Abstract
Frequency conversion between microwave and optical photons is a key enabling technology to create links between superconducting quantum processors and to realize distributed quantum networks. We propose a microwaveoptical transduction platform based on longcoherence time superconducting radiofrequency (SRF) cavities coupled to electrooptic optical cavities to mitigate the loss mechanisms that limit the attainment of high conversion efficiency. We optimize the microwaveoptical field overlap and optical coupling losses in the design while achieving long microwave and optical photon lifetime at milliKelvin temperatures. This represents a significant enhancement of the transduction efficiency up to 50% under incoming pump power of 140 μW, which allows the conversion of fewphoton quantum signals. Furthermore, this scheme exhibits high resolution for optically reading out the dispersive shift induced by a superconducting transmon qubit coupled to the SRF cavity. We also show that low microwave losses enhance the fidelity of heralded entanglement generation between two remote quantum systems. Finally, high precision in quantum sensing can be reached below the standard quantum limit.
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Introduction
In the past decade, quantum information science and technology have been greatly boosted by the advancement of superconducting circuits^{1,2,3}. However, the cryogenic temperature requirement restricts the capability to transfer quantum information between local quantum processors and, therefore, to build largescale quantum networks^{4}. Optical photons operating at a frequency of hundreds of TeraHertz (THz) are immune to thermal noise at room temperature and can be transferred distantly via the freespace and fiber communication technologies. As a result, optics has shown great promise for quantum communication and for connecting distant quantum nodes^{5}. Quantum transducers are required to mediate the huge energy gap between microwave and optical photons, about five orders of magnitude, and therefore to integrate optical communication systems with superconducting quantum devices^{6,7,8}.
Past years have witnessed extensive explorations of microwaveoptical quantum transduction based on various kinds of hybrid platforms, including electrooptomechanics^{9,10}, piezooptomechanics^{11,12,13,14}, magnonics^{15,16,17,18}, atoms^{19,20,21}, rareearth ions^{22,23}, electrooptic nonlinear materials^{24,25,26,27,28,29}, and others. Among them, the electrooptic effect provides the most direct way to couple microwave and optical fields. The optical refractive index is modulated by the microwave electric field through the secondorder electrooptic nonlinearity (χ^{(2)})^{30,31}. As a result, frequency conversion has been demonstrated based on microwave cavities integrated with nonlinear optical resonators made of electrooptic materials such as lithium niobate (LN)^{32} and aluminum nitride (AlN)^{33}. In particular, the advancement of thinfilm LN enables the fabrication of LN optical resonators with ultrahigh optical quality^{34}, offering a promising platform for microwaveoptical frequency conversion^{25,26,27,28,29}. Yet, the small mode volume leads to overheating the devices when the pump power applied to the optical resonator is high. Thus, the tradeoff between the thermal noise^{35,36} and transduction efficiency makes it hard to realize a highefficiency quantum transducer operating near the quantum threshold. Not long ago, threedimensional (3D) cavities were exploited for transduction with bulk LN crystal optical resonators embedded, where the large mode volume gives rise to larger pump power tolerance^{35,36,37,38,39}. Pioneer work shows 0.03% photon conversion efficiency and 1.1 added output noise photons per second at 1.48 mW pump power^{37}, demonstrating how threedimensional geometries are transformative in quantum transduction, reducing the detrimental effect of thermally induced quasiparticles in the frequency conversion process. So far, the transduction efficiency is limited by the lowquality factor (Q) of the microwave resonators and by the nonoptimal alignment between the microwave and optical mode distributions.
Threedimensional bulk niobium superconducting cavities provide opportunities to mitigate quantum decoherence mechanisms^{40,41,42,43}. Such cavities developed at Fermilab for particle accelerators exhibit recordlong photon lifetime at milliKelvin (mK) temperature and thus mitigate the loss of microwave photons and greatly enhance lightmatter interaction in the quantum regime. Moreover, the flexibility in cavity design and geometry offers degrees of freedom in modes selection and microwave engineering to fulfill different goals and requirements. In addition, the large vacuum space in the mode volume allows the integration with various other materials for achieving hybrid quantum systems with rich functionalities.
In this paper, we propose a highefficiency, lowpumppower quantum transducer based on a threedimensional architecture with a bulk niobium superconducting cavity integrated with an LN optical resonator. We maximize the microwaveoptical interaction by careful RF design and optimization of the overlap between the optical field in the LN crystal structure and the microwave field. Through simulations, we study and optimize the optical coupling strength between the prism coupler and the optical resonator. We demonstrate through simulations that our system can achieve 50% quantum transduction efficiency at 140 μW pump power, with an operating bandwidth of 100 kHz. Furthermore, with a superconducting qubit coupled to the microwave cavity, the transducer can read out the quantum state of the qubit with high resolution of the dispersive shift. Finally, such transducer devices can be exploited for highfidelity heralded quantum entanglement generation between two distant quantum processing units, as well as for highprecision microwave signal measurement and quantum sensing.
