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 application 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 imposed√ by quantum fluctuations and measurement backaction on conventional interferometers (δφ ∼ 1/ N) have spurred the development of schemes to circumvent these limits through quantum interference, multiparticle interactions and entanglement. A prominent example of such schemes, the so-called SU(1, 1) interferometer, has been shown to be particularly robust against particle loss and inefficient detection, and has been demonstrated with photons and ultracold atoms. Here, we realize a SU(1, 1) interferometer in a fundamentally new platform in which the interfering arms are distinct flexural modes of a millimeter-scale mechanical resonator. We realize up to 15.4(3) dB of noise squeezing and demonstrate the Heisenberg scaling of interferometric sensitivity (δφ ∼ 1/N), corresponding to a 6-fold improvement in measurement precision over a conventional interferometer. Our work extends the optomechanical toolbox for the quantum manipulation of macroscopic mechanical motion and presents new avenues for studies of optomechanical sensing and the nonequilibrium dynamics of multimode optomechanical systems.

Interferometers are an indispensable metrological tool interactions via radiation pressure [21], geometric design in the study of fundamental forces [1,2], the search for [22] or reservoir engineering [23]. This raises prospects of physics beyond the standard model [3] and the measure- manipulating macroscopic mechanical states in the quan- 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. tuations and measurement backaction has led to the con- In this work, we realize a nonlinear phonon interfer- cept of the standard quantum limit (SQL) [5]. This has ometer in a millimeter-scale mechanical resonator and spurred efforts to observe quantum effects in macroscopic demonstrate Heisenberg scaling of phase sensitivity with interferometers [6] and to circumvent the SQL via entan- phonon number. By using quantum-compatible two- glement [7,8], multiparticle interactions [9] and quantum mode nonlinearities to create strong correlations between interference [10, 11]. In a broader sense, these efforts have the two modes of the interferometer, we demonstrate up led to theoretical studies aimed at elucidating the metro- to 15.4(3) dB of noise squeezing and a 6-fold enhance- logical precision of a macroscopic quantum many-body ment in measurement sensitivity over a conventional in- system and the interplay between entanglement, many- terferometer. The features of our nonlinear interferom- body interactions, topology and nonlinearities [12–15], as eter are accurately captured by a model applicable to a well as experimental efforts to investigate these questions generic pair of parametrically coupled oscillators which in atomic, solid state and hybrid quantum systems. also shows that the achievable noise reduction is, in prin- Optomechanical systems have emerged as a promis- ciple, unbounded, enabling an unrestricted improvement arXiv:1601.02324v1 [quant-ph] 11 Jan 2016 ing arena for the investigation of these foundational as- in signal-to-noise ratio. pects of quantum metrology and the innovation of novel A schematic of the nonlinear phonon interferometer is precision measurement technologies [16]. The enormous shown in Fig.1. While this schematic highlights the nom- range of size and mass, spanning the nanoscale to the inal similarities to a Mach-Zehnder or a Ramsey interfer- macroscale, the long coherence times that compare fa- ometer, we note two key differences. First, the arms of the vorably with those realized in atomic or solid state spin interferometer consist of two distinct mechanical modes systems [17–20], and the ability to cool, probe and control of a silicon nitride (SiN) membrane resonator. The mo- mechanical motion with radiation pressure have aided tion of these modes can be spectroscopically resolved and these efforts. While optomechanical interactions have independently measured via an optical interferometer as thus far been mainly in the weak coupling regime, recent described in previous work [23]. Unlike in the optical do- work has demonstrated the possibility of realizing strong, 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 tBS PA BS weak measurement

