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arXiv:2207.07785 · Physical Review A

Mechanical cooling and squeezing using optimal control

Frederik Werner Isaksen, Ulrik L. Andersen

Published Feb 16, 2023• 11 citations
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Abstract

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.

DOI
10.1103/physreva.107.023512

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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.
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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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.
We propose a feedback control mechanism for the squeezing of the phononic mode of a mechanical oscillator. We show how, under appropriate working conditions, a simple adiabatic approach is able to induce mechanical squeezing. We then go beyond the limitations of such a working point and demonstrate the stationary squeezing induced by using repeated measurements and reinitialization of the state of a two-level system ancilla coupled to the oscillator. Our nonadaptive feedback loop offers interesting possibilities for quantum state engineering and steering in open-system scenarios.
Proportional-Integral-Derivative (PID) control is used for automatically regulating a measurable quantity to a desired setpoint. It is widely used in different types of classical control electronics. Here, we show how extending the feedback theory in quantum systems to include the derivative and integral parts influences both the transient and steady-state behavior of the amplitude and squeezing of a mechanical quadrature in an optomechanical system. We show that, in contrast to standard proportional feedback, derivative feedback affects both the conditional and unconditional squeezing. Furthermore, we demonstrate how feedback may be employed to drive a mechanical quadrature to track a desired reference signal. Our findings offer new routes for an improved quantum state control and measurement precision.