A broadband high frequency laser interferometer gravitational wave detector
aa r X i v : . [ a s t r o - ph . I M ] J u l A broadband high frequency laser interferometer gravitational wave detector
Meng-Jun Hu
1, 2, ∗ and Yong-Sheng Zhang
1, 2, † Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei 230026, China (Dated: July 31, 2020)The gravitational wave detector of higher sensitivity and greater bandwidth is required for fu-ture gravitational wave astronomy and cosmology. Here we present a new type broadband highfrequency laser interferometer gravitational wave detector utilizing polarization of light as signalcarrier. Except for Fabry-Perot cavity arms we introduce dual power recycling to further amplifythe gravitational wave signals. A novel method of weak measurement amplification is used to am-plify signals for detection and to guarantee the long-term run of detector. Equipped with squeezedlight, the proposed detector is shown sensitive enough within the window from 100Hz to severalkHz, making it suitable for the study of high frequency gravitational wave sources. With zero-areaSagnac topology, the detector has the potential to realize quantum non-demolition measurement.The detector presented here is expected to provide an alternative way of exploring the possibleground-based gravitational wave detector for the need of future research.
INTRODUCTION
The direct detection of gravitational waves from twobinary black holes by Advanced Laser InterferometerGravitational-wave Observatory (LIGO) in 2016 opensnew era for astronomy and cosmology [1]. At the endof the second observing run (O2), a total of 11 confidentgravitational waves events have been confirmed by LIGOand Virgo collaborations [2]. Among these events, thedetection of binary neutron star inspiral GW170817 [3]with a follow-up electromagnetic identification [4–8] sig-nificantly advances the development of multi-messengerastronomy [9]. In order to fulfill the requirements of re-search in next decades, the ground-based gravitationalwaves detection need to follow two development pathssimultaneously. One path is to build a global detectionnetwork such that more detections with greater accuracyare obtained. A preliminary detection network composedof LIGO, Virgo, GEO600 and KAGRA has been formed[10], in which more detectors will join in the future. Theanother path is to improve detection sensitivity by up-grading current detectors or building more advanced de-tectors [11]. New detection methods and technologiesthus have to be proposed and used for the next genera-tion gravitational wave detection.The essence of current ground-based gravitationalwave detectors is a modified Michelson interferometerin which phase differences between its two orthogonalarms caused by gravitational wave strain is measured[12]. With kilometer-scale Fabry-P´erot arms and appli-cation of other advanced technologies e.g., power recy-cling, the strain sensitivity of detectors like LIGO andVirgo in its current detection bandwidth is mainly lim- ∗ [email protected] † [email protected] ited by quantum noise [13]. quantum noise manifests intwo ways: radiation pressure noise caused by fluctuationsof photon flux impinging on mirror is dominant in lowerfrequency, while shot noise caused by statistical fluctua-tions of photons is dominant in higher frequency. It is C.M. Caves who first pointed out that both noises resultfrom the vacuum fluctuations of dark port in interferom-eter and proposed the concept of squeezed vacuum stateto surpass the quantum noise limitation [14].In the recent third observing run (O3), the squeezedlight was injected into the dark port of interferometersby both LIGO and Virgo, improving the sensitivity ofdetectors to signals by up to 3dB in shot-noise limitedfrequency range, which leads to an obvious boost for de-tection range and rate[15, 16]. Further application offrequency-dependent squeezing for brand-band reductionof quantum noise is also possible [17–19]. Besides the di-rect impact on the ability to detect astrophysical sources,the enhanced sensitivity in high frequency is also of im-portance for research of detected sources. For example,sky location is essential to multi-messenger astronomyand its accuracy highly depends on high frequency sensi-tivity of detectors [20]. Information of states of neutronstars e.g., tidal deformability and interior structure, isalso carried by high frequency gravitational wave signals[21]. In addition, gravitational waves emitted by sourcesrelating to them such as gravitational collapse, rotationalinstabilities and oscillations of the remnant compact ob-jects are expected to be within high frequency window[22–35]. There is thus a strong motivation for the de-velopment of ground-based gravitational wave detectorswith high frequency detection window [36–40].In this Letter, we propose a new type broadband laserinterferometer gravitational wave detector with high fre-quency window, in which polarization of light is utilizedas signal carrier. The antenna of detector is thus a polar-ization Michelson interferometer with Fabry-Perot cavity FIG. 1.
Schematic diagram of proposed brandband high frequency laser interferometer gravitational wavedetector.
The basic configuration of the detector can be viewed as a polarization Michelson interferometer nested in anasymmetry Mach-Zehnder interferometer. BS1 is a 50:50 beam splitter, while BS2 is slight deviation from 50:50 to realize weakmeasurement amplification. Except Fabry-Perot cavity arms, FPM1 and FPM2 with the same transmissivity and reflectivity areused as dual power recycling cavity to amplify gravitational-wave signals further. The input light is prepared in the polarizationstate | + i = ( | H i + | V i ) / √ | H i , | V i represent horizontal and vertical polarization state respectively. Gravitational wavesignals are encoded in the polarization state of light and can be extracted via a polarization analyzer placed in the dark port ofBS2. Squeezed light can be injected into the dark port of BS1 and the Michelson topology can be replaced by zero-area Sagnactopology to realize quantum non-demolition measurement. arms. In order to significantly reduce the shot noise inhigh frequency range, dual power recycling is introduced.A novel method called weak measurement amplification[42] is adopted to amplify signal for detection and toguarantee the long-term run of the detector. The calcula-tion of quantum noise spectrum shows this detector withcurrent technology is sensitive enough to gravitationalwaves within the range from above 100Hz to several kHz,making it suitable for the study of high frequency gravita-tional wave astronomy and cosmology. The possibility ofusing squeezed light and quantum non-demolition mea-surement to surpass the quantum standard limit in pro-posed detector are also discussed. Kip Throne ever saidthat “ As experimental gravity pushes toward higher andhigher precision, it has greater and greater need of newideas and technology from the quantum theory of mea-surement, quantum optics, and other branches of physics,applied physics, and engineering ” [41].
