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Bidirectional Electro-Optic Wavelength Conversion in the Quantum Ground State

William Hease, Alfredo Rueda, Rishabh Sahu, Matthias Wulf, Georg Arnold, Harald G.L. Schwefel, and Johannes M. Fink
PRX Quantum 1, 020315 – Published 23 November 2020
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  • INTRODUCTION
  • THEORY
  • DEVICE
  • BIDIRECTIONAL CONVERSION
  • ADDED NOISE
  • CONCLUSION AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    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±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 10.7MHz 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.

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    • Received 30 June 2020
    • Accepted 26 October 2020

    DOI:https://doi.org/10.1103/PRXQuantum.1.020315

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Heat transferHybrid quantum systemsOptical & microwave phenomenaOptoelectronicsQuantum cavitiesQuantum communicationQuantum fluctuations & noiseQuantum networksQuantum repeatersSecond order nonlinear optical processesSuperconducting quantum opticsSuperconductivityThermal propertiesUpconversion
    Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & OpticalGeneral PhysicsParticles & Fields

    Authors & Affiliations

    William Hease1,†, Alfredo Rueda1,2,†, Rishabh Sahu1, Matthias Wulf1, Georg Arnold1, Harald G.L. Schwefel2,3, and Johannes M. Fink1,*

    • 1Institute of Science and Technology Austria, Klosterneuburg, Austria
    • 2Department of Physics, University of Otago, Dunedin, New Zealand
    • 3The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, New Zealand

    • *jfink@ist.ac.at
    • †These authors contributed equally to this work.

    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.

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    Vol. 1, Iss. 2 — November - December 2020

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    • Figure 1
      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 ωp/(2π)≈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 Λ (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 FSR/(2π)=8.818 GHz (dashed line).

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    • Figure 2
      Figure 2

      Bidirectional microwave-optics conversion. (a) Measured photon conversion efficiency ηtot (light blue points) and inferred internal device efficiency ηint=ηtot/(ηeηoΛ2) (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 Pp=18.7μW and Po=267nW, as a function of the detuning between the optical signal frequency and ωo (blue points) together with theory, Eq. (6) (red line), indicating the conversion bandwidth B/(2π)=9.0 MHz. (b) Measured optical power spectrum for microwave-to-optical conversion at Pp=1.48mW. The weak coherent microwave tone Pe=1.0nW generates two optical sidebands (blue and red) with a suppression ratio of SR=10.7 dB. The center and sideband peaks are proportional to the intracavity pump np and converted optical signal no photon numbers, respectively. The noise floor is set by the resolution bandwidth. (c) Measured power spectrum for optical-to-microwave conversion at Pp=2.35μW. The weak optical input signal Po=161nW generates a single coherent microwave tone at ωo−ωp. In this particular example ne=1.2 intracavity microwave photons are generated with an incoherent noise floor PN,out corresponding to an added output conversion noise of Nout=0.4photonss−1Hz−1 in the center of the microwave cavity bandwidth.

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    • Figure 3
      Figure 3

      Conversion noise and mode population. (a)–(c) Measured microwave output noise spectrum (blue line) in units of photonss−1Hz−1 for (a) Pp=0.23μW, (b) 14.82μW, 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 Nsys and the dashed red line indicates the broadband noise offset Nwg. (d) External waveguide bath population Nwg (red), total output noise photons Nout (blue), microwave mode bath Nb (yellow), and mode occupancy Ne (green) as a function of Pp. The arrows indicate values extracted from panels (a), (b) and (c), as well as where the superconducting phase transition occurs. The error bars of Nwg and Nout at low Pp are dominated by systematic errors due to slow absolute variations of the baseline of ±0.03 quanta. The error bars of Nb represent the 95% confidence interval of the fit to Eq. (8), which also dominates the uncertainty of Ne since Nwg<Nb. The error bars at high Pp are dominated by the accuracy of the Nsys calibration. The shown error bars are the extrema of these absolute and relative uncertainties. The inset shows the region where the microwave bath occupancy Nb<1 on a linear scale. The dashed gray lines indicate fitted power laws, specifically Nwg∝Pp0.55 over the full range of powers, Nb∝Pp1.14 up to Pp≈2μW (see inset), and Nb∝Pp0.45 at higher powers. (e) Microwave output noise Nout measured on resonance over a 500 kHz resolution bandwidth in units of photonss−1Hz−1 (blue line) on top of the measurement system noise floor Nsys (dashed horizontal line) as a function of time. The system is excited with a rectangular optical pulse of Pp=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 dNout/dt≈1.1noise photons s−1.

