Microwave-to-optics conversion using a mechanical oscillator in its quantum groundstate

TL;DR

This paper demonstrates a highly efficient, low-noise microwave-to-optical signal conversion using a quantum groundstate mechanical oscillator, advancing quantum communication technology.

Contribution

It introduces an integrated on-chip electro-opto-mechanical device capable of coherent conversion with minimal thermal noise, operating in the quantum groundstate.

Findings

Achieved microwave-to-optical conversion with less than one phonon thermal background.

Maintained coherence of microwave signals during transduction.

Operated with an optomechanical cooperativity greater than 1.

Abstract

Conversion between signals in the microwave and optical domains is of great interest both for classical telecommunication, as well as for connecting future superconducting quantum computers into a global quantum network. For quantum applications, the conversion has to be both efficient, as well as operate in a regime of minimal added classical noise. While efficient conversion has been demonstrated using mechanical transducers, they have so far all operated with a substantial thermal noise background. Here, we overcome this limitation and demonstrate coherent conversion between GHz microwave signals and the optical telecom band with a thermal background of less than one phonon. We use an integrated, on-chip electro-opto-mechanical device that couples surface acoustic waves driven by a resonant microwave signal to an optomechanical crystal featuring a 2.7 GHz mechanical mode. We initialize the mechanical mode in its quantum groundstate, which allows us to perform the transduction process with minimal added thermal noise, while maintaining an optomechanical cooperativity >1, so that microwave photons mapped into the mechanical resonator are effectively upconverted to the optical domain. We further verify the preservation of the coherence of the microwave signal throughout the transduction process.

Figures

Equations

$$⟨\hat{n}⟩=n_{\text{coh}}+n_{\text{th}}.$$

$$g^{(2)}(0)=\frac{⟨{\hat{a}}^{†}{\hat{a}}^{†}\hat{a}\hat{a}⟩}{{⟨{\hat{a}}^{†}\hat{a}⟩}^2}=\frac{⟨{\hat{a}}^{†}\hat{a}{\hat{a}}^{†}\hat{a}⟩-⟨{\hat{a}}^{†}\hat{a}⟩}{{⟨{\hat{a}}^{†}\hat{a}⟩}^2},$$

$$g^{(2)}(0)=\frac{⟨{\hat{n}}^2⟩-⟨\hat{n}⟩}{{⟨\hat{n}⟩}^2}=1+\frac{{\sigma }^2-⟨\hat{n}⟩}{{⟨\hat{n}⟩}^2}.$$

$$g_{\text{th}}^{(2)}(0)=2=1+\frac{{\sigma }_{\text{th}}^2-n_{\text{th}}}{n_{\text{th}}^2}\to {\sigma }_{\text{th}}^2=n_{\text{th}}^2+n_{\text{th}}.$$

$$g^{(2)}(0)=1+\frac{{\sigma }_{\text{th}}^2+n_{\text{coh}}(1+2n_{\text{th}})-⟨\hat{n}⟩}{{⟨\hat{n}⟩}^2}.$$

$$g^{(2)}(0)=2-\frac{n_{\text{coh}}^2}{{\left(n_{\text{th}}+n_{\text{coh}}\right)}^2}=2-\frac{1}{{\left(\frac{n_{\text{th}}}{n_{\text{coh}}}+1\right)}^2}.$$

Full text

Research into novel quantum technologies is receiving significant attention for its potential to fundamentally transform how we receive, process and transmit information. In particular, major endeavors into building quantum processors and quantum simulators are currently underway. Many leading efforts, including superconducting qubits Kelly et al. (2015) and quantum dots Watson et al. (2018) share quantum information through photons in the microwave regime. While this allows for an impressive degree of quantum control Hofheinz et al. (2009), it also limits the distance the information can realistically travel before being lost Kurpiers et al. (2018). At the same time, the field of optical quantum communication has already seen demonstrations over distance scales capable of providing real-world applications Liao et al. (2017). In particular, by transmitting information in the optical telecom band, fiber-based quantum networks over tens or even hundreds of kilometers can be envisaged Boaron et al. (2018). In order to connect several quantum computing nodes over large distances into a quantum internet Kimble (2008), it is therefore vital to be able to convert quantum information from the microwave to the optical domain, and back.

