A quantum memory at telecom wavelengths
Andreas Wallucks, Igor Marinković, Bas Hensen, Robert Stockill, Simon Gröblacher
AA quantum memory at telecom wavelengths
Andreas Wallucks, Igor Marinkovi´c, Bas Hensen, Robert Stockill, and Simon Gr¨oblacher ∗ Kavli Institute of Nanoscience, Department of Quantum Nanoscience,Delft University of Technology, 2628CJ Delft, The Netherlands
Nanofabricated mechanical resonators are gaining significant momentum among potential quan-tum technologies due to their unique design freedom and independence from naturally occurringresonances. With their functionality being widely detached from material choice, they constituteideal tools to be used as transducers, i.e. intermediaries between different quantum systems, and asmemory elements in conjunction with quantum communication and computing devices. Their capa-bility to host ultra-long lived phonon modes is particularity attractive for non-classical informationstorage, both for future quantum technologies as well as for fundamental tests of physics. Here wedemonstrate such a mechanical quantum memory with an energy decay time of T ≈ T ∗ for qubits, whichexhibits a power dependent value between 15 and 112 µ s. This demonstration is enabled by a noveloptical scheme to create a superposition state of | (cid:105) + | (cid:105) mechanical excitations, with an arbitraryratio between the vacuum and single phonon components. Quantum memories are a core quantum technology,which are at the very heart of building quantum re-peaters enabling large quantum networks [1, 2]. Signifi-cant progress on realizing such memories has been madewith ions [3, 4], atomic ensembles [5–7], single atoms [8],NV centers [9], and erbium-doped fibers [10]. The impor-tant characteristics of a memory, besides sufficiently longstorage times, are the ability to store a true quantumstate, such as a single photon, high read-out efficiency,on-demand retrieval and operation at low-loss telecom-munication wavelengths around 1550 nm. So far, none ofthe realizations have simultaneously been able to demon-strate all of these requirements. In particular, native op-eration of memories in the telecom band has been limitedto storage times in the tens of nanoseconds [11].Recently, chip-based nanoscale mechanical resonatorshave emerged as promising components for future quan-tum technologies. Their functionality is principally basedon geometry, allowing for great flexibility in materialsand designs and creating unique opportunities for com-bining them with many other techniques such as inte-grated photonics and superconducting circuitry [12, 13].Over the past few years, experiments have demonstratedan ever increasing control over quantum states of me-chanical resonators both via optical as well as electri-cal interfaces. Experimental breakthroughs with radio-frequency drives include electromechanically induced en-tanglement [14, 15], phonon-number detection [16, 17]and single [18] as well as multi-phonon Fock state gener-ation [19]. Optical control over the modes on the otherhand has enabled the detection of non-classical optome-chanical correlations [20], single phonon Fock state cre-ation [21], mechanical entanglement [22], as well as anoptomechanical Bell-test [23]. Excitingly, many of these ∗ [email protected] chip-based devices have also been shown to host ultralong-lived mechanical modes with down to wavelength-scale footprints and low cross-talk [24–26]. Combined,these results demonstrate the key ingredients for the re-alization of a quantum optomechanical memory, poten-tially paving the way towards on-chip, integrated quan-tum transducers and repeaters, operating at telecomwavelengths [27].In this work, we demonstrate for the first time non-classical correlations from an engineered high-Q mechan-ical resonance and an optical interface in the conventionaltelecom band over the full decay time of the mechanicalmode. To achieve this, the mechanical quantum memoryis prepared in a single-phonon state, directly usable fora DLCZ-type quantum repeater scheme [28]. We showthat we can store this state for approx. 2 ms withoutdegradation due to induced thermal occupation of themode, a limitation of several previous experiments [21–23]. We study the phase fluctuations of the mechanicalmode in a classical continuous interference measurementand then by a pulsed dephasing experiment in the quan-tum regime. For the latter, we develop and experimen-tally demonstrate a scheme to optically prepare superpo-sitions of the first mechanical Fock state and the vacuum.This stored state is then optically retrieved and interferedwith a weak coherent probe to measure the coherence ofthe mechanical mode.Our device design, shown in Figure 1, is based onprevious experiments with silicon optomechanical crys-tals [20, 21, 24], which were optimized for low Q m tospeed up re-thermalization with the cryogenic environ-ment. In contrast, in our present work the mechanicalmode serves as a phononic memory for optical states andwe are hence looking for large quality factors. In par-ticular, the mechanical decay time sets a bound on thedistance over which light can travel while the stored statehas not decayed yet. For a 1 ms decay time, for example, a r X i v : . [ qu a n t - ph ] O c t a) F r e q u e n c y ( G H z ) b) a)c) min max Γ Γ
FIG. 1. (a) Scanning electron microscope image of the op-tomechanical device. Light enters from the left through thecoupling waveguide in the top part of the image. In the de-vice below, an optical and a mechanical resonance are cou-pled through radiation pressure effects. The structure is fab-ricated from a 250 nm thick silicon layer and is undercut toproduce free-standing devices. (b) Zoom in to the phononicshield region (top) and corresponding bandstructure simula-tion (bottom). Unlike the nanobeam itself, the shielding re-gion exhibits a full bandgap of the phononic crystal aroundthe resonance frequency Ω m (solid red line). (c) Finite ele-ment simulation of the isolation through the phononic shield.Also shown is the mechanical mode of interest at 5.12 GHz inthe center of the nanobeam. we can reach distances of around 200 km. The mechan-ical mode of the silicon nanobeam is confined within aphononic crystal mirror region which does not exhibit acomplete bandgap. Fabrication imperfections typicallycouple the phonon mode of interest to leaky modes withdifferent symmetries, such that the quality factor is lim-ited by radiation loss. For this experiment, we surroundthe device with a two-dimensional phononic shield whichfeatures a complete bandgap [24] and can increase the de-cay time up to several seconds [26]. Figure 1a shows thefabricated device with the additional phononic shieldingregion on the sides and an optical waveguide on top usedto probe the device. A simulation of the bandstructureof the shield region in Figure 1c shows a wide bandgapemerging between 4 GHz and 6 GHz.The actual device used in this experiment has a me-chanical resonance at Ω m / π = 5 .