Results
Device design
The proposed system consists of an LN whispering gallery mode (WGM) optical resonator enclosed by a 3D bulk niobium superconducting RF cavity (Fig. 1a). The LN resonator with a 5 mm diameter is made of a disk with the central part removed and has the crystal axis in the z direction (zcut), aligned to the electric field distribution. The electric field is enhanced by defining two symmetric interaction sections with a narrow vertical vacuum gap of 50 μm spanning 80^{∘} in the azimuthal direction (Fig. 1b). Such a design reshapes the electric field distribution. It enables the concentration of the microwave field onto the region near the rim of the LN disk where the WGM is located. This is the key to enhancing the interaction between the microwave and optical modes^{44}. It is noted that previous designs relied on electrode deposition to concentrate the microwave field onto the WGM region near the rim of the LN crystal. Such an approach requires more complicated fabrication techniques, including metal evaporation and photolithography, which could introduce additional optical losses to the WGM. In contrast, our design is mainly based on engineering the geometry of the SRF cavity and the embedded LN crystal. The region outside the thin 80^{∘} vacuum gap allows for the placement of an optical coupler, such as a diamond prism, which will be discussed further.
Bulk Nb SRF cavities are attractive for quantum applications because of the high electromagnetic field energy in the large mode volume. SRF cavities built for particle accelerators have demonstrated E_{max} = 10 MV ⋅ m^{−1}, with Q > 10^{10} ^{40}. In transduction, the high field density within the cavity enhances the lightmatter interactions and, thus, the electrooptic effect.
The LN crystal enclosed in the SRF cavity occupies most of the volume, with a participation ratio (p) in the order of 0.925, i.e., 92.5% of the microwave volume is occupied by the LN crystal. The participation ratio is defined as:
where E is the electrical field, V is the total volume, and V_{d} is the volume of the dielectric crystal. ϵ_{d,0} are the dielectric permittivities of the dielectric LN crystal and vacuum, respectively. The microwave quality factor at cryogenic temperatures is mainly determined by the losses in the dielectric crystal, which include several sources of losses, such as the twolevelsystem (TLS) loss of impurities in the LN crystal, the piezoelectric loss, the thermal quasiparticle loss, and others^{45}. For simplicity, we use a single value of loss tangent (\(\tan \delta\)), including all channels of losses in our evaluation, which has been reported to be in the order of 10^{−5} at cryogenic temperatures^{46}. The quality factor is the ratio between the energy stored in the cavity (U) and the dissipated power (P_{loss}), i.e.,
where H is the magnetic field and ω_{b} is the resonant frequency of the RF cavity. The total loss rate (γ_{b}) of the RF cavity is determined by the coupling losses of the input and output couplers (γ_{b,c}), the intrinsic Q of the RF cavity (Q_{0}), and the dielectric quality factor (Q_{d}):
where the dielectric quality factor is inversely proportional to the loss tangent:
With a high participation ratio, the dominant source of microwave loss is the dielectric loss, which is determined by the loss tangent of the electrooptic material. If Q_{0} is higher than Q_{d}, the bare microwave loss on the cavity wall is negligible. Such a requirement is satisfied by superconducting materials such as niobium (Nb) and aluminum (Al) (see Methods). The choice of cavity materials is further discussed in Supplementary Note 1.
To find the optimal parameters of the RF cavity geometry for electrooptic interactions, a tradeoff exists between the microwave quality factor, which is reduced by the large dielectric participation ratio, and the maximum electric field in the volume of WGM near the rim of the LN resonator, which is enhanced by more dielectric participation. In our design, as the microwave volume is essentially reduced to the volume of the crystal, the losses are mostly concentrated in the dielectric LN resonator, which defines the internal quality factor of the microwave cavity. The resonant frequency is tunable by adjusting the length of the lateral stub (see Supplementary Note 2).
Model of the system
The electrooptic quantum transduction is based on a threewave mixing nonlinear process between the optical pump, the optical signal, and the microwave signal (Fig. 1c). An optical mode a_{p} at frequency ω_{p} is pumped by the coherent light, enabling the coupling and bidirectional conversion between the other optical mode (a) at frequency ω_{a} and a microwave mode (b) at ω_{b}. To satisfy the energy conservation while avoiding the Stokes photon generation into the undesired optical mode, the free spectral range (FSR) of the WGMs, i.e., the frequency difference between two WGMs with consecutive azimuthal mode numbers is chosen to be slightly larger than the microwave resonance frequency (ω_{b}). Hence the Hamiltonian of the system is given by^{31}
The singlephoton electrooptic coupling strength (g_{eo}) is determined by the field overlap between the microwave mode and the WGM:
where n is the extraordinary refractive index of LN near 1550 nm, r_{33} is the linear electrooptic coefficient; V_{i} (i = p, a, b) are the mode volumes of the mode i with the dielectric constant ϵ_{i}, the frequency ω_{i}, and the singlephoton electric field \({E}_{i}(r,\theta ,\phi )=\sqrt{\hslash {\omega }_{i}/2{\epsilon }_{0}{\epsilon }_{i}{V}_{i}}{\psi }_{i}(r,\theta ,\phi )\).