Several promising approaches have been taken to realize such a microwave to optics converter, most notably by trying to either directly couple the fields inside a non-linear crystal Witmer et al. (2016); Fan et al. (2018); Wang et al. (2018); Rueda et al. (2016), by using rare earth ion doped crystals O’Brien et al. (2014), magnons Hisatomi et al. (2016) or mechanical systems as a transducer Stannigel et al. (2010); Bochmann et al. (2013); Andrews et al. (2014); Bagci et al. (2014); Laer et al. (2018); Moaddel Haghighi et al. (2018); Suchoi et al. (2014). Recent milestones include bi-directional operation Vainsencher et al. (2016), coherent coupling Balram et al. (2016), as well as efficient conversion Higginbotham et al. (2018), all of which make use of a mechanical oscillator as the transducer. While high conversion efficiency has been a particular success with some mechanically-mediated frequency converters, the demonstration of intrinsic noise sources compatible with conversion of a quantum state has remained an outstanding challenge. For quantum information protocols, particularly those that can tolerate optical loss, the requirement of subphoton added noise necessitates that the converter contains less than one thermal excitation Zeuthen et al. (2016). To this end, several experiments have recently demonstrated cooling of mechanical oscillators into the quantum groundstate of motion Chan et al. (2011); Teufel et al. (2011); Meenehan et al. (2015). The low thermal occupation forms the basis for quantum control over mechanical states, with demonstrations including quantum state preparation O’Connell et al. (2010); Hong et al. (2017); Chu et al. (2017) and entanglement between multiple mechanical degrees of freedom Lee et al. (2011); Riedinger et al. (2018); Ockeloen-Korppi et al. (2018). Reaching this occupation regime is complicated by the absorption of optical photons, while at the same time realizing sufficiently strong optomechanical cooperativity to suppress additional noise sources Meenehan et al. (2014). To date there has been no demonstration of a system with mechanically-mediated interfaces in both microwave and optical domains that operates in the quantum ground state.

In this work, we demonstrate microwave-to-optics conversion with an electro-opto-mechanical device, which contributes less than one quantum of thermal noise. We cryogenically cool a GHz frequency piezoelectric optomechanical crystal device into its quantum groundstate of motion and excite the mechanical mode using a microwave circuit through surface acoustic waves. Crucially, our system remains in the ground state while a drive of sufficient strength for upconversion from the mechanical to the optical domain is applied. This allows for conversion of a microwave pulse to the optical telecom band in a regime where we excite on average one phonon. As our converter features a noise source containing less than one photon, optimizing the electromechanical cooperativity O’Connell et al. (2010); Moores et al. (2018), the main efficiency bottleneck in our platform, will allow for the faithful transduction of a single photon from the microwave to optical domain.

Our microwave to optics converter consists of a one-dimensional optomechanical crystal (OMC) Chan et al. (2012), which is mechanically coupled to an interdigital transducer (IDT) through surface acoustic waves (see  Fig. 1A). We fabricate the devices from a 250-nm thick GaAs layer, on a 3-$\mu$m Al0.7Ga0.3As sacrificial layer, both epitaxially grown on a GaAs substrate. This material combines a large refractive index ($n_{\textrm{GaAs}}=3.37$ at $\lambda=1550$ nm) Skauli et al. (2003) and a non-zero piezoelectric coefficient ($\epsilon_{14}=$-0.16 Cm*-2*), with well established fabrication processes. The optomechanical device, shown in the lower inset of Fig. 1A, is designed using finite-element modeling, such that the patterned nanobeam confines light in the telecom band, while at the same time exhibiting a co-localized mechanical breathing mode at $\omega_{\textrm{m}}=2\pi\times 2.7$ GHz (Fig. 1B). The electro-mechanical coupling in our device is due to the piezoelectric effect that allows for the excitation of traveling acoustic waves which drive the OMC Balram et al. (2016). The opto-mechanial coupling, on the other hand, is facilitated by the parametric coupling between the mechanical excitation and the intracavity photon number owing to the combination of photoelastic coupling and moving boundary conditions Chan et al. (2012). Both effects can intrinsically operate in a bi-directional and noiseless fashion. The device fabrication consists of a two-stage lithography process to define the IDT and then pattern the nanobeams, followed by an HF etch to remove the AlGaAs sacrificial layer. A final deposition of 5 nm of AlO${}_{\textrm{x}}$ passivates the surfaces and reduces the effect of unwanted drive-laser absorption Guha et al. (2017) (see Supplementary Information for details). The lower inset of Figure 1A displays a scanning electron microscope image of an optomechanical device, and an evanescently coupled waveguide which provides optical-fiber access to the confined optical mode Hong et al. (2017).

