Efficient single sideband microwave to optical conversion using an electro-opticalwhispering gallery mode resonator

Alfredo Rueda1,2,3,+, Florian Sedlmeir1,2,3,+,∗, Michele C. Collodo1,2,4,5, Ulrich Vogl1,2,

Birgit Stiller1,2,6, Gerhard Schunk1,2,3, Dmitry V. Strekalov1, Christoph Marquardt1,2,

Johannes M. Fink4,7, Oskar Painter4, Gerd Leuchs1,2, and Harald G. L. Schwefel8,∗

1Max Planck Institute for the Science of Light, Gunther-Scharowsky-Straße 1/Building 24, 90158 Erlangen, Germany2Institute for Optics, Information and Photonics, University Erlangen-Nurnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany

3SAOT, School in Advanced Optical Technologies, Paul-Gordan-Str. 6, 91052 Erlangen, Germany4Institute for Quantum Information and Matter and Thomas J. Watson, Sr., Laboratoryof Applied Physics, California Institute of Technology, Pasadena, California 91125, USA

5currently at: Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland6currently at: Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS),

School of Physics, University of Sydney, New South Wales 2006, Australia7currently at: Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria

8Department of Physics, University of Otago, Dunedin, New Zealand∗Corresponding authors: Florian.Sedlmeir@mpl.mpg.de, Harald.Schwefel@otago.ac.nz and

+these authors contributed equally to this work

Linking classical microwave electrical circuits to the optical telecommunication band is at the coreof modern communication. Future quantum information networks will require coherent microwave-to-optical conversion to link electronic quantum processors and memories via low-loss opticaltelecommunication networks. Efficient conversion can be achieved with electro-optical modulatorsoperating at the single microwave photon level. In the standard electro-optic modulation schemethis is impossible because both, up- and downconverted, sidebands are necessarily present. Herewe demonstrate true single sideband up- or downconversion in a triply resonant whispering gallerymode resonator by explicitly addressing modes with asymmetric free spectral range. Compared toprevious experiments, we show a three orders of magnitude improvement of the electro-optical con-version efficiency reaching 0.1% photon number conversion for a 10 GHz microwave tone at 0.42 mWof optical pump power. The presented scheme is fully compatible with existing superconducting 3Dcircuit quantum electrodynamics technology and can be used for non-classical state conversion andcommunication. Our conversion bandwidth is larger than 1 MHz and not fundamentally limited.

INTRODUCTION

Efficient conversion of signals between the microwaveand the optical domain is a key feature required in clas-sical and quantum communication networks. In classicalcommunication technology, the information is processedelectronically at microwave frequencies of several giga-hertz. Due to high ohmic losses at such frequencies, thedistribution of the computational output over long dis-tances is usually performed at optical frequencies in glassfibers. The emerging quantum information technologyhas developed similar requirements with additional lay-ers of complexity, concerned with preventing loss, deco-herence, and dephasing. Superconducting qubits oper-ating at gigahertz frequencies are promising candidatesfor scalable quantum processors [1–3], while the opticaldomain offers access to a large set of very well developedquantum optical tools, such as highly efficient single-photon detectors and long-lived quantum memories [4].Optical communication channels allow for low transmis-sion losses and are, in contrast to electronic ones, notthermally occupied at room temperature due to the highphoton energies compared to kBT .To be useful for quantum information, a noiseless conver-

sion channel with unity conversion probability has to bedeveloped. For the required coupling between the vastlydifferent wavelengths, different experimental platformshave been proposed, e.g. cold atoms [5, 6], spin ensemblescoupled to superconducting circuits [7, 8], and trappedions [9, 10]. The highest conversion efficiency so farwas reached via electro-optomechanical coupling, wherea high-quality mechanical membrane provides the linkbetween an electronic LC circuit and laser light [11, 12].Nearly 10% photon number conversion from 7 GHz mi-crowaves into the optical domain was demonstrated in acryogenic environment [12]. Electro-optic modulation asa different approach has been recently discussed by Tsang[13, 14]. In this scheme the refractive index of a trans-parent dielectric is modulated by a microwave field. Theresulting phase modulation creates sidebands symmetri-cally around the optical pump frequency. In the photonpicture this can be understood as coherent superpositionof two processes. The process of sum-frequency gener-ation (SFG) combines a microwave and an optical pho-ton and creates a blue-shifted photon (anti-Stokes). Thisprocess is fundamentally noiseless and hence does notlead to decoherence if the photon conversion efficiencyapproaches unity.

arX

iv:1

601.

