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Physical Review Letters

Quantum Limit on the Michelson Interferometer used for Gravitational-Wave Detection

R. Loudon

Published Sep 21, 1981• 116 citations
Publisher

Abstract

The quantum limitations on measurements of the relative positions of the two mirrors in a Michelson gravity-wave detector are derived. The effects of intensity fluctuations in the two light beams and radiation pressure on the mirrors are modeled in a unified calculation. The radiation-pressure uncertainty in mirror position differs in general from earlier work, but gives agreement for coherent input light.

DOI
10.1103/physrevlett.47.815

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The quantum limitations on measurements of the relative positions of the two mirrors in a Michelson gravity-wave detector are derived. The effects of intensity fluctuations in the two light beams and radiation pressure on the mirrors are modeled in a unified calculation. The radiation-pressure uncertainty in mirror position differs in general from earlier work, but gives agreement for coherent input light.
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.
Laser interferometers are quantum instruments. This chapter presents the quantum theory of laser interferometer gravitational wave detectors. We show the basics for analysing the quantum noise in the detector, and for deriving the associated standard quantum limit (SQL) for the sensitivity. By providing different perspectives on the origin of the SQL, we illustrate the motivations behind different approaches for surpassing the SQL. Introduction The most difficult challenge in building a laser interferometer gravitational wave (GW) detector is isolating the test masses from the rest of the world (e.g. random kicks from residual gas molecules, seismic activities, acoustic noises, thermal fluctuations, etc.) whilst keeping the device locked around the correct point of operation (e.g. pitch and yaw angles of the mirrors, locations of the beam spots, resonance condition of the cavities, and darkport condition for the Michelson). Once all these issues have been solved, we arrive at the issue that we are going to analyse in this chapter: the fundamental noise that arises from quantum fluctuations in the system. A simple estimate (following the steps of Braginsky, 1968) will already lead us into the quantum world – as it will turn out, the superb sensitivity of gravitational wave detectors will be constrained by the standard quantum limit (SQL), which relates to the fundamental heisenberg uncertainty principle . Further improvements of detector sensitivity beyond this require us to manipulate the quantum coherence of light to our advantage. In this chapter, we will introduce how to analyse GW detectors quantum mechanically, and will describe several advanced configurations to surpass the SQL.
Abstract The measurement of minuscule forces and displacements with ever greater precision is inhibited by the Heisenberg uncertainty principle, which imposes a limit to the precision with which the position of an object can be measured continuously, known as the standard quantum limit1–4. When light is used as the probe, the standard quantum limit arises from the balance between the uncertainties of the photon radiation pressure applied to the object and of the photon number in the photoelectric detection. The only way to surpass the standard quantum limit is by introducing correlations between the position/momentum uncertainty of the object and the photon number/phase uncertainty of the light that it reflects5. Here we confirm experimentally the theoretical prediction5 that this type of quantum correlation is naturally produced in the Laser Interferometer Gravitational-wave Observatory (LIGO). We characterize and compare noise spectra taken without squeezing and with squeezed vacuum states injected at varying quadrature angles. After subtracting classical noise, our measurements show that the quantum mechanical uncertainties in the phases of the 200-kilowatt laser beams and in the positions of the 40-kilogram mirrors of the Advanced LIGO detectors yield a joint quantum uncertainty that is a factor of 1.4 (3 decibels) below the standard quantum limit. We anticipate that the use of quantum correlations will improve not only the observation of gravitational waves, but also more broadly future quantum noise-limited measurements.
In the present study we extend and generalize previous results for coherent-state and squeezed-state Michelson interferometer quantum mechanical uncertainties (or fluctuations), which are commonly referred to as `quantum noise'. The calculation of photon counting (PC) fluctuations in the squeezed-state interferometer is extended to fourth-order correlation functions used as the measured signal. We also generalize a `unified model' for treating both PC and radiation pressure fluctuations in the coherent-state interferometer, by using mathematical methods which apply to Kerr-type interactions. The results are more general than those reported previously in two ways. First, we obtain exact expressions, which lead to previous results under certain approximations. Second, we deal with cases in which the responses of the two mirrors to radiation pressure are not equal.
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.