Nonlinear Phonon Interferometry at the Heisenberg Limit H. F. H. Cheung, Y. S. Patil, L. Chang, S. Chakram and M. Vengalattore Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853∗ Interferometers operating at or close to quantum limits of precision have found wide applica- tion in tabletop searches for physics beyond the standard model, the study of fundamental forces and symmetries of nature and foundational tests of quantum mechanics. The limits impose√d by quantum fluctuations and measurement backaction on conventional interferometers (δφ ∼ 1/ N) 6 havespurredthedevelopmentofschemestocircumventtheselimitsthroughquantuminterference, 1 multiparticle interactions and entanglement. A prominent example of such schemes, the so-called 0 SU(1,1)interferometer,hasbeenshowntobeparticularlyrobustagainstparticlelossandinefficient 2 detection,andhasbeendemonstratedwithphotonsandultracoldatoms. Here,werealizeaSU(1,1) n interferometer in a fundamentally new platform in which the interfering arms are distinct flexural a modes of a millimeter-scale mechanical resonator. We realize up to 15.4(3) dB of noise squeezing J anddemonstratetheHeisenbergscalingofinterferometricsensitivity(δφ∼1/N),correspondingto 1 a 6-fold improvement in measurement precision over a conventional interferometer. Our work ex- 1 tendstheoptomechanicaltoolboxforthequantummanipulationofmacroscopicmechanicalmotion and presents new avenues for studies of optomechanical sensing and the nonequilibrium dynamics ] of multimode optomechanical systems. h p - t Interferometers are an indispensable metrological tool interactions via radiation pressure [21], geometric design n in the study of fundamental forces [1, 2], the search for [22]orreservoirengineering[23]. Thisraisesprospectsof a u physics beyond the standard model [3] and the measure- manipulatingmacroscopicmechanicalstatesinthequan- q ment of fundamental constants [4]. The realization that tum regime with techniques similar to those in quantum [ conventional interferometry is limited by quantum fluc- or atom optics. 1 tuationsandmeasurementbackactionhasledtothecon- In this work, we realize a nonlinear phonon interfer- v cept of the standard quantum limit (SQL) [5]. This has ometer in a millimeter-scale mechanical resonator and 4 spurredeffortstoobservequantumeffectsinmacroscopic demonstrate Heisenberg scaling of phase sensitivity with 2 interferometers[6]andtocircumventtheSQLviaentan- 3 phonon number. By using quantum-compatible two- glement[7,8],multiparticleinteractions[9]andquantum 2 modenonlinearitiestocreatestrongcorrelationsbetween 0 interference[10,11]. Inabroadersense,theseeffortshave the two modes of the interferometer, we demonstrate up . ledtotheoreticalstudiesaimedatelucidatingthemetro- 1 to 15.4(3) dB of noise squeezing and a 6-fold enhance- 0 logical precision of a macroscopic quantum many-body ment in measurement sensitivity over a conventional in- 6 system and the interplay between entanglement, many- terferometer. The features of our nonlinear interferom- 1 bodyinteractions,topologyandnonlinearities[12–15],as eter are accurately captured by a model applicable to a : v wellasexperimentaleffortstoinvestigatethesequestions generic pair of parametrically coupled oscillators which i in atomic, solid state and hybrid quantum systems. X alsoshowsthattheachievablenoisereductionis,inprin- ciple, unbounded, enabling an unrestricted improvement r Optomechanical systems have emerged as a promis- a in signal-to-noise ratio. ing arena for the investigation of these foundational as- pects of quantum metrology and the innovation of novel A schematic of the nonlinear phonon interferometer is precision measurement technologies [16]. The enormous showninFig.1. Whilethisschematichighlightsthenom- range of size and mass, spanning the nanoscale to the inalsimilaritiestoaMach-ZehnderoraRamseyinterfer- macroscale, the long coherence times that compare fa- ometer,wenotetwokeydifferences. First,thearmsofthe vorably with those realized in atomic or solid state spin interferometer consist of two distinct mechanical modes systems[17–20],andtheabilitytocool,probeandcontrol of a silicon nitride (SiN) membrane resonator. The mo- mechanical motion with radiation pressure have aided tionofthesemodescanbespectroscopicallyresolvedand these efforts. While optomechanical interactions have independently measured via an optical interferometer as thus far been mainly in the weak coupling regime, recent describedinpreviouswork[23]. Unlikeintheopticaldo- workhasdemonstratedthepossibilityofrealizingstrong, main, the phonons in this interferometer are necessarily quantum-compatible nonlinear or multimode mechanical confined within a cavity, i.e. the mechanical resonator, (a) (b) squeezed state preparation PA t ϕ PA coherent mixing signal mode BS t BS PA BS weak measurement idler mode 0 t f Time FIG. 1. A SU(1,1) phonon interferometer. (a) The two arms of the interferometer are distinct mechanical modes at frequences ω and ω . A parametric amplifier