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arXiv:gr-qc/0109003 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Noise in gravitational-wave detectors and other classical-force measurements is not influenced by test-mass quantization

Vladimir B. Braginsky, M. L. Gorodetsky, Farid Ya Khalili, Andrey B. Matsko, Kip S. Thorne, Sergey P. Vyatchanin

Published Apr 7, 2003• 75 citations
arXivHTMLPDF

Abstract

It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output [the photon number flux $\mathcal{N}^(t)$ entering the final photodetector] commutes with itself at different times in the Heisenberg picture, $[\mathcal{N}^(t),\mathcal{N}^{(t}^{\ensuremath{'}})]=0$ and thus can be regarded as classical. (ii) This number flux is linear to high accuracy in the test-mass initial position and momentum operators ${x}_{o}$ and ${p}_{o},$ and those operators influence the measured photon flux $\mathcal{N}^(t)$ in manners that can easily be removed by filtering. For example, in most interferometers ${x}_{o}$ and ${p}_{o}$ appear in $\mathcal{N}^(t)$ only at the test masses' $\ensuremath{\sim}1\mathrm{Hz}$ pendular swinging frequency and their influence is removed when the output data are high-pass filtered to get rid of noise below $\ensuremath{\sim}10\mathrm{Hz}.$ The test-mass operators ${x}_{o}$ and ${p}_{o}$ contained in the unfiltered output $\mathcal{N}^(t)$ make a nonzero contribution to the commutator $[\mathcal{N}^(t),\mathcal{N}^{(t}^{\ensuremath{'}})].$ That contribution is precisely canceled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in $\mathcal{N}^(t).$ This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer's standard quantum limit from test-mass considerations, and independently from photon-noise considerations, and get identically the same result. These conclusions are all true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems.

DOI
10.1103/physrevd.67.082001

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It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output [the photon number flux N(t)\mathcal{N}^(t)N(t) entering the final photodetector] commutes with itself at different times in the Heisenberg picture, [\mathcal{N}^(t),\mathcal{N}^{(t}^{\ensuremath{'}})]=0 and thus can be regarded as classical. (ii) This number flux is linear to high accuracy in the test-mass initial position and momentum operators xo{x}_{o}xo​ and po,{p}_{o},po​, and those operators influence the measured photon flux N(t)\mathcal{N}^(t)N(t) in manners that can easily be removed by filtering. For example, in most interferometers xo{x}_{o}xo​ and po{p}_{o}po​ appear in N(t)\mathcal{N}^(t)N(t) only at the test masses' \ensuremath∼1Hz\ensuremath{\sim}1\mathrm{Hz}\ensuremath∼1Hz pendular swinging frequency and their influence is removed when the output data are high-pass filtered to get rid of noise below \ensuremath∼10Hz.\ensuremath{\sim}10\mathrm{Hz}.\ensuremath∼10Hz. The test-mass operators xo{x}_{o}xo​ and po{p}_{o}po​ contained in the unfiltered output N(t)\mathcal{N}^(t)N(t) make a nonzero contribution to the commutator [\mathcal{N}^(t),\mathcal{N}^{(t}^{\ensuremath{'}})]. That contribution is precisely canceled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in N(t).\mathcal{N}^(t).N(t). This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer's standard quantum limit from test-mass considerations, and independently from photon-noise considerations, and get identically the same result. These conclusions are all true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems.
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It has long been thought that the sensitivity of laser interferometric gravitational-wave detectors is limited by the free-mass standard quantum limit, unless radical redesigns of the interferometers or modifications of their input or output optics are introduced. Within a fully quantum-mechanical approach we show that in a second-generation interferometer composed of arm cavities and a signal recycling cavity, e.g., the LIGO-II configuration, (i) quantum shot noise and quantum radiation-pressure-fluctuation noise are dynamically correlated, (ii) the noise curve exhibits two resonant dips, (iii) the standard quantum limit can be beaten by a factor of 2, over a frequency range $\ensuremath{\Delta}f/f\ensuremath{\sim}1,$ but at the price of increasing noise at lower frequencies.
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.
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The uncertainty principle, applied naively to the test masses of a laser interferometer gravitational wave detector, produces a standard quantum limit (SQL) on the interferometer's sensitivity. It has long been thought that beating this SQL would require a radical redesign of interferometers. We show that no redesign is needed at all. LIGO-II interferometers, as currently planned, can beat the SQL by as much as a factor two over a bandwidth \\Delta f \\sim f, if their thermal noise can be pushed low enough. This is due to quantum correlations between photon shot noise and radiation pressure noise, produced by the LIGO-II signal-recycling mirror.