Supplementary Information for Quantum nondemolition measurement of mechanical squeezed state beyond the 3 db limit

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1 Sulementary Information for Quantum nondemolition measurement of mechanical squeezed state beyond the db limit C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald,, F. Marquardt,, A. A. Clerk, 5 K. C. Schwab Alied Physics, California Institute of Technology, Pasadena, CA 95, USA Korea Research Institute of Standards and Science, Daejeon 5-, Reublic of Korea Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, D-958 Erlangen, Germany Max Planck Institute for the Science of Light Günther-Scharowsky-Straße /Bau, D-958 Erlangen, Germany and 5 Deartment of Physics, McGill University, Montreal, Quebec, HA T8 Canada Dated: July 6, 6) I. THEORY A. Ideal two tones otomechanical Hamiltonian The Hamiltonian of a generic otomechanical system reads Ĥ = ω c â â + ω mˆb ˆb g â â ˆb ) + ˆb + Ĥdrive, ) where â â ) is the annihilation creation) oerator of the intra-cavity field, ˆb ) ˆb is the mechanical honon annihilation creation) oerator, and g is the bare otomechanical couling between the cavity and the mechanical oscillator. Ĥ drive describes the external driving. The device studied in this work is a two orts otomechanical system. Microwave tones are alied from the left ort, which we designate L). In this section, we consider a system driven by two microwave tones. The drive Hamiltonian reads Ĥ drive = âe iω L α νt ν + â e iωνt), ) ν=± where ω ± = ω c + ± ω m + δ) and α ± are the blue and red um amlitudes at the inut ort. In the following, we aly the standard linearization i.e., we searate the cavity and the mechanical oerators, â and ˆb, into a classical art, ā or b, lus quantum fluctuations, ˆd or ˆb. E.g., â ā + ˆd. In the interaction icture with resect to Ĥ = ω c + ) â â + ω m + δ) ˆb ˆb, we find the linearized otomechanical Hamiltonian Here, Ĥ = ĤRWA + ĤCR. ) Ĥ RWA = ˆd ˆd δˆb ˆb [G +ˆb + G ˆb) ) ] ˆd + G +ˆb + G ˆb ˆd describes the resonant art of the linearized otomechanical interaction whereas [ Ĥ CR = G + e iωm+δ)tˆb ] [ iωm+δ)tˆb] + G e iωm+δ)tˆb ˆd G + e iωm+δ)tˆb + G e ˆd 5) describes off-resonant otomechanical interactions. Note that G ± = g ā ± describes the driven-enhanced otomechanical couling. Here, ā ± is the intracavity microwave amlitude due to the red and blue ums, and we have assumed ā ± R for simlicity and without loss of generality. In the following analysis, we consider the good cavity limit ω m ). At this limit, the off-resonant art of the Hamiltonian can be neglected by the rotating wave aroximation RWA). ) B. Mechanical arametric modulation In addition to the ideal otomechanical interaction, mechanical arametric modulation is observed in the exeriment. This surious mechanical arametric effect can be induced by thermal effects or nonlinearities [, ]. To take this effect into account, we henomenologically include the mechanical arametric interaction Ĥ ara = λe iψˆb + e iψˆb ), 6) where λ is the amlitude of the arametric interaction, ψ is the relative hase between the arametric drive and the squeezing um.