idler mode

0 tf Time

FIG. 1. A SU(1, 1) phonon interferometer. (a) The two arms of the interferometer are distinct mechanical modes at frequences ωs and ωi. A parametric amplifier interaction (PA) between the two modes generates strong correlations between 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 |αs, 0i prepared at t < 0. The signal and idler get correlated during the parametric amplifier pulse for time tPA. After a variable interaction period, the two modes are coherently mixed by the beam splitter pulse for time tBS , followed by a weak measurement of the output modes. and do not freely propagate. In this sense, the mechanical terferometric modes, hereafter referred to as the ‘signal’ modes are more analogous to intracavity optical fields. and ‘idler’ modes, with resonance frequencies ωs and ωi While these modes are coupled to a thermal reservoir, respectively,g ˜S, g˜D are coupling strengths between the their finite response time allows us to transiently over- two modes at the sum and difference frequencies, and come the deleterious effects of the environmental coupling XS(t),XD(t) are the amplitudes of the supporting sub- and generate strong two-mode correlations. Second, in strate modes (‘pump’) at the sum and difference frequen- 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 that results in the coherent exchange of phonons between two-mode correlations created by the parametric interac- the signal and idler modes. tion [25, 26]. Importantly, in contrast to interferometry The parametric amplifier and beamsplitter interactions 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 periments described below, the resonance frequencies and been shown to be robust to particle loss and inefficient damping rates of the signal and idler modes are ωs/2π = detection [27]. 1.233 MHz, ωi/2π = 1.466 MHz and γs/2π = 0.083(2) Hz, γi/2π = 0.108(3) Hz. As is well known in quantum The nonlinear phonon interferometer is described by optics, the parametric amplifier shows an instability when the interaction Hamiltonian (see SI) driven past a critical pump amplitude, XS,th, where the system is characterized by a divergent mechanical sus- g˜ X (t) H (t) = i 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˜DXD(t) † † + i~ (asai − asai ) (1) transition [28, 29]. When the substrate is driven beyond 2 this threshold, the signal and idler modes self-oscillate, where as, ai are the annihilation operators of the two in- achieving a steady state when their decay rate matches

2 q (a) 5 2 2 rs,i = xs,i + ys,i [30]. The cross-quadrature and ampli- 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- . 2 pendently measured damping and frequency parameters. The growth of the steady state amplitude of the modes

Std. Dev above threshold shown in Fig.2(b) is measured to have 1 9 8 a power-law growth with exponent 0.53 ± 0.03, in close 7 agreement with the theoretical prediction (1/2). Also, 6 5 the exponential growth rate of the modes’ amplitudes increases 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 izes a coherent transfer of quanta between the signal and

Amplitude

3.0 1.5 Rise Inverse Signal idler modes. By making weak optical measurements of Idler the signal and idler modes, this interaction can also be

2.0 1.0 realized in post hoc analysis. In this work, we perform

weak, independent measurements on the signal and idler

T ime [/s] ime Amplitude [/10 output modes with minimal backaction, and coherently 1.0 0.5 combine the measured quadratures to eﬀect the coherent beamsplitter. Back-action evading measurements [33] al- Inverse Rise Time Signal low for such a realization of the beamsplitter to be ex-

Normalized Idler 0 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 i by in- FIG. 2. The parametric amplifier phase diagram. (a) s i Two-mode squeezing below and above the instability thresh- dependently actuating the two modes at times t < 0. At old (µ = 1): normalized standard deviations of squeezed (red) t = 0, the parametric amplifier is actuated for a dura- −1 and amplified (blue) cross-quadratures. The solid lines are no- tion tPA γs,i by parametrically driving the substrate free-parameter predictions of our model with independently at the sum frequency ωPA = ωs + ωi. Subsequently, measured damping rates and eigenfrequencies, taking into ac- the signal mode interacts with the parameter of interest count finite measurement time and differential substrate tem- 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 γ−1 by parametri- signal and idler modes show a power-law growth of 0.53±0.03 BS s,i consistent with the prediction of 0.5. The exponential growth cally actuating the substrate at the difference frequency rate of the signal idler motions increases linearly with para- ωBS = ωs − ωi, and the two output modes are indepen- metric amplifier actuation µ. (All signal and idler motions are dently measured. normalized to their respective thermal motions.) The dynamical behavior of the phonon interferometer is governed by the equations of motion for the interfering µ √ † γs,i √ in modes given bya ˙ s,i = 2 γsγiai,s − 2 as,i + γs,ias,i, in where the input fields as,i are the coherent input states the rate of downconversion from the sum frequency para- |αs, αii with unit variance normalized to the mode’s ther- metric drive. The strength of the parametric drive can 2 2 mal motion, i.e. ∆Xs,in = ∆Ys,in = 1, and we have thus be parametrized by µ ≡ XS/XS,th with µ = 1 repre- 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 (γi 6= γs)[30], we state the results for identi- correlated, and the cross-quadratures xs√±xi , ys√±yi are 2 2 cal damping rates and note that the conclusions remain squeezed. Above threshold, the modes get correlated and essentially unaltered (see SI). For µ 1, the output their amplitude sum, difference rs√±ri are squeezed, where 2 variances are minimized for a beamsplitter mixing an-

