Quantum Measurement Theory in GravitationalWave Detectors
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Abstract
The fast progress in improving the sensitivity of the gravitationalwave detectors, we all have witnessed in the recent years, has propelled the scientific community to the point at which quantum behavior of such immense measurement devices as kilometerlong interferometers starts to matter. The time when their sensitivity will be mainly limited by the quantum noise of light is around the corner, and finding ways to reduce it will become a necessity. Therefore, the primary goal we pursued in this review was to familiarize a broad spectrum of readers with the theory of quantum measurements in the very form it finds application in the area of gravitationalwave detection. We focus on how quantum noise arises in gravitationalwave interferometers and what limitations it imposes on the achievable sensitivity. We start from the very basic concepts and gradually advance to the general linear quantum measurement theory and its application to the calculation of quantum noise in the contemporary and planned interferometric detectors of gravitational radiation of the first and second generation. Special attention is paid to the concept of the Standard Quantum Limit and the methods of its surmounting.
1 Introduction
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.
Notations and conventions, used in this review, given in alphabetical order for both, greek (first) and latin (after greek) symbols.
Notation and value  Comments 

α〉  coherent state of light with dimensionless complex amplitude α 
β = arctan γ/δ  normalized detuning 
γ  interferometer halfbandwidth 
\(\Gamma = \sqrt {{\gamma ^2} + {\delta ^2}}\)  effective bandwidth 
δ = ω_{ p } − ω_{0}  optical pump detuning from the cavity resonance frequency ω_{0} 
\({\epsilon_d} = \sqrt {{1 \over {{\eta _d}}}  1}\)  excess quantum noise due to optical losses in the detector readout system with quantum efficiency η_{ d } 
ζ = t − x/c  spacetimedependent argument of the field strength of a light wave, propagating in the positive direction of the xaxis 
η _{ d }  quantum efficiency of the readout system (e.g., of a photodetector) 
θ  squeeze angle 
ϑ, ε  some short time interval 
λ  optical wave length 
μ  reduced mass 
v = ω − ω_{0}  mechanical detuning from the resonance frequency 
\(\xi = \sqrt {{S \over {{S_{{\rm{SQL}}}}}}}\)  SQL beating factor 
ρ  signaltonoise ratio 
τ = L/c  miscellaneous time intervals; in particular, L/c 
ϕ _{LO}  homodyne angle 
φ = ϕ_{LO} − β  
\({\chi _{A \, B}}(t, t\prime) = {i \over \hbar}[\hat A(t),\;\hat B(t\prime)]\)  general linear timedomain susceptibility 
χ _{ xx }  probe body mechanical succeptibility 
ω  optical band frequencies 
ω _{0}  interferometer resonance frequency 
ω _{ p }  optical pumping frequency 
ω  mechanical band frequencies; typically, ω = ω − ω_{ p } 
ω_{0}  mechanical resonance frequency 
\({\Omega _q} = \sqrt {{{2{S_{{\mathcal F}{\mathcal F}}}} \over {\hbar M}}}\)  quantum noise “corner frequency” 
A  power absorption factor in FabryPérot cavity per bounce 
â(ω), â^{†}(ω)  annihilation and creation operators of photons with frequency ω 
\({\hat a_c}(\Omega) = {{\hat a({\omega _0} + \Omega) + {{\hat a}^\dagger}({\omega _0}  \Omega)} \over {\sqrt 2}}\)  twophoton amplitude quadrature operator 
\({\hat a_s}(\Omega) = {{\hat a({\omega _0} + \Omega)  {{\hat a}^\dagger}({\omega _0}  \Omega)} \over {i\sqrt 2}}\)  twophoton phase quadrature operator 
\(\langle {\hat a_i}(\Omega) \circ {\hat a_j}(\Omega \prime)\rangle \equiv {1 \over 2}\langle {\hat a_i}(\Omega){\hat a_j}(\Omega \prime) + {\hat a_j}(\Omega \prime){\hat a_i}(\Omega)\rangle\)  Symmetrised (cross) correlation of the field quadrature operators (i, j = c, s) 
\({\mathcal A}\)  light beam cross section area 
c  speed of light 
\({{\mathcal C}_0} = \sqrt {{{4\pi \hbar {\omega _p}} \over {{\mathcal A}c}}}\)  light quantization normalization constant 
\({\mathcal D} = {(\gamma  i\Omega)^2} + {\delta ^2}\)  Resonance denominator of the optical cavity transfer function, defining its characteristic conjugate frequencies (“cavity poles”) 
E  electric field strength 
ℰ  classical complex amplitude of the light 
\({{\mathcal E}_c} = \sqrt 2 {\rm{Re}}[{\mathcal E}],\,{{\mathcal E}_s} = \sqrt 2 {\rm{Im}}[{\mathcal E}]\)  classical quadrature amplitudes of the light 
\({\bf{\mathcal E}} = \;\,\left[ {\begin{array}{*{20}c} {{{\mathcal E}_c}} \\ {{{\mathcal E}_s}} \\ \end{array}} \right]\)  vector of classical quadrature amplitudes 
\({\hat F_{{\rm{b.a.}}}}\)  backaction force of the meter 
G  signal force 
h  dimensionless GW signal (a.k.a. metrics variation) 
\(H = \left[ {\begin{array}{*{20}c} {\cos {\phi _{{\rm{LO}}}}} \\ {\sin {\phi _{{\rm{LO}}}}} \\ \end{array}} \right]\)  homodyne vector 
\(\hat{\mathcal H}\)  Hamiltonian of a quantum system 
ℏ  Planck’s constant 
\({\mathbb I}\)  identity matrix 
\({\mathcal I}\)  optical power 
\({{\mathcal I}_c}\)  circulating optical power in a cavity 
\({{\mathcal I}_{{\rm{arm}}}}\)  circulating optical power per interferometer arm cavity 
\(J = {{4{\omega _0}{{\mathcal I}_c}} \over {McL}}\)  normalized circulating power 
k_{ p } = ω_{ p }/c  optical pumping wave number 
K  rigidity, including optical rigidity 
\({\mathcal K} = {{2J\gamma} \over {{\Omega ^2}({\gamma ^2} + {\Omega ^2})}}\)  Kimble’s optomechanical coupling factor 
\({{\mathcal K}_{{\rm{SM}}}} = {{4J\gamma} \over {{{({\gamma ^2} + {\Omega ^2})}^2}}}\)  optomechanical coupling factor of the Sagnac speed meter 
L  cavity length 
M  probebody mass 
O  general linear meter readout observable 
\({\mathbb P}[\alpha ] = \left[ {\begin{array}{*{20}c} {\cos \alpha} & { \sin \alpha} \\ {\sin \alpha} & {\cos \alpha} \\ \end{array}} \right]\)  matrix of counterclockwise rotation (pivoting) by angle α 
r  amplitude squeezing factor (e^{ r }) 
r_{dB} = 20r log_{10}e  power squeezing factor in decibels 
R  power reflectivity of a mirror 
ℝ(ω)  reflection matrix of the FabryPérot cavity 
S(ω)  noise power spectral density (doublesided) 
\({S_{{\mathcal X}{\mathcal X}}}(\Omega)\)  measurement noise power spectral density (doublesided) 
\({S_{{\mathcal F}{\mathcal F}}}(\Omega)\)  backaction noise power spectral density (doublesided) 
\({S_{{\mathcal X}{\mathcal F}}}(\Omega)\)  crosscorrelation power spectral density (doublesided) 
\({{\mathbb S}_{{\rm{vac}}}}(\Omega) = {1 \over 2}{\mathbb I}\)  vacuum quantum state power spectral density matrix 
\({{\mathbb S}_{{\rm{sqz}}}}(\Omega)\)  squeezed quantum state power spectral density matrix 
\({{\mathbb S}_{{\rm{sqz}}}}[r,\theta ] = {\mathbb P}[\theta ]\;\,\left[ {\begin{array}{*{20}c} {{e^r}} & 0 \\ 0 & {{e^{ r}}} \\ \end{array}} \right]\;\,{\mathbb P}[  \theta ]\)  squeezing matrix 
T  power transmissivity of a mirror 
\({\mathbb T}\)  transmissivity matrix of the FabryPérot cavity 
υ  testmass velocity 
\({\mathcal W}\)  optical energy 
W_{ψ〉}(X, Y)  Wigner function of the quantum state ψ〉 
X  testmass position 
\(\hat X = {{\hat a + {{\hat a}^\dagger}} \over {\sqrt 2}}\)  dimensionless oscillator (mode) displacement operator 
\(\hat Y = {{\hat a  {{\hat a}^\dagger}} \over {i\sqrt 2}}\)  dimensionless oscillator (mode) momentum operator 
2 Interferometry for GW Detectors: Classical Theory
2.1 Interferometer as a weak force probe
In order to have a firm basis for understanding how quantum noise influences the sensitivity of a GW detector it would be illuminating to give a brief description of the interferometers as weak force/tiny displacement meters. It is by no means our intention to give a comprehensive survey of this ample field that is certainly worthy of a good book, which there are in abundance, but rather to provide the reader with the wherewithal for grasping the very principles of the GW interferometers operation as well as of other similar ultrasensitive optomechanical gauges. The reader interested in a more detailed description of the interferometric techniques being used in the field of GW detectors might enjoy reading this book [12] or the comprehensive Living Reviews on the subject by Freise and Strain [59] and by Pitkin et al. [123].
