# Quantum Measurement Theory in Gravitational-Wave Detectors

- First Online:

- Accepted:

DOI: 10.12942/lrr-2012-5

- Cite this article as:
- Danilishin, S.L. & Khalili, F.Y. Living Rev. Relativ. (2012) 15: 5. doi:10.12942/lrr-2012-5

- 43 Citations
- 317 Downloads

## Abstract

The fast progress in improving the sensitivity of the gravitational-wave 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 kilometer-long 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 gravitational-wave detection. We focus on how quantum noise arises in gravitational-wave 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 more-than-ten-years-long history of the large-scale laser gravitation-wave (GW) detectors (the first one, TAMA [142] started to operate in 1999, and the most powerful pair, the two detectors of the LIGO project [98], in 2001, not to forget about the two European members of the international interferometric GW detectors network, also having a pretty long history, namely, the German-British interferometer GEO 600 [66] located near Hannover, Germany, and the joint European large-scale detector Virgo [156], operating near Pisa, Italy) can be considered both as a great success and a complete failure, depending on the point of view. On the one hand, virtually all technical requirements for these detectors have been met, and the planned sensitivity levels have been achieved. On the other hand, no GWs have been detected thus far.

The possibility of this result had been envisaged by the community, and during the same last ten years, plans for the second-generation detectors were developed [143, 64, 4, 169, 6, 96]. Currently (2012), both LIGO detectors are shut down, and their upgrade to the Advanced LIGO, which should take about three years, is underway. The goal of this upgrade is to increase the detectors’ sensitivity by about one order of magnitude [137], and therefore the rate of the detectable events by three orders of magnitude, from some ‘half per year’ (by the optimistic astrophysical predictions) of the second generation detectors to, probably, hundreds per year.

This goal will be achieved, mostly, by means of quantitative improvements (higher optical power, heavier mirrors, better seismic isolation, lower loss, both optical and mechanical) and evolutionary changes of the interferometer configurations, most notably, by introduction of the signal recycling mirror. As a result, the second-generation detectors will be *quantum noise limited*. At higher GW frequencies, the main sensitivity limitation will be due to phase fluctuations of light inside the interferometer (shot noise). At lower frequencies, the random force created by the amplitude fluctuations (radiation-pressure noise) will be the main or among the major contributors to the sum noise.

It is important that these noise sources both have the same quantum origin, stemming from the fundamental quantum uncertainties of the electromagnetic field, and thus that they obey the Heisenberg uncertainty principle and can not be reduced simultaneously [38]. In particular, the shot noise can (and will, in the second generation detectors) be reduced by means of the optical power increase. However, as a result, the radiation-pressure noise will increase. In the ‘naively’ designed measurement schemes, built on the basis of a Michelson interferometer, kin to the first and the second generation GW detectors, but with sensitivity chiefly limited by quantum noise, the best strategy for reaching a maximal sensitivity at a given spectral frequency would be to make these noise source contributions (at this frequency) in the total noise budget equal. The corresponding sensitivity point is known as the Standard Quantum Limit (SQL) [16, 22].

This limitation is by no means an absolute one, and can be evaded using more sophisticated measurement schemes. Starting from the first pioneering works oriented on solid-state GW detectors [28, 29, 144], many methods of overcoming the SQL were proposed, including the ones suitable for practical implementation in laser-interferometer GW detectors. The primary goal of this review is to give a comprehensive introduction of these methods, as well as into the underlying theory of *linear quantum measurements*, such that it remains comprehensible to a broad audience.

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

Notation and value | Comments |
---|---|

| coherent state of light with dimensionless complex amplitude |

| normalized detuning |

| interferometer half-bandwidth |

\(\Gamma = \sqrt {{\gamma ^2} + {\delta ^2}}\) | effective bandwidth |

| optical pump detuning from the cavity resonance frequency |

\({\epsilon_d} = \sqrt {{1 \over {{\eta _d}}} - 1}\) | excess quantum noise due to optical losses in the detector readout system with quantum efficiency |

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

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

| squeeze angle |

| some short time interval |

λ | optical wave length |

| reduced mass |

| mechanical detuning from the resonance frequency |

\(\xi = \sqrt {{S \over {{S_{{\rm{SQL}}}}}}}\) | SQL beating factor |

| signal-to-noise ratio |

| miscellaneous time intervals; in particular, |

| homodyne angle |

| |

\({\chi _{A \, B}}(t, t\prime) = {i \over \hbar}[\hat A(t),\;\hat B(t\prime)]\) | general linear time-domain susceptibility |

| probe body mechanical succeptibility |

| optical band frequencies |

| interferometer resonance frequency |

| optical pumping frequency |

ω | mechanical band frequencies; typically, ω = |

ω | mechanical resonance frequency |

\({\Omega _q} = \sqrt {{{2{S_{{\mathcal F}{\mathcal F}}}} \over {\hbar M}}}\) | quantum noise “corner frequency” |

| power absorption factor in Fabry-Pérot cavity per bounce |

â( | annihilation and creation operators of photons with frequency |

\({\hat a_c}(\Omega) = {{\hat a({\omega _0} + \Omega) + {{\hat a}^\dagger}({\omega _0} - \Omega)} \over {\sqrt 2}}\) | two-photon amplitude quadrature operator |

\({\hat a_s}(\Omega) = {{\hat a({\omega _0} + \Omega) - {{\hat a}^\dagger}({\omega _0} - \Omega)} \over {i\sqrt 2}}\) | two-photon 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 ( |

\({\mathcal A}\) | light beam cross section area |

| 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”) |

| electric field strength |

| 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 |

\({\hat F_{{\rm{b.a.}}}}\) | back-action force of the meter |

| signal force |

| 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 |

| 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 |

| optical pumping wave number |

| 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 |

| cavity length |

| probe-body mass |

| 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 |

| amplitude squeezing factor ( |

| power squeezing factor in decibels |

| power reflectivity of a mirror |

ℝ(ω) | reflection matrix of the Fabry-Pérot cavity |

| noise power spectral density (double-sided) |

\({S_{{\mathcal X}{\mathcal X}}}(\Omega)\) | measurement noise power spectral density (double-sided) |

\({S_{{\mathcal F}{\mathcal F}}}(\Omega)\) | back-action noise power spectral density (double-sided) |

\({S_{{\mathcal X}{\mathcal F}}}(\Omega)\) | cross-correlation power spectral density (double-sided) |

\({{\mathbb S}_{{\rm{vac}}}}(\Omega) = {1 \over 2}{\mathbb I}\) | 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 |

| power transmissivity of a mirror |

\({\mathbb T}\) | transmissivity matrix of the Fabry-Pérot cavity |

| test-mass velocity |

\({\mathcal W}\) | optical energy |

| Wigner function of the quantum state | |

| test-mass 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

*δϕ*with respect to some coherent reference of the same frequency. Having such a hypothetical tool, what would be the right way to use it, if one had a task to measure some tiny classical force? The simplest device one immediately conjures up is the one drawn in Figure 1. It consists of a movable totally-reflective mirror with mass

*M*and a coherent paraxial light beam, that impinges on the mirror and then gets reflected towards our hypothetical phase-sensitive device. The mirror acts as a

*probe*for an external force

*G*that one seeks to measure. The response of the mirror on the external force

*G*depends upon the details of its dynamics. For definiteness, let the mirror be a harmonic oscillator with mechanical eigenfrequency Ω

_{m}= 2

*πf*

_{m}. Then the mechanical equation of motion gives a connection between the mirror displacement

*x*and the external force

*G*in the very familiar form of the harmonic oscillator equation of motion:

*x*

_{0}(

*t*) = x(0) cosΩ

_{m}

*t*+

*p*(0)/(

*M*Ω

_{m})sinΩ

_{m}

*t*is the free motion of the mirror defined by its initial displacement

*x*(

*0*) and momentum

*p*(

*0*) at

*t*= 0 and

*δx*(

*t*) =

*x*(

*t*) −

*x*(0) induced by the external force

*G*. Indeed, there is a phase shift between the incident and reflected beams, that matches the additional distance the light must propagate to the new position of the mirror and back, i.e.,

*ω*

_{0}= 2

*πc*/λ

_{0}the incident light frequency,

*c*the speed of light and λ

_{0}the light wavelength. Here we implicitly assume mirror displacement to be much smaller than the light wavelength.

*G*(

*t*) can be obtained from the measured phase shift by

*post-processing*of the measurement data record \(\delta \tilde \phi (t) \propto \delta x(t)\) by substituting it into Eq. (1) instead of

*x*. Thus, the estimate of the signal force

*G*reads:

*x*(

*0*) and momentum

*p*(

*0*):

*A*(

*t*). If the expected signal spectrum occupies a frequency range that is much higher than the mirror-oscillation frequency ω

_{m}as is the case for ground based interferometric GW detectors, the oscillator behaves as a free mass and the term proportional to \(\Omega _m^2\) in the equation of motion can be omitted yielding:

#### 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 ancient-yet-die-hard 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.

*beamsplitter*, into two waves with equal amplitudes, travelling towards two highly-reflective mirrors M

_{n,e}

^{1}to get reflected off them, and then recombine at the beamsplitter. The readout is performed by a photodetector, placed in the signal port. The interferometer is usually tuned in such a way as to operate at a

*dark fringe*, which means that by default the lengths of the arms are taken so that the optical paths for light, propagating back and forth in both arms, are equal to each other, and when they recombine at the signal port, they interfere destructively, leaving the photodetector unilluminated. On the opposite, the two waves coming back towards the laser, interfere constructively. The situation changes if the end mirrors get displaced by some external force in a differential manner, i.e., such that the difference of the arms lengths is non-zero:

*δL*=

*L*

_{e}−

*L*

_{n}≠ 0. Let a laser send to the interferometer a monochromatic wave that, at the beamsplitter, can be written as

^{2}:

*δL*≪ λ

_{0}, the Michelson interferometer tuned to operate at the dark fringe has a sensitivity to ∼ (

*δL/λ*

_{0})

^{2}that yields extremely weak light power on the photodetector and therefore very high levels of dark current noise. In practice, the interferometer, in the majority of cases, is slightly detuned from the dark fringe condition that can be viewed as an introduction of some constant small bias

*δL*

_{0}between the arms lengths. By this simple trick experimentalists get linear response to the signal nonstationary displacement

*δx*(

*t*):

*δ*ϕ(

*t*) =

*4πδx*(

*t*)/

*λ*

_{0}is absolutely the same as in the case of a single moving mirror (cf. Eq. (3)). It is no coincidence, though, but a manifestation of the internal symmetry that all Michelson-type interferometers possess with respect to coupling between mechanical displacements of their arm mirrors and the optical modes of the outgoing fields. In the next Section 2.1.3, we show how this symmetry displays itself in GW interferometers.

#### 2.1.3 Gravitational waves’ interaction with interferometer

*ripples in the curvature of spacetime*that are emitted by violent astrophysical events, and that propagate out from their source with the speed of light’ [13, 110]. A weak GW far away from its birthplace can be most easily understood from analyzing its action on the probe bodies motion in some region of spacetime. Usually, the deformation of a circular ring of free test particles is considered (see Chapter 26: Section 26.3.2 of [13] for more rigorous treatment) when a GW impinges it along the

*z*-direction, perpendicular to the plane where the test particles are located. Each particle, having plane coordinates (

*x*,

*y*) with respect to the center of the ring, undergoes displacement

*δr*≡ (

*δx, δy*) from its position at rest, induced by GWs:

*h*

_{+}≡

*h*+(

*t*−

*z/c*) and

*h*

_{×}

*≡ h*

_{×}(

*t*−

*z/c*) stand for two independent polarizations of a GW that creates an acceleration field resulting in the above deformations. The above expressions comprise a solution to the equation of motion for free particles in the

*tidal acceleration field*created by a GW:

*e*

_{x}= {1, 0}

^{⊤}and

*e*

_{y}= {0, 1}

^{⊤}the unit vectors pointing in the

*x*and

*y*direction, respectively.

*x*and

*y*axes, then the mirrors will have coordinates (0,

*L*

_{n}) and (

*L*

_{e}, 0), correspondingly. For this case, the action of the GW field on the mirrors is featured in Figure 3. It is evident from this picture and from the above formulas that an

*h*

_{×}-polarized component of the GW does not change the relative lengths of the Michelson interferometer arms and thus does not contribute to its output signal; at the same time,

*h*

_{+}-polarized GWs act on the end masses of the interferometer as a pair of tidal forces of the same value but opposite in direction:

*G*

_{e}= −

*G*

_{n}= G,

*M*

_{n}=

*M*

_{e}=

*M*, and

*L*

_{e}=

*L*

_{n}=

*L*, one can write down the equations of motion for the interferometer end mirrors that are now considered free (Ω

_{m}≪ Ω

_{GW}) as:

*δL*=

*L*

_{e}−

*L*

_{n}=

*x*−

*y*, which, we have shown above, the Michelson interferometer is sensitive to, one gets the following equation of motion:

*M*. Therefore, we have proven that a Michelson interferometer has the same dynamical behavior with respect to the tidal force \(G(t) = M{\ddot h_ +}(t)L/2\) created by GWs, as the single movable mirror with mass

*M*to some external generic force

*G*(

*t*).

