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### Transducer theory

The transducer consists of one microwave resonator and one optical cavity, both parametrically coupled via the vacuum coupling rates *g*_{0,j} with *j* = *e*, *o* to the same mechanical oscillator as shown in Fig. 1a and b. The intrinsic decay rate of the optical (microwave) resonator is *κ*_{in,o} (*κ*_{in,e}), while the optical (microwave) waveguide–resonator coupling is given by *κ*_{ex,o} (*κ*_{ex,e}) resulting in a total damping rate of *κ*_{j} = *κ*_{in,j} + *κ*_{ex,j} and coupling ratios *η*_{j} = *κ*_{ex,j}/*κ*_{j}. The mechanical oscillator with intrinsic decoherence rate *γ*_{m} and frequency *ω*_{m} is shared between the optical cavity and the microwave resonator and acts as a bidirectional coherent pathway to convert the photons between the two different frequencies^{8,21,22,23}. In the interaction frame, the Hamiltonian describing the conversion process is (see Supplementary Note 1):

$${hat{H}}_{{rm{int}}}=sum _{mathrm{j} = {rm{e}},{rm{o}}}left(hslash {G}_{mathrm{j}}({hat{a}}_{mathrm{j}}^{dagger }hat{b}+{hat{a}}_{mathrm{j}}{hat{b}}^{dagger })+{hat{H}}_{{rm{CR}},mathrm{j}}right),$$

(1)

where ({hat{a}}_{mathrm{j}}), ((hat{b})) with *j* = *e*, *o* is the annihilator operator of the electromagnetic (mechanical) mode, and ({hat{H}}_{{rm{CR}},mathrm{j}}=hslash {G}_{mathrm{j}}({hat{a}}_{mathrm{j}}hat{b} {e}^{2i{omega }_{{rm{m}}}t}+,text{h.c.})) describes the counter-rotating terms which are responsible for the coherent amplification of the signal. ({G}_{mathrm{j}}=sqrt{{n}_{{rm{d}},mathrm{j}}}{g}_{0,mathrm{j}}) is the parametrically enhanced electro- or optomechanical coupling rate where *n*_{d,j} is the intracavity photon number due to the corresponding microwave and optical pump tones. For a red-detuned drive in the resolved-sideband regime 4*ω*_{m} > *κ*_{j} we neglect ({hat{H}}_{{rm{CR}},mathrm{j}}) under the rotating-wave approximation and the Hamiltonian (1) represents a beam-splitter like interaction in which the mechanical resonator mediates noiseless photon conversion between microwave and optical modes. Note that near-unity photon conversion ({zeta }_{{rm{RS}}}=4{eta }_{{rm{e}}}{eta }_{{rm{o}}}{{mathcal{C}}}_{{rm{e}}}{{mathcal{C}}}_{{rm{o}}}/{(1+{{mathcal{C}}}_{{rm{e}}}+{{mathcal{C}}}_{{rm{o}}})}^{2}) can be achieved in the limit of ({{mathcal{C}}}_{{rm{e}}}={{mathcal{C}}}_{{rm{o}}}gg 1) with ({{mathcal{C}}}_{mathrm{j}}=4{G}_{mathrm{j}}^{2}/({kappa }_{mathrm{j}}{gamma }_{{rm{m}}})) being the electro- or optomechanical cooperativity, as demonstrated between two optical^{24} and two microwave modes^{25,26}, respectively.

### Transducer design

We realize conversion by connecting an optomechanical photonic crystal zipper cavity^{27} with two aluminum coated and mechanically compliant silicon nanostrings^{28} as shown in Fig. 1c. The mechanical coupling between these two components is carefully designed (see Supplementary Note 2), leading to a hybridization of their in-plane vibrational modes into symmetric and antisymmetric supermodes. In case of the antisymmetric mode that is used in this experiment, the strings and the photonic crystal beams vibrate 180° out of phase as shown by the finite-element method simulation in Fig. 1d. The photonic crystal cavity features two resonances at telecom frequencies with similar optomechanical coupling strength. The simulated spatial distribution of the electric field component *E*_{y}(*x*, *y*) of the higher frequency mode with lower loss rate used in the experiment is shown in Fig. 1e. The lumped element microwave resonator consists of an ultra-low stray capacitance planar spiral coil inductor^{29} and two mechanically compliant capacitors with a vacuum gap of size of ~70 nm. This resonator is inductively coupled to a shorted coplanar waveguide, which is used to send and retrieve microwave signals from the device. The sample is fabricated using a robust multi-step recipe including electron beam lithography, silicon etching, aluminum thin-film deposition, and hydrofluoric vapor acid etching, as described in detail in ref. ^{30}.

