Probability distributions of continuous measurement results for conditioned quantum evolution

TL;DR

This paper develops a formalism to analyze the probability distributions of measurement outcomes in conditioned quantum evolution, revealing interference effects and non-trivial distributions in a few-state quantum system, with implications for experiments.

Contribution

It introduces a new formalism for calculating measurement outcome distributions in conditioned quantum systems, highlighting interference effects and non-trivial features.

Findings

Distributions show peaks at half-quantized measurement values.

Interference between initial and final states affects measurement statistics.

Numerical simulations extend analytic results to realistic qubit experiments.

Abstract

We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the post-selection in the end of the evolution. The statistics may drastically differ from the non-conditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of non-conditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of non-trivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extend- ing the analytic results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.