Cavity piezo-mechanics for superconducting-nanophotonic quantum interface

Hybrid quantum systems are essential for the realization of distributed quantum networks. In particular, piezo-mechanics operating at typical superconducting qubit frequencies features low thermal excitations, and offers an appealing platform to bridge superconducting quantum processors and optical telecommunication channels. However, integrating superconducting and optomechanical elements at cryogenic temperatures with sufficiently strong interactions remains a tremendous challenge. Here, we report an integrated superconducting cavity piezo-optomechanical platform where 10 GHz phonons are resonantly coupled with photons in a superconducting cavity and a nanophotonic cavity at the same time. Taking advantage of the large piezo-mechanical cooperativity (Cem ~7) and the enhanced optomechanical coupling boosted by a pulsed optical pump, we demonstrate coherent interactions at cryogenic temperatures via the observation of efficient microwave-optical photon conversion. This hybrid interface makes a substantial step towards quantum communication at large scale, as well as novel explorations in microwave-optical photon entanglement and quantum sensing mediated by gigahertz phonons.

The electromechanical coupling rate g em can be characterized by measuring the microwave reflection spectrum of the coupled modes. Since in our system, the mechanical resonance of the micro-disk is out of the frequency tuning range of the "Ouroboros", we experimentally characterize g em using a separate device, fabricated in the same way but the "Ouroboros" has slightly lower resonant frequency which can be aligned with mechanical thickness mode of the micro-disk.
As shown in Fig. 1a, when the "Ouroboros" resonance is tuned to lower frequencies by the external magnetic field, avoided crossing is observed, indicating strong electromechanical coupling. Figure 1b plots the amplitude and phase spectrum at B ext = 0.09 mT (gray dashed line in (a)), which can be fitted by using the microwave reflection coefficient without the optomechanical coupling (S1) From the fitting (red lines in Fig. 1b), we extract gem 2π = 2.7 MHz. The intrinsic microwave and mechanical Q factors and dissipation rates of this devices are Q e,i = 1.7 × 10 3 ( κe,i 2π = 6.4 MHz) and Q m = 1.4 × 10 4 ( κm 2π = 0.74 MHz). It is worth pointing out that the slightly different resonant frequencies and Q factors of the device have negligible influence on the microwave and the mechanical mode profiles and their overlap; hence won't affect the electromechanical coupling. As discussed in the main text, this directly measured g em from the characterization device agrees very well with the fitted value based on the power dependence of the photon conversion efficiency in our superconducting POM system. For a conservative estimation, we use gem 2π = 2.7 MHz and calculate C em ≈ 7.4. Supplementary Figure 1. Measurement of the electromechanical coupling rate gem. a Microwave reflection spectrum at different external magnetic fields. As the "Ouroboros" resonance is tuned across the mechanical resonance of the micro-disk, avoided crossing is observed. b Line plot of the spectrum at Bext = 0.09 mT (gray dashed line in a) with both amplitude and phase responses. The red lines are the fitting using coupled mode formula, which extracts a coupling rate gem 2π = 2.7 MHz.

Supplementary Note 3. Measurement setup
Detailed measurement setup of the pulsed microwave-optical photon conversion is shown in Fig. 2. An ultra-highfrequency lock-in amplifier (Zurich Instruments UHFLI) is used to send two low-frequency (δ 0 = 40 MHz) signals to an IQ mixer (Marki IQ-0618LXP), with their phase difference fine-tuned around 90 • for synthesizing microwave single sideband from an RF source (Anritsu MG3693C) at ∼ 10 GHz as local oscillator (LO). This upconverted singlesideband microwave tone can be swept by varying the frequency of the LO, and sent to either the converter (switch pos. 1) as a c.w. microwave input or an optical single-sideband modulator (SSBM, Fujitsu FTM7961) to generate an optical input (switch pos. 2). The optical single sideband is generated from a stable tunable laser (Santec TSL-710), which at the same time serves as the pump after amplified by an erbium-doped fiber amplifier (EDFA) and filtered by a tunable filter to reduce noise. This c.w. pump is pulsed by switching on/off an acoustic-optic modulator (AOM, Gooch & Housego T-M080). This is done by sending a 3-µs trigger pulse with a repetition period of 1 ms from a function generator to a TTL switch to modulate the 80-MHz RF carrier of the AOM. At the same time, the trigger pulse is split and sent to the Zurich UHFLI as the external trigger for temporal measurement. The output signal of the converter at ∼ 10 GHz is downconverted to 40 MHz using the same RF source and sent to the Zurich for detection in time domain.

