Antibunched photons emitted by a dc biased Josephson junction
C. Rolland, A. Peugeot, S. Dambach, M. Westig, B. Kubala, Y. Mukharsky, C. Altimiras, H. le Sueur, P. Joyez, D. Vion, P. Roche, D. Esteve, J. Ankerhold, F. Portier
AAntibunched photons emitted by a dc-biased Josephson junction
C. Rolland , ∗ A. Peugeot , ∗ S. Dambach , M. Westig , B. Kubala , Y. Mukharsky , C. Altimiras ,H. le Sueur , P. Joyez , D. Vion , P. Roche , D. Esteve , J. Ankerhold , † and F. Portier ‡ DSM/IRAMIS/SPEC, CNRS UMR 3680, CEA,Universit´e Paris-Saclay, 91190 Gif sur Yvette, France and Institute for Complex Quantum Systems and IQST,University of Ulm, 89069 Ulm, Germany (Dated: May 14, 2019)We show experimentally that a dc biased Josephson junction in series with a high-enough-impedance microwave resonator emits antibunched photons. Our resonator is made of a simplemicro-fabricated spiral coil that resonates at 4.4 GHz and reaches a 1.97 kΩ characteristic impedance.The second order correlation function of the power leaking out of the resonator drops down to 0.3 atzero delay, which demonstrates the antibunching of the photons emitted by the circuit at a rate of6 10 photons per second. Results are found in quantitative agreement with our theoretical predic-tions. This simple scheme could offer an efficient and bright single-photon source in the microwavedomain. PACS numbers: 74.50+r, 73.23Hk, 85.25Cp
Single photon sources constitute a funda-mental resource for many quantum informa-tion technologies, notably secure quantum statetransfer using flying photons. In the microwavedomain, although photon propagation is moreprone to losses and thermal photons present ex-cept at extremely low temperature, applicationscan nevertheless be considered [1, 2]. Single mi-crowave photons were first demonstrated in [3]using the standard design of single-photon emit-ters: an anharmonic atom-like quantum systemexcited from its ground state relaxes by emit-ting a single photon on a well-defined transi-tion before it can be excited again. The firstand second order correlation functions of such asource [4] demonstrate a rather low photon fluxlimited by the excitation cycle duration, butan excellent antibunching of the emitted pho-tons. We follow a different approach, where thetunnelling of discrete charge carriers through aquantum coherent conductor creates photons inits embedding circuit. The resulting quantumelectrodynamics of this type of circuits [5–11]has been shown to provide e.g. masers [12–15],simple sources of non-classical radiation [16–18],or near quantum-limited amplifiers [19]. Whenthe quantum conductor is a Josephson junc- tion, dc biased at voltage V in series with alinear microwave resonator, exactly one photonis created in the resonator each time a Cooperpair tunnels through the junction, providedthat the Josephson frequency 2 eV /h matchesthe resonator’s frequency [20].We demonstratehere that in the strong coupling regime betweenthe junction and the resonator, the presence ofa single photon in the resonator inhibits thefurther tunneling of Cooper pairs, leading tothe antibunching of the photons leaking out ofthe resonator [21, 22]. Complete antibunchingis expected when the characteristic impedanceof the resonator reaches Z c = 2 R Q /π , with R Q = h/ (2 e ) (cid:39) .
45 kΩ the superconductingresistance quantum. This regime, for which theanalogue of the fine structure constant of theproblem is of order 1, has recently attracted at-tention [23, 24], as it allows the investigationof many-body physics with photons [25, 26] orultra-strong coupling physics [27], offering newstrategies for the generation of non classical ra-diation [28].The simple circuit used in this work is rep-resented in Fig. 1a: a Josephson junction iscoupled to a microwave resonator of frequency ν R and characteristic impedance Z c , and biased a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y q e q e R h QR C RZ ,
12 3 I e R V eh a)b) FIG. 1:
Principle of the experiment: (a)
AJosephson junction in series with a resonator of fre-quency ν R and characteristic impedance Z c of theorder of R Q = h/ (2 e ) is voltage biased so that eachCooper pair that tunnels produces a photon in theresonator (1). (b) Photon creation and relaxation:A tunneling Cooper pair shifts the charge on theresonator capacitance by 2 e . The tunneling rateΓ n → n +1 starting with the resonator in Fock state | n (cid:105) is proportional to the overlap between the wave-function Ψ n ( q ) shifted by 2 e and Ψ n +1 ( q ). Thisoverlap depends itself on Z C via the curvature ofthe resonator energy. At a critical Z c , Γ → = 0and no additional photons can be created (2) un-til the photon already present has leaked out (3).The photons produced are thus antibunched, whichis revealed by measuring the g (2) function of theleaked radiation. at a voltage V smaller than the gap voltage V gap = 2∆ /e , where − e is the electron chargeand ∆ the superconducting gap, so that sin-gle electron tunneling is impossible. The time-dependent Hamiltonian H = ( a † a + 1 / hν R − E J cos[ φ ( t )] (1)of the circuit is the sum of the resonator andJosephson Hamiltonians. Here a is the photonannihilation operator in the resonator, E J isthe Josephson energy of the junction, φ ( t ) =2 eV t/ (cid:126) − √ r ( a + a † ) is the phase difference across the junction (conjugate to the numberof Cooper pairs transferred accross the junc-tion), and r = πZ c /R Q is the charge-radiationcoupling in this one-mode circuit [29]. Thenonlinear Josephson Hamiltonian thus couplesCooper pair transfer to photon creation in theresonator. This results in inelastic Cooper pairtunneling: a dc current flows in this circuitwhen the electrostatic energy provided by thevoltage source upon the transfer of a Cooperpair corresponds to the energy of an integernumber k of photons created in the resonator:2 eV = khν R . The steady state occupationnumber ¯ n in the resonator results from the bal-ance between the Cooper pair tunneling rateand the leakage rate to the measurement line.For k = 1 – the resonance condition of the ACJosephson effect – each Cooper pair transfer cre-ates a single photon. The theory of dynamicalCoulomb blockade (DCB)[29–31] predicts that,in the limit of small coupling r , the power emit-ted into an empty resonator P = 2 e E ∗ J (cid:126) Re Z ( ν = 2 eV /h ) (2)coincides with the AC Josephson expression,albeit with a reduced effective Josephson en-ergy E ∗ J = E J e − r/ renormalized by the zero-point phase fluctuations of the resonator [21–23, 34, 51–53]. In the strong-coupling regime( r (cid:39) eV = hν R for singlephoton creation. Expressed in the resonatorFock state basis {| n (cid:105)} , H reduces to H RWA = − ( E J / (cid:80) n (cid:0) h RWA n,n +1 | n (cid:105)(cid:104) n + 1 | + h . c . (cid:1) , withthe transition matrix elements h RWA n,n +1 = (cid:104) n | exp (cid:2) i √ r ( a † + a ) (cid:3) | n + 1 (cid:105) . (3)Describing radiative losses via a Lindblad super-operator, one gets the second order correlationfunction for vanishing occupation number ¯ n (cid:28) g (2) ( τ ) = (cid:10) a † (0) a † ( τ ) a ( τ ) a (0) (cid:11) (cid:104) a † a (cid:105) = (cid:104) − r − κτ / (cid:105) (4)with κ the photon leakage rate of the resonator.In the low coupling limit r (cid:28) h RWA n,n +1 scales as √ n + 1, one recovers the familiar Pois-sonian correlations g (2) (0) = 1. On the con-trary, at r = 2 ( Z c = 4 . h RWA1 , = 0 andEq. (4) yields perfect antibunching of the emit-ted photons: g (2) (0) = 0. In this regime, asillustrated by Fig. 1, a first tunnel event bring-ing the resonator from Fock state | (cid:105) to | (cid:105) can-not be followed by a second one as long as thephoton has not been emitted in the line. Thisis the mechanism involved in the Frank-Condoneffect and relies on the reduction of the matrixelement of the Josephson Hamiltonian betweenthe one and two photon states of the cavity asthe coupling parameter r increases from 0 to 2,where it vanishes. It is thus different from themechanism at work in the recent work of Grimmand coworkers [32] which relies on the charge re-laxation induced by a large on chip resistance.Standard on-chip microwave resonator de-signs yield characteristic impedances of the or-der of 100 Ω, i.e. r ∼ .
05. To appoach r ∼ − (cid:15) r (cid:39) . R t = 222 ± ν R = 5 . r = 1 . Q = 2 πν r /κ = 42 [34]. The actual values measured using the calibra-tion detailed in the Supplemental material [34]are ν r = 4.4 GHz, Q = 36 .
6, and a characteristicimpedance Z c = 1 . ± .
06 kΩ, correspondingto a coupling parameter r = 0 . ± .
03, andthus to an expected E ∗ J /E J = 0 . ± .
01. Weattribute the small difference between designand experimental values to a possible under-estimation in our microwave simulations of thecapacitive coupling of the resonator to the sur-rounding grounding box.The sample is placed in a shielded sampleholder thermally anchored to the mixing cham-ber of a dilution refrigerator at T =12 mK. Asshown in Fig. 2, the sample is connected to abias tee, with a dc port connected to a filteredvoltage divider, and an rf port connected to a90 o hybrid coupler acting as a microwave beamsplitter towards two amplified lines with an ef-fective noise temperature of 13.8 K. After band-pass filtering at room temperature, the signalsin these two channels V a ( t ) , V b ( t ) are down con-verted to the 0 - 625 MHz frequency range us-ing two mixers sharing the same local oscillatorat ν LO = 4 .
71 GHz, above the resonator fre-quency. The ouput signals are then digitized at1.25 GSamples/s to measure their two quadra-tures, and the relevant correlation functions arecomputed numerically.In Fig. 3a, the measured 2D emission map asa function of bias voltage and frequency showsthe single photon regime along the diagonal.A cut at the resonator frequency (blue line inFig. 3b) reveals an emission width of 2.9 MHz,which we attribute to low frequency fluctua-tions of the bias voltage, mostly of thermal ori-gin. Two faint lines (pointed by the obliqueyellow arrows) also appear at 2 eV = h ( ν ± ν P ),and correspond to the simultaneous emissionof a photon in the resonator and the emis-sion/absorption of a photon in a parasitic reso-nance of the detection line at ν P = 325 MHz.Comparing the weight of these peaks to themain peak at 2 eV = hν yields a 61 Ω char-acteristic impedance of the parasitic mode anda 15 mK mode temperature in good agreement V6.5M T (b) HEMTs a V t ( ) b V t ( ) LO = GHzZc = k Q = FIG. 2:
Experimental setup. (a)
Optical mi-crography of the sample showing the Al/AlOx/AlSQUID (inset) implementing the Josephson junc-tion and the resonator made of a Nb spiral inductorwith stray capacitance to ground. (b)
Schematic ofthe circuit showing the sample (green), the coil cir-cuit for tuning the Josephson energy (brown), thedc bias line (red), and the bias tee connected to themicrowave line (blue) with bandpass filters, isola-tors (not shown here), and a symmetric splitter con-nected to two measurement lines with amplifiers at4.2 K and demodulators at room temperature [34]. with the refrigerator temperature.We now set the bias at V = hν r / e = 9 . µ V,and we detect the output signals of the two am-plifiers in a frequency band of 525 MHz ( ∼ ν R . This apparently large detectionwindow – 180 times wider than the emissionline, see Fig. 3b – is actually barely enough tomeasure the fast fluctuations occuring at fre- quencies up to the inverse resonator lifetime.An even larger bandwidth would bring the mea-sured g (2) closer to the expected value of Eq.(4) but would also increase the parasitic fluctu-ations due to amplifiers’ noise and increase thenecessary averaging time. Our choice is thus acompromise, leading to a 15-day long averagingfor the lowest occupation number. From thedown-converted signals, we rebuild their com-plex envelopes S a,b ( t ) [34, 54]. We now use twoalternative methods to extract g (2) ( τ ). First,we obtain the instantaneous powers P a,b ( t ) = V ( µ V ) -3,100-0,6781-0,6200 (MHz) 0.0516 R e [ Z ] ( k ) PS D ( pho t on s ) PS D ( pho t on s ) (a) (b) FIG. 3:
Emitted microwave power andimpedance seen by the junction. (a)
2D mapof the emitted power spectral density (PSD) as afunction of the frequency ν and bias voltage V ,expressed in photon occupation number (logarith-mic color-scale). (b) Spectral line at V = 9 . µV (blue points) obtained from a cut in the 2D mapalong the horizontal white arrows and real part ofthe impedance Re[ Z ( ν )] seen by the SQUID (redpoints). The solid blue (black) line is a Gaussian(Lorentzian) fit. - 1 5 - 1 0 - 5 0 5 1 0 1 5 t ( n s ) t = 0n = 0 . 0 8 g(2)( t ) ( b ) p h o t o n n u m b e r n ( a ) FIG. 4:
Antibunching of the emitted radiation at bias V = hν R / e = 9 . µ V. (a) Experimental(dots) and theoretical (dashed line) second order correlation function g (2) as a function of delay τ for n =0 .
