A tunable High-Q millimeter wave cavity for hybrid circuit and cavityQED experiments

Aziza Suleymanzade,1 Alexander Anferov,1 Mark Stone,1 Ravi K. Naik,1 Jonathan Simon,1 and David Schuster1Department of Physics and James Franck Institute, University of Chicago, Chicago, IL

(Dated: 5 November 2019)

The millimeter wave (mm-wave) frequency band provides exciting prospects for quantum science and devices, sincemany high-fidelity quantum emitters, including Rydberg atoms, molecules and silicon vacancies, exhibit resonancesnear 100 GHz. High-Q resonators at these frequencies would give access to strong interactions between emitters andsingle photons, leading to rich and unexplored quantum phenomena at temperatures above 1K. We report a 3D mm-wave cavity with a measured single-photon internal quality factor of 3×107 and mode volume of 0.14×λ 3 at 98.2 GHz,sufficient to reach strong coupling in a Rydberg cavity QED system. An in-situ piezo tunability of 18 MHz facilitatescoupling to specific atomic transitions. Our unique, seamless and optically accessible resonator design is enabled bythe realization that intersections of 3D waveguides support tightly confined bound states below the waveguide cutofffrequency. Harnessing the features of our cavity design, we realize a hybrid mm-wave and optical cavity, designed forinterconversion and entanglement of mm-wave and optical photons using Rydberg atoms.

Cavity and circuit Quantum ElectroDynamics (QED) sys-tems provide unprecedented control over photonic quantumstates via coupling to strongly nonlinear single emitters.This effort began with pioneering works in Rydberg cavityQED, demonstrating first nonclassical micromaser radiation1,Schrodinger cat states and early EPR experiments2,3. Sincethen, cavity and circuit QED systems have become essen-tial tools for exploring quantum phenomena both in the op-tical4–7 and microwave8,9 regimes. Hybrid systems, whichcross-couple these regimes, can harness the strengths of op-tical systems for communication and microwave systems forquantum information processing, yielding a more powerfultoolset for quantum information technology10. In particular,the coherent interconversion of microwave and optical pho-tons would enable large quantum networks and robust transferof quantum information11–18.

Mm-wave frequencies provide a promising platform for hy-brid quantum science19 at less explored, but potentially ben-eficial length and energy scales. Firstly, 100 GHz resonanceswith long coherence times are abundant among commonlyused optical and microwave quantum emitters such as Ry-dberg atoms20, molecules21 and silicon vacancies22, thoughthey are rarely harnessed for quantum science due to lack ofboth high-Q resonators with tight mode confinement and ma-ture mm-wave manipulation technology. Secondly, the meanthermal photon occupation of a 100 GHz resonator at 1K is

nph = 1/(ehν

kbT − 1) = 0.009� 1. This puts such a resonatorin the quantum regime at temperatures accessible with sim-ple pumped 4He with much larger cooling powers and lowercost and complexity than the dilution refrigerators required toreach ∼ 20mK for 10 GHz experiments. Finally, the interme-diate length scale of mm-waves enables development of scal-able high Q-factor devices using both near and far field waveengineering techniques.

More broadly, the mm-wave band is gaining interest acrossmany fields of science and technology. Advances in mm-wavedetection are essential for observational cosmology and studyof the cosmic microwave background23,24. Terahertz andnear-terahertz radiation exhibits vast potential in hazardous

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FIG. 1. a. Schematic of crossed mm-wave and optical cavities. Thesuperconducting mm-wave cavity is formed by the intersection ofthree evanescent waveguides. The x-axis waveguide will be em-ployed for atom transport, y-axis waveguide for mm-wave couplingand z-axis waveguide for an optical Fabry-Pérot cavity. Each sideof the Fabry-Pérot cavity includes Invar spacers to prevent differen-tial thermal contractions and a piezo actuator for tuning and lockingthe frequency of the optical cavity. b. Photograph of the assembledcrossed mm-wave and optical cavity with wired piezos. c. Schematicreflection measurement setup for the mm-wave cavity. 100 GHzphotons are delivered into the coupling port of the cavity througha WR10 waveguide.

chemical sensing and effective medical diagnostics25,26. Re-cently, the search for higher bandwidth and lower latencycommunication brought mm-wave wave frequencies into thefocus of the telecommunication industry27,28. Once pro-hibitively limited and expensive, mm-wave technology is be-

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FIG. 2. Various seamless resonators made from intersecting 1.6mmdiameter tubes. The diameter of these tubes is chosen such that thelow-frequency cutoff of each tube is ∼ 110 GHz. Examples of ge-ometries with bound defect modes include: a. an “elbow” resonatorat 109GHz made by intersection 2 tubes. b. a 98GHz “tee”. c. 4-beam 92GHz “star”, d. a 30GHz quasi-cylindrical resonator, madeof 15 tubes, e. a 2D lattice of mm-wave “cross” resonators.

coming more available due to this progress, encouraging thedevelopment of mm-wave quantum devices.

