensation. The key assumption in deriving the QKME is a Markov approximation for the atomic collision terms. In the present paper the basic structure of the theory is developed, and approximations are stated and justified to delineate the region of validity of the theory. Limiting cases of the QKME include the quantum Boltzmann master equation and the Uehling-Uhlenbeck equation, as well as an equation analogous to the Gross-Pitaevskii equation.
Abstract This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. It discusses the relaxation to equilibrium and the multi-time structure of quantum Markov processes, as well as their irreversible nature which is characterized by an appropriate entropy functional. Microscopic derivations for various quantum master equations are presented, such as the quantum optical master equation and the master equation for quantum Brownian motion. As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. The chapter also contains a treatment of non-linear, mean field quantum master equations together with applications to laser theory and super-radiance.
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.
We consider an open quantum many-particle system in which there are dissipative processes. The evolution of this system is described by a kinetic equation for the density matrix. From the equation describing a random Markov process in this system, we obtain an equation for the single-particle statistical operator. This equation describes the evolution of a system of identical particles in a mean-field approximation. The equation for interacting particles in thermodynamic equilibrium was obtained. The distribution function of a system of interacting electrons in metals has multivalence in a certain region of wave vectors. Among many solutions one is isotropic. Other solutions have the anisotropy of the electron distribution over the wave vectors. The anisotropy arises as a result of repulsion and attraction between electrons.
Abstract Typically, the theory of open quantum systems studies the dynamics of the reduced state (density operator) of the system. However, in the early stages of evolution, it is impossible to separate the reservoir dynamics from the system dynamics. Among the consequences of this fact is the violation of the positivity of solutions of some quantum master equations for the reduced density operator. In this paper we study the joint dynamics of the system and reservoir at an early stage of evolution and the pre-relaxation of the joint state to a so-called kinetic state. A kinetic state of the system and reservoir is characterized by the fact that it is completely determined by the reduced density operator of the system alone. Only after the formation of a kinetic state, it becomes possible to describe the evolution of the reduced density operator of the system in terms of a semigroup.
We investigate the zero-temperature limit of thermodynamic quantum master equations that govern the time evolution of density matrices for dissipative quantum systems. The quantum master equations for T=0T=0T=0 and for T>0T>0T>0 possess completely different structures: (i) the equation for T=0T=0T=0 is linear in the deviation from the ground-state density matrix, whereas the equation for T>0T>0T>0, in general, is seriously nonlinear, and (ii) the Gibbs state is obtained as the steady-state solution of the nonlinear equation for T>0T>0T>0, whereas the ground state cannot be found from the equation for T=0T=0T=0. Nevertheless, the equation for T=0T=0T=0 can reproduce the behavior for $T\ensuremath{\gtrsim}0$ remarkably well. We discuss some implications of that observation for dissipative quantum field theory.