We perform an initial characterization of the device properties at room temperature with the setup depicted in Figure 1C, the results of which are presented in Figure 1D. An S11 measurement of the IDT shows a 10 MHz-wide microwave resonance centered at 2.76 GHz. The mechanical mode of the OMC is then measured by locking a laser onto the blue sideband ($\omega_{\textrm{l}}=\omega_{\textrm{c}}+\omega_{\textrm{m}}$) of the optical cavity resonance $\omega_{c}$ ($\omega_{c}=2\pi\times$194.3 THz, with a loaded optical quality factor $Q_{\textrm{o}}=3.3\times 10^{4}$, see SI), and monitoring the high-frequency noise in the reflected signal. The peak in the noise spectrum at 2.744 GHz corresponds to the thermally-occupied mechanical mode ($\sim$1$\times 10^{3}$ phonons) and has a linewidth of several MHz. The small mismatch between the IDT and mechanical resonances of $\sim$10 MHz is a result of fabrication-based inhomogeneities. As we apply an RF tone to the IDT at the mechanical frequency $\omega_{\textrm{m}}$, we observe an additional narrow peak on top of the thermal noise, corresponding to the transduced coherent signal from the IDT. The height of this peak is dependent on the RF power and the detuning from the mechanical resonance Balram et al. (2016).

The room temperature characterization highlights that the large initial thermal occupation of the mechanical mode is a significant source of noise in this conversion process. Especially at low RF drive powers, the thermal noise dominates over the transduced signal Balram et al. (2016). By placing our device in a dilution refrigerator (base temperature $\sim$20 mK), we can in principle reduce the thermal occupation of the mechanical mode to $n_{\textrm{th}}\approx 10^{-3}$. In practice, the achievable occupation is limited by residual heating through laser absorption and finite thermalization to the cryostat Hong et al. (2017). With the device cooled to Millikelvin temperatures, we measure the actual thermal occupation of the mechanical mode by monitoring cavity-enhanced Stokes (blue sideband) and anti-Stokes (red sideband) scattering rates ($\Gamma_{\textrm{B}}$ and $\Gamma_{\textrm{R}}$, respectively), when we drive the cavity with laser pulses detuned by $\pm\omega_{\textrm{m}}$ Meenehan et al. (2015). We suppress the reflected pump light through spectral filtering, such that only scattered photons on cavity resonance are detected by superconducting nanowire single-photon detectors (SNSPD) as shown in Figure 2A. Specifically, the rates we measure are set by $\Gamma_{\textrm{B}}\propto n_{\textrm{th}}+1$ and $\Gamma_{\textrm{R}}\propto n_{\textrm{th}}$ Safavi-Naeini et al. (2012). Figure 2B shows the histogram of the single photon count rates measured for 40 ns long pulses set to the two detunings with a peak-power of 107 nW at the device. For this power, we find an optomechanical cooperativity of $C=1.7$ (see SI) Hill et al. (2012). We extract a thermal occupation of $n_{\textrm{th}}=0.90\pm 0.01$, which verifies the initialization of the mechanical mode close to its quantum groundstate. This value is higher than the theoretical value set by the cryostat temperature, limited by residual heating of the structure during the laser pulse Hong et al. (2017). A sweep of the pulse power reveals that lower occupations can be achieved (e.g. $n_{\textrm{th}}=0.36\pm 0.03$, see SI), at the cost of a lower conversion efficiency.

We now proceed to verify the conversion from microwave to optical telecom signals at Millikelvin temperatures. Red-detuned ($\omega_{\textrm{l}}=\omega_{\textrm{c}}-\omega_{\textrm{m}}$) optical pulses, which realize an opto-mechanical state-swap Galland et al. (2014), are sent into the OMC to read out the state of the mechanical mode, which is coherently excited by sweeping the frequency of an RF drive tone (1 $\mu$W) across the mechanical resonance (see Fig. 2C). The data is fitted with a Lorentzian, from which we extract a mechanical linewidth of $\gamma_{\textrm{m}}/2\pi=197$ kHz, an expected improvement in mechanical quality factor of about one order of magnitude compared to the value at room temperature. Furthermore, we observe a blueshift of both the mechanical mode and the IDT resonance by about 35 MHz compared to room temperature due to the temperature dependence of the GaAs elastic constants.

