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arXiv:quant-ph/9902040 · Physical Review A

Quantum noise in the position measurement of a cavity mirror undergoing Brownian motion

Kurt A. Jacobs, Ilkka Juhani Tittonen, Howard M. Wiseman, Stephan Schiller

Published Jul 1, 1999• 60 citations
arXivHTMLPDF

Abstract

We perform a quantum theoretical calculation of the noise power spectrum for a phase measurement of the light output from a coherently driven optical cavity with a freely moving rear mirror. We examine how the noise resulting from the quantum back action appears among the various contributions from other noise sources. We do not assume an ideal (homodyne) phase measurement, but rather consider phase-modulation detection, which we show has a different shot noise level. We also take into account the effects of thermal damping of the mirror, losses within the cavity, and classical laser noise. We relate our theoretical results to experimental parameters, so as to make direct comparisons with current experiments simple. We also show that in this situation, the standard Brownian motion master equation is inadequate for describing the thermal damping of the mirror, as it produces a spurious term in the steady-state phase-fluctuation spectrum. The corrected Brownian motion master equation [L. Diosi, Europhys. Lett. 22, 1 (1993)] rectifies this inadequacy.

DOI
10.1103/physreva.60.538

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 16, 2003•0 citations
We perform a quantum theoretical calculation of the noise power spectrum for a phase measurement of the light output from a coherently driven optical cavity with a freely moving rear mirror. We examine how the noise resulting from the quantum back action appears among the various contributions from other noise sources. We do not assume an ideal (homodyne) phase measurement, but rather consider phase-modulation detection, which we show has a different shot noise level. We also take into account the effects of thermal damping of the mirror, losses within the cavity, and classical laser noise. We relate our theoretical results to experimental parameters, so as to make direct comparisons with current experiments simple. We also show that in this situation, the standard Brownian motion master equation is inadequate for describing the thermal damping of the mirror, as it produces a spurious term in the steady-state phase-fluctuation spectrum. The corrected Brownian motion master equation [L. Diosi, Europhys. Lett. 22, 1 (1993)] rectifies this inadequacy.
We consider the quantum and classical noise limits to position measurement and force detection by an atomic force microscope (AFM) with an optical readout of cantilever position. We model this by treating the cantilever as a perfectly reflecting mirror for a highly damped optical cavity. There are three sources of noise: the shot noise in the output laser measurement, the thermal noise in the cantilever, and the measurement back-action noise. This last source of noise becomes large for good measurements, measurements for which there is a high correlation between the output phase of the light and the changing position of the cantilever. The back action simultaneously drives a diffusion process in momentum and diagonalizes the cantilever state in the position basis. This latter result is ``state reduction.'' Explicit expressions for the rate of the measurement back action in terms of the device parameters are given. We also calculate the signal-to-noise ratio in the limit of a bad cavity. A comparison to recent experiments suggests that current AFMs are not quantum limited.
Compared to the quantum noise in the measurement of the translational motion of a suspended mirror using laser light, the quantum noise in the measurement of the angular motion of a suspended mirror has not been investigated intensively despite its potential importance. In this article, an expression for the quantum noise in the angular motion measurement is explicitly derived. The expression indicates that one quadrature of the vacuum field of the first-order Hermite-Gaussian mode of light causes quantum sensing noise and the other causes quantum back-action noise, or in other words the first-order vacuum field is ponderomotively squeezed. It is also shown that the Gouy phase shift the light acquires between the mirror and the position of detection of the light corresponds to the homodyne angle. Therefore, the quantum back-action noise can be canceled and the standard quantum limit can be surpassed by choosing the appropriate position of detection analogously to the cancellation of quantum radiation pressure noise by choosing an appropriate homodyne angle.
Quantum optomechanics (QOM) explores the interaction between a quantum field, often confined in a cavity, and the quantum motion of mirrors, membranes or material media, usually assumed slow enough with negligible particle production, which is the central concern of dynamical Casimir effects, and a drive laser field for effective control. This rapidly developing field has a wide range of applications, from quantum sensing to the detection of gravitational waves. Because the opto-mechanical coupling is nonlinear, most theoretical investigations assume that the amplitude of mirror motion xxx remains small. Recent years saw several papers treating x2,x3x^2, x^3x2,x3 orders in the mirror displacement. In our work we take a different route, deploying the functional perturbative method presented in [1] and enriched in [2], applicable for sufficiently weak opto-mechanical couplings. With this we can treat arbitrary displacement of the mirror, not just at a higher order in a power series expansion in xxx. We first derive the influence action of the quantum field and its non-Markovian noises back-reacting on the moving mirror with self-consistency. From it we derive a nonlinear Langevin equation driven by these back-action noises from the quantum field. We then show how results from our general modeling and treatment can be linked with the models and the Langevin equations presented in the literature, starting with the popular radiation pressure NxNxNx coupling, with NNN the photon number, up to the recent x3x^3x3 order results. We hope this new approach and the results presented here can provide useful theoretical support for high precision QOM experimentation in the future.
We discuss the use of active control to reduce mirrors position fluctuations at the quantum level. Recent experiments have shown that it is possible to reduce the thermal motion of a mirror by cold damping. The mirror motion is measured with an optomechanical sensor based on an high-finesse optical cavity, and corrected by a feedback loop. We show that this approach can be extended to lock the mirror motion at the quantum level and we propose to use this quantum locking technique to reduce the noise in interferometric measurements such as gravitational-waves detectors. We analyze the back-action effects of the optomechanical sensor and show that quantum limits can be transferred from the sensor measurement to the interferometric one. This simple technique allows one to suppress the quantum effects of radiation pressure in the interferometer and to greatly enhance its sensitivity. The effect is furthermore insensitive to losses in the interferometer.