Quantum Limits of Interferometer Topologies for Gravitational Radiation Detection
QQuantum Limits of Interferometer Topologies forGravitational Radiation Detection
Haixing Miao, Huan Yang, Rana X Adhikari, and YanbeiChen
Division of Physics, Math, and Astronomy, California Institute of Technology,Pasadena, California, 91125E-mail: [email protected]
Abstract.
In order to expand the astrophysical reach of gravitational wavedetectors, several interferometer topologies have been proposed, in the past, toevade the thermodynamic and quantum mechanical limits in future detectors.In this work, we make a systematic comparison among these topologies byconsidering their sensitivities and complexities. We numerically optimize theirsensitivities by introducing a cost function that tries to maximize the broadbandimprovement over the sensitivity of current detectors. We find that frequency-dependent squeezed-light injection with a hundred-meter scale filter cavity yieldsa good broadband sensitivity, with low complexity, and good robustness againstoptical loss. This study gives us a guideline for the near-term experimentalresearch programs in enhancing the performance of future gravitational-wavedetectors.PACS numbers: 04.80.Nn, 07.05.Dz, 07.10.Fq, 07.60.Ly, 95.55.Ym
Submitted to:
Class. Quantum Grav. a r X i v : . [ g r- q c ] J un uantum Limits of Interferometer Topologies for Gravitational Radiation Detection
1. Introduction and Summary
Over the past two decades, an international array of laser-interferometer gravitational-wave (GW) detectors has been built and operated near their theoretical sensitivitylimits [1, 2, 3, 4]. No direct detection of gravitational waves has yet been made and thisis consistent with the low event rates predicted by our knowledge of astrophysics [5].The 2 nd generation of detectors, which are now being assembled (AdvancedLIGO [6], Advanced Virgo [7], GEO-HF [8] and KAGRA [9]), are expected to improvesensitivities by a factor of ∼
10 compared with the first generation, and are expectedto make direct detections. In order to move from the era of detections to the era ofprecision GW measurement, the detector sensitivities must be further improved [10].The sensitivity of a laser interferometer gravitational-wave detector is limitedby many noise sources. Among them, the quantum noise is due to ground-statefluctuations of the electromagnetic field, which beat with the laser field to produceshot noise and radiation pressure noise. For 2 nd generation detectors, quantum noise isdominant in a large fraction of the entire observation band. Furthermore, the quantumnoise is at a level at which the Heisenberg Uncertainty Principle for kilogram scalemasses becomes important.Although many sources of noise can be regarded as having entered interferometeroutput data through “imperfections” of the interferometer, quantum noise is so tightlycoupled into the gravitational wave transduction process, that improving the quantumnoise often requires changing the optical configuration of the interferometer. In thepast decades, several types of strategies for improving the optical configuration havebeen proposed within the community [11, 12]:(i) injection of squeezed light[13, 14, 15] from the interferometer’s dark port(ii) inserting optical filters at the interferometer’s input or output port(iii) reshaping the interferometer’s optical transfer function in the frequency domain(iv) modifying the test masses’ mechanical transfer function, e.g., by using the opticalspring effect associated with detuned signal recycling(v) injecting multiple carrier fieldsThese strategies are meant to be combined with each other in order to synthesizean optimal optical configuration. In this paper, regarding (i) above, we shall alwaysassume that squeezed light will be injected; for (ii), depending on whether we useinput or output filters, we will consider two options: • frequency dependent squeeze angle —injecting squeezed light with an optical filtercavity [16, 17, 18, 19, 20]; • frequency dependent readout —filtering the output with a cavity to measureappropriate signal quadratures at different frequencies [21, 16, 22].Next, for each of the two options above for modifying the input-output optics, we willconsider one of the following four options for the interferometer itself: • keeping the signal-recycled configuration of Advanced LIGO [uses strategy (iv)above]; • speed-meter configurations [23, 24, 25, 26, 27], i.e., measuring a quantity that isproportional to the test mass speed at low frequencies, by inserting an additionallong cavity [using strategy (iii) above]; uantum Limits of Interferometer Topologies for Gravitational Radiation Detection • long signal recycling cavity —using a km scale signal recycling cavity to have afrequency-dependent response [using strategy (iii) and (iv)]; • dual-carrier scheme [28]—introducing an additional laser field to gain anotherreadout channel; in particular, we shall also consider the so-called local-readoutscheme , in which the additional field is anti-resonant in the arm cavity andresonant in the power-recycling cavity [using strategy (iii) and (iv)].When trying to evaluate these configurations, we will include the effect of realisticoptical losses and quantitatively compare these configurations against a few baselineinterferometer configuration (which includes realistic levels of non-quantum noise).This differs from the proposed Einstein Telescope (ET) [29] which assumes significantreduction of the non-quantum noises at low frequencies and entirely new infrastructurewith much longer arms, we are focusing on near-term upgrades to current detectorsusing technologies that can be deployed within the existing facilities.We will numerically optimize the sensitivity of different configurations for thisnext generation detector (which we call LIGO3), with the following cost function: C ( x ) = (cid:40)(cid:90) f max f min d(log f ) log (cid:20) h aLIGO h LIGO3 ( x ) (cid:21)(cid:41) − . (1)Here [ f min , f max ] = [4 Hz , x arethe set of parameters of the optical configuration that we optimize over; h aLIGO is thesquare root of the total noise power spectral density of the baseline design of AdvancedLIGO (aLIGO); and h LIGO3 is the square root of the total noise power spectraldensity of interferometers with various improved optical configurations. Notice thatthe integration variable is log f instead of f , which means that we want to maximizethe improvement over aLIGO in the log-log scale.The results of the numerical optimization are shown in Figures 1 and 2, wherewe plot the total noise spectra (the quantum noise + the classical noises) for differentconfigurations with frequency dependent squeezing (input filtering) and frequencydependent readout (output filtering), respectively. In producing Figure 1, we assumea moderate reduction in the thermal noise and the same mass and optical poweras those for aLIGO. In producing Figure 2, we assume a more optimistic reductionin the thermal noise, the mirror mass to be 150 kg and the maximum arm cavitypower to be 3 MW. As we can see, by adding just one filter cavity to the signal-recycled interferometer (the baseline aLIGO topology), we can already obtain asubstantial broadband improvement over aLIGO. Further low-frequency enhancementcan be achieved by applying either the speed meter or the local-readout (dual-carrierMichelson) scheme.The outline of this paper goes as follows: in Section 2, we summarize the basicsof our quantum noise calculations; in Section 3, we introduce the interferometertopologies and the features of the configurations that we compare; in Section 4,we introduce classical noise models for 3 rd generation detectors and then comparedifferent optical configurations by optimizing their parameters under the same costfunction defined in Equation 1; in Section 6, we summarize our main results. In theappendices, we present a table of the optimized interferometer parameters, describethe non-quantum noise sources, and also define the variables used here in comparisonto the previous literature on this topic. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection S t r a i n ) [ )))))))))))) ] Schemes)with)input)filtering L ong ) s i gn a l ) r ec y c li ng ) ca v it y S i gn a l Br ec y c li ng ) A dv L I GO B t yp e m ) s p ee d ) m e t e r L o ca l ) r ea dou t ) du a l ) ca rr i e r0 Frequency)[Hz] A dv L I GO Schemes)with)output)filtering S u s p e n s i on ) t h e r m a l ) K S e i s m i c ) no i s e G r a v it y ) G r a d i e n t ) )))) r e du c ti on C o a ti ng ) B r o w n i a n ) . )))) r e du c ti on Frequency)[Hz] A dv L I GO Figure 1:
The optimized total noise spectrum for different schemes assuming a moderateimprovement of the thermal noise compared with aLIGO baseline design. The left panelshows the case for schemes with input filtering, i.e., frequency-dependent squeezing, whilethe right panel shows the case for schemes with output filtering, i.e., frequency dependentreadout, which is also called variational readout in the literature. The lower panels show thelinear strain sensitivity improvement over Advanced LIGO.