12 GHz with a mea-sured decay time of the mode of 2 π/ Γ m = 1 . Q m ≈ at mK temperatures. While wehave fabricated structures with significantly better qual-ity Q m (cid:38) , this particular choice is a compromise be-tween long memory time and a low re-initialization rate,which would in turn necessitate a prohibitively long mea-surement time. The optical resonance of the device is at ω c / π = 191 . κ i / π = 460 MHz. As shown in Figure 1a,we couple to this resonance via an adjacent optical waveg-uide with a coupling rate of κ e / π = 1840 MHz. The op-tical and mechanical mode interact with a single photoncoupling rate g / π = 780 kHz. A sketch of the fiber-based optical setup is shown inFigure 2a, where the device is placed in a dilution re-frigerator with a base temperature of 15 mK. We gener-ate 40 ns long optical pulses with blue sideband detun-ing from the optical resonance ω b = ω c + Ω m , as wellas red sideband detuning ω r = ω c − Ω m . Using lin-earized optomechanical interactions [29, 30], blue driv-ing enables the optomechancial pair generation ˆ H b = − (cid:126) g √ n b ˆ a † ˆ b † +h.c., with n b the intracavity photon num-ber, (cid:126) the reduced Planck constant and ˆ a † (ˆ b † ) the opti-cal (mechanical) creation operator. Red detuning on theother hand enables a state-transfer between the opticsand the mechanics, due to the beamsplitter-type interac-tion ˆ H r = − (cid:126) g √ n r ˆ a † ˆ b +h.c., where n r is the intra-cavityphoton number for the red pulses. After interacting withthe device in the cryostat, the light passes an opticalfilter with 40 MHz bandwith which is locked to the opti-cal resonance ω c and the Stokes- / anti-Stokes fields aredetected on a superconducting nanowire single photondetector (SNPSD).As a first step, we perform a thermometry measure-ment to validate that the 5-GHz mode of the device isthermalized to its motional groundstate at the base tem-perature of the cryostat. This is performed by sendingtrains of either blue or red sideband detuned pulses to thedevice, such that an asymmetry in the scattering ratesbetween the blue and the red drives allows us to inferthe mechanical mode occupation [20, 31]. The measuredmode temperature is elevated from the bath tempera-ture due to heating caused by power-dependent opticalabsorption of the drives. We can thus use this mea-surement to determine our maximally attainable driv-ing power for a given mechanical mode occupation. Welimit the instantaneous heating caused by a single pulseto ∼ p r = 14% between phonons and photons in the anti-Stokes field by the red detuned drive using the interactionof ˆ H r .To test the performance of the device as a quantummemory, we proceed to use a two-pulse protocol witha blue-detuned excitation followed by the red-detunedreadout pulse. The first pulse enables the optomechanicalpair generation according to ˆ H b , producing entanglementbetween the optical Stokes-field and the mechanical modeof the form | ψ (cid:105) om ∝ | (cid:105) om + √ p b | (cid:105) om + O ( p b ) , (1)where o (m) indicate the optical (mechanical) modes and p b is the excitation probability. Due to the correlationsin this state, detecting a Stokes photon after the bluepulse heralds the mechanical mode in a state close to asingle phonon Fock state [21] | Ψ (cid:105) m ∝ | (cid:105) m + O ( √ p b ) . (2)Crucially, the excitation probability p b has to be keptsmall to avoid higher order excitation terms. Addition-ally, residual absorption heating causes an increased ther-mal background of the mechanical mode at delays farlonger than the pulse length. We choose an energy ofthe blue-detuned pulse of 3 fJ, which corresponds to ascattering probability of p b = 0 . g (2)om = P ( B ∧ R ) / [ P ( B ) P ( R )], for which P ( B )( P ( R )) is the probability to detect a Stokes (anti-Stokes)photon and P ( B ∧ R ) is the joint probability to detect aStokes and an anti-Stokes photon in one trial. We probethe ability of the mechanical mode to store non-classicalcorrelations by evaluating g (2)om (∆ τ ) for pulse delays ∆ τ over the energy decay time of the mechanical resonance.Generally, we expect the cross-correlation to evolve like g (2)om (∆ τ ) = 1 + exp( − Γ m ∆ τ ) /n therm (∆ τ ) [22], where Γ m is again the inverse of the decay time and n therm (∆ τ )the number of thermal phonons in the mode. We notethat this cross correlation has a classical bound givenby a Cauchy-Schwarz inequality of g (2)om ≤ g (2)om ≥ . T = 1 . ± . ∼ µ s, supporting the prospect of usingmechanical quantum memories in a device independentsetting [34].The above measurements clearly demonstrate that, al-though absorption heating is still present in the device,we can limit its effects to be able to store and retrievequantum states in the mechanical mode for its whole de-cay time. Since the scheme uses a phase-symmetricalstate of the form of Eq. (2), we are, however, not sen-sitive to frequency fluctuations of the mechanical mode.We therefore proceed to test the ability of our device topreserve non-trivial quantum mechanical states.We begin by assessing the frequency stability of themechanical mode in the classical regime. As discussed inthe SI, direct measurements of the mechanical sidebandson a fast photodector using a relatively strong continu-ous sideband drive show an inhomogeneously broadened pulsed a) optomechanicalsidebandsfilter SNSPDdrives b) n t h e r m ( ∆ τ ) c) g o m ( ∆ τ ) ( ) Delay ∆τ (s) -7 -6 -5 -4 -3 -2 ∆τ FIG. 2. (a) Laser pulses are sent to the optomechanical devicethrough an optical circulator. These pump pulses are subse-quently filtered, allowing us to measure their Stokes- and antiStokes-fields in superconducting nanowire single photon