0726

1v1

[qu

ant-

ph]

27

Jan

2016

2

In the process of difference frequency generation (DFG)the microwave signal photon can stimulate an opticalpump photon to decay into a red-shifted optical (Stokes)and an additional microwave photon. Such microwaveparametric amplification adds a minimum amount ofnoise as the process can also occur spontaneously (spon-taneous parametric down-conversion) without the pres-ence of a microwave signal [15]. This spontaneous processcan be used to generate entangled pairs of microwave andoptical photons [14]. Above threshold the spontaneousprocess can stimulate parametric oscillations generatingcoherent microwave radiation [16].Significant improvements of electro-optic modulationwere made in the recent years by developing resonantmodulators with high quality (Q) factors consisting ofa lithium niobate whispering gallery mode resonator(WGM) coupled to a microwave resonator [17, 18]. Thebest reported photon conversion efficiency so far wasof the order of 0.0001% per mW optical pump power[18, 19]. These implementations could not individuallyaddress either SFG or DFG and therefore always includeadditional noise from the spontaneous process.Electro-optic single-sideband conversion has been demon-strated in a lithium tantalate WGM resonator, where theoptical pump and the optical signal were orthogonally po-larized (type-I conversion) [20]. The highest efficiency inlithium niobate can be expected when signal and pumpare parallelly polarized (type-0 conversion). It was dis-cussed that single sideband conversion in this case canbe achieved by detuning the optical pump from its res-onance [13] (see Fig. 1(a)), which increases the requiredpump power significantly.We present a new take on the classical electro-opticalmodulation within a lithium niobate whispering gallerymode resonator, where single sideband operation is madepossible without detuning. The demonstrated photonnumber conversion efficiency of 0.2% per milliwatt opti-cal pump power is three orders of magnitude larger thanin previous works [18, 19]. The significant increase ofthe conversion efficiency is due to better optical and mi-crowave Q factors and enhanced model overlap. Embed-ding the WGM resonator within a closed three dimen-sional microwave cavity allows to strongly focus the mi-crowave field into the optical mode volume. Furthermore,our system is fully compatible with superconducting cir-cuit QED [21, 22].We are able to switch between sum- and difference fre-quency generation by employing avoided crossings of theoptical modes, which lead to an asymmetric spectral dis-tance between three neighboring resonances. Since themode crossings are temperature dependent, they can beused to tune the microwave frequency of the converterover several tens of MHz.Current state of the art electro-optomechanical conver-sion schemes have a bandwidth of several kHz [11, 12]and their bandwidth is fundamentally limited to the me-

∆− 1/2 ∆−

∆−

a) symmetric FSR - detuning of pump

ω+

microwavegeneration

microwaveannihilation

ω

FSR

ωp ωFSR−

= ω+ω−FSR+

FSR

ω

Ω Ω

ω−

microwaveannihilation

microwavegeneration

Ω Ω

b) asymmetric FSR - pump on resonance

ωp

FIG. 1. Two schemes for suppressing the DFG signal in type-0conversion are shown. In the case of equally spaced modes (a),the optical pump has to be detuned to cause the red sidebandto be off resonant. If the mode spacing is asymmetric (b),the pump can be kept fully resonant while maintaining thesuppression of the undesired sideband.

chanical resonance frequency. The electro-optical schemedoes not have this fundamental limitation. In the currentimplementation we already reach a bandwidth of 1 MHz.

ALL-RESONANT ELECTRO-OPTICS

Our conversion scheme is based on the nonlinear inter-action between resonantly enhanced microwave and opti-cal fields within a high quality WGM resonator. The op-tical resonator is made out of lithium niobate and placedwithin a three dimensional copper cavity. While the op-tical mode is guided at the rim of the WGM resonator bytotal internal reflection, the microwave field is confinedby the metallic boundary of the copper cavity which isdesigned to enforce good overlap of the optical and themicrowave modes within the lithium niobate. As lithiumniobate is an electro-optic material, the microwave fieldmodulates the refractive index and couples parametri-cally to the optical modes. Assuming weak microwavefields, maximally two sidebands, the up- and the down-converted one, will be generated next to the optical pumpfrequency. The interaction Hamiltonian describing thesystem is

Hint = ~g(ab†−c† + a†b−c︸ ︷︷ ︸DFG

+ ab†+c+ a†b+c†︸ ︷︷ ︸

SFG

), (1)

3

a) open copper cavity

z component of electrical eld (mV/m)

