Demonstration of interferometer enhancement through EPR entanglement
Jan Südbeck, Sebastian Steinlechner, Mikhail Korobko, Roman Schnabel
DDemonstration of interferometer enhancement through EPR entanglement
Jan Südbeck, Sebastian Steinlechner, Mikhail Korobko, and Roman Schnabel Institut für Laserphysik und Zentrum für Optische Quantentechnologien der Universität Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany
The sensitivity of laser interferometers usedfor the detection of gravitational waves (GWs) islimited by quantum noise of light. An improve-ment is given by light with squeezed quantumuncertainties, as employed in the GW detectorGEO 600 since 2010. To achieve simultaneousnoise reduction at all signal frequencies, however,the spectrum of squeezed states needs to be pro-cessed by 100 m-scale low-loss optical filter cavi-ties in vacuum. Here, we report on the proof-of-principle of an interferometer setup that achievesthe required processed squeezed spectrum by em-ploying Einstein-Podolsky-Rosen (EPR) entan-gled states. Applied to GW detectors, the cost-intensive cavities would become obsolete, whilethe price to pay is a 3 dB quantum penalty.Introduction — The recent series of gravitational-wave (GW) detections by the Advanced LIGO and Ad-vanced Virgo 2 nd generation observatories [1, 2] launchedthe new field of GW astronomy. A 3 rd generation (3G)of ground-based GW observatories aims at a signal-to-noise improvement by a factor of ten, resulting in anaverage increase in detection rate by three orders of mag-nitude and more precise localisations of the source. Atthese high sensitivities, binary neutron star mergers andtheir optical counterparts, acting as standard sirens inthe measurement of cosmological parameters [3], will befound on a daily basis. In addition, the ring-down phaseof such mergers will reveal the so far unknown equation ofstate of neutron stars [4]. A European design study for a3G observatory, the Einstein Telescope was already pub-lished in 2011 [5], since followed by proposals for similarefforts in the US [6, 7].To increase the measured GW signal strength, futureground-based observatories will have longer arms and willuse higher light powers. Heavier mirrors, larger laserspots on the mirrors, cryo-cooled mirrors and possiblyan underground location all help to reduce the zoo of dif-ferent noise sources. Light with a squeezed uncertaintyof its electric field [8–10] targets the reduction of inter-ferometer quantum noise, separate from the light power[11–13]. Squeezing in one of the field quadratures, in-dependently of the signal frequency, has been employedto reduce the observatory shot-noise [14, 15] and hasbeen improving the sensitivity of GEO 600 since 2010[16]. Since the third observing run O3, it also is regu-larly used in the Advanced LIGO detectors and in Ad-vanced Virgo. For a broadband reduction of both pho-ton shot-noise (quantum measurement noise) as well as C h o i c e o f r e a d o u t a n g l e (a)
10 100 10000−π/2−π/4 Sideband frequency Ω/2π (Hz)phase readoutamplitude readoutphase readoutamplitude readout10 100 10000−π/2−π/4 Sideband frequency Ω/2π (Hz) C h o i c e o f r e a d o u t a n g l e (b) FIG. 1. Simulation of the improvement in readout quantumnoise of a typical laser interferometer with squeezed light, asa function of signal frequency and readout angle, and withparameters similar to Advanced LIGO, including a loss factor40 %. In the case of injected frequency-independent squeez-ing (a) , the noise in the signal (phase) readout is reducedat high sideband frequencies. However, quantum back-action(coupled via radiation pressure) leads to a rotation of thesqueezed readout angle, worsening the readout at low frequen-cies (curved blue band). Broadband improvement (b) couldbe achieved with a frequency-dependent squeeze angle, whichwould compensate this rotation (horizontal blue band). Sofar, this necessitated a complicated hardware processing withadditional filter cavities. The EPR scheme demonstrated herecan generate such a frequency-independent spectrum withoutthe need for these filter cavities. photon radiation pressure noise (quantum back-actionnoise), the squeezed quadrature needs to change fromamplitude to phase quadrature depending on the signalfrequency [17, 18], cf. Fig. 1. The optimal, frequency-dependent