de-tectors (SNSPD). (b) Calibrated thermal mode occupancy,showing a delayed rise in the absorption heating from a blue-detuned pulse with energy 3 fJ and probed by a red-detunedpulse with energy of 280 fJ. The occupation contains a back-ground of ∼ g (2)om (see text for details). We find clear non-classicalphoton-phonon correlations up to T = 1 . ± . mechanical linewidth of several kHz. A frequency jit-ter of the mechanical mode is visible using fast scan-ning [26]. In order to determine the extent to whichoptical driving influences these dynamics, we require ameasurement scheme that employs a minimum possibleintracavity power. We devise an interferometric schemebased on coincidence detection in continuous wave driv-ing, shown in Figure 3a. We probe the device with a sin-gle optical tone on either the red or the blue sideband,such that the reflected light contains an optomechanicallygenerated sideband. This sideband is interfered with aprobe field at roughly ω c , which we create in-line fromthe reflected optical drives using an electro-optic ampli-tude modulator (EOM). To be able to detect photonsusing the SNSPDs, we remove the background with a40-MHz bandwidth filter. The main idea of the measure-ment is that the mixing of the mechanically and electro-optically generated sidebands causes intensity modula-tions at their beat frequency, which can be observed inthe coincidence rate C (2) (∆ τ ) of the detected photons.The interference can explicitly be demonstrated by de-tuning the EOM drive by δ Ω / π = 100 kHz from themechanical frequency Ω m , such that C (2) (∆ τ ) shows os-cillations of 2 π/δ Ω = 10 µ s period (see Figure 3b). Withthe mechanical mode in a thermal state, we can measurethe decay of this interference to extract a classical coher-ence time τ class by fitting an exponentially decaying sinefunction to the data (solid line). Additional details onthe data evaluation as well as measurements of the ther-mal bunching of the optomechnical photons are discussedin the SI.Figure 3c shows the dependence of the coherence time τ class for a sweep of the intracavity photon number n c .Measurements for blue detuning (blue points) and red de-tuning (red points) split for increasing photon numbersdue to optomechanical damping (see discussion in the SI).Inconsistent with residual optomechanical effects, how-ever, is the decrease of the decay constant τ class for thelowest photon numbers. A linear extrapolation to n c = 0results in τ class , min = 16 ± µ s. The behavior for n c > τ class , max = 112 ± µ s.The power dependence observed for the classical co-herence decay τ class invites an investigation of the coher-ence time in the quantum regime, where, additionally,the mode evolves in the dark. In contrast to the abovepulsing scheme (c.f. Figure 2), we are now required tomeasure the phase stability of non-symmetric mechanicalquantum states. A natural candidate are superpositionsbetween the vacuum and single phonon mechanical statesof the form | Ψ (cid:105) m = √ − n · | (cid:105) m + √ n · e iφ | (cid:105) m . (3)Here φ is an experimentally chosen phase during the statepreparation and √ n the amplitude of the single phononcomponent, with n = 1 / τ and the anti-Stokes field can be interfered with a weakcoherent state (WCS), for example. The visibility of thisinterference can then be used to assess the coherence time T ∗ over which the mechanical mode is able to preservethe phase of the superposition state. The scheme wedescribe below is conceptually similar to earlier experi-ments in quantum optics [35, 36] and enables the opticalpreparation of a massive mechanical superposition statefor the first time.We adapt the setup for the correlation decay measure-ments, adding an optical interferometer in the detectionpath (c.f. Figure 4a). The light is split by a 99:01 fiber C ( ) ( ∆ τ ) ( a . u . ) τ c l a ss ( µ s ) a)b)c) Delay ∆τ ( µ s)Intracavity photon number n c -1 amplitudeEOMcontinuous filter SNSPDdrives FIG. 3. (a) Schematics of the setup for measuring the clas-sical phase coherence with a CW laser. The amplitudeelectro-optic modulator (EOM) is driven at a detuning of δ Ω / π = 100 kHz from the mechanical frequency Ω m . (b)Interference between the optomechancially and the electro-optically generated sidebands causes an oscillatory signaturein the two-fold coincidence rate C (2) (∆ τ ). Shown is an exem-plary measured trace for ∼ τ class as a function of intracavity photon numbers n c for blue detuning (blue points) as well as for red detuning(red points). An increase in the measured coherence time, asa function of intracavity photon number, can clearly be seen(fit to the data as solid curve). For n c (cid:38) coupler and enters a Mach-Zehnder interferometer witha high transmission upper arm and a low transmissionlower arm. In the low transmission arm of the interfer-ometer, an EOM modulates the reflected blue drive toproduce a sideband at ω c (more than 20 dB bigger thanthe optomechanical one in this arm). This weak coher-ent state is then interfered with the optomechanicallygenerated sideband in the upper arm of the interferome-ter on a balanced fiber coupler. Both of its outputs pass40 MHz bandwidth filter setups to remove unwanted fre-quencies to only detect fields at ω c . A click on one ofthe SNSPDs heralds the superposition state of Eq. (3) ofthe mechanical mode, as the Stokes sideband is made in-distinguishable from the electro-optically generated weakcoherent state. In a quantum picture, the second beam-splitter removes all “which-path” information of the de-tected photons.The detection is performed after a variable delayby interfering the anti-Stokes fields from the devicewith another weak coherent