In this work, we develop technology to hybridize mm-waveand optical photons for quantum science applications. In par-ticular, we report a crossed optical and mm-wave cavity, withRydberg atoms envisioned as a transducer. Our central break-through is the optically open, tunable 3D mm-wave cavitywith Q = 3× 107 and mode volume V = 0.14λ 3 shown inFig. 1a.

In what follows, we first describe the design and manufac-ture of the seamless mm-wave cavity. We then introduce themm-wave measurement setup in a 1K cryostat and character-ize the properties of the device as a function of intra-cavitypower and temperature. We also demonstrate in-situ mechan-ical frequency control in the cryogenic environment. Finally,we discuss the application of this device in our Rydberg hy-brid experiment and more broadly, in cavity and circuit QEDsystems.

The key features of our device are its completely seamlessdesign, sub-wavelength mode volume and abundant opticalaccess to the strongly confined mm-wave mode. Typicallycomposed of two pieces, high-Q 3D cavities are vulnerable tophoton leakage through the seam between the pieces29, whichis more pronounced in cavities with shorter wavelengths. Wecreate our mode by intersecting several evanescent tubes asshown in Fig. 2. Since an intersection of any two dissimilarbodies creates a pocket with a larger cross section than each ofthem separately, this yields a bound state below the cutoff ofthe tubes. Indeed, any arbitrarily weak defect in 1D has thisproperty30, albeit with weaker localization. The number ofevanescent tubes and their diameters determines the frequencyof the resonance, while the locations of each intersection andangles between the tubes control the localization of the low-est mode and symmetries of higher order modes. Coupling

into the cavity is set by the diameter and length of the inputand output tubes attached to the WR10 waveguides. These pa-rameters fully specify the seamless cavity, leaving the internalquality factor as the sole unknown.

The flexibility of our design is demonstrated in Fig. 2: usinga single drill bit with a diameter below the cylindrical waveg-uide evanescent cutoff length 1.841λ

2π, we are able to realize

myriad resonator geometries with different spectra, dependingon design requirements. The intersection of even two evanes-cent waveguides creates a well-localized mode at the “elbow”as shown in Fig. 2a. Similarly, by intersecting more tubesin a “tee”, Fig. 2b, or “star” configuration, Fig. 2c, we traplight at lower frequencies for the same diameter of the tube.For diameter d = 1.6mm, a small number of tubes make con-veniently sized high-Q resonators at mm-wave frequencies.This approach could be applied to coupled cavities and lat-tices of resonators, both in 2D and 3D, as shown in Fig. 2e.Finally, many intersecting tubes can create lower frequencymicrowave cavities as shown in Fig. 2d., where one can ap-proach a large, seamless cylindrical or rectangular volume bydrilling out the cavity tube by tube. For the purposes of hybridexperiments, evanescent tubes have the advantage of provid-ing optical access for optical Fabry-Pérot cavities, lasers andatomic beams as shown in Fig. 1a. All of these benefits makeour design not only a powerful tool for hybrid Rydberg sys-tems, but also a flexible tool for other circuit and cavity QEDsystems.

The manufacturing of the device only involves drilling ap-propriate holes in high purity Niobium(Nb) stock. To reducesurface losses, the machined cavity is cleaned in solvents andchemically etched in a BCP bath of 2H3PO4 : HNO3 : HFfor 20 minutes at room temperature31. After rinsing and dry-ing, we immediately mount the cavity inside a Helium-4 ad-sorption cryostat to avoid surface degradation due to oxida-tion. The system is cooled down to 1K, which is well belowTc = 9.2K of Nb, where the cavity is deep in the supercon-ducting regime.