While the optical absorption can be reduced through tuning of the power of the input light as well as by pulsed operation, thus far little is known about the potential heating due to the RF tone which drives the IDT. In order to investigate the amount of heating resulting from the RF drive, we measure the second order correlation function $g^{(2)}(\tau)=\langle\hat{b}^{\dagger}(0)\hat{b}^{\dagger}(\tau)\hat{b}(0)\hat{b}(\tau)\rangle/(\langle\hat{b}^{\dagger}(0)\hat{b}(0)\rangle\langle\hat{b}^{\dagger}(\tau)\hat{b}(\tau)\rangle)$ of the mechanical resonator mode Hong et al. (2017), by swapping mechanical excitations into optical photons at the cavity resonance frequency. Here $\hat{b}$ ($\hat{b}^{\dagger}$) is the annihilation (creation) operator of the mechanical mode. We expect the coherence of the RF drive to be mapped first onto the mechanical state in our resonator and then onto the light field. Any sign of heating due to the RF drive should result in bunching of the $g^{(2)}(\tau)$ during the mechanical lifetime ($\sim$1.5 $\mu$s). For this measurement, both the laser (locked onto the red sideband) and the RF source are operated in a continuous wave (CW) mode. The scattered optical photons are detected on a Hanbury Brown and Twiss (HBT)-type setup with two SNSPDs. The time $\tau$ here is the relative delay between clicks from the two detectors. We observe a near-flat $g^{(2)}(\tau)$ over the entire range of the RF power sweep (see  Fig. 2D), clearly indicating that the coherent part of the mechanical state dominates any thermal contribution across the entire power sweep. As a reference, we also perform a measurement without an RF drive but with high enough optical power for absorption-induced heating to occur. In this case, a clear signature of photon bunching is visible, indicating, as expected, a thermal state of the mechanical resonator.

In order to demonstrate the potential of these devices as transducers of microwave to optical signals at the quantum level, we now operate both the RF driving and optical read-out in a pulsed mode Riedinger et al. (2016). We send a resonant RF pulse (1 $\mu$s long) to the IDT to excite our oscillator and access the mechanical state through a 40 ns long red-detuned state-swap pulse. This allows us to minimize the effects of heating due to optical absorption. The pulsed experiment enables us to quantify the absolute number of coherent phonons added to our initial state by comparing the scattering rate in the presence of an RF drive ($\Gamma_{\textrm{R,RF}}$) to the scattering rate we obtain from the remaining thermal population ($\Gamma_{\textrm{R}}$) (inset in Fig. 3B). By measuring the photon rate with and without a resonant RF drive, we recover an RF-phonon conversion effciency of $3.57\times 10^{-10}$ phonons per RF photon.

Using the same HBT-type setup as above, we detect the second order correlation of the scattered photons, which allows us to compare the coincidences between detection events originating from the same ($\Delta i=0$) or different ($\Delta i\neq 0$) pulse sequences. A selection of the histograms of these correlations is shown in Figure 3A for various coherent phonon occupations ($n_{\textrm{coh}}$). The full set of $g^{(2)}(0)$ values for increasing coherent phonon occupation is shown in Figure 3B. We expect the value of $g^{(2)}(0)$ to be determined by the ratio $n_{\textrm{coh}}/n_{\textrm{th}}$ (cf. the SI). We extract this number from the relative count rate we recover with and without the RF pulse, $\Gamma_{\textrm{R,RF}}/\Gamma_{\textrm{R}}$, displayed in the inset of Figure 3B, making use of $\Gamma_{\textrm{R}}\propto n$. The dashed curve in Figure 3B displays the expected $g^{(2)}(0)$ values for our RF power sweep based on the theory for a displaced thermal state (cf. the SI), which are in good agreement with our measured values. An increase in RF power results in a larger coherent displacement of the thermal state, which in turn leads to a decreased value for $g^{(2)}(0)$.