WGM

metallic tuning screw

b) numerical simulation (side view)

c) schematic of the setup d) optical pump mode in reection

e) microwave spectrum

top view

side view

coaxial pin coupler

pump reected pumpsideband signal

optical FSR

-30 -20 -10 0 10 20 300.0

0.2

0.4

0.6

0.8

1.0

8.7 8.8 8.9 9.0 9.1 9.2

-7-6-5-4-3-2-10

T

R

1.37 MHzQ ~ 1.4 x 108

screw out screw in

11 mm

13 mm

0.4 mm2 mm

170 MHzin

outabsolute frequency (GHz)

ree

cted

pow

er (d

B)no

rmal

ized

ree

ctio

n

frequency detuning (MHz)

4.8 mm

FIG. 2. a) A photograph of the bottom part of the microwave cavity with silicon coupling prism and the WGM resonator. b)Simulation of the microwave field distribution in the cavity. Only the z-component (TE) of the field is shown. c) Schematicof the cavity. An optical pump beam (red dashed) couples through a prism into the WGM and part of the light is directlyreflected. The sideband is given in blue. In the side view, the metallic tuning screw to perturb the microwave mode and thecoaxial pin coupler are shown. The pin coupler is defined as the input port of the converter, the output port is the opticaloutcoupling spot inside of the prism (solid lines). d) Reflection spectrum of the optical mode used for sum-frequency conversion(compare Fig. 5). Also shown are the linewidth and the corresponding Q factor. e) Microwave spectrum in reflection from thecoaxial pin coupler. The mΩ = ±1 mode is shown for tuning screw position inside (red) and outside (black) the cavity.

where the operators a, b+, b− denote the optical pump,the up- and downconverted optical sidebands, respec-tively, and c the microwave mode. The coupling rateg± is given by (compare e.g. [18])

g± =npn±nΩ

r

√~ωpω±Ω0

8ε0VpV±VΩ×∫

dV ΨpΨΩΨ±, (2)

where np, n±, and nΩ are the refractive indices of pump,sideband, and microwave tone and ωp, ω±, and Ω0 are therespective resonance frequencies. The normalization vol-umes Vp, V±, and VΩ are given by the integral

∫dVΨΨ∗

over the respective spatial field distributions Ψp, Ψ±, andΨΩ. The nonlinear coupling is mediated by the electro-optic coefficient r. The integral in (2) is non-zero onlyif phase-matching is fulfilled. In the case of whisperinggallery modes this means that the azimuthal mode num-bers m (number of wavelengths around the rim of theresonator) have to obey the relation m± = mp ±mΩ. Inour experiment, the optical pump Pp and the microwavesignal PΩ are both undepleted (i.e. the nonlinear con-version efficiency depends linearly on the pump and sig-nal intensities). From (1) we find the output power ofthe sidebands P± and the photon-number conversion ef-ficiency η± to be:

P± =8g2

~Ω

γ2γΩ

|Γp|2|Γ±|2|ΓΩ|2︸ ︷︷ ︸ζ±

PpPΩ and η± =Ω

ω±ζ±Pp,

(3)

where

Γp =− i(ω − ωp) + γ + γ′ (4a)

ΓΩ =− i(Ω− Ω0) + γΩ + γ′Ω. (4b)

Γ± =− i(ω ± Ω− ω±) + γ + γ′ (4c)

Expressions (4) describe the detuning of the optical pumpω, the microwave Ω and the generated sidebands ω ± Ωfrom their respective resonance frequencies. Since theoptical modes have the same polarization and are closein frequency, we can assume the same coupling and lossrates γ and γ′ for both of them. Coupling and loss ratefor the microwave mode are denoted as γΩ and γ′Ω.If the optical modes are spectrally equidistant, the sup-pression of one of the sidebands can be achieved by detun-ing the pump from its resonance frequency as schemati-cally depicted in Fig. 1(a). However, detuning decreasesthe coupling of the pump to the cavity and therefore in-creases the required pump power significantly (see be-low). We propose an alternative scheme depicted inFig. 1(b): the spectral distance, or free spectral range(FSR), can be asymmetric for the up- and downconvertedmodes. This would allow detuning of, for example thedownconverted light, by ∆− = ω − Ω − ω−, while thepump and the up-converted light remain fully resonant.We will show, that such dispersion engineering is pos-sible by exploiting avoided crossings. Both schemes inFig. 1 lead to a power suppression factor S between the

4

sidebands of

S =P+

P−=

∆2−

(γ + γ′)2+ 1. (5)

As the suppression depends only on the detuning ofthe downconverted sideband from its resonance, the twoschemes can be compared in terms of the required pumppower using (3). One can show with the help of Eqs. (3)and (5) that the detuning scheme requires (S − 1)/4 + 1times more pump power to produce the same suppressionS. For example, 30 dB suppression would require about250 times more pump power.