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 step-by-step 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].

*x*-axis. This field can be fully characterized by the strength of its electric component

*E*(

*t*−

*x/c*) that should be a sinusoidal function of its argument

*ζ*=

*t*−

*x/c*and can be written in three equivalent ways:

*ℰ*

_{0}and

*ℰ*

_{0}are called

*amplitude*and

*phase*, ℰ

_{c}and

*ℰ*

_{s}take names of cosine and sine

*quadrature amplitudes*, and complex number

*ℰ*= |

*ℰ*|

*e*

^{i argfℰ}is known as the

*complex amplitude*of the electromagnetic wave. Here, we see that our wave needs two real or one complex parameter to be fully characterized in the given location

*x*at a given time

*t*. The ‘amplitude-phase’ description is traditional for oscillations but is not very convenient since all the transformations are nonlinear in phase. Therefore, in optics, either quadrature amplitudes or complex amplitude description is applied to the analysis of wave propagation. All three descriptions are related by means of straightforward transformations:

*ζ*=

*t*−

*x/c*(for a forward propagating wave) and thus can be, without loss of generality, substituted by a time dependence of electric field in some fixed point, say with

*x*= 0, thus yielding

*E*(

*ζ*) ≡

*E*(

*t*). We will keep to this convention throughout our review.

*x*

_{1}= 0 and

*x*

_{2}=

*L*. Obviously, if nothing obscures light propagation between these two points, the value of the electric field in the second point at time

*t*is just the same as the one in the first point, but at earlier time, i.e., at

*t*′ =

*t*−

*L/c*:

*E*

^{(L)}(

*t*) =

*E*

^{(0)}(

*t*−

*L/c*). This allows us to introduce a transformation that propagates EM-wave from one spatial point to another. For complex amplitude

*ℰ*, the transformation is very simple:

*= {ℰ*

**ℰ**_{c}, ℰ

_{s}}

^{⊤}, that are:

*L*≪ λ, the above two expressions can be expanded into Taylor’s series in

*ϕL*= 2

*πL/λ*≪ 1 up to the first order:

*ϕ*

_{L}] is an

*infinitesimal*increment matrix that generate the difference between the field quadrature amplitudes vector

*ℰ*after and before the propagation, respectively.

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 two-photon 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

*ω*

_{0}, amplitude

*ℰ*

_{0}and initial phase

*ϕ*

_{0}= 0:

**Amplitude modulation**. The modulation of light amplitude is straightforward to analyze. Let us do it for pedagogical sake: imagine one managed to modulate the carrier field amplitude slow enough compared to the carrier oscillation period, i.e., Ω ≪

*ω*

_{0}, then:

*ϵ*

_{m}≪ 1 and

*ϕ*

_{m}are some constants called modulation depth and relative phase, respectively. The complex amplitude of the modulated wave equals to

*amplitude quadrature*in the literature. In our review, we shall also keep to this terminology and refer to cosine quadrature as amplitude one.

*ω*

_{0}with amplitude \({A_{w0}} = {\varepsilon _0}\) and two satellites at frequencies

*ω*

_{0}± Ω, also referred to as

*modulation sidebands*, with (complex) amplitudes \({A_{w0}}_{\pm \Omega} = {\epsilon _m}{\varepsilon _0}e{\mp ^{i{\phi _m}}}/2\). The graphical interpretation of the above considerations is given in the left panel of Figure 4. Here, carrier fields as well as sidebands are represented by rotating vectors on a complex plane. The carrier field vector has length

*ℰ*

_{0}and rotates clockwise with the rate

*ω*

_{0}, while sideband components participate in two rotations at a time. The sum of these three vectors yields a complex vector, whose length oscillates with time, and its projection on the real axis represents the amplitude-modulated light field.

*ω*

_{0}and an infinite discrete set of sideband harmonics at frequencies \({\omega _0} \pm k\Omega (h = \overline {1,\infty})\):

*ω*

_{0}and the modulation sidebands around it, whose shape retraces the modulation function spectrum

*A*(

*ω*) shifted by the carrier frequency ±

*ω*

_{0}.

**Phase modulation**. The general feature of the modulated signal that we pursued to demonstrate by this simple example is the creation of the modulation sidebands in the spectrum of the modulated light. Let us now see how it goes with a phase modulation that is more related to the topic of the current review. The simplest single-frequency phase modulation is given by the expression:

*ω*

_{0}, and the phase deviation

*δ*

_{m}is assumed to be much smaller than 1. Using Eqs. (14), one can write the complex amplitude of the phase-modulated light as:

*δ*

_{m}≪ 1), the above equations can be approximated as:

*π*/2 out-of-phase with respect to the carrier field, contains the modulation signal. That is why this sine quadrature is usually referred to as

*phase quadrature*. It is also what we will call this quadrature throughout the rest of this review.

*J*

_{k}(

*δ*

_{m}) stands for the

*k*-th Bessel function of the first kind. This looks a bit intimidating, yet for

*δ*

_{m}≪ 1 these expressions simplify dramatically, since near zero Bessel functions can be approximated as:

*δ*

_{m}, we can limit ourselves only to the terms of order \(\delta _m^0\) and \(\delta _m^1\), which yields:

*i*in front of the corresponding terms in Eq. (22)); therefore its sum is always orthogonal to the carrier field vector, and the resulting modulated oscillation vector has approximately the same length as the carrier field vector but outruns or lags behind the latter periodically with the modulation frequency Ω. The resulting oscillation of the PM light electric field strength is drawn to the right of the PM phasor diagram and is the projection of the PM oscillation vector on the real axis of the complex plane.

*t*):

*δ*

_{m}, the corresponding formulas are very cumbersome and do not give much insight. Therefore, we again consider a simplified situation of sufficiently small

*δ*

_{m}≪ 1. Then one can approximate the phase-modulated oscillation as follows:

*t*) is a periodic function: \(\Phi (t) = \sum\nolimits_{k = 1}^\infty {{\Phi _k}\,{\rm{cos}} \, k\Omega + {\phi _k}}\), and in weak modulation limit

*δ*

_{m}≪ 1, the spectrum of the PM light is apparent from the following expression:

*ω*

_{0}and shifted modulation spectra

*i*Φ(

*ω*±

*ω*

_{0}) as sidebands around the carrier peaks. The difference with the amplitude modulation is an additional ±π/2 phase shifts added to the sidebands.

#### 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 2-vector * 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].

*ê*

_{c,s}are related to

*ê*

_{n}and \({\hat \phi _n}\) in the same manner as prescribed by Eqs. (14). It is convenient to represent a noisy laser field in the Fourier domain:

*ê*

_{c,s}(Ω) is a spectral representation of a real quantity and thus satisfies an evident equality \(\hat e_{c,s}^\dagger (- \Omega) = {\hat e_{c,s}}(\Omega)\) (by † we denote the Hermitian conjugate that for classical functions corresponds to taking the complex conjugate of this function). What happens if we want to know the light field of our laser with noise at some distance

*L*from our initial reference point

*x*= 0? For the carrier field component at

*ω*

_{0}, nothing changes and the corresponding transform is given by Eq. (16), yet for the noise component

*e*

^{iΩL/c}that describes an extra phase shift acquired by a sideband field relative to the carrier field because of the frequency difference Ω =

*ω*−

*ω*

_{0}.

#### 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.

*r*and

*r*′ and transmission

*t*and

*t*′ as drawn in Figure 5. Let us now see how the ingoing and outgoing light beams couple on the mirrors in the interferometer.

**Mirrors**: From the general point of view, the mirror is a linear system with 2 input and 2 output ports. The way how it transforms input signals into output ones is featured by a 2 × 2 matrix that is known as the

*transfer matrix*of the mirror \({\mathbb M}\):

*R*= |

*r|*

^{2}and

*T*= |

*t|*

^{2}that will find extensive use throughout the rest of this review.

*coupling relations*for the quadrature amplitudes can easily be obtained from Eq. (27). Now, we have two input and two output fields. Therefore, one has to deal with 4-dimensional vectors comprising of quadrature amplitudes of both input and output fields, and the transformation matrix become 4 × 4-dimensional, which can be expressed in terms of the outer product of a 2 × 2 matrix \({{\mathbb M}_{{\rm{real}}}}\) by a 2 × 2 identity matrix \({\mathbb I}\):

In future, for the sake of brevity, we reduce the notation for matrices like \({{\mathbb M}_{{\rm{real}}}}\, \otimes {\mathbb I}\) to simply \({{\mathbb M}_{{\rm{real}}}}\).

**Beam splitters**: Another optical element ubiquitous in the interferometers is a beamsplitter (see Figure 6). In fact, it is the very same mirror considered above, but the angle of input light beams incidence is different from 0 (if measured from the normal to the mirror surface). The corresponding scheme is given in Figure 6. In most cases, symmetric 50%/50% beamsplitter are used, which imply

*R*=

*T*= 1/2 and the coupling matrix \({{\mathbb M}_{50/50}}\) then reads:

**Losses in optical elements**: Above, we have made one assumption that is a bit idealistic. Namely, we assumed our mirrors and beamsplitters to be lossless, but it could never come true in real experiments; therefore, we need some way to describe losses within the framework of our formalism. Optical loss is a term that comprises a very wide spectrum of physical processes, including scattering on defects of the coating, absorption of light photons in the mirror bulk and coating that yields heating and so on. A full description of loss processes is rather complicated. However, the most important features that influence the light fields, coming off the lossy optical element, can be summarized in the following two simple statements:

- 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.

*ϵ*:

*ϵ*∼ 10

^{−5}−10

^{−4}. Therefore, the impact of optical loss on classical carrier amplitudes is negligible. Where the noise sidebands are concerned, the transformation rule given by Eq. (31) changes a bit more:

*ϵ ≪*1 and also the fact that matrix \({{\mathbb M}_{{\rm{real}}}}\) is unitary, i.e., we redefined the noise that enters outgoing fields due to loss as \({\{\hat n_1\prime,\hat n_2\prime\} ^{\rm{T}}} = {\mathbb M} \cdot {({\hat n_1},{\hat n_2}\} ^{\rm{T}}}\), which keeps the new noise sources \(\hat n_1\prime(t)\) and \(\hat n_2\prime(t)\) uncorrelated: \(\left\langle {{{\hat n}_1}(t){{\hat n}_2}({t\prime})} \right\rangle = \left\langle {\hat n_1\prime(t)\hat n_2\prime(t)} \right\rangle = 0\).

#### 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.

*x*= 0 as drawn in Figure 8. We assume the sway of the mirror motion to be much smaller than the optical wavelength:

*x/λ*

_{0}≪ 1. The effect of the mirror displacement

*x*(

*t*) on the outgoing field \(E_{1,2}^{{\rm{out}}}(t)\) can be straightforwardly obtained from the propagation formalism. Indeed, considering the light field at a fixed spatial point, the reflected light field at any instance of time

*t*is just the result of propagation of the incident light by twice the mirror displacement taken at time of reflection and multiplied by reflectivity \(\pm \sqrt {R}\)

^{3}

*x*≪ λ

_{0}; according to Eq. (19) the mirror motion modifies the quadrature amplitudes in a way that allows one to separate this effect from the reflection. It means that the result of the light reflection from the moving mirror can be represented as a sum of two independently calculable effects, i.e., the reflection off the fixed mirror, as described above in Section 2.2.4, and the response to the mirror displacement (see Section 2.2.1), i.e., the signal presentable as a sideband vector {

**s**_{1}(Ω),

**s**_{2}(Ω)}

^{⊤}. The latter is convenient to describe in terms of the response vector {

**R**_{1},

**R**_{2}}

^{⊤}that is defined as:

*ω*

_{0}

*x*(

*t*)/

*c*= 4

*πx*(

*t*)/λ

_{0}≪ 1 by a small sideband amplitude \(\vert\hat e_{1,2}^{{\rm{in}}}(\Omega)\vert \ll \varepsilon _{1,2}^{{\rm{in}}}\vert\), the resulting contribution to the final response will be dwarfed by that of the classical fields. Moreover, the mirror motion induced contribution (35) is itself a quantity of the same order of magnitude as the noise sidebands, and therefore we can claim that the classical amplitudes of the carrier fields are not affected by the mirror motion and that the relations (30) hold for a moving mirror too. However, the relations for sideband amplitudes must be modified. In the case of a lossless mirror, relations (31) turn:

*x*(Ω) is the Fourier transform of the mirror displacement

*x*(

*t*)

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 *signal-to-noise 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

*R*= 1) moving mirror as drawn in Figure 9. The initial phase

*ϕ*

_{0}of the incident wave does not matter and can be taken as zero. Then \(\varepsilon _c^{{\rm{in}}} = {\varepsilon _0}\) and \(\varepsilon _s^{{\rm{in}}} = 0\). Putting these values into Eq. (30) and accounting for \(\varepsilon _2^{{\rm{in}}} = 0\), quite reasonably results in the amplitude of the carrier wave not changing upon reflection off the mirror, while the phase changes by

*π*:

This fact, i.e., that the mirror displacement that just causes phase modulation of the reflected field, enters only the s-quadrature, once again justifies why this quadrature is usually referred to as *phase quadrature* (cf. Section 2.2.2).

