Cavity electro-optics in thin-film lithium niobate for efficient microwave-to-optical transduction

Linking superconducting quantum devices to optical fibers via microwave-optical quantum transducers may enable large scale quantum networks. For this application, transducers based on the Pockels electro-optic (EO) effect are promising for their direct conversion mechanism, high bandwidth, and potential for low-noise operation. However, previously demonstrated EO transducers require large optical pump power to overcome weak EO coupling and reach high efficiency. Here, we create an EO transducer in thin-film lithium niobate, leveraging the low optical loss and strong EO coupling in this platform. We demonstrate a transduction efficiency of up to $2.7\times10^{-5}$, and a pump-power normalized efficiency of $1.9\times10^{-6}/\mathrm{\mu W}$. The transduction efficiency can be improved by further reducing the microwave resonator's piezoelectric coupling to acoustic modes, increasing the optical resonator quality factor to previously demonstrated levels, and changing the electrode geometry for enhanced EO coupling. We expect that with further development, EO transducers in thin-film lithium niobate can achieve near-unity efficiency with low optical pump power.


Introduction
Recent advances in superconducting quantum technology [1] have created interest in connecting these devices and systems into larger networks. Such a network can be built from relatively simple quantum interconnects [2] based on direct transmission of the few-photon microwave signals used in superconducting quantum devices [3]. However, the practical range of this approach is limited by the strong attenuation and thermal noise that microwave fields experience at room temperature. Therefore, quantum interconnects based on optical links have been explored as an alternative, because they provide long attenuation lengths, negligible thermal noise, and high bandwidth [4]. Connecting optical networks with superconducting quantum technologies requires the creation of a quantum transducer capable of converting single photons between microwave and optical frequencies [5,6]. Such a transducer offers a promising route toward both large-scale distributed superconducting quantum networks and the scaling of superconductor quantum processors beyond single cryogenic environments [7]. Furthermore, single-photon microwave-to-optical transduction can be used to create high efficiency modulators [8], detectors for individual microwave photons [9], and multiplexed readout of cryogenic electronics [10].
The desire for lower noise, higher efficiency, and faster repetition rates has motivated research into cavity-based EO transducers [24][25][26][27][28][38][39][40][41], in which microwave fields directly modulate light using an EO nonlinearity of the host material. This approach avoids any intermediate mechanical modes and may allow for lower noise owing to the strong thermal contact and spatial separation of microwave and optical resonators in these devices. Previous EO transducers have used bulk lithium niobate [24][25][26], aluminum nitride [28] and hybrid silicon-organic [41] platforms. These devices have demonstrated bidirectional operation and on-chip efficiency as high as 2% [28], yet efficiencies remain low and would require large (∼1 W) optical pump powers to reach near-unity efficiency. The efficiency of EO transducers can be improved by minimizing the loss rates of the resonators and enhancing the EO interaction strength. Toward this end, here we use a thin-film lithium niobate platform, which combines a large EO coefficient of 32 pm/V, tight confinement of the optical mode to enable a strong EO coupling [42], and the ability to realize low-loss optical resonators with demonstrated quality factors (Q) of 10 7 [43].
Specifically, we describe an EO transducer made from a thin-film lithium niobate photonic molecule [44,45] integrated with a superconducting microwave resonator and demonstrate an onchip transduction efficiency of greater than 10 −6 /µW of optical pump power for continuous-wave signals. The triple-resonance photonic molecule design of our device maximizes transduction efficiency by ensuring that both the pump light and the upconverted optical signal are resonantly enhanced by low-loss optical modes. We also reduce undesired piezoelectric coupling in the microwave resonator by engineering bulk acoustic wave resonances in the device layers. Finally, we discuss the future potential for thin-film lithium niobate cavity electro-optic transducers and show that with straightforward improvements the efficiency can be increased to near unity for ∼100 µW of optical pump power.

