Quantum back action evading measurement of motion in a negative mass reference frame
Christoffer B. Møller, Rodrigo A. Thomas, Georgios Vasilakis, Emil Zeuthen, Yeghishe Tsaturyan, Kasper Jensen, Albert Schliesser, Klemens Hammerer, Eugene S. Polzik
QQuantum back action evading measurement of motionin a negative mass reference frame
Christoffer B. Møller, Rodrigo A. Thomas, Georgios Vasilakis,
1, 2
Emil Zeuthen,
3, 1
YeghisheTsaturyan, Kasper Jensen, Albert Schliesser, Klemens Hammerer, and Eugene S. Polzik ∗ Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Institute for Electronic Structure and Laser, Foundation for Research and Technology-Hellas, Heraklion 71110, Greece Institute for Theoretical Physics & Institute for Gravitational Physics (Albert Einstein Institute),Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany
Quantum mechanics dictates that a continuous measurement of the position of an object imposesa random back action perturbation on its momentum. This randomness translates with time intoposition uncertainty, thus leading to the well known uncertainty on the measurement of motion.Here we demonstrate that the quantum back action on a macroscopic mechanical oscillator mea-sured in the reference frame of an atomic spin oscillator can be evaded. The collective quantummeasurement on this novel hybrid system of two distant and disparate oscillators is performed withlight. The mechanical oscillator is a drum mode of a millimeter size dielectric membrane and thespin oscillator is an atomic ensemble in a magnetic field. The spin oriented along the field corre-sponds to an energetically inverted spin population and realizes an effective negative mass oscillator,while the opposite orientation corresponds to a positive mass oscillator. The quantum back action isevaded in the negative mass setting and is enhanced in the positive mass case. The hybrid quantumsystem presented here paves the road to entanglement generation and distant quantum communi-cation between mechanical and spin systems and to sensing of force, motion and gravity beyond thestandard quantum limit.
Continuous measurement of an oscillator position, ˆ x ( t ) = ˆ x (0) cos(Ω t ) + ˆ p (0) sin(Ω t ) / ( m Ω) , where Ω is thefrequency and m is the mass, leads to accumulation ofthe quantum back action (QBA) of the measurementin both the position and momentum, ˆ p , non-commutingvariables [ˆ x, ˆ p ] = i ~ [1, 2]. Measurement QBA was re-cently observed for a mechanical oscillator [3] and foratomic motion [4]. Suppose, however, that the posi-tion is measured relative to an oscillator with a mass m = − m for which ˙ˆ x = − ˆ p /m . The result of ameasurement of ˆ x ( t ) − ˆ x ( t ) = (ˆ x (0) − ˆ x (0)) cos(Ω t ) +(ˆ p (0) + ˆ p (0)) sin(Ω t ) / ( m Ω) depends only on commutingvariables, [ˆ x − ˆ x , ˆ p + ˆ p ] = 0 . Hence it can be QBAfree [5, 6] and the uncertainty in the measurement of therelative position h (ˆ x − ˆ x ) i can be smaller than the un-certainty h ˆ x i . The first proposal for such a measurementbased on atomic spins [6], has been followed by a num-ber of proposals for QBA free measurements [7–10]. In[11] the negative mass approach referred to as “quantum-mechanics-free subsystems” was shown to lead to a mea-surement sensitivity approaching the Cramér-Rao bound.Earlier work on atomic spin ensembles utilized the neg-ative mass property for demonstration of entanglementof macroscopic spins [12] and for entanglement-assistedmagnetometry [13]. The back action evading measure-ment on two mechanical oscillators at room temperaturewas demonstrated in [14] in the classical regime usinglight, and recently in the quantum regime at the mil-likelvin temperature range using microwaves [15]. Waysto overcome QBA limitations for a free mass oscillatorwith squeezed light have been proposed in [16–18].Here we demonstrate QBA in a novel hybrid quantum system [19, 20] composed of a macroscopic mechanical os-cillator, a high- Q dielectric membrane [21, 22] (Fig. 1a)in a high finesse cavity, and a spin oscillator, an ensem-ble of room temperature Cesium atoms in a magneticfield contained in a spin-protecting environment [23–25](see Fig. 1b and Supplementary Information). The me-chanical oscillator Hamiltonian is ˆ H M = ( m Ω M / x +ˆ p / m = ( ~ Ω M / X M + ˆ P M ) , where we henceforth em-ploy dimensionless variables ˆ X M = ˆ x/x zpf and ˆ P M =ˆ p x zpf / ~ where x zpf = p ~ /m Ω M is the oscillator’s zeropoint position fluctuation and [ ˆ X M , ˆ P M ] = i . Comparedto a mechanical oscillator, a spin oscillator has somerather unique properties. Consider a collective atomicspin ˆ J α = P N a i =1 ˆ F iα with components α = x, y, z com-posed of a large number N a of ground state spins ˆ F i (with F = 4 in the present case). Atoms are optically pumpedto generate an energetically inverted spin population inan external magnetic field B (Fig. 1c), which we take topoint in the positive x -direction. The collective spin thusexhibits a large average projection J x = |h ˆ J x i| / ~ (cid:29) . Itsnormalized y, z quantum components form canonical os-cillator variables [ ˆ X S , ˆ P S ] = [ ˆ J z / √ ~ J x , − ˆ J y / √ ~ J x ] = i [23] in terms of which the spin Hamiltonian becomes ˆ H S = ~ Ω S ˆ J x = ~ Ω S J x − ( ~ Ω S / X S + ˆ P S ) with Ω S – the Larmor frequency. The first term is an irrelevantconstant energy offset due to the mean spin polarization.The second term is equivalent to the Hamiltonian of amechanical oscillator ˆ H M with a negative mass. Eachquantum of excitation in the negative mass spin oscil-lator physically corresponds to a deexcitation of the in-verted spin population by ~ Ω S (Fig. 1c). Preparationof the collective spin in the energetically lowest Zeeman a r X i v : . [ qu a n t - ph ] A p r ba c d Figure 1.
Mechanical and spin oscillators . A.The mechanical oscillator – the (1,2) drum mode, Ω M =2 π × .
