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Journal of Physics A Mathematical and Theoretical

Measurement and ergodicity in quantum mechanics

M. Bauer, Pier A. Mello

Published Apr 14, 2015• 1 citation
Publisher

Abstract

The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in quantum mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of the von Neumann model of measurement, coupling the system to N ‘probes’, with a strength that is at our disposal, and detecting the latter. The model parallels the procedure followed in experiments on quantum electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.

DOI
10.1088/1751-8113/48/18/185302

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The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in quantum mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of the von Neumann model of measurement, coupling the system to N ‘probes’, with a strength that is at our disposal, and detecting the latter. The model parallels the procedure followed in experiments on quantum electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.
The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in Quantum Mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of von Neumann model of measurement, coupling the system to NNN "probes" --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.
Landsberg and Farquhar have shown that the von Neumann-Fierz approach to the quantum ergodic problem does not give any criterion of ergodicity.In this paper it is shown that von Neumann's procedure of averaging over the macro-observers is sufficient by itself to "justify" at every time the use of the quantum microcanonical ensemble. It is then proved that the time behavior of the system is completely irrelevant to the demonstration of the von Neumann and von Neumann-Fierz ergodic theorems. It is concluded that the quantum ergodic problem must be attacked on entirely new lines.
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.
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables A^\hat{A}A^ and B^\hat{B}B^, with eigenvalues ana_nan​ and bmb_mbm​, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable Q2Q_2Q2​ conditioned on the first {\em probe} measurement yielding Q1Q_1Q1​ we obtain information on the statistical distribution of the {\em system} variable bmb_mbm​ conditioned on having found the system variable ana_nan​ in the interval δaδaδa around a(n)a^{(n)}a(n). The width of this statistical distribution as function of δaδaδa constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.
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.
Foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity which persists throughout an experiment and therefore it does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and it does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions like a Bose-Einstein condensate system for which a single measurement yields a real ensemble. We discuss the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from quantum field theory.