DESCRIPTION OF THE DETECTOR
The schematic diagram of proposed laser interferome-ter gravitational wave detector is shown in Fig. 1. Thebasic optical configuration of detector can be consid-ered as a polarization Michelson interferometer nestedin an asymmetry Mach-Zehnder interferometer in whichthe second beam splitter BS2 is not strictly 50:50 buthas slight deviation to realize weak measurement ampli-fication (see Appendix A for details). The polarizationMichelson interferometer with Fabry-Perot cavity arms isplaced in the middle part of another Fabry-Perot cavityconsists of mirrors FPM1 and FPM2. The FPM1 andFPM2 have the same reflection and transmission coef-ficients such that the input light from BS1 will totallyoutput from FPM2 under resonance. Since the cavitysupports two modes of polarization, we call it dual powerrecycling, which is different from the standard dual recy-cling Fabry-Perot Michelson interferometer configurationused in LIGO/Vorgo. In order to extract signals encodedin the polarization state, light that come out from thedark port of BS2 is measured via polarization analyzerin the circular basis. Most of light in the LIGO detectoris reflected back toward laser source, while in our case itcomes out of bright port of BS2 and thus the parasiticinterference in the input chain is eliminated. The highlyresemblance of optical configuration to LIGO/Virgo de-tectors implies that many advanced technologies devel-oped in LIGO/Virgo can be directly used in proposeddetector.When gravitational waves passing through the detec-tor, a tiny changes of relative length is introduced andthe signal will be first amplified by Fabry-Perot cavitiesin two interferometer arms. The cavity can be consideredas a reflection mirror with equivalent reflection coefficientof r F P ( L ) = r IT M + e − ik · L r IT M e − ik · L , (1)where r IT M represents the reflection coefficient of inputtest mass and L is the cavity length. Under the resonancecondition of e − ik · L res = −
1, we have r F P ( L res ) = − L res causes r F P ( L res + δL ) ≈ − e − ik · G arm · δL , (2)where we have omitted the higher order terms and G arm = (1 + r IT M ) / (1 − r IT M ) is the gain of arm cav-ity. The phase of reflection light caused by δL underresonance thus can be significantly amplified by choosing r IT M properly.We now consider the input-output relation of dualpower recycling cavity consist of FPM1 and FPM2. Theinput light is produced in linear polarization | + i =( | H i + | V i ) / √ | H i and | V i represent horizontaland vertical polarization state respectively. In the cavity,photons with | H i or | V i state experience different pathsas shown in Fig.1. We consider the input-output rela-tion of photons with independent polarization state firstand the final state of output photons can be obtained ac-cording to superposition law of optical field. Under thestationary condition, the light field within and outputcavity are determined by E MH/V = t · E inH/V − r r F P ( L e/n ) · E MH/V e ik · l + l ) ; E outH/V = t · r F P ( L e/n ) · E MH/V e ik ( l + l ) , (3)where t, r are coefficients of transitivity and reflectivityof FPM1/FPM2, L e/n represent length of east or northarms in Fig.1, l labels the distance between FPM1 andITM and l labels the distance betweem ITM and FPM2.Choosing proper l , l such that e ik · l + l ) = −
1, weobtain the equivalent transmission coefficient of FPM1- FPM2 cavity as t H/V = it · r F P ( L e/n )1 − r r F P ( L e/n ) . (4)A tiny deviation from resonance L e/n = L res results in t H/V ( L res + δL e/n ) ≈ − ie − ik · G P · G arm · δL e/n , (5)where G dp ≡ (1 + r ) /t gives the gain of dual power re-cycling. The signal of gravitational waves h ( t ) propagat-ing perpendicular to detector introduces relative lengthchange of ∆ L = δL e − δL n = h ( t ) L . According to su-perposition rule, the polarization state of output lightreads | ϕ i = 1 √ | H i + e iθ ( t ) | V i ) , (6)where θ ( t ) = G dp · G arm · k · h ( t ) L and global phase isomitted. The detector equipped with Fabry-Perot cavityarms and dual power recycling thus amplify gravitationalwave signals by the factor of G dp · G arm . It should benoted that power recycling used in LIGO/Virgo detectorsenhances only input light power rather than signal itself.In order to catch the possible gravitational wavesevents, detectors need to run from several months toyears in practical case. It is thus impossible to extractsignals directly at the output port of FPM2. The dif-ficulty, however, can be overcome by using weak mea-surement amplification (see Appendix for details). Inproposed detector as shown in Fig. 1, it is realized bya simple Mach-Zehnder interferometer consisting of BS1and BS2. After passing through BS1, the state of pho-tons is described as | Ψ i i = ( r | d i + t | u i ) ⊗ | + i , (7)where r , t are coefficients of reflection and transmissionof BS1 satisfying | r | + | t | = 1 and | d i , | u i representpath state of down and up arm respectively. The gravita-tional wave signals are encoded in the polarization stateof photons flying along up arm. Since global phase canbe compensated in the down arm, the state of photons,before arriving at BS2, reads | Ψ f i = r | d i ⊗ | + i + t | u i ⊗ | ψ i , (8)where | ψ i is shown in Eq. (6). When only photonsout the down port of BS2 are considered, path state | φ i = r | d i + t | u i is post-selected, where r , t are coef-ficients of reflection and transmission of BS2. The polar-ization state of post-selected photons, in the first orderapproximation, becomes | ψ i = 1 √ P h φ | Ψ i f = 1 √ | H i + e i Θ( t ) | V i ) . (9)Here P = |h φ | Ψ i f | = ( r r + t t ) is the successfulprobability of post-selection and Θ( t ) is determined bythe formula oftanΘ = sin θ cos θ + r r /t t = θ r r /t t ≡ Aθ, (10)where θ ≪