28 MHz , of a . , square silicon nitride membrane(light square in the center of the inset) supported by the sil-icon phononic crystal structure. B. The spin oscillator is anoptically pumped gas of Cesium atoms contained in squarecrossection channel inside a glass cell. Channel walls arecoated with a spin-protecting coating. The cell is placed in astatic magnetic field with the Larmor frequency Ω S tunablearound Ω M . Depending on the direction of the magnetic fieldwith respect to the direction of the atomic spin, the oscillatorcan have lower (higher) energy of the excited state, corre-sponding to the negative (positive) effective mass, as shownin C and D, respectively. state realizes instead a positive mass spin oscillator with ˆ H S = − ~ Ω S J x + ( ~ Ω S / X S + ˆ P S ) (Fig. 1d).The experiment implementing a quantum measure-ment on the hybrid system is sketched in Fig. 2a, whichdepicts the cascaded interaction between a traveling lightfield and the two oscillators (see Supplementary Informa-tion for details). A coherent optical field with a strong,classical, linearly polarized component LO (photon flux Φ ) and vacuum quantum fluctuations in the polariza-tion orthogonal to it, described by quadrature phase op-erators ˆ X L ,in and ˆ P L ,in , first interacts with the spin oscil-lator. The interaction for far-off-resonant light is of thequantum nondemolition (QND) type ˆ H int ,S ∝ ˆ X S ˆ X L ,in ,where ˆ X S ∝ ˆ J z is the projection of the collective spinon the direction of light propagation [23]. The lightoutput quadrature, ˆ P SL ,out (Ω) = ˆ P L ,in (Ω) + √ Γ S ˆ X S (Ω) ,reads out the atomic spin projection ˆ X S at the rate Γ S ∝ Φ . At the same time ˆ H int ,S implies that mea-surement QBA due to ˆ X L ,in is imprinted on the atomic ˆ P S quadrature. The atomic spin projection is drivenin addition by intrinsic spin noise ˆ F S so that ˆ X S = χ S (Ω)[ √ γ S ˆ F S + √ Γ S ˆ X L, in ] . Here and henceforth weconsider all quantities in Fourier (frequency) domainwhich is most appropriate for a continuous-time mea-surement. The atomic oscillator’s susceptibility χ S (Ω) = ± S / (Ω S − Ω − i Ω γ S ) is determined by the sign ofits effective mass ( ± ) , resonance frequency Ω S and relax- ation rate γ S (half width at half maximum convention isused throughout the paper). The physics of the QBA inthe spin system can be understood as fluctuations of theStark shift of the atomic energy levels due to fluctuationsof the angular momentum of light [23].The classical drive LO is filtered out after lightpasses through the atoms (Fig. 2a), whereas the rele-vant fluctuations in the orthogonal polarization, ˆ P SL ,out and ˆ X SL ,out = ˆ X L ,in are mixed with a classical drivefield LO (with photon flux Φ ) in the same polariza-tion and sent onto the mechanical oscillator. The phaseof LO is adjusted so that ˆ X ML ,in = ˆ X SL ,out , ˆ P ML ,in = ˆ P SL ,out .The linearized optomechanical Hamiltonian is ˆ H int ,M ∝ ˆ X M ˆ X ML ,in [26]. In analogy with the spin, the output phasequadrature of light, ˆ P L ,out = ˆ P ML ,in + √ Γ M ˆ X M , reads outthe membrane position ˆ X M at the rate Γ M ∝ Φ . Themembrane position is driven by thermal state noise ˆ F M and the QBA of light, that is ˆ X M = χ M (Ω)[ √ γ M ˆ F M + √ Γ M ˆ X ML, in ] , where the mechanical susceptibility is givenby χ M (Ω) = 2Ω M / (Ω M − Ω − i Ω γ M ) and determinedby the mechanical resonance frequency Ω M and damp-ing rate γ M . Hence ˆ X L ,in is the source of measurementQBA for both the membrane and the spin oscillator.Overall, the homodyne readout of the joint systemwith the local oscillator LO can be cast as ˆ P L ,out =ˆ P L ,in + √ Γ M ˆ X M + √ Γ S ˆ X S . The back action evadingcharacter of this measurement comes out most clearlywhen the measured light quadrature for the joint systemis expressed as ˆ P L ,out = ˆ P L ,in + √ Γ M γ M χ M (Ω) ˆ F M + √ Γ S γ S χ S (Ω) ˆ F S + [Γ M χ M (Ω) + Γ S χ S (Ω)] ˆ X L, in , withthe terms corresponding to shot noise of light, membranethermal noise, spin noise, and measurement QBA noise,respectively. Notably, the QBA term shows the inter-fering responses of the membrane and the spin oscilla-tor. Ideal broadband QBA evasion is achieved for equalreadout rates, Γ S = Γ M , and χ M (Ω) = − χ S (Ω) whichrequires Ω M = Ω S , γ M = γ S and a negative mass spinoscillator (Supplementary Information and [8]).We exploit the high level of flexibility in our mod-ular hybrid setup to fulfill these requirements: It isstraightforward to match the readout rates Γ M ’ Γ S by a proper choice of power levels Φ , , and to tunethe atomic Larmor frequency Ω S to the resonance fre-quency Ω M = 2 π × .
28 MHz of the mechanical drummode. In order to observe appreciable QBA at the mem-brane’s thermal environment of K we use a phononic-bandgap shielded membrane with high mechanical qual-ity factor Q corresponding to an intrinsic damping rateof γ M = 2 π ×
50 mHz . On the other hand, the intrin-sic spin damping rate γ S ’ π ×
500 Hz is due to powerbroadening by optical pumping and atomic collisions. Inaddition, efficient spin readout requires significant powerbroadening by the probe light, γ S (cid:29) γ S (SupplementaryInformation), impeding an adjustment of the spin to the bac d [[ [ Figure 2.
Experimental setup and observation of QBA for the spin and mechanical oscillators . A. Atomic spinensemble S in magnetic field B is probed by the field LO . The quadrature X L, in in the polarization mode orthogonal to themean polarization of LO is the back action (BA) force. The LO is filtered out with the first polarizing beam splitter (PBS ),while transmitted quantum fluctuations are superimposed with the field LO at PBS , projected into the same polarizationas LO at PBS , and become a driving force for the mechanical oscillator M . PBS and the quarter wave plate ensurethat almost all light reflected off the cavity is directed to the homodyne detection with LO . B. Amplitude noise spectrumof the optomechanical system showing frequency dependent squeezing of light. C. Phase noise spin spectrum. Black dots– spin driven with the broadband thermal light noise and thermal force, brown dots – spin driven by vacuum light noiseand thermal force, brown area – thermal noise of the spin. Striped area – quantum back action determined from the data(see Supplementary Information). D. Phase noise of optomechanical system driven by vacuum light noise and thermal force.Blue area – membrane thermal noise. Striped area – quantum back action determined from squeezing data shown in A) (seeSupplementary Information). Axes labels: (SN) – shot noise of light, x zpf – zero point fluctuations. Curves are generated bythe detailed numerical model of the experiment (Supplementary Information). See comments in the text. mechanical linewidth. Instead we optically broaden themechanical linewidth by introducing a detuning ∆ < of LO from the cavity resonance. This is a well estab-lished technique in optomechanical cooling experimentswhich exploits the dynamical back action of light on themechanical oscillator for changing the mechanical suscep-tibility in order to generate a significantly enhanced effec-tive damping rate γ M (cid:29) γ M [26]. In this way matchedlinewidths γ M ’ γ S can be achieved by a proper choiceof Φ and ∆ , cf. Fig. 2c,d. The experimental parame-ters are listed in the Extended Data section. Introduc-ing a nonzero detuning also modifies the optomechanicalinput-output relations and the QBA interference as de-tailed further below and in the Supplementary Informa-tion.Having matched the susceptibilities and readout rateswe perform a back-action limited readout of the two sys-tems as shown in Fig. 2b,c,d. The ratio of QBA from vac-uum noise of light ˆ X L, in to thermal noise due to ˆ F M ( S ) is proportional to the quantum cooperativity parameters C M ( S ) q respectively which we separately calibrate for eachsystem (Supplementary Information). We achieve an op-tomechanical cooperativity of C Mq = 2 . ± . and on theside of atoms C Sq = 1 . ± . which signifies that QBAand thermal noise contribute roughly on the same levelin both systems.Fig. 3 displays the results for the hybrid system. As areference we show the spectra of the two individual sys-tems taken separately (blue – the mechanics, brown –the spin) in Fig. 3a both measured with the LO detec-tor. Fig. 3b presents the hybrid noise for the negative(red) and positive (green) effective spin masses, corre-sponding to two opposite orientations of the DC mag-netic field relative to the spin polarization. The hybridspectra differ significantly from each other, with the areaof the spectrum for the negative (positive) spin mass be-ing significantly smaller (larger) than that for uncorre-lated systems – a clear demonstration of the destructive(constructive) interference of the QBA contributions for bac d Figure 3.
Quantum back action for the mechanical and spin oscillators with equal central frequencies.
Axeslabels: (SN) – shot noise of light, x zpf – zero point fluctuations. A. Blue – mechanical oscillator, brown – spin oscillator, dashed– the sum of the two spectra. B. Hybrid spectrum for the system with the negative (red) and positive (green) effective spinmasses. Black curve – the model for the joint noise spectrum of the hybrid system with quantum BA interference put to zero.C. Hybrid spectrum noise for the negative mass (red dots). Thermal noise of the membrane (blue shade), thermal noise of thespin (brown shade) and joint thermal noise (red dashed curve). Striped area – QBA of the hybrid system. Blue curve – modelfit to the membrane noise data (same as in A). D. Same as in C, but for the joint system with the positive mass spin oscillator.Curves – full model (Supplementary Information). the two systems. We emphasize that these data signifya QBA cancellation irrespective of theoretical modelling.For comparison, the Fig. 3a also shows the curve (dashed)obtained by adding the two noise spectra recorded in sep-arate measurements on atoms and the mechanical oscil-lator.An intriguing feature of the hybrid noise spectra is theapparent absence of interference and noise cancellationexactly at the Fourier frequency Ω = Ω S = Ω M wherethe negative joint, positive joint and the mechanics spec-tra overlap (Fig. 3b). This is due to the strong opticalbroadening of the mechanical oscillator which leads tosuppression of the spin phase noise contribution to lighton the exact joint resonance. The effect is well under-stood from the full quantum model (Supplementary In-formation) and is analogous to optomechanically inducedtransparency [27]. The solid red, green and blue curvesfor the negative joint, positive joint and mechanics, re-spectively, are generated from this model and are in ex-cellent agreement with the data. Fig. 3c presents thespectrum for the hybrid system with the negative massand the model fit (blue curve) to the spectrum of themechanics (data in Fig. 3a). The noise reduction of thehybrid spectrum (red dots) compared to the mechanicsonly (blue curve) in the wings of the spectrum is observ-able, although its effect is diminished by the added spinthermal noise which is present in the red data, but does not contribute to the blue curve. The observed variance(area) for the joint negative system is . × x zpf whichis (97 ± of the observed variance for the mechanicaloscillator, where the error is derived from the fits. Forthe positive spin mass the constructive interference of theQBA for the two systems is evident from comparing thegreen data points to the blue curve for the membraneonly (Fig. 3d).To find the reduction/enhancement of the QBA forthe hybrid system, we use the calibration of the thermalnoise described in the Supplementary Information andpresented in Fig. 2c,d. The mechanical thermal noisefound in Fig. 2d is shown as the blue shaded area inFig. 3c,d. The spin thermal noise found in Fig. 2c isused as an input to the detailed model to find its con-tribution to the observed hybrid spectra (brown shadedarea in Fig. 3c,d). Note that this noise is suppressed bythe opto-mechanical response around Ω M = Ω S by thesame mechanism as the QBA contribution of the spin isreduced to zero at this point. Subtracting the thermalnoise area from the total area, we find the QBA variancecontribution for the hybrid negative system of . × x zpf (striped area in Fig. 3c) and for the hybrid positive sys-tem, . × x zpf (striped area in Fig. 3d). Comparingthese values with the QBA of . × x zpf for the mechanicaloscillator, we conclude that the variance of the QBA forthe joint negative mass system is − . ( ± ) be- ba c Figure 4.