2 C. Quantum Langevin equations The linearized quantum Langevin equations read ) ˆd ) = i ˆd + i G ˆb + G+ˆb + ˆd in, 7) ) Γm ˆb = iδ ˆb iλe iψˆb + i G ˆd ) + G+ ˆd + Γ mˆbin. 8) Here, ˆd in = σ σ=l,r,i ˆdσ,in is the total inut noise of the cavity, where ˆd σ,in describes the inut fluctuations to the cavity from channel σ with daming rate σ. σ = L and R corresond to the left and right microwave cavity orts, while σ = I corresonds to internal losses. The noise oerator ĉ in describes quantum and thermal noise of the mechanical oscillator with intrinsic daming rate Γ m. The inut field oerators satisfy the following commutation relations: [ ˆdσ,in t), ˆd ] σ,in t ) = δ σ,σ δ t t ), 9) [ˆbin t), ˆb in )] t = δ t t ), ) ˆd σ,in t) ˆd σ,in t ) = n th σ δ σ,σ δ t t ), ) ˆb in t) ˆb in t ) = n th mδ t t ), ) where n th σ is the hoton occuation in ort σ, and n th m = / [ex ω m /k B T ) ] is the thermal occuation of the mechanical oscillator. The total occuation of the cavity is the weighted sum of the contributions from different channels: n th c = σ σ n th σ. D. Otomechanical outut sectrum and mechanical sectrum In this section, we derive the otomechanical outut sectrum and the mechanical quadrature sectrum. For this, we solve the quantum Langevin equations Eqs. 7, 8) in Fourier sace. It is convenient to define the vectors D = ˆd, ˆd, ˆb, ˆb ) T, Din = ˆdin, ˆd in, ˆb in, ˆb in) T and L = diag,, Γm, Γ m ). We then find the following solution to the quantum Langevin equations in frequency sace: ˆD [ω] = χ [ω] L ˆD in [ω], ) where χ [ω] i ω + ) ig ig + i ω ) ig + ig Γ ig ig m + i ω + δ) iλe iψ ig + ig iλe iψ Γm i ω δ). ) In the exeriment, we measure the outut microwave sectrum through the undriven right) cavity ort. One finds the outut field ˆd R,out ω) using the inut-outut relation ˆd σ,out ω) = ˆd σ,in ω) σ ˆd ω). This yields ˆd R,out ω) = ˆd R,in ω) R χ [ω]) ˆdin R χ [ω]) ˆd in R Γ m χ [ω]) ˆbin R Γ m χ [ω]) ˆb in. 5) The transmission sectrum driven resonse) is S [ω] = L R χ [ω]). 6) The symmetric noise sectral density is S R [ω] = { dt ˆd R,out [], ˆd } R,out [t] e iωt = + RS [ω], 7) where S [ω] = χ [ω]) n th c + χ [ω]) n th c + ) + Γ m χ [ω]) n th m + Γ m χ [ω]) n th m + ). 8)

3 The mechanical quadrature sectrum is S X [ω] = { dt ˆX t), ˆX } ) e iωt, 9) where ˆX = x z ˆbe i + ˆb e i ). The quadrature variance is given by the integral ˆX = dω π S X ω). ) In some um configurations, we can simlify the results. For δ =, the exressions can be simlified to L R Γ m iω) S [ω] = G + [ i ω + )] Γ m iω), ) S [ω] = Γ [ m Γm n th c + G n th m + G + n th m + )] + 6n th c ω G + + iω) [Γ m + i ω + )], ) where G = G G +. For both δ = and =, the mechanical quadrature sectra and the quadrature variances are S X, [ω] = x G G + ) n th c + ) + Γm + ω ) ) n th m + z [G + Γ m ]. ) + Γ m + 8G ) ω + 6ω ˆX, = x G G + ) n th c + ) + [ G + + Γ m ) ] Γ m n th m + ) z + Γ m ) G, ) + Γ m ) in the regime G, Γ m, Eq. ) reduced to Eq. ) in the main text. II. MEASUREMENT CIRCUIT The schematic of the measurement circuit is shown in Fig. S We cool the device down to mk with a dilution refrigerator. In the exeriment, u to four microwave drives are alied to the device. Because the excess hase noise from the microwave sources at the cavity resonance can excite the cavity and degrade the squeezing, a tunable notch filter cavity is used to rovide more than 5 db noise rejection at the cavity resonance frequency ω c. The inut microwave ums are then attenuated by about db at different temerature stages in the cryostat to dissiate the Johnson noise from higher temerature, keeing the inut microwave noise at the shot noise level. The outut signal asses through two cryocirculators at 5mK, then amlified by a cryogenic high electron-mobility transistor amlifier HEMT) at. K and a low noise amlifier at room temerature for analysis. During the measurement of the noise sectrum, we continuously monitor the hase difference between the squeezing ums and the BAE robes. The beat tones of the ums and the robes are acquired by microwave detection diodes, then fed into the sub-harmonic circuits to halve the frequencies. The relative hase between the resulting beat tones are comared and measured by the lock-in. A comuter is used to generate the error