### Transducer characterization

Standard sample characterization (see Supplementary Notes 3 and 4) reveals an optical resonance frequency of *ω*_{o}/(2*π*) = 198.081 THz with total loss rate *κ*_{o}/(2*π*) = 1.6 GHz and waveguide coupling rate *κ*_{ex,o}/(2*π*) = 0.18 GHz leading to a coupling efficiency of *η*_{o} = 0.11. When the optical light is turned off, the microwave resonance frequency is *ω*_{e}/(2*π*) = 10.5 GHz with coupling efficiency *η*_{e} = 0.4 and *κ*_{ex,e}/(2*π*) = 1.15 MHz. The mechanical resonator frequency has a value of *ω*_{m}/(2*π*) = 11.843 MHz with an intrinsic decoherence rate *γ*_{m}/(2*π*) = 15 Hz at a mode temperature of 150 mK. The achieved single-photon-phonon coupling rates are as high as *g*_{0,e}/(2*π*) = 67 Hz and *g*_{0,o}/(2*π*) = 0.66 MHz.

### Conversion measurements

To perform coherent photon conversion, red-detuned microwave and optical tones with powers *P*_{e(o)} are applied to the microwave and the optical resonator. These drive tones establish the linearized electro- and optomechanical interactions, which results in the conversion of a weak microwave (optical) signal tone to the optical (microwave) domain measured in our setup as shown in Fig. 1f. We experimentally characterize the transducer efficiency by measuring the normalized reflection ∣*S*_{jj}∣^{2} (*j* = *e*, *o*) and the bidirectional transmission *ζ* : = ∣*S*_{eo}∣∣*S*_{oe}∣ coefficients as a function of signal detuning *δ*_{j}. As shown in Fig. 2a, for drive powers *P*_{e} = 601 pW and *P*_{o} = 625 pW with drive frequencies *ω*_{d,j} and detunings Δ_{j} = *ω*_{j} − *ω*_{d,j} of Δ_{e} = *ω*_{m} and Δ_{o}/(2*π*) = 126 MHz leading to intracavity photon numbers of *n*_{d,e} (approx) 9 × 10^{5} and *n*_{d,o} (approx) 0.2 with cooperativities ({{mathcal{C}}}_{{rm{e}}}approx 0.57) and ({{mathcal{C}}}_{{rm{o}}}approx 0.9), the measured total (waveguide to waveguide) photon transduction efficiency is (approx)1.1% corresponding to 96.7% internal (resonator to resonator) photon transduction efficiency over the total bandwidth of Γ_{conv}/(2*π*) (approx) 0.37 kHz. In the case of *κ*_{o} > 4*ω*_{m} and *κ*_{e} < 4*ω*_{m}, the bandwidth is given by ({Gamma }_{{rm{conv}}}approx ({{mathcal{C}}}_{{rm{e}}}+1){gamma }_{{rm{m}}}) because the nonsideband resolved optomechanical cavity does not induce mechanical broadening. The signal tone adds 17(10^{−3}) photons to the microwave resonator (optical cavity).

Here we use a self-calibrated measurement scheme that is independent of the gain and loss of the measurement lines as described in ref. ^{31} and we only take into account transduction between the upper two sidebands at *ω*_{d,j} + *ω*_{m} as shown in Fig. 1b. Neglecting the lower optical sideband that is generated due to the nonsideband resolved situation *κ*_{o}/4*ω*_{m} (approx) 30 reduces the reported mean bidirectional efficiencies by (sqrt{2}) compared to the actually achieved total transduction efficiency between microwave and optical fields. The observed reflection peaks indicate that both resonators are undercoupled, equivalent to an impedance mismatch for incoming signal light. All scattering parameters are obtained from measured coherent tones whose linewidths are given by the chosen resolution bandwidth and the stability of the heterodyne setup. While this does not explicitly show long term phase stability of the conversion we find that these results are in excellent agreement with our coherent conversion theory model (solid lines) with *γ*_{m} as the only free fit parameter.