interaction (PA) between the two modes generates strong correlations between s i these modes. A phase shift of interest ϕ is then imparted to the signal mode. A timed and pulsed beam splitter interaction (BS) between the modes coherently mixes the two correlated arms resulting in reduced quadrature noise at the outputs. (b) The timing sequence : The input to the interferometer is the coherent state |α ,0(cid:105) prepared at t<0. The signal and idler get s correlatedduringtheparametricamplifierpulsefortimet . Afteravariableinteractionperiod,thetwomodesarecoherently PA mixed by the beam splitter pulse for time t , followed by a weak measurement of the output modes. BS anddonotfreelypropagate. Inthissense,themechanical terferometric modes, hereafter referred to as the ‘signal’ modes are more analogous to intracavity optical fields. and ‘idler’ modes, with resonance frequencies ω and ω s i While these modes are coupled to a thermal reservoir, respectively, g˜ ,g˜ are coupling strengths between the S D their finite response time allows us to transiently over- two modes at the sum and difference frequencies, and comethedeleteriouseffectsoftheenvironmentalcoupling X (t),X (t) are the amplitudes of the supporting sub- S D and generate strong two-mode correlations. Second, in stratemodes(‘pump’)atthesumanddifferencefrequen- contrast to a conventional interferometer, a nondegen- cies. The first term represents the nondegenerate para- erate parametric amplifier takes the place of the input metric oscillator that causes the correlated production of beamsplitter. As proposed in [24], such a configuration, down-converted phonons in the signal and idler modes. also referred to as a SU(1,1) interferometer, exhibits in- The second term signifies the beamsplitter interaction terferometric sensitivity surpassing the SQL due to the thatresultsinthecoherentexchangeofphononsbetween two-modecorrelationscreatedbytheparametricinterac- the signal and idler modes. tion [25, 26]. Importantly, in contrast to interferometry Theparametricamplifierandbeamsplitterinteractions with squeezed or entangled input states, the Heisenberg in our system are independently ascertained. For the ex- scaling of sensitivity in the SU(1,1) interferometer has perimentsdescribedbelow,theresonancefrequenciesand been shown to be robust to particle loss and inefficient damping rates of the signal and idler modes are ω /2π = s detection [27]. 1.233 MHz, ω /2π = 1.466 MHz and γ /2π = 0.083(2) i s Hz, γ /2π = 0.108(3) Hz. As is well known in quantum i The nonlinear phonon interferometer is described by optics,theparametricamplifiershowsaninstabilitywhen the interaction Hamiltonian (see SI) driven past a critical pump amplitude, X , where the S,th system is characterized by a divergent mechanical sus- g˜ X (t) H (t)=i(cid:126) S S (a†a†−a a ) ceptibility and critical dynamics. This instability can be int 2 s i s i regarded in terms of a nonequilibrium continuous phase g˜ X (t) +i(cid:126) D D (a†a −a a†) (1) transition [28, 29]. When the substrate is driven beyond 2 s i s i this threshold, the signal and idler modes self-oscillate, where a ,a are the annihilation operators of the two in- achieving a steady state when their decay rate matches s i 2 (cid:113) (a) 5 r = x2 +y2 [30]. Thecross-quadratureandampli- s,i s,i s,i 4 tude sum, difference squeezing phase diagram is shown 3 in Fig. 2(a), and shows excellent agreement with a no- free-parameter calculation based on the model and inde- ev. 2 pendently measured damping and frequency parameters. D d. The growth of the steady state amplitude of the modes St above threshold shown in Fig. 2(b) is measured to have 1 9 a power-law growth with exponent 0.53±0.03, in close 8 7 agreement with the theoretical prediction (1/2). Also, 6 theexponentialgrowthrateofthemodes’amplitudesin- 5 creases linearly in the drive strength µ, as predicted by 0 1 2 3 μ our parametric amplifier model. (b) The mechanical beamsplitter interaction [31, 32] real- 3]03.0 Amplitude 1.5 izes a coherent transfer of quanta between the signal and /1 Signal nI idler modes. By making weak optical measurements of e [ Idler ev the signal and idler modes, this interaction can also be mplitud2.0 1.0esiR esr wreeaalikz,eidndineppeonsdtehnotcmaenaasluyrseims.enItnstohnisthweosrikg,nwaleapnedrfiodrlemr A T output modes with minimal backaction, and coherently d mi malize1.0 Inverse Rise Time 0.5]s/[ e cboemambisnpelitttheer.mBeaascukr-aedctqiounadevraatduinregsmtoeaesffuercetmtehnetcso[h33er]eanlt- or Signal low for such a realization of the beamsplitter to be ex- N 0 Idler 0.0 tended into the quantum regime. 