squeeze spectrum can be produced by re-flecting frequency-independent squeezed light off low-lossnarrow-band filter cavities [19–21]. Even with cavity mir-rors of highest quality and operation in ultra-high vac-uum, appropriate filter cavities must have lengths of theorder of hundred meters to achieve the linewidth andloss requirements [22]. Recently, Y. Ma and co-workers a r X i v : . [ qu a n t - ph ] A ug proposed to avoid these cost-intensive filter cavities [23]by exploiting (i) the Einstein-Podolsky-Rosen (EPR) en-tanglement [24–26] between different frequency compo-nents of the injected squeezed field [27, 28] and (ii) thesignal-recycling cavity that is already part of current andfuture GW observatories. The same approach is alsouseful in detuned signal-recycled interferometers withoutradiation-pressure noise [29].Here, we report on the experimental proof of principleof the proposal made in [23]. In our experiment, a linearoptical cavity represented the GW observatory signal-recycling cavity. Onto this cavity, we mode-matched aconventional, frequency-independent squeezed field. Em-ploying bichromatic (two-frequency) balanced homodynereadout, we demonstrated that the unwanted effect onquantum noise caused by a detuning of the cavity canbe undone by conditioning the measurement at the sig-nal frequency on the measurement at the corresponding,EPR-correlated idler frequency. Our scheme correspondsto a realisation of the EPR gedanken experiment [24],where a measurement at location A instantaneously re-sults in reality of the entangled quantity at location B . Quantum measurement process — In interfero-metric gravitational-wave detectors, optomechanical in-teraction between the light and the test-mass mirrorsinduces correlations in quantum noise due to the pon-deromotive squeezing effect [18]: quantum fluctuationsin the amplitude quadrature of the light create fluctu-ating radiation pressure forces on the suspended mir-rors. The resulting motion in turn couples into thephase quadrature of the reflected light, and the ampli-tude and phase quadratures of the light become quantumcorrelated. Mathematically, this coupling defines the re-lation between the input and output quadratures in atuned interferometer (omitting the unimportant phasefactor)[19]:ˆ b (ampl)out = ˆ b (ampl)in , (1)ˆ b (phase)out = ˆ b (phase)in − K (Ω) ˆ b (ampl)in + signal , (2)where the optomechanical coupling is described by thecoupling strength (Kimble factor) K (Ω) that depends onthe signal frequency Ω: K (Ω) = 2 Jγ Ω ( γ + Ω ) , J = 8 πI c M λL . (3)The Kimble factor depends on the light power insidethe arm cavities of the detector I c and their length L ,as well as the on the optical linewidth of the detec-tor γ , the mass of the mirrors M and the laser wave-length λ . Its frequency dependence has contributionsfrom the cavity Lorentzian profile γ + Ω and from themechanical response of the free test mass Ω . The Kim-ble factor becomes larger at lower sideband frequencies,which is exactly the effect of radiation pressure noise φ LO ω pump ω LO, S ω S ω LO, I ω I FRPBS : SRCSLS BBHD
FIG. 2. Simplified representation of this work’s experimentalsetup. The frequency components emitted from a broadbandsqueezed-light source (SLS) that are reflected off a cavity ex-perience a frequency dependent phase that can compensatefor the frequency dependence of quantum back action. If theSLS is operated at particular centre frequency, the signal-recycling cavity (SRC) can be used that already exists in everyGW observatory. The EPR entangled sidebands ω I and ω S ofthe squeezed field are detected with a bichromatic balanced-homodyne detection (BBHD), employing local oscillators atfrequencies ω LO,S and ω LO,I . The greyed-out parts indicatehow the cavity of our experiment could relate to a full-scalemeasurement device (here, an Advanced LIGO-style Michel-son interferometer with arm resonators, power-, and signalrecycling). PBS, polarizing beam-splitter; FR, Faraday rota-tor. SRC length in our experiment was 2 . contaminating the low-frequency sensitivity ˆ b (phase)out ≈−K (Ω)ˆ b (ampl)in + signal. The radiation pressure noise atlow frequencies can therefore be reduced by squeezingthe amplitude quadrature ˆ b (ampl)in . At the same time theshot noise in phase quadrature dominates the high fre-quency