1. Choosing r , t , r , t properly such that r r + t t → A > θ isamplified. The amplification of signal is at the cost oflow probability of success P = ( t t ) /A . In proposeddetector, BS1 is set that t = r = 1 /
2. Choosing r = sin ( π/ δ ) , t = cos ( π/ δ ) for BS2 and no-tice that π/ A = 1 /δ . The degree of deviation of BS2 from 50:50beam splitter determines the ability of amplification.Weak measurement amplification is of importance inproposed detector for guaranteeing dark port detectionsuch that long-term run of detector is possible. Withcurrent technology, A = 1 /δ ≈ is available andthe measured light intensity will be reduced to aboutmillionth of input light from laser source. In addition,amplified signals is more robust to the calibration er-ror of polarization analyzer. A proof of principle exper-imental demonstration of weak measurement amplifica-tion has already been given [43]. Amplified signals Θ( t )is easily extracted in the circular basis of polarization {| R/L i = ( | H i ± i | V i ) / √ } as h ψ | ˆ σ | ψ i = I − I I + I = sinΘ , (11)where ˆ σ ≡ | R ih R | − | L ih L | and I , I are light intensitiesof two detection ports. The measured quantity is directlyproportional to signals, while in Advanced LIGO the DCreadout is required [44]. QUANTUM NOISE ANALYSIS
The noise spectrum determines ultimately the sensitiv-ity and bandwidth of gravitational-wave detectors. Ex-cept quantum noise, there are various of other noisessource e.g., thermal noise, seismic noise, coating noiseetc. With current state-of-art technologies, quantumnoise has been the dominant noise in LIGO detectorsand thus we only focus on it here.Quantum noise consists of radiation pressure noise andshot noise, they are dominant in different frequency do-main. Radiation pressure noise takes responsibility forlow frequency performance of detector, while shot noisedetermines the high frequency performance. Intuitively,radiation pressure noise is caused by photons impingingon mirrors and shot noise is due to counting statistics,essentially, however, they originate from quantum fluctu-ations of vacuum. Two different approaches, traditional[45] or full-quantum [46], can be used to calculate the
FIG. 2.
Quantum noise strain sensitivity of proposeddetector vs LIGO detector.
Parameters are chosen as λ L = 1064nm , P = 125W , L = 4km , M = 40Kg , G arm =150 , G dp = G P = 50. Signal recycling used in LIGO detectoris not considered here. The green line is the case that squeezedlight is used in proposed detector and purple line representsthe possible case that proposed zero-area Sagnac topologydetector equipped with squeezed light. A 3dB squeezing isused here. quantum noise spectrum, which of course give the sameresult. The calculation of quantum noise in proposed de-tector is very similar to that in LIGO detector except thatpolarization of photons should be taken into considera-tion. Notice that the minimal detectable amplified phaseΘ is proportional to 1 / √ P N with P is the successfulprobability of post-selection and N represents photonnumbers of input light during observation time. Sincethe response function of Fabry-Perot cavity arm is thesame in LIGO detector [45], the strain sensitivity due toquantum noise in our detector reads S / n ( f gw ) | shot = s π ~ λ L ct t P · πG dp · G arm · L · C ( f gw ) S / n ( f gw ) | rad = s ~ t P πλ L c · √ πG dp · G arm (2 πf gw ) · M · L · C ( f gw ) , (12)where C ( f gw ) is the frequency response function C ( f gw ) = sinc(2 πf gw L/c ) p G arm / · [1 − cos(4 πf gw L/c )] (13)and the total strain sensitivity is S / n ( f gw ) = p S n ( f gw ) | shot + S n ( f gw ) | rad . When f gw ≪ c/ πL ≈ C ( f gw ) reduces to the familiar form of1 / p f gw /f p ) with f p is the pole frequency. Weakmeasurement amplification does not improve strain sensi-tivity of detector. The intriguing thing is that it providesus a way to extract signals without need to detect all thephotons but in the meantime it maintains the sensitivity.The strain sensitivity of proposed detector due toquantum noise is shown in Fig. 2 with the LIGO de-tector as a comparison. The case of signal recycling usedin LIGO detector is not shown here since it broadens thebandwidth with sensitivity sacrifice in some frequencydomain. In both cases, the proposed detector has bet-ter sensitivity in the higher frequency area from above100Hz to several kHz. The improvement is due to the useof dual power recycling with gain G dp as clearly shownin Eq. (12). In fact, if G dp is replaced by p G p in Eq.(12) with G p is the gain of power recycling, we obtainthe strain sensitivity of LIGO detector without signalrecycling. With current technology it is a reliable as-sumption that the magnitude of G dp can be comparableto G p . The squeezed light, which has already been usedin LIGO/Virgo detectors, can also be equipped with pro-posed detector. Contrary to LIGO detector, the origin ofquantum noise in proposed detector results from vacuumfluctuations of the dark port of BS1 (see Appendix B fordetails). The location separation of squeezed light injec-tion from signal detection as shown in Fig. 1 is obviouslybeneficial for better squeezing quality and control. Along-term stable 3dB squeezing has already been achiev-able [15, 16], which further improves the strain sensitivityof high frequency and widen the bandwidth of the pro-posed detector as shown in Fig. 2. With current state-of-art squeezing technology [47–49], 10 dB squeezing inproposed detector seems plausible in practice.. The ap-plication of frequency-dependent