Device design and characterization
The operating principle of our transducer is illustrated in Fig. 1(a). Two lithium niobate optical ring resonators are evanescently coupled to create a pair of hybrid photonic-molecule modes, with a strong optical pump signal tuned to the red optical mode at ω − . A superconducting microwave resonator with resonance frequency ω m modulates the optical pump signal, upconverting photons from the microwave resonator to the blue optical mode at ω + .
This transduction process can be effectively described by a beamsplitter interaction Hamiltonian where g 0 is the single-photon EO interaction strength, n − is the number of (pump) photons in the red optical mode, while b and a + are the annihilation operators for the microwave and blue optical modes, respectively. The interaction strength g 0 is determined by the microwave resonator's total capacitance, the overlap between microwave and optical modes, as well as the EO coefficient of the host material. We use a thin-film superconducting LC resonator and an integrated lithium niobate racetrack resonator [42] to optimize g 0 . The on-chip transduction efficiency η for continuous-wave signals depends on both this interaction strength and the loss rates of the modes (see Supplement Section 1), where κ m,ex , κ m and κ +,ex , κ + are the external and total loss rates for the microwave and blue optical modes, respectively, and C = 4g 2 0 n − κ m κ + is the cooperativity. The first term in Equation (2) represents the efficiency of a photon entering and exiting the transducer. To maximize this photon coupling efficiency, the resonators in our device are strongly overcoupled.
A microscope image of our device is shown in Fig. 1(b). Light is coupled from an optical fiber array onto the chip using grating couplers with ≈10 dB insertion loss. The photonic molecule optical modes are created using evanescently coupled racetrack resonators made from 1.2 µm-wide rib waveguides in thin-film lithium niobate atop a 4.7 µm-thick amorphous silicon dioxide layer on a silicon substrate. The optical waveguides are cladded with a 1.5 µm-thick layer of amorphous silicon dioxide. The fabrication process for these optical resonators is described in detail in Ref. [44]. To create the superconducting resonator, a ≈40 nm-thick niobium nitride film is deposited on top of the cladding by DC magnetron sputtering [46] and patterned using photolithography followed by CF 4 reactive ion etching. The detuning between the optical modes can be controlled using a bias capacitor on the dark (i.e. not directly coupled to the bus waveguide) racetrack resonator.
The optical transmission spectrum displayed in Fig. 1c shows a typical pair of photonicmolecule optical modes. The internal loss rate for these optical modes are κ ±,in /2π ≈ 2π × 130 MHz, corresponding to an intrinsic quality factor of 1.4 × 10 6 . As shown in Fig. 1d, we observe a clear anticrossing between bright and dark resonator modes when tuning the bias voltage and observe a minimum optical mode splitting of 3.1 GHz. For far-detuned optical modes, the bright resonator mode has a total loss rate κ bright ≈ 2π × 1.0 GHz, indicating that the optical modes are strongly overcoupled to the bus waveguide.
Lithium niobate has a strong piezoelectric susceptibility, which gives the microwave resonator a loss channel to traveling acoustic modes [47]. To investigate this loss mechanism, we perform a two-dimensional simulation of a cross section of the waveguide and resonator capacitor (see Supplement Section 2). The simulated intrinsic microwave quality factor due to piezoelectric loss displays a strong frequency dependence, as shown in Fig. 2(a). This frequency dependence is caused by low quality-factor bulk acoustic modes -illustrated in Fig. 2(b) -that form in the thin-film layers of our device, which resonantly enhance the coupling between the microwave resonator and acoustic fields. Lower loss can be achieved by designing the microwave resonance frequency to avoid the bulk acoustic resonances. The relative orientation of the capacitor and the lithium niobate crystal axes also strongly affects the microwave loss. Figure 2(c) shows that the intrinsic microwave quality factor is maximized when the electric field produced by the capacitor is oriented close to the Z-axis of the lithium niobate crystal, which is also the condition that maximizes electro-optic response. Using these considerations, we designed a microwave resonator which has a measured intrinsic quality factor higher than 10 3 at a temperature of 1 K, as shown in Fig. 2(d).