The 100GHz measurement setup is shown in Fig. 1c. It isanalogous to traditional microwave measurement chains, butwith all components shifted to the 100GHz band. This in-cludes: a Faraday isolator Quinstar QIF-W, cryogenic am-plifier LNF-LNC65-115WB and directional coupler QJR-W-40-S. For accessing the mm-wave band we use a multiplierVNAX-WR10 for up-conversion and multiplier + mixer VDIMixAMC 296 for down-conversion. We characterize our cav-ities using S11 reflection to fit both magnitude and phase andthereby extract quality factors and resonance frequencieTotest the capabilities of our design, we have made and charac-terized several mm-wave cavities shown in Fig. 3a. By vary-ing the length and diameter of the in-coupling port as well asnumber of tubes at the intersection point, we are able to cre-ate resonances of varying frequency and coupling Q (Fig. 3b).We consistently measured internal quality factors in the tensof millions. All of these cavities have mode volumes below0.2λ 3, which allows an tight confinement of mm-wave pho-tons for tens of microseconds at 1K using evanescence of thetubes alone.

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FIG. 3. a. Photograph of various tested mm-wave cavity geometries. b. Reflection spectra from several cavities with varying frequenciesand coupling Q’s, resulting in different total Q’s. c. Internal Q as a function of number of photons for the hybrid cavity. The constant trendindicates that the limiting loss mechanism is not power-dependent. d. Internal Q (black) and fractional frequency change (red) as a functionof temperature for the hybrid cavity. The deviation from Mattis-Bardeen curve at 2.3K suggests that the resistivity of Nb does not limit thelifetime of the photons at the lowest temperatures.

Fabry-Pérot cavity for our experiments with Rydberg atomsin Fig. 1a, we created the mm-wave mode volume by inter-secting three tubes, each with diameter 1.5mm. We chose theprecise dimensions of our resonator to match the 35p to 36stransition frequency of the 85Rb (See App. B) and to avoidclipping of the laser and atomic beams transiting the modein the experiment. We measured an internal quality factor of3×107 at 98.2 GHz, which corresponds to 50µs photon life-time. To identify the loss channels limiting the lifetime ofphotons, we study how the internal quality factor depends onthe power and temperature as shown in Fig. 3c,d.

The internal quality factor of the resonator as a function ofnumber of photons inside, shown in Fig. 3c, reveals no powerdependence from single photons to ∼10000 photons, indicat-ing that our low-power Q is not limited by two-level system(TLS) absorbers, as is common in 2D resonators32,33, or thatthey remain unsaturated up to powers where other nonlinear-ities set in. We do observe a degradation in the quality factorand a nonlinear frequency response at high powers, suggest-ing the appearance of hot spots and meta-stable states on thesurface of the etched Nb (see App. A). As our primary interestlies in the performance of the cavity at single photon powers,we did not investigate this further.

Another common loss mechanism is surface resistivity dueto thermal quasi-particles in the superconductor. Because thismechanism is known to be temperature dependent34, we areable to rule it out by examining the behaviour of the resonatorQ with temperature. In Fig. 3d, we plot the temperature de-pendence of the fitted internal quality factor and fractional fre-

quency change of the resonance for∼ 1 and 9,000 photons in-side the cavity. The data is obtained by gradually heating andthermalizing the cavity using a power resistor. The internalquality factor as a function of temperature follows a Mattis-Bardeen law down to 2.3 K, where temperature-independentlosses begin to dominate resulting in a deviation from the ex-ponential trend. This indicates that we are not limited by ther-mal processes at the lowest temperatures.

In addition to these mechanisms, other potential loss chan-nels include magnetic flux pinning35 and photon leakage atthe coupling boundary. The performance of the cavity couldbe further improved by adding magnetic field shielding36 andsealing the rectangular to circular waveguide transition at thecoupling port of the cavity to avoid leakage.

To precisely match the frequency of the cavity to an atomictransition, the cavity must be tunable. To achieve this, wethin one side of the cavity and pre-load a Noliac piezo electricstack actuator as shown in Fig. 4b and c, to stress the cavityand thereby change its volume. Such stress-tuning providesseveral GHz of tunability at room-temperature, or ∼ 18 MHzat 1K, as shown in Fig. 4a. We note that while tunability ispowerful, it has drawbacks: we observe that the thinning re-quired to make the cavity tunable results in mechanical cou-pling to the pulse tube cryocooler vibrations, leading to fluc-tuations of the cavity frequency by many linewidths.