While our pulsed experiments clearly demonstrate conversion between a coherent state in the microwave and the telecom domain, they do not imply the retention of the input-state phase. In order to access the coherence of the transduction process, we use a modified version of the setup (for a detailed sketch see SI Fig. S5). We split the red-detuned excitation laser into two branches of a phase-stabilized Mach-Zehnder interferometer (MZI), one of which contains our device and the other an amplitude electro-optic modulator (EOM). We drive both the IDT and the EOM with a single RF source, such that a coherent transduction process results in a fixed phase relationship between the up-converted light in the two inteferometer arms. We then mix the light on a beamsplitter, matching the photon rate in the two arms. Figure 4A displays the count rate at one output port of the MZI when we vary the phase of the interferometer for several coherent phonon occupations. We observe a clear interference pattern with a visibility of 44$\pm$3% for powers corresponding to a coherent phonon occupation of $n_{\textrm{coh}}=1.1$, which increases to 85$\pm 7$% as the coherent contribution dominates over the small thermal background. Figure 4B displays these experimentally retrieved visibilities for several coherent phonon occupations, the expected modeled behavior assuming only thermal noise ($n_{\textrm{noise}}=n_{\textrm{th}}$, solid line), as well as additional incoherent noise sources ($n_{\textrm{noise}}=n_{\textrm{th}}+n_{\textrm{other}}$, dashed line). These respective trends are given by $\sqrt{n_{\textrm{coh}}/\left(n_{\textrm{coh}}+n_{\textrm{noise}}\right)}$ and scaled by the maximally achievable interference visibility in our setup of 90%. Here, $n_{\textrm{other}}$ represents the equivalent noise figure for any other source than the thermal occupation of the resonator, including imperfections in the measurement setup. We estimate the upper bound of these sources to be $n_{\textrm{other}}\sim 2.5$. The main contributions to this remaining part are drifts of the interferometer free spectral range over the duration of the measurement, imperfect sideband resolution ($\left(\kappa/4\omega_{\textrm{m}}\right)^{2}=0.27$), as well as mechanical decoherence Zeuthen et al. (2016); Andrews et al. (2014). With this measurement, we confirm the phase-preserving nature of the conversion process down to the single phonon level.

The total efficiency of our device is the product of two parts: the loading efficiency of the mechanical mode from the microwave side ($3.57\times 10^{-10}$, measured from the attenuator output at at the mixing chamber to the excitation of phonons in the mechanical mode) and the optical readout efficiency of the mechanical mode $1.55\times 10^{-5}$. The latter one can itself be separated into two parts: $\eta_{\textrm{ro}}=p_{\textrm{r}}\times\eta_{\textrm{det}}$, with $\eta_{\textrm{det}}=1.41\times 10^{-3}$ (see SI). The state swap probability $p_{\textrm{r}}=1.1\%$ is a function of the power with which the optical readout is performed and can be increased through improvements with respect to optical absorption. Note that the current performance of our device is already sufficient to read out a non-classical state of the mechanical mode Hong et al. (2017). The low loading efficiency of the mechanical mode can be attributed to the design and size of the electromechanical transducer. We estimate the efficiency of transferring a SAW wave from the IDT (150 $\mu$m wide) into a single, narrow ($\sim$1 $\mu$m), suspended beam to be less than $2.5\times 10^{-5}$. Additional contributions arise from the difference in the polarizations of the incoming SAW wave and the mechanical mode, the discrepancy between the IDT and mechanical frequencies, as well as the large electrical impedance of the IDT. These factors can be improved by tailoring the size and design of the electromechanical transducer specifically to the purpose of exciting the breathing mode of a single nanobeam Wu et al. (2020). While the small-scale piezo-resonator required to mode-match the nanobeam will necessitate careful electrical impedance matching, the required network falls into the range accessible with coplanar resonator technology Wu et al. (2020). The relatively small optomechanical state-swap probability and detection efficiency reported here, on the other hand, can be circumvented with post-selection techniques routinely used in quantum optics experiments Hong et al. (2017); Marinković et al. (2018).

We have demonstrated faithful conversion of a microwave to an optical signal with only a small added thermal contribution due to the groundstate occupation of the mechanical resonator. Furthermore, these measurements show our ability to detect the displacement amplitude of the initial state in our mechanical resonator down to one phonon, (corresponding to our lowest measured RF power), marking a crucial benchmark for applications in the quantum regime Zeuthen et al. (2016). The device used for this experiment is a fully integrated, on-chip hybrid electro-opto-mechanical system with a mechanical mode as the transducer. We cool this mode to its quantum groundstate using a dilution refrigerator, which allows us to operate directly at the quantum noise limit. This work allows for the on-chip integration of a single-photon RF-source, such as a superconducting qubit Gustafsson et al. (2014); Moores et al. (2018), which paves the way for building a true quantum network over large distances, based on superconducting nodes. The device we consider here is specifically suited towards heralded entanglement generation between remote superconducting qubits, a protocol which is described in Ref. Zeuthen et al. (2016).