EXPERIMENTAL REALIZATION

The challenge in this experiment is to bring two res-onant systems having largely different frequencies to in-teract. The design has to provide a good spatial overlapbetween microwave and optical modes while the involvedmodes have to fulfill phase-matching and energy conser-vation. This requires careful tailoring of the system basedon numerical simulations.For the experiment, we fabricated a lithium niobateWGM resonator with a radius of R = 2.4 mm and athickness d = 0.4 mm by single point diamond turningand subsequent polishing [23]. The resonator has a z-cut configuration where the optic axis is aligned parallelto the symmetry axis. The disk is placed in a 3D mi-crowave copper cavity as depicted in Fig. 2(a-c) whereit is clamped by two metallic rings that are designedto maximize the field overlap between the microwaveand the optical modes. A silicon prism is placed withinthe cavity touching the WGM resonator for optical cou-pling. To enable critical coupling, the coupling plane ofthe prism is coated with a thermally grown silicon ox-ide layer serving as a spacer. As optical pump we usea narrow-band tunable laser (λ ≈ 1550 nm) which caneither be swept over or locked to a resonance via thePound-Drever-Hall technique. Two holes in the coppercavity allow the optical pump light to enter and the lightreflected and emitted from the WGM resonator to leavethe cavity. This light is collected and analyzed by a pho-todiode and an optical spectrum analyzer (OSA). Themicrowave signal is coupled via a coaxial pin couplermounted close to the WGM resonator into the cavityand analyzed with a vector network analyzer (VNA) inreflection. A metallic tuning screw is used to perturbthe microwave field for fine adjustment of its resonancefrequency. The whole setup is thermally stabilized atroom temperature with mK precision by a proportional-integral-derivative (PID) controller and a thermoelectricelement attached on the outer side of the closed coppercavity.All fields are primarily polarized along the optic axisof the WGM resonator (TE type modes) as we aim

for type-0 conversion. This allows us to address thelargest electro-optic tensor element of lithium niobate(r33 ≈ 31 pm/V [24]).Fig. 2(d) shows the tuning over a typical optical mode de-tected in reflection. We find coupling efficiencies of about60%. Although the modes are critically coupled (γ = γ′),40% of the light is reflected from the cavity due to im-perfect spatial mode matching. Typically, the loaded Qis larger than 1 × 108 corresponding to linewidths lessthan 2 MHz. The optical free spectral ranges, which weremeasured with sideband spectroscopy [25], are around8.95 GHz.With the VNA the phase and amplitude of the mi-crowave signal back-reflected from the cavity was mea-sured. Fig. 2(e) shows that the tuning screw allows tochange the microwave resonance frequency from Ω0 =2π×8.90 GHz to 2π×9.07 GHz. We find a loadedQ = 246for the undisturbed mode, which decreases toQ = 174 formaximum perturbation by the tuning screw. Measure-ments of the phase response indicate that the microwavemode is undercoupled (γΩ < γ′Ω) resulting in a non-idealcoupling efficiency around 60 − 75%, depending on theposition of the tuning screw.Note, that the pin coupler in our current setup excitesa propagating (mΩ = 1) and a counter propagating(mΩ = −1) microwave mode and only the part prop-agating in the direction of the excited optical modes isconverted. Considering these geometries and the analyticknowledge of the optical modes [26] we can rewrite (2) asg± = n2

pωpr33EΩ(r)/2, where the single photon electricfield EΩ(r) is determined from the simulated microwavefield distribution in the region of the optical mode (com-pare Fig. 2(b)). From this we estimate the single photoncoupling rate for the m = 1 mode to be g ≈ 2π × 28 Hz.

