Advanced quantum techniques for future gravitationalwave detectors
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Abstract
Quantum fluctuation of light limits the sensitivity of advanced laser interferometric gravitationalwave detectors. It is one of the principal obstacles on the way towards the nextgeneration gravitationalwave observatories. The envisioned significant improvement of the detector sensitivity requires using quantum nondemolition measurement and backaction evasion techniques, which allow us to circumvent the sensitivity limit imposed by the Heisenberg uncertainty principle. In our previous review article (Danilishin and Khalili in Living Rev Relativ 15:5, 2012), we laid down the basic principles of quantum measurement theory and provided the framework for analysing the quantum noise of interferometers. The scope of this paper is to review novel techniques for quantum noise suppression proposed in the recent years and put them in the same framework. Our delineation of interferometry schemes and topologies is intended as an aid in the process of selecting the design for the nextgeneration gravitationalwave observatories.
Keywords
Gravitationalwave detectors Optomechanics Quantum measurement theory Quantum noise Standard quantum limit Fundamental quantum limit Optical rigidity Quantum speed meter Squeezed light Backaction evasion Atomic spin ensemble Whitelight cavity1 Introduction
The second generation of groundbased gravitationalwave (GW) interferometers, Advanced LIGO (Aasi et al. 2015) and Advanced Virgo (Acernese et al. 2015), with significantly improved sensitivities, superseded the initial generation in 2015, which led to a Nobel Prizewinning first direct observation of GWs from the binary black hole (BBH) coalescence on September 14, 2015 (Abbott et al. 2016c). This has marked the start of the new era of GW astronomy.
Contrary to the predictions based on the previous Xray observations (Abbott et al. 2016a), the first detected GW signal has come from an unexpectedly massive BBH with the mass of components \(\sim \,30\,M_\odot \) and the final BH with mass \(\sim \,60\,M_\odot \). The following detections (Abbott et al. 2016b, 2017c, d, g) have not only confirmed the existence of this new population of massive black holes but also highlighted the importance of sensitivity improvement at low frequencies (< 30 Hz) for better parameter estimation and more quantitative analysis of the nature of these exotic objects.
However, massive BBHs are not the only reason for lowfrequency improvement. With all three detectors of the LIGOVirgo network being online, the sky localisation is dramatically improved (see Sect. 4.2. in Abbott et al. 2018) enabling multimessenger astronomy of compact binaries (Abbott et al. 2017b). The longer lead times before the merger necessary for directing electromagnetic (EM) telescopes to the right sky location depend directly on the lowfrequency sensitivity where the spectral components of the inspiral stage of the binary evolution are most prominent (Harry and Hinderer 2018). We observed this situation when LIGO and Virgo had detected a GW signal from the final stages of evolution of the binary neutronstar (BNS) system (Abbott et al. 2017e) before the coalescence and merger that has produced a chain of followon electromagnetic (EM) counterparts detected by the EM partners of LIGO (Abbott et al. 2017b).
This fascinating discovery has also revealed the significance of enhancing the GW detector sensitivity in the relatively highfrequency band, from 1 to 5 kHz, which hosts the spectrum of the merger and the ringdown phases of the BNS system. It is the precise measurement of the GW signal shape emitted in these two phases that promise to unveil many details about the physics of nuclear matter and also to shed light on the physical mechanisms of short gammaray bursts (Abbott et al. 2017a).
And this brings us to the point of this review. As we can see from the Advanced LIGO design sensitivity shown in Fig. 1, the fundamental quantum fluctuations of light are limiting the sensitivity of the current generation of GW detectors in the most of its detection band, above \(\sim \) 10 Hz. The dominant noises below 10 Hz comprise seismic and gravity gradient fluctuations (Harms 2015) together with suspension thermal noise (LIGO Scientific Collaboration 2018), while at medium frequencies around \(\sim \) 50 Hz the mirror coating thermal fluctuations come close to the level of projected quantum noise. There is an active research going on to suppress the lowfrequency noise sources further in the next generation facilities (Cole et al. 2013). With these classical noises suppressed, we need to reduce the quantum noise to further improve the detector sensitivity. Similarly for the nextgeneration GW interferometers (Punturo et al. 2010; Hild et al. 2011; Abbott et al. 2017f), to go beyond their design sensitivity goal of at least an order of magnitude better sensitivity than in Fig. 1, we will need to incorporate the advanced techniques of quantum noise suppression that this review is about.
Hence, to reach the aforesaid objective and suppress the QN in the entire detection band, one has to suppress the uncertainties of both noncommuting observables in parallel, which seemingly violates the Heisenberg uncertainty relation. It sounds impossible, at a first glance. Yet, there are actually many approaches that seek to perhaps not violate (it’s impossible indeed), but circumvent the limitations imposed by the uncertainty principle. In this review, we will focus on those of these techniques applicable to interferometric GW detection.
The quantum noisemitigation techniques we consider in this review include (1) techniques well tested and already applied in the large scale GW detectors, such as squeezed light injection (Caves 1981; Abadie et al. 2011; Aasi et al. 2013; Schnabel 2017), (2) techniques that are at the stage of prototyping, e.g., speed meters (Braginsky and Khalili 1990; Chen 2003; Purdue 2002; Purdue and Chen 2002; Chen 2003; Danilishin 2004; Wade et al. 2012; Gräf et al. 2014; Voronchev et al. 2015) and frequencydependent squeezing (Oelker et al. 2016; Isogai et al. 2013), and (3) recently proposed ones, which would require quite some research and development, before one could implement them in a real detector, like conditional frequencydependent squeezing (Ma et al. 2017; Brown et al. 2017) or whitelightcavity based schemes (Wicht et al. 1997; Zhou et al. 2015; Ma et al. 2015; Peano et al. 2015; Korobko et al. 2017; Miao et al. 2015; Page et al. 2018; Miao et al. 2018).
Parameters for all configurations considered in the paper, unless explicitly specified otherwise
Parameter  Notation  Value 

Mirror mass (kg)  M  200 
Arm length (km)  L  20 
Laser wavelength (nm)  \(\lambda _p\)  1550 
Optical power in each arm (MW)  \(P_c/2\)  4.0 
Effective detector bandwidth (Hz)  \(\gamma \)  100 
The structure of the review is the following. In the next section, we give a brief introduction into the physics of quantum noise and how it manifests in GW interferometers. In Sect. 3, we consider the general limitations that arise in precision interferometry due to constraints that quantum mechanics imposes on the magnitude of quantum fluctuations of light. In Sect. 4, we review the concept of quantum noise mitigation using squeezed light injection, including frequencydependent squeezing. Section 5 is devoted to the suppression of quantum noise through quantum nondemolition measurement of speed and to a myriad of different ways of realising this principle in GW detectors. In Sect. 6, the enhancement of the interferometer response to GW signal by modifying test masses’ dynamics is investigated and different variations based on optical rigidity also sometimes referred to as dynamical backaction are analysed. Section 7 deals with proposals which consider active elements, such as atomic spin ensembles and unstable optomechanical filters, for the mitigation of quantum noise both at low and at high frequencies. In Sect. 8, we give some concluding remarks and outlook.