As the pump and signal optical modes have a mode number difference of 1, their amplitudes can be expressed as ψ_{p} = ϕ_{p}(r, θ)e^{−imϕ} and ψ_{a} = ϕ_{a}(r, θ)e^{−i(m+1)ϕ}. To satisfy the momentum conservation, a dipole mode is chosen for the microwave cavity whose electric field oscillates along the azimuthal direction as \({E}_{RF}(r={R}_{0},\theta =0,\phi )={\widetilde{E}}_{RF}(\phi )\cos (\phi )\), where \({\widetilde{E}}_{RF}(\phi )\) is the correction function that maps the sinusoidal function to the realistic mode distribution. Since the mode size of the WGM is relatively small, we can assume that the microwave electric field has uniform r and θ distribution inside the WGM volume^{37}, and therefore, calculate g_{eo} as
where the electric field is normalized by the photon number determined by the total stored microwave energy (W) in the RF cavity. As seen from Eq. (7), maximizing the electric field (\({\widetilde{E}}_{RF}\)) by tailoring the microwave mode distribution is essential to obtain a large electrooptic coupling rate.
To trigger the threewave mixing process efficiently, a strong coherent pump with frequency ω_{a} − ω_{b} and detuned from ω_{p} by δ_{p} = FSR − ω_{b} is coupled to the optical resonator (Fig. 1c). The Hamiltonian in the rotatingwave approximation is reduced to
where α is the amplitude of the pump field in the optical cavity.
Assuming an optical driving field \({A}_{in}{e}^{i({\omega }_{a}+\Delta )t}\) and a microwave driving field \({B}_{in}{e}^{i({\omega }_{b}+\Delta )t}\) are applied as the signals for conversion, the dynamics of the system are given by
with
where, for a generic m mode (m = a, b), γ_{m} is the total loss rate, consisting of both the internal loss and external (coupling) loss: γ_{m} = γ_{m,0} + γ_{m,c}. By solving the system’s dynamics in the steady state, we can obtain the transfer functions for conversion from the microwave to optical signals and vice versa, which are equal due to reciprocity. The bidirectional frequency conversion efficiency depends on the microwaveoptical cooperativity and losses:
where C is the cooperativity between the optical and the microwave modes, which depends on the photon number in the pump mode (n_{p}). The second term of Eq. (12) is the internal efficiency: η_{i} = 4C/(1 + C)^{2}, which reaches unity at C = 1.
As a result, the limiting factors in the attainment of high efficiency in microwaveoptical transduction are:

The quality factor of the microwave hybrid cavity (Q_{b}) – While the optical Q_{a} can reach 10^{7}, microwave Q_{b} is limited in the stateofart schemes.

The singlephoton electrooptic coupling coefficient (g_{eo}) – This parameter is determined by the overlap between microwave and optical fields and has a tradeoff with the microwave Q factor.

The pump power – High pump power leads to overheating of the device, and thermal quasiparticle poisoning, which hinders the device’s operation in the quantum regime.
Simulation of the microwave and optical modes
Using the FiniteDifference TimeDomain (FDTD) method, we simulate the electric field distribution of a 9 GHz dipole mode in the RF cavity with the LN resonator embedded (Fig. 2a, b). As noted, the electric field has a strong concentration around the rim of the LN crystal. The dipole mode property is verified by the complete oscillation of the electric field along the rim of the disk (Fig. 2c). The field distribution is distorted from a standard sinusoidal shape due to the variation in the distance between the RF cavity wall and the LN resonator. The proper choice of the size of the narrow gap between Nb walls is critical for the optimization of the field magnitude on the rim of the LN crystal (Fig. 2d). The maximum z component of the electric field is over 1 × 10^{10}V ⋅ m^{−1}, yielding a singlephoton electrooptic coupling coefficient of 2π × 46.75 Hz based on Eq. (7).
As noted before, due to the extremely low niobium loss and high intrinsic Q for the RF cavity, the microwave loss at cryogenic temperature is dominated by the loss tangent of LN crystals. Based on the reported LN loss tangent (\(\tan \delta \sim 1\times 1{0}^{5}\))^{47}, we estimate the quality factor of the microwave dipole mode to be 1.1 × 10^{5}, which is well above what was achieved in previous works. The removal of the central part of the disk allows for lower microwave losses on the redundant LN crystals where the WGM does not exist. While further optimization is possible by further enlarging the diameter of the central hole, the proposed design, with an inner diameter of 2 mm, is the tradeoff solution between the fabrication techniques, the microwave losses, and the optimal overlap between microwave and optical modes. To ensure that the FSR matches the microwave resonance frequency (ω_{b}), we tune ω_{b} by applying a deformation to the volume of the stub L (Fig. 2f). The sensitivity of 700 MHz ⋅ mm^{−1} allows the tunability of the cavity resonant frequency by several tens of MHz and with high resolution through piezo stages. Additional details of the microwave design are available in Supplementary Note 2.
We also simulate the mode distribution of an optical WGM with a frequency around 192.43 THz by the FDTD approach (Fig. 3b). Based on the eigenfrequencies of three WGMs with consecutive mode numbers, we found that the FSR is around 8.93 GHz. The curvature of the side wall is 2.5 mm, making it appear nearflat. The smooth surface enabled with stateoftheart LN fabrication techniques can ensure a high optical quality factor (Q_{a}) above 10^{7}. In addition, the upper and bottom surfaces can also be engineered to be slightly rounded depending on fabrication needs. Those minor curvatures will have negligible influence on the microwave and optical mode profile.
