Measurement-based Feedback Control of a Quantum System in a Harmonic Potential

TL;DR

This paper develops a measurement-based feedback control method for a quantum particle in a harmonic potential, enabling cooling and confinement by damping observables and modifying the Hamiltonian, with potential applications to arbitrary potentials.

Contribution

It introduces a feedback master equation and demonstrates how to cool and confine quantum systems using measurement and feedback, including exact solutions for harmonic oscillators.

Findings

Quantum feedback can cool a harmonic oscillator to its ground state.

Feedback modifies the system Hamiltonian by adding quadratic terms.

The method extends to arbitrary potentials with sufficient measurement strength.

Abstract

We present a formulation of measurement-based feedback control of a single quantum particle in one spatial dimension. An arbitrary linear combination of the position and momentum of the particle is continuously monitored, and feedback proportional to the measured signal is used to control the system. We derive a feedback master equation and discuss a general approach to computing the steady-state solutions for arbitrary potentials. For a quantum harmonic oscillator or a free particle, we show that it is possible to cool and confine the system using feedback that simultaneously damps the measured observable and its conjugate momentum, as well as compensates for noise introduced by the measurement. In addition, we demonstrate that appropriate feedback adds a quadratic term in the measured observable to the Hamiltonian of the system. For the particular case of the harmonic potential, we describe the exact dynamics of the system that can be cooled to the ground state. Moreover, we provide an argument for the possibility to cool systems with arbitrary potentials provided that the measurement is strong enough to localise the particle on an interval smaller than the characteristic length scale of the potential.