2.1.1 Light phase as indicator of a weak force
2.1.2 Michelson interferometer
Above, we assumed a direct light phase measurement with a hypothetical device in order to detect a weak external force, possibly created by a GW. However, in reality, direct phase measurement are not so easy to realize at optical frequencies. At the same time, physicists know well how to measure light intensity (amplitude) with very high precision using different kinds of photodetectors ranging from ancientyetdiehard reliable photographic plates to superconductive photodetectors capable of registering individual photons [67]. How can one transform the signal, residing in the outgoing light phase, into amplitude or intensity variation? This question is rhetorical for physicists, for interference of light as well as the multitude of interferometers of various design and purpose have become common knowledge since a couple of centuries ago. Indeed, the amplitude of the superposition of two coherent waves depends on the relative phase of these two waves, thus transforming phase variation into the variation of the light amplitude.
2.1.3 Gravitational waves’ interaction with interferometer
The foregoing conclusion can be understood in the following way: for GWs are inherently quadruple and, when the detector’s plane is orthogonal to the wave propagation direction, can only excite a differential mechanical motion of its mirrors, one can reduce a complicated dynamics of the interferometer probe masses to the dynamics of a single effective particle that is the differential motion of the mirrors in the arms. This useful observation appears to be invaluably helpful for calculation of the real complicated interferometer responses to GWs and also for estimation of its optical quantum noise, that comprises the rest of this review.
2.2 From incident wave to outgoing light: light transformation in the GW interferometers
To proceed with the analysis of quantum noise in GW interferometers we first need to familiarize ourselves with how a light field is transformed by an interferometer and how the ability of its mirrors to move modifies the outgoing field. In the following paragraphs, we endeavor to give a stepbystep introduction to the mathematical description of light in the interferometer and the interaction with its movable mirrors.
2.2.1 Light propagation
We first consider how the light wave is described and how its characteristics transform, when it propagates from one point of free space to another. Yet the real light beams in the large scale interferometers have a rather complicated inhomogeneous transverse spatial structure, the approximation of a plane monochromatic wave should suffice for our purposes, since it comprises all the necessary physics and leads to right results. Inquisitive readers could find abundant material on the field structure of light in real optical resonators in particular, in the introductory book [171] and in the Living Review by Vinet [154].
It is worthwhile to note that the quadrature amplitudes representation is used more frequently in literature devoted to quantum noise calculation in GW interferometers than the complex amplitudes formalism and there is a historical reason for this. Notwithstanding the fact that these two descriptions are absolutely equivalent, the quadrature amplitudes representation was chosen by Caves and Schumaker as a basis for their twophoton formalism for the description of quantum fluctuations of light [39, 40] that became from then on the workhorse of quantum noise calculation. More details about this extremely useful technique are given in Sections 3.1 and 3.2 of this review. Unless otherwise specified, we predominantly keep ourselves to this formalism and give all results in terms of it.
2.2.2 Modulation of light
2.2.3 Laser noise
Thus far we have assumed the carrier field to be perfectly monochromatic having a single spectral component at carrier frequency ω_{0} fully characterized by a pair of classical quadrature amplitudes represented by a 2vector ℰ. In reality, this picture is no good at all; indeed, a real laser emits not a monochromatic light but rather some spectral line of finite width with its central frequency and intensity fluctuating. These fluctuations are usually divided into two categories: (i) quantum noise that is associated with the spontaneous emission of photons in the gain medium, and (ii) technical noise arising, e.g., from excess noise of the pump source, from vibrations of the laser resonator, or from temperature fluctuations and so on. It is beyond the goals of this review to discuss the details of the laser noise origin and methods of its suppression, since there is an abundance of literature on the subject that a curious reader might find interesting, e.g., the following works [119, 120, 121, 167, 68, 76].
2.2.4 Light reflection from optical elements
So, we are one step closer to understanding how to calculate the quantum noise of the light coming out of the GW interferometer. It is necessary to understand what happens with light when it is reflected from such optical elements as mirrors and beamsplitters. Let us first consider these elements of the interferometer fixed at their positions. The impact of mirror motion will be considered in the next Section 2.2.5. One can also refer to Section 2 of the Living Review by Freise and Strain [59] for a more detailed treatment of this topic.
In future, for the sake of brevity, we reduce the notation for matrices like \({{\mathbb M}_{{\rm{real}}}}\, \otimes {\mathbb I}\) to simply \({{\mathbb M}_{{\rm{real}}}}\).
 1.Optical loss of an optical element can be characterized by a single number (possibly, frequency dependent) ϵ (usually, ϵ ≪ 1) that is called the absorption coefficient. ϵ is the fraction of light power being lost in the optical element:$${E^{{\rm{out}}}}(t) \rightarrow \sqrt {1  \epsilon} {E^{{\rm{out}}}}(t).$$
 2.
Due to the fundamental law of nature summarized by the Fluctuation Dissipation Theorem (FDT) [37, 95], optical loss is always accompanied by additional noise injected into the system. It means that additional noise field \(\hat n\) uncorrelated with the original light is mixed into the outgoing light field in the proportion of \(\sqrt \epsilon\) governed by the absorption coefficient.
2.2.5 Light modulation by mirror motion
For full characterization of the light transformation in the GW interferometers, one significant aspect remains untouched, i.e., the field transformation upon reflection off the movable mirror. Above (see Section 2.1.1), we have seen that motion of the mirror yields phase modulation of the reflected wave. Let us now consider this process in more detail.
It is important to understand that signal sidebands characterized by a vector {s_{1}(Ω), s_{2}(Ω)}^{⊤}, on the one hand, and the noise sidebands {ê_{1}(Ω), ê_{2}(Ω)}^{⊤}, on the other hand, have the same order of magnitude in the real GW interferometers, and the main role of the advanced quantum measurement techniques we are talking about here is to either increase the former, or decrease the latter as much as possible in order to make the ratio of them, known as the signaltonoise ratio (SNR), as high as possible in as wide as possible a frequency range.
2.2.6 Simple example: the reflection of light from a perfect moving mirror
This fact, i.e., that the mirror displacement that just causes phase modulation of the reflected field, enters only the squadrature, once again justifies why this quadrature is usually referred to as phase quadrature (cf. Section 2.2.2).
2.3 Basics of Detection: Heterodyne and homodyne readout techniques
Let us now address the question of how one can detect a GW signal imprinted onto the parameters of the light wave passing through the interferometer. The simple case of a Michelson interferometer considered in Section 2.1.2 where the GW signal was encoded in the phase quadrature of the light leaking out of the signal(dark) port, does not exhaust all the possibilities. In more sophisticated interferometer setups that are covered in Sections 5 and ??, a signal component might be present in both quadratures of the outgoing light and, actually, to different extent at different frequencies; therefore, a detection method that allows measurement of an arbitrary combination of amplitude and phase quadrature is required. The two main methods are in use in contemporary GW detectors: these are homodyne and heterodyne detection [36, 138, 164, 79]. Both are common in radiofrequency technology as methods of detection of phase and frequencymodulated signals. The basic idea is to mix a faint signal wave with a strong local oscillator wave, e.g., by means of a beamsplitter, and then send it to a detector with a quadratic nonlinearity that shifts the spectrum of the signal to lower frequencies together with amplification by an amplitude of the local oscillator. This topic is also discussed in Section 4 of the Living Review by Freise and Strain [59] with more details relevant to experimental implementation.
2.3.1 Homodyne and DC readout
However, in real interferometers, the implementation of a homodyne readout appears to be fraught with serious technical difficulties. In particular, the local oscillator frequency has to be kept extremely stable, which means its optical path length and alignment need to be actively stabilized by a lownoise control system [79]. This inflicts a significant increase in the cost of the detector, not to mention the difficulties in taming the noise of stabilising control loops, as the experience of the implementation of such stabilization in a Garching prototype interferometer has shown [60, 75, 74].
DC readout. These factors provide a strong motivation to look for another way to implement homodyning. Fortunately, the search was not too long, since the suitable technique has already been used by Michelson and Morley in their seminal experiment [109]. The technique is known as DCreadout and implies an introduction of a constant arms length difference, thus pulling the interferometer out of the dark fringe condition as was mentioned in Section 2.1.2. The advantage of this method is that the local oscillator is furnished by a part of the pumping carrier light that leaks into the signal port due to arms imbalance and thus shares the optical path with the signal sidebands. It automatically solves the problem of phaselocking the local oscillator and signal lights, yet is not completely free of drawbacks. The first suggestion to use DC readout in GW interferometers belongs to Fritschel [63] and then got comprehensive study by the GW community [138, 164, 79].
In the case of a small offset of the interferometer from the dark fringe condition, i.e., for ω_{0}ΔL/c = 2πΔL/λ_{0} ≪ 1, the readout signal scales as local oscillator classical amplitude, which is directly proportional to the offset itself: \(L_{{\rm{DC}}}^{(0)} \simeq 2\pi {E_0}{{\Delta L} \over {{\lambda _0}}}\). The laser noise associated with the pumping carrier also leaks to the signal port in the same proportion, which might be considered as the main disadvantage of the DC readout as it sets rather tough requirements on the stability of the laser source, which is not necessary for the homodyne readout. However, this problem, is partly solved in more sophisticated detectors by implementing power recycling and/or FabryPérot cavities in the arms. These additional elements turn the Michelson interferometer into a resonant narrowband cavity for a pumping carrier with effective bandwidth determined by transmissivities of the power recycling mirror (PRM) and/or input test masses (ITMs) of the arm cavities divided by the corresponding cavity length, which yields the values of bandwidths as low as ∼ 10 Hz. Since the target GW signal occupies higher frequencies, the laser noise of the local oscillator around signal frequencies turns out to be passively filtered out by the interferometer itself.
DC readout has already been successfully tested at the LIGO 40meter interferometer in Caltech [164] and implemented in GEO 600 [77, 79, 55] and in Enhanced LIGO [61, 5]. It has proven a rather promising substitution for the previously ubiquitous heterodyne readout (to be considered below) and has become a baseline readout technique for future GW detectors [79].