3 t [s] PA Time [s] 4 0 0.06 0.12 0.18 0.24 0.1 1 10 1000 (a) (b) (c)

2 100

20 10 0 0 PA PA Variance 1 t=0 t=0 -20 -2 0.1 -20 0 20 Thermal motion

-4 0.01 -4 -2 0 2 4 0 1/64 2/64 3/64 4/64 1/641/16 1/4 1 4 t [ 2 -1] Time [2 -1] PA γ γ

FIG. 3. Enhanced transient squeezing – phase space distribution and squeezing dynamics. By transient application of the parametric interaction, two-mode squeezing beyond the steady-state bound of 3 dB can be achieved (see main text). (a) Phase space distribution of 15.4 ± 0.3dB squeezed states – (xs, xi) (red) and (ys, yi) (blue). The thermal state (grey) is shown 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.)

γµtPA/2 2 2 gle φ =g ˜DXDtBS/2 = −π/4 and are respectively given where Ns ≈ (1/4)(Re[αs]e ) = (1/4)(Re[αs]×G) 2 1 µ −γ(1+µ)tPA 2 by h∆Xouti = 1+µ + 1+µ e and h∆Youti = is the mean phonon number in the signal mode, showing 1 + µ eγ(µ−1)tPA where X ≡ xs√−xi , Y ≡ xs√+xi the Heisenberg scaling of phase sensitivity with measure- 1−µ µ−1 out 2 out 2 are the output quadratures. The squeezed X-quadrature ment resource (Fig.4). As indicated by the dynamical reduces exponentially to a variance 1/(1 + µ) with a time equations for the interferometric modes, the degree of −1 −1 two-mode squeezing saturates as the parametric pulse constant [γ(1 + µ)] γs,i allowing for a signiﬁcant noise squeezing well beyond the 3 dB bound. By seeding duration tPA approaches the mechanical damping time γ−1 and the nonlinear interferometer reverts back to SQL the modes with only thermal motion, i.e. αs, αi = 0, we demonstrate these dynamics and a resultant noise reduc- scaling (Fig.4,inset). 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- improvement of SNR for the detection of impulsive forces 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 |αs, 0i can be evaluated from the above demonstrate a substantial degree of transient two-mode Re[αs] squeezing, achieving a noise reduction of 15.4(3) dB, well equations of motion to be hXouti = 2 [(cos φ − sin φ)e−γ(µ+1)tPA/2 + (cos φ + sin φ)eγ(µ−1)tPA/2]. For a beyond the conventional 3 dB bound, and show that this −1 transient state is long lived, surviving on the order of 106 parametric pulse duration tPA such that tPA γ , 2 1/2 mechanical periods. Our work extends the optomechani- the remnant noise in this quadrature is h∆Xouti ∝ e−γ(µ+1)tPA (see SI) and the minimum detectable phase cal toolbox for the quantum manipulation of macroscopic is mechanical motion, and enables new techniques for optomechanical sensing and the manipulation of mechanical h∆X2 i1/2 Re[α ] ﬂuctuations. Extending these techniques to the quantum ∆φ = s,out ∝ s (2) d hXs,outi /d(φ) Ns regime should allow for studies of macroscopic decoher-

4 10-5 H. F. H. C. and Y. S. P. carried out the data analysis and modelling. M.V. supervised all stages of the work. All authors contributed to the preparation of the manuscript. ~ N -1/2 Δφ s Correspondence Correspondence should be ad- 10-6 dressed to M.V. ([email protected]). 8 α 1.0 Δφ 6 0.9

t 0.6 0.8

μ) 0.7 -7 4 0.7 10 0.8 -1

(1+ 0.9 0.6 Δφ ~ N 2 s 0.5

0 20 40 60 80 100 μ 10-8 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 1 10 G 2 100

