PHYSICAL REVIEW X 9, 011053 (2019) Quantum Limit for Laser Interferometric Gravitational-Wave Detectors from Optical Dissipation Haixing Miao,1 Nicolas D. Smith,2 and Matthew Evans3 1School of Physics and Astronomy, and Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom 2California Institute of Technology, Pasadena, California 91125, USA 3Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 2 August 2018; revised manuscript received 1 January 2019; published 26 March 2019) We derive a quantum limit to the sensitivity of laser interferometric gravitational-wave detectors from optical-loss-induced dissipation, analogous to the sensitivity limit from the mechanical dissipation. It applies universally to different interferometer configurations and cannot be surpassed unless the optical properties of the interferometer are improved. This result provides an answer to the long-standing question of how far we can push the detector sensitivity given the state-of-the-art optics. DOI: 10.1103/PhysRevX.9.011053 Subject Areas: Gravitation, Optics, Quantum Physics I. INTRODUCTION together with a squeezed light source, a passive filter cavity can be used for producing the frequency-dependent squeez- Advanced gravitational-wave (GW) detectors are long- ing [9–11]. The general scheme is illustrated in Fig. 1. baseline interferometers with suspended mirrors which act The quantum Cram´er-Rao bound (QCRB) [12,13] is as test masses for probing spacetime dynamics. Quantum a sensitivity limit that, unlike the SQL, is inviolable. In noise, arising from quantum fluctuations of the optical the context of GW detection with laser interferometers, field, is one of the sources of noise that limits the sensitivity the QCRB is also called the energetic or fundamental of such instruments. In particular, the phase fluctuation quantum limit [14,15]. The spectral representation of the gives rise to the shot noise, while the amplitude fluctuation QCRB is exerts a random force on the test masses and induces the quantum radiation pressure noise. These two types of ℏ2c2 quantum noise, when uncorrelated, lead to the standard SQCRBðΩÞ¼ ; ð2Þ hh 2S ðΩÞL2 quantum limit (SQL) [1,2] of which the power spectral PP density is 8ℏ SSQLðΩÞ¼ ; ð1Þ hh MΩ2L2 where Ω is the angular frequency of the GW signal, M is the mass of the test-mass mirror, and L is the interferometer arm length. Despite its name, the SQL is not a true limit: It can be surpassed with a wide class of so-called quantum- nondemolition (QND) schemes (cf. review articles [3–5]). Intracavity These techniques usually involve modifications to the filters optical configuration of current-generation GW detectors O by introducing extra optical filters. These filters can be a I cascade of both passive Fabry-Perot cavities and active S cavities that have external energy input [6–8]. For example, O Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. FIG. 1. Illustration of the general QND scheme with optical Further distribution of this work must maintain attribution to filters added to the typical configuration of advanced GW the author(s) and the published article’s title, journal citation, detectors—a dual-recycled Michelson interferometer. (ETM: and DOI. end test mass; SRM: signal-recycling mirror.) 2160-3308=19=9(1)=011053(10) 011053-1 Published by the American Physical Society MIAO, SMITH, and EVANS PHYS. REV. X 9, 011053 (2019) in which S is the spectral density of the power fluctua- ℏ 2 Ω2 ϵ PP ϵ ¼ c ϵ þ 1 þ Titm src þ α ϵ Shh 2 arm 2 Tsrc ext : tions in the interferometer arms. As shown in Ref. [16], the 4L ω0P γ 4 QCRB can be approached in a lossless system with optimal ð Þ frequency-dependent homodyne read-out, which can be 3 realized with proper output filters [9]. This result has been generalized to laser interferometers with multiple carrier Here, P is the optical power inside each arm (assumed to be ω ϵ frequencies [17]. equal); 0 is the laser frequency; arm quantifies the internal ϵ ¼ 10−6 ϵ According to the QCRB, the sensitivity is ultimately loss of the arm cavity (e.g., arm for 1 ppm loss); src bounded by the power or, equivalently, energy fluctuation quantifies the optical loss inside the SRC, including addi- ϵ inside the arm cavities. This can be intuitively understood tional intracavity filters if any; ext denotes the external loss, from the fact that we want to measure the phase or timing which includes the loss in the output chain and also the difference between the two interferometer arms accurately, quantum inefficiency of the photodetection; γ is the ð4 Þ and a large uncertainty in the photon number or energy is bandwidth of the arm cavity and is equal to cTitm= L , α needed, due to the number-phase or energy-time uncertainty with Titm being the power transmission of ITM; is equal