*x*(

*t*). The left part of Figure 9 illustrates the aforesaid. As for laser noise, it enters the outgoing light in an additive manner and the typical (though simplified) amplitude spectrum of a noisy light reflected from a moving mirror is given in Figure 10.

### 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 radio-frequency technology as methods of detection of phase- and frequency-modulated 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 non-linearity 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

**Homodyne readout**. Homodyne detection uses local oscillator light with the same carrier frequency as the signal. Write down the signal wave as:

*S*

_{c,s}(

*t*) might contain GW signal

*G*

_{c,s}(

*t*) as well as quantum noise

*n*

_{c,s}(

*t*) in both quadratures:

*homodyne angle ϕ*

_{LO}, and laser noise

*l*

_{c,s}(

*t*):

*L*

_{0}is much larger than all other signals:

*i*

_{1,2}will be proportional to the intensities

*I*

_{1,2}of these two lights:

*A*(

*t*) over many optical oscillation periods, which reflects the inability of photodetectors to respond at optical frequencies and thus providing natural low-pass filtering for our signal. The last terms in both expressions gather all the terms quadratic in GW signal and both noise sources that are of the second order of smallness compared to the local oscillator amplitude

*L*

_{0}and thus are omitted in further consideration.

*ϕ*

_{LO}one can recover it with minimum additional noise. That is how homodyne detection works in theory.

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 low-noise 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 DC-readout 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 phase-locking 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].

*δL*of the lengths of the interferometer arms. It is also worth noting that the component of this local oscillator created by the asymmetry in the reflectivity of the arms that is always the case in a real interferometer and attributable mostly to the difference in the absorption of the ‘northern’ and ‘eastern’ end mirrors as well as asymmetry of the beamsplitter. All these factors can be taken into account if one writes the carrier fields at the beamsplitter after reflection off the arms in the following symmetric form:

*ϵ*

_{n,e}and

*L*

_{n,e}stand for optical loss and arm lengths of the corresponding interferometer arms, \(\Delta \epsilon = {{{\epsilon _n} - {\epsilon _e}} \over {2(1 - \bar \epsilon)}}\) is the optical loss relative asymmetry with \(\bar \epsilon = ({\epsilon _n} + {\epsilon _e})/2,\,\bar \varepsilon = {E_0}(1 - \bar \epsilon)\) is the mean pumping carrier amplitude at the beamsplitter, \(\Delta L = {L_n} - {L_e}\) and \(\bar t = t + {{{L_n} + {L_e}} \over {2c}}\). Then the classical part of the local oscillator light in the signal (dark) port will be given by the following expression:

*ω*

_{0}Δ

*L/c*≪ 1 and the rather small absolute value of the optical loss coefficient max [

*ϵ*

_{n},

*ϵ*

_{e}] ∼ 10

^{−4}≪ 1 available in contemporary interferometers. One sees that were there no asymmetry in the arms optical loss, there would be no opportunity to change the local oscillator phase. At the same time, the GW signal in the considered scheme is confined to the phase quadrature since it comprises the time-dependent part of Δ

*L*and thus the resulting photocurrent will be proportional to:

*n*

_{c,s}stand for the quantum noise associated with the signal sidebands and entering the interferometer from the signal port.

In the case of a small offset of the interferometer from the dark fringe condition, i.e., for *ω*_{0}Δ*L/c* = 2πΔ*L*/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 Fabry-Pérot cavities in the arms. These additional elements turn the Michelson interferometer into a resonant narrow-band 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 40-meter 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.

_{RF}∼ several megahertz. In GW interferometers with heterodyne readout, local oscillator light of different than

*ω*

_{0}frequency is produced via phase-modulation of the pumping carrier light by means of electro-optical modulator (EOM) before it enters the interferometer as drawn in Figure 12. The interferometer is tuned so that the readout port is dark for the pumping carrier. At the same time, by introducing a macroscopic (several centimeters) offset Δ

*L*of the two arms, which is known as Schnupp asymmetry [134], the modulation sidebands at radio frequency Ω

_{RF}appear to be optimally transferred from the pumping port to the readout one. Therefore, the local oscillator at the readout port comprises two modulation sidebands,

*L*

_{het}(

*t*) =

*L*

_{+}(

*t*) +

*L*

_{−}(

*t*), at frequencies

*ω*

_{0}+ Ω

_{RF}and

*ω*

_{0}− Ω

_{RF}, respectively. These two are detected together with the signal sidebands at the photodetector, and then the resulting photocurrent is demodulated with the RF-frequency reference signal yielding an output current proportional to GW-signal.

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.

*L*+(

*t*) and

*L*

_{−}(

*t*)) at frequencies

*ω*

_{0}

*±*ω

_{rf}:

^{4}and

*l*

_{(c,s)±}(

*t*) represent laser noise around the corresponding modulation frequency.

*ω*

_{0}has to be accounted for but also those rallying around twice the RF modulation frequencies

*ω*

_{0}± 2Ω

_{RF}:

*[*

**H***ϕ*] = {cos

*ϕ*, sin

*ϕ*}

^{⊤}, e.g.:

*= {*

**G***G*

_{c},

*G*

_{s}}

^{⊤}and \({n_{\omega \alpha}} = {\{n_c^\omega, n_s^\omega \} ^{\rm{T}}}\). Another useful observation, provided that

*ω*

_{0}≫ max[Ω

_{1}, Ω

_{2}], gives us the following relation:

_{RF}. It is only the term \(2\overline {({L_ +} + L -)S}\) that is linear in GW signal and thus contains useful information:

*σ*

_{y}is the 2nd Pauli matrix:

*i*

_{het}is mixed with (multiplied by) a demodulation function

*D*(

*t*) =

*D*

_{0}cos(Ω

_{RTF}+

*ϕ*

_{D}) with the resulting signal filtered by a low-pass filter with upper cut-off frequency Λ ≪ Ω

_{RF}so that only components oscillating at GW frequencies Ω

_{GW}≪ Ω

_{RF}appear in the output signal (see Figure 12).

*balanced heterodyne detection*), only single phase quadrature of the GW signal can be extracted from the output photocurrent, which is fine, because the Michelson interferometer, being equivalent to a simple movable mirror with respect to a GW tidal force as shown in Section 2.1.1 and 2.1.2, is sensitive to a GW signal only in phase quadrature. Another important feature of heterodyne detection conspicuous in the above equations is the presence of additional noise from the frequency bands that are twice the RF-modulation frequency away from the carrier. As shown in [36] this noise contributes to the total quantum shot noise of the interferometer and makes the high frequency sensitivity of the GW detectors with heterodyne readout 1.5 times worse compared to the ones with homodyne or DC readout.

For more realistic and thus more sophisticated optical configurations, including Fabry-Pérot 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 frequency-independent homodyne readout schemes can be obtained [36]. However, inherent additional frequency-independent quantum shot noise brought by the heterodyning process into the detection band hampers the simultaneous use of advanced quantum non-demolition (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 high-frequency 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: Two-photon formalism

*= (*

**r***x, y, z*) at time

*t*by a Heisenberg operator of an electric field strength

*Ê*(

*,*

**r***t*).

^{5}The electric field Heisenberg operator of a light wave traveling along the positive direction of the z-axis can be written as a sum of a positive- and negative-frequency parts:

*u*(

*x, y, z*) is the spatial mode shape, slowly changing on a wavelength λ scale, and

*annihilation*(

*creation*) operator in the mode of the field with frequency

*ω*. The meaning of Eq. (49) is that the travelling light wave can be represented by an expansion over the continuum of harmonic oscillators — modes of the electromagnetic field, — that are, essentially, independent degrees of freedom. The latter implies the commutation relations for the operators

*â*

_{ω}and \(\hat a_\omega ^\dag\):

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 two-photon 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 two-photon 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 two-photon formalism [39, 40], which is widely used in GW detectors as well as in quantum optics and optomechanics.

*ω*

_{0}in Eqs. (48), which yields:

*ω*

_{0}≫ Ω

_{GW}enables us to expand the limits of integrals to

*ω*

_{0}→ ∞. The operator expressions in front of \({e^{\pm i{\omega _0}t}}\) in the foregoing Eqs. (51) are quantum analogues to the complex amplitude

*ℰ*and its complex conjugate

*ℰ**defined in Eqs. (14):

*â*

_{c,s}(

*t*) are Hermitian and thus their frequency domain counterparts satisfy the relations for the spectra of Hermitian operator:

*â*

_{c}(

*t*),

*â*

_{c}(

*t*′)] = [

*â*

_{s}(

*t*),

*â*

_{s}(

*t*′)] ≠ 0, which imply they could not be considered for proper output observables of the detector, for a nonzero commutator, as we would see later, means an additional quantum noise inevitably contributes to the final measurement result. The detailed explanation of why it is so can be found in many works devoted to continuous linear quantum measurement theory, in particular, in Chapter 6 of [22], Appendix 2.7 of [43] or in [19]. Where GW detection is concerned, all the authors are agreed on the point that the values of GW frequencies Ω (1 Hz ≼/2π ≼ 10

^{3}Hz), being much smaller than optical frequencies

*ω*

_{0}/2π ∼ 10

^{15}Hz, allow one to neglect such weak commutators as those of Eqs. (53) in all calculations related to GW detectors output quantum noise. This statement has gotten an additional ground in the calculation conducted in Appendix 2.7 of [43] where the value of the additional quantum noise arising due to the nonzero value of commutators (53) has been derived and its extreme minuteness compared to other quantum noise sources has been proven. Braginsky et al. argued in [19] that the two-photon quadrature amplitudes defined by Eqs. (52) are not the real measured observables at the output of the interferometer, since the photodetectors actually measure not the energy flux

*u*(

*x, y, z*) since it does not influence the final result for quantum noise spectral densities. Moreover, in order to comply with the already introduced division of the optical field into classical carrier field and to the 1st order corrections to it comprising of laser noise and signal induced sidebands, we adopt the same division for the quantum fields, i.e., we detach the mean values of the corresponding quadrature operators via the following redefinition \(\hat a_{{c_s}}^{{\rm{old}}} \to {A_{c,s}} + \hat a_{c,s}^{{\rm{new}}}\) with \({A_{c,s}} \equiv \left\langle {\hat a_{c,s}^{{\rm{old}}}} \right\rangle\). Here, by \(\left\langle {\hat a_{c,s}^{{\rm{old}}}} \right\rangle\) we denote an ensemble average over the quantum state |

*ψ*〉 of the light wave: (Â) ≡ 〈

*ψ*|

*Â*|

*ψ*〉. Thus, the electric field strength operator for a plain electromagnetic wave will have the following form:

Now, when we have defined a quantum Heisenberg operator of the electric field of a light wave, and introduced quantum operators of two-photon 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

*vacuum*state |vac〉, is straightforward and is simply the direct product of the ground states |0〉

_{ω}of all modes over all frequencies

*ω*:

*ω*is the state with minimum energy

*E*

_{vac}= ℏ

*ω*/2 and no excitation:

^{6}\({\mathbb S}(\Omega)\) according to the rule:

*S*

_{ij}(Ω) (

*i,j*=

*c*,

*s*) denote (cross) power spectral densities of the corresponding quadrature amplitudes 〈

*â*

_{i}(Ω) ◦

*â*

_{j}(Ω′)〉 standing for the symmetrized product of the corresponding quadrature operators, i.e.:

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 square-wave 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.