Frequencyo[Hz] A dv L I GO Frequencyo[Hz] A dv L I GO Schemesowithoinputofiltering Schemesowithooutputofiltering L ong o s i gn a l o r ec y c li ng o ca v it y S i gn a l ec y c li ng o A dv L I GO t yp e K m o s p ee d o m e t e r L o ca l o r ea dou t o du a l o ca rr i e rK S u s p e n s i on o t h e r m a l o K K S e i s m i c o no i s e G r a v it y o G r a d i e n t o oooo r e du c ti on K C o a ti ng o B r o w n i a n o oooo r e du c ti on K S t r a i n o [ oooooooooooo ] Figure 2:
Optimization results for different schemes assuming more substantial thermalnoise improvements, increasing the mirror mass from 40 – 150 kg, and increasing the armcavity power from 800 – 3000 kW. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection
2. Basics of Quantum Noise
In this section, we will briefly review the basics for evaluating quantum noise in alaser-interferometer gravitational-wave detector by using an input-output formalism.Additionally, we will discuss the principle behind the use of filter cavities for reducingthe quantum noise. For more detail, one can refer to a recent review article [30]. !"
Figure 3:
Schematics showing the configuration of an interferometric gravitational-wave(GW) detector (left) and the block diagram illustrating the two paths (upper and lower)through which the vacuum fluctuation propagates to the output (right).
When analyzing the quantum noise of a laser interferometer, as shownschematically in Figure 3, we assume linearity and stationarity of the system;a frequency-domain analysis can therefore be applied with the noise and signalpropagating through the system via various linear transfer functions. There aretwo types of noise: (i) the shot noise, also called the readout noise, is the one thatcomes from the measurement device itself—in the context here, arising from the phasefluctuation of the light, and it usually decreases as we increase the measurementstrength (the optical power). Its propagation is denoted by the lower path of theblock diagram in Figure 3 (ii) the back-action noise, also called the radiation-pressurenoise here, is the one that disturbs the test mass due to noise in the device, and itusually increases when the measurement strength increases. Its propagation is shownby the upper path of the diagram. In general, these two types of noise are mixedwith each other. To evaluate detector sensitivity, the key is then to analyze how thenoise and signal propagate and to identify those transfer functions, which gives theinput-output relations.For these interferometers, the photocurrent output I out that we measure is linearlyproportional to a certain optical quadrature—a linear combination of the amplitudequadrature b and phase quadrature b ‡ : I out (Ω) ∝ b (Ω) sin ζ + b (Ω) cos ζ, (2) ‡ These quadratures are related to the upper b (Ω) and lower audio sideband b ( − Ω) via b =[ b (Ω) + b † ( − Ω)] / √ b = [ b (Ω) − b † ( − Ω)] / ( i √ uantum Limits of Interferometer Topologies for Gravitational Radiation Detection ζ is the readout quadrature angle and can be adjusted by the phase of the localoscillator (the optical field that beats with the interferometer output). In terms ofamplitude and phase quadratures, the input-output relation can generally be put intothe following form: (cid:20) b (Ω) b (Ω) (cid:21) = (cid:20) M (Ω) M (Ω) M (Ω) M (Ω) (cid:21) (cid:20) a (Ω) a (Ω) (cid:21) + (cid:20) v (Ω) v (Ω) (cid:21) h (Ω) . (3)Here Ω = 2 πf is the angular frequency; b ( a ) and b ( a ) are the output (input)amplitude quadrature and phase quadrature, respectively; M ij are the elements ofthe transfer matrix, which depend on the specific optical configuration; v i quantifythe detector response to the gravitational-wave strain h . More compactly, one canrewrite the input-output relation in a vector form: b (Ω) = M (Ω) a (Ω) + v (Ω) h (Ω) . Different configurations will have different transfer matrices and response functions tothe gravitational-wave signal—thus different input-output relations. In the followingsections, we will see an interesting assortment of them.Given the above input-output relation and the readout angle ζ of the homodynedetection by adjusting the local oscillator phase, the output current of the photodiodeis proportional to y (Ω) = d T ζ b (Ω), namely, y (Ω) = d T ζ M (Ω) a (Ω) + d T ζ v (Ω) h (Ω) , (4)where the readout vector is defined as d ζ ≡ (sin ζ, cos ζ ) T . The first term is thequantum noise, while the second term is the output response to the GW signal. Thedetector sensitivity is quantified by the noise power spectral density § (normalizedwith respect to the GW strain h ): S h (Ω) = d T ζ M (Ω) S (Ω) M † (Ω) d ζ | d T ζ v (Ω) | , (5)where S is the noise spectral-density matrix for the input amplitude quadrature a and the phase quadrature a — (cid:104) a i (Ω) a † j (Ω (cid:48) ) (cid:105) sym ≡ π S ij (Ω) δ (Ω − Ω (cid:48) ) ( i, j = 1 , S = S = 1and S = S = 0 (uncorrelated amplitude and phase noise).Taking a broadband, resonant sideband extraction (RSE), tuned dual-recycledinterferometer (the baseline configuration of aLIGO) for example, the input-outputrelation is given by [16]: (cid:20) b b (cid:21) = e iφ (cid:20) −K (Ω) 1 (cid:21) (cid:20) a a (cid:21) + e iφ (cid:20) (cid:112) K (Ω) (cid:21) h (Ω) h SQL . (6)Here we have introduced: φ ≡ arctan(Ω /γ ) , K (Ω) ≡ γ ι c Ω (Ω + γ ) , h SQL ≡ (cid:114) (cid:126) m Ω L (7)with γ the arm cavity bandwidth, L the arm cavity length, ι c ≡ ω P c / ( mLc ) forquantifying the measurement strength, ω the laser angular frequency, and P c thearm cavity power. If we measure the phase quadrature by choosing the readout phaseto be ζ = 0, the corresponding noise power spectral density will be: S h (Ω) = (cid:20) K (Ω) + 1 K (Ω) (cid:21) h ≥ h = 8 (cid:126) m Ω L . (8) § The single-sided power spectral density S A (Ω) for any quantity A is defined by [cf. Equation (22)in [16]]: (cid:104) A (Ω) A † (Ω (cid:48) ) (cid:105) sym = (cid:104) A (Ω) A † (Ω (cid:48) ) + A † (Ω (cid:48) ) A (Ω) (cid:105) = πS A (Ω) δ (Ω − Ω (cid:48) ). uantum Limits of Interferometer Topologies for Gravitational Radiation Detection S t r a i n [ ] r a d i a ti on - p r e ss u r e no i s e s h o t n o i s e Frequency [Hz] S Q L Figure 4:
The quantum-noise spectral density S / h for a broadband RSE interferometergiven the same specification of aLIGO— m = 40 kg and P c = 800 kW. The first term, proportional to the optical power (
K ∝ P c ), is the radiation-pressure noise and comes from the fluctuation of the input amplitude quadrature a ;the second term, inversely proportional to the optical power, is the shot noise andcomes from the fluctuation of the input phase quadrature a . In this simple scenario,the sensitivity is limited by the standard quantum limit (SQL)—the benchmark for thestrength of quantum noise [31]. In Figure 4, we plot S / h (Ω)—the radiation-pressurenoise dominates at low frequencies and the shot noise dominates at high frequencies. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection
3. Optical Topologies
In this section, we briefly describe the strategies for configuration improvements thatwill be used in Section 4. We will compute the corresponding quantum noise spectrumusing the input-output formalism introduced in Reference 2.
Inputnoise OutputTest massOptical transfer functionGWs
Filter cavity
Faraday isolator
Squeezed light filter cavity
Figure 5:
Schematic showing the frequency dependent squeezed light phase angle rotationscheme (left) and its associated block diagram (right).
