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arXiv:2503.15694 · IEEE Conference on Decision and Control (CDC)

Regulation of a continuously monitored quantum harmonic oscillator with inefficient detectors

Ralph Sabbagh, Olga Movilla Miangolarra, Tryphon T. Georgiou

Published Dec 9, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/cdc57313.2025.11312174

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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 consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements.We deduce the stochastic evolution equations for position and momentum expectation values and the covariance-matrix elements from the characteristic function of the system.By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al., 2013(Chantasri et al., , 2015) ) to this continuous-variable system, we construct its stochastic Hamiltonian and action.Action extremization gives us the equations for the most likely readout paths and quantum trajectories.For steady states of the covariance-matrix elements, the analytical solutions for these most likely paths are obtained.Using the CDJ formalism, we calculate final-state probability densities exactly starting from any initial state.We also demonstrate the agreement between the optimal-path solutions and the averages of simulated clustered stochastic trajectories.Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement-engine or refrigerator experiments.Although quantum mechanics fundamentally limits our ability to extract information via simultaneous measurements of noncommuting observables, insights into the dynamics of the system during such a process can lead to a deeper understanding of the physics of measurement.
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.
Linear--quadratic--Gaussian (LQG) control is optimal only with respect to a prescribed cost function, the choice of which dictates the physical objective of the control. We consider feedback cooling of a continuously monitored quantum harmonic oscillator by shifting the minimum of its trapping potential. In this setting, the physically relevant cooling objective can be defined as minimizing the oscillator's energy relative to the feedback-shifted potential. In contrast, conventional LQG control evaluates the energy from a fixed origin and thus fails to directly optimize this quantity. To address this problem, we introduce a redesigned cost function that explicitly accounts for the feedback-induced shift of the potential. We then derive the corresponding optimal feedback law and obtain an analytic expression for the minimum achievable steady-state phonon occupation number. The redesigned LQG control achieves a lower occupation number than low-pass-filter (LPF) feedback formulated for the same cooling objective. While this improvement is minor at detection efficiencies currently attainable in experiments---indicating that LPF feedback already delivers near-optimal cooling performance---the advantage becomes pronounced as the detection efficiency approaches unity. In this regime, the redesigned LQG control provides an increasing advantage for reaching the motional ground state at a finite measurement strength. We clarify that the conventional and redesigned cost functions represent distinct control objectives rather than different implementations of the same optimization problem.
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.
We study the evolution of a quantum particle in a harmonic potential whose position and momentum are repeatedly monitored. A back-action of measuring devices is accounted for. Our model utilizes a generalized measurement corresponding to the positive operator-valued measure. We assume that upon measurement, the particle's wavefunction is projected onto one of the possible detector states depending on the observed result. We chose these post-measurement states to be moving Gaussian wavepackets. The wavefunction quantum Monte Carlo formalism is used to simulate single quantum trajectories of the particle. We show how classical trajectories emerge in the course of observation and study in detail the dispersion of position and momentum of the particle.
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.