*X*

_{0}and \({P_0}:{\hat X_\varepsilon}(t) \equiv {\hat x_\varepsilon}/{X_0}\) and \({\hat X_\varepsilon}(t) \equiv {\hat p_\varepsilon}/{P_0}\). This fact is also justified by the value of their commutator:

*Wigner function*, a quantum version of joint (quasi) probability distribution for particle displacement and momentum (

*X*

_{ε}and

*Y*

_{ε}in our case):

*ξ*is simply the variable of integration. The above Wigner function describes a Gaussian state, which is simply the ground state of a harmonic oscillator represented by a mode with displacement

*X*

_{ε}and momentum

*Y*

_{ε}. The corresponding plot is given in the left panel of Figure 14. Gaussian states are traditionally pictured by error ellipses on a phase plane, as drawn in the right panel of Figure 14 (cf. right panel of Figure 13). Here as well as in Figure 13, a red line in both plots circumscribes all the values of

*X*

_{ε}and

*Y*

_{ε}that fall inside the standard deviation region of the Wigner function, i.e., the region where all pertinent points are within 1 standard deviation from the center of the distribution. For a vacuum state, such a region is a circle with radius \(\sqrt {{{\mathbb V}_{cc}}} = \sqrt {{{\mathbb V}_{ss}}} = 1/\sqrt 2\). The area of this circle, equal to 1/2 in dimensionless units and to

*ℏ/*2 in case of dimensional displacement and momentum, is the smallest area a physical quantum state can occupy in a phase space. This fact yields from a very general physical principle, the Heisenberg uncertainty relation, that limits the minimal uncertainty product for canonically conjugate observables (displacement

*X*

_{ε}and momentum

*Y*

_{ε}, in our case) to be less than 1/2 in

*ℏ*-units:

*ψ*〉, rather than by a density operator \(\hat \rho\). For more sophisticated Gaussian states with a non-diagonal covariance matrix \({\mathbb V}\), the Heisenberg uncertainty relation reads:

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

*coherent state*(see, e.g., [163, 136, 99, 132]). It is straightforward to introduce a coherent state |

*α*〉 of a single mode or a harmonic oscillator as a result of its ground state |0〉 shift on a complex plane by the distance and in the direction governed by a complex number

*α*= |α|

*e*

^{i arg(α)}. This can be caused, e.g., by the action of a classical effective force on the oscillator. Such a shift can be described by a unitary operator called a displacement operator, since its action on a ground state |0〉 inflicts its shift in a phase plane yielding a state that is called a coherent state:

_{ω}is the coherent state that the mode of the field with frequency

*ω*is in, and

*α*(

*ω*) is the distribution of complex amplitudes

*α*over frequencies

*ω*. Basically,

*α*(

*ω*) is the spectrum of normalized complex amplitudes of the field, i.e.,

*α*(

*ω*) ∝

*ℰ*(

*ω*). For example, the state of a free light wave emitted by a perfectly monochromatic laser with emission frequency

*ω*

_{p}and mean optical power \({{\mathcal I}_0}\) will be defined by \(\alpha (\omega) = \pi \sqrt {{{2{{\mathcal I}_0}} \over {\hbar {\omega _p}}}} \delta (\omega - \omega {- _p})\), which implies that only the mode at frequency

*ω*

_{p}will be in a coherent state, while all other modes of the field will be in their ground states.

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 Heisenberg-limited. 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 Fabry-Pérot 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].

*ω*

_{p}= 2

*ω*

_{0}give birth to pairs of correlated photons with frequencies

*ω*

_{1}and

*ω*

_{2}(traditionally called

*signal*and

*idler*) by means of the nonlinear dependence of polarization in a birefringent crystal on electric field. Such a process can be described by the following Hamiltonian, provided that the pump field is in a coherent state |

*a*〉

_{ωp}with strong classical amplitude

*|α*

_{p}| ≫ 1 (see, e.g., Section 5.2 of [163] for details):

*â*

_{1,2}describe annihilation operators for the photons of the signal and idler modes and χ =

*ρe*

^{2iϕ}is the complex coupling constant that is proportional to the second-order susceptibility of the crystal and to the pump complex amplitude. Worth noting is the meaning of

*t*in this Hamiltonian: it is a parameter that describes the duration of a pump light interaction with the nonlinear crystal, which, in the simplest situation, is either the length of the crystal divided by the speed of light c, or, if the crystal is placed between the mirrors of the optical cavity, the same as the above but multiplied by an average number of bounces of the photon inside this cavity, which is, in turn, proportional to the cavity finesse \({\mathcal F}\). It is straightforward to obtain the evolution of the two modes in the interaction picture (leaving apart the obvious free evolution time dependence \({e^{- i{\omega _{s,i}}t}}\)) solving the Heisenberg equations:

*ω*

_{0}=

*ω*

_{p}/2:

*ω*

_{1}→

*ω*

_{+}=

*ω*

_{0}+ Ω and

*ω*

_{2}→

*ω*

_{−}=

*ω*

_{0}− Ω (

*â*

_{1}→

*â*

_{+}and

*â*

_{2}→

*â*

_{−}). Then the electric field of a two-mode state going out of the nonlinear crystal will be written as (we did not include the pump field here assuming it can be ruled out by an appropriate filter):

*ω*

_{±}−

*ω*

_{0}= ±Ω should be kept in the integral, which yields:

*ϕ*= 0) case, while ℙ[

*ϕ*] stands for a counterclockwise 2D-rotation matrix by angle

*ϕ*defined by (17). Therefore, the evolution of a two-mode light quadrature amplitude vector

*in a PDC process described by the Hamiltonian (71) consists of a clockwise rotation by an angle*

**â***ϕ*followed by a deformation along the main axes (stretching along the

*a*

_{c}-axis and proportional squeezing along the a

_{s}-axis) and rotation back by the same angle. It is straightforward to show that vector \({\hat X^{{\rm{sqz}}}} = {\left\{{\hat X(t),\,\hat Y(t)} \right\}^{\rm{T}}} = {\hat a^{{\rm{sqz}}}}{e^{- i\Omega t}} + {\hat a^{{\rm{sqz*}}}}{e^{i\Omega t}}\) transforms similarly \(({\rm{here}}\,{\hat a^{{\rm{sqz*}}}} = {\left\{{\hat a_c^{{\rm{sqz}}\dagger},\,\hat a_s^{{\rm{sqz}}\dagger}} \right\}^{\rm{T}}})\).

*Ŝ*[

*ρ*,

*t*,

*ϕ*] on the vacuum state

*â±*are transformed in accordance with Eqs. (72).

*r ≡ ρt*and used a short notation for the symmetrized outer product of vector \(\hat X\) with itself:

*r*is the quantity reflecting the strength of the squeezing. This way of characterizing the squeezing strength, though convenient enough for calculations, is not very ostensive. Conventionally, squeezing strength is measured in decibels (dB) that are related to the squeezing parameter

*r*through the following simple formula:

*r*≃ 1.15.

*ϕ*clockwise as featured in Figure 16.

_{α}(

*r*,

*ϕ*)⟩ that is obtained from the squeezed vacuum state in the same manner as the coherent state yields from the vacuum state, i.e., by the application of the displacement operator (equivalent to the action of a classical force):

*Y*by an amount proportional to the magnitude of the signal force. Such a displacement has no other consequence than simply to shift the mean values of \(\hat X\) and \(\hat Y\) by some constant values dependent on shift complex amplitude

*α*:

*â*

_{c}(

*t*) and

*â*

_{s}(

*t*) for the traveling wave case. Utilizing this similarity, let us define a squeezing operator for the continuum of modes as:

*r*(Ω) and

*ϕ*(Ω) are frequency-dependent squeezing factor and angle, respectively. Acting with this operator on a vacuum state of the travelling wave yields a squeezed vacuum state of a continuum of modes in the very same manner as in Eq. (76). The result one could get in the measurement of the electric field amplitude of light in a squeezed state as a function of time is presented in Figure 18. Quadrature amplitudes for each frequency Ω transform in accordance with Eqs. (73). Thus, we are free to use these formulas for calculation of the power spectral density matrix for a traveling wave squeezed vacuum state. Indeed, substituting \({\hat a_{c,s}}(\Omega) \to \hat a_{c,s}^{{\rm{sqz}}}(\Omega)\) in Eq. (64) and using Eq. (74) one immediately gets:

*r*(Ω) and squeezing angle (Ω) are frequency dependent as is the case in all physical situations. This indicates that quantum noise in a squeezed state of light is not Markovian and this can easily be shown by calculating the the covariance matrix, which is simply a Fourier transform of \({\mathbb S}(\Omega)\) according to the Wiener-Khinchin theorem:

*r*(Ω) and

*ϕ*(Ω). Note that the noise described by \({{\mathbb V}_{{\rm{sqz}}}}(t - {t\prime})\) is stationary since all the entries of the covariance matrix (correlation functions) depend on the difference of times

*t*−

*t*′.

### 3.3 How to calculate spectral densities of quantum noise in linear optical measurement?

*(Ω) = {*

**â***â*

_{c}(Ω),

*â*

_{s}(Ω)}

^{⊤}Tto the readout quantity of a meter is linear and can be written in spectral form as:

*G*(Ω) is the spectrum of the measured quantity, \({{\mathcal Y}_{c,s}}(\Omega)\) are some complex-valued functions of Ω that characterize how the light is transformed by the device. Quantum noise is represented by the terms of the above expression not dependent on the measured quantity

*G*, i.e.,

*S*

_{Y}(Ω) that is defined by the following expression:

*ψ*〉 is the quantum state of the light wave.

_{0}(r,

*ϕ*)〉 states. Let us show how to calculate the power (double-sided) spectral density of a generic quantity \(\hat Y(\Omega)\) in a vacuum state. To do so, one should substitute Eq. (86) into Eq. (87) and obtain that:

^{7}. Similarly, one can calculate the spectral density of quantum noise if the light is in a squeezed state |sqz

_{0}(

*r*,

*ϕ*)}, utilizing the definition of the squeezed state density matrix given in Eq. (83):

*S*

_{YZ}(Ω) of \(\hat Y(\Omega)\) with some other quantity \(\hat Z(\Omega)\) with quantum noise defined as:

*S*

_{YZ}(Ω) similar to (87):

*s*=

*i*Ω.

*â*

_{i}(Ω) stand for quadrature amplitude vectors of

*N*independent electromagnetic fields, and \({\mathcal Y}_i^\dagger (\Omega)\) are the corresponding complex-valued coefficient functions indicating how these fields are transmitted to the output. In reality, the readout observable of a GW detector is always a combination of the input light field and vacuum fields that mix into the output optical train as a result of optical loss of various origin. This statement can be exemplified by a single lossy mirror I/O-relations given by Eq. (34) of Section 2.2.4.

**â**_{i}(Ω) independent from each other, the initial state will simply be a direct product of the initial states for each of the fields:

## 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 back-action 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

*M*. The position

*x*of

*M*is probed periodically with time interval

*ϑ*. In order to make our model more realistic, we suppose that each pulse reflects from the test mass

*> 1 times, thus increasing the optomechanical coupling and thereby the information of the measured quantity contained in each reflected pulse. We also assume mass*

**F***M*large enough to neglect the displacement inflicted by the pulses radiation pressure in the course of the measurement process.

*j-th*pulse, when reflected, carries a phase shift proportional to the value of the test-mass position

*x*(

*t*

_{j}) at the moment of reflection:

*k*

_{p}=

*ω*

_{p}/

*c*,

*ω*

_{p}is the light frequency,

*j*= …, −1, 0, 1,… is the pulse number and \({\hat \phi _j}\) is the initial (random) phase of the

*j-th*pulse. We assume that the mean value of all these phases is equal to zero, \(\left\langle {{{\hat \phi}_j}} \right\rangle = 0\), and their root mean square (RMS) uncertainty \(\left\langle {({{\hat \phi}^2}} \right\rangle - {\left\langle {\hat \phi} \right\rangle ^2}{/^{1/2}}\) is equal to Δ

*ϕ*.

*ϕ*. In this case, the initial uncertainty will be the only source of the position measurement error:

*j*-th pulse. The major part of this perturbation is contributed by classical radiation pressure:

*x*

_{meas}and the momentum perturbation Δ

*p*

_{b.a.}due to back action also satisfy the uncertainty relation:

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 back-action 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

*T*, which, on the one hand, is long enough to comprise a large number of individual pulses:

*x*not to change considerably during this time due to the test-mass self-evolution. Then one can use all the

*N*measurement results to refine the precision of the test-mass position

*x*estimate, thus getting \(\sqrt N\) times smaller uncertainty

*continuous*measurement of the test-mass position \(\hat x(t)\) as a result. We need more adequate parameters to characterize its ‘strength’ than Δ

*x*

_{meas}and Δ

*p*

_{b.a.}. For continuous measurement we introduce the following parameters instead:

*ϑ*:

*S*

_{x}and

*S*

_{ϕ}let us rewrite Eq. (98) in the continuous limit:

*measurement noise*, proportional to the phase \(\hat \phi (t)\) of the light beam (in the continuous limit the sequence of individual pulses transforms into a continuous beam). Then there is no difficulty in seeing that

*S*

_{x}is a power (double-sided) spectral density of this noise, and

*S*

_{ϕ}is a power double-sided spectral density of \(\hat \phi (t)\).