squeezing is also possi-ble to reduce quantum noise in low frequency.Since gravitational wave signals are extremely weak,the behaviour of test mass is actually governed byquantum theory in which Heisenberg uncertainty prin-ciple sets a minimum noise called standard quantumlimit(SQL) [50] S / SQL = p ~ /M / (2 πf gw L ) ≈ . × − /f gw . The application of squeezed light discussedabove is one way to beat the SQL. Another way is thequantum non-demolition measurement (QNM) [51], withwhich detectors will only be limited by shot noise. Laserinterferomter with zero-area Sagnac topology configura-tion as a speed meter is considered one of promising waysto realize QNM of gravitational wave signals [52–54]. Theproposed detector can be naturally redesigned with zero-area Sagnac topology as shown in block diagram of Fig.1. The application of squeezed light, of course, reducesfurther the quantum noise of a QNM detector. It is be-lieved that technical issues will not be limiting factor andthe QNM detectors shall be next generation ground-basegravitational-wave detectors [55]. DISCUSSION AND CONCLUSION
Compared with current LIGO/Virgo detectors, theproposed detector has three advantages of technical im-plementation mainly due to the application of weak mea-surement application. The first one is the spatial sepa-ration of squeezing injection and detection port. Thesecond one is that most of light coming out of detection system rather reflected back toward to the light sourceand the third one is the extraction of signals encodedin the polarization of photons without need of DC read-out. On the other hand, however, it requires extremehigh quality of manipulating polarization optics, whichis a big technical challenges. Detailed theoretical analy-sis and prototype experiment are needed for performanceassessment in the next step.In conclusion, a new type broadband laser interferom-eter gravitational-wave detector is proposed. Except forFabry-Perot cavity arms, dual power recycling is intro-duced for the amplification of signals that encoded in thepolarization of photons. A novel method weak measure-ment amplification is used to amplify signals for dark portdetection such that long-term run of detector is guar-anteed. Squeezed light can be injected in the detectorfor further reduction of quantum noise. With zero-areaSagnac topology, the detector has the potential to real-ize quantum non-demolition measurement. The proposeddetector is shown sensitivity enough in the high frequencywithin the range from above 100Hz to several kHz, mak-ing it suitable for the study of high frequency sourcessuch as gravitational collapse, merger of binary neutronstars, rotational instabilities and oscillations of the rem-nant compact objects. With this kind of detector join thecurrent global detection network, not only the detectionrange and rate can be increased but also the accuracyof sky location, which is essential to multi-messenger as-tronomy, can be significantly improved. Application ofproposed detector in fundamental physics such as prob-ing the Planck scale [56, 57] and axion dark matter [58]seems also promising.The authors thanks Jinming Cui, Tan Liu and ShuaiZha for helpful discussions. This work was supported bythe National Natural Science Foundation of China (No.11674306 and No. 61590932), the Strategic Priority Re-search Program (B) of the Chinese Academy of Sciences(No. XDB01030200) and National key R & D program(No. 2016YFA0301300 and No. 2016YFA0301700). Inmemory of those brave people who protect the world dur-ing the outbreak of COVID-19.
Appendix A: Weak Measurement Amplification
In this section, we will give a detailed introduction ofweak measurement amplification that is essential to theproposed gravitational wave detector. The concept ofweak measurement is first proposed by Aharonov, Albertand Vaidman (AAV) [59], which focuses on disturbing asystem as small as possible so that the state of systemwould not collapse after measurement.Consider a two-level system initially prepared in thesuperposition state of | ψ i i s = α | i s + β | i s with | α | + | β | = 1. For simplicity, we choose a qubit state aspointer with initial state | i p . The initial state of thecomposite system is thus | Ψ i i sp = | ψ i i s ⊗ | i p . Theinteraction Hamiltonian usually takes the form of vonNeumann-type as ˆ H = g ˆ A ⊗ ˆ σ y , where ˆ A ≡ | ih | − | ih | is the observable of system, ˆ σ y is Pauli operator acting onpointer and g represents the coupling between system andpointer. According to quantum measurement theory, thesystem and pointer are entangled after interaction andthe state of the composite system becomes | Ψ f i sp = e − ig ∆ t ˆ A ⊗ ˆ σ y | Ψ i i sp = α | i s ⊗ e − iθ ˆ σ y | i p + β | i s ⊗ e iθ ˆ σ y | i p = α | i s (cos θ | i p + sin θ | i p ) + β | i s (cos θ | i p − sin θ | i p ) , (14)where ~ ≡ θ = g ∆ t . Different eigenstates of ob-servable ˆ A cause different rotation of pointer in Blochsphere after interaction. When θ = π/