signal and feedback to the sources to kee the hase drift within half degrees. III. CALIBRATIONS A. Calibrations of the squeezing outut sectrum In the exeriment, we send an equal time interleaving measurement to measure the umed noise sectrum S meas [ω] and the unumed noise sectrum S [ω] at the outut of the measurement chain. The unumed noise sectrum S [ω] is the noise floor of the system which is dominated by the noise figure of the cryogenic HEMT amlifier. The difference of the umed and unumed noise sectra is related to the outut noise sectrum of the otomechanical system by S meas [ω] = S meas [ω] S [ω] = G [ω c ] R ω c S [ G, G +,, δ,, Γ m, n th c, n th m, ω ], 5)

4 - mk device Pum filter cavity sectrum analyzer cryostat Probe ω/ ω/ network analyzer Lock-in K 5 mk mk FIG. S. Schematic of the measurement circuit. See text for detail. c) x 9 = d) = ωc ωm G -, G + rad/s) 8 6 S f-f khz) c S f-f c khz) + ) Pm P - -5 S aw/hz) f-f m Hz) ωc ωm δ ωc + ωm + δ δ P -, P + nw) T mk) FIG. S. Calibrations of the mechanical squeezing exeriment. Pum configuration of the enhanced otomechanical couling G ) calibration. Pum configuration of the enhanced otomechanical couling G +) calibration. c) Calibrations of the enhanced otomechanical coulings G ±, the inserts are the transmission sectrums corresonding to the solid circles. d) Calibration of the normalized motional sideband ower, the insert is the sideband sectrum at the base temerature. where G [ω] is the gain of the measurement chain which is flat in the measurement bandwidth of the exeriment and R is the couling rate to the outut ort of the device. In order to fit the measured noise sectrum with the otomechanical model Eq. 8)), an indeendent measurement of the refactor G [ω c ] R ω c is required. Which can be obtained from the calibrations of the enhanced otomechanical coulings and the thermal calibration.. Calibrations of the enhanced otomechanical coulings To calibrate the enhanced otomechanical coulings G ±, we measure the transmitted ower of the drive tones at the outut of the measurement chain P ±, which is related to the intracavity um hoton number by P ± = G [ω ± ] R ω ± λ [ω ± ] n ±, 6) where λ [ω ± ] are the correction factors due to the arasitic channel []. The square of the enhanced otomechanical coulings are related linearly to the transmitted um owers by G ± = g n ± = a ± P ±, 7) where the a ± = G[ω ±] R ω ± g λ[ω ±] is the calibrations of the enhanced otomechanical coulings. We start with the calibration of the enhanced otomechanical couling G induced by the red detuned tone. To do that, a single red detuned tone is alied at ω c ω m with transmitted ower P Fig. Sa). Then, a network analyzer is used to generate a weak robe and swee it through the center of the cavity resonance to measure the transmission sectrum of the mechanical sideband. The enhanced otomechanical couling rate G can be extracted by fitting the transmission sectrum with the otomechanical model 6). By measuring the transmission sectrum

5 with various transmitted ower P and fitting with the linear relation 7) the red line in Fig. Sc), we obtain the calibration a = 7.9 ±.) 7 rad s W and the intrinsic mechanical linewidth Γ m = π 8 Hz. A similar method is used to calibrate the enhanced otomechanical couling G + induced by the blue detuned tone. In this case, a blue detuned tone is laced at ω c +ω m +δ with transmitted ower P + and δ = π khz. Because the blue detuned tone would amlify the mechanical motion and narrow the mechanical linewidth, the mechanical resonator becomes unstable when the cooerativity C + = G + Γ m aroaches to unity. In order to kee the mechanics stable, a constant red detuned tone is alied at ω c ω m δ to dam the mechanical motion Fig. Sb). Similar to the calibration of G, we measure the transmission sectrum of the mechanical sidebands with the sectrum analyzer and extract the enhanced otomechanical couling rate G + from the fit at various transmitted ower P +. The linear fit blue line in Fig. Sc) gives a + =. ±.7) 8 rad s W. 5. Thermal calibration Having calibrated the enhanced otomechanical coulings, we turn to the thermal calibration of the motional sideband noise ower. To do that, a single red detuned tone is laced at ω = ω c ω m with