*S*

_{F}, and double-sided power spectral density of \(\hat {\mathcal I}\) is \({S_{\mathcal I}}\).

*continuous linear measurement*, which nevertheless comprises the main features of a more general theory, i.e., it contains equations for the calculation of measurement noise (112) and also for back action (114). The precision of this measurement and the object back action in this case are described by the spectral densities

*S*

_{x}and

*S*

_{F}of the two meter noise sources, which are assumed to not be correlated in our simple model, and thus satisfy the following relation (cf. Eqs. (109)):

**Simple case: light in a coherent state**. Recall now that scheme of representing the quantized light wave as a sequence of short statistically-independent pulses with duration

*ε*≡

*ϑ*we referred to in Section 3.2. It is the very concept we used here, and thus we can use it to calculate the spectral densities of the measurement and back-action noise sources for our simple device featured in Figure 20 assuming the light to be in a coherent state with classical amplitude \({A_c} = \sqrt {2{{\mathcal I}_0}/(\hbar {\omega _p})}\) (we chose

*A*

_{s}=0 thus making the mean phase of light \(\left\langle {\hat \phi} \right\rangle = 0\)). To do so we need to express phase \(\hat \phi\) and energy \(\hat {\mathcal W}\) in the pulse in terms of the quadrature amplitudes

*â*

_{c,s}(

*t*). This can be done if we refer to Eq. (61) and make use of the following definition of the mean electromagnetic energy of the light wave contained in the volume \({\upsilon _\vartheta} \equiv {\mathcal A}c\vartheta\) (here, \({\mathcal A}\) is the effective cross-sectional area of the light beam):

^{8}. We used here the definition of the mean pulse quadrature amplitude operators introduced in Eqs. (67). In the same manner, one can define a phase for each pulse using Eqs. (14) and with the assumption of small phase fluctuations (Δ

*ϕ*≪ 1) one can get:

- 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 (frequency-independent) spectral densities.

The features 1 and 2, in turn, lead to characteristic fundamentally-looking sensitivity limitations, the SQL. We will call linear quantum meters, which obey these limitations (that is, with mutually non-correlated 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.

*G*(

*t*), and the meter. The action of this force on the probe causes its displacement \(\hat x\) that is monitored by the meter (e.g., light, circulating in the interferometer). The output observable of the meter

*Ô*is monitored by some arbitrary classical device that makes a measurement record

*o*(

*t*). The quantum nature of the probe-meter interaction is reflected by the back-action force \(\hat F\) that randomly kicks the probe on the part of the meter (e.g., radiation pressure fluctuations). At the same time, the meter itself is the source of additional quantum noise

*Ô*

_{fl}(

*t*) in the readout signal. Quantum mechanically, this system can be described by the following Hamiltonian:

*t*

_{0}is the arbitrary initial moment of time that can be set to −∞ without loss of generality.

*For a linear system with Hamiltonian*(124),

*for any linear observable Â of the probe and for any linear observable*\(\hat B\)

*of the meter, their full Heisenberg evolutions are given by*:

*where Â*

^{(0)}(t)

*and*\({\hat B^{(0)}}(t)\)

*stand for the free Heisenberg evolutions in the case of no coupling, and the functions χ*

_{Ax}(

*t,t′*)

*and χ*

_{BF}(

*t, t′*)

*are called (time-domain) susceptibilities and defined as*:

*χ*(

*t, t′*) =

*χ*(

*t*+

*τ,t′*+

*τ*), therefore they depend only on the difference of times:

*χ*(

*t, t′*) →

*χ*(

*t*−

*t′*). In this case, one can rewrite Eqs. (126) in frequency domain as:

*Ô*that fully characterize our linear measurement process in the scheme featured in Figure 21:

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.

*Ô*(

*t*). As we have mentioned already, the output observable in the linear measurement process should be precisely measurable at any instance of time. This implies a

*simultaneous measurability condition*[30, 41, 147, 22, 43, 33] on the observable

*Ô*(

*t*) requiring that it should commute with itself at any moment of time:

*Ô*(

*t*

_{1}) at some moment of time

*t*

_{1}shall not disturb the measurement result at any other moments of time and therefore the sample data {

*Ô*(

*t*

_{1}),

*Ô*(

*t*

_{1}),…,

*Ô*(

*t*

_{n})} can be stored directly as bits of classical data in a classical storage medium, and any noise from subsequent processing of the signal can be made arbitrarily small. It means that all noise sources of quantum origin are already included in the quantum fluctuations of

*Ô*(

*t*) [43, 33]. And the fact that due to (131) this susceptibility turns out to be zero reflects the fact that

*Ô*(

*t*) should be a classical observable.

The second equation in (129) describes how the back-action 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 back-action* of the meter and since, by construction, it is the part of the back-action 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.

*x*

_{s}(

*t*) is the probe’s response on the signal force

*G*(

*t*) and is, actually, the part we are mostly interested in. This expression also unravels the role of susceptibility

*χ*

_{xx}(

*t*−

*t*′): it is just the Green’s function of the equation of motion governing the probe’s bare dynamics (also known as impulse response function) that can be shown to be a solution of the following initial value problem:

*M*and to \({\Omega _m}\) for a harmonic oscillator with eigenfrequency Ω

_{m}. Apparently, operator

**D**is an inverse of the integral operator

**χ**_{xx}whose kernel is

*χ*

_{xx}(

*t*−

*t*′):

The second value, \({\hat x_{{\rm{b}}{\rm{.a}}{\rm{.}}}}(t)\), is the displacement of the probe due to the back-action 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].

*t*

_{0}, as per the structure of the operator

**D**governing the probe’s dynamics. It is this part of the actual displacement that bears quantum uncertainties imposed by the initial quantum state of the probe. One could argue that these uncertainties might become a source of additional quantum noise obstructing the detection of GWs, augmenting the noise of the meter. This is not the case as was shown explicitly in [19], since our primary interest is in the detection of a classical force rather than the probe’s displacement. Therefore, performing over the measured data record

*o*(

*t*) the linear transformation corresponding to first applying the operator \({\mathcal X}_{OF}^{- 1}\) on the readout quantity that results in expressing

*o*(

*t*) in terms of the probe’s displacement:

**D**that yields a force signal equivalent to the readout quantity

*o*(

*t*):

**D**

_{x}

^{(0)}(

*t*) vanishes since

*x*

^{(0)}(

*t*) is the solution of a free-evolution equation of motion. Thus, we see that the result of measurement contains two noise sources, \({\hat x_{{\rm{fl}}}}(t)\) and \({\hat F_{{\rm{b}}{\rm{.a}}{\rm{.}}}}(t)\), which comprise the sum noise masking the signal force

*G*(

*t*).

*A, B*) ⇒ (

*O, F, x*), and we omit the term \({\hat x^{(0)}}(\Omega)\) in the last equation for the reasons discussed above. The solution of these equations is straightforward to get and reads:

*Ô*(Ω) to unit signal. In GW interferometers, two such normalizations are popular. The first one tends to consider the tidal force

*G*as a signal and thus set to 1 the coefficient in front of

*G*(Ω) in Eq. (135). The other one takes GW spectral amplitude

*h*(Ω) as a signal and sets the corresponding coefficient in

*Ô*(Ω) to unity. Basically, these normalizations are equivalent by virtue of Eq. (12) as:

*Ô*(Ω) in force normalization:

\(\hat {\mathcal X}\) 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 back-action force at earlier times*t*<*t*′. In the following we will refer to \(\hat {\mathcal F}\) as the*effective back-action noise*(radiation-pressure noise, in the GW interferometer common terminology).

*Schrödinger-Robertson*uncertainty relation:

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.

*S*

_{x}(Ω) instead of

*S*

_{χχ}(Ω) and

*S*

_{F}(Ω) instead of \({S_{{\mathcal F}{\mathcal F}}}(\Omega)\) Then the uncertainty relation (148) transforms into:

*at any given frequency*Ω. To derive this limit we assume noise sources

*x*

_{fl}and

*F*

_{b.a.}to have minimal values allowed by quantum mechanics, i.e.

*achieved when contributions of measurement noise and back-action noise to the sum noise are equal to each other*, i.e., when

*h*-normalization and for

*x*-normalization. The former is obtained from (151) via multiplication by 4/(

*M*

^{2}

*L*

^{2}Ω

^{4}):

*χ*

_{xx}(Ω)

*F*(Ω):

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 brute-force methods of sensitivity improving cease working, and methods that allow one to blot out the back-action 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.

*S*

_{x}and

*S*

_{F}can be expressed through Ω

_{q}as follows:

_{q}is, the smaller

*S*

_{x}is (the higher is the measurement precision), and the larger

*S*

_{F}is (the stronger the meter back action is).

*M*,

_{q}. It is easy to see that these plots never dive under the SQL line (161), which embodies a common envelope for them. Due to this reason, the sensitivities area above this line is typically considered as the ‘classical domain’, and below it — as the ‘quantum domain’.

#### 4.3.2 Harmonic oscillator SQL

*χ*

_{xx}(Ω)) is, the smaller its force SQL at this frequency is. In the harmonic oscillator case,

_{0}standing for the oscillator mechanical eigenfrequency, and the sum quantum noise power (double-sided) spectral density equal to

_{0}. Therefore, reducing the value of Ω

_{q}, that is, using weaker measurement, it is possible to increase the sensitivity in a narrow band around Ω

_{0}. At the same time, the smaller Ω

_{q}is, the more narrow the bandwidth is where this sensitivity is achieved, as can be seen from the plots drawn in Figure 22 (right).

*ν*= Ω − Ω

_{0}be the detuning from the resonance frequency. Suppose also

_{0}, with the maximums at its edges,

*ν*= ±ΔΩ/2. The sum noise power (double-sided) spectral density at these points is equal to

*free mass*SQL (161), which reads

*oscillator*SQL, equal to

#### 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.

*h*

**-normalization**: The noise spectral density in

*h*-normalization can be obtained using Eq. (12). Where the SQM is concerned, the sum noise in

*h*-normalization reads

*χ*

_{xx}(Ω) = −1/(

*M*Ω

^{2}) the above expression transforms as:

*h*-normalization:

_{q}are given in the left panel of Figure 23.

*h*-normalization:

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 near-resonance force, as the corresponding force-normalized spectral densities (163, 172).

*x*

**-normalization**: Another normalization that is worth considering is the actual probe displacement, or

*x*-normalization. In this normalization, the sum noise spectrum is obtained by multiplying noise term \({\hat {\mathcal N}^F}(\Omega)\) in Eq. (139) by the probe’s susceptibility

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.

*h*-normalization except for the multiplication by 4/

*L*

^{2}:

*x*-normalization is given in Figure 24.

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 near-resonance 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

*χ*

_{FF}(Ω) that should be at any frequency equal to:

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 back-action 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 cross-correlations between the measurement and back-action noise. We start from the very toy example discussed in Section 4.1.1.

*homodyne angle ϕ*

_{LO}(cf. Eq. (39)):

*Ö*(

*t*

_{j})).

*p*

_{b.a.}evidently remains the same as in the uncorrelated case, and we provided its value here only for convenience), which gives the exact equality in the Schrödinger-Robertson uncertainty relation:

*virtual rigidity*\({\mathcal X}_{FF}^{{\rm{virt}}}(\Omega) \equiv - {K_{{\rm{virt}}}} = {\rm{const}}{\rm{.}}\) as one can conclude looking at Eqs. (141). Indeed,

*K*

_{virt}as ‘virtual rigidity’.

_{0}.

*S*

_{F}, which is a good measure of measurement strength according to Eqs. (156), for the virtual rigidity against the real one. For the latter, to overcome the free mass SQL by a factor

_{0}, the back-action noise spectral density has to be

*reduced*by this factor:

_{0}. Such a sensitivity gain is achieved at the expense of proportionally reduced bandwidth:

*S*

_{F}results from Eq. (167):

*ξ*

^{2}), the larger

*S*

_{F}must be and, therefore, measurement strength.

Another evident flaw of the virtual rigidity, which it shares with the real one, is the narrow-band 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

*τ*. An outgoing pulse acquires a phase shift proportional to the difference of the test-object positions at time moments separated by

*τ*, which is proportional to the test-mass average velocity \(\hat \bar \upsilon ({t_j})\) in this time interval (

*t*

_{j}indicates the time moment after the second reflection):

*τ*is much smaller than the signal force variation characteristic time (∼ Ω

^{−1}) that spills over into the following condition:

*S*

_{xF}≠ 0, allows the reduction of the sum noise spectral density to arbitrarily small values. One can easily see it after the substitution of Eq. (215) into Eq. (144) with a free mass

*χ*

_{xx}(Ω) = −1/(

*M*Ω

^{2}) in mind:

*S*

_{vp}= 0, then by virtue of the uncertainty relation, the sum sensitivity appeared limited by the SQL (161):

*S*

_{xF}∝ cot

*ϕ*

_{LO}allows one to get only a narrow-band sub-SQL sensitivity akin to that of a harmonic oscillator. This effect we called

*virtual rigidity*, and showed that for position measurement this rigidity

*K*

_{virt}=

*S*

_{xF}/

*S*

_{x}is constant. In the speed-meter case, the situation is totally different; it is clearly seen if one calculates virtual rigidity for a speed meter:

*S*

_{p}is provided; that is, if the optomechanical coupling is sufficiently strong.