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    • Figure 4
      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 Pp. 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 ωe (approximately FSR) using a single-sideband EO modulator with suppressed carrier (SSB SC) driven by a microwave source with local oscillator frequency ωe (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 Ω 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 T50Ω 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).

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    • Figure 5
      Figure 5

      Optical coupling. (a) Measured κo/(2π) as a function of piezo voltage Vdc (blue points) and exponential fit (red line). For large distances d∝−Vdc, we find that κo/(2π)=κin,o/(2π)=9.46 MHz. The lower dashed line indicates the lower limit κo=κin,o. The upper dashed line shows the critical coupling condition where κin,o=κex,o and κo=2κin,o with the corresponding Vdc (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 Λ2=0.38, an amount proportional to (1−Λ2)2 of the input power is reflected at critical coupling (dashed line).

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    • Figure 6
      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 ϕ=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 ϕ [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.

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    • Figure 7
      Figure 7

      Direct measurement of g. Measured optical mode splitting of S/(2π)=220 MHz at room temperature, obtained with an input power of 9.3 dBm applied to the microwave cavity at ωe. Here S≈4negrt is related to the measured single-photon coupling strength grt/(2π)=36.1 Hz via the calculated intracavity photon number ne=2.3×1012 [46].

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    • Figure 8
      Figure 8

      Microwave cavity properties. (a) Measured reflection spectra of the microwave resonance for the minimum (blue) and maximum (red) applied optical pump powers Pp together with Lorentzian fits (black lines). (b) Fitted microwave resonance frequency ωe−ωe,0 as a function of Pp with the error bars indicating the 95% confidence interval of the fit. Here ωe,0 is the fitted resonance frequency obtained for the minimum optical pump power of Pp=0.23μW. (c) Fitted microwave intrinsic loss rate κin,e as a function of Pp with the error bars indicating the 95% confidence interval of the fit. (d) The mixing chamber temperature sensor reading Tf of the dilution refrigerator as a function of the optical pump power Pp (blue points) and a power-law fit Tf∝Pp0.48 to the intermediate power range (black line).

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    • Figure 9
      Figure 9

      Electro-optic photon conversion. (a) Schematic representation of the microwave (a^e) and optical (a^o) modes with coherent populations ne and no and incoherent occupancies Ne and No (not shown). The electro-optic coupling strength G=npg originates from the Pockels effect in the lithium niobate WGM resonator. Both modes are coupled to an internal thermal bath Nb and Nb,o with the rates κin,i. They are also coupled to the respective coaxial and fiber waveguides with the rates κex,i. At finite temperature the microwave waveguide has a thermal bath occupancy Nwg and also hosts the total output noise Nout measured in the experiment. (b) The Sij 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 β1=−4.81 dB and β2=−5.5 dB (without EFDA), or the gain β2=+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 β3=−74.92 dB and β4=+67.05 dB, respectively. From two conversion measurements on resonance (Sij) and two reflection measurements out of resonance (Sii) where ηii=1, we infer the bidirectional photon conversion efficiency of the device according to Eq. (5) and referenced to the input and output photon numbers nin,i and nout,i.

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    • Figure 10
      Figure 10

      System noise calibration. Measured noise (circles) together with a fit to Eq. (D12) (line) shown in units of photons using Ndet−Nadd=PESA/(ℏωeβ4BW)−Nadd. The dashed line indicates the vacuum noise.

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    • Figure 11
      Figure 11

      Conversion bandwidth and bidirectionality. Measured total conversion efficiency ηeo (blue circles) and ηoe (green circles) as a function of output signal frequency ω (optics to microwave, blue) or ω=ωo−ωp (microwave to optics, green) for Pp=18.7μW and Pp=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.

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    • Figure 12
      Figure 12

      Calculated noise photon numbers only due to measured laser fluorescence noise at ωp+FSR. Shown are the broadband laser noise Nln, the resulting optical mode occupancy No, optical output noise Nout,o on resonance, microwave mode occupancy Ne, and the resulting output noise contribution on resonance Nout as a function of optical pump power Pp. The horizontal dashed line indicates a population of 1 and the vertical dashed line the pump power Pp=23.5μW where we measure Ne=1 due to optical absorption heating.

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