1.0 1.5 2.0 2.5 3.0 As shown in Fig. 1(b), the nonlinear interferometer is μ realized in the time domain. The signal and idler modes are initialized in a coherent product state |α ,α (cid:105) by in- FIG. 2. The parametric amplifier phase diagram. (a) s i dependently actuating the two modes at times t<0. At Two-mode squeezing below and above the instability thresh- t = 0, the parametric amplifier is actuated for a dura- old(µ=1): normalizedstandarddeviationsofsqueezed(red) andamplified(blue)cross-quadratures. Thesolidlinesareno- tion tPA (cid:28) γs−,i1 by parametrically driving the substrate free-parameter predictions of our model with independently at the sum frequency ω = ω + ω . Subsequently, PA s i measureddampingratesandeigenfrequencies,takingintoac- the signal mode interacts with the parameter of interest countfinitemeasurementtimeanddifferentialsubstratetem- for a variable duration. Lastly, the beamsplitter interac- perature effects (see SI). (b) Steady state amplitudes of the tion is pulsed for a duration t (cid:28) γ−1 by parametri- signalandidlermodesshowapower-lawgrowthof0.53±0.03 BS s,i cally actuating the substrate at the difference frequency consistentwiththepredictionof0.5. Theexponentialgrowth ω = ω −ω , and the two output modes are indepen- rate of the signal idler motions increases linearly with para- BS s i metricamplifieractuationµ. (Allsignalandidlermotionsare dently measured. normalized to their respective thermal motions.) The dynamical behavior of the phonon interferometer isgovernedbytheequationsofmotionfortheinterfering √ √ modes given by a˙ = µ γ γ a† − γs,ia + γ ain, s,i 2 s i i,s 2 s,i s,i s,i where the input fields ain are the coherent input states s,i therateofdownconversionfromthesumfrequencypara- |α ,α (cid:105)withunitvariancenormalizedtothemode’sther- s i metric drive. The strength of the parametric drive can mal motion, i.e. ∆X2 = ∆Y2 = 1, and we have s,in s,in thusbeparametrizedbyµ≡XS/XS,th withµ=1repre- assumed a Markovian reservoir. While the simultane- senting the critical point for the onset of self-oscillation. ous solution to these equations is straightforward for Below threshold, the motional amplitudes of the signal the experimentally relevant case of mismatched damp- and idler modes are zero while their fluctuations are ing rates (γ (cid:54)= γ ) [30], we state the results for identi- i s correlated, and the cross-quadratures xs√±xi,ys√±yi are cal damping rates and note that the conclusions remain 2 2 squeezed. Abovethreshold,themodesgetcorrelatedand essentially unaltered (see SI). For µ (cid:29) 1, the output theiramplitudesum,difference rs√±ri aresqueezed,where variances are minimized for a beamsplitter mixing an- 2 3 tPA [s] Time [s] 4 0 0.060.120.180.24 0.1 1 10 1000 (a) (b) (c) 2 100 20 e10 c 0 n 0 aria PA PA V 1 t=0 t=0 -20 -2 0.1 -20 0 20 Thermal motion -4 0.01 -4 -2 0 2 4 0 1/642/643/644/641/641/161/4 1 4 t [ 2 γ-1] Time [2 γ-1] PA FIG.3. Enhanced transient squeezing – phase space distribution and squeezing dynamics. Bytransientapplication oftheparametricinteraction,two-modesqueezingbeyondthesteady-stateboundof3dBcanbeachieved(seemaintext). (a) Phase space distribution of 15.4±0.3dB squeezed states – (x ,x ) (red) and (y ,y ) (blue). The thermal state (grey) is shown s i s i for reference. (b,c) Dynamics of growth and decay of the two-mode squeezed state in units of the mechanical damping time 2γ−1. The parametric amplifier is driven transiently with strength µ=38(5), and the quadrature variance of 236 iterations is plotted vs time. For comparison, the steady state bound of 3 dB is indicated in grey. The shaded regions represent no-free- parameter bounds due to variations in the parametric drive µ across the iterations. (All signal and idler displacements have been normalized to their respective thermal amplitudes measured at t=0.) gle φ = g˜DXDtBS/2 = −π/4 and are respectively given whereNs ≈(1/4)(Re[αs]eγµtPA/2)2 =(1/4)(Re[αs]×G)2 by (cid:104)∆X2 (cid:105) = 1 + µ e−γ(1+µ)tPA and (cid:104)∆Y2 (cid:105) = is the mean phonon number in the signal mode, showing out 1+µ 1+µ out 1−1µ + µ−µ1eγ(µ−1)tPA where Xout ≡ xs√−2xi, Yout ≡ xs√+2xi tmheenHteriesseonubrecreg(sFcaigli.n4g)o.fAphsaisnedsiceantseitdivbityytwhiethdymneaamsuicrae-l are the output quadratures. The squeezed X-quadrature equations for the interferometric modes, the degree of reducesexponentiallytoavariance1/(1+µ)withatime constant [γ(1+µ)]−1 (cid:28) γ−1 allowing for a significant two-mode squeezing saturates as the parametric pulse s,i duration t approaches the mechanical damping time noise squeezing well beyond the 3 dB bound. By seeding PA γ−1 andthenonlinearinterferometerrevertsbacktoSQL the modes with only thermal motion, i.e. α ,α = 0, we s i scaling (Fig. 4,inset). demonstrate these dynamics and a resultant noise reduc- tion by 15.4±0.3 dB, (Fig. 3). The slow decay to the In conclusion, we demonstrate a SU(1,1) phonon in- thermal state, shown in Fig. 3c, enables an unbounded terferometer capable of Heisenberg-limited phase sensi- improvementofSNRforthedetectionofimpulsiveforces tivity using parametrically coupled mechanical modes in or the detection of forces at enhanced bandwidth. a monolithic SiN membrane resonator. Owing to the Further, the output X-quadrature for a coherent large f × Q product of the interferometric modes, we input state |α ,0(cid:105) can be evaluated from the above demonstrate a