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Journal of Physics A Mathematical and General

An ergodic theorem for quantum counting processes

B K mmerer, Hans Maassen

Published Feb 12, 2003• 28 citations
Publisher

Abstract

For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean. This implies equality of time average and ensemble average for correlation functions of the detection current to all orders and with probability 1.
DOI
10.1088/0305-4470/36/8/312

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An ergodic theorem is a statement of the equality of the time averages of the physical properties of a system and the averages of these quantities obtained from the consideration of a Gibbsian ensemble of identical systems. It is shown that, in quantum statistical mechanics, a necessary and sufficient condition for the existence of an ergodic theorem can be derived for a system in weak energetic interaction with its surroundings. This necessary and sufficient condition for the equality of time and (microcanonical) ensemble averages is that all operators invariant in time shall reduce to constant multiples of the unit operator when applied to the system itself. It is shown that the conditions for ergodicity are formally analogous to those derived by von Neumann for classical statistical mechanics. No assumption of "molecular chaos" is made.
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.
In a paper by Klein, it is pointed out that for the quantum mechanical ergodic theorem (time average = average over states) to be valid, all integrals of motion R must satisfy the condition ∑ρ"(α′ρ"|R|β′ρ")=const δα′β′ (1) where α′, β′, γ, … refer to the assembly under question and α", β",…, ρ" ,… refer to the surrounding with which our assembly is in equilibrium. In this short note, it is pointed out that the argument of Klein is doubtful at one point and a slightly different approach is given. In this approach, it is shown that (1) is actually sufficient, but only after introducing an additional assumption that the different states of the surroundings have equal probabilities, and that in general, a stronger condition such as (α′ρ″|R|β′θ″)=const δα′β′δρ″θ″ (2) is needed. Cases with integrals of motion is also considered. We consider first an isolated system, write the wave function as (α1α2>, where α1 represents a set of commuting integrals of motion and α2 a set of observables not containing integrals of motion and introduce the assumption that for any integral of motion R, the matrix 1α′2Rα1"α2"> representing it is of the form φ(α′1α1")δ(α′2α2"). (3) Under such conditions, we prove that if the initial state of our system is an eigenstate of α1 corresponding to the eigenvalue α10, then the average of an observable F over time is given by ∑α210α2|F|α10α2>/∑α210α2|1|α10α2>, a result which is clearly to be expected. Extension to systems interacting with external surroundings is easily made.
Abstract The theoretical cornerstone of statistical mechanics is the ergodic assumption, i.e. the assumption that the time average of an observable equals its ensemble average. Here, we show how such a property is present when an open quantum system is continuously perturbed by an external environment effectively observing the system at random times while the system dynamics approaches the quantum Zeno regime. In this context, by large deviation theory we analytically show how the most probable value of the probability for the system to be in a given state eventually deviates from the non-stochastic case when the Zeno condition is not satisfied. We experimentally test our results with ultra-cold atoms prepared on an atom chip.