Unlike the vacuum state, for which the cross spectral density matrix for any twoorthogonal field quadratures is the identity matrix, squeezed light has the followingcross spectral density matrix for the quadratures b and b : (cid:20) S S S S (cid:21) = (cid:20) cosh 2 r − sinh 2 r cos 2 ϕ − sinh 2 r sin 2 ϕ − sinh 2 r sin 2 ϕ cosh 2 r + sinh 2 r cos 2 ϕ (cid:21) (9)where r is called the squeezing factor (10 dB squeezing means that e r = 10) and ϕ isthe squeezing angle. If we define the ζ quadrature [see Section 2 for details] as b ζ = b cos ζ + b sin ζ , (10)and if we denote by S ζ the spectral density of b ζ , then the matrix becomes S ζ = S cos ζ + 2 S cos ζ sin ζ + S sin ζ . (11)In particular, S ϕ = e − r , which means the ϕ -quadrature (often referred to as thesqueezed quadrature) has a fluctuation that is e − r the level of vacuum (in amplitude),while the π/ ϕ quadrature orthogonal to ϕ -quadrature (often referred to as the anti-squeezed quadrature) has S ϕ = e +2 r , which is e + r the level of vacuum fluctuation.The above picture exists for each audio sideband frequency Ω. Frequency-dependent squeezed light describes a state that has different squeeze factors and/orangles for each frequency. As schematically shown in Figure 5, such a frequencydependence can be realized, for example, by injecting frequency-independent squeezedlight into a Fabry-Perot cavity with a linewidth and detuning frequency comparableto the audio frequencies of interest. If we define b , as the quadrature fields of the uantum Limits of Interferometer Topologies for Gravitational Radiation Detection a , the quadratures of the input field, the cavity has an input-outputrelation of (cid:20) b b (cid:21) = e i ( α + − α − )2 (cid:20) cos α + + α − − sin α + + α − sin α + + α − cos α + + α − (cid:21) (cid:20) a a (cid:21) , (12)with α ± is defined as e i α ± ≡ iγ ∓ Ω − ∆ iγ ± Ω − ∆ , (13)where ∆ and γ are the detuning frequency and bandwidth of the filter cavity,respectively. As indicated by Equation 12, the quadratures undergo a frequencydependent rotation of ( α + + α − ) /
2. This converts a frequency independent squeezedvacuum into one with a frequency dependent squeezing angle.With the correct frequency dependence, one can rotate the squeezing angle suchthat the quantum noise is reduced by a factor e r over the entire frequency band,namely (in the case of the broadband RSE interferometer) S opt h (Ω) = e − r (cid:20) K (Ω) + 1 K (Ω) (cid:21) h . (14)This is the optimum performance that can be realized with frequency dependentsqueezed light injection.Figure 6 shows the resulting noise spectrum in the lossless case. As we can see,the squeezing angle rotates in such a way that at low frequencies the fluctuation in theamplitude quadrature is squeezed—thus reducing the radiation-pressure noise, whileat high frequencies the phase quadrature is squeezed—thus reducing the shot noise.In order to achieve the desired rotation of squeezing angle, the filter cavity needs tohave a frequency bandwidth that is near the frequency where the radiation pressurenoise is comparable to the shot noise.The frequency dependence of a series of such filter cavities as well as theconcomitant parameters required for realizing this frequency dependence has beenderived in [25]. In practice, however, the complexity of using the “optimal” number ofcavities and the performance degradation which comes from optical losses, leads oneto use a sub-optimal number of cavities; the resulting degradation of the astrophysicalsensitivity is negligible. So far, we have been considering the idealcase without optical loss. Here we provide a qualitative understanding of how lossin the filter cavity affects the sensitivity of input filtering. Basically, the optical lossintroduces additional (vacuum) noise that is uncorrelated with the input squeezedlight: a (cid:48) = √E n + √ − E a , (15) a (cid:48) = √E n + √ − E a , (16)where E quantifies the total optical loss of the filter cavity and n , are the associatednoise terms in the amplitude and phase quadratures. These noise sources will degradethe squeezing. For example, the amplitude squeezed light originally has S = e − r with r >
0. Introducing optical loss according to Equation 15, it becomes: S (cid:48) = (1 − E ) e − r + E . (17) uantum Limits of Interferometer Topologies for Gravitational Radiation Detection ! " $ % & ’ ( ’’’’’’’’’’’’ ) * ! &8’4,-++1%&5 !" * ;<3 =>4 ;<3 =>4 ;<3 =>4 Figure 6:
Noise spectrum for frequency dependent squeezing (left) and illustration of therotation of the squeezing angle (right).
For a completely lossy case with E = 1, we have S = 1 and the squeezing simplyvanishes.The squeezed light at different audio frequencies experiences different levels ofoptical loss from the filter cavity. The low frequency part enters the cavity andcirculates multiple times, while the high frequency part barely enters the cavity.Therefore, the optical loss affects the low frequency part most significantly (referto Appendix C for a detailed discussion). In terms of the noise power spectrum, weapproximately have: S h (Ω) = (cid:26) [(1 − E ) e − r + E ] K (Ω) + e − r K (Ω) (cid:27) h , (18)in contrast to Equation 14. Compared with the ideal frequency dependent squeezingcase, the low frequency radiation pressure noise increases due to the optical loss andthe high frequency shot noise remains almost the same. In Figure 7, we show the effectof optical loss. In producing the figure, we have assumed a total optical loss of 20%,which is equivalent to a round trip loss of 40 ppm given a filter cavity input mirrortransmittance of T f = 200 ppm. Apart from the opticalloss, there are other imperfections of the filter cavity that will degrade the sensitivity.In particular, here we consider the effect of variations in the parameters of the filtercavity, which make the bandwidth γ f , or equivalently the input mirror transmittance T f , and the detuning ∆ f deviate from their optimal value, i.e., T f = T opt f + δT f and∆ f = ∆ opt f + δ ∆ f .As shown in Appendix D, such a parameter variation will mainly decrease the low-frequency sensitivity (for a reason similar to the effect of loss), and we approximatelyhave: S h (Ω) ≈ S opt h (Ω) + sinh 2 r (cid:32) δT f T opt f (cid:33) + (cid:32) δ ∆ f ∆ opt f (cid:33) K (Ω) h . (19) uantum Limits of Interferometer Topologies for Gravitational Radiation Detection S t r a i n [ ] Frequency [Hz]Frequency dependent squeezing S Q L no squeezing effect of optical loss Frequency [Hz] effect of parameter variation
Figure 7:
The effect of optical loss in the filter cavity (left) and the effect of parametervariation of the filter cavity (right) in the case of input filtering. The shaded regions illustratethe degradation of sensitivity as losses are added. Here we have assumed the total opticalloss E = 20% and parameter variations of δT f /T opt f = δ ∆ f / ∆ opt f = 10%. If the relative error in the transmittance and the detuning can be as low as 100 ppm,namely δT f /T opt f ∼ − , δ ∆ f / ∆ opt f ∼ − , we have S h ≈ S opt h + 10 − K h for10 dB squeezing, which is a negligible deviation from the optimal one. In Figure 7, weillustrate this effect with an exaggerated variation of 10%. OutputTest massOptical transfer functionGWsinputnoise
Filter cavity filtercavity
Figure 8:
Schematic optical layout of the frequency dependent (or variational) readoutscheme (left) and its associated block diagram (right).