For hybrid experiments with Rydberg atoms, the seamlessmm-wave cavity enabled immediate integration with an op-tical Fabry-Pérot cavity with a waist w = 80µm at 780nmand measured optical finesse F = 10000. For quantum inter-

4

conversion between optical and mm-wave bands, large collec-tive cooperativities on both optical and mm-wave transitionsare desirable37; large single-particle cooperativity on the mm-wave transition will be an enabling technology for quantum-nonlinear optics38. We compute single-particle cooperativi-ties of Copt =

24Fπ

1(kw)2 = 0.2 and Cmm = 22000 (see App. B),

which satisfy these constraints for a moderate-density atomicensemble39.

In conclusion, we have designed and machined a tunablemm-wave cavity with a measured internal Q factor of 3×107.The seamless geometry tightly confines photons at the inter-section of evanescent waveguides in the 0.14λ 3 mode volume,while allowing multiple lines-of-sight for optical access, in-jection of cold atoms and integration with an optical cavityfor quantum interconversion experiments37,40.

Our cavity Q’s at 1K are comparable to those of state-of-the-art 3D microwave (∼ 10GHz) cavities at 20mK that areconventionally integrated with transmon qubits9,29. We antic-ipate the possibility of interfacing our high-Q, high-frequencyresonators with such 10GHz transmons via 100GHz nonlineardevices41,42 employed as mixers.

For cavity and circuit QED, the tight confinement andlong coherence of our device allows access to a strongcoupling regime between a single photon and an emitter,which is difficult to achieve in many other platforms (seeApp. C). The large cooperativity between a Rydberg atomand a mm-wave photon suggests studies of quantum many-body physics, photon-number-dependent group delay38 andvacuum-induced squeezing43 on the mm-wave transition,while the effects can also be read out optically through theFabry-Pérot cavity. In sum, this work heralds a new era ofoptical/mm-wave quantum science.

ACKNOWLEDGMENTS

The authors would like to thank Andrew Oriani for helpwith the cryogenic setup. This work was primarily supportedby the University of Chicago Materials Research Science andEngineering Center, which is funded by National ScienceFoundation under award number DMR-1420709. This workwas supported by ARO grant W911NF-15-1-0397; D.S. ac-knowledges support from the David and Lucile Packard Foun-dation.

Appendix A: Nonlinear frequency response of the cavity athigh powers

At high photon numbers, the frequency response of thecavity has a strong nonlinear character as shown in Fig. 5.The nonlinear behavior appears at maximum simulated cur-rent densities of 100 A/cm2, which is much smaller than theNb critical current density of 150 MA/cm2. This effect hasbeen observed previously both in 2D16,44,45 and in 3D46 su-perconducting resonators, and it is commonly attributed tonon-linearity caused by hotspots or “Josephon junction-like”weak-links on the granular surface of Nb. The defects on the

a

Piezo

Nb Cavityc

0 50 100 150 200 250-20

-10

0

Piezo Voltage, V

FrequencyShift,MHz

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�2 > �1<latexit sha1_base64="375+5v0IcdH8FE/wWEiqmVXKjZw=">AAACAnicbVDLSsNAFL3xWesr6krcDLaCq5LEha6k6MZlBfuANoTJZNIOnTyYmQglFDf+ihsXirj1K9z5N07bINp6YOBwzrncucdPOZPKsr6MpeWV1bX10kZ5c2t7Z9fc22/JJBOENknCE9HxsaScxbSpmOK0kwqKI5/Ttj+8nvjteyokS+I7NUqpG+F+zEJGsNKSZx5WUY/reIA9B13+cBtVPbNi1awp0CKxC1KBAg3P/OwFCckiGivCsZRd20qVm2OhGOF0XO5lkqaYDHGfdjWNcUSlm09PGKMTrQQoTIR+sUJT9fdEjiMpR5GvkxFWAznvTcT/vG6mwgs3Z3GaKRqT2aIw40glaNIHCpigRPGRJpgIpv+KyAALTJRuraxLsOdPXiQtp2af1Zxbp1K/KuoowREcwynYcA51uIEGNIHAAzzBC7waj8az8Wa8z6JLRjFzAH9gfHwDD7SVSA==</latexit>

b

FIG. 4. Cryogenic frequency tuning of a high-Q mm-wave cavity.a. Frequency shift of the cavity resonance as a function of piezovoltage. At 1K, the linear fit shows ∼ 0.1 MHz/V tunability, withmaximum frequency shift of ∼ 18MHz. At room temperature, theincreased piezo throw enables cavity tuning by ∼GHz. b. Photo-graph of the piezoelectric actuator system attached to a test cavity.c. The low-temperature (1K) tuning is accomplished by displacing apre-thinned wall of the cavity by 1-2 µm using a piezo stack actuator.This pushes the mode farther out into the evanescent waveguides, ef-fectively, increasing the wavelength and decreasing the frequency ofthe lowest mode.