Asymmetric FSR around avoided crossings

The key feature required for the single sideband con-version in a WGM resonator with type-0 phase-matchingwithout detuning the pump is an asymmetric spectraldistance between the pump and the two signal modes(compare Fig. 1). In a mm-size resonator adjacent freespectral ranges differ only by a couple of kHz due tothe combined effect of material and geometric dispersion.Hence, both sidebands are generated in the case of zeropump detuning. However, in WGM resonators differentmodes can couple to each other linearly if they are closeto degeneracy [27, 28]. Because of different thermal dis-persion of different WGM families, such modes can bebrought into resonance by changing the resonator tem-perature. This feature allows them to exchange energyand leads to avoided crossings. Such crossings appearas a mode splitting around the unperturbed resonancefrequency, where the minimal spectral separation is pro-portional to the coupling rate between the two modes and

5

frequency detuning (MHz)

tem

pera

ture

offs

et ∆

T (m

K)

0 20 40 60 80 100−50

0

50

fit data

∆T=-20 mK

∆T=20 mK

20 40 60 80detuning (MHz)

normalized reflected pow

er

10.54 MHz

0

0

0

1

1

1

FIG. 3. On the left, the resonance position of a TE modeexperiencing an avoided crossing with a TM mode is shownfor different temperature settings. On the right, the WGMspectrum around the avoided crossing is plotted for differenttemperatures.

can reach many linewidths. Essentially, the modes shiftaway from their unperturbed resonance position leadingto a change of the effective dispersion.Such a crossing is shown in Fig. 3 where the frequencyof a TE polarized mode within a sweeping window of100 MHz is plotted against a temperature change of theresonator. From the fit in Fig. 3 we extract a couplingrate of κ = 2π × 5.27 MHz and the asymptotes yield-ing the temperature dependence of the two interactingmodes. The ratio of their slopes is approximately 1.8,which corresponds to the ratio between the temperaturesensitivities of TE and TM polarized modes in a lithiumniobate WGM resonator of that size [29]. Such polariza-tion coupling has been previously observed in ring res-onators [30] and in larger magnesium fluoride resonators[31].To demonstrate the effect of the splitting on the spectraldistance between two adjacent modes of the same fam-ily we show a spectrum of the m − 1, m, and m + 1modes in Fig. 4(a). The corresponding spectral dis-tances FSR+ = νm+1 − νm and FSR− = νm − νm−1

are presented in Fig. 4(b). The free spectral rangesare strongly temperature-dependent close to the avoidedcrossings and the difference |∆+ − ∆−| can be severaltens of MHz, exceeding the linewidth of the modes sig-nificantly. We will use this temperature-dependent asym-metry of the FSR to switch between sum- and differencefrequency generation in the following.

Single sideband conversion

For conversion of a microwave signal to a single opticalsideband we use the mode triplet shown in Fig. 4(a) and(b). The optical pump laser was locked to the centralmode denoted with m (black curve) while the tempera-ture of the cavity was stabilized around 27.88 C. Thefree spectral ranges at that working point are marked by

the dashed line in Fig. 4(b). The microwave cavity reso-nance was set to Ω0 = 2π×8.960 GHz and the microwavesignal sent to the cavity was swept from 8.925 GHz to8.975 GHz in steps of 200 kHz. For each step the opticalsignal was measured with an optical spectrum analyzer(YOKOGAWA AQ6370C) (see inset in Fig. 5). The ob-tained magnitude of the generated sidebands is shown inFig. 4(c) as a function of the microwave detuning from itsresonance, which is indicated by the black dashed line.The up- and downconverted signals can be addressed sep-arately as they appear at different microwave frequenciesdue to the different FSRs of the respective modes. TheLorentzian given by (3) agrees well with the measureddata. The asymmetry of the peaks and their intensitydifference can be attributed to the detuning from the mi-crowave resonance position. The central frequencies ofthe peaks are separated by 18.1 MHz which correspondsto the difference between FSR+ and FSR−. The extinc-tion ratio for the suppressed sideband is determined bythe FSR-asymmetry and the linewidth of the microwaveand optical modes. From the fit we find a suppressionof the down-converted signal at maximum up-conversionof 23 dB. This suppression could be further increased bya stronger mode splitting and narrower bandwidths ofboth, the microwave and the optical modes.

CONVERSION EFFICIENCY

The photon number conversion efficiency can be foundby measuring the microwave power at the input and theoptical signal power at the output of the converter. From(3) we find:

η+ =Ω

ω+

P+

PΩ(6)

As input power of the microwave we take the power ar-riving at the coaxial pin coupler while cable losses arecalibrated out. As optical signal power we considerthe power leaving the WGM resonator inside the sili-con prism, see Fig. 2 c). This is justified for estimatingthe performance of the system because the prism canbe coated to prevent reflection loss. The power of thesideband is obtained by measuring its relative strengthcompared with the reflected pump power on the opti-cal spectrum analyzer. This allows us to calculate thesideband power leaving the resonator from the couplingefficiency of the pump and the absolute pump power sentto the resonator.

