A pedagogical introduction to continuously monitored quantum systems and measurement-based feedback

Abstract

In this manuscript we present a pedagogical introduction to continuously monitored quantum systems. We start by giving a simplified derivation of the Markovian master equation in Lindblad form, in the spirit of collision models and input-output theory, which describes the unconditional dynamics of a continuously monitored system. The same formalism is then exploited to derive stochastic master equations that describe the conditional dynamics. We focus on the two most paradigmatic examples of continuous monitoring: continuous photodetection, leading to a discontinuous dynamics with "quantum jumps", and continuous homodyne measurements, leading to a diffusive dynamics. We then present a derivation of feedback master equations that describe the dynamics (either conditional or unconditional) when the continuous measurement photocurrents are fed back to the system as a linear driving Hamiltonian, a paradigm known as linear Markovian feedback. In the second part of the manuscript we focus on continuous-variable Gaussian systems: we first present the equations for first and second moments describing the dynamics under continuous general-dyne measurements, and we then discuss in more detail the conditional and unconditional dynamics under Markovian and state-based feedback.

Tutorial paper based on the lecture notes for the course "Coherence and control of quantum systems" taught by MGG at the University of Milan and submitted to the special issue "Foundations and applications of Quantum Optics" of Physics Letters A related to the Central European Workshop on Quantum Optics 2023

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• Review: Quantum Metrology and Sensing with Many-Body Systems
• Experimental simulation of daemonic work extraction in open quantum batteries on a digital quantum computer
• Measurement-induced phase transitions in monitored infinite-range interacting systems
• Analysis of spin-squeezing generation in cavity-coupled atomic ensembles with continuous measurements
• Noisy atomic magnetometry with Kalman filtering and measurement-based feedback
• A tensor network approach to sensing quantum light-matter interactions
• Squeezing below the ground state of motion of a continuously monitored levitating nanoparticle
• Daemonic ergotropy of Gaussian quantum states and the role of measurement-induced purification via general-dyne detection
• Optimal limits of continuously monitored thermometers and their Hamiltonian structure
• Optimal control of a dissipative micromaser quantum battery in the ultrastrong coupling regime
• Stochastic Schrödinger Equations for Quantum Reverse Diffusion
• Quantum stroboscopy for time measurements
• Unifying quantum stochastic methods using Wick's theorem on the Keldysh contour
• Theoretical Limits of Protocols for Distinguishing Different Unravelings
• Bounding fidelity in quantum feedback control: theory and applications to Dicke state preparation
• Dynamically Optimal Unraveling Schemes for Simulating Lindblad Equations
• Boosting Work Extraction in Quantum Batteries via Continuous Environment Monitoring