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Physical Review A

Quantum limits in interferometric detection of gravitational radiation

A. F. Pace, Matthew J. Collett, Daniel F. Walls

Published Apr 1, 1993• 171 citations
Publisher

Abstract

A spectral analysis is given of the quantum fluctuations in an optical interferometer to detect gravitational radiation. Two different methods of beating the standard quantum limit are examined: directing a squeezed state into the nonlaser input port of the interferometer and placing a Kerr medium into both arms of the interferometer. For both the Kerr medium and large squeezing cases the interferometer system is limited ultimately by the damping noise in the mirrors, not by noise in the light.

DOI
10.1103/physreva.47.3173

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A spectral analysis is given of the quantum fluctuations in an optical interferometer to detect gravitational radiation. Two different methods of beating the standard quantum limit are examined: directing a squeezed state into the nonlaser input port of the interferometer and placing a Kerr medium into both arms of the interferometer. For both the Kerr medium and large squeezing cases the interferometer system is limited ultimately by the damping noise in the mirrors, not by noise in the light.
The interferometers now being developed to detect gravitational waves work by measuring the relative positions of widely separated masses. Two fundamental sources of quantum-mechanical noise determine the sensitivity of such an interferometer: (i) fluctuations in number of output photons (photon-counting error) and (ii) fluctuations in radiation pressure on the masses (radiation-pressure error). Because of the low power of available continuous-wave lasers, the sensitivity of currently planned interferometers will be limited by photon-counting error. This paper presents an analysis of the two types of quantum-mechanical noise, and it proposes a new technique---the "squeezed-state" technique---that allows one to decrease the photon-counting error while increasing the radiation-pressure error, or vice versa. The key requirement of the squeezed-state technique is that the state of the light entering the interferometer's normally unused input port must be not the vacuum, as in a standard interferometer, but rather a "squeezed state"---a state whose uncertainties in the two quadrature phases are unequal. Squeezed states can be generated by a variety of nonlinear optical processes, including degenerate parametric amplification.
Quantum noise reduction in optical systems and the determination of quasi-distribution functions, in particular Wigner functions, to describe quantum optical states, have been central issues in quantum optics for a number of years. The generation of sources of squeezed light has opened very exciting possibilities of ultrahigh precision measurements beyond the standard quantum limit, by either interferometric or spectroscopic means. In the present issue, the paper by Mundarain and Orszag discusses the interferometric detection of ultrasmall signals, in particular gravitational waves, when one injects a squeezed state in the unused port of the interferometer. The authors find that the sensitivity of the system is very strongly affected by the quantum efficiency of the non-ideal photodetectors, when the squeezed signal is injected, as opposed to the ordinary vacuum case where the effects are rather small. This imposes strong conditions on the quality of those photodetectors. The standard quantum limit and the corresponding minimum detectable gravitational amplitude is also discussed in general, for short time measurements.
The quantum noise of the light field is a fundamental noise source in interferometric gravitational wave detectors. Injected squeezed light is capable of reducing the quantum noise contribution to the detector noise floor to values that surpass the so-called Standard-Quantum-Limit (SQL). In particular, squeezed light is useful for the detection of gravitational waves at high frequencies where interferometers are typically shot-noise limited, although the SQL might not be beaten in this case. We theoretically analyze the quantum noise of the signal-recycled laser interferometric gravitational-wave detector GEO600 with additional input and output optics, namely frequency-dependent squeezing of the vacuum state of light entering the dark port and frequency-dependent homodyne detection. We focus on the frequency range between 1 kHz and 10 kHz, where, although signal recycled, the detector is still shot-noise limited. It is found that the GEO600 detector with present design parameters will benefit from frequency dependent squeezed light. Assuming a squeezing strength of -6 dB in quantum noise variance, the interferometer will become thermal noise limited up to 4 kHz without further reduction of bandwidth. At higher frequencies the linear noise spectral density of GEO600 will still be dominated by shot-noise and improved by a factor of 10^{6dB/20dB}~2 according to the squeezing strength assumed. The interferometer might reach a strain sensitivity of 6x10^{-23} above 1 kHz (tunable) with a bandwidth of around 350 Hz. We propose a scheme to implement the desired frequency dependent squeezing by introducing an additional optical component to GEO600s signal-recycling cavity.
The sensitivity in interferometric measurements such as gravitational-wave detectors is ultimately limited by quantum noise of light. We discuss the use of feedback mechanisms to reduce the quantum effects of radiation pressure. 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 a high-finesse cavity, and reduced by a feedback loop. We show that this technique can be extended to lock the mirror at the quantum level. In gravitational-waves interferometers with Fabry-Perot cavities in each arms, it is even possible to use a single feedback mechanism to lock one cavity mirror on the other. This quantum locking greatly improves the sensitivity of the interferometric measurement. It is furthermore insensitive to imperfections such as losses in the interferometer.
The performance of an interferometric gravity-wave detector is analyzed, treating both the motion of the mirrors and the optical fields quantum mechanically, in continuous time. Kerr cells in each arm of the interferometer are used to counter the effects of radiation-pressure fluctuations, a feedback loop is used to keep the interferometer operating at the proper null, and a large mean-field assumption is used to linearize the analysis. When narrow-band disturbances are being measured, the performance of this interferometer can greatly surpass the standard quantum limit. When the Kerr cells are removed, the interferometer operates at the standard quantum limit, and this analysis agrees with earlier studies.