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arXiv:1205.7035 · New Journal of Physics

Position estimation of a parametrically driven optomechanical system

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

Published Sep 27, 2012• 14 citations
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Abstract

We study the position estimation of a mechanical oscillator undergoing both detuned parametric amplification and continuous quantum measurement. This model, which can be utilised to produce squeezed states, is applied to a general optoelectromechanical system. Using a stochastic master equation formalism, we derive general formulae for the reduction in position uncertainty of one quadrature of motion. The filter for extracting the optimal position estimate from the measurement record is derived. We also find that since this scheme does not work far into the back-action dominated regime, implementing resolved-sideband cooling improves the squeezing only marginally.

DOI
10.1088/1367-2630/14/9/095026

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We study the position estimation of a mechanical oscillator undergoing both detuned parametric amplification and continuous quantum measurement. This model, which can be utilised to produce squeezed states, is applied to a general optoelectromechanical system. Using a stochastic master equation formalism, we derive general formulae for the reduction in position uncertainty of one quadrature of motion. The filter for extracting the optimal position estimate from the measurement record is derived. We also find that since this scheme does not work far into the back-action dominated regime, implementing resolved-sideband cooling improves the squeezing only marginally.
Recently it has been demonstrated that the combination of continuous position detection with detuned parametric driving can lead to significant steady-state mechanical squeezing, far beyond the 3 dB limit normally associated with parametric driving. In this work, we show the close connection between this detuned scheme and quantum non-demolition (QND) measurement of a single mechanical quadrature. In particular, we show that applying an experimentally realistic detuned parametric drive to a cavity optomechanical system allows one to effectively realize a QND measurement despite being in the bad-cavity limit. In the limit of strong squeezing, we show that this scheme offers significant advantages over standard backaction evasion, not only by allowing operation in the weak measurement and low efficiency regimes, but also in terms of the purity of the mechanical state.
After a quick review of the basic theory of quantum optomechanical systems, based largely on linearized Heisenberg–Langevin equations, this chapter focuses on selected topics related to quantum measurement and quantum optomechanics. Included are: a comprehensive discussion of the quantum limit on the added noise of a continuous position detector, following the quantum linear response approach; a detailed discussion of the role of noise correlations, and how these can be achieved in an optomechanical cavity (by using squeezed input light, or by modifying the choice of measured output quadrature); and a discussion of back-action evading measurements of a mechanical quadrature, discussing how this can be achieved in a two-tone driven cavity system. The chapter ends with a quick introduction to the theory of conditional continuous quantum measurement, and a discussion on how a back-action evading measurement can be used to produce conditional mechanical squeezed states.
Building on Chapters 3 and 4, this chapter introduces the concepts of measurement-based quantum control of mechanical systems in the linearised approximation for the optomechanical interaction, including techniques such as feedback cooling and back-action evading measurements using both stroboscopic optical driving and parametric mechanical driving. To model these techniques we introduce the stochastic master equation approach. We then discuss the application of squeezed light fields and ponderomotive optical squeezing to overcome the standard quantum limit introduced in Chapter 3. These methods are important both to improve the precision of quantum sensors and for quantum state tomography of mechanical systems.
We study the quantum measurement of a cantilever using a parametrically coupled electromagnetic cavity which is driven at the two sidebands corresponding to the mechanical motion. This scheme, originally due to Braginsky et al (Braginsky V, Vorontsov Y I and Thorne K P 1980 Science 209 547), allows a back-action free measurement of one quadrature of the cantilever's motion, and hence the possibility of generating a squeezed state. We present a complete quantum theory of this system, and derive simple conditions on when the quantum limit on the added noise can be surpassed. We also study the conditional dynamics of the measurement, and discuss how such a scheme (when coupled with feedback) can be used to generate and detect squeezed states of the oscillator. Our results are relevant to experiments in optomechanics, and to experiments in quantum electromechanics employing stripline resonators coupled to mechanical resonators.
Multiparameter quantum estimation in open optomechanical systems is fundamentally constrained by dissipation, thermal fluctuations, and measurement incompatibility. In this work, we investigate a coupled cavity optomechanical platform in which two mechanical modes interact with driven optical cavities, two mode squeezed vacuum, intracavity degenerate parametric amplification, and coherent optical feedback. Using the continuous variable Gaussian state formalism, we derive the linearized quantum Langevin dynamics and steady state covariance matrix and evaluate the quantum Fisher information matrices associated with simultaneous estimation of the optomechanical coupling strength and cavity dissipation rate. We characterize the precision bounds using the symmetric and right logarithmic derivative formalisms and employ BMI=max⁡{BS,BR}\mathcal{B}_{\rm MI}=\max\{\mathcal{B}_S,\mathcal{B}_R\}BMI​=max{BS​,BR​} as a comparative figure of merit within the SLD/RLD framework. We find that parametric amplification can substantially reduce the most informative bound, BMI\mathcal{B}_{\rm MI}BMI​, thereby enhancing multiparameter estimation precision over a broad range of operating conditions. Coherent feedback can further reduce BMI\mathcal{B}_{\rm MI}BMI​ and improve the estimation precision, particularly in the strong feedback regime. Moreover, the simultaneous presence of coherent feedback, parametric amplification, and squeezed light injection provides a combined mechanism for controlling quantum fluctuations and parameter dependent correlations, leading to enhanced and more robust multiparameter estimation. We further analyze the dynamical and steady state regimes to identify the parameter regions in which these quantum resources provide the largest metrological gain. Our results establish coupled cavity optomechanics as a flexible platform for quantum-noise engineering in multiparameter sensing.
We provide an argument to infer stationary entanglement between light and a mechanical oscillator based on continuous measurement of light only.We propose an experimentally realizable scheme involving an optomechanical cavity driven by a resonant, continuous-wave field operating in the non-sideband-resolved regime.This corresponds to the conventional configuration of an optomechanical position or force sensor.We show analytically that entanglement between the mechanical oscillator and the output field of the optomechanical cavity can be inferred from the measurement of squeezing in (generalized) Einstein-Podolski-Rosen quadratures of suitable temporal modes of the stationary light field.Squeezing can reach levels of up to 50% of noise reduction below shot noise in the limit of large quantum cooperativity.Remarkably, entanglement persists even in the opposite limit of small cooperativity.Viewing the optomechanical device as a position sensor, entanglement between mechanics and light is an instance of object-apparatus entanglement predicted by quantum measurement theory.