Quantum back action for the optimally detuned mechanical and spin oscillators.
Noise spectra ofdetected light. Axes labels: (SN) – shot noise of light, x zpf – zero point fluctuations. A. Membrane noise (blue dots), hybridsystem with the negative/positive mass spin oscillator (red/green dots), blue area – membrane thermal noise, brown area –spin thermal noise. Solid brown curve – a fit to the experimental spin spectrum taken without mechanical response. Red area– QBA for the hybrid system. Striped area – QBA for the membrane. B. Displacement sensitivity for the hybrid system withthe negative mass (red) normalized to the sensitivity for the mechanical oscillator. C. An example of the squeezed amplitudeoutput of the spin system for positive (black) and negative (brown) effective mass. low the variance for the mechanics alone, whereas for thejoint positive mass system it is . ( ± ) higher.The main contributions to the error bars are the uncer-tainties in the calibration of quantum cooperativities.Further studies reveal that a more efficient QBA eva-sion can be achieved when the two oscillator frequenciesare not exactly equal, Ω M = Ω S . Taking advantage of thestraightforward tunability of Ω S with magnetic field, werun the QBA evasion experiment with the spin oscillatorslightly detuned from the mechanical oscillator. In thiscase the best QBA evasion is obtained if the quadraturesof light between the atomic and the optomechanical sys-tems are rotated with respect to the phase of LO . Fig. 4ashows the data for the hybrid system with the negativespin mass (red dots) with Ω S − Ω M = 2 π × . kHz anda phase rotation of °, along with the noise of the me-chanical oscillator (blue dots). For this experiment wefind C Mq = 2 . . We observe the broadband QBA eva-sion which, additionally, is most pronounced at Ω = Ω M where the mechanical response is maximal. The observedtotal variance for the hybrid system with the negativespin mass, . × x zpf , is ± of the variance formembrane only, . × x zpf . Note that interference inthe hybrid system leads to suppression of the spin noise(solid brown curve) at Ω S , which is instead transformedinto efficient QBA evasion around Ω M for the negativemass hybrid system. Fig. 4b shows the improvement inthe membrane displacement sensitivity obtained by the QBA evasion calculated as the ratio of the blue and redcurves from Fig. 4a. These data signify broadband QBAevasion in a model independent way.Subtracting thermal noise contributions we find thehybrid QBA (red area in Fig. 4a) of . × x zpf , thatis − . ( ± ) suppression compared to the me-chanical QBA of . × x zpf (striped area). For the hy-brid system with the positive spin mass (green dots), theQBA is . × x zpf which is . ( ± ) above theQBA for the mechanics alone. In this detuned case theQBA reduction in case of negligible thermal noise can,in principle, overcome the limit of / valid for the caseof Ω M = Ω S (see Supplementary Information), as indi-cated by the reduction of the classical BA that wehave observed in an independent experiment with thesystem driven by classical white noise. The physics ofthe broadband QBA interference is due to the combina-tion of the frequency dependent amplitude squeezing ofthe light generated by the spin and the interference ofQBA of the two systems. An example of the amplitudesqueezed output from the spin in shown in (Fig. 4c).In conclusion, we have presented a novel hybrid quan-tum system consisting of distant mechanical and spin os-cillators linked by propagating photons. Constructive ordestructive interference of the quantum back action forthe two oscillators depending on the sign of the effectivemass of the spin oscillator is demonstrated. A detailedmodel describes the results with high accuracy. We haveshown that the back action evading measurement in thehybrid system leads to the enhanced sensitivity of thedisplacement measurement. Further improvements arerealistic with reduced propagation losses, even higher Q mechanical oscillators [28] and cavity enhanced spin sys-tems. These results pave the way for entanglement gener-ation and quantum communication between mechanicaland spin systems, and to QBA free measurements of ac-celeration, gravity and force. ∗ To whom correspondence should be addressed. Email:[email protected][1] Caves, C. M., Thorne, K. S., Drever, R. W. P., Sandberg,V. D. & Zimmermann, M. On the measurement of a weakclassical force coupled to a quantum-mechanical oscilla-tor. i. issues of principle.
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ACKNOWLEDGMENTS
We acknowledge illuminating discussions with FaridKhalili. The cells with spin-protecting coating were fab-ricated by Mikhail Balabas. This work was supported bythe European Union Seventh Framework Program (ERCgrant INTERFACE, projects SIQS and iQUOEMS), theEuropean Union’s Horizon 2020 research and innovationprogramme (ERC grant Q-CEOM, grant agreement no.638765), a Sapere Aude starting grant from the Dan-ish Council for Independent Research, and the DARPAproject QUASAR. R.A.T. is funded by the program Sci-ence without Borders of the Brazilian Federal Govern-ment. E.Z. is supported by the Carlsberg Foundation.We acknowledge help from Marius Gaudesius at the earlystage of the experimental development.