To couple the optical pump into the WGM, a highindex prism is placed close to the rim of the optical resonator (Fig. 3a). The effect of the prism and the apertures to couple the beam light to the optical resonator is negligible on the microwave loss, as discussed in Supplementary Note 3. The coupling strength between the resonator and the prism (γ_{a,c}) is controlled by their spatial gap. It is known that the pump photon number always reaches the maximum at critical coupling^{48}, i.e., γ_{a,c} is equal to the intrinsic loss rate γ_{a,0} (Fig. 3c). However, the critical coupling does not lead to optimized transduction efficiency. As shown in Fig. 3d, the cooperativity is maximized around γ_{a,c} = 0.7 × γ_{a,0}, regardless of the microwave quality factor (Q_{b}). The overall transduction efficiency, however, is optimized at different values of coupling losses for various microwave quality factors (Q_{b}) (Fig. 3e). In our case, i.e., Q_{b} ~ 10^{5}, the optimized coupling loss is around γ_{a,c} = 2.3 × γ_{a,0}. This lightly overcoupled regime is favorable for efficient readout of the transduction signal.
Figures of merit of the quantum transducer
With the device design presented above, we explore the figures of merit of the transduction process at different conditions of operation. In particular, a highperformance transducer should have: (1) high cooperativity and efficiency; (2) low operating pump power, which indicates low thermal noise; and (3) large bandwidth for signal conversion.
As noted, a high level of optical pump benefits the cooperativity and transduction efficiency but introduces more overheating of the device, leading to excessive thermal noise photons. Moreover, the stray infrared pump photons that are scattered out of the designed optical path and illuminate on the RF cavity surface can break the cooper pairs of the Nb and deteriorate the coherence of RF cavities and qubits^{49,50}. To enable a quantum operation, as shown by previous literature, the pump power needs to be in the submW level^{37}, while the exact noise photon level depends on the specific geometry and the materials of the physical systems^{36}. Therefore, we study the cooperativity and transduction efficiency as a function of the pump power for our design. We compare the cases with different microwave quality factors and find that the conversion efficiency is greatly enhanced by a high microwave quality factor (Q_{b}). Our design can achieve 0.58 cooperativity and 50% conversion efficiency at 140 μW pump power (Fig. 4a and b). This level of performance greatly exceeds the limit of stateofart technology. Current transduction designs with lower Q_{b} rely on ~ mW pump power to achieve less than 0.1 cooperativity and 2% conversion efficiency^{7}.
Moreover, we investigate the bandwidth of the transducer by assuming that the RF cavity is probed by a weak microwave signal with a certain frequency detuning (Δ) from the operating frequency ω_{b}. From the simulation, we find out that the bandwidth is enhanced by decreasing Q_{b} (Fig. 4c). This is due to the fact that a larger linewidth of the microwave mode can tolerate a wider frequency range of the input signal. Therefore, there is always a tradeoff between the transduction efficiency and the operating bandwidth. For our design, with Q_{b} ~ 10^{5}, the microwave bandwidth (Δ_{b}) is 100 kHz, with the center of the spectrum window located at ω_{b}.
The figures of merit for this design are summarized in Table 1.
Quantum transduction in a hybrid quantum electrodynamics system
We now extend the electrooptic quantum transducer to a cavity QED system where the RF cavity is dispersively coupled to a superconducting transmon qubit located within the cavity mode volume. With the hybrid QED system built upon our transducer, one can convert the microwave input signal to optical photons, and also optically read out the qubit quantum state via optical measurement^{51}. The full Hamiltonian of the hybrid QED system is given as
where \({\omega }_{q,b}^{{\prime} }\) refer to the renormalized frequencies of the qubit and the microwave cavity mode, and χ is the dispersive coupling rate. As indicated by the term \(\hslash ({\omega }_{b}^{{\prime} }+\chi {\sigma }_{z}){b}^{{\dagger} }b\), the excited state of the qubit induces a dispersive shift (χ) to the resonance frequency of the RF cavity. Conventionally, the qubit state can be measured by characterizing the frequency shift in the spectrum of the readout cavity. Based on our design, the dispersive frequency shift can be further converted to the amplitude change of the optical signal as the readout of the transducer. Thus, the optical readout and transfer of the state information of the superconducting qubit can be achieved^{51}.
We assume that the renormalized RF resonance frequency \({\omega }_{b}^{{\prime} }\) is equal to FSR − δ_{p}, and a weak microwave readout pulse with amplitude B_{in} and frequency \(\omega ={\omega }_{b}^{{\prime} }\) is added to the RF cavity. The dynamics of the system in the interaction picture are given by
where we define \(a=\widetilde{a}{e}^{i({\omega }_{a}+\chi )t}\) and \(b=\widetilde{b}{e}^{i({\omega }_{b}^{{\prime} }+\chi )t}\). The resulting frequency components of the output optical field of the hybrid QED system are shown in Fig. 5a.