Microwave to optical transduction
To measure the transduction efficiency of our device, we locked the frequency of the pump to be near-resonant with the red optical mode (side-of-fringe locking) and sent a resonant microwave signal into the device. The pump and upconverted optical signal were collected and sent to an amplified photodetector, which produced a beat note at the input microwave frequency. We inferred the transduction efficiency from this beat note by calibrating the input optical power, system losses and detector efficiency (see Supplement Section 3). During this transduction efficiency measurement, we swept the bias voltage in a triangle waveform with a period of ≈1 min to vary the splitting between the optical modes. The pump light remained locked to the red optical mode throughout the measurement. Figure 3(a) illustrates the optical modes and signals over the course of the bias voltage sweep. The results of this measurement are depicted in Fig. 3. The two maxima in transduction efficiency near ±10 V (Figs. 3(b) and (c)) correspond to the two cases in which a triple-resonance condition is met and the upconverted light is resonant with the blue optical mode. For large negative bias voltages, the blue mode is far-detuned, and most of the upconverted light is generated in the red optical mode by a double-resonance process involving just the red optical mode and the microwave mode (see Supplement Section 1.3). This process does not depend on the resonance frequency of the blue optical mode, so the transduction efficiency is nearly independent of the bias voltage in this regime. Destructive interference of upconverted light produced in the red and blue optical modes (created by the double-and triple-resonance processes, respectively) causes the transduction efficiency minimum near −20V. For large positive bias voltage, the red optical mode is undercoupled and has a narrow linewidth, so the double-resonance transduction process is weak. The measured data presented in Fig. 3 analytical model (see Supplement Section 1) based on independently measured and estimated device parameters, shown in Fig. 3 The measured transduction efficiency features strong hysteresis when varying the bias voltage, which is caused by hysteresis in the detuning of the optical modes, shown in Fig. 3(d). We observed that this hysteresis could be reduced by lowering the optical pump power and sweeping the voltage bias faster. Based on the slow timescale (seconds for −30 dBm on-chip optical pump power), we attribute the hysteresis to photoconductive and photorefractive effects in lithium niobate [48,49]. These effects are caused by optical excitation of charge carriers in the lithium niobate waveguide, which can migrate to create built-in electric fields that shift the optical resonance frequencies through the EO effect.
We measured the bandwidth of the transducer by varying the frequency of the input microwave drive and measuring the highest transduction efficiency reached during a bias voltage sweep. The inset of Fig. 3(d) shows that our transducer has a 3 dB bandwidth of 13 MHz, slightly larger than the measured 10 MHz linewidth of the microwave resonator. This discrepancy is caused by the nonlinear response of the NbN microwave resonator for high microwave power, which leads to an apparent resonance broadening [50] (see Supplement Section 4). To reduce measurement noise, here we use a relatively large microwave power (−38 dBm on-chip), which causes a small degree of nonlinear broadening. This nonlinearity also leads to reduced transduction efficiency for large input microwave powers ( Fig. 4(a)   During this measurement, we found that the detuning between the optical modes could change by several GHz when the optical pump power was varied, likely due to the same effects that cause the hysteresis described earlier. To measure the transduction efficiency at the triple-resonance point for each power level, we performed measurements on several pairs of photonic-molecule modes. For the highest optical pump powers used in this study (denoted by crosses in Fig. 4(a)), we modulated the optical pump to extinction at a rate of 20 kHz and with a 10% duty cycle. The lower average power in these modulated-pump measurements resulted in smaller power-dependent detunings and more stable resonances.
In the low-power regime (< −30 dBm) we observe that the transduction efficiency scales linearly with pump power at a rate of (1.9 ± 0.4) × 10 −6 /µW. From this and the measured loss rates of the resonators, we estimate the single-photon coupling rate of our transducer to be g 0 = 2π × 650 ± 70 Hz, comparable in magnitude to the predicted g 0 = 2π × 830 Hz (see Supplement Section 1.4), yet slightly lower than expected. This difference is likely due to variations in the as-fabricated geometry of the device. The transduction efficiency begins to saturate at (2.7 ± 0.3) × 10 −5 , the highest-measured efficiency for this transducer. This saturation is caused by optical absorption in the microwave resonator, which generates quasiparticles that shift the resonance frequency and increase the loss rate of the resonator [51]. Figure 4(b) shows the optical-power dependence of the microwave resonator's properties. We find that the quasiparticle-induced changes in the microwave resonator are independent of whether the pump laser is tuned on-or off-resonance with an optical mode, which suggests that the absorbed light does not come from the optical resonator itself. Instead, light scattered at the fiber array and grating couplers is likely the dominant contribution to the quasiparticle loss.