08 photons in the resonator. Error bars indicate ± the statistical standard deviation. (b) Experimental(dots) and theoretical (dashed line) g (2) (0) as a function of n . The solid line is the theoretical predictionnot taking into account the finite detection bandwidth. | S a,b ( t ) | , and extract g (2) ( τ ) = (cid:104) P a ( t ) P b ( t + τ ) (cid:105)(cid:104) P a ( t ) (cid:105) (cid:104) P b ( t + τ ) (cid:105) (5)from their cross-correlations. Here, the sam-ple’s weak contribution has to be extracted fromthe large background noise of the amplifiers,which we measure by setting the bias voltageto zero. To overcome this complication and geta better precision on g (2) , we compute the com-plex cross-signal C ( t ) = S a ∗ ( t ) S b ( t ), which isproportional to the power emitted by the res-onator and has a negligible background aver-age contribution. g (2) ( τ ) can then be extractedfrom the correlation function of C ( t ) and C ∗ ( t )[34]. As g (2) ( τ ) is real and the instantaneousnoise on C ( t ) is spread evenly between real andimaginary parts, this also improves the signalto noise ratio by √ g (2) values shown in Fig. 4 correspond to the average ofthe two procedures. As we decrease the pho-ton emission rate by adjusting E J with themagnetic flux threading the SQUID, g (2) (0) de-creases. For the lowest measured emission rateof 60 millions photons per second, correspond-ing to an average resonator population of 0.08photons, g (2) (0) goes down to 0.31 ± .
04, ingood agreement with the theoretical predictionof 0.27, cf. Eq. (4) for r=0.96. This is the mainresult of this work, which demonstrates a signif-icant antibunching of the emitted photons. Inagreement with Eq. (4), the characteristic timescale of the g ( τ ) variations coincides with the1.33 ns resonator lifetime deduced from the cal-ibrations. As our design did not reach r = 2,the transition from | (cid:105) to | (cid:105) is not completelyforbidden, and from then on, transitions from | (cid:105) to | (cid:105) and higher Fock states can occur. Thelarger E J , the more likely to have 2 photons andhence photon bunching. To predict the time-dependent g (2) ( τ ) for arbitrary E J , we solve thefull quantum master equation˙ ρ = − i (cid:126) [ H RWA , ρ ] + κ (cid:0) aρa † − a † aρ − ρa † a (cid:1) . (6)This approach also allows for the quantitativemodeling of the experimental measurement viaa four-time correlator [34]. Properly account-ing for filtering in the measurement chain (seeRef. [4, 54] and Supplemental Material [34]),this description accurately reproduces the ex-perimental results in Fig. 4 (lines) without anyfitting parameters.We finally probe the renormalization of E J by the zero point fluctuations of the resonatorusing Eq. (2). This requires to maintain the res-onator photon population much below 1, whichshould be obtained by reducing the Josephsonenergy using the flux through the SQUID. How-ever, magnetic hysteresis due to vortex pinningin the nearby superconducting electrodes pre-vented us from ascribing a precise flux to a givenapplied magnetic field, the only straightforwardand reliable working point at our disposal thusoccurring at zero magnetic flux and maximumJosephson energy. To ensure that the SQUIDremains in the DCB regime even at this maxi-mum E J , and ensure a low enough photon pop-ulation, we select a bias voltage V = 10 . µ Vyielding radiation at 4.91 GHz, far off the res-onator frequency. Here again, the normal cur-rent shot noise is used as a calibrated noisesource to measure in-situ G Re Z ( ν = 4 .
91 GHz).The effective Josephson energy E ∗ J = 1 . ± . µ eV extracted in this way is significantlysmaller than the Ambegaokar-Baratoff value of E J = 3 . ± . µ eV, and in good agreementwith our prediction of E ∗ J = 1 . ± . µ eV[55], taking also into account the phase fluctua-tions coming from the parasitic mode at ν p andits harmonics.In conclusion, we have explored a new regimeof the quantum electrodynamics of coherentconductors by strongly coupling a dc biasedJosephson junction to its electromagnetic en-vironment, a high-impedance microwave res- onator. This enhanced coupling first results in asizeable renormalization of the effective Joseph-son energy of the junction. Second, it pro-vides an extremely simple and bright source ofantibunched photons. Appropriate time shap-ing either of the bias voltage [56], or the res-onator frequency, or the Josephson energy [32]should allow for on-demand single photon emis-sion. This new regime that couples quantumelectrical transport to quantum electromagneticradiation opens the way to new devices forquantum microwaves generation. It also al-lows many fundamental experiments