AUTHOR CONTRIBUTIONS
E.S.P. conceived and led the project. C.B.M, R.A.Tand G.V. built the experiment with the help of K.J., Y.T.and A.S. The membrane resonator has been designed andfabricated by Y.T. C.B.M, R.A.T, G.V. and E.S.P. tookthe data. E.Z. and K.H. developed the theory with inputfrom A.S. and E.S.P. The paper was written by E.S.P.,K.H., E.Z., R.A.T., C.B.M. and G.V. with contributionsfrom other authors. A.S., K.H. and E.S.P supervised theresearch. upplementary Information for“Quantum back action evading measurement of motionin a negative mass reference frame”
Christoffer B. Møller, Rodrigo A. Thomas, Georgios Vasilakis,
1, 2
Emil Zeuthen,
3, 1
YeghisheTsaturyan, Kasper Jensen, Albert Schliesser, Klemens Hammerer, and Eugene S. Polzik Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Institute for Electronic Structure and Laser, Foundation for Research and Technology-Hellas, Heraklion 71110, Greece Institute for Theoretical Physics & Institute for Gravitational Physics (Albert Einstein Institute),Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany
I. EXPERIMENTAL SETUPA. Atomic spin oscillator
The spin ensemble consists of N a ∼ Cesium atoms contained in an anti-relaxation coated pyrex vapor cell(microcell) [S1] heated to a temperature of ◦ C . These atoms are confined in a channel of µ m × µ m × connected to an external Cesium reservoir via a ∼ µ m radius laser drilled hole, as shown in Figure 1b. They areaddressed by light of a waist (radius) size of µ m focused through the channel. The microcell is enclosed in a fourlayer magnetic shielding, protecting the spins from ambient magnetic fields and external RF sources. An inner systemof coils produces a homogeneous bias field, B , which leads to a Larmor frequency Ω S . The wall-to-wall transient timeof Cs atoms in the channel is on average ’ . µ s . Within the characteristic evolution time of the quantum state, themoving atoms cross the light beam many times and thus experience a motionally averaged interaction [S2, S3]. Thespin-protecting coating of the cell walls grants an intrinsic decoherence rate of ’ µ s . This rate is limited by spindestruction collisions of atoms with the cell walls, magnetic field inhomogeneities and spin-exchange collisions.A circularly polarized diode laser tuned to the F = 3 → F = 4 transition of the D2 line is used to spin polarizeatoms into the F = 4 ground state. The probing of the atomic ensemble is done with linearly polarized light at . (LO ), blue detuned by ∆ S ∼ π × from the F = 4 → F = 5 in the D transition. Absorptioneffects can be neglected as ∆ S (cid:29) ∆ ω HWHMDoppler ’ π ×
200 MHz , effectively eliminating the effect of spin motion. Thepolarization of the probe set by the half-waveplate (HWP in Fig. S1) allows for adjustment of the polarization axisof the linearly polarized probing light (LO ). In the experiment this axis is chosen to minimize the added broadeningof the spin oscillator, resulting in an angle θ a ≈ ◦ with respect to the direction of the atomic polarization. Weemphasize that the dominant part of the light-atoms interaction is of the QND type [S4] (see Eq. (S18)) and does notdepend on the angle θ a ; the optical rotation that the light experiences is to a very good approximation independentof the polarization orientation. The vacuum sidebands that affect the spin oscillator are in an orthogonal polarizationmode and π/ out of phase with respect to the local oscillator LO .The optical readout rate, Γ S ∝ ∆ − S Φ | J x | , is a function of the number of atoms J x = F N a , the local oscillatorphoton flux Φ and the detuning ∆ S . Its origin lies in the re-parametrization of the Faraday rotation experienced bylight due to interaction with far detuned atoms [S5, S6]. In the established language of the light-atoms interface [S2],the readout rate is related to the interaction strength κ atoms = Γ S T , with T being the temporal length of the probinglight mode. The atomic spins’ linewidth, γ S , is dominated by the optical broadening and is proportional to Φ in theregime of interest. Typical values for optical powers and γ S are presented in Table I.The Cesium spin ensemble fully polarized to F = 4 , m F = ± has projection noise variance var ( ˆ J y,z ) PN = | J x | ~ / F N A ~ / N A ~ in its ground state, whereas a completely unpolarized ensemble has var ( ˆ J y,z ) Th = F ( F +1) N A ~ / / × var ( ˆ J y,z ) PN [S2]. In the experiment var ( ˆ J y,z ) ’ . × var ( ˆ J y,z ) PN with the degree of spin polarization of ,equivalent to having . units of extra ground state noise. The negative (positive) mass configuration is achieved byoptical pumping of the atoms to the F = 4 , m F = +4 ( F = 4 , m F = − ) state, i.e, parallel (antiparallel) to themagnetic field, which we take to define the positive x -direction. Within the Holstein-Primakoff approximation [S7]this is formally equivalent to having a harmonic oscillator with Ω S < ( Ω S > ) as depicted in Figure 1c (Figure 1d)of the main text. Note that throughout the Methods, we include the sign of the effective spin mass in Ω S (whereasthis sign is stated as an explicit prefactor in the main text). For the negative (positive) mass oscillator, the firstexcited state of the spin oscillator is the one with a single atom in the F = 4 , m F = 3 ( F = 4 , m F = − ) state.Experimentally, we change the magnetic field direction to choose the sign of the oscillator’s mass. a r X i v : . [ qu a n t - ph ] A p r As presented in Figure 2c, the quantum cooperativity for the spin oscillator, C Sq , is characterized via broadbandthermal modulation of the optical driving force. This technique requires an electro-optical modulator to drive LO ’spolarization quadrature ˆ X L, in and a RF source outputing thermal voltage noise, producing a frequency independentand proportional to the driving voltage, optical power with excess n WN photon flux in the frequency band of interest.In Figure 2c, the black dots represent the spin oscillator driven with n WN = 1 . . Comparing this curve to theshot-noise limited ( n WN = 0 ) probing, in brown dots, we extract the effect of quantum noise of light on the spinoscillator and the ratio of back action to thermal noise, the quantum cooperativity. For a thorough discussion on thiswe refer to the Section I G. B. Optomechanical System
The optomechanical system is based on a near-monolithic cryogenic membrane-in-the-middle system described indetail elsewhere [S8]. The mechanical oscillator is a highly stressed,
60 nm thick SiN membrane supported by asilicon periodic structure forming a phononic bandgap. The bandgap protects the oscillator from phonon tunnelingfrom the clamping of the structure and provides a region clear of undesired phononic modes [S9]. The high stressboosts the quality factor through dissipation dilution [28]. The membrane thickness, t M , is chosen to maximizethe optomechanical single photon coupling rate, g . This introduces a tradeoff between the zero point fluctuations x zpf = p ~ / ( m eff Ω M ) ∝ t − / M and the amplitude reflection coefficient of SiN which is periodic with t M .The (1,2) drum mode of the membrane with frequency Ω M = 2 π × .
28 MHz is used as it is the lowest frequency modeto lie within the bandgap and has a high quality factor of Q = 13 × as measured by ring-downs ( γ M = 2 π ×
50 mHz ).The ∼ side length difference of the membrane breaks the degeneracy of the (1,2) and (2,1) modes significantly,with the (1,2) mode being ∼ lower in frequency than its sibling. This membrane is placed in a cavity andaligned such that the cavity TEM mode has a good overlap with the (1,2) mode and a poor overlap with the (2,1)mode. This further separates the systems as the optical spring effect (dynamical back action) pushes the (1,2) modeanother ∼ away, while having only a marginal impact on the (2,1) mode.The . long plano-concave Fabry-Pérot optical cavity with finesse F = 4500 (half bandwidth of κ = 2 π ×
13 MHz ) is mounted in a continuous flow cryostat with large windows for good optical access and a base temperature of . . The power tranmissions of the mirrors are
20 ppm and thus giving a largely one-sided cavity. Placingthe aforementioned dielectric membrane µ m from the
20 ppm mirror, forms two sub-cavities whose dynamics canbe mapped onto the canonical optomechanical formulation used in the theory section via the transfer matrix modelapproach described in [S10]. In effect, the cavity half bandwidth κ is modulated depending on the position of themembrane with respect to the standing wave in the cavity [S11]. This is due to a differential intracavity photonpopulation being built up in each sub-cavity. Having more intracavity photons populating the sub-cavity boundedon one side by the low transmission (
20 ppm ) mirror produces an overall decreased cavity loss rate. In the canonicalformulation this is equivalent to the decay rates κ and κ of the cavity ports 1 and 2 being altered asymmetrically,see Fig. S2. The membrane itself adds negligible additional loss.We position the membrane in the cavity such that the coupling rate is large and where the overall cavity bandwidthis reduced from κ = 2 π × (bare cavity) to κ = 2 π × . (finesse enhanced to ). This comes at theexpense of having a less one-sided cavity. The ratio of the cavity ports decay rates goes from κ /κ = 70 (bare cavity)to κ /κ = 25 . The reduced cavity bandwidth is advantageous as a certain degree of sideband resolution is requiredto optically broaden the mechanical oscillator significantly without requiring a too large readout rate (required tomatch the spin system).The high transmission incoupling mirror is mounted on a piezoelectric transducer, which is used to tune the cavityresonance frequency close to that dictated by the atomic probe LO . It is tuned such that LO probes the cavityred detuned by ∆ = − π × . . This is ensured using a separate beam originating from the same laser. Thisbeam is blue shifted (by − ∆ ) from LO described above by an acousto-optic modulator. It is then phase modulatedat 12MHz and probes the cavity from the undercoupled port 2, see Fig. S1. An error signal is derived using thePound-Drever-Hall technique and is used to feedback on the aforementioned piezoelectric transducer which stabilizesthe cavity such that this beam is locked on resonance. This locking beam is in the orthogonal polarization to LO and contributes to < of the intracavity power. The locking beam thus has a negligible impact on the intracavitydynamics and the final detection.The mechanical oscillator is initially only coupled to a thermal bath of temperature T bath with mean occupation ¯ n bath . Adding the probe LO alters the dynamics of the system as the oscillator is coupled to the intracavity fieldwith a rate g = g √ N , where N is the mean intracavity photon number. Dynamical back action optically broadensthe mechanical linewidth by γ M,opt (such that γ M = γ M + γ M,opt ) and the mean thermal occupation is reducedto ¯ n thM = ( γ M /γ M )¯ n bath . This (so-called) sideband cooling is due to an asymmetry in the Stokes and anti-Stokesprocesses caused by the detuning from cavity resonance. The Stokes sideband (causing heating) is never completelysuppressed which sets a minimum achievable mean occupation ¯ n minM [S12]. The effective mean occupation of themechanical oscillator is now ¯ n M = ( γ M /γ M )¯ n bath + ( γ M,opt /γ M )¯ n minM , where the contribution ¯ n minM is correlatedwith the quantum back-action.The total variance of motion will thus have contributions from both the QBA and the thermal bath; the ratio ofthese is γ M,opt (¯ n minM + 1 / γ M (¯ n bath + 1 / ≈ C Mq (cid:18) κ κ + (∆ − Ω M ) + κ κ + (∆ + Ω M ) (cid:19) , (S1)where we have introduced the quantum cooperativity C Mq = g N/ (2 κγ M ¯ n bath ) and approximated ¯ n bath +1 / ≈ ¯ n bath (in the present scenario ¯ n bath ∼ ).The total variance of the motion can be directly inferred from the area of the measured output spectrum by h ˆ X M i = R (Γ − M h ˆ P L ,out i − d Ω ≈ Γ − M R h ˆ P L ,out i d Ω . The approximation is good as the spectra used are dominated byQBA and thermal noise contributions, with a negligible SN contribution, in the frequency range shown in all Figures.By the same token, Eq. (S1) gives operational meaning to the quantum cooperativity C Mq as the ratio of QBA andthermal contributions to the observed variance (except for the Lorentzian factor of order unity, . , seen on theright-hand side).The optomechanical system is probed in reflection. The combination of a quarter-waveplate at ◦ and a polarizingbeam splitter effectively acts as an isolator transmitting the input light (in the hybrid configuration coming from thespin system) and reflecting the light emerging from the cavity. With well characterized optical losses and systemparameters, the bath temperature and C Mq can be inferred from the observed degree of ponderomotive squeezing.A spectrum (amplitude quadrature) showing squeezing of - . (- . corrected for detection efficiency of )is shown in Figure 2b. The shot noise (SN) level is verified by balanced detection and by comparison to a whitelight source to within < accuracy. Using the detailed model (outlined in the theory section) with the bathtemperature of the membrane, T bath , as the only adjustable parameter, we obtain the fit shown in Figure 2b with T bath = (7 ± .