sensitivity: ˆ b (phase)out ≈ ˆ b (phase)in + signal, and thuscalls for squeezing the phase quadrature ˆ b (phase)in . Thisfrequency dependence of the quantum noise in the in-terferometer is shown on the Fig. 1(a), which illustratesthe need for frequency-dependent squeezing. When suchfrequency-dependent squeezing is injected from the out-side, the detected noise in the phase quadrature takes theformˆ b (phase)out = ˆ b (phase)in (cid:2) cosh r − sinh r (cos 2 φ + K sin 2 φ ) (cid:3) −− K ˆ b (ampl)in (cid:2) cosh r + sinh r (cid:0) cos 2 φ − K − sin 2 φ (cid:1)(cid:3) , (4)with squeeze factor r and a frequency-dependentsqueeze angle φ ≡ φ (Ω). A straightforward optimiza-tion leads to φ (Ω) = arctan K (Ω), which results in ω I ω S ω pump Ω ΩΩδ S δ I =0ΩEPR entangled sideband fields frequency p h a s e (a) R e l a t i v e n o i s e p o w e r ( d B ) Sideband frequency Ω/2π (Hz)phase readout (0)amplitude readout (−π/2)intermediate readout (−π/4)−π/4−π/20π/4 phase readoutamplitude readout 200k 400k 600k 800k 1M 1.2M 1.4M 1.6M 1.8M 2MSideband frequency Ω/2π (Hz) ω I ω S ω pump Ω ΩΩδ S δ I =−δ S ΩEPR entangled sideband fields frequency p h a s e (b) R e l a t i v e q u a n t u m n o i s e s u pp r e ss i o n ( d B ) phase readout (0)amplitude readout (−π/2)intermediate readout (−π/4)intermediate readout intermediate readout C h o i c e o f r e a d o u t a n g l e FIG. 3. The schematics in the top row show the relative positions of signal and idler bands around ω I and ω S in our demon-stration of frequency-dependent squeeze angle rotation. Electric fields at frequencies symmetrically located around half theSLS pump frequency, ω pump /
2, are mutually EPR entangled. In the bichromatic homodyne readout, electric fields at ω S ± Ωand ω I ± Ω are combined in such a way that the result directly gives the total quantum noise of the detection scheme. Thespectrograms in the middle show the measured quantum noise variances for continuously scanned readout angle. The bottomplots present cuts of the spectrogram at phase, intermediate and amplitude readout quadratures: light traces show experimentaldata; dark traces show theoretical fits with a joint set of parameters. (a)
While the signal band around ω S is detuned fromcavity resonance by δ S = 2 π ×
460 kHz, the idler band around ω I is held exactly on the next resonance. For frequencies Ω muchlarger than the cavity linewidth and detuning, a noise level 4 dB below shot noise is achieved in a specific readout quadrature.Around the detuned cavity resonances, however, sidebands of ω S and ω I experience different phase shifts, spoiling the EPRcorrelations and resulting in an increased noise. The spectrogram shows a frequency-dependent squeezing phase at frequenciesbelow ∼ (b) When ω S and ω I are detuned from cavity resonances by the same but opposite amount δ S , EPR entangled fields at all sidebands Ω receivethe same frequency-dependent phase shift. As a result, the detrimental effect of detuning (and/or ponderomotive squeezing) onthe detection quantum noise can be cancelled and a flat, frequency-independent squeezed spectrum is obtained. This validatesthe proposed scheme for quantum-noise enhancement with EPR-entangled states. The features at around 50 kHz, 100 kHz and1 . optimally reduced noise at all frequencies, ˆ b (phase)out = e − r (cid:16) ˆ b (phase)in − K (Ω)ˆ b (ampl)in (cid:17) , compare with Eq.(2). Theeffect of the frequency-dependent squeezing on the quan-tum noise can be seen in Fig. 1(b).The frequency-dependent rotation of quadratures byradiation-pressure noise has the same effect on the quan- tum noise as detuning a cavity without movable mir-rors from its resonance by δ . In this case the cou-pling constant in Eq. (4) is replaced by K cav (Ω , δ ) =2 γδ/ (cid:0) γ − δ + Ω (cid:1) [18]. While generally the frequencydependence of the two coupling factors is different, inthe case of optimal detuning δ = γ within the cavitylinewidth γ (cid:29) Ω they become equivalent up to a scaling −3π/4-π/4 4 26 0 −2 −4 −6 −8 −10 −12Relative quantum noise suppression (dB)-π/20−π 200k 400k 600k 800k 1M 1.2M 1.4M 1.6M 1.8M 2MSideband frequency Ω/2π (Hz) ω I ω S ω pump Ω ΩΩ δ S δ I =δ S ΩEPR entangled sideband fields frequency p h a s e C h o i c e o f r e a d o u t a n g l e FIG. 4. This figure shows the results of an additional ex-perimental step demonstrating the flexibility of the setup. Inthis case both ω S and ω I are detuned from cavity resonancein the same direction and by the same amount δ S = δ I =2 π ×