While we demonstrate quantum limited noise performance in the readout of the mechanical state, the conversion efficiency is currently limited by low microwave-phonon excitation efficiency. We would like to note that this is not a fundamental limit but rather a result of design choices for this proof-of-principle demonstration. The material itself imposes some limitations on the efficiency, such as the remaining absorption heating and the relatively low piezoelectric coupling. However, demonstrations of coherent coupling between superconducting qubits and surface acoustic waves in GaAs Gustafsson et al. (2014); Moores et al. (2018) suggest the latter are highly suitable choices for noise-free carriers of quantum information. Importantly, our system is already operating in the range of optical drive strengths expected for mediating efficient conversion. In particular, the opto-mechanical cooperativity of $C\approx 1.7$ (see SI) is sufficient for efficiently converting GHz phonons into telecom photons.

During the submission process we became aware of related work demonstrating GaAs optomechanical crystal in the low thermal occupation regime Ramp et al. (2019).

I Supplementary Information

I.1 Device fabrication

The samples were fabricated on an epitaxial material stack consisting of a GaAs $\langle 100\rangle$ substrate, a 3 $\mu$m thick sacrificial layer of Al0.7Ga0.3As, followed by a 250 nm thick device layer of GaAs. The devices are aligned with the $\langle 110\rangle$ plane to allow propagation of the SAWs into the nanobeam. As a first step, the IDT structures, waveguides and alignment markers are defined using electron beam lithography. A stack of 5 nm Cr, 15 nm Pt, and 30 nm Au Balram et al. (2016) is then evaporated onto the chip, with any excess removed in a lift-off process. The coplanar waveguide is impedance matched to the 50$\Omega$ coaxial cables in the setup. The OMC pattern is then aligned to the IDT structure using the markers from the previous lithography step in the second electron beam lithography step. The pattern is transferred into the device layer using a reactive ion etch process in an Alcatel GIR-300 etcher using a N2/BCl3/Cl2 chemistry. In order to remove all organic residues, a digital etching step is performed which consists of 1 min submersion in 37% H2O2, followed by thorough rinsing, 2 min submersion in 7:1 BOE, and a second round of thorough rinsing. Finally, the OMC structures are suspended by selectively removing the sacrificial layer using a 10% HF solution, which is followed by a second round of the digital etching to remove residue from the underetch process.

I.2 ALD passivation

First low temperature tests of our OMCs fabricated from bare GaAs resulted in devices that were very prone to absorption of laser pulses, with varying results for $n_{\textrm{th}}$ of typically many thermal phonons. We attribute the variance between different devices to a variation in the native oxide layer. Previous research Guha et al. (2017) has indicated that a thin layer of ALD deposited aluminium oxide can reduce these absorption effects in GaAs microdiscs. We therefore proceeded to strip the native oxide off the GaAs using 7:1 BOE, followed by a 5 nm deposition of ALD AlOx at 300 *∘*C. Using this method, we observe a reduction in the initial thermal occupation of our mechanical resonator from $n_{\textrm{th}}\sim$ 2 – 10 to $n_{\textrm{th}}<1$.

I.3 Optical characterization

Optical characterization of our device is done by measuring the reflected power as we scan a tunable laser across its resonance, which is shown Fig. S1. The loaded linewidth is $\kappa=2\pi\times 5.8$ GHz, corresponding to a quality factor of about $Q_{\textrm{o}}=3.3\times 10^{4}$. Light is evanescently coupled from a central waveguide into the devices. The waveguide itself is tapered towards the edge of the chip (see Fig. 1), where we position a lensed fiber with a focal length of 14 $\mu$m. In order to calibrate the coupling efficiency from the fiber to this waveguide, we compare the off-resonant reflected power to the power reflected from a fiber mirror. This ratio gives $\eta_{\textrm{fc}}^{2}$, resulting in $\eta_{\textrm{fc}}=33\%$. In addition, we determine our optomechanical coupling rate to be $g_{0}/2\pi=1.3$ MHz Hong et al. (2017), which for 0.5 $\mu$W optical power (intra-cavity photon number $n_{\textrm{c}}=280$) results in a cooperativity of $C=\frac{4g_{0}^{2}n_{\textrm{c}}}{\kappa\gamma_{\textrm{m}}}=1.73$.