5) K . From T bath we thus obtain the value of C Mq = 2 . ± . for the data presented in Figure 2d. Thethermal noise contribution is shown as the blue area and can easily be found using Eq. (S1).For the two phase quadrature data sets shown the mechanical oscillator was optically broadened to γ M = 2 π × . . This required the input LO power to be adjusted as the cavity half bandwidth and single photon couplingrate varied slightly between runs. This was due to a varying membrane position with respect to the intra-cavityfield on different days caused by overnight temperature cycles. The experimentally realised parameters for these aredisplayed in Table I. C. Hybrid system
A detailed schematic of the experimental setup is shown in Fig. S1: a travelling light field interacts with the atomicspin and the mechanical oscillator in a cascaded way. The driving beams for the system, LO and LO , as well asthe local oscillator LO for the homodyne measurement are generated by a Ti-Sapphire laser. The light is shot noiselimited for the relevant powers and Fourier frequencies of interest in the quadratures that matter for both systems.The spin-mechanics interface requires filtering of the spins’ output field. The atomic spins respond to modulationout of phase and in the orthogonal polarization to its local oscillator (denoted by the quantum field operator ˆ X L, in in the main text). The mechanical oscillator responds to modulation in phase and in the same polarization mode asits own driving local oscillator (represented by the field operator ˆ X ML, in ). Therefore, LO should be in an orthogonalpolarization mode and π/ phase shifted with respect to LO1. Two filtering stages are required, one for polarizationand one for phase, both of them being depicted in Fig. S1 and described in the following. The polarization filtering isdone using HWP and PBS right after the microcell, decoupling LO from the quantum fluctuations of interest in theorthogonal polarization quadrature; the phase filtering is realized using a Mach-Zehnder interferometer with outputat PBS , setting a variable phase for the spin sideband fields { ˆ X SL, out , ˆ P SL, out } in respect to LO . The driving localoscillator and the sidebands are then projected in the same polarization mode with HWP and PBS and coupled tothe optomechanical cavity. To detect the optical quadrature of interest, balanced polarimetry with the local oscillatorLO (with the aid of another Mach-Zehnder interferometer) is performed when the phase quadrature is of interest; tomeasure of the amplitude quadrature, the local oscillator is removed and all light is directed to a single photodiode.In the experiment, PBS extinguishes LO better than : from the optical path with little loss of the modulationsidebands or those carrying information about the spin oscillator. [[[ FIG. S1. Detailed schematic of the experimental setup. The atomic spin system is pictured in the black-brown dashed box,along with its B-field and optical pumping; in the blue lined box, the optomechanical membrane-in-the-middle setup. Thehybrid system is probed via a travelling optical mode. The atomic system, driven by LO with linear polarization angle setby 0 (cid:13) has its output polarization filtered in 1 (cid:13) and is recombined with the correct phase with LO in 2 (cid:13) , set electronically viasuitable detection in D ; 3 (cid:13) ensures that both local oscillator and the filtered atomic response have the same polarization. Theoptomechanical system is probed in reflection and frequency stabilized via PDH locking in the unused port. Phase sensitivedetection is done via homodyning with LO in D . An electro-optic modulator (EOM) in LO is used for locking the phase filtering interferometer. The phase and axisof the EOM are adjusted in such a way so that a voltage modulation (small with respect to the π voltage) results in asmall modulation predominantly in the degree of circular polarization of light (quantified by the ˆ S z Stokes component).The ratio of circular polarization modulation to linear polarization modulation introduced by the EOM is typically ∼ in power. Sinusoidal modulation sidebands at , far from both oscillators’ responses, provide the phasereference. These sidebands are combined with LO in the output of the interferometer, PBS . A half-waveplate allowsfor an adjustable fraction of the sideband power to be used for locking (typically ∼ ). The demodulated result ofthe balanced polarimetry measurement of the locking signal in D is proportional to cos δφ LO1,2 , where δφ LO , is thephase difference between LO and LO . Feedback on a piezoelectric transducer proportional to this signal allows usto lock the new local oscillator in phase with the sideband quadrature that drove the spin oscillator.The half-waveplate HWP and PBS project a small portion of LO ( ∼ ) and most of the sideband signal( ∼ ) into the same polarization mode. Suitable optics direct the beam onto the optomechanical cavity with atotal optical power transmission for the spin system response sidebands in the − range. These sidebandsare in the same spatial mode as LO , which is modematched to the cavity with an efficiency of η mm ’ . Themodematching is defined as the fraction of incident LO power going into the TEM compared to all TEM modes.When the characterization of the atomic spin oscillator is performed, a function generator provides a white noise(WN) modulation over the interesting frequency range, from to ; typical values for the added modulatedWN photons (in units of SN) ranges from 0.5 to 100.The spectra of ˆ P L, out are measured by balanced homodyning of the field reflected off the optomechanical cavitywith LO , with power in the order of . , which is locked to the DC zero of the interference fringe with LO ,thus ensuring that the phase quadrature is being measured.The model fits and knowledge of all relevant system parameters provide a reliable reference point from which wecalibrate the spectra in units of the mechanical zero point fluctuations. For example, the right vertical axis in Fig. 1dshows the spectral density of motion in units of x zpf / kHz , calibrated by dividing the thermal noise by n th . Integratingthe power spectral density data we find the observed membrane variance of . × x zpf . Subtracting the thermal noisevariance . × x zpf we obtain the QBA variance . × x zpf .Losses in the system are due to the finite transmission coefficient between the spin and the optomechanical systemsof η = 0 . , which includes the finite optomechanical coupling efficiency . , and the transmission coefficient betweenthe optomechanical system and the detection of η = 0 . which includes the quantum efficiency of the photodetectorof . . These values vary within a few percentage points from experiment to experiment. THEORETICAL MODELD. Optomechanical System