**Simple case: light in a coherent state**. Let us consider how the spectral density of a speed meter will appear if the light field is in a coherent state. The spectral densities of phase and power fluctuations are given in Eqs. (122), hence the sum noise power (double-sided) spectral density for the speed meter takes the following form:

_{q}for our scheme is defined in Eq. (157). This formula indicates the ability of a speed meter to have a sub-SQL sensitivity in all frequencies provided high enough optical power and no optical loss.

#### 4.5.2 QND measurement of a free mass velocity

*β*(

*t*) is the coupling factor, which has the form of two short pulses with the opposite signs, separated by the time

*τ*, see Figure 27. We suppose for simplicity that the evolution of the meter observable \({\mathcal N}\) can be neglected during the measurement (this is a reasonable assumption, for in real schemes of the speed meter and in the gedanken experiment considered above, this observable is proportional to the number of optical quanta, which does not change during the measurement). This assumption allows one to omit the term \({{\mathcal L}_{{\rm{meter}}}}\) from consideration.

*Û*(0,

*τ*) is the evolution operator of probe-meter dynamics from the initial moment

*t*

_{start}= 0 when the measurement starts till the final moment

*t*

_{end}=

*τ*when it ends. Basically, the latter condition guarantees that the value of \(\hat x\) before the measurement will be equal to that after the measurement, but does not say what it should be in between (see Section 4.4 of [22] for details).

^{9}, the Lagrangian (225) can be converted to the form, satisfying the simple condition of [28]:

*β*(

*t*) guarantees that the coupling factor

*α*(

*t*) becomes equal to zero when the measurement ends. The canonical momentum of the mass

*M*is equal to

*Ô*of the meter. The Heisenberg equations of motion for these observables are the following:

*α*≠ 0), yet restore their initial values after the measurement, and (iii) the output signal of the meter \(\hat \Phi\) carries the information about this perturbed value of the velocity.

*α*(

*t*) to be of a simple rectangular shape:

*τ*), the

*perturbed*value of velocity \(\bar \upsilon\) is measured with an imprecision

*v*

_{init}=

*p*/

*m*yielding from this measurement is thus equal to:

*velocity measurement*SQL:

*v*

_{init}with arbitrarily high precision. Such a cross-correlation can be achieved by measuring the following combination of the meter observables

## 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 Fabry-Pérot 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 real-life 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

*ω*

_{p}, and mean power values \({{\mathcal I}_1}\) and \({{\mathcal I}_2}\). In terms of the general linear measurement theory of Section 4.2 we have two meters represented by these two incident light waves. The two arbitrary quadratures of the reflected waves are deemed as measured quantities

*Ô*

_{1}and

*Ô*

_{2}. Measurement can be performed, e.g., by means of two independent homodyne detectors. Let us analyze quantum noise in such a model keeping to the scheme given by Eqs. (133).

#### 5.1.1 Optical transfer matrix of the movable mirror

*ê*

_{1,2}(Ω) by their dimensionless counterparts as introduced by Eq. (61) of Section 3.1:

_{1s}= 0 and \({{\rm{A}}_{1c}} = \sqrt {2{{\mathcal I}_1}/(\hbar {\omega _p})}\). Then factoring in the constant phase difference between the left and the right beams equal to Φ

_{0}, one would obtain for the left light \(\{{A_{2c}},{A_{2s}}\} = \sqrt {2{{\mathcal I}_2}/(\hbar {\omega _p})} \{{\rm{cos}}\,{\Phi _0},\,\sin \,{\Phi _0}\}\).

*[*

**H***ϕ*] was first introduced in Section 2.3.2 after Eq. (47) as:

#### 5.1.2 Probe’s dynamics: radiation pressure force and ponderomotive rigidity

*G*:

*χ*

_{xx}(Ω) = −1/(

*M*Ω

^{2}). The term \({\hat F_{{\rm{b}}{\rm{.a}}{\rm{.}}}}(t)\) stands for the radiation pressure force imposed by the light that can be calculated as

*k*

_{p}≡

*ω*

_{p}/

*c*

^{10}. It is constant and thus can be compensated by applying a fixed restoring force of the same magnitude but with opposite direction, which can be done either by employing an actuator, or by turning the mirror into a low-frequency pendulum with

*ω*

_{m}≪ Ω

_{GW}by suspending it on thin fibers, as is the case for the GW interferometers, that provides a necessary gravity restoring force in a natural way. However, it does not change the quantum noise and thus can be omitted from further consideration. The latter term represents a quantum correction to the former one

**â**_{1}(Ω) and

**â**_{2}(Ω) with coefficients given by vectors:

*χ*

_{FF}(Ω) = −

*K*.

#### 5.1.3 Spectral densities

*G*according to Eq. (140):

*K*into the effective mechanical susceptibility:

*ϕ*

_{1}→

*ϕ*

_{LO}.

#### 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 back-action 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.

*full transfer matrix*of our system. To do so, let us first consider the readout observables separately. We start with

*Ô*

_{1}and rewrite it as follows:

*= {*

**α***α*

_{1},

*α*

_{2}}

^{T}and

*= {*

**β***β*

_{1},

*β*

_{2}}

^{T}written in short notation as:

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

*η*

_{d}< 1 that indicates how many photons absorbed by the detector give birth to photoelectrons, i.e., it is the measure of the probability for the photon to be transformed into the photoelectron. As with any other dissipation, this loss of photons gives rise to an additional noise according to the FDT that we should factor in. We have shown in Section 2.2.4 that this kind of loss can be taken into account by means of a virtual asymmetric beamsplitter with transmission coefficients \(\sqrt {{\eta _d}}\) and \(\sqrt {1 - {\eta _d}}\) for the signal light and for the additional noise, respectively. This beamsplitter is set into the readout optical train as shown in Figure 8 and the

*i*-th readout quantity needs to be modified in the following way:

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 Fabry-Pérot cavity.

### 5.2 Fabry-Pérot cavity

A Fabry-Pérot 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}.

*τ*=

*L*/

*c*for the light travel time between the mirrors.

The frequency domain version of the above equations and their solutions are derived in Appendix A.1. We write these I/O-relations 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 *single-mode approximation*.

**Single-mode approximation**. We note that:

- (i)in GW detection, rather high-finesse 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}= (2*τ*)^{−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)

*n*is even or odd, the final results do not. Therefore, we assume for simplicity that

*n*is even.

*τ*and keeping only the first non-vanishing terms, we obtain that

#### 5.2.1 Optical transfer matrix for a Fabry-Pérot 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$$\begin{array}{*{20}c} {\hat \beta (\omega) = f(\Omega)\hat \alpha (\omega)\; \rightarrow \;{{\hat \beta}_ +} \equiv \hat \beta ({\omega _p} + \Omega) = f({\omega _p} + \Omega)\hat \alpha ({\omega _p} + \Omega) \equiv {f_ +}{{\hat \alpha}_ +}\;{\rm{and}}\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\hat \beta _ - ^\dagger \equiv {{\hat \beta}^\dagger}({\omega _p} - \Omega) = {f^{\ast}}({\omega _p} - \Omega){{\hat \alpha}^\dagger}({\omega _p} - \Omega) \equiv f_ - ^{\ast}\hat \alpha _ - ^\dagger \,;} \\ \end{array}$$(288)*f*(Ω) is an arbitrary complex-valued function of sideband frequency Ω; - 2.use the definition (57) to get the following relations for two-photon 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)

_{1,2}and \({\mathbb T}\) satisfy the following unitarity relations:

#### 5.2.2 Mirror dynamics, radiation pressure forces and ponderomotive rigidity

*χ*

_{xx,i}are the mechanical susceptibilities of the mirrors,

*G*

_{i}stand for any external classical forces that could act on the mirrors (for example, a signal force to be detected),

_{i}, f

_{i}, etc. The signs for all forces are chosen in such a way that the positive forces are oriented outwards from the cavity, increasing the corresponding mirror displacement

*x*

_{1,2}.

*x*. Therefore, combining Eqs. (301), we obtain for this mode:

*M*is some (in general, arbitrary) mass. Typically, it is convenient to make it equal to the reduced mass

*μ*.

### 5.3 Fabry-Pérot-Michelson 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 Fabry-Pérot 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 Fabry-Pérot 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 proportionally-amplified 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 Fabry-Pérot cavities, for a given laser power, and provide the means for fine-tuning 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 Fabry-Pérot 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 non-signal-recycled configurations).

#### 5.3.1 Optical I/O-relations

_{N,E}now describe the noise sources due to optical losses in the arm cavities.

*T*

_{arm}is the input mirrors power transmittance,

*A*

_{arm}is the arm cavities power losses per bounce,

*δ*

_{arm}is the arm cavities detuning,

*R*=

*T*= 1/2. Let

*l*

_{W}=

*cτ*

_{W}be the power recycling cavity length (the optical distance between the power recycling mirror and the input test masses) and

*l*

_{S}=

*cτ*

_{S}— power recycling cavity length (the optical distance between the signal recycling mirror and the input test masses). In this case, the light propagation between the recycling mirrors and the input test masses is described by the following equations for the classical field amplitudes:

*ω*

_{p}passing through the power and signal recycling cavities, and the similar equations:

*R*

_{W},

*T*

_{W}and

*R*

_{S},

*T*

_{S}are the reflectivities and transmissivities of the power and signal recycling mirrors, respectively. These equations, being linear and frequency independent, are valid both for the zeroth-order classical amplitudes and for the first-order quantum ones.

#### 5.3.2 Common and differential optical modes

*x*

_{−}makes part of the pumping carrier energy stored in the common mode pour into the differential mode, which means a non-linear parametric coupling between these modes.

#### 5.3.3 Interferometer dynamics: mechanical equations of motion, radiation pressure forces and ponderomotive rigidity

*μ*=

*M*/2 is the effective mass of these modes and

*G*

_{N,E}are the external classical forces acting on the cavities end mirrors. The differential mechanical mode equation of motion (338) taking into account Eq. (334) reads:

*M*. For the same reason Eq. (334) implies for the effective optical power a value twice as high as the power of light, circulating in the arm cavities:

#### 5.3.4 Scaling law theorem

*γ*

_{1}and

*δ*with the effective parameters

*γ*

_{1W}and

*δ*

_{W}. The only difference is an additional phase shift

*α*

_{W}. Thus, we have shown that the power recycling cavity formed by the PRM and the ITMs can be treated as a single mirror with some effective parameters defined implicitly by Eqs. (350), complemented by light propagation over length

*α*

_{W}

*/k*

_{p}. Note also that the phase shift

*α*

_{W}can be absorbed into the field amplitudes simply by renaming

*δ*≠ 0.

*e*

^{iωτS}that describes a frequency-dependent phase shift the sideband fields acquire on their pass through the signal recycling cavity. It is due to this frequency-dependent phase shift that the differential mode cannot be reduced, strictly speaking, to a single effective cavity mode, and a more complicated two-cavity model of Figure 31 should be used instead. The reduction to a single mode is nevertheless possible in the special case of a short signal-recycling cavity, i.e., such that:

*ϕ*

_{S}can be approximated by the frequency-independent value:

The mechanical equations of motion for the effective cavity are absolutely the same as for an ordinary Fabry-Pérot cavity considered in Section 5.2 except for the new values of the effective mirrors’ mass 2*M* 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 Fabry-Pérot-Michelson interferometer

The scaling law we have derived above enables us to calculate spectral densities of quantum noise for a dual-recycled Fabry-Pérot-Michelson featured in Figure 30 as if it were a bare Fabry-Pérot cavity with movable mirrors pumped from one side, similar to that shown in Figure 32.

_{1}, \({\mathbb T}\) are defined by Eqs. (294) and (296),

**â**^{vac}corresponds to the vacuum state.

#### 5.3.6 Full transfer matrix approach to calculation of the Fabry-Pérot-Michelson 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).

_{1,2}(Ω) can be computed using the fact that

*h*(Ω). This can easily be done using the simple rule given in Eq. (138) and taking into account that in our case

*G*(Ω) →

*G*(Ω)/2:

*h*

_{GW}(Ω) is the spectrum of the GW signal and the second term

*ĥ*

_{n}(Ω) stands for the sum quantum noise expressed in terms of metrics variation spectrum units, i.e., in Hz

^{−1/2}.