substantial degree of transient two-mode s equations of motion to be (cid:104)X (cid:105) = Re[αs][(cosφ − squeezing,achievinganoisereductionof15.4(3)dB,well out 2 sinφ)e−γ(µ+1)tPA/2 +(cosφ+sinφ)eγ(µ−1)tPA/2]. For a beyond the conventional 3 dB bound, and show that this parametric pulse duration t such that t (cid:28) γ−1, transientstateislonglived,survivingontheorderof106 PA PA the remnant noise in this quadrature is (cid:104)∆X2 (cid:105)1/2 ∝ mechanical periods. Our work extends the optomechani- out e−γ(µ+1)tPA (see SI) and the minimum detectable phase caltoolboxforthequantummanipulationofmacroscopic is mechanical motion, and enables new techniques for op- tomechanicalsensingandthemanipulationofmechanical (cid:104)∆X2 (cid:105)1/2 Re[α ] fluctuations. Extendingthesetechniquestothequantum ∆φ= s,out ∝ s (2) d(cid:104)X (cid:105)/d(φ) N regime should allow for studies of macroscopic decoher- s,out s 4 10-5 H.F.H.C.andY.S.P.carriedoutthedataanalysisand modelling. M.V. supervised all stages of the work. All authorscontributedtothepreparationofthemanuscript. Δφ~ N-1/2 Correspondence Correspondence should be ad- s 10-6 dressed to M.V. ([email protected]). 8 φ α Δ 1.0 10-7 1+μ)tPA64 000...867 000...987 Δφ~ N-1 (2 0.9 0.6 s 0.5 0 20 40 60 80 100 10-8 μ 2 3 4 5 6789 2 3 4 5 6789 2 3 1 10 100 G 2 FIG. 4. The Heisenberg scaling of phase sensing in the non- linear phonon interferometer is shown vs the phonon number gain G2. The shot noise limit for conventional interferome- try, i.e. in the absence of two-mode correlations, is shown for comparison (blue). The data correspond to the experi- mental parameters of Fig. 3. The shaded regions represent no-free-parameterboundsduetovariationsintheparametric drive µ across the iterations. Inset: The phonon interferom- eter’s estimated scaling exponent α for the phase sensitivity, δφ∼1/Nα, is shown as a function of the parametric drive µ andtheparametricpulsedurationt ,indicatingthetransi- PA tionfromSQLscaling(α=1/2)toHeisenbergscaling(α=1) as t is reduced (see text). PA ence in highly correlated phononic states. Even in the classical regime, we note that the emergence and decay of two-mode correlations and the ensuing thermalization dynamicsareintimatelytiedtothenatureofthereservoir that couples to the interferometric modes. As such, ul- traprecise phonon interferometers such as demonstrated inthisworkenablethestudyofnon-equilibriumoptome- chanical dynamics, the interferometric detection of non- Markovian dynamics [34] arising from non-Ohmic reser- voirs, and the harnessing of such reservoirs for the cre- ationandstablizationofmacroscopicnon-classicalstates [35]. Acknowledgments This work was supported by the DARPA QuASAR program through a grant from the ARO, the ARO MURI on non-equilibrium Many- bodyDynamics(63834-PH-MUR)andanNSFINSPIRE award. M. V. acknowledges support from the Alfred P. Sloan Foundation. Contributions H. F. H. C., Y. S. P., L. C. and S. C. performed the experimental work and data acquisition. 5 SUPPLEMENTARY INFORMATION for the mechanical thermal motion degrades to 1 as the mechanical amplitude approaches 150 times the room- temperature thermal amplitude. This restricts the dy- Optical detection and stabilization of mechanical modes namic range of membrane amplitudes over which we can study the mechanical interferometer. In this work, we overcomethislimitationbyartificiallyincreasingtheme- The mechanical resonators studied in this work are chanical thermal noise (and effective temperature of the the eigenmodes of a square silicon nitride membrane res- mechanical modes), by driving each mode of the phonon onator fabricated by NORCADA Inc. The membranes interferometer with gaussian noise centered at each me- have lateral dimensions of 5 mm and a thickness of 100 chanicaleigenfrequencyoverabandwidthof10Hz,much nm, with typical mechanical quality factors in the range larger than the respective mechanical linewidths. Tech- of107 [17]. Thedisplacementofthemembranemodesare niques of active RAM control [36] can be used to extend detected using a Michelson optical interferometer with a our measurements to the quantum regime of mechanical position sensitivity of 0.1 pm/Hz1/2 for typical powers of motion. 200 µW incident on the membrane. An external cavity diodelaseroperatingatawavelengthof795nmprovides the light for this interferometer. Parametric amplifier and beam splitter dynamics Due to the differential thermal expansion of the mem- brane and the supporting substrate, the mechanical Thetwo-modenonlinearitiesaredescribedbyaninter- modes of the resonators are susceptible to large fre- action Hamiltonian [23] quency drifts due to temperature fluctuations. Thus, the precise measurement of thermomechanical motion and H (t)=−g X (t)x x (3) int S,D S,D s i non-thermal two-mode correlations requires active sub- linewidth stabilization of the mechanical eigenfrequen- where g parametrize the strength of the interactions, S,D cies. Thisisaccomplishedbyphotothermalcontrolofthe X are the amplitudes of the