Quantum Fokker-Planck Master Equation for Continuous Feedback Control

TL;DR

This paper introduces a quantum Fokker-Planck master equation formalism for continuous measurement and feedback control, capable of handling both linear and nonlinear protocols, with applications to quantum and classical information engines.

Contribution

It develops a comprehensive formalism for continuous quantum measurement and feedback, including a master equation for finite bandwidth detectors and a Markovian approximation for fast measurements.

Findings

Derived a quantum Fokker-Planck master equation for joint system-detector dynamics.

Obtained a Markovian master equation for the system under fast measurement conditions.

Applied the formalism to analyze quantum and classical information engines.

Abstract

Measurement and feedback control are essential features of quantum science, with applications ranging from quantum technology protocols to information-to-work conversion in quantum thermodynamics. Theoretical descriptions of feedback control are typically given in terms of stochastic equations requiring numerical solutions, or are limited to linear feedback protocols. Here we present a formalism for continuous quantum measurement and feedback, both linear and nonlinear. Our main result is a quantum Fokker-Planck master equation describing the joint dynamics of a quantum system and a detector with finite bandwidth. For fast measurements, we derive a Markovian master equation for the system alone, amenable to analytical treatment. We illustrate our formalism by investigating two basic information engines, one quantum and one classical.