4, the two pointerstates become |±i p = ( | i p ± | i p ) / √ {| + i , |−i} we can definitely know thestate of system after measurement, which corresponds toprojective measurement. In general cases that θ = π/ θ ≪ | ψ f i = γ | i + η | i after system-pointer interaction. The state of pointer, after post-selection, becomes | ϕ i p = 1 N h ψ f | Ψ f i sp = 1 N h ψ f | e − iθ ˆ A ⊗ ˆ σ y | ψ i i s | i p , (15)where N is normalization factor that N = |h ψ f | Ψ f i sp | .Since θ ≪ | ϕ i p = h ψ f | ψ i i N e − iθ h ˆ A i w ˆ σ y | i p . (16)Here so-called weak value of observable ˆ A is defined as h ˆ A i w = h ψ f | ˆ A | ψ i i / h ψ f | ψ i i . Intuitively, post-selection ofthe system causes about θ h ˆ A i w rotation of pointer onBloch sphere. When proper measurement is performedon pointer, information about θ or weak value < ˆ A > w is obtained. Specifically, we have h ˆ σ + i p = 2 θ Re h ˆ A i w , h ˆ σ R i p = 2 θ Im h ˆ A i w , (17)where ˆ σ + ≡ | + ih + | − |−ih−| , ˆ σ R ≡ | R ih R | − | L ih L | with |±i = ( | i ± | i ) / √ , | R/L i = ( | i ± i | i ) / √ h ψ f | ψ i i approaches to zero, h ˆ A i w becomes ex-tremely large, signal θ is thus amplified. This is, of course, at the cost of extremely low success probabilityof |h ψ f | ψ i i| . Weak value amplification (WVA) has beenextensively studied since the observation of the spin halleffect of light [60]. Due to the low successful probabil-ity of post-selection in the WVA, controversy on whetherWVA measurement outperforms conventional measure-ment in parameter estimation has been last for decades.In most precision measurement experiments e.g., gravi-tational wave detection we actually care only about sen-sitivity, which cannot be simply improved by amplifica-tion. The fact that WVA has the potential to magnifyultra-small signals to the measurable level undoubtedlymakes it useful. This is because the measurement appa-ratus in practice are limited to finite precision. The mostvivid example is microscope, which is used to distinguishtwo spots that are invisible to human eyes. Microscopeamplifies the spots so that we can direct see them for dis-tinguish. However, the resolution of microscope is deter-mined by scattering limit of light. Amplification helps usto distinguish objects with eyes but will not improve theresolution. More works related to weak measurementsand its application can be found in references [61–66]and references there.Based on above discussion, application of WVA togravitational wave detection will not improve the sen-sitivity of detector. Besides, how to amplify general lon-gitudinal phase signal as in gravitational wave detectionvia WVA has not been resolved completely. We will showhere that both of concerns can be eliminated. It shouldbe realized that the definition of weak value is due to thechoice of von Neumann-type interaction. Without thisrestriction we consider the control-rotation evolutionˆ U = | i s h | ⊗ ˆ I p + | i s h | ⊗ ( | i p h | + e iθ | i p h | ) (18)with θ be the phase signal to be measured. Choosingthe initial state of pointer as | + i p then the state of thecomposite system, after interaction, becomes | Ψ f i sp = α | i s ⊗ | + i p + β | i s ⊗ ( | i p + e iθ | i p ) / √ . (19)When the system is in state of | i s nothing happens topointer, while in state of | i s the state of pointer is ro-tated θ along the equator of Bloch sphere. With post-selection of the system into state | ψ f i , the pointer statereads (unnormalized) | ϕ i p = h ψ f | Ψ f i sp = ( αγ + βη ) | i p + ( αγ + βηe iθ ) | i p (20)with α, β, γ, η are taken real numbers here. αγ + βηe iθ can be recast as p α γ + β η + 2 αγβη cos θe iφ withtan φ = βη sin θβη cos θ + αγ . (21)Since phase signal θ ≪ p α γ + β η + 2 αγβη cos θ = αγ + βη in the first order approximation and the state ofpointer thus becomes | ϕ i p = 1 √ | i p + e iφ | i p ) . (22)Post-selection of the system after weak interaction resultsin φ rotation of the pointer state along equator of Blochsphere. Amplification is realized when we choosing post-selection state | ψ f i properly such that φ > θ is satisfied.In general, we can set α = β = 1 / √ γ = cos χ, η =sin χ , in the case of θ ≪
1, Eq. (21) reduces totan φ = θ χ . (23)Suppose that χ = − ( π/ δ ) with δ ≪
1, then cot χ = − δ in first order approximation and we have tan φ = θ/δ . The factor of amplification in this case is h = φθ = arctan( θ/δ ) θ . (24)The amplified phase φ can be easily extracted by per-forming ˆ σ R measurement on pointer h ˆ σ R i = p h ϕ | ˆ σ R | ϕ i p = sin φ. (25)The above amplification protocol is within weak mea-surement framework but different from WVA and thusnamed as weak measurement amplification (WMA). Theconcept of weak value is not required in WMA. In anal-ogy with micrometer that transforms a small displace-ment into a larger rotation of circle, WMA transformsa ultra-small phase signal into a larger rotation of thepointer state along equator of Bloch sphere. The WMAis a general amplification protocol and applied to anyphysical qubit systems.We have figured out how to realize longitudinal phaseamplification within weak measurement framework. Thenext natural question is how to apply WMA to gravi-tational wave detection or more specifically how to con-struct detection system based on WMA protocol. Fortu-nately, it is not hard to construct an optical system real-ize WMA. As shown in Fig. 3, the path and polarizationdegrees of freedom of photons are chosen as system andpointer respectively and polarization Michelson interfer-ometer is used to realize control-rotation evolution.The WMA optical system consists of five parts i.e.,laser source, initial state preparation, control-rotationevolution, post-selection and pointer measurement. Thelaser source produces stabilized photons in the linear po-larization state | + i = ( | H i + | V i ) / √
2. The initial statepreparation is fulfilled by the beam splitter (BS1). Thestate of photons, after passing through the BS1, becomes | Ψ i i sp = | ψ i i s ⊗ | + i p = ( r | d i + t | u i ) ⊗ | + i p , (26)where r , t are coefficients of reflection and transmissionof the BS1 satisfying | r | + | t | = 1 and | u i , | d i repre-sent path state of up arm and down arm respectively. FIG. 3. Optical realization of weak measurement amplifica-tion. This optical configuration is the foundation of proposeddetector in main text.