state, generated from thered drive with the EOM in the lower arm. Thequantum interference of the two can be detectedthrough oscillations of the correlation coefficient E = (cid:16) g (2)om , sym − g (2)om , asy (cid:17) / (cid:16) g (2)om , sym + g (2)om , asy (cid:17) of the cross-correlations g (2)om , sym for detection events in the same and g (2)om , asy for different detectors. While we have to ensurephase stability of the Stokes and anti-Stokes fields andthe respective weak coherent states, we only have tostabilize the Mach-Zehnder interferometer, since path-length fluctuations from the device to the first fiber cou-pler are canceled due the fields propagating in commonmode. Similar to the classical coherence measurement,the requirement to actively change the interferometerphase to detect the interference can be alleviated byslightly detuning the EOM drive frequency from Ω m . Wechose a relative detuning of δω/ π = 1 MHz, which isagain much smaller than the bandwidth of the opticalpulses of ∼
25 MHz but enables a full 2 π phase sweep of E for delays of 1 µ s.The amplitude n of the superposition state can be cho-sen experimentally by adjusting the ratio of optomechan-ically to electro-optically generated photons in the side-bands. Without a thermal background on the mechan-ical mode, it is possible to prepare states of the formof Eq. (3) with a given single phonon amplitude n bymatching the count rate due to the WCS C WCS to theStokes count rate C b according to C WCS /C b = (1 − n ) /n .In our measurements the amplitude of the WCS is setto optimize for maximum visibility and detection ratesat the same time, which allows us to perform the me-chanical coherence measurements as quickly as possible.In the SI, we provide further numerical studies how thechoice of amplitude affects the expected interference vis-ibility in the presence of a thermal background on themechanical mode. Experimentally, we choose the samepulse energies for the red and the blue drives as in thecorrelation measurement (c.f. Figure 2). We adjust theEOM drive power such that C WCS /C b ≈
7, which, fora thermal background of n therm = 0 .
1, results in a sin-gle Fock amplitude n = 0 .
56. The second weak coherentstate is matched in power to the anti-Stokes field. Fig-ure 4b shows our experimentally measured correlationcoefficient E . A clear oscillation in the signal with theexpected period of 2 π/δω = 1 µ s can be seen for small de-lays ∆ τ around 1 µ s, demonstrating the successful inter-ference of the superposition state in the anti-Stokes fieldwith the WCS. We obtain a visibility of V = 58 ±
5% byfitting the signal between 1 µ s and 2 µ s with a sinusoidalfunction. This visibility exceeds the classical threshold of V class = 50% for the second order interference visibilityof two coherent states and is in good agreement with thetheoretically expected value of 63% (see SI).We plot the decay of the interference visibility for ex-tended delays in Figure 4c. The quantum coherence time C o rr e l a t i o n c o e ffi c i e n t E V i s i b i l i t y V single settingfitted a)b)c) Delay ∆τ ( µ s) Delay ∆τ ( µ s) ∆τ Ω - Ω reddrive bluedrive 50% Δ ω Ω - Ω Δ ω Ω - Ω Δ ω Ω - Ω Δ ω FIG. 4. (a) Sketch of the experimental setup for the coherencemeasurement of the quantum memory. After being reflectedfrom the device, the optical pulses are sent to an imbalancedMach-Zehnder interferometer. In the lower, highly attenu-ated arm, an electro-optic modulator (EOM) creates a weakcoherent state at the cavity frequency ω c by sideband mod-ulation of the reflected drives. The WCS is then interferedwith the optomechanical Stokes field from the upper, weaklyattenuated interferometer arm. All “which-path” informationis erased, such that a detection event on either of the detectors(SNSPDs) heralds a superposition state of the form shown inEq. (3). The state is retrieved from the device using a red-detuned drive and the interference of the anti-Stokes field witha second WCS generated with the same EOM is measured.Insets schematically depict the frequency components presentin the optical pulses as well as the filter transmission function.(b) Interference of the anti-Stokes field with a WCS after adelay from the blue pulse of ∆ τ ≈ µ s. The phase evolu-tion over the delay is enabled by detuning the EOM drive by δω/ π = 1 MHz. A sinusoidal fit to the data (solid curve)yields a visibility V = 58 ± of the state that we obtain from fitting an exponential de-cay is T ∗ = 15 ± µ s. This decay happens much fasterthan the measured T for the symmetrical state. Theobtained value is in very good agreement with the clas-sical coherence time τ class , min = 16 ± µ s, consideringthe intracavity photon number averaged over the dutycycle in the pulsed experiment is n pulse , avg ≈ − (c.f.Figure 3c). As shown before, the power dependence ofthe coherence time is not consistent with optomechani-cal effects. The observed saturation at τ class , max indicatesdispersive coupling of the mode to defect states such astwo-level fluctuators in the host material. In particularthe surface of silicon is known to host a variety of statesthat couple to mechanical modes [26, 37, 38]. Opticaldriving of our particular device causes a saturation ofthe frequency jitter imposed on the mechanical mode. Amore detailed study of the dynamics and prospects on animproved τ q will be the focus of future work. Similar two-level fluctuator noise is already known for a wide varietyof systems, for which saturation driving of the states bydifferent experimental means could offer a way to reducethe induced noise on the mode of interest [39, 40].In conclusion, we have measured the quantum decay T and the coherence time