The optomechanical system is described by the standard linearized Hamiltonian ˆ H = Ω M (cid:16) ˆ X M + ˆ P M (cid:17) − ∆ ˆ a † ˆ a − g (cid:0) ˆ ae iφ + ˆ a † e − iφ (cid:1) ˆ X M , where [ ˆ X M , ˆ P M ] = i are the dimensionless canonical operators for the mechanical system, and [ˆ a, ˆ a † ] = 1 are an-nihilation/creation operators for cavity photons. ∆ = ω L − ω c is the detuning of the driving laser from the cavityresonance ω c . The linearized optomechanical coupling rate g = g | α | depends on the single photon coupling rate g of the optomechanical system and the intracavity amplitude α . It is linked to the optomechanical readout rate Γ M introduced in the main text by Γ M = 2 g κ , (S2)where κ is the cavity half linewidth. The phase φ = arctan(∆ /κ ) denotes the phase of the intracavity field amplitude α relative to driving field, as is discussed further below in Eq. (S32). Here we take the incoming field as the phasereference instead of the intracavity field (as is usually done in cavity optomechanics) since we eventually interestedin the transfer matrix for the incoming/outgoing amplitudes resulting from this Hamiltonian. Including decay andLangevin noise forces the equations of motion corresponding to the Hamiltonian are ˙ˆ a ( t ) + ( κ − i ∆)ˆ a ( t ) − ige − iφ ˆ X M ( t ) = √ κ ˆ a in ( t ) + √ κ ˆ v in ( t )¨ˆ X M ( t ) + Ω M ˆ X M ( t ) + 2 γ M ˙ˆ X M ( t ) − Ω M g (cid:0) ˆ a ( t ) e iφ + ˆ a † ( t ) e − iφ (cid:1) = p γ M Ω M ˆ f ( t ) , where ˆ a in ( t ) and ˆ v in ( t ) are incoming quantum fields driving the cavity through port and , respectively, cf. Fig. S2;their commutation relations are [ˆ a in ( t ) , ˆ a † in ( t )] = δ ( t − t ) = [ˆ v in ( t ) , ˆ v † in ( t )] . The partial decay rates κ fulfill κ = κ + κ . The linewidth of the mechanical resonance (excluding optical broadening) is γ M , and the thermal Langenvinforce is ˆ f ( t ) . In the high temperature limit we can take h ˆ f ( t ) ˆ f ( t ) i = δ ( t − t )(¯ n + 1 / where ¯ n ’ k B T bath / ( ~ Ω M ) . FIG. S2. Schematic of the setup: The atomic spin is driven by light noise X L,in and spin noise F S . Output light of the the spinsystem X SL,out is channeled to the atomic system, and experiences losses characterized by a transmissivity η associated withadditional light noise V ,in and a phase rotation by an angle ϕ , resulting in a driving field X ML,in of the optomechanical system.The optomechanical cavity is two-sided with decay rates κ and κ . The optomechanical system is driven in addition by lightnoise V in and a thermal force F . The output field of the optomechanical system experiences further losses with transmissivity η associated with additional light noise V ,in . In the frequency domain the equations of motion read (cid:0) κ − i (∆ + Ω) (cid:1) ˆ a (Ω) − ige − iφ ˆ X M (Ω) = √ κ ˆ a in (Ω) + √ κ ˆ v in (Ω) D M (Ω) ˆ X M (Ω) − g Ω M (cid:0) ˆ a (Ω) e iφ + ˆ a † ( − Ω) e − iφ (cid:1) = p γ M Ω m ˆ f (Ω) , where D M (Ω) = Ω M − Ω − i Ω γ M . (S3)We define field quadratures as ˆ X L (Ω) = 12 (ˆ a (Ω) + ˆ a † ( − Ω)) ˆ P L (Ω) = 12 i (ˆ a (Ω) − ˆ a † ( − Ω)) (S4)and similar definitions are used for quadratures of incoming/outgoing fields. In terms of these the equations of motionin frequency domain are κ − i Ω ∆ − g sin φ − ∆ κ − i Ω − g cos φ − g Ω M cos φ g Ω M sin φ D M (Ω) ˆ X L (Ω)ˆ P L (Ω)ˆ X M (Ω) = √ κ ˆ X ML,in (Ω) + √ κ ˆ V x,in (Ω) √ κ ˆ P ML,in (Ω) + √ κ ˆ V p,in (Ω) √ γ M Ω M ˆ f (Ω) which can be conveniently written in terms of block matrices (cid:18) O Tφ
00 1 (cid:19) (cid:18)
A BC D M (Ω) (cid:19) (cid:18) O φ
00 1 (cid:19) (cid:18) ˆX L (Ω)ˆ X M (Ω) (cid:19) = √ κ ˆX ML,in (Ω) + √ κ ˆV in (Ω) √ γ M Ω M ˆ f (Ω) ! where O φ = (cid:18) cos( φ ) − sin( φ )sin( φ ) cos( φ ) (cid:19) , A = (cid:18) κ − i Ω ∆ − ∆ κ − i Ω (cid:19) ,B = (cid:18) − g (cid:19) , C = (cid:0) − g Ω M (cid:1) , ˆX L (Ω) = (cid:18) ˆ X L (Ω)ˆ P L (Ω) (cid:19) , etc.The equations of motion are solved by (cid:18) ˆX L (Ω)ˆ X M (Ω) (cid:19) = (cid:18) O Tφ
00 1 (cid:19) (cid:18)
A BC D M (Ω) (cid:19) − (cid:18) O φ
00 1 (cid:19) √ κ ˆX ML,in (Ω) + √ κ ˆV in (Ω) √ γ M Ω M ˆ f (Ω) ! where the inverse Block matrix can be expressed in two equivalent forms (cid:18) A BC D M (Ω) (cid:19) − = (cid:18) A − + A − BS − CA − − A − BS − − S − CA − S − (cid:19) (S5) = T − − T − BD − M − D − M CT − D − M + D − M CT − BD − ! (S6)by means of the Schur complements S = D M (Ω) − CA − B = D M (Ω) + Γ M κ Ω M ∆( κ − i Ω) + ∆ =: D M (Ω) , (S7) T = A − BD − M (Ω) C = (cid:18) κ − i Ω ∆ − ∆ − Γ M κ Ω M D M (Ω) κ − i Ω (cid:19) . (S8)The effective mechanical susceptibility including optically induced shift and broadening is χ M (Ω) = 2Ω M D − M (Ω) . (S9)The intracavity quadratures following from Eqs. (S5) and (S6) are ˆX L (Ω) = O φ T − O Tφ (cid:16) √ κ ˆX ML,in (Ω) + √ κ ˆV in (Ω) (cid:17) + √ γ M κ Γ M Ω m D M (Ω) O φ A − ˆF (Ω) , in which ˆF (Ω) := (cid:18) f (Ω) (cid:19) , and the field reflected off the cavity in port is ˆX ML,out (Ω) = − ˆX ML,in (Ω) + √ κ ˆX L (Ω)= O φ (2 κ T − − ) O Tφ ˆX ML,in (Ω) + √ κ κ O φ T − O Tφ ˆV in (Ω)+ 2 √ Γ M κκ γ M Ω m D M (Ω) O φ A − ˆF (Ω)=: M (Ω) ˆX ML,in (Ω) + V (Ω) ˆV in (Ω) + F (Ω) ˆF (Ω) (S10)The optomechanical transfer matrix M (Ω) is explicitly given by M (Ω) = 2 κ ( κ − i Ω) + ∆ (cid:16) ∆ + Γ M κ Ω M D M (Ω) (cid:17) O φ (cid:18) κ − i Ω − ∆∆ + Γ M κ Ω M D M (Ω) κ − i Ω (cid:19) O Tφ − = 1 D c (Ω) 2 κ D M (Ω) D M (Ω) O φ (cid:18) κ − i Ω − ∆∆ + Γ M κ Ω M D M (Ω) κ − i Ω (cid:19) O Tφ − (S11)where D c (Ω) = ( κ − i Ω) + ∆ . In the form given in the second line the dependence on the effective mechanicalsusceptibility becomes evident.We note that for a broadband cavity ( κ (cid:29) ∆ , Ω M , Ω ) and neglecting losses ( κ = 0 ) one recovers from Eq. (S10)the simple optomechanical input-output relation stated in the main text, ˆ X ML,out (Ω)ˆ P ML,out (Ω) ! = (cid:18) M χ M (Ω) 1 (cid:19) ˆ X ML,in (Ω)ˆ P ML,in (Ω) ! + p Γ M γ M χ M (Ω) (cid:18) f (Ω) (cid:19) . (S12)In the limit considered here the susceptibility corresponds to the one of the bare mechanical system (without shiftand broadening).For nonzero detuning and taking into account effects of a finite cavity linewidth, the more involved input-outputrelations described by Eqs. (S10) have to be considered in general. However, in the unresolved-sideband regime( κ (cid:29) Ω M , Ω ) we may obtain a simplified expression for the optomechanical transfer matrix (S11). To this end, wenote that the cavity response to the individual sideband components at ± Ω of the light quadratures (S4) is governedby the complex Lorentzian L (Ω) = κκ − i (∆ + Ω) =: | L (Ω) | e iθ (Ω) , | L (Ω) | = κ p κ + (∆ + Ω) , θ (Ω) = arctan (cid:18) ∆ + Ω κ (cid:19) , (S13)where we have introduced its polar decomposition. In terms of this, Eq. (S11) can be reexpressed as (again neglectingcavity losses for simplicity, κ = 0 ) M (Ω) = e i [ θ (Ω) − θ ( − Ω)] O φ +[ θ (Ω)+ θ ( − Ω)] / [1 + i Γ M χ M (Ω)4 ( | L (Ω) | − | L ( − Ω) | )] + Γ M χ M (Ω)4 (cid:18) − ( | L (Ω) | − | L ( − Ω) | ) ( | L (Ω) | + | L ( − Ω) | ) (cid:19) ! O Tφ − [ θ (Ω)+ θ ( − Ω)] / , (S14)where φ = θ (0) (see discussion of Eq. (S32) below) and χ M (Ω) is the effective mechanical susceptibility (S9). Toobtain a simpler expression for Eq. (S14) in the regime κ (cid:29) Ω M , Ω , we expand | L (Ω) | and θ (Ω) to linear order aroundthe carrier frequency ( Ω = 0 in the rotating frame), | L (Ω) | ≈ L + δL (Ω) , L := | L (0) | = κ √ κ + ∆ , δL (Ω) := − Ω∆ κ ( κ + ∆ ) / (S15) θ (Ω) ≈ φ + δθ (Ω) , δθ (Ω) := Ω κκ + ∆ , (S16)resulting in the optomechanical scattering matrix M (Ω) ≈ e iδθ (Ω) O φ [1 + i Γ M χ M (Ω) L δL (Ω)] + Γ M χ M (Ω) L (cid:18) (cid:19) ! , (S17)to leading order in δθ, δL (the phase prefactor does not affect the resulting spectra and will be suppressed for brevityhenceforth).The transfer matrix in Eq. (S17) interpolates between the simple result in Eq. (S12), which is valid in the limit κ → ∞ , and the general result in Eq. (S11).When considering the field reflected off the cavity in port 1 in Eq. (S10), the finite modematching η mm of the inputquadratures ˆX L,in (Ω) to the cavity quadratures ˆX L (Ω) is treated as equivalent to the input port having higher loss,i.e. κ → η mm κ . The total cavity loss remains fixed κ = κ + κ and we simply treat κ → κ + (1 − η mm ) κ as theinput for the additional vacuum noise. E. Atomic Spins System