For demonstration of the absolute conversion efficiency,we choose the optical mode depicted in Fig. 2(d) andpump it on resonance. The working temperature is setsuch, that the avoided crossing occurs on the red detunedside of the mode. This ensures that we can obtain a singlesum-frequency generated sideband which is not disturbed

6

-9.50 -9.25 -9.00 -0.50 -0.25 0.00 8.50 8.75 9.0027.75

27.85

27.95

28.05

28.15

27.80

27.90

28.00

28.10

28.20

FSR+FSR−

8.90 8.95 9.00 9.05

FSR+FSR-

(a) spectral positions of FSR spaced modes (b) free spectral ranges (c) up- and down-conversion

Ω0

-25 -20 -15 -10 -5 0 5 10 150.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0microwave modeup-converteddown-converted

-23 dB

18.1 MHz

frequency detuning (GHz) free spectral range (GHz) detuning of microwave pump Ω0 - Ω (MHz)

tem

pera

ture

(°C)

rela

tive

pow

er

m - 1 m m + 1

FIG. 4. (a) Resonance frequencies of three TE modes that are separated by one azimuthal mode number m corresponding tothe free spectral range FSR± ≈ 9 GHz. Each mode experiences avoided crossings of different strength for some temperatures.(b) Frequency difference of the m+ 1 and m− 1 mode from the m mode. One can see that FSR+ and FSR− are functions ofthe temperature and differ by up to 50 MHz. This allows selective up- and downconversion as depicted in (c): The temperaturewas set to 27.88 C, indicated by the dashed line in (a) and (b), where FSR+ −FSR− = 18.1 MHz. The optical pump is lockedto the m mode. A microwave signal sweeping over the microwave resonance which is indicated by the gray Lorentzian, whilethe generated sidebands are measured with an optical spectrum analyzer (see Fig. 5). The shown sidebands are separated by18.1 MHz and the suppression of the down-conversion at the maximum of the up-conversion is about 23 dB.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

5

10

15

20

25

30

35

40

45

linearregime

192.93 192.96 192.99 193.02-80

-70

-60

-50

-40

-30

frequency (THz)

pow

er (d

Bm)

microwave power (µW)

up-c

onve

rted

opt

ical

sid

eban

d (µ

W)

reected pump

signal

FIG. 5. Power of the up-converted sideband as a functionof the microwave power sent to the cavity. With increas-ing microwave power, the optical pump mode depletes andthe conversion efficiency decreases. A linear fit in the unde-pleted regime (grey) yields a slope of P+/PΩ = 23.68 ± 0.46corresponding to a photon number conversion efficiency ofη+ = (1.09 ± 0.02) × 10−3. The inset shows an OSA spec-trum (resolution 2.5 GHz) of the reflected pump power andthe generated upconverted sideband.

by the linear mode coupling causing the avoided cross-ing. We measure FSR+ = 8.941 GHz and adjust themicrowave resonance frequency Ω0 to that value withthe tuning screw. In this configuration we find for the

undercoupled microwave mode γΩ = 2π × 3.6 MHz andγ′Ω = 2π×16.2 MHz and for the critically coupled opticalmode γ = γ′ = 2π × 346 kHz. The laser is locked to theresonator and 1 mW optical power is sent into the cav-ity. Considering the imperfect coupling and the resultingreflection from the prism (see Fig. 2(d)), 0.42 mW are ac-tually coupled into the WGM resonator. The microwavesignal frequency is set on resonance and increased from−54 dBm to −22 dBm in steps of 1 dBm.The results are presented in Fig. 5. The inset showsa typical OSA spectrum, where the single up-convertedoptical sideband is highlighted in blue. The plot showsthe optical signal power leaving the resonator as a func-tion of the input microwave signal power. In the un-depleted pump regime, the optical signal scales linearlywith the input microwave power according to (3). Atapproximately 1 µW microwave power, the pump startsto deplete and the up-converted signal saturates. Fromfitting the linear regime we find a power conversionefficiency of P+/PΩ = 23.68 ± 0.46 which transformsinto an absolute photon number conversion efficiency ofη+ = (1.09± 0.02)× 10−3 using (6). Inserting the cavityparameters into (3), we find the single photon couplingrate to be geff = 2π×(7.43±0.02) Hz. In order to comparethis value with the theoretically derived one, we have tomultiply it with

√2 since the theory does not take into

account that we experimentally excite propagating andcounterpropagating modes at the same time.This effective coupling rate is about three times smallerthan the expected one from numerical simulations, cor-responding to almost ten times decreased conversion ef-

7

ficiency. This is likely caused by air gaps between theWGM resonator and the metallic rings clamping the res-onator. Due to the high dielectric constant of lithiumniobate in the microwave regime even very small air gapslead to a significant field drop in air. Our simulationsshow that an air gap of 20 µm caused by imperfect ma-chining of the cavity and polishing of the upper and lowerside of the WGM resonator lowers g already by a factorof two.