A closely related counterpart to the frequency dependent squeezed light injectionis the frequency dependent readout angle and, as shown schematically in Figure 8,it uses an optical cavity to filter the detector output allowing one to measuredifferent optical quadratures at different frequencies. The filter cavity has the same uantum Limits of Interferometer Topologies for Gravitational Radiation Detection b ζ (Ω) = b (Ω) sin ζ + b (Ω) cos ζ = e iφ [sin ζ − K (Ω) cos ζ ] a (Ω) + e iφ cos ζ a (Ω)+ e iφ cos ζ (cid:112) K (Ω) h (Ω) h SQL . (20)Here the first term, proportional to a , is the radiation pressure noise; the secondterm, proportional to a , is the shot noise; the third term is the signal. As we can see,if the quadrature angle ζ has the following frequency dependence:tan ζ = K (Ω) , (21)the radiation pressure term would be canceled, and give rise to a shot noise only noisefloor. Since the phase for the local oscillator is usually fixed, before beating with thelocal oscillator we need to rotate the output quadratures with a filter cavity to achievesuch a frequency-dependent quadrature readout.The resulting noise spectrum for this scheme is simply: S h (Ω) = 1 K (Ω) h . (22)If we simultaneously inject phase squeezed light, we will have: S opt h (Ω) = e − r K (Ω) h . (23)In Figure 9, we plot the noise spectrum in the ideal lossless case with the low frequencyradiation pressure noise completely evaded. In reality, due to optical losses, sucha cancellation cannot be perfect. In the numerical optimization, we will take intoaccount the optical loss and optimize the parameters for the filter cavity. For the frequency dependent readoutscheme, the additional noise introduced by optical loss influences the output andmodifies the input-output relation in the following way: (cid:20) b (cid:48) b (cid:48) (cid:21) = √E (cid:20) n n (cid:21) + √ − E e iφ (cid:20) −K (cid:21) (cid:20) a a (cid:21) + √ − E e iφ (cid:20) √ K (cid:21) hh SQL . (24)Due to the presence of uncorrelated noise, the condition in Equation 21 no longerprovides radiation-pressure noise cancellation. By optimizing the quadrature angle ζ for the tuned interferometer with phase squeezed light injection, one can find that theoptimal sensitivity, in contrast to Equation 23, reads: S h (Ω) = (cid:20) η e r K (Ω) η + e r + e − r + η K (Ω) (cid:21) h η ≡ E / (1 − E ) ≈ E . The effect of loss is illustrated in Figure 10. As we can see,the low-frequency performance is very fragile, and we end up with a sensitivity similarto the input filtering case, given the same level of loss. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection S t r a i n [ ] Frequency [Hz]frequency-dependentreadout S Q L Ideal case without optical loss additional phase squeezing
Figure 9:
Noise spectra for the frequency dependent readout scheme without (red curve)and with (purple curve) additional phase squeezed light injection. S t r a i n [ ] Frequency [Hz]frequency-dependent readout + frequency-independent squeezing S Q L effect of optical loss Frequency [Hz] effect of parameter variation
Figure 10:
The effect of optical loss in the filter cavity (left) and the effect of parametervariation of the filter cavity (right) in the case of frequency-dependent readout (outputfiltering). Similar to the case shown in Figure 7, the shaded areas denote the degradationof sensitivity. We have used a total optical loss of E = 20%. In contrast, the parametervariation is chosen to be only δT f /T opt f = δ ∆ f / ∆ opt f = 10 − in order to produce reasonablesensitivity, as it is much more sensitive than input filtering. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection As shown in AppendixD, the parameter variation of the filter cavity results in the following sensitivity S h ≈ S opt h + (2 e r K + e r K ) (cid:32) δT f T opt f (cid:33) + (cid:32) δ ∆ f ∆ opt f (cid:33) h . (26)Since K (cid:29)
1, by comparing Equation 19 with Equation 26, we can see that the outputfiltering is more susceptible to parameter variation than input filtering, which isillustrated in Figure 10.
OutputTest massOptical transfer functionGWsinputnoise signalrecycling
Closed port
Figure 11:
Schematic optical layout showing the long signal recycling cavity scheme (left)and its associated block diagram (right). The signal recycling mirror coherently reflects backthe signal, forming a feedback loop as indicated in the block diagram.
In this subsection, we discuss the idea of long signal-recycling cavity, and one canrefer to Ref. [32] and references therein for an overview of different recycling techniquesapplied in the context of GW detectors. In the usual case when the beam splitterand the signal recycling mirror are close to each other, the signal recycling cavity isrelatively short (order of 10 meters) and one can ignore the phase accumulated in thiscavity by the audio sidebands: Ω L sr /c ≈ L sr being the length of the signal-recycling cavity. We can therefore treat the signal-recycling cavity as an effectivecompound mirror with complex transmissivity and reflectivity, which is the approachapplied in [33]. With a long signal recycling cavity, however, Ω L sr /c is not negligibleand different sidebands pick up different phase shifts. Specifically, the transfer functionmatrix for the quadratures due to the free propagation in the signal recycling cavityis given by: e i Ω τ sr (cid:20) cos ∆ τ sr − sin ∆ τ sr sin ∆ τ sr cos ∆ τ sr (cid:21) (27)with τ sr ≡ L sr /c and ∆ being the detuning frequency of the signal recycling cavity.One can then apply the standard procedure to derive the input-output relation forthis scheme. In general, the final expression is quite lengthy and not illuminating,so we will not show it here, and will evaluate its noise spectrum numerically. Thereis one interesting special case which allows an intuitive understanding. It is whenthe signal-recycling detuning phase is equal to π/
2. In this case, the coupled cavity, uantum Limits of Interferometer Topologies for Gravitational Radiation Detection et al. [34] and experimentally demonstrated by Gr¨af et al. [35], which are motivated by the above mentioned two advantages.
The motivation for the speed meter originates from the perspective of viewingthe gravitational-wave detector as a quantum measurement device. Normally, wemeasure the test mass position at different times to infer the gravitational-wave signal.However, position is not a conserved dynamical quantity of a free mass. Accordingto quantum measurement theory [31], such a measurement process will inevitablyintroduce additional back action on to the test mass. In the context here, the backaction is the radiation-pressure noise. In order to evade the back action, one needsto measure the conserved dynamical quantities of the test mass: the momentum orthe energy. Since the momentum is proportional to the speed, the speed meter cantherefore detect gravitational wave without being limited by the radiation-pressurenoise [23].
OutputTest massOptical transfer functionGWsinputnoise closed port sloshingcavity sloshing cavity
Figure 12:
Schematics showing the speed-meter configuration (left) and its block diagram(right).
There are several speed-meter configurations, e.g., the Sagnac interferometer [36,37, 26, 27] and a recent proposed scheme by using different polarizations [38]. InFigure 12, we show one particular variant, which is proposed in [25], by using a sloshingcavity. We can gain a qualitative understanding of how such a scheme allows us tomeasure the speed of the test mass. Basically, the information of test mass position atan early moment is stored in the sloshing cavity, and it is coherently superimposed (butwith a minus sign due to the phase shift in the tuned cavity) with the output of theinterferometer which contains the current test mass position. The sloshing happens ata frequency that is comparable to the detection frequency, and the superposed outputis, therefore, equal to the derivative of the test-mass position, i. e., the speed. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection S t r a i n [ ] Frequency [Hz] S Q L Ideal case without optical loss L o w op ti ca l po w e r H i gh op ti ca l po w e r Figure 13:
Noise spectrum for the speed-meter configuration with two different levels ofcirculating optical power.
The details of this scheme have been presented in [25], in particular the input-output relation which will be used in the numerical optimization. At this moment, wejust show the resulting quantum-noise spectrum for this scheme: S h (Ω) = (tan ζ − K sm ) + 12 K sm (Ω) h (Ω) (28)with K sm (Ω) = 16 ω γP c mcL [(Ω − ω s ) + γ Ω ] . (29)Because K sm has a flat frequency response, by properly choosing the homodyne angle ζ , we can remove the low-frequency radiation pressure noise, and the sensitivity isonly limited by the amount of optical power that we have. This noise spectrum isshown in Figure 13. The low-frequency spectrum has the same slope as the standardquantum limit, which is a unique feature of speed meter. When the optical power ishigh enough, we can surpass the standard quantum limit.One important characteristic frequency for this type of speed meter is the sloshingfrequency ω s , and it is defined as ω s = c (cid:114) T s L L s , (30)where T s is the power transmissivity for the front mirror of the sloshing cavity and L s is the cavity length. To achieve a speed response in the detection band, this sloshingfrequency needs to be around 100 Hz. For a 4 km arm cavity— L = 4000 m and 100 msloshing cavity— L s = 100 m, it requires the transmittance of the sloshing mirror tobe T s ≈
30 ppm . (31) uantum Limits of Interferometer Topologies for Gravitational Radiation Detection ω s only depends on theratio between the transmissivity of the sloshing mirror and the cavity length and wecan therefore increase the cavity length.In addition, it seems that no filter cavity is needed for speed meter configuration,as the radiation pressure noise at low frequencies is canceled. However, such acancellation is achieved by choosing the homodyne detection angle ζ = arctan K sm | Ω → , (32)and a high optical power means a large K sm and therefore ζ deviates from 0 (the phasequadrature), decreasing sensitivity at high frequencies. With frequency-dependentsqueezing, we can reduce K sm , or equivalently, ζ , at low frequencies, which allows usto enhance the high-frequency sensitivity. Similarly, the frequency dependent readoutallows us to cancel the low-frequency radiation pressure noise without sacrificing thehigh-frequency sensitivity by rotating the readout angle to the phase quadrature athigh frequencies. OutputTest massOptical transfer functionGWsinputnoise
Figure 14:
Schematics showing the dual-carrier scheme (left) and its block diagram (right).
