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New Journal of Physics

Measurement-based feedback control of a quantum system in a harmonic potential

Amy Rouillard, Humairah Bassa, Shamik Maharaj, Anirudh Reddy, Lajos Diósi, Thomas Konrad

Published Feb 23, 2026• 0 citations
Publisher

Abstract

Abstract We present a formulation of measurement-based feedback control of a single quantum particle in one spatial dimension to consider arbitrary linear combinations of the position and momentum of the particle used as observables for monitoring and as generators of unitary feedback with strength proportional to the measured signal. 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. Our general approach allows to identify a combination of measurement and feedback variables that, under certain circumstances, completely mitigates the noise induced by the measurement. The resulting deterministic evolution of expectation values in each realisation of the measurement resembles the dynamics of a damped classical oscillator, leading to a stationary state centred at the minimum of the potential, which becomes the ground state in the weak measurement limit. Remarkably, this stabilisation can be achieved using a fixed generator of feedback that is determined by the asymptotic values of the second-order moments of the steady state. In addition, we demonstrate that appropriate feedback adds a quadratic term in the measured observable to the Hamiltonian of the system. 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.

DOI
10.1088/1367-2630/ae4922

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Abstract We present a formulation of measurement-based feedback control of a single quantum particle in one spatial dimension to consider arbitrary linear combinations of the position and momentum of the particle used as observables for monitoring and as generators of unitary feedback with strength proportional to the measured signal. 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. Our general approach allows to identify a combination of measurement and feedback variables that, under certain circumstances, completely mitigates the noise induced by the measurement. The resulting deterministic evolution of expectation values in each realisation of the measurement resembles the dynamics of a damped classical oscillator, leading to a stationary state centred at the minimum of the potential, which becomes the ground state in the weak measurement limit. Remarkably, this stabilisation can be achieved using a fixed generator of feedback that is determined by the asymptotic values of the second-order moments of the steady state. In addition, we demonstrate that appropriate feedback adds a quadratic term in the measured observable to the Hamiltonian of the system. 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.
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.
Abstract Feedback control is an essential component of many modern technologies and provides a key capability for emergent quantum technologies. We extend existing approaches of direct feedback control in which the controller applies a function directly proportional to the output signal (P feedback), to strategies in which feedback determined by an integrated output signal (I feedback), and to strategies in which feedback consists of a combination of P and I terms. The latter quantum PI feedback constitutes the analog of the widely used proportional-integral feedback of classical control. All of these strategies are experimentally feasible and require no complex state estimation. We apply the resulting formalism to two canonical quantum feedback control problems, namely, generation of an entangled state of two remote qubits, and stabilization of a harmonic oscillator under thermal noise under conditions of arbitrary measurement efficiency. These two problems allow analysis of the relative benefits of P, I, and PI feedback control. We find that for the two-qubit remote entanglement generation the best strategy can be a combined PI strategy when the measurement efficiency is less than one. In contrast, for harmonic state stabilization we find that P feedback shows the best performance when actuation of both position and momentum feedback is possible, while when only actuation of position is available, I feedback consistently shows the best performance, although feedback delay is shown to improve the performance of a P strategy here.
In quantum mechanics, standard or strong measurement approaches generally result in the collapse of an ensemble of wavefunctions into a stochastic mixture of eigenstates. On the other hand, continuous or weak measurements have the propensity to dynamically control the evolution of quantum states over time, guiding the trajectory of the state into non-trivial superpositions and maintaining state purity. This kind of measurement-induced state steering is of great theoretical and experimental interest for the harnessing of quantum bits or "qubits", which are the fundamental unit of the emerging quantum computer. We explore continuous measurement-based quantum state stabilization through linear feedback control for a single quantum bit. By applying a time-varying Rabi drive that includes a linear feedback term, we show that the fixed points of the continuous measurement may be relocated. Analytical derivations of Itô Stochastic Master Equations being ideal models, we employ them to derive the projected ensemble average while utilizing numerical simulations to characterize the stability of the set of possible fixed points, as well as their modified collapse time-scales. We include the effects of realistic experimental non-idealities, such as environmental energy relaxation, dephasing, time-delay, and inefficient measurement. Ultimately, we discuss potential experimental implementations from collaborating universities.
We study the control problem of regulating the purity of a quantum harmonic oscillator in a Gaussian state via weak measurements. Specifically, we assume time-invariant Hamiltonian dynamics and that control is exerted via the back-action induced from monitoring the oscillator’s position and momentum observables; the manipulation of the detector measurement strengths regulates the purity of the target Gaussian quantum state. After briefly drawing connections between Gaussian quantum dynamics and stochastic control, we focus on the effect of inefficient detectors and derive closed-form expressions for the transient and steady-state dynamics of the state covariance. We highlight the degradation of attainable purity that is due to inefficient detectors, as compared to that dictated by the Robertson-Schrödinger uncertainty relation. Our results suggest that quantum correlations can enhance the purity at steady-state. The quantum harmonic oscillator represents a basic system where analytic formulae may provide insights into the role of inefficient measurements in quantum control; the gained insights are pertinent to measurement-based quantum engines and cooling experiments.
We study measurement-based feedback control of a damped harmonic oscillator, motivated by optomechanical and interferometric position measurements, in which feedback is synthesized from a displacement measurement. For a band-limited derivative controller, the loop shapes the mechanical susceptibility and reinjects measurement noise, creating a tradeoff between stabilization and noise amplification. We present a compact classical formulation that isolates this tradeoff in the closed-loop displacement spectrum and identifies the parameters governing resonance suppression versus noise-driven actuation. We then extend the same topology to a quantum-limited position detector, enforcing the imprecision-backaction constraint and introducing finite measurement bandwidth that regularizes high-frequency noise reinjection and adds phase lag in the feedback path. In the high-QQQ regime we obtain an analytic design rule for the optimal feedback gain at fixed controller and measurement bandwidths and compare it with full-spectrum calculations. The comparison shows that a high mechanical quality factor alone does not guarantee quantitative agreement: the accuracy of the near-resonant approximation depends strongly on the controller and measurement bandwidths. Because finite measurement bandwidth changes both the magnitude and phase of the feedback loop, the analysis separates the near-resonant damping contribution from imprecision-noise reinjection and shows that positive near-resonant feedback damping requires the product of the controller cutoff and measurement bandwidth to exceed the square of the mechanical resonance frequency. Finally, we present a dimensionless map of the gain-optimized high-QQQ occupation and compare its predictions with full-spectrum optimization. All analytical gains evaluated in this map satisfy the full closed-loop stability criterion for these parameters.
Quantum technologies and experiments often require preparing systems in low-temperature states. Here we investigate cooling schemes using feedback protocols modeled with a quantum Fokker-Planck master equation (QFPME) recently derived by Annby-Andersson et al. [Phys. Rev. Lett. 129, 050401 (2022)0031-900710.1103/PhysRevLett.129.050401]. This equation describes systems under continuous weak measurements, with feedback based on the outcome of these measurements. We apply this formalism to study the cooling and trapping of a harmonic oscillator for several protocols based on position and/or momentum measurements. We find that the protocols can cool the oscillator down to, or close to, the ground state for suitable choices of parameters. Our analysis provides an analytically solvable case study of quantum measurement and feedback and illustrates the application of the QFPME to continuous quantum systems.