T ∗ of a high-Q mechanicalsystem and demonstrated its suitability as a mechani-cal quantum memory. This was possible by employinga two-dimensional phononic shield for mechanical isola-tion, small absorption heating in the optomechanical de-vice compared to previous experiments and by devisinga novel way to create and characterize mechanical super-position states. We design our device to operate directlyin the low-loss telecommunication band, with several or-ders of magnitude larger coherence times than competingsystems [11], while allowing an on-demand read-out. Thememory is prepared in both a superposition and a singlephonon state, which will allow it to be directly used ina DLCZ-type quantum network architecture. Our mea-surements show a clear power dependence of the coher-ence time, which is indicative of two-level systems on thesurface of our structure. Future experiments will deter-mine the nature and detailed properties of these surfacestates, which should lead the way to significant improve-ments of the coherence time of our devices.We would like to thank Moritz Forsch and MichailVlassov for experimental support and also gratefully ac-knowledge assistance from the Kavli Nanolab Delft. Thiswork is further supported by the Foundation for Funda-mental Research on Matter (FOM) Projectruimte grants(15PR3210, 16PR1054), the European Research Council(ERC StG Strong-Q, 676842), and by the NetherlandsOrganization for Scientific Research (NWO/OCW), aspart of the Frontiers of Nanoscience program, as well asthrough a Vidi grant (680-47-541/994). 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We perform initial measurements of the mechanicallinewidth in the cryostat with a continuous blue sidebanddrive and detection of the optomechanical sideband on afast photodiode such that it can be evaluated on a real-time spectrum analyzer. We find the center frequency atΩ with a visible jitter when scanning for around 50 msas shown in the bottom panel of Figure S1a. We pro-ceed to generate a histogram (top panel Figure S1a) bytaking the frequency differences between each two suc-cessive measurements. The fit (solid line) to a Gaus-sian lineshape results in an inhomogeneously broadenedFWHM linewidth of about 1.6 kHz. As can be seen fromthe plot, we additionally observe slower drifts of the cen-ter frequency in the single digit kHz range, which canreduce the inferred coherence time during long measure-ment runs. For the pulsed coherence experiment of Fig-ure 4c, the center frequency was monitored during themeasurement for the delay of ∆ τ = 50 µ s. While thisdrift does in principle limit the coherence of the device,it was found to be small enough not to have a significantinfluence on our data.We proceed to measure the mechanical quality factorusing a pulsed pump-probe experiment [21] with two red-detuned pulses of 40 ns length each. While the device isassumed to be thermalized upon arrival of the first pulsesuch that the optomechanical interaction is largely sup-pressed, absorption of the optical drive in the materialcauses a rise in the occupancy of the 5 GHz mechanicalmode. The heating dynamics can then be studied usingthe second equally strong pulse to read out the tempera-ture of the mode at a variable delay (c.f. Figure S1b). Un-like Figure 2b of the main text, this data is not calibratedin terms of phonon numbers of the mechanical mode.The decrease in the count rate nevertheless shows theexpected decrease in the amount of absorption inducedheating, which is what ultimately enables us to performthe measurements in the main text. At relatively highoptical powers, the absorption causes delayed heating ef-fects which reaches a maximum occupation around sev-eral µ s before the device re-thermalizes to the bath. Sur-prisingly, the dynamics of the delayed mode occupancychanges as well. For low powers, where the initial tem-perature rise around ∼ µ s is reduced, a slower dynamicbecomes apparent such that the temperature starts topeak around a few hundreds of µ s. The mechanical de-cay on the other hand seems largely unaffected by thisnew feature.While further investigation of this change in heat-ing dynamics is required, we demonstrate our ability tolimit the total induced occupation to below 0.2 phononsthrough a sufficiently weak blue drive in the main text, Time (hrs) O cc u r e n c e Delay (s) -1 -2 -2 -6 -5 -3 -4 -7 C o u n t r a t e ( a . u . ) a) b) δΩ / 2π (kHz) δ Ω / π ( k H z ) FIG. S1. (a) Direct measurements of the mechanical line showa jitter δ Ω around the center mechanical frequency (bottom),that results in a Gaussian lineshape of FWHM ∼ enabling the quantum experiments with the device overits full lifetime. Setup for pulsed experiments
A sketch of the fiber-based setup used in the main textis shown in Figure S2. The pulsed optical drives are gen-erated from two tunable diode lasers, which are stabi-lized using a wavelength meter. Both light sources arefiltered with fiber filters of 50 MHz bandwidth to reducethe amount of classical laser noise at GHz frequencies.The 40 ns Gaussian shaped pulses used in the experimentare created by two 110 MHz accousto-optic modulators(AOMs 1 and 2), which, after the lines are combined, areadditionally gated by AOM3 for a better pulse on-off ra-tio. All AOMs can be operated in CW mode to allow fordevice characterization and the CW coherence measure-ment. An electro-optical modulator (EOM2) is used totest the longterm stability of the setup (see text below).The light is routed into the dilution refrigerator and tothe device via an optical circulator, whereas coupling tothe device waveguide is achieved with a lensed fiber tip.For the coherence measurements, we use an imbalancedMach-Zehnder interferometer with a 99:01 and a 50:50fiber coupler. The free spectral range of the interferome-ter is measured to be 5 . lockinglight A-EOM EOM2fiber stretcher filter set 1 SNSPD 1AOM2AOM1AOM3AOM4 switch filter (fiber) filter set 2 SNSPD 2CW coherencemeasurement λ wavemeterredblueEOM1 s e t up s t ab ili t y