2.3.2 Heterodyne readout
Up until recently, the only readout method used in terrestrial GW detectors has been the heterodyne readout. Yet with more and more stable lasers being available for the GW community, this technique gradually gives ground to a more promising DC readout method considered above. However, it is instructive to consider briefly how heterodyne readout works and learn some of the reasons, that it has finally given way to its homodyne adversary.
This method was proposed and studied in great detail in the following works [65, 134, 60, 75, 74, 104, 116] where the heterodyne technique for GW interferometers tuned in resonance with pumping carrier field was considered and, therefore, the focus was made on the detection of only phase quadrature of the outgoing GW signal light. This analysis was further generalized to detuned interferometer configurations in [36, 138] where the full analysis of quantum noise in GW dualrecycled interferometers with heterodyne readout was done.
For more realistic and thus more sophisticated optical configurations, including FabryPérot cavities in the arms and additional recycling mirrors in the pumping and readout ports, the analysis of the interferometer sensitivity becomes rather complicated. Nevertheless, it is worthwhile to note that with proper optimization of the modulation sidebands and demodulation function shapes the same sensitivity as for frequencyindependent homodyne readout schemes can be obtained [36]. However, inherent additional frequencyindependent quantum shot noise brought by the heterodyning process into the detection band hampers the simultaneous use of advanced quantum nondemolition (QND) techniques and heterodyne readout schemes significantly.
3 Quantum Nature of Light and Quantum Noise
Now is the time to remind ourselves of the word ‘quantum’ in the title of our review. Thus far, the quantum nature of laser light being used in the GW interferometers has not been accounted for in any way. Nevertheless, quantum mechanics predicts striking differences for the variances of laser light amplitude and phase fluctuations, depending on which quantum state it is in. Squeezed vacuum [163, 99, 136, 38, 90] injection that has been recently implemented in the GEO 600 detector and has pushed the highfrequency part of the total noise down by 3.5 dB [151, 1] serves as a perfect example of this. In this section, we provide a brief introduction into the quantization of light and the typical quantum states thereof that are common for the GW interferometers.
3.1 Quantization of light: Twophoton formalism
In GW detectors, one deals normally with a close to monochromatic laser light with carrier frequency ω_{0}, and a pair of modulation sidebands created by a GW signal around its frequency in the course of parametric modulation of the interferometer arm lengths. The light field coming out of the interferometer cannot be considered as the continuum of independent modes anymore. The very fact that sidebands appear in pairs implies the twophoton nature of the processes taking place in the GW interferometers, which means the modes of light at frequencies ω_{1,2} = ω_{0} ± Ω have correlated complex amplitudes and thus the new twophoton operators and related formalism is necessary to describe quantum light field transformations in GW interferometers. This was realized in the early 1980s by Caves and Schumaker who developed the twophoton formalism [39, 40], which is widely used in GW detectors as well as in quantum optics and optomechanics.
Now, when we have defined a quantum Heisenberg operator of the electric field of a light wave, and introduced quantum operators of twophoton quadratures, the last obstacle on our way towards the description of quantum noise in GW interferometers is that we do not know the quantum state the light field finds itself in. Since it is the quantum state that defines the magnitude and mutual correlations of the amplitude and phase fluctuations of the outgoing light, and through it the total level of quantum noise setting the limit on the future GW detectors’ sensitivity. In what follows, we shall consider vacuum and coherent states of the light, and also squeezed states, for they comprise the vast majority of possible states one could encounter in GW interferometers.
3.2 Quantum states of light
3.2.1 Vacuum state
It is instructive to discuss the meaning of these matrices, \({\mathbb S}\) and \({\mathbb V}\), and of the values they comprise. To do so, let us think of the light wave as a sequence of very short squarewave light pulses with infmitesimally small duration ε → 0. The delta function of time in Eq. (66) tells us that the noise levels at different times, i.e., the amplitudes of the different square waves, are statistically independent. To put it another way, this noise is Markovian. It is also evident from Eq. (65) that quadrature amplitudes’ fluctuations are stationary, and it is this stationarity, as noted in [39] that makes quadrature amplitudes such a convenient language for describing the quantum noise of light in parametric systems exemplified by GW interferometers.
Note the difference between Figures 13 and 14; the former features the result of measurement of an ensemble of oscillators (subsequent light pulses with infmitesimally short duration ε), while the latter gives the probability density function for a single oscillator displacement and momentum.
3.2.2 Coherent state
Operator \(\hat D[\alpha ]\) is unitary, i.e., \({\hat D^\dagger}[\alpha ]\hat D[\alpha ] = \hat D[\alpha ]{\hat D^\dagger}[\alpha ] = \hat I\) with Î the identity operator, while the physical meaning is in the translation and rotation of the Hilbert space that keeps all the physical processes unchanged. Therefore, one can simply use vacuum states instead of coherent states and subtract the mean values from the corresponding operators in the same way we have done previously for the light wave classical amplitudes, just below Eq. (60). The covariance matrix and the matrix of power spectral densities for the quantum noise of light in a coherent state is thus the same as that of a vacuum state case.
3.2.3 Squeezed state
One more quantum state of light that is worth consideration is a squeezed state. To put it in simple words, it is a state where one of the oscillator quadratures variance appears decreased by some factor compared to that in a vacuum or coherent state, while the conjugate quadrature variance finds itself swollen by the same factor, so that their product still remains Heisenberglimited. Squeezed states of light are usually obtained as a result of a parametric down conversion (PDC) process [92, 172] in optically nonlinear crystals. This is the most robust and experimentally elaborated way of generating squeezed states of light for various applications, e.g., for GW detectors [149, 152, 141], or for quantum communications and computation purposes [31]. However, there is another way to generate squeezed light by means of a ponderomotive nonlinearity inherent in such optomechanical devices as GW detectors. This method, first proposed by Corbitt et al. [47], utilizes the parametric coupling between the resonance frequencies of the optical modes in the FabryPérot cavity and the mechanical motion of its mirrors arising from the quantum radiation pressure fluctuations inflicting random mechanical motion on the cavity mirrors. Further, we will see that the light leaving the signal port of a GW interferometer finds itself in a ponderomotively squeezed state (see, e.g., [90] for details). A dedicated reader might find it illuminating to read the following review articles on this topic [133, 101].
3.3 How to calculate spectral densities of quantum noise in linear optical measurement?
4 Linear Quantum Measurement
In Section 3, we discussed the quantum nature of light and fluctuations of the light field observables like phase and amplitude that stem thereof and yield what is usually called the quantum noise of optical measurement. In GW detection applications, where a sensitivity of the phase measurement is essential, as discussed in Section 2.1.3, the natural question arises: is there a limit to the measurement precision imposed by quantum mechanics? A seemingly simple answer would be that such a limit is set by the quantum fluctuations of the outgoing light phase quadrature, which are, in turn, governed by the quantum state the outgoing light finds itself in. The difficult part is that on its way through the interferometer, the light wave inflicts an additional backaction noise that adds up to the phase fluctuations of the incident wave and contaminates the output of the interferometer. The origin of this back action is in amplitude fluctuations of the incident light, giving rise to a random radiation pressure force that acts on the interferometer mirrors along with the signal GW force, thus effectively mimicking it. And it is the fundamental principle of quantum mechanics, the Heisenberg uncertainty principle, that sets a limit on the product of the phase and amplitude uncertainties (since these are complementary observables), thus leading up to the lower bound of the achievable precision of phase measurement. This limit appears to be a general feature for a very broad class of measurement known as linear measurement and is referred to as the SQL [16, 22].
In this section, we try to give a brief introduction to quantum measurement theory, starting from rather basic examples with discrete measurement and then passing to a general theory of continuous linear measurement. We introduce the concept of the SQL and derive it for special cases of probe bodies. We also discuss briefly possible ways to overcome this limit by contriving smarter ways of weak force measurement then direct coordinate monitoring.
4.1 Quantum measurement of a classical force
4.1.1 Discrete position measurement
This example represents a simple particular case of a linear measurement. This class of measurement schemes can be fully described by two linear equations of the form (98) and (100), provided that both the measurement uncertainty and the object backaction perturbation \(({\hat x_{{\rm{fl}}}}({t_j})\) and \({\hat p^{{\rm{b}}{\rm{.a}}}}({t_j})\) in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case).
4.1.2 From discrete to continuous measurement
 1.
energy and phase fluctuations in each of the light pulses uncorrelated: \(\left\langle {\hat {\mathcal W}({t_j})\hat \phi ({t_j})} \right\rangle = 0\);
 2.
all pulses to have the same energy and phase uncertainties \(\Delta {\mathcal W}\) and Δ_{ ϕ }, respectively;
 3.
the pulses statistically independent from each other, particularly taking \(\left\langle {\hat {\mathcal W}({t_j})\hat {\mathcal W}({t_j})} \right\rangle = \left\langle {\hat \phi ({t_i})\hat \phi ({t_j})} \right\rangle = \left\langle {\hat {\mathcal W}({t_i})\hat \phi ({t_i})} \right\rangle = 0\) with t_{ i } ≠ t_{ j }.
 1.
these noise sources are mutually not correlated;
 2.
they are stationary (invariant to the time shift) and, therefore, can be described by spectral densities S_{ x } and S_{ F };
 3.
they are Markovian (white) with constant (frequencyindependent) spectral densities.
The features 1 and 2, in turn, lead to characteristic fundamentallylooking sensitivity limitations, the SQL. We will call linear quantum meters, which obey these limitations (that is, with mutually noncorrelated and stationary noises \({\hat x_{{\rm{fl}}}}\) and \({\hat F_{{\rm{b}}{\rm{.a}}{\rm{.}}}}\)), Simple Quantum Meters (SQM).