Notations and conventions, used in this review
Notation and value  Comments 

L  Length of the arms of the interferometer 
\(\tau =L/c\)  Light travel time at distance L 
\(\omega \)  Optical frequencies 
\(\omega _0\)  Interferometer resonance frequency 
\(\omega _p\)  Optical pumping frequency (laser frequency) 
\(\varOmega = \omega \omega _p\)  Modulation sideband frequency w.r.t. laser frequency \(\omega _p\) 
\(\varDelta =\omega _p\omega _0\)  Optical pump detuning from the cavity resonance frequency \(\omega _0\) 
\(\mathcal {E}_{0} = \sqrt{\dfrac{4\pi \hbar \omega _p}{\mathcal {A}c}}\)  Normalisation constant of the second quantisation of a monochromatic light beam 
\(A^{in} = \sqrt{\dfrac{2P^{in}}{\hbar \omega _p}}\)  Classical quadrature amplitude of the incident light beam with power \(P^{in}\) 
\(T\,(R)\)  Power transmissivity (reflectivity) of the mirror 
\(\gamma _{\mathrm{arm}} = cT/4L\)  Arm cavity halfbandwidth for input mirror transsmissivity T and perfect end mirror 
\(\delta _{\mathrm{arm}}\)  Arm cavity detuning/differential detuning of the arms of Fabry–Perot–Michelson interferometer 
\(\gamma \)  Interferometer effective halfbandwidth 
\(\beta (\varOmega )\)  Phase shift acquired by sidebands in the interferometer 
\(\mathcal {K}(\varOmega )\)  Optomechanical coupling factor (Kimble factor) of the interferometer 
\(P^{in}\)  Incident light beam power 
\(P_c = 2P_{\mathrm{arm}}\)  Total power, circulating in both arms of the interferometer (at the test masses) 
M  Mass of the mirror 
m  Reduced mass of the signal mechanical mode of the interferometer (e.g., dARM mode)\(^\mathrm{a}\) 
\(\varTheta = \dfrac{4\omega _p P_c}{mcL}\)  Normalised intracavity power 
\(h_{\mathrm{SQL}} = \sqrt{\dfrac{8\hbar }{mL^2\varOmega ^2}}\)  Standard quantum limit of a free mass for GW strain 
\(x_{\mathrm{SQL}} = \sqrt{\dfrac{2\hbar }{m\varOmega ^2}}\)  Standard quantum limit of a free mass for displacement 
2 Quantum noise
In a nutshell, every interferometer is a device that uses interference to measure the relative phase of one beam to the other. It detects variations of intensity of the interference pattern caused by this phase shift. The precision of this procedure is dependent on many factors, which can be decomposed by source in a noise budget (cf. Fig. 1). The one, which we are focusing on, in this review is rooted in the very nature of light as a quantum field, i.e., the quantum fluctuation of optical phase and amplitude.
2.1 Twophoton formalism and input–output relations
As shown by Caves and Schumaker (1985) and Schumaker and Caves (1985), the quantum noise of light in any linear optical device can be conveniently described within the framework of the twophoton formalism. Namely, noise can be considered as tiny stochastic variations in the quadratures of the optical field travelling through the device. Any variations of interferometer parameters induced by the signal, e.g., differential arm length change, also lead to variations of the quadratures of the outgoing field, which can be described using the same formalism.
2.1.1 Case of multiple input/output channels
2.2 Transfer functions of the quantumnoiselimited interferometer

mirrors can move when subject to the action of an external force, thus making the interferometer sensitive to the GW, and^{1}

light interacts with the mirrors, which manifests in two ways, i.e., the mirror motion modulating the phase of light and the light exerting a radiation pressure force on the mirror.
 1.
Forcetodisplacement TF is described by the mechanical susceptibility, \(\chi _{m}\) of the centre of mass motion of the test mass mirror;
 2.
Displacementtofield TF, \(\varvec{R}_{x}\equiv \{\partial a_{c}/\partial x,\,\partial a_{s}/\partial x\}\), reflects how much the two quadratures of the outgoing field are changed by the displacement of the mirror x, and
 3.
Fieldtoforce TF, \(\varvec{F} \equiv \{\partial \hat{F}_{\mathrm{r.p.}}/\partial a_{c},\,\partial \hat{F}_{\mathrm{r.p.}}/\partial a_{s}\}\), describes how much the radiation pressure force depends on the sine and cosine quadrartures of the ingoing field;
 4.
Displacementtoforce TF, \(\varvec{K} \equiv \partial F/\partial x\), describes the dynamic backaction or optical spring that manifests as restoring force created by the part of the optical field dependent on the mirror displacement x.
The other end of the optomechanical coupling is given by the fieldtoforce TF. It stems from the radiation pressure (RP) that light exerts on the mirrors. Thus the TF in question is a vector of coefficients at the corresponding quadratures of the input fields in the expression for a backaction force, \(F_{\mathrm{BA}}\). This force contributes to the actual displacement of the mirrors and thus mimics the signal displacement. Noteworthy is that the radiation pressure may depend on the displacement of the mirror, if the interferometer is detuned. This creates a feedback loop and results in a restoring force. This lightinduced restoring force is known as dynamical backaction or optical rigidity, represented by a violet box in Fig. 3.
Finally, there is also the fieldtofield TF that describes how the input light fields would be transformed by the interferometer, were its mirrors fixed. This is an optical TF shown as a yellow block in the flowchart.
Note that all these considerations apply equally to a system with an arbitrary number of inputs and outputs.
2.3 I/Orelations for tuned interferometers
2.4 Quantum noise of a tuned Michelson interferometer
The approximate expressions above are obtained assuming that cavity linewidth and signal frequency are much smaller than the cavity free spectral range \(FSR=c/2L\), which is known as a singlemode approximation. For the next generation GW detectors with longer arms where FSR may be close to the detection band, one normally needs to use the exact expressions, although the effect of factor \(D(\varOmega )\) is usually stronger and covers up any effects of departure of the interferometer response from the ones written in the singlemode approximation.
From Eq. (20) one can immediately notice that setting homodyne angle \(\phi _{\mathrm{LO}}\) such that \(\mathcal {K} = \cot \phi _{\mathrm{LO}}\), the second term in the brackets vanishes, which means one evades the backaction noise this term is standing for. This is the manifestation of the principle of variational readout, first proposed in Vyatchanin and Matsko (1996) and later generalised in Kimble et al. (2002) that prescribes to read out not the phase quadrature of the outgoing light where GW signal strength is maximal, rather the one that does not contain backaction noise. This technique, in an absence of loss, allows to completely get rid of the back action noise where the above match of homodyne phase to OM coupling strength could be satisfied. However, since \(\mathcal {K}_{\mathrm{MI}}\) is strongly frequency dependent, the total back action cancellation is only possible at a single frequency, as demonstrated by a series of thin dashdotted traces in Fig. 5a with an envelope of these curves being the quantum shot noiselimited sensitivity. We show in Sect. 4 a fundamental relation of this shot noiselimited sensitivity and variational readout concept to the fundamental quantum limit for precision interferometry.
2.5 Quantum backaction and ponderomotive squeezing
Ponderomotive squeezing is the direct consequence of quantum back action, since it is through this nonlinear mechanism amplitude fluctuations of light are transformed into the additional fluctuations of phase with the frequency dependent gain given by the OM coupling factor \(\mathcal {K}\). Understanding quantum backaction in terms of squeezing of the state of light leaving the interferometer comes very useful when one tries to figure out why one needs frequency dependent squeezing injection to achieve broadband quantum noise suppression, and why injection of phasesqueezed light in the readout port does not suffice. We discuss these topics in Sect. 4. One can also gain additional understanding of noise transformations in more complicated schemes, like, e.g., the scheme of the EPRspeed meter that we consider in Sect. 5 (Fig. 25).
In Appendix B.1, we consider a more general case of a detuned interferometer and derive general formulas for ponderomotive squeezing.