Based on the above analysis and our design parameters, we simulate the transduction efficiency, which determines the level of optical output, as a function of the dispersive shift χ (Fig. 5b). The transduction efficiency shows quick decay as the magnitude of the dispersive shift increases. This is mainly due to the fact that, when the microwave frequency is detuned from both the optical FSR and the readout pulse, the threewave mixing process becomes increasingly less efficient. As the Q_{b} increases, the resolution to distinguish the dispersive shift becomes higher, which can reduce the error rate in the qubit state readout.
Figures 4c and 5b show similar behaviors that lead to a tradeoff between the input detuning bandwidths and the dispersive shift resolution. At Q_{b} = 10^{5}, we find that our system can tolerate ± 50 kHz detuning for the input signal around the RF resonance frequency while being able to fully resolve ± 98 kHz (η < 0.1) dispersive shift induced by the qubit. With typical superconducting qubitcavity coupling strength up to ~ MHz level^{52}, this scheme can therefore work for a variety of QED systems to optically read out qubit information and connect superconducting qubits in remote quantum processors^{51}.
For successful dispersive measurements, the readout rate should be much faster than the decay of the qubit state. The lifetime of stateoftheart transmon qubits (T_{1}) is at a level of 100 μs^{53}. The transduction readout process occurs at a rate of \(\max \left(\frac{1}{{g}_{eo} \alpha  },\frac{1}{{\gamma }_{a,c}}\right)\,\, \sim \,\,3\,\mu {{{\rm{s}}}}\, < \, {T}_{1}\). Further improving the efficiency of the transducer can lead to a higher signaltonoise ratio and fidelity in qubit readout^{3}.
Heralded entanglement generation based on transducers
We discuss an application of our highefficiency quantum transducer to quantum entanglement generation. One of the key challenges for realizing quantum networks is creating entangled quantum states between two distant quantum nodes. It has been proposed that the optically heralded quantum entanglement can be generated with highefficiency quantum transducers^{54,55}. Such a scheme can work in both the regimes of blue sideband pumping and red sideband pumping, with the latter offering higher fidelity due to its protection against multiphoton generation.
Here, we analyze the entanglement generation features based on the optically heralded approach using our electrooptic transducer. The setup consists of two quantum transducers spatially separated. The optical cavities of the transducers are coupled to optical waveguides, i.e., fiber optics, which send the generated optical photons to a beam splitter followed by two detectors without distinguishing whichpath information. The heralded signal indicates the generation of remote entangled microwave photon pairs in superconducting cavities. In entanglement generation, the target state is a state in the two distant microwave cavities ∣ψ > = c_{1}∣01 > + c_{2}∣10>. Such a state can be generated by either the bluesideband pumping or redsideband pumping approach, as discussed below.
As for the bluesideband pumping case, a pump optical photon can be converted into a microwave and an optical photon. The photon generation in one transducer follows a Poissonian process with a photon generation rate r_{0}. Therefore, the probability that a photon is generated in each microwave cavity is expressed as
while, on the other hand, the probability that no photon is generated is
Therefore, the probability that one photon is generated in each microwave cavity is given as
Furthermore, there is a possibility that more than one photon is generated in one microwave cavity. The probability associated with this event is
Moreover, there can also be more than one photons generated in both cavities, with the probability
One can write the final state of the system as a superposition of twophoton Fock states:
where we neglect the phase information in the coefficients. Assuming fixed duration time (Δt) of the transduction process, the entangled microwave photon pair is generated at a rate \(2{r}_{0}{e}^{{r}_{0}\Delta t}\frac{\Delta t}{\Delta t+{t}_{r}}\), where t_{r} is the microwave reset time after each generation event^{55}. The infidelity corresponds to the probability that each microwave cavity contains one photon, or that more than one photon is generated in at least one cavity. Therefore, the fidelity is expressed as
In the redsideband pumping case, the remote entanglement is enabled by first preparing both microwave cavities at the onephoton state while leaving the optical cavity at the zerophoton state^{56,57}. Upon the simultaneous onset of the transduction processes in both transducers, a microwave photon can be converted to an optical photon that triggers the “click" of the detector, with the probability
As such, the probability that no optical photon is generated is instead given by
As there cannot be more than one photon generated in a microwave cavity, the generated state of the whole system, including two quantum units, is simply \(\left\vert {\psi }_{f}\right\rangle =\sqrt{{P}_{00}}\left\vert 00\right\rangle +\sqrt{{P}_{10}}\left\vert 10\right\rangle +\sqrt{{P}_{01}}\left\vert 01\right\rangle +\sqrt{{P}_{11}}\left\vert 11\right\rangle\). Moreover, the infidelity is only affected by \({P}_{11}={P}_{1}^{2}\), which leads to
Here, we analyze the rate and infidelity of entanglement generation based on the parameters in our design of the transducers. While the entanglement generation rate is optimized at a pump power higher than 1 mW, the increase in pump power reduces the fidelity, due to the additional possibility of generating more than one microwave photon (Fig. 6). This tradeoff limits the amount of pump power that can be applied, as well as the speed of entangled state generation for distributed quantum communication. For our design, a 20 kHz entanglement rate can be realized at a pump power of 18 μW with an infidelity of about 0.01.