sufficiently small um ower P such that the otomechanical daming effect is negligible Γ ot = G Γ m ). We then measure the noise ower of the u-converted motional sideband Pm over a range of calibrated cryostat temerature T Fig. Sd). Due to the weak temerature deendence of the cavity linewidth, we monitor the cavity linewidth at each measurement temerature. The resulting normalized sideband ower is given by + ) Pm P = b ) k B T, 8) ω m where = ω ω c + ω m which is equal to zero in this case, is the average value of the cavity linewidth over the resective temeratures and b = G[ωc]ωc G[ω ]ω g λ[ω ] is the thermal calibration. The linear fit in Fig. Sd gives b =.5 ±.7) 5 rad/s), which enables us to convert the normalized noise ower into quanta. The refactor G [ω c ] R ω c is given by the ratio of the thermal calibration b and the calibration of the enhanced otomechanical couling a i.e. G [ω c ] R ω c = b /a ), which allow us to relate the measured noise sectrum to the otomechanical model S meas [ω] = b a S [ G, G +,, δ,, Γ m, n th c, n th m, ω ]. 9). Fitting rocedure In order to extract the cavity occuation n th c and the honon bath occuation n th m from the outut sectrum. We need to determine the enhanced otomechanical coulings G ±, the detunings and δ, the cavity linewidth and the intrinsic mechanical linewidth Γ m. To obtain these arameters, we measure the transmitted um ower of the microwave drives at each um configuration, from which we obtain the enhanced otomechanical coulings G ± with the calibrations a ±. Since the resonance frequencies of the cavity and the mechanical mode change with the owers of the microwave drives due to thermal effects [] and nonlinearties [], to recisely align the frequencies of the drive tones at ω c ± ω m, we measure the transmission sectrum to extract the detunings and correct the frequencies of the drives. Fig. S shows examles of the measured driven resonses dots) and the corresonding fits solid curves) at various um hoton ratios. By fitting the transmission sectrum with Eq. 6), we can extract the detunings δ, ) and the cavity linewidth, which enable us to correct the frequencies of the drive tones. By iterating this rocedures, we can recisely set the owers and the frequencies of the microwave drives. Together with the intrinsic mechanical linewidth Γ m obtained from the calibration, we can fit the outut noise sectrum with the otomechanical model and extract the occuation factors n th c, n th m) with Eq. 9). Fig. S shows examles of the noise sectra and the corresonding fits. B. Calibrations of the backaction evasion sectrum In our exeriment, we erform an additional BAE measurement away from the cavity resonance to directly and indeendently measure the mechanical quadratures. Since the detuning of the BAE sideband = π 6kHz

6 S ω-ωc )/ -. = ω-ωc )/...5 = = = n+ /n - = FIG. S. Amlitude of the transmission sectra at ntot =.85 and um hoton ratio n /n = blue),. green),. yellow),.6 urle),.8 red). The black curves are the corresonding fits with Eq. 6) Amlitude of the transmission sectra in cavity san. Amlitude of the transmission sectra near the cavity resonance. S out - Sout aw/hz) n+ /n - = ω-ωc )/ =. = = ω-ωc )/.5. = FIG. S. Outut noise sectra at ntot =.85 5 and um hoton ratio n+ /n = blue),. green),. yellow),.6 urle),.8 red). The black curves are the corresonding fits with Eq. 9) and 8) Outut noise sectra in cavity san. Outut noise sectra near the cavity resonance. is comarable to the cavity linewidth, to recisely balance the BAE tones and correctly interret the BAE noise sectrum, an indeendent calibrations of the enhanced otomechanical couling rate G± and the normalized sideband ower are necessary. We follow the same rocedures in the last section to calibrate the BAE measurement, the only difference is the frequency of the drive tones. Fig. S5a and S5b are the um configurations in the calibrations of the enhanced otomechanical coulings G and G+. The results of the calibrations are shown in Fig. S5c. From the linear