As discussed in Ref. [S2, S6], in the limit of low saturation and large detuning from the atomic resonance, theHamiltonian affecting the atomic spin and light polarization observables can be written in the form: ˆ H int = α ˆ S z ˆ J z , (S18)where ˆ J z is the dimensionless ( ~ = 1 ) collective spin component along the direction of light propagation (taken hereto coincide with the z axis in the lab frame) and ˆ S z is the Stokes component of light that measures the degree ofcircular polarization. The parameter α depends on the detuning from the resonance ∆ S , on the area A of interactionand on physical constants: α = Γ sp A ∆ λ π α (∆) , (S19)where Γ sp is the spontaneous emission rate associated with the optical transition, λ is the wavelength of light and α (∆) is a numerical factor that depends on the specific atomic structure and for detunings much larger than theexcited state hyperfine structure can be approximated to be unity.For an ensemble of a large number of atoms, highly polarized along the direction of a static magnetic field ( x direction), the Holstein-Primakoff transformation can be performed and map the collective spin operators to positionand momentum operators of an effective (spin) oscillator: ˆ X S = ˆ J z p | J x | ; ˆ P S = − sgn ( J x ) ˆ J y p | J x | . (S20)Here we represent the macroscopic mean polarization by its x -projection J x = h ˆ J x i (including its sign) rather thanmerely its magnitude (as done in the main text for simplicity). As described in the main text and above in theMethods section, the relative sign of J x and B can be either positive or negative, reflecting whether the macroscopicspin is aligned or anti-aligned with respect to the applied magnetic field. For the case of this work, where Cesiumatoms are polarized in the F = 4 hyperfine manifold of the ground electronic state, positive sgn ( J x /B ) corresponds toa negative mass oscillator (energy should be extracted to remove the ensemble from the fully polarized state), whereasnegative sgn ( J x /B ) corresponds to a positive mass oscillator.The presence of the static magnetic field adds the Hamiltonian term ˆ H S = µ B g F B ˆ J x (S21)with µ B being the Bohr magneton and g F = 1 / the Landé factor for the F = 4 manifold. In the language of spinoscillators this Hamiltonian term affects the evolution of ˆ X S and ˆ P S in the following way: ˙ˆ X S (cid:12)(cid:12)(cid:12) B = Ω S ˆ P S ; ˙ˆ P S (cid:12)(cid:12)(cid:12) B = − Ω S ˆ X S , (S22)where Ω S = − sgn ( J x ) µ B g F B so that Ω S > < refers to the positive (negative) mass scenario.A similar mapping can be performed with the Stokes components of light. For linearly polarized light in the x direction with Stokes component S x = sgn ( S x )Φ / , where Φ is the photon flux, the mapping can be written in theform: ˆ X L = ˆ S z p | S x | ; ˆ P L = − sgn ( S x ) ˆ S y p | S x | . (S23)From Eqs. (S18), (S20), (S21) (S23) and (S27) we can write the Hamiltonian as ˆ H = p Γ S ˆ X S ˆ X L + Ω S ( ˆ X S + ˆ P S ) / , (S24)The input-output relationships for the Stokes components ˆ S z, out ( t ) = ˆ S z, in ( t ); ˆ S y, out ( t ) = ˆ S y, in ( t ) + αS x ˆ J z ( t ) , (S25)are mapped into: ˆ X L, out ( t ) = ˆ X L, in ( t ); ˆ P L, out = ˆ P L, in ( t ) + p Γ S ˆ X S ( t ) , (S26)where the readout rate Γ S is Γ S = 12 α Φ | J x | . (S27)The atomic spin dynamics, including the effects of its interaction with a Markovian reservoir, is dd t ˆ P S ( t ) = − Ω S ˆ X S ( t ) − γ S ˆ P S ( t ) + p γ S ˆ F S ( t ) + p Γ S ˆ X L ( t ) , (S28) dd t ˆ X S ( t ) = Ω S ˆ P S ( t ) , (S29)with ˆ F S ( t ) being the random Langevin force acting on the spin; this force is the analogous of the thermal noise ˆ f thatacts on the mechanical oscillator. Its correlation function is h ˆ F S ( t ) ˆ F S ( t ) i = δ ( t − t )( n S + 1 / , where a thermal spinoccupancy n S > reflects the excess noise induced by imperfect polarization of the ensemble. In the above analysis,the effect of tensor polarizability in the evolution of the light and spin state was neglected. For the detuning used inthe experiment ( ∆ S ∼ ) the effect of the tensor polarizability is estimated to be on the few percent level [S12].In frequency space, the spin system is structurally identical to the one of the simple limit considered in Eq. (S12)for the optomechanical system, that is, Fourier transforming and solving Eqs. (S26,S28,S29), one obtains the matrixrelationship X SL,out (Ω) = S (Ω) X L,in (Ω) + p Γ S γ S χ S (Ω) F S (Ω) . S (Ω) = (cid:18) S χ S (Ω) 1 (cid:19) (S30)where the spin oscillator susceptibility is χ S (Ω) = 2Ω S / (Ω S − Ω − i Ω γ S ) . The spin thermal noise is representedby F S (Ω) = [0 , ˆ F S (Ω)] T . Here we adopt a phenomenological model for the susceptibility of the spin oscillator. Amicroscopic derivation along the lines of [S13] would result in a slightly different susceptibility with corrections to thepresent one scaling as Q − S where Q S (cid:29) is the quality factor of the atomic oscillator. F. Hybrid System
The two systems are connected such that X ML,in (Ω) = X MS,out (Ω) , as shown schematically in Fig. S2. Takinginto account losses and further phase shifts as indicated in the figure the compound transfer matrix for the hybridoptomechanical-spin system is ˆX L,out (Ω) = √ η η M (Ω) O ϕ S (Ω) X L,in (Ω) (vacuum noise of light transduced through S and M ) + √ η η M (Ω) O ϕ p Γ S γ S χ S (Ω) F S (Ω) (spin noise transduced through M ) + p (1 − η ) η M (Ω) ˆV ,in (Ω) (vacuum noise of light from losses btw S and M ) + √ η V (Ω) ˆV in (Ω) (vacuum noise of light from losses in optomechanical cavity) + √ η F (Ω) F (Ω) (thermal noise from M ) + p − η ˆV ,in (Ω) (vacuum noise from losses between M and detector) (S31)0where η and η denote the transmission efficiencies from the spin system to the optomechanical cavity and from theoptomechanical cavity to the detector, respectively. Vacuum noises incurred through these losses are described by ˆV ,in (Ω) . An optional phase shift ϕ introduced deliberately in between the two systems is accounted for by therotation matrix O ϕ .Finally, the homodyne detection is performed in the frame of the classical field after the optomechanical systemwhere it has acquired a phase shift relative to the field before the optomechanical cavity. This phase is found asfollows: The classical intracavity amplitude α is connected to the incoming amplitude α in by α = √ κ κ − i ∆ α in = α in √ κ + ∆ e iφ , φ = arctan(∆ /κ ) . (S32)where