like inves-tigating high photon number processes, para-metric transitions in the strong coupling regime[21, 22, 33, 57], the stabilization of a Fock stateby dissipation engineering [56], or the develop-ment of new type of Qbit based on the Lamb-shift induced by the junction [58].We thank B. Huard, S. Seidelin and M.Hofheinz for useful discussions. This work re-ceived funding from the European ResearchCouncil under the European Unions Pro-gramme for Research and Innovation (Horizon2020)/ERC Grant Agreement No. [639039].We gratefully acknowledge partial support fromLabEx PALM (ANR-10-LABX-0039-PALM),ANR contracts ANPhoTeQ and GEARED,from the ANR-DFG Grant JosephCharli, andfrom the ERC through the NSECPROBE grant,from IQST and the German Science Foundation(DFG) through AN336/11-1. S.D. acknowl-edges financial support from the Carl-Zeiss-Stiftung. ∗ These two authors contributed equally. † Electronic address: email:[email protected] ‡ Electronic address: email: [email protected][1] Z.-L. Xiang, M. Zhang, L. Jiang, andP. Rabl, Phys. Rev. X , 011035 (2017),URL https://link.aps.org/doi/10.1103/PhysRevX.7.011035 .[2] X. Gu, A. F. Kockum, A. Miranowicz, Y.- X. Liu, and F. Nori, Physics Report ,1-102 (2017).[3] A. A. Houck, D. I. Schuster, J. M. Gambetta,J. A. Schreier, B. R. Johnson, J. M. Chow,L. Frunzio, J. Majer, M. H. Devoret, S. M.Girvin, et al., Nature , 328 (2007).[4] D. Bozyigit, C. Lang, L. Steffen, J. M. Fink,C. Eichler, M. Baur, R. Bianchetti, P. J. Leek,S. Filipp, M. P. da Silva, et al., Nat Phys ,154 (2011), ISSN 1745-2473, URL http://dx.doi.org/10.1038/nphys1845 .[5] A. Cottet, T. Kontos, and B. Dou¸cot,Phys. Rev. B , 205417 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.205417 .[6] O. Dmytruk, M. Trif, C. Mora, and P. Si-mon, Phys. Rev. B , 075425 (2016),URL https://link.aps.org/doi/10.1103/PhysRevB.93.075425 .[7] C. Mora, C. Altimiras, P. Joyez, andF. Portier, Phys. Rev. B , 125311(2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.125311 .[8] C. Altimiras, F. Portier, and P. Joyez, Phys.Rev. X , 031002 (2016), URL https://link.aps.org/doi/10.1103/PhysRevX.6.031002 .[9] A. L. Grimsmo, F. Qassemi, B. Reulet,and A. Blais, Phys. Rev. Lett. , 043602(2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.116.043602 .[10] J. Lepp¨akangas, G. Johansson, M. Marthaler,and M. Fogelstr¨om, New Journal of Physics ,015015 (2014), URL http://stacks.iop.org/1367-2630/16/i=1/a=015015 .[11] J. Lepp¨akangas, G. Johansson, M. Marthaler,and M. Fogelstr¨om, Phys. Rev. Lett. ,267004 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.110.267004 .[12] Y.-Y. Liu, K. Petersson, J. Stehlik, J. Taylor,and J. Petta, Phys. Rev. Lett. , 036801(2014), URL http://link.aps.org/doi/10.1103/PhysRevLett.113.036801 .[13] J.-C. Forgues, C. Lupien, and B. Reulet,Phys. Rev. Lett. , 130403 (2015),URL https://link.aps.org/doi/10.1103/PhysRevLett.114.130403 .[14] F. Chen, J. Li, A. D. Armour, E. Brahimi,J. Stettenheim, A. J. Sirois, R. W. Simmonds,M. P. Blencowe, and A. J. Rimberg, Phys. Rev.B , 020506 (2014), URL http://link.aps.org/doi/10.1103/PhysRevB.90.020506 .[15] M. C. Cassidy, A. Bruno, S. Rubbert,M. Irfan, J. Kammhuber, R. N. Schouten, A. R. Akhmerov, and L. P. Kouwenhoven,Science , 939 (2017), ISSN 0036-8075,http://science.sciencemag.org/content/355/6328/939.full.pdf,URL http://science.sciencemag.org/content/355/6328/939 .[16] J.-C. Forgues, C. Lupien, and B. Reulet,Phys. Rev. Lett. , 043602 (2014),URL http://link.aps.org/doi/10.1103/PhysRevLett.113.043602 .[17] M. J. Gullans, J. Stehlik, Y.-Y. Liu, C. Eichler,J. R. Petta, and J. M. Taylor, Phys. Rev. Lett. , 056801 (2016), URL http://link.aps.org/doi/10.1103/PhysRevLett.117.056801 .[18] M. Westig, B. Kubala, O. Parlavecchio,Y. Mukharsky, C. Altimiras, P. Joyez,D. Vion, P. Roche, D. Esteve, M. Hofheinz,et al., Phys. Rev. Lett. , 137001(2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.137001 .[19] S. Jebari, F. Blanchet, A. Grimm, D. Hazra,R. Albert, P. Joyez, D. Vion, D. Estve,F. Portier, and M. Hofheinz, Nature Electron-ics , 223 (2018), URL https://doi.org/10.1038/s41928-018-0055-7 .[20] M. Hofheinz, F. Portier, Q. Baudouin,P. Joyez, D. Vion, P. Bertet, P. Roche, andD. Esteve, Phys. Rev. Lett. , 217005(2011), URL http://link.aps.org/doi/10.1103/PhysRevLett.106.217005 .[21] V. Gramich, B. Kubala, S. Rohrer, andJ. Ankerhold, Phys. Rev. Lett. , 247002(2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.111.247002 .[22] S. Dambach, B. Kubala, V. Gramich, andJ. Ankerhold, Phys. Rev. B , 054508(2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.054508 .