Supplementary Note 4. Measurement repeatability and efficiency calibration
The bidirectional conversion signals in our experiment are highly repeatable within the largest applied pump power at 17.2 dBm. As indicated as cyan stars in upper panel of Fig. 3a, we repeat the S oe measurement at low powers after sweep up to highest pump power. It can be seen that even after a few hours of delay between measurements, the results still overlap very well with previous data, confirming no hysteresis effects. This indicates that during our experiment, the "Ouroboros" remains well in its superconducting state with stable and repeatable resonant frequency even in presence of the pump induced heating and the external magnetic bias field. In addition, we plot the measured S eo at different pump powers in the lower panel of Fig. 3a to show consistent power dependence as the S oe (|S eo | 2 is manually shifted by 30.5 dB for comparison purpose), revealing the bidirectional and reciprocal nature of the M-O photon conversion.
The conversion efficiency is calibrated by measuring the full spectra of the scattering matrix elements, following the procedure described in [1]. The principle of the calibration is illustrated in Fig. 3b. The directly measured scattering matrix elements S ij are proportional to the intrinsic elements of the converter up to a constant gain or loss factor. Although it is difficult to calibrate individual gain or loss factor along each path, they can be canceled out together to obtain the intrinsic conversion efficiency since conversion is reciprocal. The on-chip efficiency (peak of the spectrum at ω = ω m ) can be obtained by where S oo,bg and S ee,bg denote the background of the reflection spectra S oo and S ee without resonance, namely, α 1 β 1 and α 2 β 2 . By fitting the reflection and conversion spectra at the optimal pump power of 16.2 dBm, we extract 15±0.02)×10 −6 . Using Eq. (S2) and the propagation of uncertainty, we can calibrate the highest on-chip conversion efficiency to be (7.3 ± 0.2) × 10 −4 , corresponding to an internal efficiency of (5.8 ± 0.2)%.
In our experiment, the microwave and the optical input powers are set to be low enough to avoid any nonlinear effects. As an example, we give the typical signal power levels that are used when the optical pump power is 16 dBm with the highest conversion efficiency. For microwave-to-optical conversion, the typical microwave input power is around −70 dBm (photon number rate ∼ 10 13 Hz), and the converted optical power is around −55 dBm (photon number rate ∼ 10 10 Hz). For optical-to-microwave conversion, since an optical single-sideband modulator was used to generate the optical input, the sideband power is typically controlled to be ∼ 50 dB lower than the pump. So at 16 dBm pump power, the optical input power is around −35 dBm (photon number rate ∼ 10 12 Hz), and the converted microwave power is around −110 dBm (photon number rate ∼ 10 9 Hz). We first calculate the response of the classical intra-cavity field α(t) under a pulse pump input α in (t), which is related to the input pump power via P in (t) = ω o |α in (t)| 2 . To first order approximation, the optomechanical backaction can be neglected and α simply followsα If the rising edge of the input is a perfect step function, namely, α in (t) = 0 when t < 0 and α in (t) = α 0 is constant when t ≥ 0, the intra-cavity photon number can be solved as (t ≥ 0) which reaches steady state with a time constant 1/κ o . Similarly, the falling edge will have the same response time.
The cosine oscillation term in solution Eq. (S4) is due to the perfectly sharp step edge of the input which doesn't exist in experiment. Therefore, in numerical calculation in Fig. 3b of the main text, we used a Gauss error function for a smoother input, instead of a step function, to reduce the artificial oscillation.