I.4 RF-Setup

In order to send the microwave tone into the cryostat to our device, we attenuate the signal by several dB at the respective temperature stages in the fridge. This serves to both thermalize the coaxial cable and to reduce the noise temperature of the signal seen by the device. The specific attenuation steps are shown in Fig. S2. The coaxial cables connecting the stages with a temperature of $<$1 K are made from aluminium and are thus superconducting when the cryostat is cold. In order to be able to perform the S11 measurement of our IDT device despite the strong attenuation of the input signal, we use two RF circulators at the bottom of the refrigerator. The first one connects to the device and the second one acts as an isolator sending any thermal noise coming from higher the temperature stages to the 50 $\Omega$ terminated $3^{rd}$ port of the circulator. The reflected signal from the device is then amplified using a cryogenic low-noise amplifier and sent back to the VNA. For the pulsed experiments, we use an RF source which replaces the VNA at the input of the cryostat. To generate these pulses, we send the signal through an RF switch, which is gated using a pulse generator.

I.5 Efficiency calibration

The calibration of the overall efficiency consists of two steps. The first part concerns the RF to phonon conversion efficiency. We send 1 $\mu$s-long RF pulses into the dilution refrigerator at a nominal power of -14 dBm, which results in, on average, 1 added coherent phonon to the resonator. Combined with the 39 dB of attenuation of the coaxial line inside the cryostat, this means we have around $2.8\times 10^{9}$ RF photons on the input of the device for every phonon in the resonator. This low efficiency is mostly due to the IDT not being optimized for a single OMC device, but rather being 160 $\mu$m wide, which corresponds to about 300 times the width of a single nanobeam device. Furthermore, the structure is not designed to be impedance matched to the OMC, which leads to reflections and additional losses.

We calibrate the total efficiency of our setup by sending a few-photon off-resonant laser pulse to the device. This gives the product: $\eta_{\textrm{tot}}=\eta_{\textrm{fc}}^{2}\times\eta_{\textrm{trans}}\times\eta_{\textrm{QE}}$, where $\eta_{\textrm{fc}}$ is the fiber coupling efficiency, $\eta_{\textrm{trans}}$ the transmission efficiency of the detection path (filter cavities, fiber components, etc.) , and $\eta_{\textrm{QE}}$ the quantum efficiency of the detectors. We measure this total efficiency to be $8.43\times 10^{-4}$. In addition we determine the coupling between the center waveguide and the nanobeam cavity, $\eta_{\textrm{dev}}=\kappa_{\textrm{e}}/\kappa$, to be 0.65. With the previously determined fiber-coupling efficiency of 0.33, we get a total detection efficiency $\eta_{\textrm{det}}=\eta_{\textrm{fc}}\times\eta_{\textrm{dev}}\times\eta_{\textrm{trans}}\times\eta_{\textrm{QE}}=\eta_{\textrm{tot}}\times\eta_{\textrm{dev}}/\eta_{\textrm{fc}}=1.41\times 10^{-3}$. We estimate the state readout probability (using a red-detuned pulse) to be $p_{\textrm{r}}=$1.1$\%$. Within the 80 ns window considered in the correlation measurements, we find a total detector dark count rate for both detectors of 0.5 Hz.

I.6 Sideband asymmetry

At the 20 mK base temperature of the dilution refrigerator, we predict a thermal population of the mechanical mode of around $10^{-3}$ phonons. The occupation is increased, however by absorption of the laser pulses required to convert the microwave-frequency signal to the optical domain. Owing to this effect, we operate at an elevated occupation of $\sim$0.9 phonons for the pulsed experiments presented in the main text. Notably, the heating occurs on a rapid timescale. Absorption heating is also an important limitation for the operation of coherent opto-mechanical devices fabricated from silicon, however the slower internal dynamics compared to GaAs allow for multiple operations before the mode temperature is raised Meenehan et al. (2015).

In order to further illustrate the effect of absorption induced heating on our optomechanical devices, Figure S3 displays additional sideband asymmetry measurements taken for 40 ns long Gaussian pulses with different peak powers. The upper panel displays the count rates from driving the blue and red mechanical sidebands of the optical resonance, while the lower panel displays the extracted thermal population of the mechanical resonator. Notably, for a peak power of $0.1~{}\mu$W, we extract a mechanical occupation of only 0.36$\pm$0.03 phonons. While operating at this power would ensure coherent transduction of even lower power microwave pulses, it also reduces the associated conversion efficiency further and increases the total measurement time beyond reasonable time scales.