[23] J. Puertas Martnez, S. Lger, N. Gheeraert,R. Dassonneville, L. Planat, F. Foroughi,Y. Krupko, O. Buisson, C. Naud, W. Hasch-Guichard, et al., npj Quantum Information , 19 (2019), URL https://doi.org/10.1038/s41534-018-0104-0 .[24] R. Kuzmin, R. Mencia, N. Grabon, N. Mehta,Y.-H. Lin, and V. E. Manucharyan, Quantumelectrodynamics of a superconductor-insulatorphase transition (2018), arXiv:1805.07379.[25] K. Le Hur, Phys. Rev. B , 140506(2012), URL https://link.aps.org/doi/10.1103/PhysRevB.85.140506 .[26] M. Goldstein, M. H. Devoret, M. Houzet, andL. I. Glazman, Phys. Rev. Lett. , 017002(2013), URL https://link.aps.org/doi/10. .[27] B. Peropadre, D. Zueco, D. Porras, and J. J.Garc´ıa-Ripoll, Phys. Rev. Lett. , 243602(2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.111.243602 .[28] N. Gheeraert, S. Bera, and S. Florens,New Journal of Physics , 023036 (2017),URL http://stacks.iop.org/1367-2630/19/i=2/a=023036 .[29] G.-L. Ingold and Y. V. Nazarov, in Singlecharge tunneling , edited by H. Grabert andM. H. Devoret (Plenum, 1992).[30] D. Averin, Y. Nazarov, and A. Odintsov, Phys-ica B , 945 (1990).[31] T. Holst, D. Esteve, C. Urbina, and M. H. De-voret, Phys. Rev. Lett. , 3455 (1994).[32] A. Grimm, F. Blanchet, R. Albert,J. Lepp¨akangas, S. Jebari, D. Hazra, F. Gus-tavo, J.-L. Thomassin, E. Dupont-Ferrier,F. Portier, et al., Phys. Rev. X , 021016(2019), URL https://link.aps.org/doi/10.1103/PhysRevX.9.021016 .[33] C. Padurariu, F. Hassler, and Y. V. Nazarov,Phys. Rev. B , 054514 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.86.054514 .[34] More details about the fabrication process andmeasurement procedure can be found in the on-line Supplementary Material, which containsrefs [35-50].[35] G.-L. Ingold and H. Grabert, Europhys. Lett. , 371 (1991).[36] B. Kubala, V. Gramich, and J. Anker-hold, Phys. Scr. T165 , 014029 (2015), URL http://stacks.iop.org/1402-4896/2015/i=T165/a=014029 .[37] A. D. Armour, B. Kubala, and J. Anker-hold, Phys. Rev. B , 214509 (2017),URL https://link.aps.org/doi/10.1103/PhysRevB.96.214509 .[38] D. Walls and G. Milburn, Quantum Op-tics (Springer Berlin Heidelberg, 2009), ISBN9783540814887.[39] S. Boutin, D. M. Toyli, A. V. Venka-tramani, A. W. Eddins, I. Siddiqi, andA. Blais, Phys. Rev. Applied , 054030(2017), URL https://link.aps.org/doi/10.1103/PhysRevApplied.8.054030 .[40] E. del Valle, A. Gonzalez-Tudela, F. P.Laussy, C. Tejedor, and M. J. Hartmann,Phys. Rev. Lett. , 183601 (2012),URL https://link.aps.org/doi/10.1103/PhysRevLett.109.183601 . [41] E. del Valle, New Journal of Physics ,025019 (2013), URL http://stacks.iop.org/1367-2630/15/i=2/a=025019 .[42] C. Dory, K. A. Fischer, K. M¨uller, K. G.Lagoudakis, T. Sarmiento, A. Rundquist,J. L. Zhang, Y. Kelaita, N. V. Sapra,and J. Vuˇckovi´c, Phys. Rev. A , 023804(2017), URL https://link.aps.org/doi/10.1103/PhysRevA.95.023804 .[43] C. M. Caves, Phys. Rev. D , 1817 (1982).[44] J. M. Fink, M. Kalaee, A. Pitanti, R. Norte,L. Heinzle, M. Davan¸co, K. Srinivasan, andO. Painter, Nature Communications , 12396EP (2016), article, URL http://dx.doi.org/10.1038/ncomms12396 .[45] M.-C. Harabula, T. Hasler, G. F¨ul¨op,M. Jung, V. Ranjan, and C. Sch¨onenberger,Phys. Rev. Applied , 054006 (2017),URL https://link.aps.org/doi/10.1103/PhysRevApplied.8.054006 .[46] T. Hasler, M. Jung, V. Ranjan, G. Puebla-Hellmann, A. Wallraff, and C. Sch¨onenberger,Phys. Rev. Applied , 054002 (2015),URL https://link.aps.org/doi/10.1103/PhysRevApplied.4.054002 .[47] C. Altimiras, O. Parlavecchio, P. Joyez,D. Vion, P. Roche, D. Esteve, and F. Portier,Applied Physics Letters , 212601 (2013),https://doi.org/10.1063/1.4832074, URL https://doi.org/10.1063/1.4832074 .[48] T. Holmqvist, M. Meschke, and J. P.Pekola, Journal of Vacuum Science &Technology B: Microelectronics andNanometer Structures Processing, Mea-surement, and Phenomena , 28 (2008),https://avs.scitation.org/doi/pdf/10.1116/1.2817629,URL https://avs.scitation.org/doi/abs/10.1116/1.2817629 .[49] I. Wolff, Coplanar Microwave Integrated Cir-cuits (Wiley, 2006).[50] G. J. Dolan, Applied Physics Letters , 337(1977), https://doi.org/10.1063/1.89690, URL https://doi.org/10.1063/1.89690 .[51] G. Schn and A. Zaikin, Physics Reports , 237 (1990), ISSN 0370-1573, URL .[52] H. Grabert, G.-L. Ingold, and B. Paul,EPL (Europhysics Letters) , 360 (1998),URL http://stacks.iop.org/0295-5075/44/i=3/a=360 .[53] P. Joyez, Phys. Rev. Lett. , 217003(2013), URL https://link.aps.org/doi/10. .[54] M. P. da Silva, D. Bozyigit, A. Wallraff,and A. Blais, Phys. Rev. A , 043804(2010), URL https://link.aps.org/doi/10.1103/PhysRevA.82.043804 .[55] The flux jumps due to vortex depinning wereslow enough that we could compensate forthem manually. We could thus obtain the totalflux dependance of the emitted power, whosefit with a sinusoidal law yields the same valuefor E J within experimental errors and a negli-gible asymmetry.