κ = κ + κ . The outgoing field is α out = − α in + √ κ α = κ − κ + i ∆ κ + κ − i ∆ α in = [ κ − κ + i ∆] [ κ + κ + i ∆]( κ + κ ) + ∆ α in ∼ e i ( ψ + φ ) α in where ψ = arctan(∆ / ( κ − κ )) . Accordingly, the measured field quadrature ˆ P L,meas is determined by (cid:18) ˆ X L,meas ˆ P L,meas (cid:19) = O Tψ + φ ˆX L,out (Ω) . This relation is used to determine the measured noise spectral densities shown in the main text. For simplicity ofnotation, the measured quadrature ˆ P L,meas is referred to as ˆ P L, out in the main text and other parts of the Methods.We will now use the transfer matrix of the hybrid system to analyse the QBA contribution to the optical output field[Eq. (S31), 1st line]. For the case when the opto-mechanical damping dominates the membrane response, γ M (cid:29) γ M ,and in the sideband unresolved limit, Ω M (cid:28) κ , we can apply the approximate optomechanical scattering matrix (S17)to find (ignoring optical losses η = 0 = η , κ = 0 for simplicity and setting ϕ = 0 ) ˆ P L,meas = (cid:2) Γ M L χ M (Ω) + Γ S χ S (Ω) { i Γ M χ M (Ω) L δL } (cid:3) ˆ X L,in , (S33)where L is the empty cavity Lorentzian response and δL is the difference in cavity response at frequencies ± Ω (cid:28) κ .Only with δL = 0 (LO tuned to cavity resonance) the spin QBA and the mechanical QBA add/subtract in ˆ P L,meas .From Eqs. (S3,S7,S9) one finds that L δL = ( γ M − γ M ) / Γ M , that is the distortion of the QBA due to δL = 0 hasthe same origin as the optomechanical broadening. In the relevant case of strong optomechanical cooling γ M (cid:29) γ M ,there is no back action cancellation at the exact joint resonance frequency since iχ M (Ω = Ω M ) γ M = − . In thisregime the QBA power spectum of the hybrid system S P L,meas corresponding to Eq. (S33) becomes S P L,meas = (Γ M δ S + Γ S δ M ) + Γ M γ S ( δ M + γ M )( δ S + γ S ) S X L,in , (S34)with δ M,S = Ω − Ω M,S and S X L,in being the power spectral density of the input light amplitude fluctuations. Formatched responses ( Γ S = Γ M L ) , Ω M = Ω S , Γ M = Γ S , γ M = γ S , the ratio of the hybrid QBA spectrum to the QBAspectrum of the mechanics becomes γ M / ((Ω − Ω M ) + γ M ) , thus QBA evasion is indeed expected everywhere, exceptfor Ω = Ω M . The minimal variance of the hybrid QBA is / of the QBA of the mechanical oscillator alone. G. Calibration of quantum back action for the spin system
To characterize the quantum cooperativity, C Sq , and the readout rate, Γ S , one can use the fact that a singlelight quadrature is coupled to the oscillator: by suitable modulation of ˆ X L, in one can boost the contribution of themeasurement-induced back action.As thoroughly discussed in the previous sections and summarized by equation (S30), the input-output relations forthe continuous readout of a harmonic oscillator are (cid:18) X L, out P L, out (cid:19) = (cid:18) X L, in P L, in (cid:19) + Γ v T Lv (cid:18) (cid:19) (cid:18) X L, in P L, in (cid:19) + p Γ γv T LF T h (cid:18) (cid:19) , (S35)1in which Γ is the readout rate, γ the decay rate and L = ( iω − M ) − M = (cid:18) ω − ω − γ (cid:19) v = (cid:18) (cid:19) v = (cid:18) (cid:19) . In a more straightforward language, equation (S35) becomes (cid:18) X L, out P L, out (cid:19) = (cid:18) X L, in P L, in (cid:19) + R BA (cid:18) (cid:19) (cid:18) X L, in P L, in (cid:19) + R Th (cid:18) (cid:19) , (S36)with R BA and R Th being the response functions of the oscillator to the back action and thermal forces.The effect of losses is also important, as there is an admixture of uncorrelated vacuum, indicated by the subscript v , with the signal of interest; therefore (cid:18) X L, out P L, out (cid:19) → √ η (cid:18) X L, out P L, out (cid:19) + p − η (cid:18) X L, v P L, v (cid:19) . (S37)The PSD for both light quadratures are calculated from the absolute square of the equation (S36): S XX = η h X L, in X † L, in i + (1 − η ) h X L, v X † L, v i S P P = η h h P L, in P † L, in i + R BA h X L, in X † L, in i + R Th i + (1 − η ) h P L, v P † L, v i . (S38)Therefore, it is explicit that to boost the back action component of the oscillator readout in comparison to the othernoise contributions, one should modulate the in-phase quadrature of light, ˆ X L . Doing so, the input spectral densitiesare h X L, in X † L, in i → ( n W N + 1) h X L, in X † L, in i , and h X L,i ( ω ) X † L,j ( − ω ) i = δ ( ω − ω ) δ ij , in which i, j represent the different sources of fluctuations. Therefore, theinput-output relations from (S38) are S XX = ηn W N + 1 S P P = η (cid:2) R BA ( n W N + 1) + R Th (cid:3) + 1 . (S39)Experimentally, to be able to calculate the back action to thermal noise ratio, one needs to measure the (i) responseof the system to SN drive and (ii) the response of the system with some known modulation n W N , for a given probepower. Calibrating the curves in shot noise units, the measured spectral on-resonance heights after the subtractionof the white noise contribution, defined here as A and B, are S W NP P − B = η (cid:2) R BA ( n W N + 1) + R Th (cid:3) S SNP P − A = η (cid:2) R BA + R Th (cid:3) , therefore, the desired ratio is: R BA R Th = B − A ( n W N + 1) A − B . (S40)This technique was used to calibrate C Sq , the quantum cooperativity of the spin oscillator. For this measurement, η = 0 . is the detection efficiency, S WN is the spectral density of added white noise in units of vacuum noise and S QBA and S TH are the back action and thermal spectral densities, respectively. The measurements of the phasenoise presented in (Figure 2c) and of S WN are performed with polarization interferometry using LO calibrated to theshot noise of LO . From the phase noise S PP , and S PP ,N measured for n WN = 0 (vacuum input) and S WN = 1 . ,respectively (Figure 2c), we find S QBA = ( S PP ,N − S PP , ) / (1 . × η A ) and C Sq = ( S PP ,N − S PP , ) / (2 . × S PP ,N − S PP , ) =1 . ± . .2 [S1] Corsini, E. P., Karaulanov, T., Balabas, M. & Budker, D. Hyperfine frequency shift and zeeman relaxation in alkali-metal-vapor cells with antirelaxation alkene coating. Phys. Rev. A , 022901 (2013).[S2] Hammerer, K., Sørensen, A. S. & Polzik, E. S. Quantum interface between light and atomic ensembles. Rev. Mod. Phys. , 1041–1093 (2010).[S3] Borregaard, J. et al. Scalable photonic network architecture based on motional averaging in room temperature gas.
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Intrinsic decoherence rate γ S π × Total decoherence rate γ S π × (2 . , . LO driving power (1 . , . Detuning from the D F = 4 → F = 5 transition ∆ S GHz
Quantum cooperativity C Sq . ± . Spin Polarization 60%Microcell optical losses η microcell ◦ C Mechanical oscillator
Effective mass m eff Zero point fluctuations x zpf Intrinsic mechanical frequency Ω M π × . Intrinsic damping rate γ M π × Cavity detuning ∆ 2 π × − . Total cavity half linewidth κ π × (8 . , . LO2 drive power (54 , µ W Intracavity photons N (5 . , . × Single photon coupling rate g π × Thermal bath temperature T bath Bath occupancy n bath × Quantum cooperativity C Mq (2.6, 2.2)Mechanical linewidth γ M π × . Mean thermal occupancy ¯ n thM Hybrid & detection
Quantum efficiency in between systems
Detection efficiency , Homodyning visibility
Cavity mode-matching η mm90%