460 kHz, leading to a frequency-dependent squeeze an-gle. The spectrogram shows the measured quantum noisevariances for continuously scanned readout angle. The ac-quired frequency dependence in squeeze angle sweeps acrossthe phase range of about 3 π/
4. This is larger than neededin gravitational-wave detector for cancelling the ponderomo-tive squeezing effect, but shows the potential of the setup intailoring the frequency dependence of squeeze angle to theneeds of quantum-optical experiment. The exact frequencydependence is ultimately a function of the cavity linewidthand the detunings. The features at around 50 kHz, 100 kHzand 1 . factor: K (Ω) ≈ J/ ( γ Ω ) = K cav (Ω , γ ) J/γ . This sym-metry between a detuned cavity and the ponderomotivesqueezing effect explains the need for a detuned filtercavity for producing the frequency-dependent squeezingthat reduces the quantum noise at all frequencies. Whensqueezing is reflected off a detuned filter cavity, the fre-quency dependence that its angle acquires can be exactlyopposite to the one caused by the radiation pressure in-side the detector, and the two cancel each other. Thisresults in the frequency-independent sensitivity enhance-ment, as illustrated in Fig. 1(b). Such a frequency de-pendence is possible to create with EPR-entangled statesof light, following the proposal by Ma et al. [23]. Whilewe do not demonstrate the ponderomotive squeezing ef- fect in our experiment, the above relations allow us toemulate it with a detuned cavity. Generation of EPR entanglement — We pro-duced light with a squeezed quantum uncertainty by sub-threshold pumping of a non-degenerate cavity-enhancedparametric amplifier. It naturally consisted of mutuallyentangled electric fields at lower ( signal ω S ) and higher( idler ω I ) sidebands of the central frequency, which washalf the pump frequency ω pump / ω I + ω S . Measuringthe quadrature noise at ω S,I ± Ω individually with twoseparate homodyne detectors leads to an increased noiseover the vacuum uncertainty. However, the measure-ment outcome at one detector could be conditioned (i.e.in post-processing) on the measurement outcome of theother detector with an optional scaling factor g . Then,for sufficiently high squeeze factors, the conditional vari-ances ˆ a (phase) s + g ˆ a (phase) i and ˆ a (ampl) s + g ˆ a (ampl) i could bereduced to below the vacuum uncertainty ( conditionalsqueezing ), certifying Einstein-Podolsky-Rosen entangle-ment [25]. Experimental realization — For our experimentaldemonstration of interferometer enhancement throughEPR entanglement, we employed a setup as shown inFig. 2. A squeezed-light source provided a broadbandsqueeze spectrum with a full-width half-maximum of140 MHz. Inside this linewidth, fields at ω pump / ± ∆ ω show mutually strong EPR correlations, of which we se-lect the signal and idler bands around ω S and ω I for ourdiscussion. A 2 . .
73 MHz mimicked the signalrecycling cavity of a GW detector. The cavity could belocked to arbitrary detunings with an auxiliary field ω lock that was orthogonally polarized such as to not contam-inate the measurement output. The squeezed field ω S,I was mode-matched onto the linear cavity and the back-reflected light was separated with a Faraday rotator anda polarizing beam-splitter. We then performed bichro-matic balanced homodyne detection [30, 31] by overlap-ping the output field with two bright fields ω LO,I and ω LO,S at a 50:50 beam splitter and detecting the differ-ence in photo current of the two outputs. For technicaldetails on the setup and its implementation, see Methods.In the first step, see Fig. 3(a), we adjusted the EPRentangled field ω S,I such that ω S was slightly detunedby an offset frequency δ S = 460 kHz from a resonanceof the linear cavity, while ω I was exactly on the nextlongitudinal resonance peak, one FSR ∆ away. In thisconfiguration, measurement sidebands ± Ω around ω S re-ceived a different (unequal and opposite) phase shift,leading to a frequency-dependent quadrature rotation viathe coupling factor K cav (Ω , δ S ). On the other hand, thesidebands around ω I received equal and opposite