4.2 General linear measurement
In this section, we generalize the concept of linear quantum measurement discussed above and give a comprehensive overview of the formalism introduced in [22] and further elaborated in [33, 43]. This formalism can be applied to any system that performs a transformation from some unknown classical observable (e.g., GW tidal force in GW interferometers) into another classical observable of a measurement device that can be measured with (ideally) arbitrarily high precision (e.g., in GW detectors, the readout photocurrent serves such an observable) and its value depends on the value of unknown observable linearly. For definiteness, let us keep closer to GW detectors and assume the continuous measurement of a classical force.
The meaning of the above equations is worth discussing. The first of Eqs. (129) describes how the readout observable Ô(t) of the meter, say the particular quadrature of the outgoing light field measured by the homodyne detector (cf. Eq. (39)), depends on the actual displacement \(\hat x(t)\) of the probe, and the corresponding susceptibility χ_{ OF }(t − t′) is the transfer function for the meter from \(\hat x\) to Ô. The term Ô^{(0)}(t) stands for the free evolution of the readout observable, provided that there was no coupling between the probe and the meter. In the case of the GW detector, this is just a pure quantum noise of the outgoing light that would have come out were all of the interferometer test masses fixed. It was shown explicitly in [90] and we will demonstrate below that this noise is fully equivalent to that of the input light except for the insignificant phase shift acquired by the light in the course of propagation through the interferometer.
The second equation in (129) describes how the backaction force exerted by the meter on the probe system evolves in time and how it depends on the probe’s displacement. The first term, \({\hat F^{(0)}}(t)\), meaning is rather obvious. In GW interferometer, it is the radiation pressure force that the light exerts on the mirrors while reflecting off them. It depends only on the mean value and quantum fluctuations of the amplitude of the incident light and does not depend on the mirror motion. The second term here stands for a dynamical backaction of the meter and since, by construction, it is the part of the backaction force that depends, in a linear way, from the probe’s displacement, the meaning of the susceptibility χ_{ FF }(t − t′) becomes apparent: it is the generalized rigidity that the meter introduces, effectively modifying the dynamics of the probe. We will see later how this effective rigidity can be used to improve the sensitivity of the GW interferometers without introducing additional noise and thus enhancing the SNR of the GW detection process.
The second value, \({\hat x_{{\rm{b}}{\rm{.a}}{\rm{.}}}}(t)\), is the displacement of the probe due to the backaction force exerted by the meter on the probe. Since it enters the probe’s response in the very same way the signal does, it is the most problematic part of the quantum noise that, as we demonstrate later, imposes the SQL [16, 22].

\(\hat {\mathcal X}\) is the effective output fluctuation of the meter not dependent on the probe. Henceforth, we will refer to it as the effective measurement noise (shot noise, in the GW interferometer common terminology);

\(\hat {\mathcal F}\) is the effective response of the output at time t to the meter’s backaction force at earlier times t < t′. In the following we will refer to \(\hat {\mathcal F}\) as the effective backaction noise (radiationpressure noise, in the GW interferometer common terminology).
The general structure of quantum noise in the linear measurement process, comprising two types of noise sources whose spectral densities are bound by the uncertainty relation (148), gives a clue to several rather important corollaries. One of the most important is the emergence of the SQL, which we consider in detail below.
4.3 Standard Quantum Limit
Recall the SQM in Section 4.1.2.
These limits look fundamental. There are no parameters of the meter (only ħ as a reminder of the uncertainty relation (116)), and only the probe’s dynamics is in there. Nevertheless, this is not the case and, actually, this limit can be beaten by more sophisticated, but still linear, position meters. At the same time, the SQL represents an important landmark beyond which the ordinary bruteforce methods of sensitivity improving cease working, and methods that allow one to blot out the backaction noise \(\hat {\mathcal F}(t)\) from the meter output signal have to be used instead. Due to this reason, the SQL, and especially the SQL for the simplest test object — free mass — is usually considered as a borderline between the classical and the quantum domains.
4.3.1 Free mass SQL
In the rest of this section, we consider in more detail the SQLs for a free mass and for a harmonic oscillator. We also assume the minimal quantum noise requirement (150) to hold.
The free mass is not only the simplest model for the probe’s dynamics, but also the most important class of test objects for GW detection. Test masses of GW detectors must be isolated as much as possible from the noisy environment. To this end, the design of GW interferometers implies suspension of the test masses on thin fibers. The real suspensions are rather sophisticated and comprise several stages slung one over another, with mechanical eigenfrequencies f_{ m } in ≲ 1Hz range. The sufficient degree of isolation is provided at frequencies much higher than f_{ m }, where the dynamics of test masses can be approximated with good precision by that of a free mass.
4.3.2 Harmonic oscillator SQL
4.3.3 Sensitivity in different normalizations. Free mass and harmonic oscillator
Above, we have discussed, in brief, different normalizations of the sum noise spectral density and derived the general expressions for the SQL in these normalizations (cf. Eqs. (153) and (154)). Let us consider how these expressions look for the free mass and harmonic oscillator and how the sensitivity curves transform when changing to different normalizations.
The corresponding plots are drawn in the right panel of Figure 23. Despite a quite different look, in essence, these spectral densities are the same force spectral densities as those drawn in Figure 22, yet tilted rightwards by virtue of factor 1/Ω^{4}. In particular, they are characterized by the same minimum at the resonance frequency, created by the strong response of the harmonic oscillator on a nearresonance force, as the corresponding forcenormalized spectral densities (163, 172).
It looks rather natural at a first glance; however, as we have shown below, it is less heuristic than the force normalization and could even be misleading. Nevertheless, for completeness, we consider this normalization here.
Note that the curves display a sharp upsurge of noise around the resonance frequencies. However, the resonance growth of the displacement due to signal force have a long start over this seeming noise outburst, as we have shown already, leads to the substantial sensitivity gain for a nearresonance force. This sensitivity increase is clearly visible in the force and equivalent displacement normalization, see Figures 22 and 23, but completely masked in Figure 24.
4.4 Beating the SQL by means of noise cancellation
The simplest way is to make the relation (4.4) hold at some fixed frequency, which can always be done either (i) by preparing the meter in some special initial quantum state that has measurement and backaction fluctuations correlated (Unruh [148, 147] proposed to prepare input light in a squeezed state to achieve such correlations), or (ii) by monitoring a linear combination of the probe’s displacement and momentum [162, 159, 158, 160, 161, 51, 53] that can be accomplished, e.g., via homodyne detection, as we demonstrate below.
We consider the basic principles of the schemes, utilizing the noise cancellation via building crosscorrelations between the measurement and backaction noise. We start from the very toy example discussed in Section 4.1.1.
Another evident flaw of the virtual rigidity, which it shares with the real one, is the narrowband character of the sensitivity gain it provides around Ω_{0} and that this band shrinks as the sensitivity gain rises (cf. Eq. (199)). In order to provide a broadband enhancement in sensitivity, either the real rigidity \(K = M\Omega _0^2\), or the virtual one K_{virt} = S_{ xF }/S_{ F } should depend on frequency in such a way as to be proportional (if only approximately) to Ω^{2} in the frequency band of interest. Of all the proposed solutions providing frequency dependent virtual rigidity, the most well known are the quantum speed meter [21] and the filter cavities [90] schemes. Section 4.5, we consider the basic principles of the former scheme. Then, in Section 6 we provide a detailed treatment of both of them.
4.5 Quantum speed meter
4.5.1 The idea of the quantum speed meter
4.5.2 QND measurement of a free mass velocity
5 Quantum Noise in Conventional GW Interferometers
In Section 4, we have talked about the quantum measurement, the general structure of quantum noise implied by the quantum mechanics and the restrictions on the achievable sensitivity it imposes. In this section, we turn to the application of these general and lofty principles to real life, i.e., we are going to calculate quantum noise for several types of the schemes of GW interferometers and consider the advantages and drawbacks they possess.
To grasp the main features of quantum noise in an advanced GW interferometer it would be elucidating to consider first two elementary examples: (i) a single movable mirror coupled to a free optical field, reflecting from it, and (ii) a FabryPérot cavity comprising two movable mirrors and pumped from both sides. These two systems embody all the main features and phenomena that also mold the advanced and more complicated interferometers’ quantum noise. Should one encounter these phenomena in reallife GW detectors, knowledge of how they manifest themselves in these simple situations would be of much help in successfully discerning them.
5.1 Movable mirror
5.1.1 Optical transfer matrix of the movable mirror
5.1.2 Probe’s dynamics: radiation pressure force and ponderomotive rigidity
5.1.3 Spectral densities
5.1.4 Full transfer matrix approach to the calculation of quantum noise spectral densities
It was easy to calculate the above spectral densities by parts, distinguishing the effective measurement and backaction noise sources and making separate calculations for them. Had we considered a bit more complicated situation with the incident fields in the squeezed states with arbitrary squeezing angles, the calculation of all six of the above individual spectral densities (264) and subsequent substitution to the sum spectral densities expressions (263) would be more difficult. Thus, it would be beneficial to have a tool to do all these operations at once numerically.
Thus, we obtain the formula that can be (and, actually, is) used for the calculation of quantum noise spectral densities of any, however complicated, interferometer given the full transfer matrix of this interferometer.
5.1.5 Losses in a readout train
Now, when we have considered all the stages of the quantum noise spectral densities calculation on a simple example of a single movable mirror, we are ready to consider more complicated systems. Our next target is a FabryPérot cavity.
5.2 FabryPérot cavity
A FabryPérot cavity consists of two movable mirrors that are separated by a distance L + x_{1} + x_{2}, where L = cτ is the distance at rest with standing for a single pass light travel time, and x_{1} and x_{2} are the small deviations of the mirrors’ position from the equilibrium. Each of the mirrors is described by the transfer matrix \({{\mathbb M}_{1,2}}\) with real coefficients of reflection \(\sqrt {{R_{1,2}}}\) and transmission \(\sqrt {{T_{1,2}}}\) according to Eq. (243). As indicated on the scheme, the outer faces of the mirrors are assumed to have negative reflectivities. While the intermediate equations depend on this choice, the final results are invariant to it. The cavity is pumped from both sides by two laser sources with the same optical frequency ω_{ p }.