2.6 Losses and imperfections
In a real experiment, the idealised situation where the interferometer can be described solely by the I/Orelations (5) with one input and one output channel can never work. According to the FluctuationDissipation Theorem of Callen and Welton (1951), in a lossy system, there are always additional channels through which a part of the signalcarrying light field leaves the interferometer unobserved, while the incoherent vacuum fields from the environment enter and admix with the nonclassical light travelling through the interferometer, thereby curtailing quantum correlations contained therein and increasing noise. Generally, there are many places in the interferometer where loss can occur and therefore, there are many loss channels and vacuum fields associated with them.
These vacuum fields propagate through the interfrometer and couple to the readout channel very similar to the input field \(\hat{\varvec{a}}\) with the only difference in the frequency dependence of the optical transfer matrix \(\mathbb {N}_k\) that reflects the fact that the optical path of loss vacuum fields differs from that of \(\hat{\varvec{a}}\) (see, e.g., treatment of a lossy Fabry–Perot–Michelson interferometer in Appendix B.2.1).
As for the imperfections, by which we mean here departure of the parameters of key components of the interferometer from the assumed uniformity (e.g., perfect overlap of the signal and local oscillator beams, perfect mode matching on the beam splitter etc.) and symmetry (e.g. perfect 50/50 beam splitting ratio, equal mass of all test masses, equal length/tuning of the arms, equal absorption and photon loss in the arms etc.), it is hard to give a general recipe how to account for their influence on quantum noise. However, these studies are crucial for the design of the next generation GW interferometers, and there are several studies that attempted rigorous treatment of imperfections for selected configurations. Nonideal FPMI with frequency dependent squeezing injection (see Sect. 4) was studied in Miao et al. (2014). An indepth comparison of FPMI and Sagnac speed meters (see Sect. 5 with account for imperfections was done in Voronchev et al. (2015). Influence of imperfections on Sagnac speed meter performance was the topic of Danilishin et al. (2015). Impact of optical path stability and mode matching in balanced homodyne readout was the topic of Steinlechner et al. (2015), Zhang et al. (2017).
2.6.1 Losses in the readout train
The sources of loss in the readout train are quite diverse, ranging from nonunity quantum efficiency of the photodiodes to the imperfect mode matching of the local oscillator beam with the signal beam in the balanced homodyne detector (Zhang et al. 2017). In most cases loss may be reduced to a single, frequency independent coefficient of an effective quantum efficiency, \(\eta _d = 1  \epsilon _d < 1\), where \(\epsilon _d < 1\) can be thought of as a fractional photon loss at the photodetector (Kimble et al. 2002; Miao et al. 2014; Danilishin and Khalili 2012). Frequency dependence can be safely omitted here, for any resonant optical element in the readout train, including output mode cleaners (OMC), has bandwidth much larger than the detection band of the main interferometer.
2.6.2 Optical loss in the arms and in filter cavities
Optical loss in Fabry–Perot cavities, such as arm cavities and filter cavities, is known to have frequency dependence with the major impact at low sideband frequencies within the cavity optical bandwidth. A very illuminating discussion on this subject is given in Miao et al. (2014) where optical loss in filter cavities is studied in detail. The main source of such loss in large suspended cavities is the scattering of light off the mirror surface imperfections of microscopic (microroughness) and relatively macroscopic (“figure error”) size (Isogai et al. 2013; Miao et al. 2014).
3 Quantum limits
3.1 Standard quantum limit
The standard quantum limit (SQL) was firstly pointed out by Braginsky when studying the quantum limit of continuous position measurements (Braginsky and Khalili 1992). In the context of laser interferometric GW detectors, it constraints the detector sensitivity in the nominal operation mode with tuned optical cavities and phase quadrature measurement. It comes from a tradeoff between the shot noise and radiation pressure noise—the former is inversely proportional to the optical power while the latter is proportional to the power. There is an optimal power for achieving the maximum sensitivity at each frequency which defines the SQL.
3.2 Fundamental quantum limit
Worthy of highlighting, there are two noticeable dips in the FQL for the detuned case. The highfrequency one simply coincides with the detuning frequency which defines the optical resonance of the interferometer. The lowfrequency one, as discussed by Buonanno and Chen (2002), is attributable to the socalled optical spring effect which shifts the testmass centreofmass frequency due to the positiondependent radiation pressure. In sight of the FQL, we can provide an alternative point of view: the sensitivity is better around such a frequency implies that the optical power fluctuation is significantly larger than other frequencies, according to Eq. (41). It can be explained using the positive feedback as illustrated in Fig. 9. The quantum fluctuation in the amplitude quadrature is converted into that of phase quadrature due to the ponderomotive squeezing (amplification) effect. In the presence of nonzero detuning and the signalrecycling mirror, the phase quadrature fluctuation is feeding back to the amplitude one. With the roundtrip feedback gain approaching unity, the amplitude quadrature fluctuation, or equivalently the power fluctuation, is significantly enhanced and leads to the dip in the sensitivity curve that we observe.
4 Interferometers using nonclassical light
4.1 Squeezed vacuum injection
4.2 Frequencydependent squeezing
As we have learnt from the previous section, a fixed squeezing angle only improves the sensitivity for some frequencies but not all. This is because the fluctuation in the amplitude quadrature and the phase quadrature contribute to the quantum noise differently at different frequencies. We can, therefore, optimise the sensitivity by making the squeezing angle frequency dependent.
For example, in the tuned dualrecycled Michelson, two filter cavities are needed to achieve the optimal squeezing angle, as the highest order of \(\varOmega \) in \(\tan \theta _s\) is four, cf. Eq. (50). When the detector bandwidth \(\gamma \) is much larger than the frequency for the transition from the radiationpressurenoise dominated to the shotnoise dominated, one filter cavity can approximately realise the optimal squeezing angle. This is the case for the resonantsidebandextraction mode of detectors. Given the default parameters that we assumed, the bandwidth is of the order of a few hundred Hz and the transition frequency is around 30 Hz. Indeed, as shown in the left panel of Fig. 12, the difference between the optimal angle and the one realised with one filter cavity is less than one milliradian, and the projection noise from the antisqueezing is smaller than 0.4 dB for 10 dB squeezing. However, when the detector bandwidth is narrow, e.g., around 100 Hz, as shown in the right panel of Fig. 12, one filter cavity is not able to produce the optimal angle which has a steeper change than the case of having a broad detector bandwidth.
4.3 Conditional frequencydependent squeezing via EPR entanglement
As mentioned in the previous section, the canonical setup for realising the frequencydependent squeezing for a broadband detector involves at least one additional filter cavity. In contrast, the recently proposed idea based upon the Einstein–Podolsky–Rosen (EPR) entanglement of light shows a new approach without a need of the external long filter cavity (Ma et al. 2017). This idea takes advantage of the entanglement (correlation) between fields around the half of the frequency \(\omega _p\) of the pump field that drives the nonlinear crystal.
Fig. 16 illustrates how the interferometer affects the signal field and the idler field by only looking at the differential mode from the dark port (the interferometer is mapped into a coupled cavity). For the former, the signalrecycling cavity (SRC) formed by SRM and ITM is tuned on resonance with respect to \(\omega _0\) in the resonant sideband extraction case. The strong carrier inside the arm cavity mixes with the signal field and interacts with the test mass mediated by the radiation pressure. This process makes the signal field at the output squeezed, which is the ponderomotive squeezing effect mentioned earlier. It introduces the radiation pressure noise by converting the fluctuation of the amplitude quadrature into that of the phase quadrature. For the latter, there is no strong carrier at \(\omega _0+\varDelta \) and there is no radiation pressure effect associated with the idler field. The interferometer behaves as a passive filter cavity that imprints frequencydependent rotation on the quadratures of the idler field. Since measuring \(\phi \) quadrature of the idler field will make \(\phi \) quadrature of the signal field squeezed, cf. Eq. (62), the frequency dependence will be transferred to the squeezing of the signal field. As shown in Ma et al. (2017), we can achieve the desired frequencydependent squeezing by choosing a proper value of \(\varDelta \) and fine tuning the length of SRC. One may as well use the I/Orelations formalism of Sect. 2.1 to arrive to the above described result. However, since the two modes of squeezed light are entangled and thus ought to be considered together, as manifested by Eq. (58), the dimensions of the corresponding transfer matrix \(\mathbb {T}\) and the response vector \({\mathbf {t}}\) should be expanded to \(4\times 4\) and \(4\times 1\), respectively.
