Probability distributions of continuous measurement results for two non-commuting variables and conditioned quantum evolution

TL;DR

This paper develops a formalism to analyze the probability distributions of measurement outcomes for two non-commuting variables in a conditioned quantum system, revealing interference effects and peculiarities in various measurement regimes.

Contribution

It introduces a new formalism for evaluating measurement outcome distributions in conditioned quantum measurements involving non-commuting variables, with analytical and numerical predictions.

Findings

Interference effects manifest in measurement outcome distributions.

Analytical predictions of peculiarities at specific measurement outcome circles.

Numerical simulations confirm interference effects even in small regimes.

Abstract

We address the statistics of a simultaneous CWLM of two non-commuting variables on a few-state quantum system subject to a conditioned evolution. Both conditioned quantum measurement and that of two non-commuting variables differ drastically for either classical or quantum projective measurement, and we explore the peculiarities brought by the combination of the two. We put forward a proper formalism for the evaluation of the distributions of measurement outcomes. We compute and discuss the statistics in idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle ${\cal O}^2_1+{\cal O}^2_2=1$ in the distribution of measurement outcomes in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically anomalously large values of the time-integrated output cumulants in the limit of short measurement times(sudden jump) and zero overlap between initial and final states, and give the detailed distributions. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the conditioned measurement can be accurately predicted even if they are small.