[56] J.-R. Souquet and A. A. Clerk, Phys. Rev. A , 060301 (2016), URL https://link.aps.org/doi/10.1103/PhysRevA.93.060301 .[57] S. Meister, M. Mecklenburg, V. Gramich,J. T. Stockburger, J. Ankerhold, andB. Kubala, Phys. Rev. B , 174532 (2015),URL https://link.aps.org/doi/10.1103/PhysRevB.92.174532 .[58] J. Est`eve, M. Aprili, and J. Gabelli,arXiv:1807.02364 (2018). ntibunched photons emitted by a dc-biased Josephson junction:Supplementary Material C. Rolland , ∗ A. Peugeot , ∗ M. Westig , B. Kubala , S. Dambach , Y. Mukharsky , C. Altimiras ,H. Lesueur , P. Joyez , D. Vion , P. Roche , D. Esteve , J. Ankerhold , † and F. Portier ‡ SPEC (UMR 3680 CEA-CNRS), CEA Paris-Saclay, 91191 Gif-sur-Yvette, France and Institute for Complex Quantum Systems and IQST,University of Ulm, 89069 Ulm, Germany
I. DERIVATION OF EQ. 2 OF THE MAIN TEXT USING P ( E ) THEORY
The spectral density of the emitted radiation is given by [1]: γ ( V, ν ) = 2Re[ Z ( ν )] R Q π ~ E J P (2 eV − hν ) , (1)where Z ( ν ) is the impedance across the junction, R Q is the superconducting resistance quantum R Q = h/ e , E J is the Josephson energy of the junction, and P ( E ) represents the probabilitydensity for a Cooper pair tunneling across the junction to dissipate the energy E into the electro-magnetic environment described by Z ( ν ) [2]. P ( E ) is a highly nonlinear transform of Z ( ν ): P ( E ) = π ~ R ∞−∞ exp[ J ( t ) + iEt/ ~ ] dtJ ( t ) = R + ∞−∞ d ωω Z ( ω ) R Q e − iωt − − e − β ~ ω , (2)where β = 1 /k B T . For an LC oscillator of infinite quality factor at zero temperature, P ( E ) is givenby P ( E ) = e − r X n r n n ! δ ( eV − n ~ ω ) (3)where r = π q LC /R Q and ω = 1 / √ LC .Here, we consider the case of a mode of finite linewidth, so that near the resonance the real partof the impedance can be approximated as Z ( ω ) R Q ’ r L ( ω, ω , Q ) . (4)where L ( ω, ω , Q ) ≡ π Q Q (cid:16) ωω − (cid:17) ∗ These two authors contributed equally. † Electronic address: email:[email protected] ‡ Electronic address: email:[email protected] denotes a Lorentzian function centered at ω with a maximum value π Q and a quality factor Q = ω ∆ ω . Note that R L ( ω, ω , Q ) dω = ω .For such a finite- Q mode, we aim to get a formula similar to Eq. 3, i.e. we look for an expansion P ( E ) = P ( E ) + P ( E ) + P ( E ) + . . . + P n ( E ) + . . . (5)where each P n ( E ) ∝ r n . However, from the integral expressions (2), accessing the different mul-tiphoton peaks, i.e. calculating P ( E ’ n ~ ω ) is not straight-forward. Such an expansion can beobtained using the so-called Minnhagen equation [2], which is an exact integral relation obeyed by P ( E ), valid for any impedance. We first establish the Minnhagen equation starting from e J ( t ) − e J ( ∞ ) = R t −∞ dτ J ( τ ) e J ( τ ) , which, using the definition (2) of J can be recast as e J ( t ) − e J ( ∞ ) = − i Z + ∞−∞ dω h ( ω ) Z ∞−∞ dτ e − iω τ e J ( τ ) θ ( t − τ )where θ is the Heaviside function, h ( ω ) = − e − β ~ ω Z ( ω ) R Q and using the fact that J ( −∞ ) = J ( ∞ ).The rightmost integral being the Fourier transform of a product, we replace it by the convolutionproduct of the Fourier transforms and use the detailed balance property of h and P to simplify ther.h.s.: e J ( t ) − e J ( ∞ ) = − i Z + ∞−∞ dω h ( ω ) Z du πδ ( u ) + ie it u u ! P ( − ω − u )= Z + ∞−∞ dω h ( ω ) Z du e itu u P ( − ω − u ) . Finally, we take the Fourier transform on both sides and rearrange, which yields the Minnhagenequation P ( E ) = ~ E R P ( E − ~ ω ) − e − β ~ ω Z ( ω ) R Q dω + δ ( E ) e Re J ( ∞ ) . (6)At zero temperature − e − β ~ ω → θ ( ω ) and P ( E ) is zero for negative energies, so that the Minnhagenequation is most frequently found written as P ( E ) = ~ E R E P ( E − ~ ω ) Z ( ω ) R Q dω + δ ( E ) e Re J ( ∞ ) . (7)Plugging the expansion (5) into Eq. 6, one immediately gets P ( E ) = δ ( E ) e J ( ∞ ) P ( E ) = 1 E Z ∞−∞ P ( E − ~ ω ) r L ( ω, ω , Q )1 − e − β ~ ω d ~ ω ’ e J ( ∞ ) ~ ω r L (cid:18) E ~ , ω , Q (cid:19) where the approximation of the last line was obtained assuming that k B T (cid:28) ~ ω and taking thevalue of the denominator at E = ~ ω –where L (and P ) peak– which is reasonable if the Q is largeenough. By repeated replacement in Eq. 6 and with similar approximations, one systematicallyobtains the higher orders terms of (5) as shifted Lorentzians of constant QP n > ( E ) ’ e J ( ∞ ) r n nn ! L ( E/ ~ , nω , Q ) ~ ω whose value at each peak are P n > ( E = n ~ ω ) = 2 π e J ( ∞ ) r n nn ! Q ~ ω yielding a tunneling rate at the peaksΓ e ( eV = n ~ ω ) = 1 ~ E J e J ( ∞ ) ~ ω r n n ! Qn .