Enhanced quantum sensing with highcoherence transduction devices
The highefficiency microwaveoptical transducer can also enhance the measurement precision of weak microwave signals in quantum sensors. The microwave signal measurement is of critical importance in various fundamental studies and applications, including, for instance, single photon detection, highly sensitive axion and dark photon haloscope measurement, new physics particle searches in the THz range, and so on^{42}. In the upconversion operation performed with this hybrid transduction system, the microwave information is converted to the optical regime through selfheterodyne techniques, which have the advantage of eliminating the local oscillator (LO) used in conventional superheterodyne mixers, reducing, therefore, complexity, power consumption, pump power, and thermal quasiparticle poisoning.
One issue associated with the transducer working for precise measurement is that the conversion process introduces additional singlephoton shot noise and backaction noise. The standard noise spectrum of microwave detection using singlequadrature measurement is given by:
which is ultimately limited by both the standard quantum limit (SQL) and the RF noise floor as S(Δ) ≥ S_{RF} + S_{SQL}, where S_{RF} = 2κ_{b}(2n_{T} + 1) and \({S}_{SQL}=\sqrt{{\kappa }_{b}^{2}+{\Delta }^{2}}\).
The enhancement of microwave photons measurement at the quantum threshold through transduction methods and with the assistance of backaction noise cancellation techniques have been reported in ref. ^{58}. In particular, a judicious choice of combination of independent quadrature measurements can lead to the cancellation of the backaction noise in the transduction process^{58}. In the backaction evading approach, the detection noise density is rather given by:
where n_{T} is the number of thermally excited photons and C is the cooperativity for transduction following Eq. (12). Therefore, with the backaction scheme, there is a possibility to break the standard detection limit with reference to the SQL. Figure 7 shows the noise spectral densities as a function of pump power for different microwave Q’s. The noise level of the sensor can be lowered down to the RF limit breaking the SQL limit by increasing the microwave coherence time or increasing the pump power. Notably, the large cavity coherence can significantly reduce the required pump power to obtain a high precision for detection, making it meaningful to increase the coherence time of cavities further.
Discussion
We design a threedimensional electrooptic system for highefficiency and lowpumppower microwaveoptical quantum transduction based on SRF technology. The transduction process is optimized by calculating the figures of merit versus various degrees of freedom in the SRF cavity geometry, LN crystal geometry, optical coupling, and optical pump power. Through microwave and optical analysis and simulations, we show transduction performance and up to 50% frequency conversion at very low pump power. This level of efficiency is critical for achieving frequency conversion with meaningful capacity in the quantum regime^{59}. Compared to the previous works, this design overcomes the conventional challenge of achieving high transduction efficiency and low incoming pump power simultaneously.
This platform holds great potential for the nearfuture realization of several applications. The electrooptic quantum transducer allows for optical readout of quantum information in superconducting QED systems inside the dilution refrigerators^{60}. It provides a critical interface for achieving huge scaling capabilities for quantum computers, going beyond the limit of local dilution refrigerators, building links between superconducting quantum processors, and realizing remote entanglement over optical fibers.
High conversion efficiency can also be leveraged in quantum sensing techniques breaking the standard quantum limit (SQL) by applying backaction evading schemes or squeezing techniques for the detection of microwave photons. Thus, these transduction techniques can serve as a platform for fundamental physics experiments such as dark matter detection over a wide frequency range.
Methods
Material requirement for the microwave cavity
In our design, the LN crystal has a filling factor above 90%, and the LN loss tangent at cryogenic temperatures is \({Q}_{d} \sim \tan \delta =1{0}^{5}\) ^{46}. Therefore, the quality factor associated with the properties of the dielectric part is Q_{d} ~ 10^{5}. The intrinsic quality factor of the bare cavity without LN crystal (Q_{0}) should be at least in the order of 10^{6} to avoid additional microwave losses and make a minimal impact on the total Q_{b}. Such a requirement is satisfied by superconducting materials, such as niobium or aluminum. However, it is not easily satisfied by a normal conducting metal with higher losses (see Supplementary Note 1). Among superconducting materials, niobium is preferable to aluminum, as the critical temperature in aluminum is lower than in niobium.
RF design
Electromagnetic (EM) simulations are performed using commercially available CST Studio Suite and COMSOL Multiphysics software packages. The RF design is optimized to achieve a high electric field on the rim of the LN resonator and, therefore, a significant interaction with the optical field. Results from EM simulations are used to model the quantum transducer and to compute the figures of merit, such as the singlephoton electrooptic coupling coefficient, the cooperativity, and the transduction efficiency. Furthermore, the parameters derived from the simulations are applied to qubit readout in QED systems, heralded entanglement generation, and quantum sensing.
Data availability
The numerical data generated in this work is available from the authors upon reasonable request.
Code availability
The code generated in this work is available from the authors upon reasonable request.
References
Schoelkopf, R. & Girvin, S. Wiring up quantum systems. Nature 451, 664–669 (2008).
Wendin, G. Quantum information processing with superconducting circuits: a review. Rep. Prog. Phys. 80, 106001 (2017).
Blais, A., Grimsmo, A. L., Girvin, S. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 (2021).