In this section, we will introduce the multiple carrier scheme, and in particular, wewill focus on the dual-carrier case as shown schematically in Figure 14. The additionalcarrier field provides us with another readout channel. As these two fields can have avery large frequency separation, we can, in principle, design the optics in such a waythat they have different optical power and see different detuning and bandwidth. Inaddition, they can be independently measured at the output. This allows us to gaina lot flexibilities and effectively provides multiple interferometers but within the sameset of optics.These two optical fields are not completely independent, and they are coupledto each other as both act on the test masses and sense the test-mass motion (shownpictorially by the block diagram in Figure 14). More explicitly, we can look at theinput-output relation for this scheme in the simple case when both fields are tuned uantum Limits of Interferometer Topologies for Gravitational Radiation Detection b ( A )1 b ( A )2 b ( B )1 b ( B )2 = −K A −√K A K B
00 0 1 0 −√K A K B −K B a ( A )1 a ( A )2 a ( B )1 a ( B )2 + √ K A √ K B hh SQL , (33)where we have ignored the uninteresting phase factor e iφ and we have introduced K A = 16 ω ( A )0 γ A P ( A ) c mLc Ω (Ω + γ A ) , K B = 16 ω ( B )0 γ B P ( B ) c mLc Ω (Ω + γ B ) . (34)The term −√K A K B in the transfer function matrix indicates the coupling betweenthese two optical fields, and it comes from the fact that the radiation-pressure noisefrom the first one is sensed by the second one and vise versa.As mentioned earlier, because the frequency separation between these two fieldsis much lager than the detection band, they can be measured independently and givetwo outputs b ( A ) ζ and b ( B ) ζ : b ( A ) ζ = b ( A )1 sin ζ A + b ( A )2 cos ζ A , b ( B ) ζ = b ( B )1 sin ζ B + b ( B )2 cos ζ B . (35)To achieve the optimal sensitivity, we need to combine them with ”optimal” filters C A (Ω) and C B (Ω), obtaining b tot ζ (Ω) = C A (Ω) b ( A ) ζ (Ω) + C B (Ω) b ( B ) ζ (Ω) . (36)In Ref. [28], the authors have shown the procedure for obtaining the optimal sensitivityand the associated optimal filters in the general case with multiple carrier fields. Giventhe input-output relation: b = M a + v h —a simplified vector form of Equation 33, thenoise spectrum that gives the optimal sensitivity is: S h (Ω) = (cid:104) v † M † hd ( M hd M M † M † hd ) − M hd v (cid:105) − , (37)where we have defined: M hd = (cid:20) sin ζ A cos ζ A ζ B cos ζ B (cid:21) . (38)This result is used for our numerical optimization in Section 4. Here we will discuss a special case of the multiple-carrier scheme—the local-readoutscheme, as shown schematically in Figure 15. In this scheme, the second carrier fieldis only resonant in the power-recycling cavity and is anti-resonant in the arm cavity(barely enters the arm cavity). Why we single this scheme out of the general dual-carrier scheme and give it a special name is more or less due to a historic reason.This scheme was first proposed in [28] and was motivated by trying to enhance thelow-frequency sensitivity of a detuned signal-recycling interferometer, which is not asgood as the tuned signal-recycling due to the optical-spring effect. The name “local uantum Limits of Interferometer Topologies for Gravitational Radiation Detection OutputEnd test massOptical transfer functionGWsinputnoise Input test mass
Figure 15:
Schematics showing the local-readout topology (left) and the correspondingblock diagram (right). readout” originates from the fact that the second carrier field only measures the motionof the input test mass (ITM) which is local in the proper frame of the beam splitterand does not contain the gravitational-wave signal. One might ask: “how can werecover the detector sensitivity if the second carrier measures something that does notcontain the signal?” Interestingly, even though no signal is measured by the secondcarrier, it measures the radiation-pressure noise of ITM introduced by the first carrierwhich has a much higher optical power due to the amplification of the arm cavity, asshown schematically by the block diagram of Figure 15. By combining the outputsof two carriers optimally, we can cancel some part of the radiation-pressure noiseand enhance the sensitivity—the local-readout scheme can therefore be viewed as anoise-cancellation scheme. The cancellation efficiency is only limited by the radiation-pressure noise of the second carrier field.To evaluate the sensitivity for this scheme rigorously, one has to treat the inputtest mass (ITM) and end test mass (ETM) individually, instead of combining theminto an single effective mass as we did for those schemes mentioned earlier. One canread [28] for details. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection
4. Numerical Optimization
To arrive at the optimum sensitivity (as defined by Equation 1, our cost function), weuse a simplex numerical optimization to vary the optical parameters for each of thepreviously described topologies. For this optimization, we also take into account thevarious classical noise sources (to be distinguished from the quantum noise). Thesenoise sources are described in Appendix B. C o a ti ng m B r o w n i a n m L . mmmm r e du c ti on A S t r a i n m [ mmmmmmmmmmmm ] Frequencym[Hz] S u s p e n s i on m t h e r m a l m L K A G r a v it y m G r a d i e n t m L mmmm r e du c ti on A S e i s m i c m no i s e Frequencym[Hz]
Highmthermalmnoisemmodel Lowmthermalmnoisemmodel A dv a n ce d m L I GO m C o a ti ng m B r o w n i a n m L mmmm r e du c ti on A S u s p e n s i on m t h e r m a l m L K A G r a v it y m G r a d i e n t m L mmmm r e du c ti on A S e i s m i c m no i s e Figure 16:
Spectra of the high classical noise model (left) and the low classical noise model(right).
The final optimization result critically depends on the cost function. In the literature,optimizations have been carried out by using a cost function that is source-oriented—trying to maximize the signal-to-noise ratio for particular astrophysical sources. Herewe apply a rather different cost function, as shown in Eq. 1, that tries to maximizethe broadband improvement over aLIGO.
For the optimization, we separate the configurations into two groups: (i) the frequency-dependent squeezing (input filtering) group, in which we consider adding input filtercavities to those configurations mentioned in Section 3; (ii) the variational-readout(output filtering) group , in which we consider adding output filter cavities. Notethat for those multiple-carrier schemes, e.g., the local-readout scheme, the number offilter cavities is equal to the number of carrier fields, and the number of optimizationparameters therefore increases proportionally. In real implementations, we mightspecifically design one filter cavity that is able to simultaneously filter several carrierfields with different filtering parameters, and we can then reduce the number of optics.
The optimization result for the high classical noise model was shown at the very beginning (cf. Figure 1). Notice that, in that plot, we did notshow the dual-carrier scheme with both carrier fields resonant in the arm cavities, and uantum Limits of Interferometer Topologies for Gravitational Radiation Detection low classical noise model was shown in Figure 2.It is clear that the general features are identical to the input-filtering one. The onlyprominent difference comes from the low-frequency sensitivities. This is attributableto the susceptibility to loss of the frequency dependent readout scheme, as mentionedearly in Appendix C. Again, we can see that the speed meter and the local-readoutscheme both allow significant improvements at low frequencies.In Appendix A, we have listed the optimal values for the different parameters. S t r a i n F [ FFFFFFFFFFFF ] L ong F s i gn a l F r ec y c li ng F ca v it y S i gn a l -r ec y c li ng F ( A dv L I GO - t yp e ) m F s p ee d F m e t e r L o ca l F r ea dou t F ( du a l F ca rr i e r) FrequencyF[Hz] A dv L I GO FrequencyF[Hz] A dv L I GO SchemesFwithFinputFfiltering SchemesFwithFoutputFfiltering
Figure 17:
Plot showing the spectra for the quantum noise that contributes to the totalnoise shown in Figure 1.
