Bidirectional Electro-Optic Wavelength Conversion in the Quantum Ground State

Microwave photonics lends the advantages of fiber optics to electronic sensing and communication systems. In contrast to nonlinear optics, electro-optic devices so far require classical modulation fields whose variance is dominated by electronic or thermal noise rather than quantum fluctuations. Here we demonstrate bidirectional single-sideband conversion of $X$ band microwave to $C$ band telecom light with a microwave mode occupancy as low as $0.025\pm 0.005$ and an added output noise of less than or equal to $0.074$ photons. This is facilitated by radiative cooling and a triply resonant ultra-low-loss transducer operating at millikelvin temperatures. The high bandwidth of $10.7\mathrm{MHz}$ and total (internal) photon conversion efficiency of 0.03% (0.67%) combined with the extremely slow heating rate of 1.1 added output noise photons per second for the highest available pump power of 1.48 mW puts near-unity efficiency pulsed quantum transduction within reach. Together with the non-Gaussian resources of superconducting qubits this might provide the practical foundation to extend the range and scope of current quantum networks in analogy to electrical repeaters in classical fiber optic communication.

Popular Summary

We are witnessing rapid progress in the fields of quantum computing with superconducting circuits on the one hand, and long-distance optical quantum communication on the other. Meanwhile, there is currently no solution to interface these two domains of quantum technology in analogy to fiber optic modems in classical communication systems. Apart from close to unity efficiency and high bandwidth, such a quantum interface also needs to operate close to its quantum ground state with hardly any excess noise on either the electrical or the optical output—an important milestone that we demonstrate in this work.

We realize the electro-optic wavelength converter based on a mechanically polished crystalline lithium niobate whispering gallery mode resonator. In contrast to traditional modulators, the interaction is resonantly enhanced using a superconducting microwave cavity that matches the free spectral range and leads to an extremely efficient bidirectional conversion process. We show that this conversion works well despite the relatively high optical pump powers required. The microwave mode remains close to the quantum ground state at millikelvin temperatures where superconducting qubits operate.

The centimeter-sized device benefits from a large heat capacity and a good thermalization to the cold environment, resulting in an extremely slow observed heating rate compared to on-chip devices. Based on this, we estimate that pulsing the pump can boost the conversion efficiency by another 4 orders of magnitude without a significant increase of added noise. This would open the way for long-distance quantum networks utilizing superconducting processors for secure communication and distributed quantum computing.

Figure 1

(a) Exploded-view rendering of the electro-optic converter. The WGM resonator (light blue disc) is clamped between two aluminum rings (blue shaded areas) belonging to the top and bottom parts of the aluminum microwave cavity, respectively. Two gradient index (GRIN) lenses are used to focus the optical input and output beams (red) on a diamond prism surface in close proximity to the optical resonator. The microwave tone is coupled in and out of the cavity using a coaxial pin coupler at the top of the cavity (gold). The prism, both lenses, and the microwave tuning cylinder (gold shaded area inside the lower ring) positions can be adjusted with eight linear piezo positioners. (b) Optical reflection spectrum of the WGM resonator at base temperature (approximately $7$ mK). The optical pump mode at ${\omega }_p/(2\pi )\approx 193.5$ THz (green) and the signal mode (blue) are critically coupled and separated by one free spectral range (FSR, dashed lines). On resonance 38% of the optical power is reflected without entering the WGM resonator due to imperfect optical mode overlap $\mathrm{\Lambda }$ (horizontal dotted line). The lower sideband mode (red) is chosen to couple to a mode family of different polarization, which splits it and facilitates the single-sideband selectivity of the converter. (c) Reflection spectrum of the microwave cavity at base temperature (approximately $7$ mK) for the tuning cylinder in its up position (blue line) and in its down position (red line). With a tuning range of approximately $0.5$ GHz we can readily match the cavity frequency with that of the optical free spectral range $\mathrm{FSR}/(2\pi )=8.818$ GHz (dashed line).