DISCUSSION

The three main results of this paper are the following.The first result is a three order of magnitude improve-ment of resonant electro-optic conversion of microwavephotons into the optical regime compared to the so farbest reported values [18, 32]. The second result is ahighly efficient suppression of either the up- or down-converted sideband by engineering the dispersion of theresonator. The third result is a MHz bandwidth for theconversion. All three are important steps towards coher-ent and bidirectional microwave photon conversion basedon direct electro-optic modulation. For quantum opera-tions, the conversion efficiency will have to be pushed to-wards unity, which is possible with some engineering im-provements. Tsang [14] introduces the electro-optic co-operativity G0 = |gα|2/(Γ+ΓΩ), where |α|2 is the numberof pump photons within the resonator. For our systemwe find G0 ≈ 4 × 10−3, which is almost three orders ofmagnitude smaller than required for a device suitable forconverting non-classical photon states requiring G0 ≈ 1.To enhance the cooperativity the pump power can beincreased only slightly, as photo- and thermorefractiveeffects in lithium niobate cause the system to becomeunstable. While the optical Q factors are already closeto the material limitation [33], it was shown recently thatthe intrinsic microwave absorption in lithium niobate atmillikelvin temperatures allows for QΩ ≈ 105 [34]. Thisalone would increase our G0 by three orders of magni-tude. Furthermore, simulations show, that g scales in-versely with the resonator thickness. Assuming a real-istic resonator thickness of 50 µm, together with closingany air gaps deteriorating g, can improve the coopera-tivity additionally by two to three orders of magnitude.Recently, a theoretical study showed further routes toincrease g [35]. Putting these optimizations together,G0 1 can be achieved in a realistic scenario.Electro-optomechanical schemes work in the resolvedsideband regime where two pump tones, an optical anda microwave one, are detuned from their respective reso-nance frequencies by the mechanical resonance frequency.The consequence is that the bandwidth of the process isultimately limited to the resonance frequency of the me-chanical resonator which is typically below 1 MHz. Inpractice, only tens of kilohertz have so far been achieved

in efficient conversion experiments [12]. Coupling suchsystems to on demand single microwave photon sources[36] is challenging as the temporal signature of these pho-tons is in the order of MHz [37]. Furthermore, the achiev-able suppression of the red-detuned sideband is limitedas the pump detuning is also determined by the mechan-ical resonance frequency. Depending on the system pa-rameters this can lead to contamination of the convertedsignal with spontaneous emission.Our system does not have these restrictions as the medi-ator between the two regimes is the non-resonant, non-linear polarizability of lithium niobate. Hence, the band-width of the electro-optic scheme is solely determined byintrinsic losses and the nonlinear as well as external cou-pling rates. Furthermore, the described dispersion man-agement of the optical modes allows us to choose thedegree of suppression freely without detuning the opticalpump.These advantages justify the further investigation andoptimization of the electro-optical approach, even thoughthe electro-optomechanical presently has the efficiencyadvantage. As we have discussed, the single photonregime is within reach provided an optimized system.This would not only allow interfacing microwave qubitswith the optical domain, but also electro-optic cooling ofthe microwave mode. With even lesser requirements, thegeneration of entangled pairs of microwave and opticalphotons utilizing parametric down conversion is possible,which has recently been shown in the optical domain ina WGM based system [38].

ACKNOWLEDGMENTS

We would like to acknowledge stimulating discussionswith Konrad Lehnert and Alessandro Pitanti. MCCwould like to acknowledge support of the Studienstiftungdes deutschen Volkes.

[1] R. J. Schoelkopf and S. M. Girvin, Nature 451, 664(2008).

[2] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169(2013).

[3] D. Riste, S. Poletto, M.-Z. Huang, A. Bruno, V. Vester-inen, O.-P. Saira, and L. DiCarlo, Nature Communica-tions 6, 6983 (2015).

[4] A. I. Lvovsky, B. C. Sanders, and W. Tittel, NaturePhotonics 3, 706 (2009).

[5] M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L.Lev, and J. M. Taylor, Physical Review A 85, 020302(2012).

[6] J. Verdu, H. Zoubi, C. Koller, J. Majer, H. Ritsch, andJ. Schmiedmayer, Physical Review Letters 103, 043603(2009).