*δ*= 0 was analyzed, and then by Buonanno and Chen in [32, 34], who considered a more general detuned case. Thus, treading their steps, we have shown that the quantum noise of the Fabry-Pérot-Michelson interferometer (as well as the single cavity Fabry-Pérot one) has the following distinctive features:

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 back-action noise \(\hat {\mathcal F}\), often referred to as*quantum radiation-pressure 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 signal-recycling 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 Sub-SQL Sensitivity

### 6.1 Noise cancellation by means of cross-correlation

#### 6.1.1 Introduction

In this section, we consider the interferometer configurations that use the idea of the cross-correlation of the shot and the radiation pressure noise sources discussed in Section 4.4. This cross-correlation allows the measurement and the back-action 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 cross-correlation 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 frequency-independent 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 narrow-band 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 frequency-dependent 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 frequency-dependent 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 frequency-dependent squeezing angle (called *pre-filtering*), or between the main interferometer and the homodyne detector, creating the effective frequency-dependent squeezing angle (*post-filtering*). As we show below, in principle, both pre- and post-filtering 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 back-action 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 filter-cavities-based interferometer topologies, we limit ourselves here by the case of the resonance-tuned 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 cross-correlation 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 high-reflectivity mirrors, the losses per bounce do not exceed *A*_{arm} ≲ 10^{−4}, and the arms lengths of the large-scale 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 narrow-band 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 Frequency-dependent 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.

*γ*of the interferometer. If

*ϵ*

_{d}= 0 and

*γ*≫ Ω, then Eqs. (173) and (394) become identical, with the evident correspondence

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}.

*J*=

*J*

_{aLIGO}≠ (2

*π*× 100)

^{3}s

^{−3}corresponds to the circulating power of \({{\mathcal I}_{{\rm{arm}}}} = 840\,{\rm{kW,}}\,L = 4{\rm{km}}\), and

*M*= 40 kg; the interferometer bandwidth

*γ*= 2

*π*× 500 s

^{−1}is close to the one providing the best sensitivity for Advanced LIGO in the presence of technical noise [93]; 10 dB squeezing (

*e*

^{2r}= 10), which corresponds to the best squeezing available at the moment (2011) in the low-frequency band [102, 150, 151]);

*η*

_{d}= 0.95 can be considered a reasonably optimistic estimate for the real interferometer quantum efficiency.

Noteworthy is the proximity of the plots for the interferometer with 10 dB input squeezing and the one with 10-fold increased optical power. The noticeable gap at higher frequencies is due to optical loss.

**Frequency dependent squeezing angle**. Now suppose that the homodyne angle can be frequency dependent, and calculate the corresponding minimum of the sum noise spectral density (391). The first term in square brackets in this equation can be rewritten as:

**V**^{⊤}

*V*

*e*

^{−2r}. Therefore,

*e*

^{−2r}in comparison with the vacuum input case. Note, however, that the noise contribution due to optical losses remains unchanged. Concerning the homodyne angle

*ϕ*

_{lo}, we use again the classical optimization, setting

*η*

_{d}= 0.95 (dotted line). In both cases, the optical power and the squeezing factor are equal to

*=*

**J***J*

_{aLIGO}and

*e*

^{2r}= 10, respectively.

**Frequency dependent homodyne angle**. Suppose now that the squeezing angle corresponds to the classical optimization:

*ϕ*

_{LO}. The minimum is provided by the following dependence

*η*

_{d}= 0.95 (dashed lines).

Compare this spectral density with the one for the frequency-dependent squeezing angle (pre-filtering) case, see Eq. (403). The shot noise components in both cases are exactly equal to each other. Concerning the residual back-action noise, in the pre-filtering case it is limited by the available squeezing, while in the post-filtering case — by the optical losses. In the latter case, were there no optical losses, the back-action noise could be removed completely, as shown in Figure 35 (left). For the parameters of the noise curves presented in Figure 35 (right), the post-filtering still has some advantage of about 40% in the back-action 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 pre-filtering is preferable if good squeezing is available, and the optical losses are relatively large, and vice versa. In particular, post-filtering can be used even without squeezing, *r* = 0.

**Frequency dependent homodyne and squeezing angles**. And, finally, consider the most sophisticated configuration: double-filtering with both the homodyne angle *ϕ*_{LO} and the squeezing angle *θ* being frequency dependent.

*θ*

_{LO}corresponds to

It is easy to see that in the ideal lossless case the double-filtering configuration reduces to a post-filtering one. Really, if *ϵ*_{d} = 0, the spectral density (410) becomes exactly equal to that for the post-filtering case (408), and the frequency dependent squeezing angle (411) degenerates into a constant value (405). However, if *ϵ*_{d} > 0, then the additional pre-filtering allows one to decrease more the residual back-action term. For example, if *e*^{2r} = 10 and *η*_{d} = 0.95 then the gain in the back-action noise amplitude \(\sqrt S\) is equal to about 25%.

We have plotted the sum quantum noise spectral density (410) in Figure 34, right (dash-dots). 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.

*ϵ*

_{d}= 0.95 that we use for our estimates corresponds to

*ϵ*

_{d}≈ 0.23. It means that without squeezing (

*r*= 0) one is only able to beat the SQL in amplitude by

*r*→ ∞ then, in principle, arbitrarily high sensitivity can be reached. But

*ξ*depends on

*r*only as

*e*

^{−r/2}, and for the 10 dB squeezing, only a modest value of

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 cross-correlation of the measurement and back-action noises and thus limiting the performance of the quantum measurement schemes, which rely on this cross-correlation.

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 frequency-independent 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 narrow-band gain in sensitivity, similar to the one provided by the harmonic oscillator (thus the term ‘virtual rigidity’).

This narrow-band gain could be more interesting not for the full-scale GW detectors (where broadband optimization of the sensitivity is required in most cases) but for smaller devices like the 10-m Hannover prototype interferometer [7], designed for the development of the measurement methods with sub-SQL 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*.

*S*

^{h}(Ω)with the following values of homodyne and squeezing angles

*ηd*= 1) and no squeezing (

*r*= 0); (lower left) no losses (

*η*

_{d}= 1) and 10 dB squeezing (

*e*

^{2r}= 10); (upper right) with losses (

*η*

_{d}= 0.95) and no squeezing (

*r*= 0); (lower right) with losses (

*η*

_{d}= 0.95) and 10 dB squeezing(

*e*

^{2r}= 10). In each pane, the family of plots is shown that corresponds to different values of the ratio Ω

_{0}/Ω

_{q}, ranging from 0.1 to \(\sqrt {10}\).

*ϵ*

_{d}= 0, there is no limitation on the SQL beating factor, provided a sufficiently small ratio of Ω

_{0}/Ω

_{q}:

*ϵ*

_{d}> 0, then function (420) has a minimum in Ω

_{q}at

#### 6.1.3 Filter cavities in GW interferometers

**Input/output relations for the filter cavity**. In essence, a filter cavity is an ordinary Fabry-Pérot 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 high-quality mirrors, the mirrors losses appear in the final expressions inflated by the filter cavity finesse).

_{c,s}= 0 (there is no classical pumping in the filter cavity and, therefore, there is no displacement sensitivity) and by some changes in the notations:

*T*

_{f}is the power transmittance of the input/output mirror,

*A*

_{f}is the factor of power loss per bounce,

*L*

_{f}is the filter cavity length,

*δ*

_{f}is its detuning.

*θ*and squeezing factor

*r*and thus can be described by the following two-photon quadrature vector

*θ*

_{f}(Ω).

*θ*

_{f}(Ω).

It is easy to see that the necessary frequency dependencies of the homodyne and squeezing angles (404) or (407) (with the second-order 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 post-filtering 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 pre-filtering and post-filtering, respectively.

**Single-filter cavity-based schemes**. The schemes under consideration are shown in Figure 37. In the pre-filtering scheme drawn in the left panel of Figure 37, a squeezed light source emits frequency-independent squeezed vacuum towards the filter cavity, where it gets reflected, gaining a frequency-dependent phase shift

*θ*

_{f}(Ω), and then enters the dark port of the main interferometer. The light going out of the dark port is detected by the homodyne detector with fixed homodyne angle

*ϕ*

_{LO}in the usual way.

*θ*of the input squeezed vacuum must be zero. Combining Eqs. (390) and (423) taking these assumptions into account, we obtain the following equation for the sum quantum noise of the pre-filtering scheme:

*π*/2, and that the phase shift introduced by the filter cavity goes to zero at high frequencies, we obtain that the real homodyne angle must also be

*π*/2. Assuming that the squeezing angle is defined by Eq. (405) and again using Eqs. (390) and (423), we obtain that the sum quantum noise and its power (double-sided) spectral density are equal to

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.

*J*= J

_{aLIGO},

*γ*= 2

*π ×*500 s

^{−1},

*η*

_{d}= 0.95), we obtain

*η*

_{f0}≈ 280 s

^{−1}, and there should be

*A*

_{f}= 10

^{−5}).

*γ*

_{f2}should be small compared to the filter cavity bandwidth

*γ*

_{f}:

*J*and

*γ*as above, we obtain:

*A*

_{f}= 10

^{−5}or 25 m cavity with

*A*

_{f}= 10

^{−4}).

**Numerical optimization of filter cavities**. In the experiments devoted to detection of small forces, and, in particular, in the GW detection experiments, the main integral sensitivity measure is the probability to detect some calibrated signal. This probability, in turn, depends on the matched filtered SNR defined as

*h*

_{s}(Ω) the spectrum of this calibrated signal.

*k*

_{0}is a factor that does not depend on the interferometer parameters, and

*f*

_{max}is the cut-off frequency that depends on the binary system components’ masses. In particular, for neutron stars with masses equal to 1.4 solar mass,

*f*

_{max}≈ 1.5 kHz.

*γ*

_{f1}and

*δ*

_{f}, providing this gain, we choose to normalize the SNR by the value corresponding to the ordinary interferometer (without the filter cavities):

We optimized numerically the ratio \(^{{\rho ^2}\rho _0^2}\), with filter cavity half-bandwidth *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* = 2*π* × 500 s^{−1}, and *η*_{d} = 0.95.

*γ*

_{f1}and

*δ*

_{f}are plotted, and in the right one the corresponding optimized values of the SNR are. It follows from these plots that the optimal values of

*γ*

_{f1}and

*δ*

_{f}are virtually the same as

*γ*

_{f0}, while the specific loss factor

*A*

_{f}/

*L*

_{f}satisfies the condition (448), and starts to deviate sensibly from

*γ*

_{f0}only when

*A*

_{f}/

*L*

_{f}approaches the limit (451). Actually, for such high values of specific losses, the filter cavities only degrade the sensitivity, and the optimization algorithm effectively turns them off, switching to the ordinary frequency-independent squeezing regime (see the right-most part of the right pane).

It also follows from these plots that post-filtering provides slightly better sensitivity, if the optical losses in the filter cavity are low, while the pre-filtering has some advantage in the high-losses scenario. This difference can be explained in the following way [87]. The post-filtration 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 back-action 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 pre-filtering 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).

*A*

_{f}/

*L*

_{f}≲ 10

^{−8}), the single filter cavity based schemes can provide virtually the same result as the abstract ones with the ideal frequency dependence for squeezing or homodyne angles.

### 6.2 Quantum speed meter

#### 6.2.1 Quantum speed-meter 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 cross-correlation tailoring with filter cavities, considered above. Here, instead of fitting the quantum noise spectral dependence to the Fabry-Pérot-Michelson 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 frequency-independent in the low- and medium-frequency range, thus making the frequency-dependent cross-correlation 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 \({{\mathcal K}_{{\rm{SM}}}}(\Omega)\).

*sloshing-cavity speed meter*, was proposed. This version uses two coupled resonators (e.g., microwave ones), as shown in Figure 40 (left), one of which (2), the

*sloshing cavity*, is pumped on resonance through the input waveguide, so that another one (1) becomes excited at its eigenfrequency

*ω*

_{e}. The eigenfrequency of resonator 1 is modulated by the position

*x*of the test mass and puts a voltage signal proportional to position

*x*into resonator 2, and a voltage signal proportional to velocity

*dx*/

*dt*into resonator 1. The velocity signal flows from resonator 1 into an output waveguide, from which it is monitored. One can understand the production of this velocity signal as follows. The coupling between the resonators causes voltage signals to slosh periodically from one resonator to the other at frequency Ω. After each cycle of sloshing, the sign of the signal is reversed, so the net signal in resonator 1 is proportional to the difference of the position at times

*t*and

*t*+ 2

*π*/

*Ω*, thus implementing the same principle of the double position measurement.

Later, the optical version of the sloshing-cavity speed-meter scheme suitable for large-scale 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 kilometer-scale Fabry-Pérot cavity — as the resonator 2, thus making a practical interferometer configuration.