Pirandola, S. & Braunstein, S. L. Physics: Unite to build a quantum internet. Nature 532, 169–171 (2016).
Chen, Y.A. et al. An integrated spacetoground quantum communication network over 4600 kilometres. Nature 589, 214–219 (2021).
Lauk, N. et al. Perspectives on quantum transduction. Quantum Sci. Technol. 5, 020501 (2020).
Han, X., Fu, W., Zou, C.L., Jiang, L. & Tang, H. X. Microwaveoptical quantum frequency conversion. Optica 8, 1050–1064 (2021).
Lambert, N. J., Rueda, A., Sedlmeir, F. & Schwefel, H. G. Coherent conversion between microwave and optical photonsan overview of physical implementations. Adv. Quantum Technol. 3, 1900077 (2020).
Andrews, R. W. et al. Bidirectional and efficient conversion between microwave and optical light. Nat. Phys. 10, 321–326 (2014).
Arnold, G. et al. Converting microwave and telecom photons with a silicon photonic nanomechanical interface. Nat. Commun. 11, 4460 (2020).
Mirhosseini, M., Sipahigil, A., Kalaee, M. & Painter, O. Superconducting qubit to optical photon transduction. Nature 588, 599–603 (2020).
Jiang, W. et al. Efficient bidirectional piezooptomechanical transduction between microwave and optical frequency. Nat. Commun. 11, 1166 (2020).
Forsch, M. et al. Microwavetooptics conversion using a mechanical oscillator in its quantum ground state. Nat. Phys. 16, 69–74 (2020).
Han, X. et al. Cavity piezomechanics for superconductingnanophotonic quantum interface. Nat. Commun. 11, 3237 (2020).
Zhang, X., Zhu, N., Zou, C.L. & Tang, H. X. Optomagnonic whispering gallery microresonators. Phys. Rev. Lett. 117, 123605 (2016).
Hisatomi, R. et al. Bidirectional conversion between microwave and light via ferromagnetic magnons. Phys. Rev. B 93, 174427 (2016).
Zhang, X., Zou, C.L., Jiang, L. & Tang, H. X. Strongly coupled magnons and cavity microwave photons. Phys. Rev. Lett. 113, 156401 (2014).
Zhu, N. et al. Waveguide cavity optomagnonics for microwavetooptics conversion. Optica 7, 1291–1297 (2020).
Bartholomew, J. G. et al. Onchip coherent microwavetooptical transduction mediated by ytterbium in yvo4. Nat. Commun. 11, 3266 (2020).
Adwaith, K., Karigowda, A., Manwatkar, C., Bretenaker, F. & Narayanan, A. Coherent microwavetooptical conversion by threewave mixing in a room temperature atomic system. Opt. Lett. 44, 33–36 (2019).
Vogt, T. et al. Efficient microwavetooptical conversion using rydberg atoms. Phys. Rev. A 99, 023832 (2019).
O’Brien, C., Lauk, N., Blum, S., Morigi, G. & Fleischhauer, M. Interfacing superconducting qubits and telecom photons via a rareearthdoped crystal. Phys. Rev. Lett. 113, 063603 (2014).
FernandezGonzalvo, X., Chen, Y.H., Yin, C., Rogge, S. & Longdell, J. J. Coherent frequency upconversion of microwaves to the optical telecommunications band in an er: Yso crystal. Phys. Rev. A 92, 062313 (2015).
Holzgrafe, J. et al. Cavity electrooptics in thinfilm lithium niobate for efficient microwavetooptical transduction. Optica 7, 1714–1720 (2020).
JaverzacGaly, C. et al. Onchip microwavetooptical quantum coherent converter based on a superconducting resonator coupled to an electrooptic microresonator. Phys. Rev. A 94, 053815 (2016).
Fu, W. et al. Cavity electrooptic circuit for microwavetooptical conversion in the quantum ground state. Phys. Rev. A 103, 053504 (2021).
Xu, Y. et al. Bidirectional interconversion of microwave and light with thinfilm lithium niobate. Nat. Commun. 12, 4453 (2021).
McKenna, T. P. et al. Cryogenic microwavetooptical conversion using a triply resonant lithiumniobateonsapphire transducer. Optica 7, 1737–1745 (2020).
Soltani, M. et al. Efficient quantum microwavetooptical conversion using electrooptic nanophotonic coupled resonators. Phys. Rev. A 96, 043808 (2017).
Tsang, M. Cavity quantum electrooptics. Phys. Rev. A 81, 063837 (2010).
Tsang, M. Cavity quantum electrooptics. II. Inputoutput relations between traveling optical and microwave fields. Phys. Rev. A 84, 043845 (2011).
Wang, C. et al. Integrated lithium niobate electrooptic modulators operating at cmoscompatible voltages. Nature 562, 101–104 (2018).
Fan, L. et al. Superconducting cavity electrooptics: a platform for coherent photon conversion between superconducting and photonic circuits. Sci. Adv. 4, eaar4994 (2018).
Zhang, M., Wang, C., Cheng, R., ShamsAnsari, A. & Lončar, M. Monolithic ultrahighQ lithium niobate microring resonator. Optica 4, 1536–1537 (2017).