FIG. 4. The Heisenberg scaling of phase sensing in the nonlinear phonon interferometer is shown vs the phonon number gain G2. The shot noise limit for conventional interferometry, i.e. in the absence of two-mode correlations, is shown for comparison (blue). The data correspond to the experimental parameters of Fig.3. The shaded regions represent no-free-parameter bounds due to variations in the parametric drive µ across the iterations. Inset: The phonon interferometer’s estimated scaling exponent α for the phase sensitivity, δφ ∼ 1/N α, is shown as a function of the parametric drive µ and the parametric pulse duration tPA, indicating the transition from SQL scaling (α = 1/2) to Heisenberg scaling (α = 1) as tPA is reduced (see text).

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 dynamics are intimately tied to the nature of the reservoir that couples to the interferometric modes. As such, ul- traprecise phonon interferometers such as demonstrated in this work enable the study of non-equilibrium optomechanical dynamics, the interferometric detection of non- Markovian dynamics [34] arising from non-Ohmic reservoirs, and the harnessing of such reservoirs for the cre- ation and stablization of macroscopic non-classical states [35]. Acknowledgments This work was supported by the DARPA QuASAR program through a grant from the ARO, the ARO MURI on non-equilibrium Many- body Dynamics (63834-PH-MUR) and an NSF INSPIRE 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- Optical detection and stabilization of mechanical temperature thermal amplitude. This restricts the dy- modes namic range of membrane amplitudes over which we can study the mechanical interferometer. In this work, we overcome this limitation by artificially increasing the me- 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 chanical eigenfrequency over a bandwidth of 10 Hz, much nm, with typical mechanical quality factors in the range larger than the respective mechanical linewidths. Tech- of 107 [17]. The displacement of the membrane modes are 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 diode laser operating at a wavelength of 795 nm provides the light for this interferometer. Parametric amplifier and beam splitter dynamics Due to the differential thermal expansion of the membrane and the supporting substrate, the mechanical The two-mode nonlinearities are described by an inter- 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 Hint(t) = −gS,DXS,D(t)xsxi (3) non-thermal two-mode correlations requires active sub- linewidth stabilization of the mechanical eigenfrequen- where gS,D parametrize the strength of the interactions, cies. This is accomplished by photothermal control of the XS,D are the amplitudes of the Silicon substrate ex- silicon substrate. As described in previous work [23], we citations at the sum and difference frequencies, and implement active stabilization by continuously monitor- xs,i are the amplitudes of the individual membrane res- ing the mechanical eigenfrequency of a high-Q membrane onator modes. Defining as,i as the annihilation opera- 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 xs,i = x(s,i),0(as,i + as,i) where 1/2 erates an error signal with an on-resonant phase slope x(s,i),0 = (~/2ms,iωs,i) are their respective zero point 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˜S,D = gS,Dxs,0xi,0/2~, the interaction Hamiltonian un- µW, generated by a diode laser at 830 nm. Under this ac- der the rotating wave approximation simplifies to tive stabilization, the rms frequency fluctuations of this ‘thermometer mode’ are measured to be below 2 mHz, g˜SXS(t) † † Hint(t) = i~ (asai − asai) equivalent to temperature fluctuations of the substrate of 2 less than 2 µK. For the modes of the interferometer, this g˜DXD(t) † † + i~ (asai − asai ) (4) translates to frequency fluctuations less than 0.002 × γ. 2 At low frequencies (< 3 Hz), the optical interferometer 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 tPA, the evolu- 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, as,i evolve as as,i(t) = Gas,i(0) + gai,s(0), where In our experiments, this low frequency amplitude noise G = cosh(˜gSXStPA/2) and g = sinh(˜gSXStPA/2) (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 tBS is governed by the second term, and as,i membrane displacement and the two-mode correlations. evolve at final time tf into as,i(tf ) = cos φ as,i(tPA) ± Due to this low-frequency RAM, the signal-to-noise ratio sin φ ai,s(tPA), where the mixing angle φ =g ˜DXDtBS/2