4.4 Optical losses in interferometers with nonclassical light
The performance of the described interferometers with squeezed vacuum injection depends rather strongly on how well the quantum correlations generated by the squeezer are transmitted to the interferometer to counteract the corresponding quantum correlations created by optomechanics (i.e., ponderomotive squeezing discussed earlier). As shown by Kimble et al. (2002), this effect is quite significant and detrimental. There are several mechanisms that cause deterioration of the QNLS of the interferometers using squeezing injection, which we consider below.
4.4.1 Optical loss in a squeezing injection optics
4.4.2 Squeezing angle fluctuations
Another source of noise is known as ‘phase quadrature noise’, or ‘squeezing angle jitter’ (Dooley et al. 2015). It comes from the random fluctuations of the optical path length between the squeezer and the dark port of the interferometer.
4.4.3 Losses in filter cavities
4.5 Summary and outlook
After years of developments and researches, squeezing now becomes an indispensable quantum technique for enhancing the detector sensitivity. We can now produce a high level of squeezing, more than 10 dB, at the audio band for both 1064 nm and 1550 nm with the goal of expanding to other wavelengths (Schnabel 2017). To fully take advantage of the squeezing, efforts are being put into minimisation of the optical loss, due to scattering and mode mismatch, in between the squeezed light source and the interferometer output. The frequencydependent squeezing with a filter cavity has already been demonstrated in a tabletop experiment (Oelker et al. 2016), and the large scale filter cavity, of the order of hundred meter, will be implemented in the near term upgrades of current advanced detectors (LIGO Scientific Collaboration 2018). The EPR squeezing idea is at an early stage and requires tabletop demonstrations, which have been started by several experimental groups. Since reducing the shot noise of a detuned interferometer also requires the frequencydependent squeezing, this idea equally applies there, which has been modelled in details in Brown et al. (2017). Indeed, the ongoing experimental demonstrations all use this fact.
Looking further into the future, more complex frequencydependent squeezing might be needed to optimise the sensitivity of detectors operating beyond the current broadband operation. This may require a cascade of filter cavities with parameters that can be tuned in situ. For passive optics (without external energy input), one can achieve the tunability by using compound mirrors. The active optomechanical filter cavity idea provides an alternative approach and also can achieve narrow cavity bandwidth with a short cavity length (Ma et al. 2014). However, it has not yet been investigated experimentally as systematically as the passive filter cavity, and more researches are needed.
5 Speedmeter interferometers
5.1 Speed meters as GW detectors
The more or less complete chart of configurations developed so far is shown in Fig. 18. All schemes are classified in 3 types—(i) Sagnaclike speed meters (Chen 2003; Khalili 2002; Danilishin 2004; Wang et al. 2013; Danilishin et al. 2018), (ii) sloshing speed meters (Braginsky et al. 2000a; Purdue 2002; Purdue and Chen 2002; Wade et al. 2012; Huttner et al. 2017) and (iii) the EPRtype speed meter (Knyazev et al. 2018) by the mechanism the speed measurement is arranged. In Sagnac speed meter, signal sidebands interact with the interferometer twice and copropagate all the time with the carrier light. Sloshing speed meters use an additional not pumped sloshing cavity to store the signal sidebands between the two interactions with the interferometer arms and thus have an extra parameter, the sloshing frequency (defined by the sloshing cavity length and the input coupler mirror reflectivity), that discerns its response function from that of a Sagnac speed meters. And finally, the EPRtype speed meter uses two optically independent positionsensitive interferometers and devise the speed information by combining their outputs into sum and difference combinations with a beamsplitter and then adding the so obtained correlated photocurrents with optimal weights. Let us see how it works in individual schemes.
5.2 Sloshing speed meter
Two possible implementations of such a scheme are shown in Fig. 20. The left panel shows the variant with space separation of optical beams used for sequential measurement of arms’ differential displacement (Braginsky et al. 2000a; Purdue and Chen 2002), whereas the right one, proposed by Wade et al. (2012) employs two orthogonal polarisations to separate the beams. The latter also gets rid of an extra sloshing cavity by using the orthogonal not pumped polarisation mode of the interferometer.
One can immediately see that the QNLS of speed meter has the same frequency dependence as the SQL at low frequencies, where quantum backaction noise dominates, which is a unique feature of the speed meters in general. It results from the backaction suppression, as expected from the QND speed measurement. However, it does not go parallel to the frequency axis, like, for instance, the frequencydependent variational readout and the FQL do (see Fig. 8 in Sect. 3). One can see why on the two right panels of Fig. 21, where on the top plot, the quantum noise of the outgoing light phase quadrature (\(\phi _{\mathrm{LO}} = \pi /2\)) is plotted (the numerator of Eq. (10)), whereas on the lower panel we see the response of the interferometer to the signal variation of GW strain (the numerator of Eq. (10)). Hence the QNLS plot to the left is simply the ratio of the upper and lower plots to the right.
So, one can see in Fig. 21b that quantum backaction noise of speed meter is indeed heavily suppressed as compared to the Michelson interferometer and has the same constantlike frequency dependence as quantum shot noise. The 1 / fslope in QNLS is coming from the speed response that rolls off as \(\propto f\) towards the DC as shown in Fig. 21c.
Another consequence of the peculiar behaviour of \(\mathcal {K}_{\mathrm{SSM}}\) for a speed meter is the ability to surpass the SQL at low frequencies if the right quadrature is selected for readout. Indeed, the general expression for the SSM QNLS has a term \(\propto [\mathcal {K}_{\mathrm{SSM}}\cot \phi _{\mathrm{LO}}]^{2}\) that can be made zero, were \(\cot \phi _{\mathrm{LO}}]=\mathcal {K}_{\mathrm{SSM}}\). This is quite easy to achieve, as \(\mathcal {K}_{\mathrm{SSM}} = const\) below the cavity pole. The resulting sensitivity at these low frequencies is the FQL for the speed meter, as mentioned in Sect. 3. For instance, at the threshold power where \(\mathcal {K}_{\mathrm{SSM}}(0)=1\) the optimal readout quadrature will equal \(\phi _{\mathrm{LO}} = \pi /4\). This case is plotted as a thin dashed line in Fig. 21a.
5.3 Sagnactype speed meters
To avoid this problem, a few polarisationbased variants of speedmeter schemes were proposed (Danilishin 2004; Wang et al. 2013; Danilishin et al. 2018), which relaxed the need for modifications of the main interferometer significantly. The most recent proposal (Danilishin et al. 2018), depicted in Fig. 22b, no changes to the infrastructure of the main interferometer. It requires, however that all reflective coatings of the core optics have the same properties for both polarisations of light. This is a tough, though not impossible requirement, and some research in this direction is under way already (Hild 2017; Krocker 2017).