Silicon substrate ex- S,D silicon substrate. As described in previous work [23], we citations at the sum and difference frequencies, and implement active stabilization by continuously monitor- x are the amplitudes of the individual membrane res- s,i ingthemechanicaleigenfrequencyofahigh-Qmembrane onator modes. Defining a as the annihilation opera- s,i mode at 2.736 MHz - far from the modes of interest in tors of the signal and idler modes, their motional am- this work. Phase sensitive detection of this mode gen- plitudes are given by x = x (a + a† ) where s,i (s,i),0 s,i s,i erates an error signal with an on-resonant phase slope x = ((cid:126)/2m ω )1/2 are their respective zero point (s,i),0 s,i s,i of 5.91 radians/Hz. Active photothermal stabilization of motions. On normalizing the coupling constants as the substrate is accomplished with typical powers of 600 g˜ = g x x /2(cid:126), the interaction Hamiltonian un- S,D S,D s,0 i,0 µW,generatedbyadiodelaserat830nm. Underthisac- der the rotating wave approximation simplifies to tive stabilization, the rms frequency fluctuations of this g˜ X (t) ‘thermometer mode’ are measured to be below 2 mHz, H (t)=i(cid:126) S S (a†a†−a a ) equivalenttotemperaturefluctuationsofthesubstrateof int 2 s i s i g˜ X (t) less than 2 µK. For the modes of the interferometer, this +i(cid:126) D D (a†a −a a†) (4) translates to frequency fluctuations less than 0.002×γ. 2 s i s i Atlowfrequencies(<3Hz),theopticalinterferometer as in the main text. When only the parametric am- used for the detection of mechanical displacement is sus- plifier coupling is pulsed on for time t , the evolu- PA ceptible to residual amplitude modulation (RAM) due tion of the fields is governed by the squeezing Hamil- to gradual temperature fluctuations and temperature- tonian given by the first term. In the Heisenberg pic- dependent birefringence of the various optical elements. ture, a evolve as a (t) = Ga (0)+ga† (0), where s,i s,i s,i i,s In our experiments, this low frequency amplitude noise G = cosh(g˜ X t /2) and g = sinh(g˜ X t /2) S S PA S S PA (measured to be around 10 ppm with respect to the car- parametrize the pulsed parametric amplifier gains. The rier peak) convolves with the mechanical displacement dynamics when the beam splitter coupling is pulsed on signal leading to a 0.75% contamination of the detected for time t is governed by the second term, and a BS s,i membrane displacement and the two-mode correlations. evolve at final time t into a (t ) = cosφa (t )± f s,i f s,i PA Duetothislow-frequencyRAM,thesignal-to-noiseratio sinφa (t ), where the mixing angle φ = g˜ X t /2 i,s PA D D BS 6 characterizesthecoherentmixingbetweenthesignaland belowthreshold(µ<1), theirvaliditywellabovethresh- idler modes due to the beam splitter interaction. old,asinthiswork,isrestrictedtosmalltimest <τ PA s,i (see“PumpDepletion”below). Importantly,becausethe The output variance of the X-quadrature of the sig- nal mode ∆X2 (or of the Y-quadrature of the idler squeezed quadrature variances reduce exponentially to mode ∆Y2 )sd,oeuptends on the mixing angle φ, where the 1/(1+µ)withatimeconstant[γ(1+µ)]−1 (cid:28)τs,i,thereis i,out ampletimeforthesqueezedvariancestobreakthesteady quadraturesaredefinedsuchthata =X +iY . For s,i s,i s,i state squeezing bound of 3 dB [23, 30]. thecoherentstates|α ,α (cid:105)whichformtheinputtothein- s i terferometer, the variances are (∆Xin )2 =(∆Yin )2 = (s,i) (s,i) 1, where the motion of each mode is normalized to its thermal or quantum zero-point motion. For G,g (cid:29) 1, the output variances are minimized for φ = −π/4 Pump Depletion and equals ∆X2 (t ) = ∆Y2 (t ) = 1/(G+g)2 = s,out f i,out f exp(−g˜ X t ). This improves the SNR of measuring S S PA the phase φ by an exponential factor exp(−g˜ X t /2) The assumption that µ(t) = µ is a constant over the S S PA as compared to that achieved using a conventional inter- course of evolution of the parametric amplifier pulse is ferometer,forwhich∆X2 (t )=∆Y2 (t )=1. This invalid for µ ≡ µ(t = 0) > 1. As the signal and idler s,out f i,out f 0 improvement is, in principle, unbounded - the paramet- amplitudes grow exponentially, the absolute downcon- ric amplifier gains G≈g =sinh(g˜ X t /2) diverge on verted phonon loss rate increases before balancing out S S PA increasing the argument. with the absolute downconversion rate, thereby decreas- Unlike in the optical domain, phononic fields are nec- ingµ(t)= XS(t). Thiseffect,whicharisesfromtheinter- XS,th essarilyconfinedinacavity–themechanicalresonator– ference of the signal, idler and pump modes, is referred and do not propagate freely without losses. The lossy to as pump depletion. system can be formally modelled as a linear coupling To quantify the effect of pump depletion, we first con- of the mechanical modes to an environmental bath of sider the equations of motion of the signal, idler and harmonic oscillators. For a Markovian bath, the dy- sum-frequency pump modes. Within the rotating wave namics can be evaluated using