The photons, which fly along path of up arm, enters apolarization Michelson interferometer (PMI) that usedfor signal collection. The PMI, which consists of a po-larizing beam splitter (PBS1), two quarter wave plates(QWP) and two end masses, outputs the state of photons( | H i + e iθ | V i ) / √ | H i + | V i ) / √ θ is the phase signal to be measured. The function ofQWP, which is fixed at π/
4, is to transform polarizationstates | H i and | V i into its orthogonal states | V i and | H i respectively when photons pass through it twice. Hencethe transmitted photons with the state | H i is convertedby QWP to the state | V i , which is reflected by the PBS1,and the reflected photons with the state | V i is convertedto the state | H i by the other QWP and is thus transmit-ted by the PBS1 such that the photons will not come outfrom the input port. There is no change of state whenphotons flying along the down arm path. The control-rotation interaction is fulfilled when photons come out ofPMI with state | Ψ f i sp = r | d i ⊗ | + i + t | u i ⊗ ( | H i + e iθ | V i ) / √ . (27)The process of signal amplification, which depends onpost-selection, can be fulfilled by the BS2 with coeffi-cients of reflection and transmission of r and t , whichsatisfy | r | + | t | = 1. The post-selection is completedwhen we focus only on the photons come out from thedown port of BS2 with the post-selected path state | ψ f i s = r | d i + t | u i . (28)The polarization state of post-selected photons, in thefirst order approximation, thus becomes | ϕ i p = 1 √ | H i + e iφ | V i ) (29)with φ determined bytan φ = sin θ cos θ + r r /t t . (30)The amplified phase signal φ thus can be obtained byproperly choosing r , r , t , t such that s h ψ f | ψ i i s = r r + t t →
0. The HWP, QWP, PBS2 and two de-tectors complete the detection of amplified phase signalas a polarization analyser.
Appendix B: Squeezing Quantum Noise
In this section, we will show that the quantum noise inproposed detector results from the vacuum fluctuationsin the dark port of BS1. The analysis follows the way ofCaves [14]. The quantized light field propagating in thedirection of z with | + i = ( | H i + | V i ) / √ E + = r π ~ ωAc [ˆ a + e − i ( ωt − kz ) + ˆ a † + e i ( ωt − kz ) ] (31)where A is cross-section of light field and ˆ a + , ˆ a † + representannihilation and creation operator of photons with | + i polarization. Clearly, we have ˆ a + = (ˆ a H +ˆ a V ) / √
2. First,consider the dark port output of BS2 as shown in Fig. 4.For BS1, the input-output relations areˆ a , + = t ˆ a , + + r ˆ a , + ˆ a , + = t ˆ a , + + r ˆ a , + . (32)Here it is assumed that all optical elements are ideal andoptical losses can be neglected. For simplicity of discus-sion, we assume that there exists no signals and othernoises such that the polarization state of photons is un-changed after passing through the polarization Michelsoninterferometer. The dark port output of BS2 is thusˆ b + = t ˆ a , + + r ˆ a , + = ( t t + r r )ˆ a , + + ( t r + r t )ˆ a , + (33)Suppose that there are certain N photons are input fromthe port 1 of BS1, then the initial state of the wholesystem is | N i , + ⊗| i , + . The average number of photonscoming out of dark port of BS2 is given as h ˆ n i = h ˆ b † ˆ b i = | t t + r r | N = P N. (34)It can be seen that P is the success probability of post-selection. The direct calculation gives the fluctuation ofphotons number as∆ n = p h ˆ n i − h ˆ n i = p | t t + r r | | t r + r t | · √ N. (35)Note that only terms like ˆ a † ˆ a ˆ a † ˆ a and ˆ a † ˆ a ˆ a ˆ a † makecontributions. In our case, t = 1 / √ , r = i/ √ t ≈ / √ , r ≈ i/ √
2, then | t r + r t | ≈
1, we obtain δn = √ P N as expected. The shot noise is indeed due tothe vacuum fluctuation of dark port of BS1.
FIG. 4. input-output of quantized optical field.
We now consider the radiation pressure noise that orig-inates from photons impinging on mirrors. The opticalfield in the two arms ˆ a ,R , ˆ a ,L is related to ˆ a asˆ a , + = 1 √ a ,H + ˆ a ,V ) = 1 √ a ,R + ˆ a ,L ) . (36)According to Eq. (32), we haveˆ a ,R = t ˆ a ,H + r ˆ a ,H ˆ a ,L = t ˆ a ,V + r ˆ a ,V . (37)What we care about is difference of photons numberin two arms δ ˆ n = ˆ a † ,R ˆ a ,R − ˆ a † ,L ˆ a ,L . Notice that h ˆ a † ,H ˆ a ,H i = h ˆ a † ,V ˆ a ,V i = N/
2. The average value ofnumber difference is h δ ˆ n i = 0. The fluctuation ∆( δ ˆ n ) ofnumber difference is thus determined by∆( δ ˆ n ) = p h ( δ ˆ n ) i = t √ N . (38)We have shown that quantum noise in the proposed de-tector results from the vacuum fluctuations of dark portof BS1 and thus usage of squeezed light in dark port ofBS1 will certainly reduce shot noise or radiation pressurenoise. [1] B. P. Abbott et al ., Observation of Gravitational Wavesfrom a Binary Black Hole Merger, Phys. Rev. Lett. ,061102 (2016).[2] B. P. Abbott et al ., GWTC-1: A Gravitational-WaveTransient Catalog of Compact Binary Mergers Observedby LIGO and Virgo during the First and Second Observ-ing Runs, Phys. Rev. X , 031040 (2019).[3] B. P. Abbott et al ., GW170817: Observation of Gravita-tional Waves from a Binary Neutron Star Inspiral, Phys.Rev. Lett. , 161101 (2017).