phaseshifts, resulting in the absence of quadrature rotation, K cav (Ω , δ I = 0) = 0. At the balanced-homodyne detec-tor, quantum noise components at ω S ± Ω and ω I ± Ω weredetected with frequency-independent gain and for a full2 π sweep of the local oscillator angle φ LO . The resultingphoto-current variance is shown in Fig. 3(a), normalizedto the vacuum noise of both local oscillators. In accor-dance with the EPR gedanken experiment , a noise powerbelow 0 dB demonstrates that it was possible to infer thenoise components around ω S by conditioning on a mea-surement at ω I . Here, this was the case for sidebandfrequencies Ω that were much larger than the linewidthof the linear cavity. In effect, the original squeeze fac-tor was restored. Around the cavity linewidth, however,the squeeze factor was lost at all readout angles, sincethe frequency rotation around ω S meant that the quan-tum correlations at the signal and idler wavelengths wereno longer aligned for optimal inference. As explained inMa et al. [23], an optimal Wiener filter would still allowto recover (most of) the correlations, but it cannot beimplemented in our bichromatic BHD readout scheme.In the second step, Fig. 3(b), we detuned ω I away fromthe cavity resonance as well, by δ I = − δ S . In this con-figuration, the noise sidebands around ω I experiencedequal, but opposite phase shifts to the noise sidebandsaround ω S . As a result, the two measurement bases wereoptimally aligned at all frequencies, thus cancelling thefrequency-dependent rotation everywhere. The result-ing spectrum shows a constant squeeze factor for a sin-gle readout quadrature, demonstrating near-perfect in-ference of the quantum noise components around ω S fromthe components around ω I .Finally, we performed an additional step, not moti-vated by the gravitational-wave detection directly, butdemonstrating the flexibility and the potential of the ex-perimental setup. For that we detuned ω I away from thecavity resonance in another direction, by δ I = δ S , seeFig. 4. As a result, the noise sidebands around ω I werephase shifted by the same amount as the ones around ω S .The resulting squeezing spectrum acquired frequency de-pendence, spanning a larger phase range than in the firststep of the experiment. This demonstrates the possibilityto create a desired frequency-dependent squeezing spec-trum by carefully tuning the filter cavities. Conclusion — Since the third observation run, theAdvanced LIGO and Advanced Virgo observatories ben-efit from frequency-independently squeezed states of lightto reduce the quantum shot noise at high signal frequen-cies. Due to the quantum measurement process, bothobservatories will suffer from increased quantum back-action noise at low signal frequencies. Additional long-baseline, low-loss filter cavities are considered for resolv-ing this issue. Here, we provide the proof of principlethat the same effect can be achieved with the signal-recycling cavity, which exists already in GW observato-ries. Following the proposal in [23], we utilized Einstein-Podolsky-Rosen entanglement. As pointed out in [23],the approach suffers from a 3 dB noise penalty on theinitial squeezing and an increased sensitivity to opticalloss, both of which are due to the measurement of an additional pair of sidebands. However, in light of thehigh additional costs for low-loss, narrow-linewidth fil-ter cavities, the demonstrated broadband enhancementof interferometer sensitivity through EPR entanglementcan still be a viable alternative.This work was supported by the Deutsche Forschungs-gemeinschaft (DFG), project SCHN757/6-1. It has theLIGO document number P1900226.
Methods — Squeezed-light source:
We used amonolithic cavity-enhanced OPA to generate broadbandsqueezed states of light via parametric down-conversionin PPKTP. The cavity was pumped by 100 mW of lightat ω pump = 2 πc/ (1064 nm / Linear cavity:
We set up a 2 . .
73 MHz. Themirror on the back of the cavity had a highly-reflectivecoating and the coupling mirror had a reflectivity of 97 %resulting in a linewidth of ∼
150 kHz (HWHM). The cav-ity was locked with the Pound-Drever-Hall method us-ing an auxiliary field in the orthogonal polarization. Weused the zero-crossing of one sideband of the resultingerror signal to allow for an easy shift of the resonancefrequency.