6 characterizes the coherent mixing between the signal and below threshold (µ < 1), their validity well above thresh- idler modes due to the beam splitter interaction. old, as in this work, is restricted to small times tPA < τs,i The output variance of the X-quadrature of the sig- (see “Pump Depletion” below). Importantly, because the nal mode ∆X2 (or of the Y -quadrature of the idler squeezed quadrature variances reduce exponentially to s,out −1 mode ∆Y 2 ) depends on the mixing angle φ, where the 1/(1+µ) with a time constant [γ(1+µ)] τs,i, there is i,out ample time for the squeezed variances to break the steady quadratures are defined such that as,i = Xs,i +iYs,i . For state squeezing bound of 3 dB [23, 30]. the coherent states |αs, αii which form the input to the in- in 2 in 2 terferometer, the variances are (∆X(s,i)) = (∆Y(s,i)) = 1, where the motion of each mode is normalized to its thermal or quantum zero-point motion. For G, g 1, the output variances are minimized for φ = −π/4 2 2 2 Pump Depletion and equals ∆Xs,out(tf ) = ∆Yi,out(tf ) = 1/(G + g) = exp(−g˜SXStPA). This improves the SNR of measuring the phase φ by an exponential factor exp(−g˜SXStPA/2) The assumption that µ(t) = µ is a constant over the as compared to that achieved using a conventional inter- course of evolution of the parametric amplifier pulse is 2 2 ferometer, for which ∆Xs,out(tf ) = ∆Yi,out(tf ) = 1. This invalid for µ0 ≡ µ(t = 0) > 1. As the signal and idler improvement is, in principle, unbounded - the paramet- amplitudes grow exponentially, the absolute downcon- ric amplifier gains G ≈ g = sinh(˜gSXStPA/2) diverge on verted phonon loss rate increases before balancing out increasing the argument. with the absolute downconversion rate, thereby decreas- ing µ(t) = XS (t) . This effect, which arises from the inter- Unlike in the optical domain, phononic fields are nec- XS,th essarily confined in a cavity – the mechanical resonator – 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 † √ µ γ in 1 gS rates is governed bya ˙ s,i = γai,s − as,i + γas,i, 2 ∗ 2 2 x¨s,i + γs,ix˙ s,i + ωs,ixs,i = (Fs,i(t) + XSxi,s) where µ = XS/XS,th parametrizes the parametric pump ms,i 2 drive strength at the sum frequency. Defining the cross- √ ¨ ˙ 2 1 gS quadrature modes d = (a ± a )/ 2, the equations of XS + γSXS + ωSXS = (FS(t) + xsxi) (5) ± s i √ mS 2 motions are rewritten as X˙ = γ (µ − 1)X + γXin d+ 2 d+ d+ γ √ in where xs,i,XS denote the complex displacement of each and Y˙d = − (µ + 1)Yd + γY , and similarly for + 2 + 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 on each of the modes. These coupled equations of motion to be h∆Y 2 (t)i = h∆X2 (t)i = 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 h∆X2 (t)i = h∆Y 2 (t)i = d+ d− [37] and simplify to the first order coupled equations, 1 + µ eγ(µ−1)t, which are amplified. 1−µ µ−1 h g i 2A˙ = γ −A + i S χ A∗A + iχ F˜ (t) For a beam splitter interaction with mixing angle φ, the s s s 2 s i S s s signal output quadratures are expressed in terms of the ˙ h gS ∗ ˜ i 2Ai = γi −Ai + i χiAsAS + iχiFi(t) cross-quadratures as Xout = sin(φ + π/4)Xd+ + cos(φ + 2 h g i π/4)Xd and Yout = sin(φ + π/4)Yd + cos(φ + π/4)Yd . S − + − 2A˙ S = γS −AS + i χSAsAi + iχSF˜S(t) (6) The variance of Xout is minimized for a mixing angle 2 φ = −π/4, i.e. when the quadratures are X = X out d− where x = A e−iωkt, k ∈ [i, s, S]; F˜ are the slowly and Y = Y , and the dynamics of their variances k k k out d− varying (complex) amplitudes of the external forces on 2 1 µ −γ(1+µ)t 2 is h∆X i = + e PA and h∆Y i = −1 out 1+µ 1+µ out the individual modes, and χk = (mkωkγk) are their 1 µ γ(µ−1)tPA in 1−µ + µ−1 e , where the inputs as,i are assumed to on-resonant susceptibilities. We ignore terms such as be coherent states |αs,ii. While these dynamics are exact A¨k, γiA˙ k in the slow time approximation. These are fur-

7 ther simpliﬁed to with feedback and other control systems.