Discussion and Conclusion
The transduction efficiency demonstrated here falls well below the requirements for a useful quantum transducer. However, several straightforward improvements can be made to the transducer to increase this figure of merit (see Supplement Section 5). First, optical quality factors above 10 7 have been demonstrated in thin-film lithium niobate [43], suggesting that the optical loss rates seen here can be reduced by roughly 10-fold, leading to a 100-fold improvement in transduction efficiency. Second, the microwave resonator loss rate can be reduced through improved engineering of the bulk acoustic waves to which the microwave resonator couples. For example, simulations suggest that suspending the lithium niobate layer can reduce the microwave loss by more than 10-fold. The quasiparticle losses caused by stray light absorption for higher optical pump powers can be made negligible by changing the design of the sample mount and optical fiber coupling. By using these and other (see Supplement Section 5) interventions, we predict that near-unity transduction efficiency can be achieved for optical pump powers of ∼100 µW. Although we did not demonstrate optical-to-microwave transduction (only microwave-tooptical) due to a low signal-to-noise ratio, we note that the transduction process described here is fully bidirectional [28].
In this work, we have demonstrated transduction between microwave and optical frequencies using a thin-film lithium niobate device. The photonic molecule design of our transducer enables straightforward tuning of the optical modes using a bias voltage, ensures strong suppression of the downconverted light that acts as a noise source, and takes full advantage of the large electro-optic coefficient in lithium niobate. We have described how the piezoelectric coupling of the microwave resonator to traveling acoustic waves can be engineered to minimize loss in the microwave resonator. The advantages of an EO transducer -namely the system simplicity and the possibility for low-noise operation -and the opportunities for improved transduction efficiency suggest that further development of thin-film lithium niobate cavity electro-optics is warranted.
During the preparation of this manuscript we became aware of a similar lithium niobate quantum transducer device reported by McKenna et al. [52].

Optical modes
The optical modes used in our device are delocalized between two evanescently coupled ring resonators. These optical modes are governed by the Hamiltonian where a 1 and a 2 are respectively the annihilation operators for the bright and dark ring resonator modes, 2δ is the detuning between ring resonator modes, and µ is the evanescent coupling rate. This Hamiltonian can be diagonalized by the Bologliubov transformation Note that a + and a − must obey a bosonic commutation relation a i , a j = δ i j , which requires that u 2 + v 2 = 1. For convenience, we set u = cos θ 2 and v = sin θ 2 , where θ is a hybridization parameter. The Hamiltonian will be diagonalized for tan θ = µ δ , giving where the resonance frequencies are ω ± = ω 0 ± δω 2 + µ 2 . These optical system eigenmodes at frequency ω ± will be used to calculate the interaction Hamiltonian. The loss rates of the photonic molecule modes change with the hybridization parameter θ. In the resolved sideband approximation (2µ κ, where κ is the typical optical mode loss rate), Eq. S2 also diagonalizes the open system, and the internal (i) and external (e) loss rates for the hybrid modes, κ ±, {i,e} , are given by where κ {1,2}, {i,e} are the intrinsic and extrinsic loss rates for the bright and dark ring resonator modes.

Triple-resonance transduction
Our electro-optic transducer has three resonant modes. The two optical photonic-molecule modes Here ω {±,m} are the resonance frequencies, is the dielectric permittivity, V ± are effective mode volumes, a ± and b are annihilation operators for the optical and microwave modes, ì ψ {±,m} are the field profiles1, C is the total capacitance of the microwave resonator, and d eff is a constant with dimensions of length that relates the voltage on the microwave resonator's capacitor to the electric field at the center of the optical waveguide. For our device geometry, all three modes are polarized approximately along the Z-axis of the nonlinear crystal in the region of interest, so we can use a scalar interaction approximation. The three-wave mixing process used in our device can be described by a nonlinear energy density [53] The interaction Hamiltonian for this process can be obtained from Eq. S5 by inserting our expressions for the mode fields and considering only terms that vary slowly near the triple-resonance condition ω m = ω + − ω − , which yields: The single-photon interaction strength is where n is the optical refractive index and the integral is taken over the nonlinear material. If a strong pump laser is tuned to the red optical mode, we can replace the a − operator with its is the total loss rates of the red optical mode, ω l is the pump laser frequency, ∆ − = ω l − ω − is the pump detuning, and P is the pump power. Moving to a frame where the optical modes rotate at the pump laser frequency, the full Hamiltonian for our system is where ∆ + = ω l − ω + is the detuning of the pump from the blue optical mode, and g = g 0 √ n − is the pump-enhanced coupling rate. We now use the above Hamiltonian to estimate the bidirectional transduction efficiency between continuous-wave (CW) optical and microwave signals. Consider two signal fields incident on the transducer: an optical signal a in detuned from the pump by ω p , and a microwave signal b in with frequency ω bin . The semi-classical Heisenberg-Langevin equations of motion governing the interaction between the microwave and blue optical modes are where κ m and κ m,e are the total and external coupling rates for the microwave mode. The resonator modes couple to propagating output fields a out and b out via the input-output relations In the steady state, Eqs. S9 and S10 yield the frequency-domain transduction scattering matrix where The conversion is symmetric, and the on-chip transduction efficiency is where ω is the excitation frequency, and C = 4g 2 n − κ + κ m is the electro-optic cooperativity. At the triple-resonance condition, where ω = −∆ + = ω m , this efficiency takes the form (S13) The first term represents the efficiency associated with getting a photon into and out of the converter, while the second term gives the transduction efficiency inside the converter.