Bichromatic LO:
The bichromatic local oscillator wascreated by a strongly phase modulated light of the fre-quency ω pump / Theoretical modeling:
We compute the input-outputrelations for the light fields propagating in our setup in asimilar way to [23, 29]. The resulting spectral density ofthe shot noise with frequency-dependent squeezing em-bedded is described by: S (Ω) = 1 − η + η cosh 2 r ++ 2 η √ αβ sinh 2 rα + β ( K (Ω) cos 2 ζ + K (Ω) sin 2 ζ ) , (5)where we defined the readout efficiency η , squeeze pa-rameter r , readout angle ζ , the power of local oscillators α, β , and two coupling coefficients K , (Ω): K (Ω) = C (Ω) D (Ω) (cid:0) C (Ω) − γ ( δ + δ ) (cid:1) , (6) K (Ω) = C (Ω) D (Ω) (cid:0) γ ( δ + δ ) (cid:0) γ − δ δ + Ω (cid:1)(cid:1) , (7) C (Ω) = (cid:0) δ − Ω (cid:1) (cid:0) δ − Ω (cid:1) + γ (cid:0) γ + δ + δ + 2Ω (cid:1) , (8) D (Ω) = Y i =1 , (cid:16) γ + ( δ i − Ω) (cid:17) (cid:16) γ + ( δ i + Ω) (cid:17) . (9)This equation can be used to describe the detected spec-tra, as shown in Fig.3. Author Contributions — JS, SS and RS plannedthe experiment. JS and SS built and performed the ex-periment. MK provided the theoretical analysis. JS, SS,MK and RS prepared the manuscript. [1] The LIGO Scientific Collaboration and the Virgo Collab-oration, Physical Review Letters , 061102 (2016).[2] The LIGO Scientific Collaboration and the Virgo Col-laboration, arXiv:1811.12907 [astro-ph, physics:gr-qc](2018), 1811.12907.[3] The LIGO Scientific Collaboration and The Virgo Col-laboration, The 1M2H Collaboration, The Dark EnergyCamera GW-EM Collaboration and the DES Collab-oration, The DLT40 Collaboration, The Las CumbresObservatory Collaboration, The VINROUGE Collabora-tion, and The MASTER Collaboration, Nature , 85(2017).[4] LIGO Scientific Collaboration and Virgo Collaboration,Physical Review Letters , 161101 (2017).[5] ET Science Team, “Einstein gravitational wave telescopeconceptual design study,” (2011).[6] The LIGO Scientific Collaboration, “The LSC-virgowhite paper on instrument science (2018 edition),”(2018).[7] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber-nathy, K. Ackley, et al. , Classical and Quantum Gravity , 044001 (2017).[8] D. F. Walls, Nature , 141 (1983). [9] G. Breitenbach, S. Schiller, and J. Mlynek, Nature ,471 (1997).[10] R. Schnabel, Physics Reports , 1 (2017).[11] C. M. Caves, Physical Review D , 1693 (1981).[12] R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K.Lam, Nature Communications , 121 (2010).[13] D. E. McClelland, N. Mavalvala, Y. Chen, and R. Schn-abel, Laser & Photonics Reviews , 677 (2011).[14] The LIGO Scientific Collaboration, Nature Physics ,962 (2011).[15] The LIGO Scientific Collaboration, Nature Photonics ,613 (2013).[16] H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel,J. Slutsky, and H. Vahlbruch, Physical Review Letters , 181101 (2013).[17] M. T. Jaekel and S. Reynaud, Europhysics Letters ,301 (1990).[18] S. L. Danilishin and F. Y. Khalili, Living Reviews inRelativity , 5 (2012).[19] H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, andS. P. Vyatchanin, Physical Review D , 022002 (2001).[20] S. Chelkowski, H. Vahlbruch, B. Hage, A. Franzen,N. Lastzka, K. Danzmann, and R. Schnabel, PhysicalReview A , 013806 (2005).[21] F. Y. Khalili, Physical Review D , 122002 (2010).[22] L. Barsotti, J. Harms, and R. Schnabel, Reports onProgress in Physics , 016905 (2018).[23] Y. Ma, H. Miao, B. H. Pang, M. Evans, C. Zhao,J. Harms, R. Schnabel, and Y. Chen, Nature Physics , 776 (2017).[24] A. Einstein, B. Podolsky, and N. Rosen, Physical Review , 777 (1935).[25] M. D. Reid, Physical Review A , 913 (1989).[26] W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph,Physical Review A , 012304 (2005).[27] C. Schori, J. L. Sørensen, and E. S. Polzik, PhysicalReview A , 033802 (2002).[28] B. Hage, A. Samblowski, and R. Schnabel, Physical Re-view A , 062301 (2010).[29] D. D. Brown, H. Miao, C. Collins, C. Mow-Lowry,D. Töyrä, and A. Freise, Physical Review D , 062003(2017).[30] A. M. Marino, C. R. Stroud Jr., R. S. Bennink, andR. W. Boyd, JOSA B , 335 (2007).[31] W. Li, Y. Jin, X. Yu, and J. Zhang, Physical Review A96