surface of granular superconductor have both superconduct-ing and normal states, which drive the device into multiplemetastable states: one with superconducting defects and onewith normal metal defects at lower Q and different frequen-cies16,44. The magnetic flux penetration into the surface alsoplays an important role in this mechanism, and can enhancethis effect making it more dramatic at high powers. Since ourcavity is not shielded from stray B fields, we expect enhance-ment of the nonlinear behavior.

In Fig. 5, at low intra-cavity powers around P =−130dBmor photon occupation of n = 104 the cavity exhibits a lin-ear response with a Lorentzian lineshape centered on f1 =98.218508 with the Q1 = 3× 107. As power increases, thisstate becomes metastable, and the frequency response exhibitsa Duffing-like behavior, shifting up in frequency. At a powerof P ≈ −80dBm, the frequency shift saturates and the res-onator falls into a second metastable state at f2 = f1 +24kHzwith Q2 = 1.3×107, again exhibiting a Lorentzian lineshape.It is important to note that even though the Duffing effect iscommon, a nonlinear coefficient resulting in a positive fre-quency shift with power is unusual.

An understanding of the mechanism behind the frequencyshifts, nonlinear behaviour and Q degradation in differentpower regimes is an ongoing topic of both experimental andtheoretical research. The observed response at high powersis not prohibitive for our experiments in the quantum regime,since we will work at very low photon occupation number.If high power operation were required, shielding of the mag-netic fields could inhibit flux pinning and other magnetic fieldeffects contributing to loss.

5

-1 0 1 2 3 4

0

-5

-10

-15

-20

-25

f-f0, kHz

S11,dB

Power, dB

-130

-120

-110

-100

-90

-80

-70

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FIG. 5. The frequency response of the mm-wave cavity at differentpowers. There are two clear meta-stable states with linear frequencyresponses: 1 - at low powers with f1 = 98.218508 GHz and Q1 =3× 107, and 2 - at high powers with f2 = f1 + 24kHz and Q2 =1.3×107. For intermediate powers, the resonators exhibits Duffing-like behavior, suggesting nonlinearity caused by surface heating.

Appendix B: Calculation of hybrid system cooperativities

In this section we perform a rough estimate of the resonantcoupling parameters for both the microwave and optical tran-sitions of Rydberg atoms trapped at the center of the opticaland mm-wave mode for the cavity shown in Fig. 1. Fig. 6a.shows the level diagram of four energy states of 85Rb we pro-pose to utilize for both entangling and inter-converting singleoptical and mm-wave photons. The 780nm optical photon inthe Fabry-Pérot cavity is resonant with ground

∣∣5S1/2⟩

to firstexcited state

∣∣5P3/2⟩

transition. The coupling strength and co-operativity between one Rb atom and a single optical photonare shown in Eq. B1 and B2:

gopt

2π=

dopt ·Eopt

2π h≈ 600 kHz, (B1)

Copt =24F

π

1(kw0)2 = 0.2 (B2)

where dopt is the dipole moment 〈5s|er |5p〉, Eopt is theelectric field strength of one optical photon at the location ofthe electron, F is the finesse of the optical Fabry-Pérot cavity,k is the wavevector and w0 is the waist of the optical cavity.The single atom interaction can be boosted by

√Natoms, due

to coherent interaction between the cloud of Natoms cold atomsand a single photon39.

On the mm-wave transition, the cooperativity between aRydberg atom and a single mm-wave photon of the supercon-ducting cavity is much higher due to strong confinement of100GHz field in the cavity, which is shown in Eq. B3 and B4.

gmm

2π=

dmm ·Emm

2π h= 460 kHz, (B3)

Cmm =4g2

mm

Γκ= 22000 (B4)

where Γ here is the linewidth of the Rydberg energy state|36S〉 and κ = f0

Q is the linewidth of the mm-wave cavity. Thehigh strength of the interaction is the result of the large Ryd-berg dipole moment dmm = 〈35P|er |36S〉 and a tight confine-ment of the mm-wave photon that our device provides.