5.4 EPRtype speed meters
Knyazev et al. (2018) proposed a third distinct way to realise speed measurement in GW laser interferometer, using twoposition meters (Fabry–Perot–Michelson interferometers, see Fig. 24a that have rigidly connected test masses (or simply share them) but have contrasting light storage times (bandwidths satisfy condition \(\gamma _1\gg \gamma _2\)). The information about the differential motion of the arms thus comes of the two interferometers at a very different rate given by respective bandwidths. Hence, combining the readout beams of the two interferometers on a beamsplitter and reading out the “−”channel thereof one gets the difference of the two position signals at different times that is, in fact, velocity. There is an additional backaction noise associated with the vacuum fields entering the “\(+\)”port of the beamsplitter that however can be subtracted from the readout, if one measures the amplitude quadrature at the “\(+\)”channel and subtracts it, with optimal filter, from the readout of the “−”channel. As the two output channels of the readout beamsplitter get entangled, when the two ponderomotively squeezed output fields, \(\hat{b}_{1}\) and \(\hat{b}_{2}\), of the two position meters get overlapped on it, and this entanglement is used to remove the excess backaction noise from the output, this speed meter was dubbed an EPRspeed meter.
As the design of Fig. 24a is obviously a nightmare to implement in a real GW detector (it was never intended to be), another one, based on orthogonal polarisation modes of light was proposed in Knyazev et al. (2018) and is shown in Fig. 24b. The key element here is the quarterwave plate (QWP) that acts as a \(\pi /2\)phase retarder between the two orthogonal polarisation modes of the main interferometer. The QWP placed between the main IFO and the signalrecycling mirror, which position with respect to the arm’s ITMs is chosen so that the resulting SR cavity (with the QWP) is tuned resonantly for one of the polarisation modes. The orthogonally polarised light sees the SR cavity as antiresonant due to the \(\pi /2\) phase shift given to it by the QWP. As a consequence of the “scaling law” (Buonanno and Chen 2003), the polarisation mode that is in resonance with the SRC sees the interferometer with a very narrow effective bandwidth \(\gamma _2\) (tuned SR regime, see Eq. (24a) with \(\phi _\mathrm{SR} = 0\)), whereas for the orthogonal one the effective bandwidth \(\gamma _2\gg \gamma _1\) is greatly increased (resonant sideband extraction (RSE) regime, see Eq. (24a) with \(\phi _\mathrm{SR} = \pi /2\)). The polarisation beam splitter (PBS) with a polarisation plane rotated by \(45^{\circ }\) angle with respect to the s and ppolarised modes of the main interferometer creates the EPRtype correlations in the “\(+\)” and “−” readout channels. The optimal distribution of circulating powers among the two effective position meters is organised by the proper choice of the angle \(\vartheta \) of the carrier light polarisation plane to the vertical direction (see the blue box in Fig. 24b).
5.5 Imperfections and loss in speedmeter interferometers
Speedmeter interferometers suffer in general from the same sources of noise due to loss as Fabry–Perot–Michelson interferometers, since in most of the speedmeter configurations presented in Fig. 18 FPMI itself is an integral part. However, loss influence is different from the FPMI and there are also some specific features of speed meters worth mentioning.
Firstly, cancellation of backaction noise in speed meters comes from the coherent subtraction of back action forces created by the same light beam in two consequent interactions with the test mass. Therefore any admixture of incoherent vacuum due to loss between the two interactions, e.g., the loss in the arms, creates an unbalanced backaction force. This lossassociated back action leads to a positionmeterlike rise of quantum noise at low frequencies, where it starts to dominate over the suppressed quantum back action of the speed meter.
Secondly, as any scheme where balancing of the noise contributions between the arms is essential for the noise cancellation, speed meters are very sensitive to the asymmetry of the arms, as discussed in detail in Danilishin et al. (2015). Asymmetry of the beam splitter in Sagnac interferometers, for instance, creates a coupling of laser fluctuations to the readout port through an excess radiation pressure they create on the mirrors which is quite strong as laser noise in the lowfrequency range is far from the shot noise limit. This excess noise, however can be cancelled by a wise choice of local oscillator in the balanced homodyne readout as shown in the recent study by Zhang et al. (2018).
In general, speedmeter interferometers show higher robustness to intracavity loss than Michelson ones due to lower backaction component of quantum noise, which means lower ponderomotive squeezing as discussed in Sect. 2.5. This reduces the effect of loss vacuum fields on the internal squeezing of light since quantum correlation between phase and amplitude fluctuations is already suppressed by the speed meter. For the same reason the requirements on tolerable filter cavity loss and bandwidth in case of frequencydependent squeezing injection are significantly relaxed for speed meters. The detailed study of loss influence on speedmeter quantum noise is given in Voronchev et al. (2015).
5.6 Summary and outlook
Speedmeter interferometers are arguably the most elaborate and well studied concept alternative to the Michelson interferometer based on position measurement. Their main advantage is a greatly reduced backaction noise that potentially allows to increase the rate of detection of massive binary blackhole systems by up to 2 orders of magnitude compared to the equivalent position meter (Danilishin et al. 2018) if only quantum noise is considered. Although there is an obvious penalty of vanishing response at low frequencies, the reduction of backaction is still greater to make the overall increase of the SNR worth it. The progress in development of new, more practical topologies of speed meters shown in Fig. 18 has led to designs that allow to keep the main interferometer intact, yet this comes at a price of using polarisation optics that is prone to imperfections and even more importantly, it requires development of the new allpolarisation type mirror coatings.
We have considered here all three main genera of speed meters and gave the comparison of their performance. All studies done so far indicate the superiority of speed meters’ performance over that of the conventional Michelson interferometers even in the presence of losses and imperfections (Wang et al. 2013; Miao et al. 2014; Voronchev et al. 2015; Danilishin et al. 2018). However, a thorough and systematic study of losses and imperfections in all the speedmeter schemes is needed as well as experimental prototyping, before any final conclusion can be made.
In the context of FQL, the speedmeter configuration is an approach to shaping the power fluctuation inside the arm cavity. The FQL can be reached at low frequencies, where the optomechanical coupling strength is approximately constant, by using the frequencyindependent readout rather than the frequencydependent readout as in the case of a position meter (Michelson interferometer). In the tuned case, the price we paid is that the power fluctuation gets reduced at low frequencies and the resulting FQL is parallel to the SQL rather than flat for the position meter.
An interesting future direction is to investigate detuned speedmeter configurations with additional intra and external filters. Since the optomechanical coupling strength is approximately constant at low frequencies, this means the resulting ponderomotive squeezing is frequency independent at these frequencies. With detuning, the optical feedback, illustrated in Fig. 9, could result in a broadband enhancement of the power fluctuation. Or equivalently, this can be viewed as a broadband enhancement of the mechanical response of the test mass, similar to the idea of negative inertia to be discussed in the section that follows.
6 Interferometers with optomechanically modified dynamics
6.1 Introduction
At the same time, the SQL, normalized to the signal force, decreases as the test object susceptibility increases. Because this approach does not require any precise mechanisms for mutual compensation of measurement noise and backaction noise (and, in particular, the SQL is not evaded), it is much more robust with respect to optical losses, than quantum noise cancellation.
A trivial example is just the use of smaller inertial mass \(m_{\mathrm{inert}}\). This method can be used, for example, in atomic force microscopes. However, when detecting forces of a gravitational nature, particularly in gravitationalwave experiments, the signal force is proportional to the testobject gravitational mass \(m_{\mathrm{grav}}\). Taking into account that, due to the equivalence principle, \(m_{\mathrm{inert}}=m_{\mathrm{grav}}\), the overall sensitivity decreases with the mass, which can be seen, for example, from the expressions for SQL in the hnormalization (39) (see, however, Sect. 6.4).