Figure 2

Bidirectional microwave-optics conversion. (a) Measured photon conversion efficiency ${\eta }_{\mathrm{tot}}$ (light blue points) and inferred internal device efficiency ${\eta }_{\mathrm{int}}={\eta }_{\mathrm{tot}}/({\eta }_e{\eta }_o{\mathrm{\Lambda }}^2)$ (dark blue points) together with theory (red lines), i.e., Eq. (4) taking into account measured cavity linewidth changes. The dashed lines are linear fits for the ten lowest power data points, respectively. The arrow marks the input power where the aluminum cavity goes from super to normal conducting. The inset shows the measured and normalized coherent optics-to-microwave conversion power ratio for $P_p=18.7\mu \mathrm{W}$ and $P_o=267\mathrm{nW}$, as a function of the detuning between the optical signal frequency and ${\omega }_o$ (blue points) together with theory, Eq. (6) (red line), indicating the conversion bandwidth $B/(2\pi )=9.0$ MHz. (b) Measured optical power spectrum for microwave-to-optical conversion at $P_p=1.48\mathrm{mW}$. The weak coherent microwave tone $P_e=1.0\mathrm{nW}$ generates two optical sidebands (blue and red) with a suppression ratio of $\mathrm{SR}=10.7$ dB. The center and sideband peaks are proportional to the intracavity pump $n_p$ and converted optical signal $n_o$ photon numbers, respectively. The noise floor is set by the resolution bandwidth. (c) Measured power spectrum for optical-to-microwave conversion at $P_p=2.35\mu \mathrm{W}$. The weak optical input signal $P_o=161\mathrm{nW}$ generates a single coherent microwave tone at ${\omega }_o-{\omega }_p$. In this particular example $n_e=1.2$ intracavity microwave photons are generated with an incoherent noise floor $P_{N,\mathrm{out}}$ corresponding to an added output conversion noise of $N_{\mathrm{out}}=0.4\mathrm{photons}{\mathrm{s}}^{-1}{\mathrm{Hz}}^{-1}$ in the center of the microwave cavity bandwidth.

Figure 3

Conversion noise and mode population. (a)–(c) Measured microwave output noise spectrum (blue line) in units of $\mathrm{photons}{\mathrm{s}}^{-1}{\mathrm{Hz}}^{-1}$ for (a) $P_p=0.23\mu \mathrm{W}$, (b) $14.82\mu \mathrm{W}$, and (c) 1.48 mW together with a fit to Eq. (8) (black line). In all three panels the dashed black line indicates the measurement system noise floor $N_{\mathrm{sys}}$ and the dashed red line indicates the broadband noise offset $N_{\mathrm{wg}}$. (d) External waveguide bath population $N_{\mathrm{wg}}$ (red), total output noise photons $N_{\mathrm{out}}$ (blue), microwave mode bath $N_b$ (yellow), and mode occupancy $N_e$ (green) as a function of $P_p$. The arrows indicate values extracted from panels (a), (b) and (c), as well as where the superconducting phase transition occurs. The error bars of $N_{\mathrm{wg}}$ and $N_{\mathrm{out}}$ at low $P_p$ are dominated by systematic errors due to slow absolute variations of the baseline of $\pm 0.03$ quanta. The error bars of $N_b$ represent the 95% confidence interval of the fit to Eq. (8), which also dominates the uncertainty of $N_e$ since $N_{\mathrm{wg}}<N_b$. The error bars at high $P_p$ are dominated by the accuracy of the $N_{\mathrm{sys}}$ calibration. The shown error bars are the extrema of these absolute and relative uncertainties. The inset shows the region where the microwave bath occupancy $N_b<1$ on a linear scale. The dashed gray lines indicate fitted power laws, specifically $N_{\mathrm{wg}}\propto P_p^{0.55}$ over the full range of powers, $N_b\propto P_p^{1.14}$ up to $P_p\approx 2\mu \mathrm{W}$ (see inset), and $N_b\propto P_p^{0.45}$ at higher powers. (e) Microwave output noise $N_{\mathrm{out}}$ measured on resonance over a 500 kHz resolution bandwidth in units of $\mathrm{photons}{\mathrm{s}}^{-1}{\mathrm{Hz}}^{-1}$ (blue line) on top of the measurement system noise floor $N_{\mathrm{sys}}$ (dashed horizontal line) as a function of time. The system is excited with a rectangular optical pulse of $P_p=1.48$ mW and a length of 68 s (dashed vertical lines). The arrow indicates the time of approximately 30 s when superconductivity breaks, the cavity quality factor degrades, and the frequency detunes from the measurement frequency. This process is reversed at the end of the pulse when the cavity tunes back and the detected noise increases temporarily. The inset shows the fastest timescale, i.e., the initial heating rate of $d{N}_{\mathrm{out}}/dt\approx 1.1\mathrm{noise\; photons }{s}^{-1}$.