The hybrid mm-wave/optical cavity in Fig. 1 allows fora cloud of cold atoms trapped in an optical lattice to enterthrough one of the tubes into, simultaneously, the center ofthe mm-wave cavity mode and the waist of Fabry-Pérot cav-ity. Here, the atoms can interact efficiently with both mm-wave and optical photons. To facilitate this interaction, we useElectromagnetically-Induced Transparency (EIT) with a bluelaser at 481nm, which couples the |5P〉 and |36S〉 states. Asshown in Fig 6b., by weakly probing the optical cavity, we an-ticipate a vacuum Rabi splitting47 of the optical transition dueto presence of the cold atomic cloud, cavity EIT48 in the pres-ence of the blue light and finally, a splitting of the EIT peaksproportional to the square root of number of photons in themm-wave cavity, akin to proposed photon-number dependentgroup velocity experiments in free space38. The strong cou-pling between single optical and mm-wave photons throughinteractions with atoms can open doors for entanglement andmanipulation of mm-wave photons using optical light and viceversa. For interconversion of mm-wave and optical photons,we need a UV light at 297nm between |35s〉 and |5s〉 statesto allow coherent and bidirectional conversion and quantuminformation transfer.

Appendix C: Comparison with other resonators

Table. I contains a summary of different resonators typesemployed in cavity and circuit QED experiments, in both op-tical and microwave regimes. To evaluate the performance ofour device for quantum systems, we compared the mode vol-ume and finesse of the resonators as parameters which are im-portant for reaching strong-coupling regime. In cases of sub-wavelength microwave cavities, Q and finesse are the same.

1Rempe, G., Schmidt-Kaler, F. & Walther, H. Observation of sub-Poissonianphoton statistics in a micromaser. Physical Review Letters 64, 2783–2786(1990).

2Brune, M. et al. Quantum Rabi Oscillation: A Direct Test of Field Quanti-zation in a Cavity. Physical Review Letters 76, 1800–1803 (1996).

3Raimond, J. M., Brune, M., Haroche, S. & Brossel, L. K. Colloquium:Manipulating quantum entanglement with atoms and photons in a cavity.Rev. Mod. Phys. 73, 18 (2001).

4Birnbaum, K. M. et al. Photon blockade in an optical cavity with onetrapped atom. Nature 436, 87–90 (2005).

5Boca, A. et al. Observation of the Vacuum Rabi Spectrum for One TrappedAtom. Physical Review Letters 93, 233603 (2004).

6Thompson, R. J., Rempe, G. & Kimble, H. J. Observation of normal-modesplitting for an atom in an optical cavity. Physical Review Letters 68, 1132–1135 (1992).

6

TABLE I. Common resonators in cavity and circuit QED systems

System V, λ 3 f, GHz Finesse2D superconducting resonators49 10−6 1.53 106−3×107

3D superconducting resonators for qubits29 0.1 11 7×108

Our mm-wave cavity 0.14 98.2 3×107

Accelerator Cavity50 1 1-10 1011

MM-wave Fabry-Pérot Cavity51 260 51 4.6×109

Optical Fabry-Pérot microcavity52 6000 4.6×105 105

Microsphere cavity52,53 9000 3×105 106

-20 -10 0 10 20Detuning, MHz

CavityTransmission

Cavity Resonance

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-20 -10 0 10 20Detuning, MHz

CavityTransmission

Vacuum Rabi Splitting

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-20 -10 0 10 20Detuning, MHz

CavityTransmission

Rydberg EIT

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Detuning, MHz

-20 -10 0 10 20Detuning, MHz

CavityTransmission

Mm-waves

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Mm-wave splitting

Rydberg EIT

Vacuum Rabi Splitting

Bare cavity transmission

a b

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FIG. 6. The proposed hybrid cavity QED experiment with Rydbergatoms: a The energy levels of the 85Rb atoms involved in the hy-brid experiments for interacting and coherently inter-converting mm-wave and optical photons. A blue laser is used to couple single mm-wave and optical photons, whereas an additional UV laser provides away to bidirectionally convert between two bands. b The simulationof the optical cavity transmission in the hybrid experiment to ob-serve the nonlinear interaction between single optical and mm-wavephotons (top to bottom): starting with a single Lorentian peak corre-sponding to bare optical cavity transmission, then vacuum Rabi split-ting of the cavity transmission due to presence of the atomic cloud,then EIT while the high power blue beam is on inside of the cavitymode, and, finally, mm-wave splitting due to presence of mm-wavephotons inside of the resonator.

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