8

[7] A. Imamoglu, Physical Review Letters 102, 083602(2009).

[8] D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D.Lukin, and A. S. Sørensen, Physical Review Letters 105,210501 (2010).

[9] L. A. Williamson, Y.-H. Chen, and J. J. Longdell, Phys-ical Review Letters 113, 203601 (2014).

[10] X. Fernandez-Gonzalvo, Y.-H. Chen, C. Yin, S. Rogge,and J. J. Longdell, Physical Review A 92, 062313 (2015).

[11] T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva,E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Us-ami, A. Schliesser, and E. S. Polzik, Nature 507, 81(2014).

[12] R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak,R. W. Simmonds, C. A. Regal, and K. W. Lehnert,Nature Physics 10, 321 (2014).

[13] M. Tsang, Physical Review A 81, 063837 (2010).[14] M. Tsang, Physical Review A 84, 043845 (2011).[15] C. M. Caves, Physical Review D 26, 1817 (1982).[16] A. A. Savchenkov, A. B. Matsko, M. Mohageg, D. V.

Strekalov, and L. Maleki, Optics Letters 32, 157 (2007).[17] D. Cohen, M. Hossein-Zadeh, and A. Levi, Electronics

Letters 37, 300 (2001).[18] V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and

L. Maleki, Journal of the Optical Society of America B20, 333 (2003).

[19] D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, andN. Yu, Optics Letters 34, 713 (2009).

[20] A. A. Savchenkov, W. Liang, A. B. Matsko, V. S.Ilchenko, D. Seidel, and L. Maleki, Optics Letters 34,1300 (2009).

[21] C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde,J. M. Chow, A. D. Corcoles, J. A. Smolin, S. T. Merkel,J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen,and M. Steffen, Physical Review B 86, 100506 (2012).

[22] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair,G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor,L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. De-voret, and R. J. Schoelkopf, Physical Review Letters107, 240501 (2011).

[23] I. S. Grudinin, A. B. Matsko, A. A. Savchenkov,D. Strekalov, V. S. Ilchenko, and L. Maleki, Optics Com-munications 265, 33 (2006).

[24] K. Wong, Properties of Lithium Niobate, EMIS datare-views series (INSPEC/Institution of Electrical Engineers,2002).

[25] J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, OpticsExpress 20, 26337 (2012).

[26] I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel,and K. Buse, Opt. Express 21, 30683 (2013).

[27] T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow,A. D. Stone, and K. J. Vahala, Physical Review Letters100, 103905 (2008).

[28] J. Wiersig, Phys. Rev. Lett. 97, 253901 (2006).[29] U. Schlarb and K. Betzler, Physical Review B 50, 751

(1994).[30] S. Ramelow, A. Farsi, S. Clemmen, J. S. Levy, A. R.

Johnson, Y. Okawachi, M. R. E. Lamont, M. Lipson,and A. L. Gaeta, Optics Letters 39, 5134 (2014).

[31] W. Weng and A. N. Luiten, Optics Letters 40, 5431(2015).

[32] D. V. Strekalov, H. G. L. Schwefel, A. A. Savchenkov,A. B. Matsko, L. J. Wang, and N. Yu, Physical ReviewA. 80, 033810 (2009).

[33] M. Leidinger, S. Fieberg, N. Waasem, F. Kuhnemann,K. Buse, and I. Breunig, Optics Express 23, 21690(2015).

[34] M. Goryachev, N. Kostylev, and M. E. Tobar, PhysicalReview B 92, 060406 (2015).

[35] C. Javerzac-Galy, K. Plekhanov, N. Bernier, L. D. Toth,A. K. Feofanov, and T. J. Kippenberg, arXiv:1512.06442[physics, physics:quant-ph] (2015), arXiv: 1512.06442.

[36] D. Bozyigit, C. Lang, L. Steffen, J. M. Fink, C. Eich-ler, M. Baur, R. Bianchetti, P. J. Leek, S. Filipp, M. P.da Silva, A. Blais, and A. Wallraff, Nature Physics 7,154 (2011).

[37] R. W. Andrews, A. P. Reed, K. Cicak, J. D. Teufel, andK. W. Lehnert, Nature Communications 6, 10021 (2015).

[38] G. Schunk, U. Vogl, D. V. Strekalov, M. Fortsch,F. Sedlmeir, H. G. L. Schwefel, M. Gobelt, S. Chris-tiansen, G. Leuchs, and C. Marquardt, Optica 2, 773(2015).