15 mK φ device FIG. S2. Schematic of the setup used in the experiment fea-turing all active elements, including acousto-optic modulators(AOM), electro-optic modulators (EOM) and one amplitudemodulator (A-EOM). Elements for polarization alignment arenot shown. The setup is based on low loss SMF-28 opti-cal fibers, with the only exception being the filter setups inthe detection path, which consist of two linear, optical free-space cavities each, for improved insertion losses comparedto commercial fiber based components. The superconductingnanowire single photon detectors (SNSPDs) are placed at the800 mK stage in the same cryostat as the device. achieve a locking bandwidth of 10 kHz. The light usedfor the locking is injected into the open port of the 99:01fiber coupler. It is derived from a third laser, which islocked ∼
10 GHz away from the blue drive to operateat the maximum suppression point of the detection fil-ters (see below). We use continuous light for locking theinterferometer and blank it for 1 µ s around the measure-ment pulses with AOM4. The amplitude modulator usedfor the continuous coherence measurement (c.f. Figure 3of the main text) is shown in the dotted box in the upperinterferometer arm.The detection part of the setup consists of two lineswith two filter setups each to suppress the reflected drivesfrom the devices and two SNSPDs. The filter setupsare free-space, with two ∼
40 MHz linear cavities each.The free spectral range of these home built cavities isdesigned to be between 17 GHz and 19 GHz in each line.The measurements are paused every 4 s and continuouslight is injected to the filter lines via optical switches (notshown, see also [20]) such that the filters can be stabilizedon resonance with the optical cavity. This re-locking stepgenerally consumes about 10-15% of the measurementtime, whereas the filter transmission stays within ∼ ∼
84 dB for filter set 1 and ∼
89 dB for set 2. We further measure ourdetection efficiency by using an off-resonant weak pulsefrom the device [21]. We achieve overall efficiencies ofthe filter setups and the SNSPDs of approximately 34%for setup 1 and 32% for setup 2 including the efficien-cies of the respective detectors. The collection efficiencyof the optomechanically generated sidebands consists ofthe device to waveguide coupling κ e /κ ≈
80% as givenin the main text and fiber coupling efficiency of 56%.The measurement also suffers additional loss of ∼ ∼ Continuous wave coherence measurement
Data evaluation
For the data evaluation of Figure 3b of the main text,we take ∼ ∼
500 Hz for each of the driving powers by attenuating inthe detection path if necessary. This is a tradeoff allowingus to detect on the relevant timescales of the mechanicalmode while enabling a good signal to noise ratio by sur-passing the optical background. We assume Poissonianstatistics for the delayed clicks, such that we can accountfor the decay by fitting an exponential function to thedata (orange line in Figure S3a).A second decay of the count rate can be seen in thedata (yellow line in Figure S3a). This is due to bunch-ing of the optomechanically scattered photons since thedevice is in a thermal state. While rather small in the dis-played trace, it can be fairly pronounced for higher driv-ing powers (see also below). We account for this by fittinga second, faster exponential to the data. Accounting forboth of these effects, we get the normalized two-fold co-incidence rate C (2) (∆ τ ) for delays ∆ τ which is shown inFigure S3b. We fit an exponentially decaying sine func-tion to the processed data (solid line), from which we canconfirm the expected period of the oscillation as well asextract the time-constant of the visibility decay. Devi-ations from the exponential decay of the visibility yieldinformation on the decay time of the proposed two-level0 Delay (s) N u m . r e c o r d s C ( ) ( Δ τ ) ( a . u . ) Delay (μs) a)b)
FIG. S3. (a) Example of the CW coherence measurement.Beating of an electro-optically generated and the mechanicalsideband can be seen in the statistics of the time difference be-tween successive detection events. We process the raw data bytaking an exponentially decaying background due to the finitecount rate in the experiment (solid red line) into account, aswell as an exponential count rate decay due to thermal bunch-ing of the optomechanically scattered photons (solid yellowline). (b) Processed data showing the two-fold coincidencerate C (2) (∆ τ ) after background correction, in which we fitan exponentially decaying sinusoidal function (solid line) toextract the time-constant of the dephasing. fluctuators as the physical mechanism of the decay. Wefind that the simple exponential fit captures the decaywell for all measured powers.We further fit the increase of the visibility decay τ class in Figure 3c of the main text using a phenomenologi-cally motivated model. We find a good agreement toa linear dependence at the lowest powers with the zeropower offset τ class , min . This dependence is only visible upto n c ≈