I.7 IDT - Mechanics detuning

In Fig. 2c of the main text, we plot the mechanical resonance of the optomechanical device, which we access by monitoring the rate of anti-Stokes scattering as we vary the frequency of the coherent RF drive. This measurement can be supported by the dynamics of the mechanical mode, which we present in Fig. S4. For this measurement we continuously drive the red-sideband of the optical resonance at low power, and pulse the IDT with a $1.5~{}\mu$s-long tone, such that we continuously record the occupation of the mechanical mode. The figure displays the count-rate we record correlated to the pulse arrival time for a variety of microwave-pulse detunings from the mechanical resonance. The detuning-dependent oscillatory signal we measure during the pulse is the transient response of the mechanical oscillator to the detuned surface-acoustic wave. After the end of the pulse at $1.5~{}\mu$s, the mechanical resonance continuously decays to the thermal background level.

I.8 Displaced thermal states

In the main text we have used a displaced thermal state model to fit our data (cf. Fig. 3). Here we briefly derive the second order correlation function for such a state. We start with an initial thermal occupancy $n_{\textrm{th}}$ of our mechanics and displace it coherently with an amplitude $\sqrt{n_{\textrm{coh}}}$. Thus, our resulting mode occupation is given by

[TABLE]

To derive the expected $g^{(2)}(0)$ values for a given ratio of $n_{\textrm{coh}}/n_{\textrm{th}}$, we follow Oz-Vogt et al. (1991) and start with the general expression for $g^{(2)}(0)$:

[TABLE]

where $\hat{a}$ and $\hat{a}^{\dagger}$ are annihilation and creation operators,respectively, and we used the commutation relation $[\hat{a},\hat{a}^{\dagger}]=1$. Substituting $\hat{a}^{\dagger}\hat{a}=n$ and $\sigma^{2}=\left\langle\hat{n}^{2}\right\rangle-\left\langle\hat{n}\right\rangle^{2}$, we can simplify this to

[TABLE]

For a purely thermal state, this results in

[TABLE]

Using $\sigma^{2}=\sigma_{\textrm{th}}^{2}+n_{\textrm{coh}}(1+n_{\textrm{th}})$ Oz-Vogt et al. (1991) and substituting it in Eq. (S3), we get

[TABLE]

Substituting $\sigma_{\textrm{th}}^{2}$ from Eq. (S4), $\left\langle\hat{n}\right\rangle$ from Eq. (S1) and simplifying yields

[TABLE]

This result is also consistent with the one obtained by Marian et al. Marian and Marian (1993). We directly extract the ratio $\frac{n_{\textrm{coh}}}{n_{\textrm{th}}}$ from $\Gamma_{\textrm{R,RF}}/\Gamma_{\textrm{R}}=\left(n_{\textrm{coh}}+n_{\textrm{th}}\right)/n_{\textrm{th}}$, where $\Gamma_{\textrm{R}}$ is the count rate resulting from a red-detuned pulse alone and $\Gamma_{\textrm{R,RF}}$ is the count rate resulting from a red-detuned pulse, which is overlapped with the RF drive pulse, displayed in the inset of Figure 3b in the main text.

I.9 Phase Sensitive Detection Setup

In order to confirm the coherence of the transduction process from our RF source all the way to the telecom-band photons, we use an interferometer with our device in one arm and an electro-optic modulator (EOM) in the other (cf. Fig. S5). By driving both the IDT and the EOM with the same RF source, we generate optical sidebands of the drives in both interferometer arms with a well defined phase. This allows us to verify the coherence of the transduction process by measuring the interference between the two signals. We use a strong second, far detuned laser pulse (blue in Fig. S5) to stabilize the interferometer. This locking-drive is reflected from the photonic crystal mirror in the waveguide without resulting in intracavity photons and is delayed from the red measurement tone. The stabilization is done by measuring the transmission of this strong laser pulse at both outputs of the interferometer and feeding back the signal to a fiber stretcher ($\phi$ in the figure). We change the interferometer phase for the red (and the upconverted) light by sweeping the frequency of the locking laser over the free spectral range (approx. 20 GHz) of the interferometer. Both lasers, as well as other sidebands which are not resonant with the cavity, are filtered out before the SNSPDs.