Abstract
Practical quantum networks require lowloss and noiseresilient optical interconnects as well as nonGaussian 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 femtometerscale motion of two silicon nanobeams reaching a V_{π} as low as 16 μV for subnanowatt 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 noisefree operation on this qubitcompatible platform.
Introduction
Large 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 longdistance 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, electrooptic, magnetooptic, 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 piezoelectricity, 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 electrooptic 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 silicononinsulator 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 electrooptomechanical design is optimized for very strong field confinements and radiation pressure couplings, which enable internal efficiencies exceeding unity for ultralow 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 noisefree conversion in the future. Such a powerefficient, ultrasensitive, 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^{8,21,22,23}. In the interaction frame, the Hamiltonian describing the conversion process is (see Supplementary Note 1):
where \({\hat{a}}_{\mathrm{j}}\), \((\hat{b})\) with j = e, o is the annihilator operator of the electromagnetic (mechanical) mode, and \({\hat{H}}_{{\rm{CR}},\mathrm{j}}=\hslash {G}_{\mathrm{j}}({\hat{a}}_{\mathrm{j}}\hat{b}\ {e}^{2i{\omega }_{{\rm{m}}}t}+\,\text{h.c.})\) describes the counterrotating terms which are responsible for the coherent amplification of the signal. \({G}_{\mathrm{j}}=\sqrt{{n}_{{\rm{d}},\mathrm{j}}}{g}_{0,\mathrm{j}}\) is the parametrically enhanced electro or optomechanical coupling rate where n_{d,j} is the intracavity photon number due to the corresponding microwave and optical pump tones. For a reddetuned drive in the resolvedsideband regime 4ω_{m} > κ_{j} we neglect \({\hat{H}}_{{\rm{CR}},\mathrm{j}}\) under the rotatingwave approximation and the Hamiltonian (1) represents a beamsplitter like interaction in which the mechanical resonator mediates noiseless photon conversion between microwave and optical modes. Note that nearunity photon conversion \({\zeta }_{{\rm{RS}}}=4{\eta }_{{\rm{e}}}{\eta }_{{\rm{o}}}{{\mathcal{C}}}_{{\rm{e}}}{{\mathcal{C}}}_{{\rm{o}}}/{(1+{{\mathcal{C}}}_{{\rm{e}}}+{{\mathcal{C}}}_{{\rm{o}}})}^{2}\) can be achieved in the limit of \({{\mathcal{C}}}_{{\rm{e}}}={{\mathcal{C}}}_{{\rm{o}}}\gg 1\) with \({{\mathcal{C}}}_{\mathrm{j}}=4{G}_{\mathrm{j}}^{2}/({\kappa }_{\mathrm{j}}{\gamma }_{{\rm{m}}})\) being the electro or optomechanical cooperativity, as demonstrated between two optical^{24} and two microwave modes^{25,26}, respectively.
Transducer design
We realize conversion by connecting an optomechanical photonic crystal zipper cavity^{27} with two aluminum coated and mechanically compliant silicon nanostrings^{28} as shown in Fig. 1c. The mechanical coupling between these two components is carefully designed (see Supplementary Note 2), leading to a hybridization of their inplane 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 finiteelement 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 ultralow 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 multistep recipe including electron beam lithography, silicon etching, aluminum thinfilm deposition, and hydrofluoric vapor acid etching, as described in detail in ref. ^{30}.
Transducer characterization
Standard sample characterization (see Supplementary Notes 3 and 4) reveals an optical resonance frequency of ω_{o}/(2π) = 198.081 THz with total loss rate κ_{o}/(2π) = 1.6 GHz and waveguide coupling rate κ_{ex,o}/(2π) = 0.18 GHz leading to a coupling efficiency of η_{o} = 0.11. When the optical light is turned off, the microwave resonance frequency is ω_{e}/(2π) = 10.5 GHz with coupling efficiency η_{e} = 0.4 and κ_{ex,e}/(2π) = 1.15 MHz. The mechanical resonator frequency has a value of ω_{m}/(2π) = 11.843 MHz with an intrinsic decoherence rate γ_{m}/(2π) = 15 Hz at a mode temperature of 150 mK. The achieved singlephotonphonon coupling rates are as high as g_{0,e}/(2π) = 67 Hz and g_{0,o}/(2π) = 0.66 MHz.
Conversion measurements
To perform coherent photon conversion, reddetuned 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} \(\approx\) 9 × 10^{5} and n_{d,o} \(\approx\) 0.2 with cooperativities \({{\mathcal{C}}}_{{\rm{e}}}\approx 0.57\) and \({{\mathcal{C}}}_{{\rm{o}}}\approx 0.9\), the measured total (waveguide to waveguide) photon transduction efficiency is \(\approx\)1.1% corresponding to 96.7% internal (resonator to resonator) photon transduction efficiency over the total bandwidth of Γ_{conv}/(2π) \(\approx\) 0.37 kHz. In the case of κ_{o} > 4ω_{m} and κ_{e} < 4ω_{m}, the bandwidth is given by \({\Gamma }_{{\rm{conv}}}\approx ({{\mathcal{C}}}_{{\rm{e}}}+1){\gamma }_{{\rm{m}}}\) because the nonsideband resolved optomechanical cavity does not induce mechanical broadening. The signal tone adds 17(10^{−3}) photons to the microwave resonator (optical cavity).
