1 Introduction

The more-than-ten-years-long history of the large-scale laser gravitation-wave (GW) detectors (the first one, TAMA [142Jump To The Next Citation Point] started to operate in 1999, and the most powerful pair, the two detectors of the LIGO project [98Jump To The Next Citation Point], 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 [66Jump To The Next Citation Point] located near Hannover, Germany, and the joint European large-scale detector Virgo [156Jump To The Next Citation Point], 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 [143Jump To The Next Citation Point, 64Jump To The Next Citation Point, 4Jump To The Next Citation Point, 169Jump To The Next Citation Point, 6, 96Jump To The Next Citation Point]. 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 [137Jump To The Next Citation Point], 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 [38Jump To The Next Citation Point]. 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, 22Jump To The Next Citation Point].

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 [28Jump To The Next Citation Point, 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.

Table 1: Notations and conventions, used in this review, given in alphabetical order for both, greek (first) and latin (after greek) symbols.
Notation and value


|α ⟩

coherent state of light with dimensionless complex amplitude α

β = arctan γβˆ•δ

normalized detuning


interferometer half-bandwidth

∘ -2----2 Γ = γ + δ

effective bandwidth

δ = ωp − ω0

optical pump detuning from the cavity resonance frequency ω0

∘ ------- 1-- πœ–d = ηd − 1

excess quantum noise due to optical losses in the detector readout system with quantum efficiency ηd

ζ = t − x βˆ•c

space-time-dependent argument of the field strength of a light wave, propagating in the positive direction of the x-axis


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


squeeze angle


some short time interval


optical wave length


reduced mass

ν = Ω − Ω0

mechanical detuning from the resonance frequency

∘ ------ S ξ = ----- SSQL

SQL beating factor


signal-to-noise ratio

τ = Lβˆ•c

miscellaneous time intervals; in particular, L βˆ•c


homodyne angle

φ = Ο•LO − β

i χAB (t,t′) = ℏ[A ˆ(t), ˆB(t′)]

general linear time-domain susceptibility


probe body mechanical succeptibility


optical band frequencies


interferometer resonance frequency


optical pumping frequency


mechanical band frequencies; typically, Ω = ω − ωp


mechanical resonance frequency

∘ ------ 2Sβ„±β„±-- Ωq = ℏM

quantum noise “corner frequency”


power absorption factor in Fabry–Pérot cavity per bounce

† ˆa(ω),ˆa (ω)

annihilation and creation operators of photons with frequency ω

ˆa(ω0-+-Ω-) +-ˆa†(ω0-−-Ω-) ˆac(Ω ) = √2--

two-photon amplitude quadrature operator

ˆa(ω + Ω ) − ˆa†(ω − Ω) ˆas(Ω ) = ----0-----√------0----- i 2

two-photon phase quadrature operator

′ ⟨aˆi(Ω ) ∘ ˆaj(Ω )⟩ ≡

Symmetrised (cross) correlation of the field quadrature operators (i,j = c,s)

1 ′ ′ 2⟨ˆai(Ω )ˆaj(Ω ) + ˆaj(Ω )ˆai(Ω )⟩


light beam cross section area


speed of light

∘ ------- 4π-ℏωp- π’ž0 = π’œc

light quantization normalization constant

π’Ÿ = (γ − iΩ )2 + δ2

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


electric field strength


classical complex amplitude of the light

√ -- √ -- β„°c = 2Re[β„°],β„°s = 2Im [β„° ]

classical quadrature amplitudes of the light

[ ] β„°β„°β„° = β„°c β„°s

vector of classical quadrature amplitudes


back-action force of the meter


signal force


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

[ ] cosΟ•LO H = sin Ο• LO

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

4ω0 ℐc J = -M-cL-

normalized circulating power

kp = ωpβˆ•c

optical pumping wave number


rigidity, including optical rigidity

----2Jγ----- 𝒦 = Ω2(γ2 + Ω2)

Kimble’s optomechanical coupling factor

4J γ 𝒦SM = --2----2-2- (γ + Ω )

optomechanical coupling factor of the Sagnac speed meter


cavity length


probe-body mass


general linear meter readout observable

[ ] cosα − sinα β„™[α] = sin α cosα

matrix of counterclockwise rotation (pivoting) by angle α


amplitude squeezing factor (r e)

rdB = 20rlog10e

power squeezing factor in decibels


power reflectivity of a mirror

ℝ(Ω )

reflection matrix of the Fabry–Pérot cavity

S(Ω )

noise power spectral density (double-sided)


measurement noise power spectral density (double-sided)

Sβ„±β„± (Ω)

back-action noise power spectral density (double-sided)


cross-correlation power spectral density (double-sided)

π•Švac(Ω ) = 1𝕀 2

vacuum quantum state power spectral density matrix

π•Šsqz(Ω )

squeezed quantum state power spectral density matrix

[ ] er 0 π•Šsqz[r,πœƒ] = β„™ [πœƒ] 0 e−r β„™ [− πœƒ]

squeezing matrix


power transmissivity of a mirror


transmissivity matrix of the Fabry–Pérot cavity


test-mass velocity


optical energy

W |ψ⟩(X, Y )

Wigner function of the quantum state |ψ ⟩


test-mass position

ˆX = ˆa+√ˆa†

dimensionless oscillator (mode) displacement operator

dimensionless oscillator (mode) momentum operator

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