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Repeated Interaction Quantum Systems

Laurent Bruneau

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

We consider a quantum system S interacting sequentially with independent systems Em, m = 1, 2,... Before interacting, each Em is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics. We solve exactly a model in which S and all the Em are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the Em vary randomly. We analyze a model in which S is a spin and the Em are thermal fermion baths and obtain the asymptotic state by rigorous perturbation theory, for random interaction times varying slightly around a fixed mean, and for small values of a coupling constant. 1

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We consider a quantum system S interacting sequentially with independent systems Em, m = 1, 2,... Before interacting, each Em is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics. We solve exactly a model in which S and all the Em are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the Em vary randomly. We analyze a model in which S is a spin and the Em are thermal fermion baths and obtain the asymptotic state by rigorous perturbation theory, for random interaction times varying slightly around a fixed mean, and for small values of a coupling constant. 1
A quantum system $\s$ interacts in a successive way with elements $\ee$ of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τττ and is governed by a fixed coupling between $\s$ and $\ee$. We show that the system, initially in any state close to a reference state, approaches a {\it repeated interaction asymptotic state} in the limit of large times. This state is τττ--periodic in time and does not depend on the initial state. If the reference state is chosen so that $\s$ and $\ee$ are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.
von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of NNN identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials VVV.
A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.
Any quantum process is represented by a sequence of quantum channels. We consider ergodic processes, obtained by sampling channel valued random variables along the trajectories of an ergodic dynamical system. Examples of such processes include the effect of repeated application of a fixed quantum channel perturbed by arbitrary correlated noise, or a sequence of channels drawn independently and identically from an ensemble. Under natural irreducibility conditions, we obtain a theorem showing that the state of a system evolving by such a process converges exponentially fast to an ergodic sequence of states depending on the process, but independent of the initial state of the system. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk. Further applications and physical implications of our results are discussed in the companion paper [11].
A small system in contact with a macroscopic environment usually approaches an asymptotic state, determined only by some macroscopic properties of the environment, such as the temperature or the chemical potential. In the long-time limit, the state of the small system is thus expected to be independent of its initial state. In some situations, however, the asymptotic state of the system is influenced by its initial state, and some information about the initial state is kept for all times. Motivated by this finding, we propose a measure to quantify the influence of the initial state of an open system on its dynamics. Using this measure we derive conditions under which the asymptotic state exists and is unique and discuss how this can be used to characterize the asymptotic behavior of an open system. We demonstrate our concepts for the dynamics of the spin-boson model, identify three qualitatively different long-time behaviors, and discuss how they can be distinguished based on the proposed measure.
A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρS(t) approaches an equilibrium state ρ(eq)(S), the latter depending only on macroscopic characteristics of the bath (e.g., temperature) but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present paper we concentrate on one aspect of thermalization, namely, on the system initial state independence (ISI) of ρ(eq)(S). A necessary condition for the system ISI is derived in the quantum framework. We use the derived condition to prove the absence of the system ISI in a specific class of models. Namely, we consider a single spin coupled to a large bath, the interaction term commuting with the bath self-Hamiltonian (but not with the system self-Hamiltonian). Although the model under consideration is nontrivial enough to exhibit the decoherence and the approach to equilibrium, the derived necessary condition is not fulfilled and thus ρ(eq)(S) depends on the initial state of the spin.