The frequency domain version of the above equations and their solutions are derived in Appendix A.1. We write these I/Orelations given in Eqs. (545) in terms of complex amplitudes instead of 2 photon amplitudes, for the expressions look much more compact in that representation. However, one can simplify them even more using the singlemode approximation.
 (i)in GW detection, rather highfinesse cavities are used, which implies low transmittance coefficients for the mirrors$${T_{1,2}} \ll 1;$$(275)
 (ii)the cavities are relatively short, so their Free Spectral Range (FSR) f_{FSR} = (2τ)^{−1} is much larger than the characteristic frequencies of the mirrors’ motion:and (iii) the detuning of the pump frequency from one of the cavity eigenfrequencies:$$\vert\Omega \vert\tau \ll 1;$$(276)is also small in comparison with the FSR:$$\delta = {\omega _p}  {{\pi n} \over \tau}\quad (n\;{\rm{is\, an\, integer}})$$(277)$$\vert\delta \vert\tau \ll 1.$$(278)
5.2.1 Optical transfer matrix for a FabryPérot cavity
 1.change frequency ω → ω_{ p } ± Ω and rewrite the relations between the input \(\hat \alpha (\omega)\) and output operators \(\hat \beta (\omega)\) in the form:where f(Ω) is an arbitrary complexvalued function of sideband frequency Ω;$$\begin{array}{*{20}c} {\hat \beta (\omega) = f(\Omega)\hat \alpha (\omega)\; \rightarrow \;{{\hat \beta}_ +} \equiv \hat \beta ({\omega _p} + \Omega) = f({\omega _p} + \Omega)\hat \alpha ({\omega _p} + \Omega) \equiv {f_ +}{{\hat \alpha}_ +}\;{\rm{and}}\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\hat \beta _  ^\dagger \equiv {{\hat \beta}^\dagger}({\omega _p}  \Omega) = {f^{\ast}}({\omega _p}  \Omega){{\hat \alpha}^\dagger}({\omega _p}  \Omega) \equiv f_  ^{\ast}\hat \alpha _  ^\dagger \,;} \\ \end{array}$$(288)
 2.use the definition (57) to get the following relations for twophoton quadrature operators:$$\left[ {\begin{array}{*{20}c} {{{\hat \beta}_c}(\Omega)} \\ {{{\hat \beta}_s}(\Omega)} \\ \end{array}} \right] = {1 \over 2}\left[ {\begin{array}{*{20}c} {({f_ +} + f_  ^{\ast})} & {i({f_ +}  f_  ^{\ast})} \\ { i({f_ +}  f_  ^{\ast})} & {({f_ +} + f_  ^{\ast})} \\ \end{array}} \right] \cdot \left[ {\begin{array}{*{20}c} {{{\hat \alpha}_c}(\Omega)} \\ {{{\hat \alpha}_s}(\Omega)} \\ \end{array}} \right].$$(289)
5.2.2 Mirror dynamics, radiation pressure forces and ponderomotive rigidity
5.3 FabryPérotMichelson interferometer
This scheme works similar to the ordinary Michelson interferometer considered briefly in Section 2.1.2. The beamsplitter BS distributes the pump power from the laser evenly between the arms. The beams, reflected off the FabryPérot cavities are recombined on the beamsplitter in such a way that, in the ideal case of perfect symmetry of the arms, all the light goes back to the laser, i.e., keeping the signal (‘south’) port dark. Any imbalance of the interferometer arms, caused by signal forces acting on the end test masses (ETMs) makes part of the pumping light leak into the dark port where it is monitored by a photodetector.
The FabryPérot cavities in the arms, formed by the input test masses (ITMs) and the end test masses, provide the increase of the optomechanical coupling, thus making photons bounce many times in the cavity and therefore carry away a proportionallyamplified mirror displacement signal in their phase (cf. with the F factor in the toy systems considered in Section 4). The two auxiliary recycling mirrors: the PRM and the signal recycling (SRM) allow one to increase the power, circulating inside the FabryPérot cavities, for a given laser power, and provide the means for finetuning of the quantum noise spectral density [103, 155], respectively.
It was shown in [34] that quantum noise of this dual (power and signal) recycled interferometer is equivalent to that of a single FabryPérot cavity with some effective parameters (the analysis in that paper was based on earlier works [112, 128], where the classical regime had been considered). Here we reproduce this scaling law theorem, extending it in two aspects: (i) we factor in optical losses in the arm cavities by virtue of modeling it by the finite transmissivity of the ETMs, and (ii) we do not assume the arm cavities tuned in resonance (the detuned arm cavities could be used, in particular, to create optical rigidity in nonsignalrecycled configurations).
5.3.1 Optical I/Orelations
5.3.2 Common and differential optical modes
5.3.3 Interferometer dynamics: mechanical equations of motion, radiation pressure forces and ponderomotive rigidity
5.3.4 Scaling law theorem
The mechanical equations of motion for the effective cavity are absolutely the same as for an ordinary FabryPérot cavity considered in Section 5.2 except for the new values of the effective mirrors’ mass 2M and effective circulating power \({{\mathcal I}_c} = 2{{\mathcal I}_{{\rm{arm}}}}\). Bearing this in mind, we can procede to the quantum noise spectral density calculation for this interferometer.
5.3.5 Spectral densities for the FabryPérotMichelson interferometer
The scaling law we have derived above enables us to calculate spectral densities of quantum noise for a dualrecycled FabryPérotMichelson featured in Figure 30 as if it were a bare FabryPérot cavity with movable mirrors pumped from one side, similar to that shown in Figure 32.
5.3.6 Full transfer matrix approach to calculation of the FabryPérotMichelson interferometer quantum noise
In order to compute the sum quantum noise spectral density one has to first calculate S_{ χχ }(Ω), \({S_{{\mathcal F}{\mathcal F}}}(\Omega)\) and \({S_{{\mathcal X}{\mathcal F}}}(\Omega)\) using Eqs. (376), (377), and (378) and then insert them into the general formula (144).

It comprises two effective noise sources as in any quantum linear measurement device. These are measurement noise \({\hat {\mathcal X}^{{\rm{loss}}}}\), more frequently called quantum shot noise in the GW community, and the backaction noise \(\hat {\mathcal F}\), often referred to as quantum radiationpressure noise.

These noise sources are correlated and this correlation depends not only on the homodyne angle Õ_{LO} or the correlations in the input light (e.g., squeezing angle θ in case of squeezed input), but also on the interferometer effective detuning δ, which, according to the scaling law theorem, can be changed by varying signalrecycling cavity parameters.

The scaling law theorem also shows that changing the arm cavities’ detuning is equivalent to the modification of the signal recycling cavity parameters in terms of effective detuning and bandwidth of the interferometer.

Another important corollary of the scaling law is that the effective bandwidths and detunings for the common and differential optical modes can be chosen independently, thus making it possible to tune the former in resonance with the pumping laser to keep as high a value of the circulating optical power in the arms as possible, and to detune the latter one to modify the test masses dynamical response by virtue of the introduction of optical rigidity that arises in the detuned cavity as we have shown.
All of these features can be used to decrease the quantum noise of the interferometer and reach a sensitivity below the SQL in a decent range of frequencies as we show in Section 6.
6 Schemes of GW Interferometers with SubSQL Sensitivity
6.1 Noise cancellation by means of crosscorrelation
6.1.1 Introduction
In this section, we consider the interferometer configurations that use the idea of the crosscorrelation of the shot and the radiation pressure noise sources discussed in Section 4.4. This crosscorrelation allows the measurement and the backaction noise to partially cancel each other out and thus effectively reduce the sum quantum noise to below the SQL.
As we noted above, Eq. (378) tells us that this crosscorrelation can be created by tuning either the homodyne angle ϕ_{LO}, the squeezing angle θ, or the detuning δ. In Section 4.4, the simplest particular case of the frequencyindependent correlation created by means of measurement of linear combination of the phase and amplitude quadratures, that is, by using the homodyne angle Õ_{LO} ≠ π/2, has been considered. We were able to obtain a narrowband sensitivity gain at some given frequency that was similar to the one achievable by introducing a constant rigidity to the system, therefore such correlation was called effective rigidity.
However, the broadband gain requires a frequencydependent correlation, as it was first demonstrated for optical interferometric position meters [148], and then for general position measurement case [82]. Later, this idea was developed in different contexts by several authors [81, 118, 159, 90, 70, 69, 149, 9]. In particular, in [90], a practical method of creation of the frequencydependent correlation was proposed, based on the use of additional filter cavities, which were proposed to be placed either between the squeeze light source and the main interferometer, creating the frequencydependent squeezing angle (called prefiltering), or between the main interferometer and the homodyne detector, creating the effective frequencydependent squeezing angle (postfiltering). As we show below, in principle, both pre and postfiltering can be used together, providing some additional sensitivity gain.
It is necessary to note also an interesting method of noise cancellation proposed by Tsang and Caves recently [146]. The idea was to use matched squeezing; that is, to place an additional cavity inside the main interferometer and couple the light inside this additional cavity with the differential mode of the interferometer by means of an optical parametric amplifier (OPA). The squeezed light created by the OPA should compensate for the ponderomotive squeezing created by backaction at all frequencies and thus decrease the quantum noise below the SQL at a very broad frequency band. However, the thorough analysis of the optical losses influence, that as we show later, are ruinous for the subtle quantum correlations this scheme is based on, was not performed.