*Sagnac speed meter*will be used for them below.

*τ*

_{arm}:

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 Fabry-Pérot-Michelson 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 Speed-meter sensitivity, no optical losses

*J*here is still defined by Eq. (369), but the circulating power is now twice as high as that of the position meter, for the given input power, because after leaving the beamsplitter, here each of the “north” and “east” beams visit both arms sequentially.

*γ*, \({{\mathcal K}_{{\rm{SM}}}}\) is approximately constant and reaches the maximum there:

*frequency-independent*readout quadrature optimized for low frequencies can be used:

*γ*), and \(S_{{\rm{SMLF}}}^h\) can beat the SQL in a broad frequency band.

#### 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 lossy-filter-cavity 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 near-resonance 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 kilometers-scale distances, ordinary solid-state springs cannot be used due to unacceptably high levels of mechanical loss and the associated thermal noise.

At the same time, in detuned Fabry-Pérot cavities, as well as in the detuned configurations of the Fabry-Pérot-Michelson 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 radio-frequency systems [26]. Then its existence was predicted for the optical Fabry-Pérot 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 second-order resonance will exist.

In the last decade, the optical rigidity has been observed experimentally both in the table-top 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 optical-rigidity-based regimes, we limit ourselves to the vacuum-input case only, setting \({{\mathbb S}_{{\rm{sqz}}}}[r,\theta ] = {\mathbb I}\) in Eq. (375).

*unified quantum efficiency*, which accounts for optical losses both in the interferometer and in the homodyne detector.

*η*<

*η*

_{d}. Now we can write down explicit expressions for the interferometer quantum noises (376), (377) and (378), which can be calculated using Eqs. (552):

#### 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 resonance-tuned interferometer case. This approximation, in addition to its importance for the smaller-scale prototype interferometers, provides a bridge between our idealized harmonic oscillator consideration of Section 4.3.2 and the frequency-dependent rigidity case specific to the large-scale GW detectors, which will be considered below, in Section 6.3.4.

*Γ*≫ Ω, the Eqs. (473) for the interferometer quantum noises, as well as the expression (374) for the optical rigidity can be significantly simplified:

*K*

_{eff}). Following the reasoning of Section 4.3.2, it is easy to see that this spectral density allows for narrow-band sensitivity gain equal to

*η*= 1),

*η*< 1, then the bandwidth, for a given

*ξ*lessens gradually as the homodyne angle

*φ*goes down. Therefore, the optimal case of the broadest bandwidth, for a given

*ξ*, corresponds to

*φ*=

*π*/2, and, therefore, to

*S*

_{xF}= 0 [see Eqs. (475)], that is, to the pure ‘real’ rigidity case with non-correlated radiation-pressure and shot noises. This result naturally follows from the above conclusion concerning the amenability of the quantum noise sources cross-correlation to the influence of optical loss.

*φ*=

*π*/2 in Eq. (479) and taking into account that

_{0}, similar to one discussed in Section 6.1.2. Now, the optimization parameter is

*β*, that is, the detuning

*δ*of the interferometer. It is easy to show that the optimal

*β*is given by the following equation:

*β*cannot be solved in radicals. However, in the most interesting case of Ω

_{0}≪ Ω

_{q}, the following asymptotic solution can easily be obtained:

*β*defined by the condition (486), is plotted in Figure 43 for several values of the normalized detuning. We assumed in these plots that the unified quantum efficiency is equal to

*η*= 0.95. In the ideal lossless case

*η*=1, the corresponding curves do not differ noticeably from the plotted ones. It means that in the real rigidity case, contrary to the virtual one, the sensitivity is not affected significantly by optical loss. This conclusion can also be derived directly from Eqs. (485) and (488). It stems from the fact that quantum noise sources cross-correlation, amenable to the optical loss, has not been used here. Instead, the sensitivity gain is obtained by means of signal amplification using the resonance character of the effective harmonic oscillator response, provided by the optical rigidity.

_{q}, it can be approximated as follows:

_{q}). It follows from this equation that in order to obtain a sensitivity significantly better than the SQL level, the interferometer should be detuned far from the resonance,

*δ*≫

*γ*.

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.

*i*Ω, describes an optical friction, and the positive sign of this term (if

*δ*> 0) means that this friction is negative.

*J*is limited for technological reasons, the only way to get a more stable configuration is to decrease

*ξ*, that is, to

*improve*the sensitivity by means of increasing the detuning. Another way to vanquish the instability is to create a stable optical spring by employing the second pumping carrier light with opposite detuning as proposed in [48, 129]. The parameters of the second carrier should be chosen so that the total optical rigidity must have both positive real and imaginary parts in Eq. (490):

*J*

_{1,2}, γ

_{1,2}and δ

_{1,2}(δ

_{1}δ

_{2}< 0).

#### 6.3.4 General case

**Frequency-dependent rigidity**. In the large-scale laser GW detectors with kilometer-scale 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.

*γ*= 0, the roots of this equation are equal to

*mechanical resonance*(Ω

_{m}) and the

*optical resonance*(Ω

_{0}) of the interferometer [32]. In order to clarify their origin, consider an asymptotic case of the weak optomechanical coupling,

*J*≪

*δ*

^{3}. In this case,

_{m}originates from the ordinary resonance of the mechanical oscillator consisting of the test mass

*M*and the optical spring

*K*[compare with Eq. (476)]. At the same time, Ω

_{o}, in this approximation, does not depend on the optomechanical coupling and, therefore, has a pure optical origin — namely, sloshing of the optical energy between the carrier power and the differential optical mode of the interferometer, detuned from the carrier frequency by

*δ*.

*γ*≠ 0, the characteristic equation roots are complex. For small values of

*γ*, keeping only linear in

*γ*terms, they can be approximated as follows:

*J/δ*

^{3}, together with the analytical approximate solution (498), for the particular case of

*γ*/

*δ*= 0.03. These plots demonstrate the peculiar feature of the parametric optomechanical interaction, namely, the

*decrease*of the separation between the eigenfrequencies of the system as the optomechanical coupling strength goes up. This behavior is opposite to that of the ordinary coupled linear oscillators, where the separation between the eigenfrequencies increases as the coupling strength grows (the well-known avoided crossing feature).

*γ*= 0, the eigenfrequencies become equal to each other:

*γ*> 0, then some separation retains, but it gets smaller than 2

*γ*, which means that the corresponding resonance curves effectively merge, forming a single, broader resonance. This

*second-order pole regime*, described for the first time in [83], promises some significant advantages for high-precision mechanical measurements, and we shall consider it in more detail below.

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 (non-quantum) noise in the interferometer.

*L*= 4 km, and

*M*= 40 kg, which translates to

*J*=

*J*

_{aLIGO}= (2

*π*× 100)

^{3}s

^{−3}, and the planned technical noise). The optimization performed in [93] gave the quantum noise spectral density, labeled as ‘Broadband’ in Figure 45. It is easy to notice two (yet not discernible) minima on this plot, which correspond to the mechanical and the optical resonances.

Another example is the configuration suitable for detection of the narrow-band 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 ‘High-frequency’ in Figure 45. Here, despite one order of magnitude less optical power used (*J* = 0.1*J*_{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.

**The second-order pole regime**. In order to clarify the main properties of the second-order pole regime, start with the asymptotic case of

*γ*→ 0. In this case, the optical rigidity and the mechanical susceptibility (496) read

_{0}(see Eq. (500)):

_{0}(thus the name of this regime).

_{0}, than in the ordinary harmonic oscillator case. Consider, for example, the resonance force

*F*

_{0}sin Ω

_{0}

*t*. The response of the ordinary harmonic oscillator with eigenfrequency Ω

_{0}on this force increases linearly with time:

_{0}by contrast to the harmonic oscillator.

Consider the quantum noise of the system, consisting of this test object and the SQM (that is, the Heisenberg’s-uncertainty-relation-limited quantum meter with frequency-independent and non-correlated measurement and back-action noises; see Section 4.1.1), which monitors its position. Below we show that the real-life long-arm interferometer, under some assumptions, can be approximated by this model.

_{0}:

_{q}is defined by Eq. (155).

_{q}is equal to

*ξ*

^{2}), than the harmonic oscillator, or, alternatively, much broader bandwidth ΔΩ for a given value of

*ξ*

^{2}. It is noteworthy that the factor (513) can be made smaller than the normalized oscillator SQL 2|ν|Ω

_{0}(see Eq. (171)), which means beating not only the free mass SQL, but also the harmonic oscillator one.

*ξ*

^{2}for the harmonic oscillator (170) and of the second-order pole system (511) are plotted for the same value of the normalized back-action noise spectral density (Ω

*q*/Ω

_{0})

^{2}= 0.01, as well as the normalized oscillator SQL (171).

_{0}is defined by Eq. (500), and assume that

*ν*

^{2}, Λ

^{2}, and

*γ*in Eq. (514), we obtain that

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 non-zero value of A allows one to further increase the sensitivity.

**Optimization of the signal-to-noise ratio**. The peculiar feature of the second-order pole regime is that, while being, in essence, narrow-band, it can provide an arbitrarily-high SNR for the broadband signals, limited only by the level of the additional noise of non-quantum (technical) origin. At the same time, in the ordinary harmonic oscillator case, the SNR is fundamentally limited.

*ξ*

^{2}is.

*ξ*

^{2}is proportional to (ΛΩ)

^{−2}(see Eq. (513)) and, therefore, it is possible to expect that the SNR will be proportional to

*S*

_{tech}(Ω). Suppose also that this spectral density does not vary much within our frequency band of interest ΛΩ. Then the factor

*σ*

^{2}can be approximated as follows:

_{0}. In order to simplify our calculations, we neglect the contribution from optical loss into the sum spectral density (we show below that it does not affect the final sensitivity much). Thus, as follows from Eq. (517), one gets

_{q}. The optimization gives the best sensitivity, for a given value of \(\xi _{{\rm{tech}}}^2\), is provided by

_{q}, provides slightly worse sensitivity:

_{q}= 0.1Ω

_{0}. In Figure 46 (right), the optimal SNR (530) is plotted as a function of the normalized technical noise \(\xi _{{\rm{tech}}}^2\).

*S*

^{h}is the sum quantum noise of the interferometer defined by Eq. (514). The only assumption we have made here is that the technical noise power (double-sided) spectral density

*η*= 1, and the realistic case of the interferometer with

*J*= (2

*π*× 100)

^{3}s

^{−3},

*η*

_{d}= 0.95 and

*γ*

_{2}= 1.875 s

^{−1}(which corresponds to the loss factor of

*A*

_{arm}= 10

^{−4}per bounce in the 4 km long arms; see Eq. (322)). The typical optimized quantum noise spectral density (for the particular case of \(\xi _{{\rm{tech}}}^2 = 0.1\)) is plotted in Figure 45.

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 second-order 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.

_{0}∼ 10

^{3}s

^{−1}). Therefore, the loss-induced part of the total bandwidth

*γ*

_{2}, which has no noticeable effect on the unified quantum efficiency

*η*[see Eq. (469)] for the ‘normal’ broadband values of γ ∼ 10

^{3}s

^{−1}, degrades it in this narrow-band case. However, it has to be emphasized that the degradation of

*σ*

^{2}, for the reasonable values of \(\xi _{{\rm{tech}}}^2\), is only about a few percent, and even for the quite unrealistic case of \(\xi _{{\rm{tech}}}^2 = 0.01\), does not exceed ∼ 25%.

## 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 step-by-step 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) multiple-carrier 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 swiftly-developing 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 quantum-measurement 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 space-time and matter appeals for unprecedentedly precise measurement. And almost at the same time, Braginsky realized that the expected amplitude of the GW-induced 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 back-action evasion via properly constructed cross correlation between the measurement and back-action noise sources [162, 159, 158, 160, 161, 51, 53], find a use in the optomechanical experiments with micro-and 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 non-quantum 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 km-scale LIGO detector has been cooled down to *T*_{eff} = 1.4 *µ*K without any special arrangement, just by modifying the standard feedback kernel of the actuators to provide a virtual rigidity, shifting the 10-kg suspended mirrors oscillation frequency from Ω_{m}/2*π* = 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 several-ton 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 close-to-pure squeezed quantum state [114], entangle the differential and common motion of the kg-scale mirrors in the EPR-like fashion [56, 115], or even prepare it in a highly non-classical Schrödinger-cat state [135, 88].

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.

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.

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*_{*}.

In the resonance-tuned 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 RF-sideband fields are transformed differently, which influences both their amplitudes and phases at the readout port.

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 operator-valued function *Ê*(*x, y, z, t*).

*ô*(

*t*):

Hereafter we will omit, for the sake of brevity, the factor 2*πδ*(Ω − Ω′) in equations that define the power (double-sided) 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).

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.

## 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 LIGO-Virgo 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.