Double-resonance transduction
When the converter is operated far-detuned from the triple-resonance condition, a doubleresonance transduction process can become a significant contribution to the total transduction efficiency. In this process, optical photons in the red optical mode can be scattered between the pump frequency and a blue-shifted sideband. This sideband field is far-detuned from the red optical mode relative the mode's linewidth in our experiments, so the transduction efficiency for this process is low, but it can be larger than that of the triple-resonance process when the splitting between red and blue optical modes is much larger than the microwave frequency. This double-resonance process is the origin of the bias-voltage independent response for large negative voltages in Figs. 3(b) and 3(c) of the main text. The nonlinear energy density that describes this double-resonance process is [53] which produces the Hamiltonian where Following the usual linearization procedure for the strongly pumped a − mode [54], we approximate a − ≈ a − + δa − , where δa − is a small fluctuating perturbation to the field in the red optical mode. Keeping terms of order a − , the linearized interaction Hamiltonian is where g dr = g 0,dr a − . This Hamiltonian contains both the desired beam-splitter terms and parametric amplification terms which cause optical down conversion, and since the pump is nearly resonant with the red optical mode, both types of terms are significant. In a frame where the optical mode rotates along with the laser, the semi-classical Heisenberg-Langevin equations of motion for double-resonance microwave-to-optical transduction are For simplicity, we assume that the double resonance process operates in the weak coupling regime, so that back-action of the optical fields on the microwave field can be neglected2. We take the ansatz solution and find The transmitted optical sideband field due to double-resonance transduction is δa out = − √ κ −,e δa − , and hence the total apparent transduction efficiency3, including both double-and triple-resonance transduction, is

Estimating the electro-optic interaction strength
The triple-resonance interaction strength g 0 (Eq. S7) can be cast in a form more useful for designing the transducer. Assuming that the electric field created by the capacitor is oriented along the lithium niobate's Z crystal axis and uniform across the optical mode (a good approximation for our device geometry), and that the microwave resonator drives the optical resonators with opposite phase and behaves as a lumped-element system, we find 2i.e. the term −ig dr δa − + δa † − can be dropped 3The existence of multiple optical sidebands in regimes where double-resonance transduction is significant means that transduction efficiency must be carefully defined. In our experiments, we measure only the transmission of a microwave signal from the transducer's input to the photoreceiver's output, and we cannot differentiate multiple optical sidebands. As such, we define transduction efficiency for multiple sidebands as the apparent transduction efficiency: i.e. the equivalent single-sideband transduction efficiency which would produce the observed signal. Note that this distinction between apparent and true transduction efficiency is significant only in far-detuned regimes of bias voltage sweeps, not near the triple-resonance condition where maximum transduction efficiency occurs. here r 33 = 2 χ (2) /n 4 is the relevant electro-optic coefficient, n e is the extraordinary refractive index of lithium niobate, Γ is an optical mode confinement factor, α is an electrode coverage parameter with a maximum value of 2 for full coverage of both optical resonators, and θ is the optical mode mixing parameter described above4. From this equation, it is clear that the interaction strength can be maximized by creating a microwave resonator with closely spaced electrodes, low total capacitance, and full coverage of the optical resonators.
The calculated values for key device parameters are given in Table S1. Using these results, we estimate g 0 (θ = π) = 2π × 1.0 kHz.

Simulating piezoelectric loss
We use a two-dimensional finite element model to simulate the piezoelectric loss of the microwave resonator. In this frequency-domain simulation, a voltage is applied to the capacitor electrodes at frequency ω, and the time-averaged electrostatic energy E electrostatic and acoustic power absorbed by the perfectly-matched layer P acoustic are calculated. The quality factor set by piezoelectric-loss is then given by The two-dimensional nature of the simulation means that acoustic modes with out-of-plane (i.e. along the waveguide) propagation or strain are neglected. Modes with an out-of-plane propagation direction couple weakly to the microwave resonator because the capacitor is much longer than the acoustic wavelength at the relevant ∼GHz frequencies. Modes with out-of-plane stress also couple weakly to the microwave resonator for X-cut lithium niobate because of lithium niobate's piezoelectric coefficients. For example, when applying the electric field along the Z crystal axis in our device the d 33 piezoelectric coefficient, which creates in-plane stress, dominates over other components.

