Sequential nonideal measurements of quantum oscillators: Statistical characterization with and without environmental coupling

TL;DR

This paper analyzes the statistical behavior of sequential quantum measurements on oscillators, revealing convergence in isolated systems and divergence when coupled with an environment, thus distinguishing intrinsic and extrinsic quantum randomness.

Contribution

It provides a rigorous statistical characterization of quantum measurement sequences for oscillators, including effects of environmental coupling, with implications for quantum system monitoring.

Findings

Measurement sequences converge to a Gaussian limit in isolated systems.

Sequences diverge under environmental coupling.

Dependence of quantum track properties on system parameters.

Abstract

A one-dimensional quantum oscillator is monitored by taking repeated position measurements. As a first con- tribution, it is shown that, under a quantum nondemolition measurement scheme applied to a system initially at the ground state, (i) the observed sequence of measurements (quantum tracks) corresponding to a single experiment converges to a limit point, and that (ii) the limit point is random over the ensemble of the experiments, being distributed as a zero-mean Gaussian random variable with a variance at most equal to the ground-state variance. As a second contribution, the richer scenario where the oscillator is coupled with a frozen (i.e., at the ground state) ensemble of independent quantum oscillators is considered. A sharply different behavior emerges: under the same measurement scheme, here we observe that the measurement sequences are essentially divergent. Such a rigorous statistical analysis of the sequential measurement process might be useful for characterizing the main quantities that are currently used for inference, manipulation, and monitoring of many quantum systems. Several interesting properties of the quantum tracks evolution, as well as of the associated (quantum) threshold crossing times, are discussed and the dependence upon the main system parameters (e.g., the choice of the measurement sampling time, the degree of interaction with the environment, the measurement device accuracy) is elucidated. At a more fundamental level, it is seen that, as an application of basic quantum mechanics principles, a sharp difference exists between the intrinsic randomness unavoidably present in any quantum system, and the extrinsic randomness arising from the environmental coupling, i.e., the randomness induced by an external source of disturbance.