Figure 4

Measurement setup. A tunable laser is equally split (50:50) into two paths at the optical coupler OC1. The upper path is used as the optical pump and it goes through a variable optical attenuator VOA1 that allows us to vary $P_p$. The optical pump can then be either sent directly to the cryostat fiber, or it can go first through an electro-optic modulator (EOM) in order to create sidebands for spectroscopy calibration. The second path (horizontal) is used to generate the optical signal. It also goes through a variable optical attenuator and is then frequency up-shifted by ${\omega }_e$ (approximately FSR) using a single-sideband EO modulator with suppressed carrier (SSB SC) driven by a microwave source with local oscillator frequency ${\omega }_e$ (S1). A small fraction (1%) of this signal is picked up and sent directly to an optical spectrum analyzer (OSA) for sideband and carrier suppression ratio monitoring. The rest (99%) is recombined with the pump at OC2, sent to the fridge input fiber, and the total power is monitored with a power meter (PM). The optical tones are focused on the prism with a GRIN lens that then feeds the WGM resonator via evanescent coupling. Polarization controllers PC2 and PC3 are set to achieve maximum coupling to a TE polarized cavity mode. The reflected (or created) optical sideband signal and the reflected pump are collected with the second GRIN lens and coupled to the cryostat output fiber. The optical signal is then split: 90% of the power goes to the OSA and 10% is sent to a photodiode (PD), which is used for mode spectroscopy and to lock on the optical mode resonance during the conversion measurement. The 90% arm is either sent directly to the OSA, or goes through an erbium doped fiber amplifier (EDFA) for amplification, depending on the microwave-to-optics converted signal power. On the microwave side, the signal is sent from the microwave source S2 (or from the VNA for microwave mode spectroscopy) to the fridge input line via the microwave combiner (MC1). The input line is attenuated with attenuators distributed between 3 K and 10 mK with a total of 60 dB in order to suppress room-temperature microwave noise. Circulator C1 redirects the reflected tone from the cavity to the amplified output line, while C2 redirects noise coming in from the output line to a matched 50 $\mathrm{\Omega }$ termination. The output line is amplified at 3 K by a HEMT amplifier and then at room temperature again with a low noise amplifier (LNA). The output line is connected to switch MS1, to select between an ESA or a VNA measurement. Lastly, microwave switch MS2 allows us to swap the device under test (DUT) for a temperature $T_{\mathrm{50}\mathrm{\Omega }}$ controllable load, which serves as a broadband noise source in order to calibrate the output line’s total gain and added noise (see Appendix pp4 16).

Figure 5

Optical coupling. (a) Measured ${\kappa }_o/(2\pi )$ as a function of piezo voltage $V_{\mathrm{dc}}$ (blue points) and exponential fit (red line). For large distances $d\propto -V_{\mathrm{dc}}$, we find that ${\kappa }_o/(2\pi )={\kappa }_{\mathrm{in},o}/(2\pi )=9.46$ MHz. The lower dashed line indicates the lower limit ${\kappa }_o={\kappa }_{\mathrm{in},o}$. The upper dashed line shows the critical coupling condition where ${\kappa }_{\mathrm{in},o}={\kappa }_{\mathrm{ex},o}$ and ${\kappa }_o=2{\kappa }_{\mathrm{in},o}$ with the corresponding $V_{\mathrm{dc}}$ (vertical arrow). (b) Measured optical reflection spectrum around 1550 nm at base temperature for critical coupling [arrow in panel (a)]. Because of imperfect optical mode matching quantified by ${\mathrm{\Lambda }}^2=0.38$, an amount proportional to $(1-{\mathrm{\Lambda }}^2{)}^2$ of the input power is reflected at critical coupling (dashed line).

Figure 6

Microwave cavity design. (a) Computer-aided design drawing of the top part of the aluminum microwave cavity. (b) FEM simulation of the single-photon electric field distribution of the $m=1$ mode of the microwave cavity. Shown is the $z$ component of the field along the radial axis at $z=0$ (in the center of the WGM) and $\varphi =0$ [yellow dashed line in panel (a)]. The vertical dashed line marks the position of the edge of the WGM resonator. (c) FEM simulation of the $z$ component of the single-photon electric field taken at the position of the optical mode maximum and plotted as a function of the azimuthal angle $\varphi$ [yellow dashed circle in panel (a)]. The red line is a sinusoidal function as a guide to the eye. (d) FEM simulation of the $z$ component of the single-photon electric field in the $z=0$ plane for the $m=1$ microwave resonance.

Figure 7

Direct measurement of $g$. Measured optical mode splitting of $S/(2\pi )=220$ MHz at room temperature, obtained with an input power of 9.3 dBm applied to the microwave cavity at ${\omega }_e$. Here $S\approx 4\sqrt{n_e}{g}_{\mathrm{rt}}$ is related to the measured single-photon coupling strength $g_{\mathrm{rt}}/(2\pi )=36.1$ Hz via the calculated intracavity photon number $n_e=2.3\times {10}^{12}$ [46].