to 1 if we use the internal squeezing to maximize the power relation. Note, however, that increasing SPP is only advanta- geous if a minimum uncertainty state, or at least nearly fluctuation and α is equal to 1=4 if instead the internal minimum uncertainty state, can be maintained. Minimum squeezing is negligible, which has to do with the effect of uncertainty states are, however, very delicate and easily internal squeezing on the signal response [24,25] destroyed by optical losses which lead to decoherence. [cf. Eq. (20) and also Appendix A]; Tsrc is the effective transmissivity of SRC, which may be frequency dependent. The power fluctuation SPP can be enhanced, and thus the QCRB can be reduced, with a variety of approaches. The The arm-cavity loss sets a flat limit across different most obvious approach is to increase the optical power in frequencies, as the additional vacuum fluctuation is directly the interferometer, but this is limited by thermal lensing mixed with the signal inside the arm. The SRC loss is [18], alignment stability [19], and parametric instabilities suppressed by ITM transmission at low frequencies. [20,21]. A second approach is to introduce squeezed states However, due to a finite arm-cavity bandwidth, which of light generated by nonlinear optical processes [11] at reduces the signal response, this effect becomes important the read-out port (cf. Fig. 1). In addition to squeezing the at high frequencies. The effect of the external loss depends vacuum fluctuations that enter the interferometer, the on the transmission of SRC, which can be frequency ϵ radiation pressure coupling between the optical field and dependent. For Advanced LIGO, arm is of the order of 10−4 ϵ 10−3 the test masses produces squeezing internally. This process, (100 ppm), src is around , which mainly comes ϵ known as “ponderomotive squeezing” [9,22], is actually the from the beam splitter, and ext is around 0.1 (coming from cause of the radiation pressure noise. We can turn it into a the mode mismatch to the output mode cleaner and ¼ resource for enhancing the detector sensitivity by detuning photodiode quantum inefficiency [26]). Given Titm 0 014 ≈ 0 14 the signal-recycling cavity (SRC) formed by input test mass . and Tsrc . in the default broadband detection (ITM) and SRM away from the perfect resonance. This is mode, we show the resulting sensitivity bound imposed by the so-called optical spring effect that was studied exten- current loss values in Fig. 2. sively by focusing on the dynamics of the test masses [23]. We can map this result to the one obtained in the In Appendix A, we illustrate an equivalent picture men- quantum metrology community [27–31] with a few sim- tioned in Ref. [16] from the perspective of the internal plifications. The quantum metrology result considers the ponderomotive squeezing. effect of optical loss on the optimal phase estimation in In principle, we can combine the above-mentioned Michelson or equivalent Mach-Zehnder interferometers. approaches to increase SPP and make the QCRB vanish- We can match this if we ignore the internal squeezing or ingly small, which implies an unbounded sensitivity. focus at high frequencies where the internal optomechan- A clear example of this would be the injection of a very ical squeezing is weak, such that α ¼ 1=4, and also ignore strongly squeezed state, which could increase SPP by more the SRC loss. The corresponding sensitivity limit from the than an order of magnitude. In the presence of optical arm-cavity loss and the external loss is reduced to losses, a highly squeezed state will cease to be a minimum ℏ 2 uncertainty state due to decoherence, and the QCRB in the ϵ ¼ c ϵ þ Tsrc ϵ ð Þ Shh 2 arm ext : 4 lossless case will not be a relevant bound. 4L ω0P 4 In this paper, we present a new general limit to GW 4 ϵ detectors, which cannot be surpassed and is more con- The extra factor of Tsrc= in front of ext can be intuitively straining than the QCRB for realistic interferometer con- understoodas the suppressionofthe loss effect from the signal figurations with optical losses. As we will show, the optical recycling. Furthermore, the impact of the losses discussed losses lead to the following sensitivity limit, to the first here is regularly included in the numerical computation of order of the loss parameter ϵ: quantum noise in GW detectors [32,33]. These numerical 011053-2 QUANTUM LIMIT FOR LASER INTERFEROMETRIC … PHYS. REV. X 9, 011053 (2019) frequency and coincides with the GW signal frequency. Because the system is linear and time invariant, different frequency components are independent of each other. At each frequency, the effects of different optical elements on the quadratures can be quantified by 2 × 2 matrices, 0 which act on the vector aˆ ¼ðaˆ 1; aˆ 2Þ .