Here we use a selfcalibrated 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} \(\approx\) 30 reduces the reported mean bidirectional efficiencies by \(\sqrt{2}\) compared to the actually achieved total transduction efficiency between microwave and optical fields. The observed reflection peaks indicate that both resonators are undercoupled, equivalent to an impedance mismatch for incoming signal light. All scattering parameters are obtained from measured coherent tones whose linewidths are given by the chosen resolution bandwidth and the stability of the heterodyne setup. While this does not explicitly show long term phase stability of the conversion we find that these results are in excellent agreement with our coherent conversion theory model (solid lines) with γ_{m} as the only free fit parameter.
Figure 2b shows the total transduction efficiency for different pump power combinations with microwave and optical pump powers ranging from 30 to 953 pW and 48 to 1561 pW, respectively. Figure 2c, d shows the efficiency versus P_{o} (P_{e}) for fixed microwave (optical) pump power P_{e} = 601 (P_{o} = 625) pW. As expected, the transduction efficiency rises with increasing pump powers and reaches a maximum of ζ = 1.2%. The internal transduction efficiency is significantly higher (ζ/(η_{o}η_{e}) ≤ 135%) because both the microwave resonator as well as the optical cavity are highly undercoupled with coupling ratios of η_{o} = 0.11 and η_{e} ranging between 0.07 and 0.18 when both pumps are on. The increase in the intrinsic loss rate of microwave κ_{in,e} and mechanical resonator γ_{m} at higher pump powers are shown in Fig. 2e and f caused by considerable heating related to (especially optical) photon absorption. This results in the degradation of the microwave and mechanical quality factors and consequently reduces the waveguide coupling efficiency, the cooperativities and the total transduction efficiency (see Supplementary Note 5).
Sideband resolution and amplification
In the nonsideband resolved limit the contribution of the counterrotating term of the Hamiltonian \({\hat{H}}_{{\rm{CR}},{\rm{o}}}\) is nonnegligible, resulting in a transduction process that cannot be fully noise free. This interesting effect can be correctly described by introducing an amplification of the signal tone with (in the absence of thermal noise) quantum limited gain \({{\mathcal{G}}}_{{\rm{o}}}\) (see Supplementary Note 1). In contrast, the microwave resonator is in the resolvedsideband condition 4ω_{m} > κ_{e}, so that the signal tone amplification due to electromechanical interaction is negligible \({{\mathcal{G}}}_{{\rm{e}}}\simeq 1\). This results in the total, power independent, bidirectional conversion gain of \({\mathcal{G}}={{\mathcal{G}}}_{{\rm{e}}}{{\mathcal{G}}}_{{\rm{o}}}\simeq {{\mathcal{G}}}_{{\rm{o}}}\), which turns out to be directly related to the minimum reachable phonon occupation:
induced by optomechanical quantum backaction when the mechanical resonator is decoupled from its thermal bath^{32}. Due to this amplification process the measured transduction efficiency in Fig. 2a is about 110 times larger than one would expect from a model that does not include gain effects for the chosen detuning, and adds the equivalent of at least one half of a vacuum noise photon to the input of the transducer in our case of heterodyne detection (for η_{j} = 1 and \({\mathcal{G}}\gg 1\)). However, it turns out that this noise limitation, which might in principle be overcome with efficient feedforward^{9}, sideband suppression^{33,34}, or sideband resolution^{35}, accounts for only about 0.1% of the total conversion noise observed in our system. The total transduction (including gain) can be written in terms of the susceptibilities of the electromagnetic modes \({\chi }_{\mathrm{j}}^{1}(\omega )=i({\Delta }_{\mathrm{j}}\omega )+{\kappa }_{\mathrm{j}}/2\) and the mechanical resonator \({\chi }_{{\rm{m}}}^{1}(\omega )=i({\omega }_{{\rm{m}}}\omega )+{\gamma }_{{\rm{m}}}/2\) as:
where \(\tilde{{\chi }_{\mathrm{j}}}(\omega )={\chi }_{\mathrm{j}}{(\omega )}^{* }\).
