The morethantenyearslong history of the largescale laser gravitationwave (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 GermanBritish interferometer GEO 600 [66] located near Hannover, Germany, and the joint European largescale 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 secondgeneration 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 secondgeneration 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 (radiationpressure 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 radiationpressure 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 solidstate GW detectors [28, 29, 144], many methods of overcoming the SQL were proposed, including the ones suitable for practical implementation in laserinterferometer 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, nonquantum) 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 fullscale 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 
coherent state of light with dimensionless complex amplitude 

normalized detuning 

interferometer halfbandwidth 

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 

spacetimedependent argument of the field strength of a light wave, propagating in the positive direction of the axis 

quantum efficiency of the readout system (e.g., of a photodetector) 

squeeze angle 

some short time interval 

optical wave length 

reduced mass 

mechanical detuning from the resonance frequency 

SQL beating factor 

signaltonoise ratio 

miscellaneous time intervals; in particular, 

homodyne angle 



general linear timedomain susceptibility 

probe body mechanical succeptibility 

optical band frequencies 

interferometer resonance frequency 

optical pumping frequency 

mechanical band frequencies; typically, 

mechanical resonance frequency 

quantum noise “corner frequency” 

power absorption factor in Fabry–Pérot cavity per bounce 

annihilation and creation operators of photons with frequency 

twophoton amplitude quadrature operator 

twophoton phase quadrature operator 

Symmetrised
(cross)
correlation
of
the
field
quadrature
operators
()




light beam cross section area 

speed of light 

light quantization normalization constant 

Resonance denominator of the optical cavity transfer function, defining its characteristic conjugate frequencies (“cavity poles”) 

electric field strength 

classical complex amplitude of the light 

classical quadrature amplitudes of the light 

vector of classical quadrature amplitudes 

backaction force of the meter 

signal force 

dimensionless GW signal (a.k.a. metrics variation) 

homodyne vector 

Hamiltonian of a quantum system 

Planck’s constant 

identity matrix 

optical power 

circulating optical power in a cavity 

circulating optical power per interferometer arm cavity 

normalized circulating power 

optical pumping wave number 

rigidity, including optical rigidity 

Kimble’s optomechanical coupling factor 

optomechanical coupling factor of the Sagnac speed meter 

cavity length 

probebody mass 

general linear meter readout observable 

matrix of counterclockwise rotation (pivoting) by angle 

amplitude squeezing factor () 

power squeezing factor in decibels 

power reflectivity of a mirror 

reflection matrix of the Fabry–Pérot cavity 

noise power spectral density (doublesided) 

measurement noise power spectral density (doublesided) 

backaction noise power spectral density (doublesided) 

crosscorrelation power spectral density (doublesided) 

vacuum quantum state power spectral density matrix 

squeezed quantum state power spectral density matrix 

squeezing matrix 

power transmissivity of a mirror 

transmissivity matrix of the Fabry–Pérot cavity 

testmass velocity 

optical energy 

Wigner function of the quantum state 

testmass position 

dimensionless oscillator (mode) displacement operator 

dimensionless oscillator (mode) momentum operator 

http://www.livingreviews.org/lrr20125 
Living Rev. Relativity 15, (2012), 5
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