Converting microwave and telecom photons with a silicon photonic nanomechanical interface

Practical quantum networks require low-loss and noise-resilient optical interconnects as well as non-Gaussian resources for entanglement distillation and distributed quantum computation. The latter could be provided by superconducting circuits but existing solutions to interface the microwave and optical domains lack either scalability or efficiency, and in most cases the conversion noise is not known. In this work we utilize the unique opportunities of silicon photonics, cavity optomechanics and superconducting circuits to demonstrate a fully integrated, coherent transducer interfacing the microwave X and the telecom S bands with a total (internal) bidirectional transduction efficiency of 1.2% (135%) at millikelvin temperatures. The coupling relies solely on the radiation pressure interaction mediated by the femtometer-scale motion of two silicon nanobeams reaching a Vπ as low as 16 μV for sub-nanowatt pump powers. Without the associated optomechanical gain, we achieve a total (internal) pure conversion efficiency of up to 0.019% (1.6%), relevant for future noise-free operation on this qubit-compatible platform.

L arge scale quantum networks will facilitate the next level in quantum information technology, such as the internet did for classical communication, enabling, e.g., secure communication and distributed quantum computation 1 . Some of the most promising platforms to process quantum information locally, such as superconducting circuits 2 , spins in solids 3 , and quantum dots 4 , operate naturally in the gigahertz frequency range, but the long-distance transmission of gigahertz radiation is relatively lossy and not resilient to ambient noise. This limits the length of supercooled microwave waveguides in a realistic scenario to tens of meters 5 . In contrast, the transport of quantum information over distances of about 100 km is nowadays routinely achieved by sending optical photons at telecom frequency through optical fibers.
There is a variety of platforms, which in principle have shown to be able to merge the advantages of both worlds ranging from mechanical, piezoelectric, electro-optic, magneto-optic, rareearth, and Rydberg atom implementations 6,7 . So far, the optomechanical approach 8 has been proven to be most efficient, reaching a record high photon conversion efficiency of up to 47% with an added noise photon number of only 38 9 . But this composite device is based on a Fabry-Perot cavity that has to be hand assembled and utilizes a membrane mode that is restricted to relatively low mechanical frequencies. Using piezo-electricity, coherent conversion between microwave and optical frequencies has been shown uni-and bidirectional at room temperature 10,11 , and at low temperatures 12,13 with integrated devices, so far with either low efficiency or unspecified conversion noise properties. The electro-optic platform has shown promising photon conversion efficiencies 14 , recently up to 2% at 2 K 15 , but generally requires very large pump powers in the milliwatt range 16 .
In this work we present a device that converts coherent signals between 10.5 GHz and 198 THz at millikelvin temperatures via the radiation pressure interaction. Due to the need of only picowatt range pump powers, both the heat load to the cryostat and local heating of the integrated device is minimized. The chipscale device is fabricated from CMOS compatible materials on a commercial silicon-on-insulator wafer over an area of~200 μm × 120 μm. It is compact, versatile and fully compatible with silicon photonics 17 and superconducting qubits 18 . The unique electroopto-mechanical design is optimized for very strong field confinements and radiation pressure couplings, which enable internal efficiencies exceeding unity for ultra-low pump powers. We find that the conversion noise so far precludes a quantum limited operation and we present a comprehensive theoretical and experimental noise analysis to evaluate the potential for scalable and noise-free conversion in the future. Such a power-efficient, ultra-sensitive, and highly integrated hybrid interconnect might find applications ranging from quantum communication 8 and RF receivers 19 to magnetic resonance imaging 20 .