Note that the Cooper pair rates at different orders scale with an extra
Q/n compared to the naiverates obtained from Eq. 3.In the main text, E J e J ( ∞ ) is called E ∗ J . This renormalization of the Josephson energy is obtainedfrom the zero point phase correlator J ( ∞ ) = −h ϕ (0) ϕ (0) i = − Z + ∞ dωω Z ( ω ) R Q coth βω k B T = 0 and for an RLC parallel resonator (it is important that Re Z ( ω ∼ ∝ ω for proper convergence) yields J ( ∞ ) = − Qr (cid:18) π atan Q − √ Q − (cid:19)p Q − − r (cid:18) − πQ + O (cid:18) Q (cid:19)(cid:19) , in agreement with the expression E ∗ J = E J e − r/ used in the main text (The finite- Q correction tothis renormalization is of order of 1%, beyond the precision of our measurements). In ref. [1], E ∗ J was given with an approximate first-order expansion of the phase correlator valid for small phasefluctuations (and which was correct for the small r value in that paper).We can use the above expressions to calculate the total emitted power via the single photonprocesses by two different ways. First, we use Eq. 1 at lowest order, to get the spectral density ofthe emitted radiation: γ ( V, ν ) ’ Z ( ν )] R Q π ~ E J P ( E = 2 eV − hν ) = e J ( ∞ ) Z ( ν )] R Q π ~ E J δ (2 eV − hν ) , (8)which, upon integrating over ν , gives Eq. 2 of the main text. Alternatively, one can calculate theCooper pair tunneling rate using P , and get the photon emission rate from energy conservation,yielding the same result.In Figure I we compare the exact P ( E ) result and the approximate formula, for the experimentalparameters. (cid:144) (cid:209) Ω (cid:72) E (cid:76) Figure 1: Comparison the exact P ( E ) result obtained by numerical evaluation of Eqs. (2) and the approx-imate sum of Lorentzians, evaluated for the experimental parameters ( Q = 36 . r = 0 . II. FRANCK-CONDON BLOCKADE IN THE JOSEPHSON-PHOTONICSHAMILTONIAN
The starting point of our theoretical description, the time-dependent Hamiltonian, see Eq. 1 ofthe main text, H = ( a † a + 1 / hν R − E J ( φ ) cos[2 eV t/ ~ − √ r ( a + a † )] , (9)describes a harmonic oscillator with an unusual, nonlinear drive term. Going into a frame rotatingwith the driving frequency, ω J = 2 eV / ~ , the oscillator operators, a and a † , acquire phase termsrotating with the same frequency. The cosine term of the Hamiltonian can then be rewritten inJacobi-Anger form so that Bessel functions of order k appear as prefactors of terms rotating withinteger multiples of the driving frequency, kω J .A rotating-wave approximation neglects time-dependent terms and, taking proper account of thecommutation relations of oscillator operators, results in the RWA Hamiltonian (on resonance), H RWA = iE J e − r/ : ( a † − a ) J ( √ rn ) √ n : , (10)where : . . . : prescribes normal ordering. While the appearance of a Bessel function highlightsthe nonlinear-dynamical aspects of the system, the Hamiltonian (10) is completely equivalent toexpression Eq. 2 of the main text, given in the main text, using the displacement operator, whichemphasizes the connection to Franck-Condon physics.From either of the two equivalent forms of the RWA Hamiltonian, explicit expressions for thetransition matrix elements in terms of associated Laguerre polynomials, h RWA n,n +1 = ie − r/ √ r √ n + 1 L n ( r ) , (11)can easily be found. Normal ordering reduces the power series of the Bessel function to a low-orderpolynomial in r (with order n for h RWA n,n +1 ) and a universal prefactor, describing renormalizationof the Josephson coupling. Transition matrix elements thus vanish at the roots of the associatedLaguerre polynomials (which in the semiclassical limit of small r and large n approach zeros of theBessel function J ).Some simple results can be directly read off from the transition matrix elements; such asthe zero-delay correlations that for weak driving measure the probability of two excitations, g (2) (0) = h n ( n − i / h n i ≈ P /P ≈ (cid:12)(cid:12) h RWA1 , /h RWA0 , (cid:12)(cid:12) . The last approximate equality expressesthe probabilities P / by transition matrix elements, as found by considering the transition ratesfor the corresponding two-stage excitation process and decay from the Fock states. As mentionedin the main text, in the harmonic limit, r (cid:28)
1, where the matrix elements scale with √ n + 1,this would result in the familiar Poissonian correlations and g (2) , H0 (0) = 1. This contrasts tothe antibunching found in our experiment relying on the fact that the experimental parameter r ∼
1, while not quite close to the zero of the transition matrix element h RWA1 , ∝ L ( r ) = 2 − r ,is sufficiently large for a considerable suppression of excitations beyond a single photon in theresonator. Coincidentally, the actual value of r is very close to one of the roots, r ≈ .
93 of L ( r ) = ( − r + 12 r − r + 24) ∝ h RWA3 , , so that the system closely resembles a four-level system.In the idealized model description by the approximated Hamiltonian Eq. 3 of the main text andthe quantum master equation Eq. 6 of the main text, the vanishing of a transition matrix elementimplies a strict cut-off of the system’s state space at the corresponding excitation level. Variouscorrection terms discussed in the next subsection can lift such a complete blockade and thereforegain relevance once the system is closer to a root than for our r ∼ III. CORRECTION TERMS TO HAMILTONIAN AND QUANTUM MASTEREQUATION
The possible impact of various terms and processes not included in RWA Hamiltonian Eq. 3 andquantum master equation Eq. 6 of the main text were carefully checked and found to be completelynegligible compared to the error bars due to other experimental uncertainties.Specifically, the impact of rotating-wave corrections to the time-independent RWA Hamiltonianis sufficiently reduced by the quality factor, Q = 36 .
