Deterministic Equations for Feedback Control of Open Quantum Systems III: Full counting statistics for jump-based feedback

TL;DR

This paper develops a comprehensive theoretical framework for analyzing jump-based feedback control in open quantum systems, enabling full counting statistics characterization and demonstrating how feedback can convert measurement information into work.

Contribution

It introduces a Lindblad master equation in a hybrid classical-quantum space for jump-based feedback, allowing detailed statistical analysis of quantum trajectories.

Findings

Framework characterizes counting statistics under feedback

Feedback can convert measurement data into work in quantum thermal machines

Analytical tools for current, noise, and correlation analysis

Abstract

In this work, we consider a general feedback protocol based on quantum-jump detections, where the last detected jump channel is stored in a memory and subsequently used to implement a feedback action, such as modifying the system Hamiltonian conditioned on the last jump. We show that the time evolution of this general protocol can be described by a Lindblad master equation defined in a hybrid classical-quantum space, where the classical part encodes the stored measurement record (memory) and the quantum part represents the monitored system. Moreover, we show that this new representation can be used to fully characterize the counting statistics of a system subject to a general jump-based feedback protocol. We apply the formalism to a three-level system coupled to two thermal baths operating as a thermal machine, and we show that jump-based feedback can be used to convert the information obtained from the jump detections into work. Our framework provides analytical tools that enable the characterization of key statistical properties of any counting observable under jump-based feedback, such as the average current, noise, correlation functions, and power spectrum.