Results
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 8,21-23 . In the interaction frame, the Hamiltonian describing the conversion process is (see Supplementary Note 1): whereâ j , ðbÞ with j = e, o is the annihilator operator of the electromagnetic (mechanical) mode, andĤ CR;j ¼ _G j ðâ jb e 2iω m t þ h:c:Þ describes the counter-rotating terms which are responsible for the coherent amplification of the signal. G j ¼ ffiffiffiffiffiffi ffi n d;j p g 0;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 4ω m > κ j we neglectĤ CR;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 j =ðκ j γ m Þ 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 . 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 /(2π) = 126 MHz leading to intracavity photon numbers of n d,e % 9 × 10 5 and n d,o % 0.2 with cooperativities C e % 0:57 and C o % 0:9, the measured total (waveguide to waveguide) photon transduction efficiency is %1.1% corresponding to 96.7% internal (resonator to resonator) photon transduction efficiency over the total bandwidth of Γ conv /(2π) % 0.37 kHz. In the case of κ o > 4ω m and κ e < 4ω m , the bandwidth is given by Γ conv % ðC e þ 1Þγ 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 /4ω m % 30 reduces the reported mean bidirectional efficiencies by ffiffi ffi 2 p 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. Sideband resolution and amplification. In the nonsideband resolved limit the contribution of the counter-rotating term of the HamiltonianĤ CR;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 G o (see Supplementary Note 1). In contrast, the microwave resonator is in the resolved-sideband condition 4ω m > κ e , so that the signal tone amplification due to electromechanical interaction is negligible G e ' 1. This results in the total, power independent, bidirectional conversion gain of G ¼ G e G o ' G o , which turns out to be directly related to the minimum reachable phonon occupation: 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 G ) 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 χ À1 j ðωÞ ¼ iðΔ j À ωÞ þ κ j =2 and the mechanical resonator χ À1 m ðωÞ ¼ iðω m À ωÞ þ γ m =2 as: whereχ j ðωÞ ¼ χ j ðÀωÞ Ã . Equation (3) can be decomposed into a product of the conversion gain G and the pure conversion efficiency θ, i.e., ζ :¼ G θ, 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 /â yb y 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 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 = ζS 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 n 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 C e ' C 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.

Discussion
In conclusion, we demonstrated an efficient bidirectional and chip-scale microwave-to-optics transducer using pump powers orders of magnitude lower than comparable all-integrated 11,13,15 approaches. Low pump powers are desired to limit the heat load of the cryostat and to minimize on-chip heating, which is a b Pure and-in the absence of thermal noise-noiseless conversion θ (green dots) as well as the total conversion gain G (blue dots) that gives rise to amplified vacuum noise. θ is extracted from (a) by dividing the measured total transduction by the calculated gain using Supplementary Eqs. (16) and (17). While G o diverges for Δ o → 0, the optomechanical damping rate drops to zero and leads to a vanishing pure conversion θ. The arrows in both panels indicate the detuning used in Figs. 2 and 4.  particularly important for integrated devices because of their limited heat dissipation at millikelvin temperatures. Due to the standard material choice involving only silicon and aluminum, our device can be easily integrated with other elements of superconducting circuits as well as silicon photonic and phononic devices in the future. The two main challenges ahead are the reduced pure conversion efficiency and the optical heating that adds incoherent noise to the converted signal. We expect that both can be solved with design improvements in combination with new measurement techniques. Specifically, starting from the observed pure efficiency of 0.019% a factor of up to nearly two orders of magnitude could be gained with better waveguide coupling geometries in combination with fabrication optimization, e.g., by using surface cleaning and the reduction of humidity 37 . Improving the sideband resolution by increasing the mechanical frequency 35 could yield another factor of up to 25 assuming the same cooperativities can be achieved. Going to the high cooperativity limit would then yield the remaining fraction needed for unity total conversion efficiency. This will certainly require a very effective mitigation of the optical pump power dependent mechanical heating and the associated linewidth degradation that is also required for noisefree conversion. Nevertheless, with better chip thermalization, reduced optical absorption and low duty cycle pulsed measurements this should be feasible. Moreover, it has already been shown that pulsed pump-probe type experiments together with high efficiency heralding measurements can be used for postselecting rare successful conversion or entanglement generation events for low-noise low-efficiency devices 12,38 .
In terms of near-term classical receiver and modulation applications, an important figure of merit is the voltage required to induce an optical phase shift of π. We are able to reach a value as low as V π = 16 μV (see Supplementary Note 6), comparable with typical zero point fluctuations in superconducting circuits, nearly a factor 9 lower than the previously reported record 19 , and almost 10 12 times more power efficient than commercial passive and wide-band unidirectional electro-optic modulators at X band gigahertz frequencies.

Data availability
The data and code used to produce the results of this paper are available at https://doi. org/10.5281/zenodo.3961562.