Another possibility is to use a harmonic oscillator instead of a free test mass. The susceptibility of a harmonic oscillator rapidly increases near its resonance frequency \(\varOmega _0\), which improves the \(S_{\mathrm{SQL}}\) by a factor of \(\varOmega _0/\varDelta \varOmega \) in the frequency band \(\varDelta \varOmega \) centered at \(\varOmega _0\) (see Sect. 4.3.2 of Danilishin and Khalili 2012). This method was demonstrated in several “tabletop” experiments with mechanical nanooscillators (Teufel et al. 2009; Anetsberger et al. 2009; Westphal et al. 2012). In laser gravitationalwave detectors, the characteristic eigenfrequencies of the test mirror pendulum modes are close to 1 Hz, and in the operating frequency range these mirrors can be considered as almost free masses. Evidently, it is technically impossible to turn the differential mechanical mode of laser detector test mirrors into an oscillator with a frequency in the operating frequency range by using “ordinary” springs. However, the optical spring which arises in detuned interferometer configurations and possesses excellent noise properties can be used for this purpose instead.
The optical spiring is a particular case of the more general electromagnetic rigidity (e.m.) effect, which takes place in any detuned e.m. resonator. This effect, together with the associated e.m. damping, were most probably first discovered and explained in the very early work by Braginsky and Minakova (1964), where the lowfrequency (subHerz) torsional pendulum was used as the mechanical object and the radiofrequency capacitor transducer—as the position sensor. Few years later, existence of these effects in the optical Fabry–Perot cavities (that is the optical spring proper) was predicted theoretically (Braginsky and Manukin 1967). After that, the e.m. damping was observed in the microwave Fabry–Perot type cavity (Braginskiǐ et al. 1970). In the beginning of 1980s, the first truly optical experiment was done (Dorsel et al. 1983).
Much later, quantum noise properties of the optical spring and the optical damping were analyzed in Braginsky et al. (1997), Braginsky et al. (2001), Braginsky and Khalili (1999), Braginsky and Vyatchanin (2002) and it was shown that the noise temperature of the optical damping can be very close to zero. This stimulated a series of experimental works where the optical rigidity was observed both in tabletop optical setups (Bilenko and Samoilenko 2003; Sheard et al. 2004; Corbitt et al. 2006b, 2007a, b) and in largerscale Caltech 40 m interferometer devoted to prototyping of future GW detectors (Miyakawa et al. 2006).
It have to be mentioned also that the very low noise temperature of the e.m. damping stimulated also a bunch of optomechanical and electromechanical experiments aimed at preparation of mechanical resonators in the ground state using this cold damping, see e.g.,Teufel et al. (2011), Chan et al. (2011) and the reviews by Aspelmeyer et al. (2014) and Khalili and Danilishin (2016).
Specifically in the context of the largescale gravitationalwave detectors the optical rigidity was analyzed in Braginsky et al. (1997), Buonanno and Chen (2001), Khalili (2001), Buonanno and Chen (2002), Buonanno and Chen (2003). Most notably, it was shown in these works that in very long cavities with the bandwidth \(\gamma \) comparable with the or smaller than the characteristic mechanical frequencies \(\varOmega \), the optical spring has sophisticated frequency dependence which enables some interesting applications, see below.
6.2 Optical rigidity
The e.m. rigidity and the e.m. damping effects were correctly explained in Braginsky and Minakova (1964) by respectively, dependence of the e.m. eigen frequency and therefore of the energy \(\mathcal {E}\) stored in the e.m. resonator on the mechanical position x and by the time lag between the variation of x and the variation of \(\mathcal {E}\). We reproduce below the semiqualitative, but simple and transparent reasoning of that paper.
6.3 Characteristic regimes of the optical spring
Nontrivial frequency dependences of the optical rigidity (99) and of the quantum noise components of the detuned interferometers [see Eqs. (376–378) of Danilishin and Khalili 2012] lead to very sophisticated shape of the corresponding sum quantum noise spectral density, see Eq. (385) of Danilishin and Khalili (2012). This shape can be tuned flexibly by varying the interferometer bandwidth \(\gamma \) and detuning \(\delta \), homodyne and squeezing angles, and the squeezing amplitude, with the optimal tuning depending on many factors, such us the available optical power, intensity of nonquantum (“technical”) noise sources, optical losses etc. The corresponding exhaustive optimization exceeds the scope of this paper (as well as probably any single paper). Broad set of examples covering the most typical scenarios can be found e.g.,in the articles Buonanno and Chen (2001, 2003), Kondrashov et al. (2008), Danilishin and Khalili (2012). Therefore, here we concentrate specifically on the modification of the mechanical probe dynamics by the optical spring.
In Fig. 28, quantum noise spectral densities for this setup are plotted for the same three characteristic values of the detuning of the narrowband carrier as in Fig. 27. It is instructive to compare these plots with the corresponding ones for the case of the single carrier detuned carrier, see e.g., Fig. 45 of Danilishin and Khalili (2012). It is easy to see that while in the latter case the use of the detuned regime leads to sharp degradation of sensitivity at higher frequencies, in the former one the highfrequency sensitivity remains intact.
In these plots, we assumed good but not very high value of the overall quantum efficiency of the interferometer \(\eta =0.8\) (note that in “ordinary” interferometers without squeezed light injection, all optical losses can be absorbed into this unified factor, see Sect. 6.3.2 of Danilishin and Khalili 2012). This resulted only in the barelyvisible sensitivity degradation in the shot noise dominated highfrequency area, confirming the above statement about tolerance of the optical spring based schemes to optical losses.
6.4 Cancellation of mechanical inertia
In the interferometer configurations with two or more optical carriers, more deep modification of the mechanical dynamics is possible, allowing, in some sense, to make the mechnaical inertial mass \(m_{\mathrm{inert}}\) smaller than the gravitational one \(m_{\mathrm{grav}}\) by attaching a negative optical inertia to the former one (Khalili et al. 2011; Danilishin and Khalili 2012). Existence of this effect immediately follows from the frequency dependence of the optical spring (99).
The resulting mechanical response function is plotted in Fig. 29 (solid line). It can be seen from this plot that indeed below some threshold frequency (it can be shown that it is equal to the smaller detuning \(\delta _1\)) the value of \(\chi ^{1}\) is noticeably suppressed (that is, the mechanical probe is more responsive) in comparison with the free mass. The “residual” lowfreqeuncy value of \(\chi ^{1}\) is created by optical damping, and for the parameters values used in this example, the gain is limited.
This scheme has also another disadvantage, namely it is dynamically unstable, and this instability could be significant. In Khalili et al. (2011), two methods of damping this instability were proposed. First, partial compensation of the mechanical inertia is possible, with the remaining nonzero inertia stabilizing the system and making the instability time long enough to be damped by an outofband feedback system.
Similar to the previous (single optical spring) case, two strategies of implementation of the negative inertia are possible. In the first one, two carriers have to be used in order to both create the negative inertia and also measure the test mirrors motion. This strategy was analyzed in detail in Danilishin and Khalili (2012) in the context of the Advanced LIGO parameters set. The second one require three dedicated carriers: one for the measurement and additional two for creation of the negative inertia. Unfortunately, in both cases the results can not be considered as satisfactory ones. Within the optical power constrains of existing and planned GW detectors, they can provide only very moderate low frequency sensitivity gain which accompanied by strong sensitivity degradation at higher frequencies. The reason for this is simple: indeed the negative inertia strongly increase the mechanical response, but only in the frequency band where the radiation pressure dominates and the therefore the sensitivity does not depend on the mechanical susceptibility.