6. Close to the complete suppression of resonant g (2) (0) contributions at r = 2, for very weak driving, and for a bad cavity such processes can becomemore relevant, as discussed in some detail in Ref. [3]. The limit of extremely strong driving, where E J & hν R , not reached here, is discussed in Appendix D of Ref. [4].Access to higher Fock-states cut-off by vanishing transition matrix elements could, in principle,also be provided by thermal excitations, i.e., by Lindblad terms not included in the T = 0 limit ofthe quantum master equation Eq. 6 of the main text. The latter, however, is safe to use for theexperimentally determined mode temperature of ∼
15 mK in our device.Finally, there are low-frequency fluctuations of the bias voltage, causing the spectral broadening ofthe emitted radiation (as argued in the main text) that are not accounted for by the quantum masterequation Eq. 6 of the main text. Their effect can be modeled, either by an additional Lindblad-dissipator term acting on a density matrix in an extended JJ-resonator space (as described, forinstance, in the supplementals to [5]), or by employing the quantum master equation Eq. 6 of themain text and average the results over a (Gaussian) bias-voltage distribution centered around thenominal biasing on resonance.As argued above, the principal antibunching effect can be understood from transition rate ar-guments so that it is not sensitive to the phase of the driving, which is becoming undetermineddue to the fluctuating voltage bias. In consequence, the measured g (2) (0) is nearly insensitive tolow-frequency fluctuations. Residual effects of detuning on g (2) ( τ ) entering g (2) (0) via the filteringare negligible due to the large ratio between inverse resonator lifetime and spectral width, cf. Fig. 3of the main text. IV. ACCOUNTING FOR FILTERING
A theoretical approach based on the quantum master equation Eq. 6 of the main text givesdirect access to any properly time-/anti-time-ordered products of multiple system operators, whichare evaluated by the quantum-regression method. Using input-output theory [6], any arbitrarilyordered product of multiple output operators can readily be expressed in such system-operatorobjects.The measured signals, however, do not immediately correspond to output operators but containoperators at the end of the microwave output chain, hence, undergoing additional filtering. Anend-of-chain operator acting at a certain time is consequently linked to output operators at allpreceding times via a convolution with the filter-response function in the time domain, see thediscussion in [7]. Specifically, the measured two-time correlator G (2) ( t , t ) = h a † t a † t a t a t i , wheretwo operators each are acting at two different times t / , is related to a four-operator object witheach operator acting at a different time.To simulate the measured G (2) ( t , t ), it is necessary to calculate corresponding four-operatorobjects and then average each instance of time with a probability distribution given by the filter-response function in the time domain. An explicit, worked out example for a three-time object canbe found in Appendix E of Ref. [8]. For the special case of a Lorentzian filter-response function asimpler scheme has been put forward [9–11].For numerical efficiency, here, we calculate the various four-time objects by evaluating the timeevolution governed by the exponential of the Liouville superoperator using Sylvester’s formula andFrobenius covariants. This approach is completely equivalent to time-evolving the quantum masterequation with any standard differential-equation solver. In a final step, three-dimensional temporalintegrals have to be numerically evaluated, wherein the uncertainty of time differences (betweenpoints at which the different operators act) is linked to the experimental filter function, see above.The multiple integrals over differently ordered operator objects hamper an intuitive understand-ing of the effects of filtering. Therefore, it may be helpful to compare the complex effects of filteringhere to a more conventionally encountered scheme describing detection-time uncertainties. In Fig. 2,we show the results of a simple, incomplete filtering description, which only allows for variations inthe time difference, τ = t − t , but artificially keeps annihilation and creation operators belongingto the same pair at equal times. Apparently, deviations from the correct, complete filtering scheme,cf. Fig. 2, are reasonably small, so that important effects of the filtering are correctly captured;except for the regime of very strong driving, where Rabi-like oscillations in the time-dependencegain strong influence on the measured g (2) ( τ = 0). The simple scheme suggests an intuitive under-standing of the effect of filtering as a simple convolution of the unfiltered G (2) ( τ = t − t ) withthe distribution function for the time difference τ due to the filtering effect on t / . Note, thatthe time-difference distribution function is itself gained by convoluting the filter function in timedomain with itself, so that τ is, in general, not distributed identical to the detection times t / , butonly for special cases of the filtering function. - 1 5 - 1 0 - 5 0 5 1 0 1 50 , 00 , 20 , 40 , 60 , 81 , 0 g(2)( t ) d e l a y t ( n s ) Figure 2: g (2) ( τ ) at n = 0 . V. MEASUREMENT OF THE g (2) ( τ ) FUNCTIONA. Principle of the measurement
Our measurement scheme is to process the small signals leaking out of the sample with stan-dard microwave techniques (filtering, amplification and heterodyning), to digitize them with anacquisition card and to compute numerically the correlation functions relevant to characterize oursingle-photon source – the most important of them being the second-order coherence function: g (2) ( t, τ ) = h ˆ a † ( t )ˆ a † ( t + τ )ˆ a ( t + τ )ˆ a ( t ) ih ˆ a † ( t )ˆ a ( t ) ih ˆ a † ( t + τ )ˆ a ( t + τ ) i Filtering and amplification are performed in multiple stages but can nonetheless be described asthe action of a single effective amplifier of gain G , which adds a noise mode ˆ h in a thermal stateat temperature T N to the input signal mode ˆ a . The output of such an amplifier is then √ G ˆ a + √ G − h † [12].After digitization of this amplified signal we demodulate numerically its I - Q quadratures. The n -thorder moment of the complex enveloppe S ( t ) = I ( t ) + jQ ( t ) is then a bilinear function of all themoments of ˆ a and of ˆ h up to order n [13]. By setting the bias voltage across the emitting squidto zero (the off position) and thus putting ˆ a in the vacuum state, we can measure independentlythe moments of ˆ h . We then iteratively substract them from the moments of S measured when thebias voltage is applied ( on position) to reconstruct the moments of ˆ a . Similarly, we can reconstruct g (2) ( τ ) from the on-off measurements of all the correlation functions of S ( t ) up to order 4.In addition to this, by splitting the signal from the sample over two detection chains (channels”1” and ”2”) in a Hanburry Brown-Twiss setup, we can cross-correlate the outputs S , S of thetwo channels to reduce the impact of the added noise on correlation functions. A model of thisnoise is thus needed to determine which combination of S and S is best suited to measure g (2) ( τ )accurately. B. Model for the detection chain
The input-output formalism links the cavity operator ˆ a to the ingoing and outgoing transmissionline operators ˆ b in , ˆ b out by: √ κ ˆ a ( t ) = ˆ b in ( t ) + ˆ b out ( t ), with κ = 2 πν R /Q the energy leak rate (seeFig. 1). In our experimental setup, ˆ b in describes the thermal radiation coming from the 50 Ω loadon the isolator closest to the sample. This load being thermalized at 15 mK (cid:28) hν R /k B , the modesimpinging onto the resonator can be considered in their ground state and the contribution of ˆ b in toall the correlation functions vanishes. We thus take ˆ b out as being an exact image of ˆ a , and all theirnormalized correlation functions as being equal.As described before, the emitted signals are split between two detection chains, filtered, amplified,and mixed with a local oscillator before digitization. Each one of these steps adds a noise mode tothe signal (FIG. 3). The beam-splitter right out of the resonator is implemented as a hybrid couplerwith a cold 50 Ω load on its fourth port, which acts as an amplifier of gain 1 / h † bs to the signal[ ? ]. The different amplifying stages are summed up intoone effective amplifier for each channel, with noise temperatures T N = 13 . T N = 14 . V i ( t ) on channel i by the acquisition card, is harder tomodel to the quantum level. After digitization, we process chunks of signal of length 1024 samplesto compute the analytical signal S i ( t ) = V i ( t ) + H ( V i )( t ), with H the discrete Hilbert transform. Ascomputing the analytical signal from V i ( t ) accounts to measuring its two quadratures, which arenon-commuting observables, quantum mechanics imposes again an added noise mode. We sum upthis digitization noise with the heterodyning noise into a single demodulation noise ