The more-than-ten-years-long history of the large-scale laser gravitation-wave (GW) detectors
(the first one, TAMA [142] started to operate in 1999, and the most powerful pair, the two
detectors of the LIGO project [98
], in 2001, not to forget about the two European members of the
international interferometric GW detectors network, also having a pretty long history, namely,
the German-British interferometer GEO 600 [66
] located near Hannover, Germany, and the
joint European large-scale detector Virgo [156
], operating near Pisa, Italy) can be considered
both as a great success and a complete failure, depending on the point of view. On the one
hand, virtually all technical requirements for these detectors have been met, and the planned
sensitivity levels have been achieved. On the other hand, no GWs have been detected thus
far.
The possibility of this result had been envisaged by the community, and during the same last ten years,
plans for the second-generation detectors were developed [143, 64
, 4
, 169
, 6, 96
]. Currently (2012), both
LIGO detectors are shut down, and their upgrade to the Advanced LIGO, which should take about three
years, is underway. The goal of this upgrade is to increase the detectors’ sensitivity by about one order of
magnitude [137
], and therefore the rate of the detectable events by three orders of magnitude, from some
‘half per year’ (by the optimistic astrophysical predictions) of the second generation detectors to, probably,
hundreds per year.
This goal will be achieved, mostly, by means of quantitative improvements (higher optical power, heavier mirrors, better seismic isolation, lower loss, both optical and mechanical) and evolutionary changes of the interferometer configurations, most notably, by introduction of the signal recycling mirror. As a result, the second-generation detectors will be quantum noise limited. At higher GW frequencies, the main sensitivity limitation will be due to phase fluctuations of light inside the interferometer (shot noise). At lower frequencies, the random force created by the amplitude fluctuations (radiation-pressure noise) will be the main or among the major contributors to the sum noise.
It is important that these noise sources both have the same quantum origin, stemming from the
fundamental quantum uncertainties of the electromagnetic field, and thus that they obey the Heisenberg
uncertainty principle and can not be reduced simultaneously [38]. In particular, the shot noise can (and
will, in the second generation detectors) be reduced by means of the optical power increase. However, as a
result, the radiation-pressure noise will increase. In the ‘naively’ designed measurement schemes, built on
the basis of a Michelson interferometer, kin to the first and the second generation GW detectors, but with
sensitivity chiefly limited by quantum noise, the best strategy for reaching a maximal sensitivity at a given
spectral frequency would be to make these noise source contributions (at this frequency) in the total noise
budget equal. The corresponding sensitivity point is known as the Standard Quantum Limit
(SQL) [16, 22
].
This limitation is by no means an absolute one, and can be evaded using more sophisticated
measurement schemes. Starting from the first pioneering works oriented on solid-state GW
detectors [28, 29, 144], many methods of overcoming the SQL were proposed, including the ones
suitable for practical implementation in laser-interferometer GW detectors. The primary goal
of this review is to give a comprehensive introduction of these methods, as well as into the
underlying theory of linear quantum measurements, such that it remains comprehensible to a broad
audience.
The paper is organized as follows. In Section 2, we give a classical (that is, non-quantum) treatment of the problem, with the goal to familiarize the reader with the main components of laser GW detectors. In Section 3 we provide the necessary basics of quantum optics. In Section 4 we demonstrate the main principles of linear quantum measurement theory, using simplified toy examples of the quantum optical position meters. In Section 5, we provide the full-scale quantum treatment of the standard Fabry–Pérot–Michelson topology of the modern optical GW detectors. At last, in Section 6, we consider three methods of overcoming the SQL, which are viewed now as the most probable candidates for implementation in future laser GW detectors. Concluding remarks are presented in Section 7. Throughout the review we use the notations and conventions presented in Table 1 below.
Notation and value |
Comments |
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coherent state of light with dimensionless complex
amplitude |
![]() |
normalized detuning |
![]() |
interferometer half-bandwidth |
![]() |
effective bandwidth |
![]() |
optical pump detuning from the cavity resonance
frequency |
![]() |
excess quantum noise due to optical losses in the
detector readout system with quantum efficiency
|
![]() |
space-time-dependent argument of the field
strength of a light wave, propagating in the
positive direction of the |
![]() |
quantum efficiency of the readout system (e.g., of a photodetector) |
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squeeze angle |
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some short time interval |
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optical wave length |
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reduced mass |
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mechanical detuning from the resonance frequency |
![]() |
SQL beating factor |
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signal-to-noise ratio |
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miscellaneous time intervals; in particular, |
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homodyne angle |
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|
![]() |
general linear time-domain susceptibility |
![]() |
probe body mechanical succeptibility |
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optical band frequencies |
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interferometer resonance frequency |
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optical pumping frequency |
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mechanical band frequencies; typically,
|
![]() |
mechanical resonance frequency |
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quantum noise “corner frequency” |
![]() |
power absorption factor in Fabry–Pérot cavity per bounce |
![]() |
annihilation and creation operators of photons
with frequency |
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two-photon amplitude quadrature operator |
![]() |
two-photon phase quadrature operator |
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Symmetrised
(cross)
correlation
of
the
field
quadrature
operators
(
![]() |
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|
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light beam cross section area |
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speed of light |
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light quantization normalization constant |
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Resonance denominator of the optical cavity transfer function, defining its characteristic conjugate frequencies (“cavity poles”) |
![]() |
electric field strength |
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classical complex amplitude of the light |
![]() |
classical quadrature amplitudes of the light |
![]() |
vector of classical quadrature amplitudes |
![]() |
back-action force of the meter |
![]() |
signal force |
![]() |
dimensionless GW signal (a.k.a. metrics variation) |
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homodyne vector |
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Hamiltonian of a quantum system |
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Planck’s constant |
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identity matrix |
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optical power |
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circulating optical power in a cavity |
![]() |
circulating optical power per interferometer arm cavity |
![]() |
normalized circulating power |
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optical pumping wave number |
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rigidity, including optical rigidity |
![]() |
Kimble’s optomechanical coupling factor |
![]() |
optomechanical coupling factor of the Sagnac speed meter |
![]() |
cavity length |
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probe-body mass |
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general linear meter readout observable |
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matrix of counterclockwise rotation (pivoting) by
angle |
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amplitude squeezing factor ( |
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power squeezing factor in decibels |
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power reflectivity of a mirror |
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reflection matrix of the Fabry–Pérot cavity |
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noise power spectral density (double-sided) |
![]() |
measurement noise power spectral density (double-sided) |
![]() |
back-action noise power spectral density (double-sided) |
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cross-correlation power spectral density (double-sided) |
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vacuum quantum state power spectral density matrix |
![]() |
squeezed quantum state power spectral density matrix |
![]() |
squeezing matrix |
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power transmissivity of a mirror |
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transmissivity matrix of the Fabry–Pérot cavity |
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test-mass velocity |
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optical energy |
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Wigner function of the quantum state |
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test-mass position |
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dimensionless oscillator (mode) displacement operator |
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dimensionless oscillator (mode) momentum operator |
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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