B. Conversion signal response
Now we analyze the pulse response of the conversion photon. Due to the fast response of the n cav (t), the optomechanical coupling g om (t) ≡ g om,0 n cav (t) can be simply treated as a step function. In other words, the pump pulse serves as a fast switch to quickly turn on and off the optomechanical coupling. In the resolved-sideband limit (ω m κ o ), the Heisenberg equations of motion of the intra-cavity field a(t) = (â,ĉ,b) T can be written as [2] a(t) = Aa(t) + Ba in (t), where a in (t) = (a in1 , a in2 , c in1 , c in2 , b in2 ) T is the input term. The subscript (1, 2) denotes the coupling or the intrinsic (noise) port, respectively. Matrices A and B are given by Combining with the input-output relation and setting the noise input terms to be zero, we can numerically calculate the temporal profiles of the conversion signals as plotted in Fig. 3b of the main text. The physical understanding of the transient conversion response is as follows. In general, characteristic response time of a cavity is determined by its total energy exchange rate (including all dissipation and coupling channels). For the mechanical mode, besides its own intrinsic dissipation κ m , the electromechanical coupling provides an effective energy transfer rate ∼ 4g 2 em κo = C em κ m . When g om is turned on by the pump pulse, the effective optomechanical energy transfer rate is ∼ C om κ m . Therefore, the mechanical total energy exchange rate is Γ m ∼ (1 + C em + C om )κ m , which results in a response time ∼ 1/Γ m 1 µs. Similarly, the microwave total rate is Γ e ∼ (1 + C em )κ e and 1/Γ e 1 µs. Therefore, as confirmed by the numerical results, the conversion signals can quickly reach steady state during the 3-µs pump pulse. It is worth pointing out that, due to the pump induced heating, the temperature of system will gradually increase during the pulse. However, since the thermal dynamics is a much slower process compared with the photon conversion, it is a good approximation to treat the temperature and resonance frequencies as constants when analyzing the conversion dynamics. In our experiment, the measurement time constant of each temporal data point is set to be as short as 30 ns. During such short period of time, the temperature change of the system can be neglected, and hence, the conversion process can be treated as a quasi-steady state.
where I 3 and I 5 are 3×3 and 5×5 identity matrix, respectively. The (1,3) and the (3,1) elements of S correspond to the microwave-to-optical and the optical-to-microwave conversion coefficients (t oe and t eo in the main text), respectively; namely, S 1, which reduces to Eq. (2) in the main text when ∆ o = −ω m and ω = ω m .

Supplementary Note 6. Noise analysis
The added noise during the photon conversion process can be analyzed by calculating the power spectrum density (PSD) of the output fields. For a fieldx(t), the power spectrum density is given by S withn th,m = 1 e ωm/kBTm − 1 ,n th,e = 1 e ωe/kBTe − 1 . (S11) Here, T m and T e are the temperatures of the intrinsic mechanical and microwave thermal baths, respectively. Note that we have neglected the thermal noise of the optical mode since it will be at the quantum ground state even at room temperature.
For microwave-to-optical conversion (upconversion), the input fields are the microwave signalĉ in1 , the microwave noiseĉ in2 , and the mechanical noiseb in2 . Then the optical output PSD can be expressed as (S12) The first term is the converted optical signal, the second and the third terms are noise. The added noise of the converter is defined as the noise referred to the input. Plugging in Eq. (S10), we can get the microwave added noise In our experiment, the microwave thermal bath is estimated to have only a few photons (n th,e ≈ 1.6 at 1 K, and 5.6 at 3 K). Therefore n up,e can be easily suppressed to below one when the microwave coupling port is very over-coupled (κ e,i κ e,c ). For the mechanical added noise, when ω = ω e = ω m , we have n up,m = 1 Cem κe κe,cn th,m . Namely, a large C em and the over-coupling condition can suppress the mechanical added noise even if device is physically in a "hot" thermal bath. These theoretical analyses, in fact, reveal the effect of radiative cooling-that is by over coupling the microwave mode to a cold bath, the electromechanical modes can be cooled to quantum ground state despite the large thermal occupancy of their physical environment.
For optical-to-microwave conversion (downconversion), the input fields are the optical signalâ in1 , the mechanical noiseb in2 , and the microwave noises from both the intrinsic bathĉ in2 and the coupling portĉ in1 . Hence, the microwave output PSD is (S15) The first term is the converted microwave signal, the second and the third terms are the mechanical and the microwave noises. The last term is the noise coming in from the microwave input port, which then gets reflected and comes out as part of the microwave output. For practical quantum operation, a circulator is needed to separate the input and the output fields of the microwave port. The microwave input should be thermalized at ∼ mK environment to make sure this last term in Eq. (S15) contributes zero noise. This is also consistent with our idea and experimental implementation of the radiative cooling [3]. Similarly, the microwave and the mechanical added noise during the downconversion can be obtained as n down,e = |S 3,4 | 2 |S 3,1 | 2n th,e = g 2 om + κo We see that the thermal noises contribute differently in different conversion directions. When ω = ω e = ω m = −∆ o , the microwave added noise simplifies to n down,e = (Com+1) 2 ComCem κo κo,c κe,i κen th,e . As we have discussed in the main text, a high conversion efficiency requires large and matched C om and C em , so the first factor would be around one. An over-coupled optical port will reduce the second factor to close to one. Therefore, it is again very crucial to over couple the microwave port (κ e,i /κ e 1) to suppressn th,e . The mechanical added noise at ω = ω m = −∆ o becomes n down,m = 1 Com κo κo,cn th,m . Hence, it is important to have a large C om to suppress the mechanical noise during the optical-to-microwave photon conversion.