To compare the quantum noise contribution tothe total noise spectrum, we show only the quantum noise spectrum in Figure 17 andFigure 18. It is clear that only at low frequencies do these schemes differ from eachother distinctively. The low-frequency classical noise masks any difference. Therefore,unless significant changes can be made to reduce the low-frequency thermal noise,a sound reasoning—for choosing one advanced configuration over the other as acandidate for upgrade—should be based on the additional complexity involved, asdifferent schemes do not perform drastically different after taking into account thethermal noise of the suspensions and mirrors. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection Frequencyo[Hz] A dv L I GO Frequencyo[Hz] A dv L I GO Schemesowithoinputofiltering Schemesowithooutputofiltering S t r a i n o [ oooooooooooo ] L ong o s i gn a l o r ec y c li ng o ca v it y S i gn a l -r ec y c li ng o ( A dv L I GO - t yp e ) m o s p ee d o m e t e r L o ca l o r ea dou t o ( du a l o ca rr i e r) Figure 18:
Spectra for the quantum noise shown in Figure 2.
5. Future Studies
In the current study, we only cover a few topologies among those that have beenproposed in the literature. To proceed, one approach is to further expand the list ofconfigurations, but this is a rather daunting task given the huge number of possiblecombinations. An alternative that we shall apply in the future is viewing opticaland mechanical components as linear filters, and seeking the answer to the followingquestion: “What is the optimal filter that we should place in between the test mass andthe photodetector such that a specific cost function is minimized or if we know the signalwaveform?”
Similar techniques are employed in the design of electronic circuits andoptimal search algorithms for finding signals in noisy data. The only subtlety is that weare dealing with quantum and classical fluctuations—there are certain constraints onthe filters that one must apply in order to preserve the quantum coherence, especiallyin cases of amplitude filtering.To be concrete, let us look at the structure of the detection process more carefully.The test mass, which contains the GW signal, turns ingoing optical fields into outgoingfields which in turn are detected by the photodetector. In between the test massand the photodetector, the most generic filter we can apply is a four-port filter, asillustrated in Figure 19. The transfer functions of such a four-port filter— T a (Ω), T b (Ω), R a (Ω) and R b (Ω) (cid:107) —are not independent and need to satisfy the Stokes relation dueto energy conservation and time-reversal symmetry. Specifically, if we separate theiramplitude and phase as follows: T a (Ω) = | T a (Ω) | e i φ a (Ω) , T b (Ω) = | T a (Ω) | e i φ b (Ω) ,R a (Ω) = | R a (Ω) | e i ϕ a (Ω) , R b (Ω) = | R b (Ω) | e i ϕ b (Ω) , (39) (cid:107) For simplicity, here we use the sideband picture instead of quadrature, otherwise these transferfunctions will be transfer matrices. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection GWsTest massInputPhoto-detector
Figure 19:
Schematics illustrating the generic four-port filter that can be applied in betweenthe test mass and the photodetector. Here we are considering one sideband frequency Ω; τ arm = L/c is the time delay by the interferometer arm. the Stokes relation dictates the following constraints: | T a (Ω) | = | T b (Ω) | , | R a (Ω) | = | R b (Ω) | , | T (Ω) | + | R (Ω) | = 1 ,e i φ a (Ω) = e i φ b (Ω) , e i ϕ a (Ω)+ i ϕ b (Ω) = − e i φ a (Ω) . (40)In order to obtain the optimal four-port filter given a certain cost function, wecan either (i) parameterize those transfer functions in terms of zeros and poles andoptimize them — this requires a mapping between zeros and poles to the physicalsetup, which is highly nontrivial, or (ii) insert a number of cavities and optimize theparameters — this is more transparent in terms of finding out the physical scheme.As a first attack, we will apply the latter approach, as illustrated in Figure 20. Notonly do we consider input filtering T (Ω) and output filtering T (Ω), we also includethe intra-cavity filtering T (Ω) and T (Ω) — the filters sit inside the signal-recyclingcavity (the sloshing cavity in the speed-meter configuration is one special example ofthe intra-cavity filtering). These filters are different cascades of cavities that can eitherhave fixed mirrors (the passive cavity) or a movable end mirror (the opto-mechanicalcavity). The usual passive optical cavity only allows us to create a frequency-dependent phase shift on the sidebands, or equivalently, frequency-dependent rotationof the amplitude and phase quadratures. By adding control light and allowing the endmirror to be movable, we can also create frequency-dependent amplitude modulation,similar to the ponderomotive squeezer proposed in [39]. Recently such active cavitieswith opto-mechanical interactions have triggered interesting discussion within the GWcommunity, as it allows us to filter the audio-band signal with table-top scale setups.However, to realize it experimentally, the mirror thermal noise needs to be low enoughsuch that the quantum coherence shall not be destroyed. This probably requirescryogenic temperatures which is somewhat challenging to realize. In future numericaloptimization, we will study the influence of thermal noise in the opto-mechanical cavityon the sensitivity.
6. Conclusions
We have optimized the quantum noise spectrum for a few different interferometerconfigurations that are candidates for the 3 rd generation LIGO. In particular, wehave considered the frequency dependent squeezing (input filtering) and frequencydependent readout (output filtering); introducing additional filter cavities either atthe input or the output ports. Limited by thermal noise at low frequencies, thedifference among these configurations is not very prominent. This leads us to the uantum Limits of Interferometer Topologies for Gravitational Radiation Detection GWsInput Arm cavitySignal-recycling cavityPhoto-detector intra-cavity filteringinput filteringoutput filtering
Figure 20:
Schematics illustrating the scheme that we will numerically optimize (top). Eachof these transfer functions corresponds to a cascade of (opto-mechanical) cavities in series(bottom). conclusion that adding one input filter cavity to Advanced LIGO seems to be themost feasible approach for upgrading in the near term, due to its simplicity comparedwith other schemes. If the low-frequency thermal noise can be reduced in the future,the speed meter and the multiple-carrier scheme can provide significant low-frequencyenhancement of the sensitivity. This extra enhancement will, for some low enoughthermal noise, be enough to compensate for the extra complexity.
Acknowledgments
We would like to thank our colleagues in the LIGO Scientific Collaboration’s AdvancedInterferometer Configurations working group for fruitful discussions. R. X. A. issupported by NSF grant PHY-0757058. H. M., H. Y., and Y. C. are supported by NSFgrants PHY-0555406, PHY-0653653, PHY-0601459, PHY-0956189, PHY-1068881, aswell as the David and Barbara Groce startup fund at Caltech.
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Appendix A. Optimal Parameters
Optimal parameters for the configurations with frequency dependent squeeze anglein the high thermal noise model are listed in Table A1. The nominal parameterscommon among different configurations are: m = 50 kg, T PRM = 0 . T ITM = 0 . P (W) T sr T f (ppm) ∆ f (Hz) FOMSignal-recycling 125.0 0.12 249.2 − − − − − Table A1:
Optimal parameters for different configurations with frequency-dependentsqueezing in the high thermal noise model. Here P is the input optical power, T sr is thepower transmittance of the signal recycling mirror, T f is the power transmittance for thefront mirror of the filter cavity, ∆ f is the detune frequency of the filter cavity, and figure ofmerit (FOM) is equal to 10 / C with C the value of the cost function defined in Eq. 1—thelarger the figure of merit is, the better the broadband sensitivity is. The optimal parameters in the high thermal noise model for differentconfigurations with frequency dependent (variational) readout quadrature are listedin the Table A2. The common parameters are the same as those for the frequencydependent squeezing. The transmissivity of the sloshing mirror for the speed meter isequal to 900 ppm.The optimal parameters for different configurations with frequency dependentsqueezing in the low thermal noise model are listed in the Table A3. The commonparameters for different configurations are: m = 150 kg, T PRM = 0 . T ITM = 0 . uantum Limits of Interferometer Topologies for Gravitational Radiation Detection P (W) T sr T f (ppm) ∆ f (Hz) FOMSignal-recycling 125.0 0.1 233.3 − − − − .
001 858.1 − Table A2:
Optimal parameters for different configurations with frequency dependentreadout in the high thermal noise model. low-noise model case—10 dB squeezing with 5% injection loss. The optimal sloshingmirror transmittance for the speed meter is equal to 970 ppm.Configurations P (W) T sr T f (ppm) ∆ f (Hz) FOMSignal-recycling 500.0 0.12 256.9 − − − − . − Table A3:
Optimal parameters for different configurations with frequency-dependentsqueezing in the low thermal noise model.