Coming back to the filtercavitiesbased interferometer topologies, we limit ourselves here by the case of the resonancetuned interferometer, δ = 0. This assumption simplifies all the equations considerably, and allows one to clearly separate the sensitivity gain provided by the quantum noise cancellation due to crosscorrelation from the one provided by the optical rigidity, which will be considered in Section 6.3.
We also neglect optical losses inside the interferometer, assuming that γ_{2} = 0. In broadband interferometer configurations considered here, with typical values of γ ≳ 10^{3} s^{−1}, the influence of these losses is negligible compared to those of the photodetector inefficiency and the losses in the filter cavities. Indeed, taking into account the fact that with modern highreflectivity mirrors, the losses per bounce do not exceed A_{arm} ≲ 10^{−4}, and the arms lengths of the largescale GW detectors are equal to several kilometers, the values of γ_{2} ≲ 1 s^{−1}, and, correspondingly, γ_{2}/γ ≲ 10^{−3}, are feasible. At the same time, the value of photodetector quantum inefficiency \(\epsilon _d^2 \approx 1  {\eta _d} \approx 0.05\) (factoring in the losses in the interferometer output optical elements as well) is considered quite optimistic. Note, however, that in narrowband regimes considered in Section 6.3, the bandwidth γ can be much smaller and influence of γ_{2} could be significant; therefore, we take these losses into account in Section 6.3.
In Section 6.1.2 we consider the optimization of the spectral density (391), assuming that the arbitrary frequency dependence of the homodyne and/or squeezing angles can be implemented. As we see below, this case corresponds to the ideal lossless filter cavities. In Section 6.1.3, we consider two realistic schemes, taking into account the losses in the filter cavities.
6.1.2 Frequencydependent homodyne and/or squeezing angles
Classical optimization. As a reference point, consider first the simplest case of frequency independent homodyne and squeezing angles. We choose the specific values of these parameters following the classical optimization, which minimizes the shot noise (386) without taking into account the back action. Because, the shot noise dominates at high frequencies, therefore, this optimization gives a smooth broadband shape of the sum noise spectral density.
The spectral density (394) was first calculated in the pioneering work of [38], where the existence of two kinds of quantum noise in optical interferometric devices, namely the measurement (shot) noise and the back action (radiation pressure) noise, were identified for the first time, and it was shown that the injection of squeezed light with θ = 0 into the interferometer dark port is equivalent to the increase of the optical pumping power. However, it should be noted that in the presence of optical losses this equivalence holds unless squeezing is not too strong, e^{−r} > ϵ_{ d }.
Noteworthy is the proximity of the plots for the interferometer with 10 dB input squeezing and the one with 10fold increased optical power. The noticeable gap at higher frequencies is due to optical loss.
Compare this spectral density with the one for the frequencydependent squeezing angle (prefiltering) case, see Eq. (403). The shot noise components in both cases are exactly equal to each other. Concerning the residual backaction noise, in the prefiltering case it is limited by the available squeezing, while in the postfiltering case — by the optical losses. In the latter case, were there no optical losses, the backaction noise could be removed completely, as shown in Figure 35 (left). For the parameters of the noise curves presented in Figure 35 (right), the postfiltering still has some advantage of about 40% in the backaction noise amplitude \(\sqrt S\).
Note that the required frequency dependences (404) and (407) in both cases are similar to each other (and become exactly equal to each other in the lossless case ϵ_{ d } = 0). Therefore, similar setups can be used in both cases in order to create the necessary frequency dependences with about the same implementation cost. From this simple consideration, it is possible to conclude that prefiltering is preferable if good squeezing is available, and the optical losses are relatively large, and vice versa. In particular, postfiltering can be used even without squeezing, r = 0.
Frequency dependent homodyne and squeezing angles. And, finally, consider the most sophisticated configuration: doublefiltering with both the homodyne angle ϕ_{LO} and the squeezing angle θ being frequency dependent.
It is easy to see that in the ideal lossless case the doublefiltering configuration reduces to a postfiltering one. Really, if ϵ_{ d } = 0, the spectral density (410) becomes exactly equal to that for the postfiltering case (408), and the frequency dependent squeezing angle (411) degenerates into a constant value (405). However, if ϵ_{ d } > 0, then the additional prefiltering allows one to decrease more the residual backaction term. For example, if e^{2r} = 10 and η_{ d } = 0.95 then the gain in the backaction noise amplitude \(\sqrt S\) is equal to about 25%.
We have plotted the sum quantum noise spectral density (410) in Figure 34, right (dashdots). This plot demonstrates the best sensitivity gain of about 3 in signal amplitude, which can be provided employing squeezing and filter cavities at the contemporary technological level.
In our particular case, the fact that the additional noise associated with the photodetector quantum inefficiency ϵ_{ d } > 0 does not correlate with the quantum fluctuations of the light in the interferometer gives rise to this limit. This effect is universal for any kind of optical loss in the system, impairing the crosscorrelation of the measurement and backaction noises and thus limiting the performance of the quantum measurement schemes, which rely on this crosscorrelation.
Noteworthy is that Eq. (410) does not take into account optical losses in the filter cavities. As we shall see below, the sensitivity degradation thereby depends on the ratio of the light absorption per bounce to the filter cavities length, A_{ f }/L_{ f }. Therefore, this method calls for long filter cavities. In particular, in the original paper [90], filter cavities with the same length as the main interferometer arm cavities (4 km), placed side by side with them in the same vacuum tubes, were proposed. For such long and expensive filter cavities, the influence of their losses indeed can be small. However, as we show below, in Section 6.1.3, for the more practical short (up to tens of meters) filter cavities, optical losses thereof could be the main limiting factor in terms of sensitivity.
Virtual rigidity for prototype interferometers. The optimization performed above can be viewed also in a different way, namely, as the minimization of the sum quantum noise spectral density of an ordinary interferometer with frequencyindependent homodyne and squeezing angles, yet at some given frequency Ω_{0}. In Section 4.4, this kind of optimization was considered for a simple lossless system. It was shown capable of the narrowband gain in sensitivity, similar to the one provided by the harmonic oscillator (thus the term ‘virtual rigidity’).
This narrowband gain could be more interesting not for the fullscale GW detectors (where broadband optimization of the sensitivity is required in most cases) but for smaller devices like the 10m Hannover prototype interferometer [7], designed for the development of the measurement methods with subSQL sensitivity. Due to shorter arm length, the bandwidth γ in those devices is typically much larger than the mechanical frequencies Ω. If one takes, e.g., the power transmissivity value of T ≳ 10^{−2} for the ITMs and length of arms equal to L ∼ 10 m, then \(\gamma \gtrsim {10^5}\,{{\rm{s}}^{ 1}}\), which is above the typical working frequencies band of such devices. In the literature, this particular case is usually referred to as a bad cavity approximation.
6.1.3 Filter cavities in GW interferometers
Input/output relations for the filter cavity. In essence, a filter cavity is an ordinary FabryPérot cavity with one partly transparent input/output mirror. The technical problem of how to spatially separate the input and output beam can be solved in different ways. In the original paper [90] the triangular cavities were considered. However, in this case, an additional mirror in each cavity is required, which adds to the optical loss per bounce. Another option is an ordinary linear cavity with additional optical circulator, which can be implemented, for example, by means of the polarization beamsplitter and Faraday rotator (note that while the typical polarization optics elements have much higher losses than the modern highquality mirrors, the mirrors losses appear in the final expressions inflated by the filter cavity finesse).
It is easy to see that the necessary frequency dependencies of the homodyne and squeezing angles (404) or (407) (with the secondorder polynomials in Ω^{2} in the r.h.s. denominators) cannot be implemented by the rotation angle (432) (with its first order in Ω^{2} polynomial in the r.h.s. denominator). As was shown in the paper [90], two filter cavities are required in both these cases. In the double pre and postfiltering case, the total number of the filter cavities increases to four. Later it was also shown that, in principle, arbitrary frequency dependence of the homodyne and/or squeezing angle can be implemented, providing a sufficient number of filter cavities [35].
Following this reasoning, we consider below two schemes, each based on a single filter cavity that realize prefiltering and postfiltering, respectively.
It is easy to show that substitution of the conditions (441) and (447) into Eqs. (440) and (445), respectively, taking Condition (437) into account, results in spectral densities for the ideal frequency dependent squeezing and homodyne angle, see Eqs. (403) and (408).
In the general case of lossy filter cavities, the conditions (404) and (407) cannot be satisfied exactly by a single filter cavity at all frequencies. Therefore, the optimal filter cavity parameters should be determined using some integral sensitivity criterion, which will be considered at the end of this section.
We optimized numerically the ratio \(^{{\rho ^2}\rho _0^2}\), with filter cavity halfbandwidth γ_{f1} and detuning δ_{ f } as the optimization parameters, for the values of the specific loss factor A_{ f }/L_{ f } ranging from 10^{−9} (e.g., very long 10 km filter cavity with A_{ f } = 10^{−5}) to 10^{−5} (e.g., 10 m filter cavity with A_{ f } = 10^{−4}). Concerning the main interferometer parameters, we used the same values as in all our previous examples, namely, J = J_{aLIGO}, γ = 2π × 500 s^{−1}, and η_{ d } = 0.95.
It also follows from these plots that postfiltering provides slightly better sensitivity, if the optical losses in the filter cavity are low, while the prefiltering has some advantage in the highlosses scenario. This difference can be explained in the following way [87]. The postfiltration effectively rotates the homodyne angle from ϕ_{lo} = γ/2 (phase quadrature) at high frequencies to ϕ_{lo} → 0 (amplitude quadrature) at low frequencies, in order to measure the backaction noise, which dominates the low frequencies. As a result, the optomechanical transfer function reduces at low frequencies, emphasizing all noises introduced after the interferometer [see the factor sin^{2} ϕ_{lo}(Ω) in the denominator of Eq. (445)]. In the prefiltering case there is no such effect, for the value of ϕ_{LO} = π/2, corresponding to the maximum of the optomechanical transfer function, holds for all frequencies (the squeezing angle got rotated instead).