Figure 8

Microwave cavity properties. (a) Measured reflection spectra of the microwave resonance for the minimum (blue) and maximum (red) applied optical pump powers $P_p$ together with Lorentzian fits (black lines). (b) Fitted microwave resonance frequency ${\omega }_e-{\omega }_{e,0}$ as a function of $P_p$ with the error bars indicating the 95% confidence interval of the fit. Here ${\omega }_{e,0}$ is the fitted resonance frequency obtained for the minimum optical pump power of $P_p=0.23\mu \mathrm{W}$. (c) Fitted microwave intrinsic loss rate ${\kappa }_{\mathrm{in},e}$ as a function of $P_p$ with the error bars indicating the 95% confidence interval of the fit. (d) The mixing chamber temperature sensor reading $T_f$ of the dilution refrigerator as a function of the optical pump power $P_p$ (blue points) and a power-law fit $T_f\propto P_p^{0.48}$ to the intermediate power range (black line).

Figure 9

Electro-optic photon conversion. (a) Schematic representation of the microwave $({\hat{a}}_e)$ and optical $({\hat{a}}_o)$ modes with coherent populations $n_e$ and $n_o$ and incoherent occupancies $N_e$ and $N_o$ (not shown). The electro-optic coupling strength $G=\sqrt{n_p}g$ originates from the Pockels effect in the lithium niobate WGM resonator. Both modes are coupled to an internal thermal bath $N_b$ and $N_{b,o}$ with the rates ${\kappa }_{\mathrm{in},i}$. They are also coupled to the respective coaxial and fiber waveguides with the rates ${\kappa }_{\mathrm{ex},i}$. At finite temperature the microwave waveguide has a thermal bath occupancy $N_{\mathrm{wg}}$ and also hosts the total output noise $N_{\mathrm{out}}$ measured in the experiment. (b) The $S_{ij}$ coefficients are defined between the microwave and optical input and output ports outside the cryostat (gray circles). Attenuation for the optical input and output paths are ${\beta }_1=-4.81$ dB and ${\beta }_2=-5.5$ dB (without EFDA), or the gain ${\beta }_2=+30.8$ dB (with EFDA). An observed gain saturation at high pump powers is measured and corrected. Attenuation and gain of the microwave input and output paths are ${\beta }_3=-74.92$ dB and ${\beta }_4=+67.05$ dB, respectively. From two conversion measurements on resonance ($S_{ij}$) and two reflection measurements out of resonance ($S_{ii}$) where ${\eta }_{ii}=1$, we infer the bidirectional photon conversion efficiency of the device according to Eq. (5) and referenced to the input and output photon numbers $n_{\mathrm{in},i}$ and $n_{\mathrm{out},i}$.

Figure 10

System noise calibration. Measured noise (circles) together with a fit to Eq. (D12) (line) shown in units of photons using $N_{\det }-N_{\mathrm{add}}=P_{\mathrm{ESA}}/(\hslash {\omega }_e{\beta }_4\mathrm{BW})-N_{\mathrm{add}}$. The dashed line indicates the vacuum noise.

Figure 11

Conversion bandwidth and bidirectionality. Measured total conversion efficiency ${\eta }_{eo}$ (blue circles) and ${\eta }_{oe}$ (green circles) as a function of output signal frequency $\omega$ (optics to microwave, blue) or $\omega ={\omega }_o-{\omega }_p$ (microwave to optics, green) for $P_p=18.7\mu \mathrm{W}$ and $P_p$=1.48 mW. Microwave-to-optics conversion (green) is displaced by 10 MHz for better visibility. Dashed lines indicate the maxima of the theory curves in agreement with the results reported in the main text.

Figure 12

Calculated noise photon numbers only due to measured laser fluorescence noise at ${\omega }_p+\mathrm{FSR}$. Shown are the broadband laser noise $N_{\ln }$, the resulting optical mode occupancy $N_o$, optical output noise $N_{\mathrm{out},o}$ on resonance, microwave mode occupancy $N_e$, and the resulting output noise contribution on resonance $N_{\mathrm{out}}$ as a function of optical pump power $P_p$. The horizontal dashed line indicates a population of 1 and the vertical dashed line the pump power $P_p=23.5\mu \mathrm{W}$ where we measure $N_e=1$ due to optical absorption heating.