the input-output formal- approximation, the Hamiltonian (3) gives ism, and for the case of matched frequencies and loss √ wrahteerseiµs =goXveSrn/XedS,tbhypaa˙sr,aim=etriµ2zeγsa†it,hse−paγ2raasm,iet+ric pγuamisn,ip, x¨s,i+γs,ix˙s,i+ωs2,ixs,i = m1s,i(Fs,i(t)+ g2SXSx∗i,s) drive strength at the sum frequency. Defining the cross- 1 g √ X¨ +γ X˙ +ω2X = (F (t)+ Sx x ) (5) quadrature modes d± = (as ±ai)/ 2, the equat√ions of S S S S S mS S 2 s i motions are rewritten as X˙ = γ(µ−1)X + γXin and Y˙ = −γ(µ + 1)Y d++ √γ2Yin, and ds+imilarly fdo+r where xs,i,XS denote the complex displacement of each d+ 2 d+ d+ mode and ms,i,S, ωs,i,S, γs,i,S and Fs,i,S denote respec- X ,Y . Assuming an initial thermal or quantum vac- d− d− tively the masses, frequencies, damping rates and forces uum seed, the dynamics of their variances is evaluated oneachofthemodes. Thesecoupledequationsofmotion to be (cid:104)∆Y2 (t)(cid:105) = (cid:104)∆X2 (t)(cid:105) = 1 + µ e−γ(µ+1)t, d+ d− 1+µ 1+µ can be solved using two timescale perturbation theory which are squeezed, and (cid:104)∆Xd2+(t)(cid:105) = (cid:104)∆Yd2−(t)(cid:105) = [37] and simplify to the first order coupled equations, 1 + µ eγ(µ−1)t, which are amplified. 1−µ µ−1 2A˙ =γ (cid:104)−A +igSχ A∗A +iχ F˜ (t)(cid:105) Forabeamsplitterinteractionwithmixingangleφ,the s s s 2 s i S s s signal output quadratures are expressed in terms of the 2A˙ =γ (cid:104)−A +igSχ A∗A +iχ F˜(t)(cid:105) cross-quadratures as X = sin(φ+π/4)X +cos(φ+ i i i 2 i s S i i Tπ/h4e)Xvadr−iaanncde YoofutX=sinois(uφtm+inπi/m4i)zYedd+f+orcoas(mφd+i+xiπn/g4)aYndg−le. 2A˙S =γS(cid:104)−AS +ig2SχSAsAi+iχSF˜S(t)(cid:105) (6) out φan=d Y−π/4=, i.Ye. w,haenndththeeqduyandaramtuicrsesofarteheXirouvta=riaXncde−s where xk = Ake−iωkt, k ∈ [i,s,S]; F˜k are the slowly out d− varying (complex) amplitudes of the external forces on is (cid:104)∆Xo2ut(cid:105) = 1+1µ + 1+µµe−γ(1+µ)tPA and (cid:104)∆Yo2ut(cid:105) = the individual modes, and χk = (mkωkγk)−1 are their 1 + µ eγ(µ−1)tPA,wheretheinputsain areassumedto on-resonant susceptibilities. We ignore terms such as 1−µ µ−1 s,i becoherentstates|α (cid:105). Whilethesedynamicsareexact A¨ ,γ A˙ in the slow time approximation. These are fur- s,i k i k 7 ther simplified to with feedback and other control systems. 2A˜˙ =γ [−A˜ +iA˜∗A˜ +iF (t)] s s s i S s 2A˜˙ =γ [−A˜ +iA˜∗A˜ +iF (t)] Heisenberg-scaling of phase sensitivity i i i s S i 2A˜˙ =γ [−A˜ +iA˜ A˜ +iF (t)] (7) S S S s i S The interferometer output X-quadrature for a coher- where the motion of the pump A˜ is normalized with re- ent input state |αs,0(cid:105) can be calculated from the above S equations of motion to be specttoA –thecriticalvaluewhichdefinestheinsta- S,th tbhileiitrycthharersahcoteldrisµti=c s1t,eAa˜ds,yi-astraetneomrmoatiloiznedabwoitvhertehsrpeeschtotldo (cid:104)Xs,out(cid:105)= R√e[αs](cid:18)cosφ√−sinφe−γ(µ+1)tPA/2 2 2 mgSo√dχ2eSsχ,iw,si,tahnFdF(tk)a=reµn(otr).mLalaiszteldy,fowreceasssounmtheethreesepmecptiirvie- + cosφ√+sinφeγ(µ−1)tPA/2(cid:19) (10) S 2 calfactthatthepumpmode,whichisaSiliconsubstrate (cid:18) d(cid:104)X (cid:105) Re[α ] −sinφ−cosφ mode,hasamuchlargerdampingratethanthesignaland s,out = √ s √ e−γ(µ+1)tPA/2 idlerresonatormodes,i.e. γ ∼(103−104)γ ,γ ,thereby d(φ) 2 2 S s i (cid:19) ensuring that the pump motion adiabatically follows the −sinφ+cosφ signal and idler motions, i.e. A˜S =iA˜sA˜i+iFS(t). + √2 eγ(µ−1)tPA/2 (11) A˜ and A˜ increase exponentially as exp(γ(µ −1)t/2) s i 0 For large phonon gain of the parametric amplifier, the after the parametric amplifier pulse is switched on, caus- effectsofpumpdepletioncauseadecreaseinnoisesqueez- ing µ(t) to deplete and settle to a steady state of ing and a reversion of the interferometer to SQL scaling t→∞ √ µ(t) −−−→ 1 [30]. For µ0 > 1, the transient squeezing (δφ∼1/ Ns). Assuch,thechoiceoftheparametricam- expression derived assuming constant µ(t) = µ0 is valid plifier duration tPA is not independent of the parametric only for small times tPA < τs,i (see also [38, 39]). Con- drive µ. To take this mutual dependence into account, sidering a case where the seed motions are A˜s0 and A˜i0, we parametrize the amplifier duration as tPA = γlo(g1[+kµµ)], a pump depletion by a factor η occurs when where k > 1/µ is a dimensionless parameter. With this parametrization,theremnantnoiseandtheminimumde- |A˜ A˜ |=η|F |=ηµ (8) s i S 0 tectable motion ∆X is thus given by D √ Dropping the absolute modulus sign for clarity, this oc- (cid:104)∆X2 (cid:105)1/2 = k+1e−γ(µ+1)tPA/2 s,out curs when (cid:104)∆X2 (cid:105)1/2 Re[α ] A˜s0A˜i0eγ(µ0−1)tPA =ηµ0 ∆φ= d(cid:104)Xss,o,ouutt(cid:105)/dφ ∝ Nss (12) FortimetPA ∼ γln((11+0µµ00)),thesqueezedvariancesreduceto where Ns ≈ (Re[αs]eγµtPA/2)2/4 = G2 ×(Re[αs]/2)2 is within 10% of 1 , and for this time t , themeanphononnumberinthesignalmode,whichisthe 1+µ0 PA measurement resource. Here, we assume that Re[α ] (cid:29) s η ∼(10µ0)µµ00−+11A˜sµ0A˜i0 −µ−0−(cid:29)−→1 10A˜s0A˜i0 (9) 1phaonndoneγnµutmPAbe(cid:29)r ga1i,n.and define G2 = eγµtPA to be the 0 For the