Sahu, R. et al. Quantumenabled operation of a microwaveoptical interface. Nat. Commun. 13, 1276 (2022).
Mobassem, S. et al. Thermal noise in electrooptic devices at cryogenic temperatures. Quantum Sci. Technol. 6, 045005 (2021).
Hease, W. et al. Bidirectional electrooptic wavelength conversion in the quantum ground state. PRX Quantum 1, 020315 (2020).
Rueda, A., Hease, W., Barzanjeh, S. & Fink, J. M. Electrooptic entanglement source for microwave to telecom quantum state transfer. Npj Quantum Inf. 5, 1–11 (2019).
Rueda, A. et al. Efficient microwave to optical photon conversion: an electrooptical realization. Optica 3, 597–604 (2016).
Romanenko, A. et al. Threedimensional superconducting resonators at T < 20 mk with photon lifetimes up to τ= 2 s. Phys. Rev. Appl. 13, 034032 (2020).
Berlin, A. et al. Searches for new particles, dark matter, and gravitational waves with srf cavities. arXiv preprint arXiv:2203.12714 (2022).
Alam, M. S. et al. Quantum computing hardware for hep algorithms and sensing. arXiv preprint arXiv:2204.08605 (2022).
Romanenko, A. & Schuster, D. Understanding quality factor degradation in superconducting niobium cavities at low microwave field amplitudes. Phys. Rev. Lett. 119, 264801 (2017).
Lambert, N., Trainor, L. & Schwefel, H. An ultrastable microresonatorbased electrooptic dual frequency comb. arXiv preprint arXiv:2108.11140 (2021).
McRae, C. R. H. et al. Materials loss measurements using superconducting microwave resonators. Rev. Sci. Instrum. 91, 091101 (2020).
Goryachev, M., Kostylev, N. & Tobar, M. E. Singlephoton level study of microwave properties of lithium niobate at millikelvin temperatures. Phys. Rev. B 92, 060406 (2015).
Yang, R.Y., Su, Y.K., Weng, M.H., Hung, C.Y. & Wu, H.W. Characteristics of coplanar waveguide on lithium niobate crystals as a microwave substrate. J. Appl. Phys. 101, 014101 (2007).
Cai, M., Painter, O. & Vahala, K. J. Observation of critical coupling in a fiber taper to a silicamicrosphere whisperinggallery mode system. Phys. Rev. Lett. 85, 74 (2000).
Serniak, K. et al. Hot nonequilibrium quasiparticles in transmon qubits. Phys. Rev. Lett. 121, 157701 (2018).
Zmuidzinas, J. Superconducting microresonators: Physics and applications. Annu. Rev. Condens. Matter Phys. 3, 169–214 (2012).
Delaney, R. et al. Superconductingqubit readout via lowbackaction electrooptic transduction. Nature 606, 489–493 (2022).
Koch, J. et al. Chargeinsensitive qubit design derived from the cooper pair box. Phys. Rev. A 76, 042319 (2007).
Place, A. P. et al. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nat. Commun. 12, 1779 (2021).
Zhong, C. et al. Proposal for heralded generation and detection of entangled microwave–opticalphoton pairs. Phys. Rev. Lett. 124, 010511 (2020).
Krastanov, S. et al. Optically heralded entanglement of superconducting systems in quantum networks. Phys. Rev. Lett. 127, 040503 (2021).
Heeres, R. W. et al. Cavity state manipulation using photonnumber selective phase gates. Phys. Rev. Lett. 115, 137002 (2015).
Chakram, S. et al. Seamless highQ microwave cavities for multimode circuit quantum electrodynamics. Phys. Rev. Lett. 127, 107701 (2021).
Nazmiev, A. I., Matsko, A. B. & Vyatchanin, S. P. Back action evading electrooptical transducer. JOSA B 39, 1103–1110 (2022).
Zhong, C., Han, X. & Jiang, L. Quantum transduction with microwave and optical entanglement. arXiv preprint arXiv:2202.04601 (2022).
Lecocq, F. et al. Control and readout of a superconducting qubit using a photonic link. Nature 591, 575–579 (2021).
Acknowledgements
This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DEAC0207CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. This work is funded by Fermilab’s Laboratory Directed Research and Development (LDRD) program. This research used resources of the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under contract number DEAC0207CH11359. The NQI Research Center SQMS contributed by supporting the design of SRF cavities and access to facilities. The authors would like to thank Johannes Fink for the insightful discussions and feedback on topics related to this paper.
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C.W. carried out the numerical analysis and optical simulations, coordinated the microwaveoptical design, and drafted the manuscript. S.Z. conceived the project, coordinated the research, and helped draft and edit the manuscript. I.G. conducted microwave simulations and analysis. S.K. performed calculations and, along with V.Y., contributed to the conceptual design. A.G. and A.R. contributed to the final version of the manuscript. All authors gave their final approval for publication.
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Wang, C., Gonin, I., Grassellino, A. et al. Highefficiency microwaveoptical quantum transduction based on a cavity electrooptic superconducting system with long coherence time. npj Quantum Inf 8, 149 (2022). https://doi.org/10.1038/s41534022006647
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DOI: https://doi.org/10.1038/s41534022006647
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