The optimal parameters for the low-noise model for different configurations withfrequency dependent readout quadrature are listed in the Table A4. The optimalsloshing mirror transmittance for the speed meter is equal to 0.0013.Configurations P (W) T sr T f (ppm) ∆ f (Hz) FOMSignal-recycling 500.0 0.087 247.5 − − − − Table A4:
Optimal parameters for different configurations with frequency dependentreadout quadrature in the low thermal noise model.
Appendix B. Other Noise Sources
In addition to the quantum noise arising from the fluctuations in the ground stateof the electromagnetic field, the sensitivity of the interferometers is also limitedby Brownian thermal noise in the mirror suspensions [40, 41], seismic vibrationspropagating to the mirror [42], terrestrial gravitational fluctuations [43, 44], andBrownian thermal fluctuations of the mirror (and mirror coating) surface [45, 46].In Table B1, we show the physical interferometer parameters used in the optimization. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection M (kg) T mir (K) T sus (K) w beam (cm) φ highcoat × − φ lowcoat × − N N FF aLIGO 40 295 295 5.9 23 4 . . . . Table B1:
Physical parameters used for computing the non-quantum noise sources duringthe configuration optimization. Here M is the mass of a single test mass; T mir and T sus arethe temperature of the mirrors (test masses) and the suspension wire, respectively; w beam isthe average beam spot radius on the arm cavity mirrors; φ highcoat and φ lowcoat are the mechanicalloss angle for the high (low) refractive index materials, respectively; NN FF is the Newtoniannoise subtraction factor achieved by feed-forward cancellation. Appendix C. Optical loss and optimal filter cavity length
It is essential to gain a full understanding—through experiments and numericalmodeling—of how the optical loss scales with the cavity length, and this will determinethe cavity length for achieving the optimal sensitivity. Here we will provide aqualitative estimate of the dependence of sensitivity on the optical loss and the cavitylength, the connection between which is left for future work.
Appendix C.1. Qualitative picture
Given a filter cavity, the detector sensitivity, is affected by the total loss of the cavity E which is equal to the round-trip loss (cid:15) multiplied by the number of round trips N ∼ /T f with T f being the transmittance of the cavity input mirror (assuming atotally reflected end mirror), namely E ≈ (cid:15)T f . (C.1)In addition, since the filter cavity bandwidth γ f needs to be comparable to thedetection bandwidth γ of the interferometer in order to reduce the quantum noise, werequire γ f = cT / (4 L f ) ≈ γ, where L f is the filter cavity length. It follows that T f ≈ γL f c . (C.2)Therefore, the total optical loss is given by: E ≈ c (cid:15) γL f ∝ (cid:15)L f , (C.3)which means that the total optical loss depends on the ratio between the round-triploss and the filter cavity length.If the optical loss were independent of the cavity length, the above scaling wouldimply that the longer the cavity, the smaller the total loss. However, this is usuallynot the case. Due to the roughness of the mirror surface, there will be an opticalloss associated with scattering off the surface. One signifcant contribution comes fromthe “figure error” which corresponds to roughness with a spatial scale larger than uantum Limits of Interferometer Topologies for Gravitational Radiation Detection
33a few millimeters; another is from micro-roughness due to small-scale imperfectionswhich induce large-angle scattering. This scattering critically depends on the beamsize which in turn depends on the cavity length [47]. To be more specific, the beamsize w f for a confocal cavity ¶ is given by: w f = (cid:18) L f λ π (cid:19) / . (C.4)The dependence of scattering loss on the beam size is more involved. Here we onlydiscuss the scattering associated with micro-roughness (the one from “figure errorrequires a numerical simulation and normally does not have a compact analyticalexpression).The corresponding scattering loss for an isotropic surface topologyreads [49]: (cid:15) = (cid:18) πλ (cid:19) (cid:90) f max s f min s S ( f s ) d f s . (C.5)Here f s has the units of m − and is the spatial frequency; f min s is the minimum spatialfrequency, or equivalently, the maximum length scale of the beam, and it is inverselyproportional to the beam size: f min s ≈ ( w f / − (C.6)and the maximum spatial frequency f max s is taken to be 1 /λ . The one dimensionalscattering power spectral density S ( f s ) depends on specific optics that we use. Forinitial LIGO mirrors, the measured power spectral density approximately satisfies [50]: S ( f s ) = 7 × − f − . s . (C.7)In Figure C1, we show how the optical loss depends on the filter cavity length(numerically integrating the above integral). It can be approximated by the followingpower law: (cid:15) | micro − roughness ≈ . L . f . (C.8)In this case, the total optical loss E [cf. Equation C.3] scales as L − . f . This seemsto imply the longer the cavity the better. However, this ignores scattering loss fromfigure errors and also diffraction. In reality, to determine the optimal cavity length,we need to experimentally investigate the dependence of the round-trip optical losson the cavity length. Appendix C.2. Detailed analysis
Here we can provide a more detailed analysis to elaborate on the qualitative picturethat we showed in the previous section concerning the magnitude of the loss. Toaccount for the optical loss of filter cavities carefully, we need to inject vacuumfluctuations at every port of all optics + , summarizing the effect from scattering andabsorption. The simple linear lossy cavity is illustrated in Figure C2, and we canderive its input-output relation from which we can determine quantitatively how theloss influences the quantum coherence of the squeezed light (for input filtering) or theoutput quadratures (for output filtering). ¶ A confocal cavity allows the minimal beam size on the mirror (see e.g. Chapter 19 in [48]). + Note that in the literature, normally one introduces a so-called lossy mirror to account for all theoptical loss and assumes other mirrors are lossless. This works generally, but may fail when theoptical path is complicated. Also there is some ambiguity in determining the loss for such a effectivemirror. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection L f @ m D e @ pp m D Figure C1:
Dependence of scattering loss on the length of the filter cavity due to micro-roughness of the mirror surface. Figure errors at large spatial frequencies are neglected here.
Figure C2:
Figure illustrating the propagating fields in a single lossy filter cavity. The reddots are the lossy ports where vacuum fluctuations n α s that are uncorrelated with the mainfields α s enter ( α s = a, b, c, d, f, g ) the optical paths. To derive the input-output relation, we use the continuity condition of the fields,which goes as follows: b = (1 − (cid:15)/ − (cid:112) R f a + (cid:112) T f d ) + (cid:112) ( (cid:15)/ − (cid:15)/ − (cid:112) R f n a + (cid:112) T f n d ) + (cid:112) (cid:15)/ n b , (C.9) c = (1 − (cid:15)/ (cid:112) R f d + (cid:112) T f a ) + (cid:112) ( (cid:15)/ − (cid:15)/ (cid:112) R f n d + (cid:112) T f n a ) + (cid:112) (cid:15)/ n c , (C.10) f = M rot c , (C.11) g = (1 − (cid:15)/ f + (cid:112) ( (cid:15)/ − (cid:15)/ n f + (cid:112) (cid:15)/ n g , (C.12) d = M rot g . (C.13)Here α = ( α , α ) T is the vector for the amplitude and phase quadratures; we have uantum Limits of Interferometer Topologies for Gravitational Radiation Detection (cid:15)/ M rot is the rotation matrix of the quadratures due to free propagation in vacuum, andit is given by M rot = e i Ω τ f (cid:20) cos ∆ f τ f − sin ∆ f τ f sin ∆ f τ f cos ∆ f τ f (cid:21) (C.14)with τ f = L f /c being the time delay and L f being the filter cavity length. From theseequalities, we can obtain the corresponding input-output relation between a and b .Given small loss: η (cid:28)