[4] A. Goldstein et al ., An Ordinary Short Gamma-RayBurst with Extraordinary Implications: Fermi-GBM De-tection of GRB 170817A, Astrophys. J. Lett. , L14(2017). [5] D. Coulter et al ., Swope supernova survey 2017a(SSS17a), the optical counterpart to a gravitational wavesource, Science , 1556 (2017).[6] E. Troja et al ., The x-ray counterpart to thegravitational-wave event GW170817, Nature , 71(2017).[7] G. Hallinan et al., A radio counterpart to a neutron starmerger, Science , 1579 (2017).[8] L. Hu et al ., Optical observations of LIGO source GW170817 by the antarctic survey telescopes at dome a,antarctica, Sci. Bull. , 1433 (2017).[9] B. P. Abbott et al ., Multi-messenger observations of abinary neutron star merger, Astrophys. J. Lett. , L12(2017).[10] B. P. Abbott et al ., GW170814: A three-detector obser-vation of gravitational waves from a binary black holecoalescence, Phys. Rev. Lett. , 141101 (2017).[11] M. Punturo et al ., The Einstein Telescope: A thirdgen-eration gravitational wave observatory, Class. QuantumGravity , 194002 (2010).[12] G. M. Harry, L. S. Collaboration, et al ., Advanced LIGO:The next generation of gravitational wave detectors,Class. Quantum Gravity , 084006 (2010).[13] H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne,and S. P. Vyatchanin, Conversion of conventionalgravitational-wave interferometers into quantum nonde-molition interferometers by modifying their input and/oroutput optics, Phys. Rev. D , 022002 (2001).[14] C. M. Caves, Quantum-mechanical noise in an interfer-ometer, Phys. Rev. D , 1693 (1981).[15] M. Tse et al ., Quantum-enhanced Advanced LIGO de-tectors in the era of gravitational-wave astronomy, Phys.Rev. Lett. , 231107 (2019).[16] F. Acernese et al ., Increasing the astrophysical reachof the Advanced Virgo detector via the application ofsqueezed vacuum states of light, Phys. Rev. Lett. ,231108 (2019).[17] E. Oelker, T. Isogai, J. Miller, M. Tse, L. Barsotti, N.Mavalvala, and M. Evans, Audio-band frequencydepen-dent squeezing for gravitational-wave detectors, Phys.Rev. Lett. , 041102 (2016).[18] Y. Zhao, N. Aritomi, E. Capocasa, M. Leonardi, M.Eisenmann, Y. Guo, E. Polini, A. Tomura, K. Arai,Y. Aso, et al., Frequency-dependent squeezed vacuumsource for broadband quantum noise reduction in ad-vanced gravitational-wave detectors, Phys. Rev. Lett. , 171101 (2020).[19] L. McCuller, C. Whittle, D. Ganapathy, K. Komori,M. Tse, A. Fernandez-Galiana, L. Barsotti, P. Fritschel,M. MacInnis, F. Matichard, et al., Frequency-dependentsqueezing for advanced ligo, Phys. Rev. Lett. , 171102(2020).[20] R. Lynch, S. Vitale, L. Barsotti, S. Dwyer, and M. Evans,Effect of squeezing on parameter estimation of gravita-tional waves emitted by compact binary systems, Phys.Rev. D , 044032 (2015).[21] J. S. Read, C. Markakis, M. Shibata, K. Ury¯u, J. D.Creighton, and J. L. Friedman, Measuring the neutronstar equation of state with gravitational wave observa-tions, Phys. Rev. D , 124033 (2009).[22] K. D. Kokkotas, High-frequency sources of gravitationalwaves, Class. Quantum Gravity , S501 (2004).[23] E. M¨uller, M. Rampp, R. Buras, H. T. Janka, and D.H. Shoemaker, Toward gravitational wave signals from realistic core-collapse supernova models, Astrophys. J. , 221 (2004).[24] N. Andersson and K. D. Kokkotas, Gravitational waveastronomy: The high frequency window, in The physicsof the early universe (Springer, 2005) pp. 255276.[25] P. Covas and A. M. Sintes, First all-sky search for con-tinuous gravitational-wave signals from unknown neu-tron stars in binary systems using advanced ligo data,arXiv:2001.08411 (2020).[26] K. C. Pan, M. Liebend¨orfer, S. M. Couch and F.K. Thielemann, Equation of state dependent dynamicsmulti-messenger signals from steller-mass black hole for-mation, Astrophys. J. , 13 (2018).[27] V. Morozova, D. Radice, A. Burrows, and D. Vartanyan,The gravitational wave signal from core-collapse super-novae, Astrophys. J. , 10 (2018).[28] D. Radice, V. Morozova, A. Burrows, D. Vartanyan, andH. Nagakura, Characterizing the gravitational wave sig-nal from core-collapse supernovae, Astrophys. J. , L9(2019).[29] C. D. Ott et al. , Dynamics and gravitational wave signa-ture of collapsar formation, Phys. Rev. Lett. , 161103(2011).[30] E. R. Most et al. , Signatures of quark-hadron phase tran-sitions in general-relativistic neutron-star mergers, Phys.Rev. Lett. , 061101 (2019).[31] A. Bauswein and H. T. Janka, Measuring neutron-star properties via gravitational waves from neutron-starmergers, Phys. Rev. Lett. , 011101 (2012)[32] A. Bauswein et al. , Identifying a first-order phase transi-tion in neutron-star mergers through gravitational waves,Phys. Rev. Lett. , 061102 (2019).[33] S. Zha, E. P. O’Connor, M. C. Chu, L. M. Lin,and S. M. Couch, Gravitational-wave signature of afisrt-order quantum chromodynamics phase transition incore-collapse supernovae, arXiv:2007.04716, accepted byPhys. Rev. Lett.[34] C. L. Fryer and K. C. B. New, Gravitational waves fromgravitational collapse, Living Rev. Relativity, , 2 (2003).