Stochastic path integral formalism for continuous quantum measurement

TL;DR

This paper extends the stochastic path integral formalism for continuous quantum measurement, providing analytical and perturbative solutions for qubit dynamics, including feedback control, and demonstrating the method's effectiveness in analyzing quantum trajectories.

Contribution

It develops a generalized stochastic path integral approach with exact and perturbative methods for analyzing continuous quantum measurement, including feedback control, for the first time.

Findings

Analytic solutions for average qubit trajectories and variances with zero Hamiltonian.

Perturbative computation of expectation values and correlations under unitary evolution.

Demonstration of qubit state stabilization via linear feedback in an ideal scenario.

Abstract

We generalize and extend the stochastic path integral formalism and action principle for continuous quantum measurement introduced in [A. Chantasri, J. Dressel and A. N. Jordan, Phys. Rev. A {\bf 88}, 042110 (2013)], where the optimal dynamics, such as the most-likely paths, are obtained by extremizing the action of the path integral. In this work, we apply exact functional methods as well as develop a perturbative approach to investigate the statistical behaviour of continuous quantum measurement, with examples given for the qubit case. For qubit measurement with zero qubit Hamiltonian, we find analytic solutions for average trajectories and their variances while conditioning on fixed initial and final states. For qubit measurement with unitary evolution, we use the perturbation method to compute expectation values, variances, and multi-time correlation functions of qubit trajectories in the short-time regime. Moreover, we consider continuous qubit measurement with feedback control, using the action principle to investigate the global dynamics of its most-likely paths, and finding that in an ideal case, qubit state stabilization at any desired pure state is possible with linear feedback. We also illustrate the power of the functional method by computing correlation functions for the qubit trajectories with a feedback loop to stabilize the qubit Rabi frequency.