˜˙ ˜ ˜∗ ˜ 2As = γs[−As + iAi AS + iFs(t)] ˜˙ ˜ ˜∗ ˜ Heisenberg-scaling of phase sensitivity 2Ai = γi[−Ai + iAsAS + iFi(t)] ˜˙ ˜ ˜ ˜ 2AS = γS[−AS + iAsAi + iFS(t)] (7) The interferometer output X-quadrature for a coherent input state |αs, 0i can be calculated from the above where the motion of the pump A˜S is normalized with re- equations of motion to be spect to AS,th – the critical value which defines the insta- ˜ bility threshold µ = 1, As,i are normalized with respect to Re[α ]cos φ − sin φ s −γ(µ+1)tPA/2 hXs,outi = √ √ e their characteristic steady-state motion above threshold 2 2 √ 2 g χ χ , and Fk are normalized forces on the respective S S i,s cos φ + sin φ γ(µ−1)t /2 modes, with F (t) = µ(t). Lastly, we assume the empiri- + √ e PA (10) S 2 cal fact that the pump mode, which is a Silicon substrate d hXs,outi Re[αs] − sin φ − cos φ mode, has a much larger damping rate than the signal and = √ √ e−γ(µ+1)tPA/2 3 4 d(φ) idler resonator modes, i.e. γS ∼ (10 −10 )γs, γi, thereby 2 2 − sin φ + cos φ ensuring that the pump motion adiabatically follows the γ(µ−1)tPA/2 ˜ ˜ ˜ + √ e (11) signal and idler motions, i.e. AS = iAsAi + iFS(t). 2 ˜ ˜ As and Ai increase exponentially as exp(γ(µ0 − 1)t/2) For large phonon gain of the parametric amplifier, the after the parametric amplifier pulse is switched on, caus- effects of pump depletion cause a decrease in noise squeezing 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). As such, the choice of the parametric am- 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, log[kµ] sidering a case where the seed motions are A˜s0 and A˜i0, we parametrize the amplifier duration as tPA = γ(1+µ) , a pump depletion by a factor η occurs when where k > 1/µ is a dimensionless parameter. With this parametrization, the remnant noise and the minimum de- |A˜sA˜i| = η|FS| = ηµ0 (8) tectable motion ∆XD is thus given by √ Dropping the absolute modulus sign for clarity, this oc- 2 1/2 −γ(µ+1)tPA/2 h∆Xs,outi = k + 1e curs when 2 1/2 h∆Xs,outi Re[αs] A˜ A˜ eγ(µ0−1)tPA = ηµ ∆φ = ∝ (12) s0 i0 0 d hXs,outi /dφ Ns

ln(10µ0) For time tPA ∼ , the squeezed variances reduce to γµtPA/2 2 2 2 γ(1+µ0) where Ns ≈ (Re[αs]e ) /4 = G × (Re[αs]/2) is 1 within 10% of , and for this time tPA, the mean phonon number in the signal mode, which is the 1+µ0 measurement resource. Here, we assume that Re[αs] γµt 2 γµt µ −1 ˜ ˜ PA PA 0 As0Ai0 µ01 1 and e 1, and define G = e to be the µ +1 ˜ ˜ η ∼ (10µ0) 0 −−−−→ 10As0Ai0 (9) phonon number gain. µ0 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˜s0, A˜i0, (ii) idler motion readout, all data are acquired by artificially 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 couplingsg ˜S (see also perature, corresponding to an effective thermal motion [39]). We note that the deleterious effects of pump deple- around 40-50 times the room temperature amplitude, by tion can be avoided by compensating the pump depletion driving the two modes with a gaussian noise source with

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