6.5 Summary and outlook
The method of increasing the GW detectors sensitivity by means of optical modification of the test masses dynamics was proposed two decades ago and looks very simple and elegant. It does not require any sophisticated quantum states of light or radical alterations in the GW detectors core optics and also tolerant to the optical losses. However, as long as we aware, no specific plans of implementing this method in future GW detectors exist. This probably can be attributed to the following two reasons: first, technical problems associated with the detuned regime of GW interferometers, and second, the optical power constraints. The ruleofthumb estimates show that in broadband configurations, in order to shift the mechanical resonance up to some frequency \(f_m\) by means of the optical spring, about the same optical power is required as make the back action noise equal to the shot noise at this frequency \(f_m\). This means that using a single carrier, it is impossible to shift \(f_m\) into the shotnoise dominated area where the increase of the mechanical response could provide a significant effect, and in the twocarriers configuration, the carrier which create the optical spring has to be more powerful than the one which do the measurement. Taking into account the tight optical power budget of the contemporary GW detectors and even more tight of the future ones (with much more heavy test masses and longer arms), implementation of this regime could be problematic.
A possible solution to this problem was proposed recently in Somiya et al. (2016); Korobko et al. (2018). In was shown in these works, that using the parametric amplification of the optical field inside the interferometer, it is possible to amplify the optical spring without increase of the optical power. This approach, in principle, can be combined with other applications of the intracavity parametric amplification (whitelight cavity, back action evasion), see in particular Sect. 7.2.
7 Hybrid schemes
In this section, we review a relatively novel approach that seeks to enhance the sensitivity of the GW interferometer by coupling it to another, generally nonlinear, quantum system. Depending on the nature of the nonlinearity and on the way it is coupled to the interferometer, one can suppress backaction noise or reshape the optomechanical response of the interferometer so as to increase its bandwidth without sacrificing peak sensitivity.
The first effect, known as coherent quantum noise cancellation (CQNC) was pioneered by Tsang and Caves (2010). They suggested to use a combination of a nonlinear Kerr crystal and an unbalanced beamsplitter to couple the optomechanichal system under study (a GW interferometer, in our case) and an ancilla optical mode, where the frequency offset of the ancilla to the main interferometer, the splitting ratio of the beamsplitter and the nonlinear gain of the crystal are tailored so as to perfectly counteract the effect of ponderomotive squeezing due to optomechanical backaction. In this work, it was also shown that an all optical ancilla system interacts with the signal light as if it was an optomechanical system with negative mass mechanical oscillator. Wimmer et al. (2014) have developed this idea further to the level of a practical experiment that is currently being built at the University of Hannover. They also performed a thorough analysis of imperfections and their influence on this system ability for coherent cancellation of quantum backaction noise. This analysis has shown that it is problematic to realise this scheme in a GW detector due to stringent constraints on the ancilla’s optical bandwidth and frequency offset that must both be much smaller than the mechanical resonance frequency, which is \(\sim \,1\) Hz for Advanced LIGO mirrors. However, another physical implementation of the negative mass oscillator principle based on the interaction of the collective spin of caesium vapours in magnetic field with light was proposed by Polzik and Hammerer (2015), and the backaction cancellation effect in such systems was demonstrated experimentally by Møller et al. (2017). As we discuss in the following Sect. 7.1, such spinbased systems might be used in GW detectors.
Another way to use nonlinear system coupled to the optical degree of freedom is for creation a socalled whitelightcavity (WLC) effect (Wicht et al. 1997), that is to introduce in the interferometer an active element that compensates the positive dispersion of the arm cavities by its own negative dispersion and thereby increase the effective band of a high response to the GW signal. Original idea by Wicht et al. (1997) proposed to use atomic medium with electromagnetically induced transparency effect providing the desired negative dispersion, which suffered from the internal loss in the gas cell. In the following Sect. 7.2, we discuss more promising variants based on active nonlinear optical and optomechanical negative dispersion elements. These solutions are less lossy and thus stand a good chance to be a part of the next generation GW detectors, which might benefit from the additional astrophysical output the improved highfrequency sensitivity of such schemes may offer (Miao et al. 2018).
7.1 Negativemass spin oscillator
7.1.1 The negativefrequency system
Multiatomic spin ensembles proposed in Duan et al. (2000) and demonstrated experimentally in Julsgaard et al. (2001) (see also the review papers Hammerer et al. 2010; Polzik and Hammerer 2015) possess a set of unique features which make them attractive for use in quantum optomechanical experiments. Under certain conditions (see below), the dynamics of collective spin of such a system with high precision models the one of the ordinary harmonic oscillator, which eigen frequency can be made both positive and negative. Moreover, interaction of this spin system with light can be made similar to the ordinary pondermotive interaction of a movable mirror with the probing light.
7.1.2 Sequential scheme
The sketch of this class of measurement schemes is shown in Fig. 30. Here the probing light interacts first with the atomic spin system and then is injected into the main interferometer, which measures the probe object position. The light leaving the interferometer is measured by the homodyne detector. It is easy to note similarity of this scheme with the one which uses the frequencydependent squeezed light prepared by means of an additional filter cavity, see Kimble et al. (2002), Sect. 6.1 of Danilishin and Khalili (2012), and Sect. 4.2 of this paper. Another option is to put the atomic system system after the main interferometer, similar to the variationaloutput scheme of Kimble et al. (2002). In the case of the atomic spin system (and opposite to the filter cavity based schemes), both layouts provide identical results in the ideal (lossfree) case. Therefore, we consider here only the one shown in Fig. 30.
In order to demonstrate the basic features of this scheme while keeping the equations length within reasonable limit, we ignore the optical losses both in the atomic spin system and in the main interferometer. The full analysis with account of the optical losses can be found in Khalili and Polzik (2018). We would like to mention however, that while the problem of optical losses is a very serious one, it is generic for all interferometric schemes which use nonclassical light, see Sect. 4.4. At the same time, the atomic spin system introuces a new source of imperfection, namely the noise associated with the imaginary part of its effective succeptibility. We take into account this noise source here.
7.1.3 Parallel (or EPR) scheme
This problem can be avoided by using another “parallel” optical layout, see Fig. 32. It relies on high degree of crosscorrelation between quantum fluctuations in the two entangled “signal” and “idler” light beams generated in the parametric downconversion conversion (PDC) process. These two beams could have different wavelengths (the nondegenerate case), which should match the working frequency of the GW detector and the atomic transition frequency. Each of the beam has to interact with the respective subsystem, as shown in see Fig. 32. Then both output signals have to be combined using optimal weight factors. Due to the abovementioned crosscorrelation, both the shot noise and the radiation pressure noise contributions will be suppressed in the combined output signal.
Note that a similar scheme was proposed initially in Ma et al. (2017) for another purposes, namely, as a method of generation of effective frequency dependent squeezing without the use of an additional filter cavity (as in Kimble et al. 2002); see details in Sect. 4.3.
Spectral density (152) optimized by the condition (142) is plotted in Fig. 31 for the same parameters (143) as in the previous case. It is easy to see that (in accord with the above reasoning) in the major part of the frequency band the sensitivity is worse by 3 db than in the sequential scheme (for the same squeeze factor r).
In order to reveal the influence of the internal damping in the atomic spin system, the case with \(\gamma _S=2\pi \times 30\,\mathrm{s}^{1}\) also is presented in Fig. 31. It can be seen that this tenfold increase of \(\gamma _S\) noticeably degrade the lowfrequency sensitivity, preventing from overcoming the SQL.
It is interesting, that in the lowfrequency band, where the condition (141) starts to deviate form the simplified one (142). the parallel scheme provide noticeably better sensitivity, than the sequential one (for the same value of \(\gamma _S\)). This result can be attributed to the fact, that the “software” summing of the photodetectors outputs using the optimized frequencydependent factor (151) is more flexible procedure that simple “hardware” subtraction of the back actions.