Figure 2b shows the total transduction efficiency for different pump power combinations with microwave and optical pump powers ranging from 30 to 953 pW and 48 to 1561 pW, respectively. Figure 2c, d shows the efficiency versus *P*_{o} (*P*_{e}) for fixed microwave (optical) pump power *P*_{e} = 601 (*P*_{o} = 625) pW. As expected, the transduction efficiency rises with increasing pump powers and reaches a maximum of *ζ* = 1.2%. The internal transduction efficiency is significantly higher (*ζ*/(*η*_{o}*η*_{e}) ≤ 135%) because both the microwave resonator as well as the optical cavity are highly undercoupled with coupling ratios of *η*_{o} = 0.11 and *η*_{e} ranging between 0.07 and 0.18 when both pumps are on. The increase in the intrinsic loss rate of microwave *κ*_{in,e} and mechanical resonator *γ*_{m} at higher pump powers are shown in Fig. 2e and f caused by considerable heating related to (especially optical) photon absorption. This results in the degradation of the microwave and mechanical quality factors and consequently reduces the waveguide coupling efficiency, the cooperativities and the total transduction efficiency (see Supplementary Note 5).

### Sideband resolution and amplification

In the nonsideband resolved limit the contribution of the counter-rotating term of the Hamiltonian ({hat{H}}_{{rm{CR}},{rm{o}}}) is nonnegligible, resulting in a transduction process that cannot be fully noise- free. This interesting effect can be correctly described by introducing an amplification of the signal tone with (in the absence of thermal noise) quantum limited gain ({{mathcal{G}}}_{{rm{o}}}) (see Supplementary Note 1). In contrast, the microwave resonator is in the resolved-sideband condition 4*ω*_{m} > *κ*_{e}, so that the signal tone amplification due to electromechanical interaction is negligible ({{mathcal{G}}}_{{rm{e}}}simeq 1). This results in the total, power independent, bidirectional conversion gain of ({mathcal{G}}={{mathcal{G}}}_{{rm{e}}}{{mathcal{G}}}_{{rm{o}}}simeq {{mathcal{G}}}_{{rm{o}}}), which turns out to be directly related to the minimum reachable phonon occupation:

$${langle nrangle }_{min }=frac{{({Delta }_{{rm{o}}}-{omega }_{{rm{m}}})}^{2}+{kappa }_{{rm{o}}}^{2}/4}{4{Delta }_{{rm{o}}}{omega }_{{rm{m}}}}={{mathcal{G}}}_{{rm{o}}}-1,$$

(2)

induced by optomechanical quantum backaction when the mechanical resonator is decoupled from its thermal bath^{32}. Due to this amplification process the measured transduction efficiency in Fig. 2a is about 110 times larger than one would expect from a model that does not include gain effects for the chosen detuning, and adds the equivalent of at least one half of a vacuum noise photon to the input of the transducer in our case of heterodyne detection (for *η*_{j} = 1 and ({mathcal{G}}gg 1)). However, it turns out that this noise limitation, which might in principle be overcome with efficient feedforward^{9}, sideband suppression^{33,34}, or sideband resolution^{35}, accounts for only about 0.1% of the total conversion noise observed in our system. The total transduction (including gain) can be written in terms of the susceptibilities of the electromagnetic modes ({chi }_{mathrm{j}}^{-1}(omega )=i({Delta }_{mathrm{j}}-omega )+{kappa }_{mathrm{j}}/2) and the mechanical resonator ({chi }_{{rm{m}}}^{-1}(omega )=i({omega }_{{rm{m}}}-omega )+{gamma }_{{rm{m}}}/2) as:

$$zeta ={left|frac{sqrt{{kappa }_{{rm{ex}},{rm{e}}}{kappa }_{{rm{ex}},{rm{o}}}}{G}_{{rm{e}}}{G}_{{rm{o}}}{chi }_{{rm{e}}}{chi }_{{rm{o}}}left[-{chi }_{{rm{m}}}+{tilde{chi }}_{{rm{m}}}right]}{1+[{chi }_{{rm{m}}}-{tilde{chi }}_{{rm{m}}}]left[{G}_{{rm{e}}}^{2}({chi }_{{rm{e}}}-{tilde{chi }}_{{rm{e}}})+{G}_{{rm{o}}}^{2}({chi }_{{rm{o}}}-{tilde{chi }}_{{rm{o}}})right]}right|}^{2},$$

(3)

where (tilde{{chi }_{mathrm{j}}}(omega )={chi }_{mathrm{j}}{(-omega )}^{* }).

