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When does resolved-sideband cooling beat measurement-based feedback cooling? Analytical results in the regime of ground-state cooling

Kurt A. Jacobs, Hendra I. Nurdin, Frederick W. Strauch, M. R. James

Published Jul 18, 2014• 1 citation
OpenAlex

Abstract

We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing --- that is, for the same interaction strength --- and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.

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We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling---an example of coherent feedback control---and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing---that is, for the same interaction strength---and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.

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Optimal measurement-based feedback control for a single qubit: a candidate protocol

Ashkan Balouchi, Kurt Jacobs

Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.

arXiv:1407.8172•New Journal of Physics•Sep 30, 2014•5 citations
We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing --- that is, for the same interaction strength --- and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.
We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling---an example of coherent feedback control---and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing---that is, for the same interaction strength---and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.
Coherent feedback control, which does not destroy the coherence of quantum signals by measurement, outperforms measurement-based feedback control in quantum systems. Nevertheless, systematic methods for designing coherent feedback controllers are absent so far. Thus, expectations for controller design methods are growing from control engineering perspectives. This paper proposes a method for designing a coherent feedback controller based on frequency analysis. Previous studies have rarely considered frequency analysisbased approaches. The paper theoretically reveals that groundstate cooling in the unresolved sideband regime is possible using a simple feedback structure with only a single double-sided cavity as the controller. While the proposed design method is a natural extension of the method based on classical control theory, it has rarely been used in coherent feedback control, thus offering new possibilities for coherent controller design methods.
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.
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.
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.
Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.