1, where the increase in coherence time slowsdown. We use a model function for the decay constant τ ( n c ) that is consistent with two-level fluctuators as thephysical origin of the power dependence. We assumethat coupling to fluctuating defects in the dark causesfrequency fluctuations of the mechanical mode result-ing in τ class , min . With the optical drives present, thesefrequency fluctuations take on a different, much smallermagnitude, resulting in τ class , max . Optical driving cantransition between these two regimes, which we will referto as states for simplicity. With the drive present, thetransition occurs proportional to the relative populationin the states, such that the rate is g def n c ( τ class , max − τ ),where g def is an unknown optical defect coupling and n c is the intracavity photon number. In the absence of theoptical drive, we assume that the system exponentiallydecays towards the state corresponding to the the zeropower decay constant. The relaxation rate is thus − Γ def τ with the unknown defect decay constant Γ def . Overall, N u m . r e c o r d s g ( ) ( Δ τ ) a)b) Delay (s)
Delay (s)
FIG. S4. (a) Bunching data from the device measured in theanti-Stokes photons without EOM modulation. The raw data(inset) is processed in the same way as for the continuous co-herence measurement. The fit (solid line) is an exponentialdecay to extract the decay timescale 2 π/ Γ bunch of the bunch-ing. (b) The measured Γ bunch for a sweep of the intracavitypower. The solid line is a theoretical power dependence basedon optomechanical damping with a zero power offset of Γ m . we have ˙ τ = g def n c ( τ class , max − τ ) − Γ def τ, (S1)which has a steady-state solution of τ ( n c ) = τ class , min + τ class , max − τ class , min Γ def g def n c . (S2)We apply this function to the background in Figure 3cin the main text, extracting estimates for τ class , min and τ class , max with Γ def g def being the final free fit parameter. Wenote that optomechanical effects are expected to cause asymmetric deviation from this background for blue andred detuned measurements (linearly with n c ). This op-tomechanical damping effect is seen in the data pointsdeviating from the solid line in Figure 3c for n c > C (2) (∆ τ ) is proportional to the second ordercoherence function g (2) (∆ τ ). To this end we perform anexperiment in which we do not mix the sidebands withan electro-optically generated probe, but rather detectscattered light from the device only using a red sidebanddrive. We show an example measurement in Figure S4afor an intracavity photon number of n c ≈ .
75. The the-oretically expected value of g (2) (0) = 2 is reduced, whichcan easily be explained by dark counts in our detectionas well as residual leakage of the sideband drive thoughthe filter setup. Both of these have flat statistics over the1shown range of delays. An exponential fit to the data letsus infer the time constant 2 π/ Γ bunch of the decay.Its inverse relates to the mechanical linewidth and assuch is expected to be subject to optomechanical damp-ing Γ opt . Since we are driving on the red sideband, weexpect the damping to cause a linear increase accord-ing to Γ bunch , theory = Γ m + Γ opt with Γ opt = 4 n c g /κ and an intrinsic zero power value Γ m . In Figure S4b,we plot the measured values (crosses) together the ex-pected effect (solid line) based on the device parametersand the damping rate Γ m from the pulsed measurements(c.f. Figure S1). While a linear extrapolation of the dat-apoints to zero power does approach the expected valueof Γ m , the broadening of the line is much bigger thanoptomechanical effects can explain. We suspect that ab-sorption heating of the continuous driving of the devicecauses an increased temperature of the device which re-sults in a broadening of the line [38]. As we do not havea simple model for this effect we abstain from taking itinto account in the data in Figure 3c of the main text.We note that due to its presence, we expect the highpower maximum coherence time τ class , max inferred fromthe model to be a lower bound on the real value. We alsonote, however, that the effect cannot be responsible thelow power drop towards τ class , min in Figure 3c. With themeasurement in Figure S4 we show that the magnitudeof the effect is too low at small intracavity powers to af-fect the coherence measurement and that in any case, itcould only cause an increase in coherence time with loweroptical power rather than a decrease. Pulsed coherence measurement
Predicted visibility
We perform numerical studies using QuTiP [41, 42]to predict the interference visibility of the mechanicalsuperposition states including a thermal background onthe mechanical mode. We start from a thermal state ofthe mechanical mode with given initial occupation n init and the optical vacuum state. For this simulation, welump all thermal occupation of the mode as seen in theexperiment into this single term, regardless whether itis present from the beginning or induced by the blue orthe red pulse. The two-mode squeezed state is generatedby applying the optomechanical pair generation Hamil-tonian ˆ H b = −√ p b ˆ a † ˆ b † + h.c., with ˆ a † (ˆ b † ) being theoptical (mechanical) creation operator and the experi-mentally chosen scattering probability p b = 0 . c † , whichwe prepare in a coherent state with variable amplitude α < θ = 0, representing the experimental in-terferometer phase. We then use a beamsplitter matrixˆ H BS = i/π ˆ a † ˆ c + h.c. to mix the two modes with equalsplitting ratio. After this operation, our density matrix n WCS1 / n
Stokes n WCS1 / n