Equation (3) can be decomposed into a product of the conversion gain ({mathcal{G}}) and the pure conversion efficiency *θ*, i.e., (zeta :={mathcal{G}}times theta), for frequencies in the vicinity of *ω*_{m} (see Supplementary Note 1). Equation (2) shows that the signal amplification depends only on the resonator linewidth and the detuning and is not directly related to the (propto {hat{a}}^{dagger }{hat{b}}^{dagger }) interaction term or the pump power^{31}. This can be understood by the alternative interpretation that the gain represents the ratio of the transduced upper sideband to the difference between upper and lower sideband at each cavity. Therefore, it is instructive to measure the transducer parameters as a function of optical pump detuning as shown in Fig. 3a. While changing the optical detuning, we also vary the pump power in order to keep the optical intracavity photon number constant at *n*_{d,o} = 0.185 ± 0.015. This way it is possible to investigate the influence of Δ_{o} at a constant optomechanical coupling ({G}_{{rm{o}}}={g}_{{rm{0,o}}}sqrt{{n}_{{rm{d}},{rm{o}}}}). The measured total transduction efficiency is shown in Fig. 3a and reaches ζ(,approx,)1% at Δ_{o} (approx) 0 for the chosen pump powers in agreement with Fig. 2c, d. We can now separate the measured transduction (Eq. (3)) into conversion gain and pure conversion, as shown in Fig. 3b. The gain shows the expected steep increase at Δ_{o} → 0 where the pure conversion *θ* approaches zero for equal cooling and amplification rates. Around Δ_{o} = *κ*_{o}/2 on the other hand, where ({langle nrangle }_{min }) reaches its minimum of roughly *κ*_{o}/4*ω*_{m} (approx) 30, also the gain reaches its minimum and the noiseless part (at zero temperature) of the total (internal) conversion process shows its highest efficiency of *θ* = 0.019% (*θ*/(*η*_{e}*η*_{o}) = 1.6%).

### Added noise

Another important figure of merit, not only for quantum applications, is the amount of added noise quanta^{36}, usually an effective number referenced to the input of the device. For clarity with regards to the physical origin and the actual measurement of the noise power, in the following we define the total amount of added noise quanta *n*_{add,j} added to the input signal *S*_{in,j} after the transduction process as *S*_{out,j} = *ζ**S*_{in,j} + *n*_{add,j}. Figure 4a, b shows the measured conversion noise *n*_{add,j} as a function of frequency *δ*_{j} for the same powers and detunings as in Fig. 2a. At these powers our device adds *n*_{add,o(e)} = 224(145) noise quanta to the output of the microwave resonator (optical cavity), corresponding to an effective input noise of *n*_{add,j}/*ζ*. The noise floor originates from the calibrated measurement system and in case of the microwave port to a small part also from an additional broadband resonator noise, cf. Fig. 4b. The solid lines are fits to the theory with the mechanical bath occupation ({bar{n}}_{{rm{m}}}) as the only fit parameter (see Supplementary Note 4).

The fitted effective mechanical bath temperature as a function of pump powers is shown in Fig. 4c. It reveals the strong optical pump dependent mechanical mode heating (blue), while the microwave pump (red) has a negligible influence on the mechanical bath. Fig. 4d shows the measured total added noise at the output of the microwave resonator and optical cavity as a function of optical pump power. The noise added to the optical output (blue) increases with pump power due to absorption heating and increasing optomechanical coupling rate *G*_{o}, while the degradation of the resonator-waveguide coupling efficiency *η*_{e} explains the decreasing *n*_{add,e} at higher optical powers for the microwave output noise (red), see Fig. 2e. The intersection of the two noise curves occurs at ({{mathcal{C}}}_{{rm{e}}}simeq {{mathcal{C}}}_{{rm{o}}}) with cooperativities *C*_{j} as defined above, and shows that the optical and microwave resonators share the same mechanical thermal bath. The power dependence is in full agreement with theory (solid lines) and demonstrates that the thermal mechanical population is the dominating origin of the added transducer noise.

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