1, we can keep the input-output relation up to the lowest orderof (cid:15) , and get b = (cid:104) − (cid:112) R f I + T f ( I − (cid:112) R f M ) − M (cid:105) a + (cid:113) (cid:15)T f / I − (cid:112) R f M ) − ( M n c + M rot n f + M rot n g + n d ) − (cid:113) (cid:15)R f / n a + (cid:112) (cid:15)/ n b . (C.15)The term on the first line gives the cavity response in the lossless case—the quadraturerotation has a significant frequency dependence; the four terms on the second line arethe losses inside the cavity, and they are amplified by the cavity around the detuningfrequency ∆ f ; the two terms on the last line are the losses outside the cavity, and theyare not amplified by the cavity response.To gain an intuitive understanding, we can use the fact that the sidebandfrequency Ω, the cavity bandwidth γ f and the detune frequency ∆ f are much smallerthan the free spectral range, namelyΩ τ f , γ f τ f , ∆ f τ f (cid:28) , (C.16)with τ f ≡ L f /c being the propagation time. We can therefore make a Taylor expansionin terms of series of these small quantities, and obtain (in the sideband picture): b (Ω) = − Ω + ∆ f − i ( γ f − γ (cid:15) )Ω + ∆ f + i ( γ f + γ (cid:15) ) a (Ω) − i √ γ f γ (cid:15) Ω + ∆ f + i ( γ f + γ (cid:15) ) [ n c (Ω) + n d (Ω) + n f (Ω) + n g (Ω)] − (cid:112) (cid:15)/ n a (Ω) + (cid:112) (cid:15)/ n b (Ω) . (C.17)Here we have defined the effective bandwidth due to loss: γ (cid:15) ≡ c (cid:15)L . (C.18)As we can see, the optical loss has two effects on the performance of the filter cavity:(i) introducing uncorrelated vacuum fluctuation that degrades the sensitivity; (ii)broadening the cavity bandwidth that prevents the use of very short cavity for filteringwith desired frequency band. For a round-trip loss of order of tens of ppm, (cid:15) ∼
30 ppm,the bandwidth due to loss can be estimated as γ (cid:15) ≈ π ×
100 s − (cid:18) L
15 m (cid:19) , (C.19)where the loss limits the bandwidth to be larger than 100Hz for a cavity length around15m. For the numerical optimization to be discussed, we choose the filter cavity to beof the order of hundreds meter, and in this case, the bandwidth is mainly determinedby the transmittance T f of the input mirror, or equivalently by γ f . The loss is mainly uantum Limits of Interferometer Topologies for Gravitational Radiation Detection − at high frequencies, and the total effective loss at lowfrequencies is given by E ≡ γ f γ (cid:15) ( γ f + γ (cid:15) ) ≈ (cid:15)T f (C.20)for filter cavity bandwidth γ f (cid:29) γ (cid:15) , which recovers what has been shown inEquation C.1. Appendix D. Tolerance to parameter variations in the filter cavity
Here we compare input filtering and output filtering in terms of tolerance to parameteruncertainties in the filter cavity. The outline of this section goes as follows: (i) we firstanalyze the deviation from the ideal frequency-dependent quadrature rotation due toparameter uncertainties of the filter cavity; (iii) we then show how these uncertaintiesinfluence the sensitivity for both input filtering (frequency-dependent squeezing) andoutput filtering (frequency dependent readout).For a single filter cavity, the output quadratures ( b , b ) are related to the inputquadratures ( a , a ) by [cf. Equation 12]: (cid:20) b b (cid:21) = e i ( α + − α +)2 (cid:20) cos α + + α − − sin α + + α − sin α + + α − cos α + + α − (cid:21) (cid:20) a a (cid:21) , (D.1)with α ± is defined as e i α ± ≡ iγ ∓ Ω − ∆ iγ ± Ω − ∆ , (D.2)where ∆ and γ are the detuning frequency and bandwidth of the filter cavity,respectively. If we have a chain of N filter cavities, the total rotation angle φ tot is given by φ tot (Ω) = N (cid:88) i =1 α ( i )+ (Ω) + α ( i ) − (Ω)2 (D.3)with α ( i ) from the i -th filter cavity. As shown in the Appendix A of [25], byproperly choosing the parameters for each cavity, one can realize any desired frequency-dependent rotation angle, as long as tan ( φ tot (Ω)) is a rational function.Suppose both the detuning frequency and bandwidth have uncertainties δ ∆ ( i ) and δγ ( i ) : ∆ ( i ) → ∆ ( i ) + δ ∆ ( i ) , γ ( i ) → γ ( i ) + δγ ( i ) . (D.4)This will induce a change δφ tot in the total rotation angle of δφ tot = N (cid:88) i =1 δα ( i )+ + δα i − N (cid:88) i =1 − γ ( i ) f ( i )+ (Ω) δ ∆ ( i ) + 2∆ ( i ) f ( i ) − (Ω) δγ ( i ) (D.5)where f ( i ) ± (Ω) ≡ γ ( i )2 + ∆ ( i )2 ± Ω (cid:2) γ ( i )2 + (∆ ( i ) − Ω) (cid:3) (cid:2) γ ( i )2 + (∆ ( i ) + Ω) (cid:3) . (D.6) uantum Limits of Interferometer Topologies for Gravitational Radiation Detection (cid:46) γ ( i ) and for γ ( i ) ∼ ∆ ( i ) , we have δφ tot | low freq . = N (cid:88) i =1 − δ ∆ ( i ) ∆ ( i ) + δγ ( i ) γ ( i ) = N (cid:88) i =1 − δ ∆ ( i ) ∆ ( i ) + δT ( i ) T ( i ) . (D.7)Basically, the total rotation angle is just modified by the relative error in the detuningand bandwidth. In reality, such a relative error can be controlled to a level of 10 − .This means that δφ tot ∼ − N . (D.8)Now we look at how such an error in the quadrature rotation influences thesensitivity for input and output filtering. We focus on the tuned configuration with theinput-output relation shown in Equation 6. For input filtering , the resulting spectraldensity is given by (cf. Equations 46-48 in [16]): S h (Ω) = h (cid:18) K + K (cid:19) (cosh 2 r − cos[2( ϕ + Φ)] sinh 2 r ) . (D.9)with Φ = arccot K . When we choose the optimal squeezing angle ϕ opt = − Φ(Ω) = − arccot K (Ω) , (D.10)we obtain S opt h (Ω) = h (cid:18) K + K (cid:19) e − r . (D.11)The uncertainties in the parameters of the filter cavity will make ϕ deviate fromthe optimal squeezing angle ϕ opt , i.e., ϕ = ϕ opt + δϕ resulting in S h = S opt h + δS h .Specifically, we have δS h (Ω) = h (cid:18) K + K (cid:19) sinh(2 r ) δϕ . (D.12)At low frequencies, we have δS h (Ω) | low freq . ≈ h sinh(2 r ) δφ / K .For output filtering , the spectral density is given by [cf. Equations 56-58 in [16]]with frequency-independent phase squeezing: S h (Ω) = h K (cid:2) e − r + (tan ζ − e r K ) (cid:3) . (D.13)When we choose the optimal frequency-dependent homodyne detection angle ζ (Ω) = arctan( e r K ) , (D.14)we obtain the optimal sensitivity that is only limited by the shot noise: S opt h (Ω) = h K e − r . (D.15)Similarly, variation of the readout quadrature due to uncertainties of the filter cavityparameters will induce the following change in the sensitivity: δS h (Ω) = h K (1 + e r K ) δζ . (D.16)Since K (cid:29) δS h (Ω) from parameter variations is much larger for the output filteringcase than for input filtering. uantum Limits of Interferometer Topologies for Gravitational Radiation Detection Appendix E. Tolerance to anti-squeezing
Here we analyze the influence of anti-squeezing (an impure squeezed state withimpurity from classical noise during its preparation) on input and output filtering.In contrast to a pure squeezed state, the determinant of the noise spectral densitymatrix for an a state with some anti-squeezing [cf. Equation 9] is: S S − S S = ξ ≥ , (E.1)This determinant is unity for a pure squeezed state. One example of anti-squeezingis S = ξ e r , S = S = 0 and S = e − r , which means that fluctuations inthe phase quadrature are still the same as the usual pure phase squeezed state butthe amplitude quadrature is larger than e r by a factor of ξ . In the ideal case withperfect filter cavities and no loss, such anti-squeezing will not influence the optimalsensitivity. However, when the loss and the above mentioned uncertainties in the filtercavity detuning and bandwidth are taken into account, the anti-squeezing will playan important role, and in particular, will degrade the low-frequency sensitivity.Specifically, for the input filtering, we have δS h (Ω) = h (cid:18) K + K (cid:19) ( e − r + e r ξ ) δϕ , (E.2)At low frequencies, we can approximate this as δS h (Ω) ∼ h K e r ξ δϕ /
2. For theoutput filtering, we have δS h (Ω) = h K (1 + ξ e r K ) δζ ..