motional amplitudes in the present study (data in Figs(3,4) of the main text), A˜s0 ≈ A˜i0 ≈ 0.03, and Effect of finite substrate temperature η ≈0.01, i.e. there is a mere 1% pump depletion even at almost saturated squeezing. To reduce the contribution of residual amplitude mod- Clearly, the degradation of squeezing due to pump de- ulation in the optical interferometer to the signal and pletion sets in sooner for (i) larger seeds A˜ ,A˜ , (ii) idler motion readout, all data are acquired by artificially s0 i0 larger damping of the signal, idler and pump modes, and driving the signal and idler modes to an elevated tem- (iii) larger nonlinear signal-idler couplings g˜ (see also perature, corresponding to an effective thermal motion S [39]). Wenotethatthedeleteriouseffectsofpumpdeple- around 40-50 times the room temperature amplitude, by tioncanbeavoidedbycompensatingthepumpdepletion driving the two modes with a gaussian noise source with 8 a bandwidth about 100 times larger than the respective mismatch dependent, but agress with the expression in mechanical linewidths. The substrate mode, however, is the main text to within 10% for asymmetry parameters not artificially driven to a larger temperature. |δω − δγ| < 0.20 where δω = ωωss−+ωωii, δγ = γγss−+γγii [30], The substrate mode fluctuations couple equally to the and (iii) the time constants of transient evolution of the signal and idler mode, and thus the amplitude-difference squeezed and amplified quadratures are respectively al- (suxbs√−s2txria)tesqfluuecetzuinagtioisnsi.ndHepoewnedveenr,t soufbasntrdatreobfluuscttutaotiaonnys tered to |γs+2γi(1±µ(cid:113)1−δγ2)|−1 for µ(cid:29) √1δ−γδγ2. couple to the amplitude-sum quadrature (xs√+xi). Thus 2 with equal substrate, signal, and idler temperatures, the amplitude-sum variance is bounded at 1/2 in the limit of large parametric drive strengths, µ(cid:29)1. However, if the ∗ [email protected] [1] K. S. Thorne, C. M. Caves, V. D. Sandberg, M. Zim- substratetemperatureismuchlower thanthesignaland merman, and R. W. P. Drever. The Quantum limit for idler temperatures, the substrate fluctuations are neg- Gravitational wave detectors and methods of circumvent- ligible in comparison, and the amplitude-sum squeezing ing it. Cambridge University Press, 1979. beatstheconventional3dBlimit–infact,theamplitude- [2] S. Dimopoulos, P. Graham, J. Hogan, and M. Kasevich. sum variance equals 1 . This difference in the signal Testinggeneralrelativitywithatominterferometry.Phys. 2(µ−1) andidlerandsubstratetemperatureshasbeenaccounted Rev. Lett., 98:111102, 2007. [3] G. M. Tino and M. A. Kasevich, editors. Atom In- for in Fig. 2a of the main text. terferometry, Proceedings of the International School of Physics ”Enrico Fermi”. IOS Press, Inc., 2014. [4] A.D.Cronin,J.Schmiedmayer,andD.E.Pritchard.Op- Effect of finite measurement time tics and interferometry with atoms and molecules. Rev. Mod. Phys., 81:1051, 2009. To accurately model the measurements described in [5] C. M. Caves. Quantum-mechanical radiation-pressure thiswork,theeffectsoffinitemeasurementdurationneed fluctuationsinaninterferometer. Phys.Rev.Lett.,45:75, to be considered. This is particularly relevant for the 1980. [6] C.M.Caves,K.S.Thorne,R.W.P.Drever,V.D.Sand- high-Q resonators used in this work. In the vicinity of berg, and M. Zimmermann. On the measurement of a the critical point µ = 1, the mechanical parametric am- weakclassicalforcecoupledtoaquantum-mechanicalos- plifier exhibits a divergent response time that results in cillator. 1. issues of principle. Rev. Mod. Phys., 52:341, extremely long thermalization times (∼ 104 − 105 sec- 1980. onds). Thus, for typical measurement durations in this [7] D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, work (∼ 100 seconds), the measured squeezing spectra J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and can deviate appreciably from those computed in steady D.J.Wineland. Towardheisenberg-limitedspectroscopy with multiparticle entangled states. Science, 304:1476, state. 2004. Inourmodel, wetakethefinitemeasurementduration [8] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, into account by computing the variances measured over M. V. Balabas, and E. S. Polzik. Quantum noise limited a measurement time τm using the truncated integral of and entanglement-assisted magnetometry. Phys. Rev. the relevant spectral density as indicated below [30], Lett., 104:133601, 2010. [9] M.Napolitano,M.Koschorreck,B.Dubost,N.Behbood, (cid:90) ∞ R.J.Sewell,andM.W.Mitchell.Interaction-basedquan- σ =2 S (ω)dω (13) α,β α,β tum metrology showing scaling beyond the heisenberg 2π/τm limit. Nature, 471:486, 2011. [10] K.Hammerer,M.Aspelmeyer,E.S.Polzik,andP.Zoller. Establishing einstein-podolsky-rosen channels between Transient squeezing in the presence of mismatched nanomechanics and atomic ensembles. Phys. Rev. Lett., damping rates 102:020501, 2009. [11] Mankei Tsang and Carlton M. Caves. Evading quantum In the case of mismatched frequencies and damping mechanics: