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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

The quantum trajectory approach to quantum feedback control of an oscillator revisited

Andrew C. Doherty, Alex Szorkovszky, Glen I. Harris, Warwick P. Bowen

Published Oct 22, 2012• 38 citations
Publisher

Abstract

We revisit the stochastic master equation approach to feedback cooling of a quantum mechanical oscillator undergoing position measurement. By introducing a rotating wave approximation for the measurement and bath coupling, we can provide a more intuitive analysis of the achievable cooling in various regimes of measurement sensitivity and temperature. We also discuss explicitly the effect of backaction noise on the characteristics of the optimal feedback. The resulting rotating wave master equation has found application in our recent work on squeezing the oscillator motion using parametric driving and may have wider interest.

DOI
10.1098/rsta.2011.0531

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We revisit the stochastic master equation approach to feedback cooling of a quantum mechanical oscillator undergoing position measurement. By introducing a rotating wave approximation for the measurement and bath coupling, we can provide a more intuitive analysis of the achievable cooling in various regimes of measurement sensitivity and temperature. We also discuss explicitly the effect of backaction noise on the characteristics of the optimal feedback. The resulting rotating wave master equation has found application in our recent work on squeezing the oscillator motion using parametric driving and may have wider interest.
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Abstract We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground-state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturbance can be understood in terms of Heisenberg’s microscope and becomes particularly relevant when the measurement and feedback processes happen to be fast within the quantum coherence time of the system to be cooled. This corresponds to the regime of large quantum cooperativity C q ≳ 1 , which was achieved in recent experiments on feedback cooling. Our method provides a simple path to further pushing the limits of current state-of-the-art experiments in quantum optomechanics.
We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturbance can be understood in terms of Heisenberg's microscope and becomes particularly relevant when the measurement and feedback processes happen to be fast within the quantum coherence time of the system to be cooled. This corresponds to the regime of large quantum cooperativity Ctextqgtrsim1C_{\\text{q}}\\gtrsim1Ctextq​gtrsim1, which was achieved in recent experiments on feedback cooling. Our method provides a simple path to further pushing the limits of current state-of-the-art experiments in quantum optomechanics.
A mechanical system can be optimally controlled through continuous measurements of its position followed by feedback. We revisit the complete formalism for predicting the performance of such a system without invoking the standard rotating-wave approximation and the adiabatic approximation. Using this formalism, we deduce both the conditional and unconditional states of a mechanical oscillator using the optimal control and feedback that leads to mechanical cooling and mechanical squeezing. We find large discrepancies between the exact solutions and the approximate solutions, stressing the importance of using the complete model. We also highlight the importance of distinguishing between the conditional and unconditional states by demonstrating that these two cannot coincide in a typical control scheme, even with infinite feedback strength.
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.
Continuously measuring the position of a mechanical resonator affects its motion given the right conditions as a result of quantum mechanics. We revisit the quantum trajectories approach for describing the dynamics of such system without introducing a rotating wave approximation for the measurement and obtain its governing equations, assuming an initial Gaussian state. We apply them to assess if we can induce mechanical squeezing on the position of an optomechanical membrane-in-the middle setup through continuous measurement. We further extend them to a multiple-mode case to include the effects of the resonator?s residual modes and, finally, we study and numerically test a feedback cooling method to damp the residual modes and the resonant one simultaneously through means of filtering the measured position.