6.2 Quantum speed meter
6.2.1 Quantum speedmeter topologies
A quantum speed meter epitomizes the approach to the broadband SQL beating, in some sense, opposite to the one based on the quantum noises crosscorrelation tailoring with filter cavities, considered above. Here, instead of fitting the quantum noise spectral dependence to the FabryPérotMichelson interferometer optomechanical coupling factor (389), the interferometer topology is modified in such a way as to mold the new optomechanical coupling factor \({{\mathcal K}_{{\rm{SM}}}}(\Omega)\) so that it turns out frequencyindependent in the low and mediumfrequency range, thus making the frequencydependent crosscorrelation not necessary.
The general approach to speed measurement is to use pairs of position measurements separated by a time delay τ ≲ 1/Ω, where Ω is the characteristic signal frequency (cf. the simplified consideration in Section 4.5). Ideally, the successive measurements should be coherent, i.e., they should be performed by the same photons. In effect, the velocity υ of the test mass is measured in this way, which gives the necessary frequency dependence of the \({{\mathcal K}_{{\rm{SM}}}}(\Omega)\).
Later, the optical version of the sloshingcavity speedmeter scheme suitable for largescale laser GW detectors was developed [20, 126, 127]. The most elaborated variant proposed in [127] is shown in Figure 40 (right). Here, the differential mode of a Michelson interferometer serves as the resonator 1 of the initial scheme of [21], and an additional kilometerscale FabryPérot cavity — as the resonator 2, thus making a practical interferometer configuration.
Both versions of the optical speed meter, the sloshing cavity and the Sagnac ones, promise about the same sensitivity, and the choice between them depends mostly on the relative implementation cost of these schemes. Below we consider in more detail the Sagnac speed meter, which does not require the additional long sloshing cavity.
We will not present here the full analysis of the Sagnac topology similar to the one we have provided for the FabryPérotMichelson one. The reader can find it in [42, 50]. We limit ourselves by the particular case of the resonance tuned interferometer (that is, no signal recycling and resonance tuned arm cavities). It seems that the detuned Sagnac interferometer can provide a quite interesting regime, in particular, the negative inertia one [113]. However, for now (2011) the exhaustive analysis of these regimes is yet to be done. We assume that the squeezed light can be injected into the interferometer dark port, but consider only the particular case of the classical optimization, θ = 0, which gives the best broadband sensitivity for a given optical power.
6.2.2 Speedmeter sensitivity, no optical losses
6.2.3 Optical losses in speed meters
This spectral density is plotted in Figure 42 (right), together with the lossy variants of the same configurations as in Figure 42 (left), for the same moderately optimistic value of η_{ d } = 0.95, the losses part of the bandwidth and for γ_{2} = 1.875 s^{−1} [which corresponds to the losses A_{arm} = 10^{−4} per bounce in the 4 km length arms, see Eq. (322)]. These plots demonstrate that the speed meter in more robust with respect to optical losses than the filter cavities based configuration and is able to provide better sensitivity at very low frequencies.
It should also be noted that we have not taken into account here optical losses in the filter cavity. Comparison of Figure 42 with Figure 39, where the noise spectral density for the more realistic lossyfiltercavity cases are plotted, shows that the speed meter has advantage over, at least, the short and medium length (tens or hundred of meters) filter cavities. In the choice between very long (and hence expensive) kilometer scale filter cavities and the speed meter, the decision depends, probably, on the implementation costs of both configurations.
6.3 Optical rigidity
6.3.1 Introduction
We have seen in Section 4.3 that the harmonic oscillator, due to its strong response on nearresonance force, is characterized by the reduced values of the effective quantum noise and, therefore, by the SQL around the resonance frequency, see Eqs. (165, 172) and Figure 22. However, practical implementation of this gain is limited by the following two shortcomings: (i) the stronger the sensitivity gain, the more narrow the frequency band in which it is achieved; see Eq. (171); (ii) in many cases, and, in particular, in a GW detection scenario with its low signal frequencies and heavy test masses separated by the kilometersscale distances, ordinary solidstate springs cannot be used due to unacceptably high levels of mechanical loss and the associated thermal noise.
At the same time, in detuned FabryPérot cavities, as well as in the detuned configurations of the FabryPérotMichelson interferometer, the radiation pressure force depends on the mirror displacement (see Eqs. (312)), which is equivalent to the additional rigidity, called the optical spring, inserted between the cavity mirrors. It does not introduce any additional thermal noise, except for the radiation pressure noise \({\hat F_{{\rm{b}}{\rm{.a}}{\rm{.}}}}\), and, therefore, is free from the latter of the above mentioned shortcomings. Moreover, as we shall show below, spectral dependence of the optical rigidity K(Ω) alleviates, to some extent, the former shortcoming of the ‘ordinary’ rigidity and provides some limited sensitivity gain in a relatively broad band.
The electromagnetic rigidity was first discovered experimentally in radiofrequency systems [26]. Then its existence was predicted for the optical FabryPérot cavities [25]. Much later it was shown that the excellent noise properties of the optical rigidity allows its use in quantum experiments with macroscopic mechanical objects [17, 23, 24]. The frequency dependence of the optical rigidity was explored in papers [32, 83, 33]. It was shown that depending on the interferometer tuning, either two resonances can exist in the system, mechanical and optical ones, or a single broader secondorder resonance will exist.
In the last decade, the optical rigidity has been observed experimentally both in the tabletop setup [48] and in the larger prototype interferometer [111].
6.3.2 The optical noise redefinition
In detuned interferometer configurations, where the optical rigidity arises, the phase shifts between the input and output fields, as well as between the input fields and the field, circulating inside the interferometer, depend in sophisticated way on the frequency Ω. Therefore, in order to draw full advantage from the squeezing, the squeezing angle of the input field should follow this frequency dependence, which is problematic from the implementation point of view. Due to this reason, considering the opticalrigiditybased regimes, we limit ourselves to the vacuuminput case only, setting \({{\mathbb S}_{{\rm{sqz}}}}[r,\theta ] = {\mathbb I}\) in Eq. (375).
6.3.3 Bad cavities approximation
We start our treatment of the optical rigidity with the “bad cavity” approximation, discussed in Section 6.1.2 for the resonancetuned interferometer case. This approximation, in addition to its importance for the smallerscale prototype interferometers, provides a bridge between our idealized harmonic oscillator consideration of Section 4.3.2 and the frequencydependent rigidity case specific to the largescale GW detectors, which will be considered below, in Section 6.3.4.
For comparison, we reproduce here the common envelopes of the plots of ξ^{2}(Q) for the virtual rigidity case with η = 0.95; see Figure 36 (the dashed lines). It follows from Eqs. (489) and (420) that in absence of the optical loss, the sensitivity of the real rigidity case is inferior to that of the virtual rigidity one. However, even a very modest optical loss value changes the situation drastically. The noise cancellation (virtual rigidity) method proves to be advantageous only for rather moderate values of the SQL beating factor of ξ ≳ 0.5 in the absence of squeezing and ξ ≳ 0.3 with 10 dB squeezing. The conclusion is forced upon you that in order to dive really deep under the SQL, the use of real rather than virtual rigidity is inevitable.
6.3.4 General case
Frequencydependent rigidity. In the largescale laser GW detectors with kilometerscale arm cavities, the interferometer bandwidth can easily be made comparable or smaller than the GW signal frequency Ω. In this case, frequency dependences of the quantum noise spectral densities (376), (377) and (378) and of the optical rigidity (374) influence the shape of the sum quantum noise and, therefore, the detector sensitivity.
If J < J_{crit}, then two resonances yield two more or less separated minima of the sum quantum noise spectral density, whose location on the frequency axis mostly depends on the detuning δ, and their depth (inversely proportional to their width) hinges on the bandwidth γ. The choice of the preferable configuration depends on the criterion of the optimization, and also on the level of the technical (nonquantum) noise in the interferometer.
Another example is the configuration suitable for detection of the narrowband GW radiation from millisecond pulsars. Apparently, one of two resonances should coincide with the signal frequency in this case. It is well to bear in mind that in order to create an optical spring with mechanical resonance in a kHz region in contemporary and planned GW detectors, an enormous amount of optical power might be required. This is why the optical resonance, whose frequency depends mostly on the detuning δ, should be used for this purpose. This is, actually, the idea behind the GEO HF project [169]. The example of this regime is represented by the curve labeled as ‘Highfrequency’ in Figure 45. Here, despite one order of magnitude less optical power used (J = 0.1J_{aLIGO}), several times better sensitivity at frequency 1 kHz, than in the ‘Broadband’ regime, can be obtained. Note that the mechanical resonance in this case corresponds to 10 Hz only and therefore is virtually useless.
Consider the quantum noise of the system, consisting of this test object and the SQM (that is, the Heisenberg’suncertaintyrelationlimited quantum meter with frequencyindependent and noncorrelated measurement and backaction noises; see Section 4.1.1), which monitors its position. Below we show that the reallife longarm interferometer, under some assumptions, can be approximated by this model.
It is evident that the spectral density (517) represents a direct generalization of Eq. (510) in two aspects. First, it factors in optical losses in the interferometer. Second, it includes the case of Λ ≠ 0. We show below that a small yet nonzero value of A allows one to further increase the sensitivity.
Optimization of the signaltonoise ratio. The peculiar feature of the secondorder pole regime is that, while being, in essence, narrowband, it can provide an arbitrarilyhigh SNR for the broadband signals, limited only by the level of the additional noise of nonquantum (technical) origin. At the same time, in the ordinary harmonic oscillator case, the SNR is fundamentally limited.