7.1.4 Summary
Using the additional spin systems with negative effective mass, it is possible to suppress the quantum noise in GW detectors across the almost entire frequency bandwidth relevant for gravitational wave observation. In comparison to the most of the other proposals for reducing the quantum noise, the spin system based approach has a significant advantage of being completely compatible with existing and planning GW interferometers thus not requiring complex alterations in the interferometers’ core optics. In both “sequential” and “parallel” variants of this scheme, the only additional elements are the spin system itself, the source of the singlemode or twomode squeezed light, and the optical scheme of injection the nonclassical light into the interferometer. This setup strongly resembles the scheme of injection of “ordinary” squeezed light into the interferometer and evidently should has about the same level of complexity and cost.
It worth to be noted also that this scheme paves the road towards generation of an entangled state of the multikilogram GWD mirrors and atomic spins which would be of fundamental interest due to the sheer size of the objects involved.
7.2 Negative dispersion and whitelightcavity schemes
To overcome the bandwidthpeaksensitivity tradeoff, there are two approaches. One is keeping the bandwidth and increasing the peak sensitivity with the squeezed light, as discussed in Sec. 4. The other is broadening the bandwidth while keeping the peak sensitivity, which is the idea of socalled white light cavity—a cavity that resonates “all” frequencies. It is motivated by the physical origin of the tradeoff, and has to do with the extra phase \(\phi =\varOmega L/c\) picked up by the GW sidebands at \(\omega _0\pm \varOmega \) when propagating inside the arm cavity that is tuned on resonance with respect to the carrier frequency \(\omega _0\). Such a positive dispersion with \(\mathrm{d}\phi /\mathrm{d}{\varOmega } > 0\) implies that higher the sideband frequency is, the more phase it is accumulated and thus is far away from the resonance, which leads to a degradation of the signal response.
7.3 Summary and outlook
The atomic based active filter for broadening the detector bandwidth is not suffering the from same thermal noise issue as the optomechanical filter; the atomic transition involved happens at the optical frequency and the thermal environment can be viewed effectively as in the vacuum state. The main issue for the atomic system has to with the wavelength being tied to the transition of some specific species of atoms, which is different from those used in the current and proposed GW detectors. Exploring atomic systems with compatible wavelength or studying coherent frequency conversion scheme will be needed. The same issue of optical loss also applies.
8 Discussion and conclusion
We made an attempt to overview in this article the vast body of quantum techniques for suppression of quantum noise that are developed specifically for the field of gravitationalwave astronomy. We are standing now at the moment of inception of the concepts for the next generation of gravitational wave detectors that must have at least 10 times better sensitivity than the existing Advanced LIGO and Advanced Virgo instruments, which are about to be limited by quantum fluctuations of light in the almost entire detection band. The task of building the detector with the best astrophysical output justifies the need to bring some order into the massive collection of quantum noisemitigation techniques that has beed developed so far. This was the goal of this work along with the aim to put all of those techniques in the same context and measure their merits and downsides against the common ruler. This pushed us towards the unified set of parameters for all considered schemes, taking the approach suggested by the GWIC 3G R&D Committee in LIGOT1800221 and summarised by Table 1.
As an outlook, with the recent understanding of the fundamental quantum limit (FQL), which only depends on the power fluctuation inside the arm cavity, it seems to lead to a unified picture of different techniques: (1) the external squeezing injection is a direct approach to increasing the power fluctuation; (2) Modifying dynamics with the optical spring effect can be viewed as using the internal ponderomotive squeezing for enhancing the power fluctuation; (3) The whitelightcavity idea is to extend the enhancement over a broad frequency range; (4) The speed meter is an approach to shaping the power fluctuation at different frequencies such that the FQL can be reached using a frequencyindependent readout quadrature at those frequencies; (5) The optimal frequencydependent readout is in general needed to attain the FQL at different frequencies. Instead of comparing techniques against each other, as in the case for nearterm upgrades of existing detectors, we may now start to think how we can coherently combine different techniques to enhance the power fluctuation at frequency of interest, i.e., lowering the FQL, and reach the limit. We can then study the susceptibility of different realisations to optical loss and other realistic imperfections. Eventually, we may obtain new configurations with high sensitivity that goes beyond what can be achieved with the current paradigm of design.
Footnotes
 1.
Strictly speaking, there are two possible ways of looking at the action of GW on the light in the interferometer. In this review, we will follow the point of view that the test masses move in a Local Lorentz (LL) frame of a central beam splitter, and GWs act akin to tidal forces on the test masses of the interferometer making them move w.r.t. the defined LLframe of the detector (Blandford and Thorne 2008). Another way to describe GW action is to consider the interferometer in a socalled transversetraceless (TT) gauge, where test masses are assumed to remain at rest and GW action leads to the modulation of the effective index of refraction of the space interval between the test masses. Interested readers are invited to read an excellent course book by Blandford and Thorne.
 2.
The rigorous mathematical treatment of the linear quantum measurement and of all transfer functions is given in Sect. 4.2 of Danilishin and Khalili (2012)
 3.
In fact, the symplectic nature of \(\mathbb {T}\) requires a more restrictive Bloch–Messiah Decomposition (Cariolaro and Pierobon 2016) that ensures singular values which include their own reciprocals.
 4.
In some sense, Eq. (62) is another way to derive a Wiener filter for a 2channel interferometer, as described in Sect. 2.1. In this case the 2 quadratures of the signal field are combined with the idlerchannel readout multiplied by a frequencydependent coefficients \({\mathbf {K}} = \{K_{c}(\varOmega ),\,K_{s}(\varOmega )\}\) that minimise the spectral density of the difference: \((\hat{\varvec{a}}{\mathbf {K}}\hat{b}_{\phi })\), i.e., \(\min \limits _{{\mathbf {K}}}\Bigl [\left\langle (\hat{\varvec{a}}{\mathbf {K}}\hat{b}_{\phi })\circ (\hat{\varvec{a}}{\mathbf {K}}\hat{b}_{\phi })^\dag \right\rangle \Bigr ]\)
 5.
It might seem counter intuitive, how a resonantly tuned signalrecycling cavity could result in a broader bandwidth of the combined effective cavity of the arms and the SRC. The reason for that is the sign flip (\(\pi \) phase shift) experienced by the light reflected off the resonancetuned arm cavities. If combined with the SR mirror placed at a distance of an integer number of halfwavelengths of carrier light, it will result in an effectively antiresonance tuned SRC and therefore will lead to a virtually lower finesse of the combined cavity.
 6.
Here, the negative dispersion may appear a positive one if one assumes a different sign convention as compared to Eq. (4). In that case a special care has to be taken to do ALL the calculations consistent with the chosen sign convention in the definition of the Fourier transform.
Notes
Acknowledgements
The authors are particularly grateful to Jan Harms for his careful and meticulous reading of the manuscript and for very helpful feedback. We also thank our colleagues from the LIGOVirgo Scientific Collaboration (LVC) for illuminating discussions and suggestions on how to improve the paper. SLD would like to thank Lower Saxonian Ministry of Science and Culture that supported his research within the frame of the program “Research Line” (Forschungslinie) QUANOMET – Quantum and NanoMetrology. The FYK was supported by the Russian Foundation for Basic Research Grants 140200399 and 165210069. FYK was also supported by the LIGO NSF Grant PHY1305863. HM is supported by UK STFC Ernest Rutherford Fellowship (Grant No. ST/M005844/1).
Supplementary material
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