Engineering a classical subsystem within a rates for the signal and idler modes, the main differ- quantumenvironment.Phys.Rev.X,2:031016,Sep2012. ences are (i) a small deviation of the optimal beam split- [12] S.Boixo,A.Datta,M.J.Davis,S.T.Flammia,A.Shaji, andC.M.Caves. Quantummetrology: Dynamicsversus ter mixing angle away from g X t = −π/4, (ii) the D D BS entanglement. Phys. Rev. Lett., 101:040403, 2008. minimum variance of the squeezed quadrature becomes 9 [13] A.Negretti,C.Henkel,andK.Molmer.Quantum-limited [29] K. Dechoum, P. D. Drummond, S. Chaturvedi, and position measurements of a dark matter-wave soliton. M. D. Reid. Critical fluctuations and entanglement in Phys. Rev. A, 77:043606, 2008. the nondegenerate parametric amplifier. Phys. Rev. A, [14] M. J. W. Hall and H. M. Wiseman. Does nonlinear 70:053807, 2004. metrologyofferimprovedresolution? answersfromquan- [30] S Chakram, Y S Patil, and M Vengalattore. Multimode tum information theory. Phys. Rev. X, 2:041006, 2012. phononiccorrelationsinanondegenerateparametricam- [15] M. J. Woolley, G. J. Milburn, and C. M. Caves. Non- plifier. New Journal of Physics, 17(6):063018, 2015. linearquantummetrologyusingcouplednanomechanical [31] T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and resonators. New J. Phys., 10:125018, 2008. E.M.Weig. Coherentcontrolofaclassicalnanomechan- [16] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt. icaltwo-levelsystem. Nature Physics,9:485–488,August Cavity optomechanics. Rev. Mod. Phys., 86:1391, 2014. 2013. [17] S.Chakram,Y.S.Patil,L.Chang,andM.Vengalattore. [32] H. Okamoto, A. Gourgout, C. Chang, K. Onomitsu, Dissipationinultrahighqualityfactorsinmembraneres- I. Mahboob, E. Y. Chang, and H. Yamaguchi. Coherent onators. Phys. Rev. Lett., 112:127201, 2014. phonon manipulation in coupled mechanical resonators. [18] M. Yuan, M. A. Cohen, and G. Steele. Silicon nitride Nature Phys., 9:480, 2013. membrane resonators at millikelvin temperatures with [33] E.E.Wollman,C.U.Lei,A.J.Weinstein,J.Suh,A.Kro- quality factors exceeding 108. arXiv:1510.07468, 2015. nwald, F. Marquardt, A. A. Clerk, and K. C. Schwab. [19] C. Reinhardt, T. Mu¨ller, A. Bourassa, and J. C. Sankey. Quantumsqueezingofmotioninamechanicalresonator. Ultralow-noise sin trampoline mems for sensing and op- Science, 349:952, 2015. tomechanics. arXiv:1511.01769, 2015. [34] S. Gro¨blacher, A. Trubarov, N. Prigge, G. D. Cole, [20] R. A. Norte, J. P. Moura, and S. Gr¨oblacher. Mechan- M. Aspelmeyer, and J. Eisert. Observation of non- ical resonators for quantum optomechanics experiments markovian micromechanical brownian motion. Nature at room temperature. arXiv:1511.06235, 2015. Comm., 6:7606, 2015. [21] A. Pontin, M. Bonaldi, A. Borrielli, L. Marconi, [35] T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. F. Marino, G. Pandraud, G. A. Prodi, P. M. Sarro, Kippenberg. Nonlinear quantum optomechanics via in- E. Serra, and F. Marin. Dynamical two-mode squeez- dividual intrinsic two-level defects. Phys. Rev. Lett., ing of thermal fluctuations in a cavity optomechanical 110:193602, 2013. system. arXiv:1509.02723, 2015. [36] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, [22] I. Mahboob, H. Okamoto, K. Onomitsu, and H. Yam- C.Hagemann,T.Legero,U.Sterr,F.Riehle,G.D.Cole, aguchi. Two-modesqueezinginanelectromechanicalres- and M. Aspelmeyer. Reduction of residual amplitude onator. Phys. Rev. Lett., 113:167203, 2014. modulation to 1×10−6 for frequency modulation and [23] Y.S.Patil,S.Chakram,L.Chang,andM.Vengalattore. laser stabilization. Opt. Lett., 39:1980, 2014. Thermomechanicaltwo-modesqueezinginanultrahighq [37] R. Lifshitz and M. C. Cross. Nonlinear dynamics of membraneresonator. Phys.Rev.Lett.,115:017202,2015. nanomechanical and micromechanical resonators. In [24] B. Yurke, S. L. McCall, and J. R. Klauder. Su(2) and H. G. Schuster, editor, Reviews of nonlinear dynamics su(1,1) interferometers. Phys. Rev. A, 33:4033, 1986. and complexity, page 1, 2008. [25] C. Gross, T. Zibold, E. Nicklas, J. Est`eve, and M. K. [38] M. Wolinsky and H.J. Carmichael. Squeezing in the de- Oberthaler. Nonlinear atom interferometer surpasses generate parametric oscillator. Optics Communications, classical precision limit. Nature, 464:1165, 2010. 55(2):138 – 142, 1985. [26] F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, [39] H.J. Carmichael and M. Wolinsky. Squeezing in the and W. Zhang. Quantum metrology with parametric degenerate parametric oscillator using the positive p- amplifier-based photon correlation interferometers. Na- representation. In JohnD. Harvey and DanielF. Walls, ture Comm., 5:3049, 2014. editors, Quantum Optics IV, volume 12 of Springer Pro- [27] A. M. Marino, N. V. Corzo Trejo, and P. D. Lett. Effect ceedings in Physics, pages 208–220. Springer Berlin Hei- of losses on the performance of a su(1,1) interferometer. delberg, 1986. Phys. Rev. A, 86:023844, 2012. [28] C.C.Gerry.Ground-statephasetransitionsofthedegen- erate parametric amplifier. Phys. Rev. A, 37:3619, 1988. 10