Stokes V i s i b i l i t y V n WCS2 / n anti-Stokes V i s i b i l i t y V E x c i t a t i o n s n t o t a l n therm = 0.00 n therm = 0.05 n therm = 0.15 n therm = 0.30 a)b)c) FIG. S5. Numerical calculations on the pulsed coherence mea-surement of the main text. The color coding for different num-bers of thermal excitations n therm on top of the figure appliesto all plots. (a) Numerical calculations of the expected num-ber of excitations n total in the superposition state of Eq. (3) ofthe main text. This number includes both the single phononpart for the superposition as well as the thermal backgroundcontribution. The horizontal axis indicates the ratio of exci-tations in the Stokes field n Stokes and the weak coherent state n WCS1 with which it is overlapped for the state preparation.Experimentally, this ratio can be chosen by the respectivecount rates on the detectors, regardless of detection efficiency.The vertical dashed line indicates the experimental settingused in the main text. (b) Expected interference visibility V for the superposition states of (a) with a second weak coher-ent state of equal amplitude to the anti-Stokes field. Here weassume optimal settings in the readout step. (c) Decrease ofthe expected visibility of the interference from mismatchingthe number of excitations n WCS2 of the second weak coherentstate from the anti-Stokes field n anti − Stokes . The calculationshere assume parameters for the state preparation correspond-ing to the vertical dashed line in (b). describes the modes of the beamsplitter output as well asthe mechanical mode. Heralding is modeled by applyinga projection matrix onto the Fock-state to one of the op-tical modes. Afterwards we trace out both of the opticalmodes, such that the remaining density matrix describesthe mechanical mode in the superposition state. At thisstage of the calculation we have prepared the state ofEquation 3 of the main text including a thermal back-2ground.To emulate the detection of the state, we define a sec-ond WCS mode with operator ˆ d † and coherent ampli-tude β < φ . We skip the optomechan-ical readout step and treat the mode ˆ b † as the anti-Stokes field. We apply another beamsplitter operationˆ H BS = i/π ˆ b † ˆ d +h.c. to model the interference of the anti-Stokes field and the WCS and finally calculate the expec-tation value for the number of excitations in the output.Sweeping the WCS phase φ , we find the intensity to fluc-tuate periodically between the two modes, from whichwe can infer the expected visibility of the interference.The results of these calculations are shown in Fig-ure S5. We find the analytically expected result that,without any thermal occupation, an equal match of theexcitations in the optomechancially generated state andthe first weak coherent state results in an equal superpo-sition | ψ (cid:105) = 1 / √ | (cid:105) m + e iφ | (cid:105) m ). We can further seethat the numerical calculation predicts that only a statewithout any thermal background can have perfect inter-ference visibility with a coherent state, and it can do soonly in the limit of vanishing amplitude. We calculate amaximum visibility for a 50:50 superposition state with-out thermal background of V = 1 / √
2, which reproducesthe analytical solution. Furthermore we see that for anincreasing thermal background, the maximally attainablevisibility is reduced and the optimal superposition ratiois shifted slightly to states with a stronger single phononcomponent. The drop-off in the expected visibility withan increase in the WCS amplitude is only gradual. Inthe experiment, we therefore choose to exceed the ampli-tude of the Stokes field with the amplitude of the weakcoherent state by a factor of ∼ β of thesecond weak coherent state used in the detection. Here,we need to match the anti-Stokes field from the device. Setup stability
We test the stability of our setup during the long in-tegration times in the coherence measurement by em-ulating the optomechanical device with the EOM1 rightbefore the circulator in Figure S1. The lasers are detunedby 1 nm from the device, such that they do not interactwith the optical mode of the device but are still efficientlyreflected in the coupling waveguide. To emulate the op-tomechanical scattering from the device we drive EOM1at the mechanical frequency to create weak sidebands at ω c . We otherwise recreate the pulsed coherence measure-ment of the main text by creating weak coherent stateswith EOM2 from the reflected drives and measuring theinterference in the coincidence counts. Since all detected V i s i b ili t y V Delay (ns)Measurement time (hrs) C o rr e l a t i o n c o e ff . E a)b) FIG. S6. (a) Setup visibility calibration. The lasers are de-tuned from the optical resonance of the device and we useEOM1 (c.f. Figure S1) to modulate sidebands on the pulsedoptical drives. The rest of the experiment is performed likethe pulsed coherence measurement of the main text. Thesecond order interference we expect from two weak coher-ent states, generated with EOM1 and EOM2, is bounded to50%. Fitting the correlation coefficient E just as in the maintext, we achieve visibilities in good agreement to this valuefor all relevant delays (shown is an example for a delay of ∼ µ s). (b) Longterm stability of the interference visibilityin (a). Drifts of our setup, in particular polarization varia-tions, slowly reduce the achievable visibility in the setup. Inthe main experiment, the polarizations and optical powerswere manually readjusted about every 24 hrs. fields are in electro-optically generated coherent states,the expected visibility in this case is 50%.This measurement allows us verify the maximally at-tainable visibility of our setup (see Figure S6a). We mea-sure a visibility of about 50%, regardless of the actualdelay of red and blue pulses, which we test up to 1 ms.We can further test the longterm stability of the setupon the timescale of hours. Figure S6b shows the evo-lution of the fitted visibility for a delay around 100 µ sbetween the two pulses over more than 12 hrs. We see aslow decrease of the system visibility, with a reduction to ∼ ∼∼