[35] N. Andersson, V. Ferrari, D. I. Jones, K. D. Kokkotas,B. Krishnan, J. Read, L. Rezzolla, and B. Zink, Gravita-tional waves from neutron stars: promises and challenges,Gen. Rel. Grav. , 409 (2011).[36] H. Miao, Y. Ma, C. Zhao, and Y. Chen, Enhancing thebandwidth of gravitational-wave detectors with unsta-ble optomechanical filters, Phs. Rev. Lett. , 211104(2015).[37] H. Miao, H, Yang, and D. Martynov, Towards the designof gravitational-wave detectors for probing neutron-starphysics, Phys. Rev. D , 044044 (2018).[38] M. Page, J. Qin, J. L. Fontaine, C. Zhao, L. Ju, andD. Blair, Enhanced detection of high frequency gravita-tional waves using optically diluted optomechanical fil-ters, Phys. Rev. D , 124060 (2018).[39] M. Korobko, Y. Ma, Y. Chen, and R. Schnabel,Quantum expander for gravitational-wave observatories,Light:Science & Applications, : 118 (2019).[40] M. A. Page et al. , Gravitational wave detectors withbroadband high frequency sensitivity, arXiv:2007.08766(2020).[41] K. S. Thorne, in Quantum Optics, Experimental Gravity,and Measurement Theory, edited by E. P. Wigner, P.Meystre, and M. O. Scully (Springer US, New York andLondon, 1983) p. 344. [42] M. J. Hu and Y. S. Zhang, Gravitational wave detectionvia weak measurements amplification, arXiv:1707.00886(2017).[43] M. J. Hu, X. M. Hu, B. H. Liu, Y. F. Huang, C. F. Li,G. C. Guo, and Y. S. Zhang, Experimental ultra-smalllongitudinal phase estimation via weak measurement am-plification, arXiv:1803.07746 (2018).[44] A. Weinstein, Advanced ligo optical configuration andprototyping effort, Class. Quantum Gravity , 1575(2002).[45] M. Maggiore, Gravitational waves: Volume 1: Theoryand experiments, Oxford University Press, Vol. 1, pp.516524 (2008).[46] A. Buonanno and Y. Chen, Quantum noise in sec-ond generation, signal-recycled laser interferometricgravitational-wave detectors, Phys. Rev. D , 042006(2001)[47] H. Vahlbruch et al. , Observation of squeezed light with10-dB quantum noise reduction, Phys. Rev. Lett. ,033602 (2008).[48] T. Eberle et al. , Quantum enhancement of the zero-areasagnac interferometer topology for gravitational wave de-tection, Phys. Rev. Lett. , 251102 (2010).[49] M. Mehmet, S. Ast, T. Eberle, S. Steinlechner, H.Vahlbruch, and R. Schnabel, Squeezed light at 1550 nmwith a quantum noise reduction of 12.3 dB, Opt. Express (25) 25763-25772 (2011).[50] C. M. Caves, K. S. Thorne, R. W. Drever, V. D. Sand-berg, and M. Zimmermann, On the measurement of aweak classical force coupled to a quantum-mechanical os-cillator. i. issues of principle, Rev. Mod. Phys. , 341(1980).[51] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne,Quantum nondemolition measurements, Science ,547 (1980).[52] Y. Chen, Sagnac interferometer as a speed-metertype,quantum-nondemolition gravitational-wave detector,Phys. Rev. D , 122004 (2003).[53] M. Wang, C. Bond, D. Brown, F. Br¨uckner, L. Carbone,R. Palmer, and A. Freise, Realistic polarizing sagnactopology with dc readout for the einstein telescope, Phys.Rev. D , 096008 (2013).[54] T. Eberle, S. Steinlechner, J. Bauchrowitz, V. H¨andchen, H. Vahlbruch, M. Mehmet, H. M¨uller-Ebhardt, and R.Schnabel, Quantum enhancement of the zero-area sagnacinterferometer topology for gravitational wave detection,Phys. Rev. Lett. , 251102 (2010).[55] Y. Chen, S. L. Danilishin, F. Y. Khalili and H. M¨uller-Ebhardt, QND measurements for future gravitational-wave detectors, Gen. Relativ. Gravit. , 671 (2011).[56] A. S. Chou et al ., The Holometer: an instrument to probePlanckian quantum geometry, Class. Quantum Gravity, , 065005 (2017).[57] A. S. Chou et al ., First measurements of high frequencycross-spectra from a pair of large Michelson interferome-ters, Phys. Rev. Lett. , 111102 (2016).[58] K. Nagano, T. Fujita, Y. Michimura and I. Obata, Ax-ion dark matter search with interferometric gravitationalwave detectors, Phys. Rev. Lett. , 111301 (2019).[59] Y. Aharonov, D. Z. Albert, and L. Vaidman, How theresult of a measurement of a component of the spin of aspin-1/2 particle can turn out to be 100, Phys. Rev. Lett. , 1351 (1988).[60] O. Hosten and P. Kwiat, Observation of the spin halleffect of light via weak measurements, Science , 787(2008).[61] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, andC. Bamber, Direct measurement of the quantum wave-function, Nature , 188 (2011).[62] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, andR. W. Boyd, Colloquium: Understanding quantum weakvalues: Basics and applications, Rev. Mod. Phys. , 307(2014).[63] N. Brunner and C. Simon, Measuring small longitudinalphase shifts: Weak measurements or standard interfer-ometry? Phys. Rev. Lett. , 010405 (2010).[64] A. N. Jordan, J. Martnez-Rincn, and J. C. Howell, Tech-nical advantages for weak-value amplification: When lessis more, Phys. Rev. X , 011031 (2014).[65] X. Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C. F. Li,and G. C. Guo, Phase Estimation with Weak Measure-ment Using a White Light Source, Phys. Rev. Lett. ,033604 (2013).[66] A. Nishizawa, Weak-value amplification beyond the stan-dard quantum limit in position measurements, Phys.Rev. A92