Measurement setup and transduction efficiency calibration
Details of the measurement setup are shown in Figure S1. To calibrate the transduction efficiency, we perform the following procedure before every set of measurements.
First, with the laser frequency detuned far from the optical resonance, we measure the optical power into and out of the DUT. After correcting for measured asymmetric losses in the optical fibers going into the cryostat, we assume the loss at both input and output grating couplers to be symmetric. Based on measurements of a large number of grating couplers, we estimate the coupler-to-coupler variation in insertion loss to be less than 0.4 dB. Next, we measure the optical 4Note that the values of g 0 used in the main text are quoted for the optical mode splitting (and hence the value of the hybridization parameter θ) which maximizes transduction efficiency. The maximum transduction efficiency is obtained for θ ≈ 0.7π in our device. The MZM is arranged for either GHz-frequency optical single-sideband modulation using a phase-shifted (PS) dual drive through a high-frequency port, or low-frequency amplitude modulation through a bias port, controlled by an arbitrary waveform generator (AWG). Focusing grating couplers (≈10 dB insertion loss) couple light from optical fibers into the device under test (DUT), which is cooled to T ≈ 1 K inside a closed-cycle cryostat. The light collected from the DUT is split into an analysis arm (90%) and a 1 kHz photoreceiver (10%), whose signal is used to lock the laser frequency to an optical mode. The analysis arm passes through several optical switches (dotted blue lines) which allow for optional and repeatable insertion of an erbium-doped fiber amplifier (EDFA) and optical filter (F, 0.2nm bandwidth). The analysis arm can be sent to an optical spectrum analyzer (OSA) for sideband calibration, a DC optical power meter for power calibration, a 100 MHz photoreceiver for measuring transmission spectra, or a 10 GHz photoreceiver for detecting transduction. The bias voltage of the DUT is controlled by a sweep generator through a bias tee. A vector network analyzer (VNA) can be connected to microwave port V1A, which is protected by a DC block (DCB) capacitor, to excite the DUT. The upconverted optical signal can be detected at port V2. In an alternative measurement setup, the transmission of an optical sideband can be monitored by connecting the VNA to the optical single-sideband modulator (port V1).
power arriving at the output of the analysis arm using the DC power meter. These measurements allow us to estimate the optical insertion loss from the DUT to the end of the analysis arm η optical = η coupler · η fiber , as well as the on-chip optical power. Next, we calibrate the response of the 10 GHz-bandwidth photoreceiver by using port V1 to generate a single optical sideband. We measure the signal in the analysis arm using the high-resolution optical spectrum analyzer (which allows us to directly measure the relative power of the sideband and carrier P sideband /P pump ), the calibrated DC power meter (which measures the total power P sideband + P pump ), and the 10 GHz photoreceiver. These measurements allow us to estimate the detector response parameter A det. , defined so that P det. = A det. P sideband P pump .
During the transduction measurement, when the laser frequency is locked to an optical mode, measurements of the total optical power and the 10 GHz photoreceiver response allow us to infer the power in the upconverted optical sideband P sideband , based on the above photoreceiver calibration. The gain provided by the erbium-doped fiber amplifier (EDFA), if in use, can be estimated by measuring the photoreceiver response with and without the EDFA in the optical path. Finally, the calibrated transduction efficiency is given by where P in is the input microwave power at port V1A and η cable is the measured insertion loss from port V1A to the DUT. Fig. S2. Effect of microwave power on transmission spectrum. Microwave input powers above about −40 dBm at port V1A (∼ −48 dBm on-chip) produce distorted transmission spectra due to nonlinear dynamics. [50] The superconducting NbN film is deposited using DC magnetron sputtering at room temperature with an RF bias on the substrate holder. The film has a thickness of ∼44 nm, room-temperature sheet resistance of 52 Ω/square, and a transition temperature T c of ∼10 K. At high microwave powers, the superconducting resonator undergoes nonlinear oscillations [50], as shown in Fig. S2. In the actual experiment, the drive power is kept below -30 dBm, and the nonlinear dynamics are therefore small. Table S2 lists several relatively straightforward interventions that can be made to improve the performance of our transducer. The predicted efficiency enhancement for each intervention in the table assumes the transducer operates in the low cooperativity limit.