7 7 c) x = π 6 khz = π 6 khz ωc ωm ωc ωm Δ ωc + ωm G -, G + rad/s) 5 S f-f c khz) S f-f c khz) d) ) + Pm P - -5 Δ P -, P + nw) T mk) FIG. S5. Calibrations of the backaction evading measurement. Pum configuration of the enhanced otomechanical couling G ) calibration. Pum configuration of the enhanced otomechanical couling G +) calibration. c) Calibrations of the enhanced otomechanical coulings G ±, the inserts are the transmission sectrums corresonding to the solid circles. d) Calibration of the normalized motional sideband ower.. S aw/hz) freq khz) FIG. S6. Examle of the BAE noise sectrum. The red line is a background fit with a quadratic olynomial. fits, we get a BAE = 7.85 ±.6) 7 rad s W and a BAE + =. ±.) 8 rad s W. Fig. S5d is the thermal calibration of the normalized motional sideband ower. The linear fit red line in Fig. S5d) gives b BAE =.77 ±.) 5 rad/s). Similar to the measurement of the squeezing outut sectrum, we send an equal time interleaving measurement to measure the umed and unumed noise sectrum. After subtracting the unumed noise sectrum to remove the noise floor, the outut noise sectrum of the BAE measurement is given by BAE BAE S meas [ω] = S meas [ω] S [ω] = S c [ω] + S BAE [ω], ) the first term S c [ω] is the noise sectrum of the cavity resonance due to the non-zero cavity occuation. The second term S BAE [ω] is the noise sectrum of the BAE sideband, which is given by g S BAE [ω] = G [ω c ] R ω c n S X [ω] +, ) where S X [ω] is the mechanical quadrature sectrum. Because the BAE sideband is detuned from the cavity resonance with detuning comarable to the cavity linewidth, over the bandwidth of the BAE measurement, the cavity noise aears as a frequency deendent noise background. An examle of the sectrum is given by Fig. S6, a quadratic olynomial is emloyed to fit the cavity noise background, as shown by the red curve in Fig. S6. The BAE sideband sectrum S BAE [ω] is given by subtracting the noise sectrum from the fitted cavity noise background. The noise ower of the BAE sideband is given by integrating Eq. ), which gives P BAE m = G [ω c ] R ω c n g + X x z x z. ) With the thermal calibtration factor b BAE, we can convert the normalized BAE sideband ower to the quadrature variance X + x z = b BAE ) P BAE m P. )

8 8 5 c) 5 S X /x z x Γm / S X /x z x Γm / Γ m / π khz) -6-6 freq khz) FIG. S7. The redicted mechanical quadrature sectra calculated with Eq. 9) circles) and the corresonding Lorentzian fits solid curves). The redicted mechanical quadrature sectra and Lorentzian fits corresond to the dashed red curve in Fig. c in the main text. The redicted mechanical quadrature sectra and Lorentzian fits corresond to the dashed blue curve in Fig. c in the main text. c) Quadrature linewidth as a function of the robe hase. The red blue) circles are the measured quadrature linewidth of the weakly strong) squeezed state using the BAE technique. The curves are the fits with the two-tone otomechanical model including the mechanical arametric effect. The solid curves are the fits with the assumtion ψ = ψ. The dashed curves are the fits with the assumtion ψ = + ψ. IV. ANALYSIS OF THE MECHANICAL PARAMETRIC EFFECT In this section, we will describe the rocedures to extract the mechanical arametric interaction from the measured quadrature linewidth data. As shown in the theory section, for given um configurations, δ, n ± ), thermal occuations n th m, n th c ) and mechanical arametric interaction λ, ψ), the mechanical quadrature sectrum S X [ω] can be calculated by Eq. 9). The mechanical quadrature linewidth is given by fitting the redicted mechanical quadrature sectrum with a Lorentzian curve, as shown in Fig. S7a. Using the um configurations and thermal occuations extracted from the corresonding outut sectra Fig. f,g in the main text), the quadrature linewidth can be written as a function of the robe hase, the amlitude λ and the hase ψ of the arametric drive i.e. Γ m, λ, ψ)). Then we can extract the mechanical arametric interaction λ, ψ) by fitting the measured mechanical quadrature linewidth in the BAE measurement with the function Γ m, λ, ψ). As discussed in the main text, we assume the hase of the arametric drive ψ follows the hase of the BAE robe i.e. ψ = ψ +, where ψ is a constant hase shift). The amlitude λ and the constant hase shift ψ can be extracted by fitting the quadrature linewidth data with the function Γ m, λ, ψ + ), the fit results are shown by the dashed curves in Fig. S7c. From the fit, we extract λ = π ± )Hz, ψ = ± 5 for the weakly squeezed state red dashed curve) and λ = π. ±.)khz, ψ = 9 ± 5 for the strong squeezed state blue dashed curve). Under this assumtion, the model catures the observed hase deendence behavior of the quadrature linewidth in the BAE measurement. With the extracted mechanical arametric drive, the corresonding quadrature variances can be calculated by Eq. ), as shown by the dashed curves in Fig. b in the main text. On the other hand, if we assume the mechanical arametric drive is induced by the squeezing um i.e. ψ = ψ ), the function Γ m, λ, ψ ) doesn t cature the observed hase deendence behavior of the mechanical quadrature linewidth, as shown by the solid curves in Fig. S7c. These results imly that the observed mechanical arametric interaction is induced by the BAE robes instead of the squeezing ums. V. ENGINEER AND OPTIMIZE THE MECHANICAL PARAMETRIC DRIVE Having established a henomenological model to understand the effects of the mechanical arametric drive to the squeezed states obtained from the reservoir engineering technique, the next ste is to engineer and otimize it to further increase the squeezing. As described in the main text the arametric drive in our exeriment is induced by the uncontrollable nonlinear rocesses []. Therefore, the amlitude and hase of the arametric drive cannot be controlled indeendently in the current exeriment. In order to imlement a tunable mechanical arametric drive, one can add an additional electrode to drive the mechanical oscillator [, 5]. By alying an ac drive V t) = V +V sin ω m t + ψ ) to the electrode, it generates a time varying sring constant k t) = k sin ω m t + ψ ) with k = C/ x ) V V.

9 9 l/ Γ eff y =7 o ˆ X / x min z / db) ˆ X λ min min ˆ X y..6.9 l/γ eff FIG. S8. Minimum quadrature variance as a function of hase ψ. Different curves reresent different arametric drive amlitude λ. The Minimum quadrature variance at the otimum hase ψ = 7 ) as a function of the arametric drive amlitude λ. This in turn yields the tunable mechanical arametric interaction Ĥ ara = λe iψˆb + e iψˆb ), ) where λ = kx z/. The amlitude and hase of the arametric drive can be controlled by the gate voltage V and the hase of the ac drive ψ. The uncontrollable mechanical arametric drive induced by the BAE robe can be diminished by reducing the ower of the BAE robes. Given the linearity of the mechanical dynamics, the mechanical arametric drive in Eq. ) could be used to directly augment the squeezing generated otomechanically. One simle chooses the hase ψ such that the arametric drive referentially dams the squeezed X mechanical quadrature. As one aroaches the arametric instability, the daming rate of this quadrature is doubled, leading to an additional db of squeezing beyond what would be achieved just via the otomechanical scheme. To illustrate this, we show below exlicit theoretical results including both the arametric drive and otomechanical interaction. Fig. S8a shows the minimum mechanical quadrature variance at different hase ψ and amlitude λ. In this lot, we use the same squeezing um configuration and thermal occuations in Fig. g in the main text. As shown in the figure, the minimum mechanical quadrature variance is very sensitive to the hase. The additional mechanical arametric effect can worsen or imrove the squeezing deending on the hase. The black line is the result with the reservoir engineering squeezing alone λ = ). At the otimum hase ψ = 7 ), the amount of squeezing increases with the arametric drive amlitude Fig. S8b). The arametric drive amlitude is limited to λ = Γ eff / due to the onset of the arametric instability, which lace a fundamental limit to the additional squeezing. As shown in Fig. S8b, the mechanical arametric drive can rovide additional db squeezing to the squeezed state generated by the reservoir engineering scheme. [] J. Suh, A. J. Weinstein, and K. C. Schwab, Al. Phys. Lett. ). [] J. Suh, M. D. Shaw, H. G. LeDuc, A. J. Weinstein, and K. C. Schwab, Nano Lett., 66 ). [] A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A. Metelmann, A. A. Clerk, and K. C. Schwab, Phys. Rev. X, ). [] D. Rugar and P. Grütter, Phys. Rev. Lett. 67, ). [5] R. Andrews, A. Reed, K. Cicak, J. Teufel, and K. Lehnert, Nat. Commun. 6, 5).