It is easy to see that the approximations (528) work very well, even if ξ_{tech} ∼ 1 and, therefore, the assumptions (516) cease to be valid. One can conclude, looking at these plots, that optical losses do not significantly affect the sensitivity of the interferometer, working in the secondorder pole regime. The reason behind it is apparent. In the optical rigidity based systems, the origin of the sensitivity gain is simply the resonance increase of the probe object dynamical response to the signal force, which is, evidently, immune to the optical loss.
7 Conclusion and Future Directions
In this review, our primary goal was to tell in a clear and understandable way what is meant by quantum measurement in GW detectors. It was conceived as a comprehensive introduction to the quantum noise calculation techniques that are employed currently for the development of advanced interferometric detectors. The target audience are the young researchers, students and postdocs, who have just started their way in this field and need a guide that provides a stepbystep tutorial into the techniques and covers all the current achievements in the field. At the same time, we tried to make this manuscript interesting to all our colleagues from the GW community and, perhaps, from other branches of physics, who might be interested in getting themselves familiar with this area, not necessarily close to their own research field.
However, the reality is crude and such a lofty ambition is always a pot of gold at the end of the rainbow. Thus, we could not claim this review to be a complete and comprehensive description of the field of quantum measurement. We present here a pretty detailed analysis of the quantum noise features in the first and second generation of GW interferometers, contemplating the techniques considered robust and established. However, many hot topics, related to the planned third generation of GW interferometers [44, 125, 80] remained uncovered. Here are only some of them: (i) xylophone configurations [78], (ii) multiplecarrier detectors [130, 129], (iii) negative optical inertia [89], (iv) intracavity detection schemes [18, 17, 84, 86, 52], etc. It is our determined intention to enjoy the great advantages of the format of living reviews and include those topics in future revisions of this review.
We would like to conclude our review by pointing out how the new swiftlydeveloping areas of modern science and technology, not directly related to GW astronomy and detector science, turn out to be deeply rooted in the quantum measurement theory developed by the GW community. It is amazing how sinuous the ways of scientific progress are. The history of how GW detection and quantummeasurement theory developed and interwove might serve as an example thereof. Indeed, from the very first steps towards the experimental observation of GWs made by Weber in the early 1960s [165, 166], it was realized that the extreme weakness of interaction between the ripples of spacetime and matter appeals for unprecedentedly precise measurement. And almost at the same time, Braginsky realized that the expected amplitude of the GWinduced oscillations of the bar detector signal mode would be on the order of the zero point oscillations of this mode, as predicted by quantum mechanics; that is, in order to observe GWs, one has to treat a detector quantummechanically and as a consequence there will be a quantum back action, setting a limitation on the achievable sensitivity, the SQL [16].
This serendipity had a powerful impact on the quantum measurement theory development, for it set an objective to contrive some ways to overcome this limitation. For decades up to this point, it was a purely theoretical discipline having little in common with experimental science, and, fancy that, become a vital necessity for GW astronomy. And again, for several decades, GW detection has been perhaps the only field where the results of quantum measurement theory were applied, mainly in the struggle with quantum noise, considered as a hindrance towards the noble goal of the detection of GWs. And only recently, the same optomechanical interaction, begetting quantum noise and the SQL in the interferometric GW detectors, has aroused a keen interest among wide circles of researchers studying the quantum behavior of macroscopic objects and testing the very foundations of quantum mechanics in the macroscopic world [91, 10].
All the techniques and concepts developed in the GW community turn out to be highly sought by this new field [45]. Such methods, initially developed for future GW detectors, as backaction evasion via properly constructed cross correlation between the measurement and backaction noise sources [162, 159, 158, 160, 161, 51, 53], find a use in the optomechanical experiments with microand nanoscale mechanical oscillators [46, 114, 108, 105, 106, 170, 62]. It turns out that GW detectors themselves fit extremely well for testing the fundamental principles of quantum mechanics just for the record low values of the noise, having nonquantum origin, that owes to the ingenuity, patience and dedication by an entire generation of experimental physicists [131]. The very fact that the mechanical differential mode of the kmscale LIGO detector has been cooled down to T_{eff} = 1.4 µK without any special arrangement, just by modifying the standard feedback kernel of the actuators to provide a virtual rigidity, shifting the 10kg suspended mirrors oscillation frequency from Ω_{ m }/2π = 0.74 Hz to 150 Hz, where the GW detector is most sensitive [2], tells its own tale. Noteworthy also is the experiment on cooling a severalton AURIGA bar detector mechanical oscillation mode to T_{eff} = 0.17 mK [153]. In principle, some dedicated efforts might yield even cooling to ground state of these definitely macroscopic oscillators [54, 107].
One might foresee even more striking, really quantum phenomena, to be demonstrated experimentally by future GW detectors, whose sensitivity will be governed by quantum noise and not limited by the SQL. It is possible, e.g., to prepare the mechanical degree of freedom of the interferometer in a closetopure squeezed quantum state [114], entangle the differential and common motion of the kgscale mirrors in the EPRlike fashion [56, 115], or even prepare it in a highly nonclassical Schrödingercat state [135, 88].
Footnotes
 1.
Here, we adopt the system of labeling parts of the interferometer by the cardinal directions, they are located with respect to the interferometer central station, e.g., M_{ n } and M_{ e } in Figure 2 stand for ‘northern’ and ‘eastern’ end mirrors, respectively.
 2.
Here and below we keep to a definition of the reflectivity coefficient of the mirrors that implies that the reflected wave acquires a phase shift equal to π with respect to the incident wave if the latter impinged the reflective surface from the less optically dense medium (air or vacuum). In the opposite case, when the incident wave encounters reflective surface from inside the mirror, i.e., goes from the optically more dense medium (glass), it is assumed to acquire no phase shift upon reflection.
 3.
In fact, the argument of x should be written as t_{*}, that is the moment when the actual reflection takes place and is the solution to the equation: c(t − t_{*}) = x(t_{*}), but since the mechanical motion is much slower than that of light one has δx/c ≪ 1. This fact implies t ≃ t_{*}.
 4.
In the resonancetuned case, the phase modulation of the input carrier field creates equal magnitude sideband fields as discussed in Section 2.2.2, and these sideband fields are transmitted to the output port thanks to Schnupp asymmetry in the same state, i.e., they remain equal in magnitude and reside in the phase quadrature. In detuned configurations of GW interferometers, the upper and lower RFsideband fields are transformed differently, which influences both their amplitudes and phases at the readout port.
 5.
Insofar as the light beams in the interferometer can be well approximated as paraxial beams, and the polarization of the light wave does not matter in most of the considered interferometers, we will omit the vector nature of the electric field and treat it as a scalar field with strength defined by a scalar operatorvalued function Ê(x, y, z, t).
 6.Herein, we make use of a doublesided power spectral density defined on a whole range of frequencies, both negative and positive, that yields the following connection to the variance of an arbitrary observable ô(t):It is worth noting that in the GW community, the sensitivity of GW detectors as well as the individual noise sousces are usually characterized by a singlesided power spectral density \({\hat a_{1,2}}\), that is simply defined on positive frequencies Ω ≽ 0. The connection between these two is straightforward: \({{\mathbb M}_{{\rm{real}}}}\) and 0 otherwise.$${\rm{Var}}\left[ {\hat o(t)} \right] \equiv \left\langle {{{\hat o}^2}(t)} \right\rangle  {\left\langle {\hat o(t)} \right\rangle ^2} = \,\int\nolimits_{ \infty}^\infty {{{d\Omega} \over {2\pi}}{S_o}(\Omega)\,,}$$
 7.
Hereafter we will omit, for the sake of brevity, the factor 2πδ(Ω − Ω′) in equations that define the power (doublesided) spectral densities of relevant quantum observables, as well as assume Ω = Ω′, though keeping in mind that a mathematically rigorous definition should be written in the form of Eq. (87).
 8.
Here, we omitted the terms of \(s_0^ + (\Omega)\) proportional to \(S_0^ + (\Omega) = 2{S_o}(\Omega)\,{\rm{for}}\,\Omega \geqslant {\rm{0}}\) since their contribution to the integral is of the second order of smallness in â_{ c,s }/A_{ c } compared to the one for the first order term.
 9.
Personal communication with Yanbei Chen.
 10.Note the second term proportional to \(\delta \hat {\mathcal W}\), which owes its existence to the interference of the two traveling waves running in opposite directions. An interesting consequence of this is that the radiation pressure does not vanish even if the two waves have equal powers, i.e., \(\hat a_{c,s}^2(t)\), that is, in order to compensate for the radiation pressure force of one field on the semitransparent mirror, the other one should not only have the right intensity but also the right phase with respect to the former one:$$\cos {\Phi _0} = {1 \over 2}\sqrt {{R \over T}} {{{{\mathcal I}_1}  {{\mathcal I}_2}} \over {\sqrt {{{\mathcal I}_1}{{\mathcal I}_2}}}}\,.$$
Notes
Acknowledgements
This review owes its existence to the wholehearted support and sound advice of our colleagues and friends. We would like to express our special thanks to our friends, Yanbei Chen and Haixing Miao for enlightening discussions, helpful suggestions and encouragement we enjoyed in the course of writing this review. Also we are greatly thankful to the referees and to our younger colleagues, Mikhail Korobko and Nikita Voronchev, who went to the trouble of thoroughly reading the manuscript and pointing out many imperfections, typos, and misprints. We greatly acknowledge as well our fellow researchers from LIGOVirgo Scientific Collaboration for all the invaluable experience and knowledge they shared with us over the years. Especially, we want to say thank you to Gregg Harry, Innocenzo Pinto and Roman Schnabel for consulting with us on literature in the areas of their expertise. And finally, we would like to thank Living Reviews in Relativity and especially Bala Iyer for the rewarding opportunity to prepare this manuscript.
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