Equation (3) can be decomposed into a product of the conversion gain \({\mathcal{G}}\) and the pure conversion efficiency θ, i.e., \(\zeta :={\mathcal{G}}\times \theta\), for frequencies in the vicinity of ω_{m} (see Supplementary Note 1). Equation (2) shows that the signal amplification depends only on the resonator linewidth and the detuning and is not directly related to the \(\propto {\hat{a}}^{\dagger }{\hat{b}}^{\dagger }\) interaction term or the pump power^{31}. This can be understood by the alternative interpretation that the gain represents the ratio of the transduced upper sideband to the difference between upper and lower sideband at each cavity. Therefore, it is instructive to measure the transducer parameters as a function of optical pump detuning as shown in Fig. 3a. While changing the optical detuning, we also vary the pump power in order to keep the optical intracavity photon number constant at n_{d,o} = 0.185 ± 0.015. This way it is possible to investigate the influence of Δ_{o} at a constant optomechanical coupling \({G}_{{\rm{o}}}={g}_{{\rm{0,o}}}\sqrt{{n}_{{\rm{d}},{\rm{o}}}}\). The measured total transduction efficiency is shown in Fig. 3a and reaches ζ\(\,\approx\,\)1% at Δ_{o} \(\approx\) 0 for the chosen pump powers in agreement with Fig. 2c, d. We can now separate the measured transduction (Eq. (3)) into conversion gain and pure conversion, as shown in Fig. 3b. The gain shows the expected steep increase at Δ_{o} → 0 where the pure conversion θ approaches zero for equal cooling and amplification rates. Around Δ_{o} = κ_{o}/2 on the other hand, where \({\langle n\rangle }_{\min }\) reaches its minimum of roughly κ_{o}/4ω_{m} \(\approx\) 30, also the gain reaches its minimum and the noiseless part (at zero temperature) of the total (internal) conversion process shows its highest efficiency of θ = 0.019% (θ/(η_{e}η_{o}) = 1.6%).
Added noise
Another important figure of merit, not only for quantum applications, is the amount of added noise quanta^{36}, usually an effective number referenced to the input of the device. For clarity with regards to the physical origin and the actual measurement of the noise power, in the following we define the total amount of added noise quanta n_{add,j} added to the input signal S_{in,j} after the transduction process as S_{out,j} = 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 \({\bar{n}}_{{\rm{m}}}\) as the only fit parameter (see Supplementary Note 4).
The fitted effective mechanical bath temperature as a function of pump powers is shown in Fig. 4c. It reveals the strong optical pump dependent mechanical mode heating (blue), while the microwave pump (red) has a negligible influence on the mechanical bath. Fig. 4d shows the measured total added noise at the output of the microwave resonator and optical cavity as a function of optical pump power. The noise added to the optical output (blue) increases with pump power due to absorption heating and increasing optomechanical coupling rate G_{o}, while the degradation of the resonatorwaveguide coupling efficiency η_{e} explains the decreasing n_{add,e} at higher optical powers for the microwave output noise (red), see Fig. 2e. The intersection of the two noise curves occurs at \({{\mathcal{C}}}_{{\rm{e}}}\simeq {{\mathcal{C}}}_{{\rm{o}}}\) with cooperativities C_{j} as defined above, and shows that the optical and microwave resonators share the same mechanical thermal bath. The power dependence is in full agreement with theory (solid lines) and demonstrates that the thermal mechanical population is the dominating origin of the added transducer noise.
Discussion
In conclusion, we demonstrated an efficient bidirectional and chipscale microwavetooptics transducer using pump powers orders of magnitude lower than comparable allintegrated^{11,13,15} approaches. Low pump powers are desired to limit the heat load of the cryostat and to minimize onchip heating, which is 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 pumpprobe type experiments together with high efficiency heralding measurements can be used for postselecting rare successful conversion or entanglement generation events for lownoise lowefficiency devices^{12,38}.
In terms of nearterm 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 wideband unidirectional electrooptic 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.
Change history
01 October 2020
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
09 October 2020
The original version of this Article was updated shortly after publication following an error that resulted in the ORCID ID of A. Rueda being incorrectly assigned to F Hassani.
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Acknowledgements
We thank Yuan Chen for performing supplementary FEM simulations and Andrew Higginbotham, Ralf Riedinger, Sungkun Hong, and Lorenzo Magrini for valuable discussions. This work was supported by IST Austria, the IST nanofabrication facility (NFF), the European Union’s Horizon 2020 research and innovation program under grant agreement no. 732894 (FET Proactive HOT) and the European Research Council under grant agreement no. 758053 (ERC StG QUNNECT). G.A. is the recipient of a DOC fellowship of the Austrian Academy of Sciences at IST Austria. W.H. is the recipient of an ISTplus postdoctoral fellowship with funding from the European Union’s Horizon 2020 research and innovation program under the Marie SklodowskaCurie grant agreement no. 754411. J.M.F. acknowledges support from the Austrian Science Fund (FWF) through BeyondC (F71), a NOMIS foundation research grant, and the EU’s Horizon 2020 research and innovation program under grant agreement no. 862644 (FET Open QUARTET).
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G.A., M.W., and S.B. performed and analyzed the measurements. S.B. and G.A. contributed to the theoretical model. G.A. designed the transducer device. M.W., G.A., A.R., and W.H. built the experimental setup. M.W., E.R., and G.A. contributed to sample fabrication. F.H. tapered optical fibers used for optomechanical characterization tests. G.A., M.W., S.B., and J.M.F. wrote the paper. J.M.F. supervised the research.
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Arnold, G., Wulf, M., Barzanjeh, S. et al. Converting microwave and telecom photons with a silicon photonic nanomechanical interface. Nat Commun 11, 4460 (2020). https://doi.org/10.1038/s4146702018269z
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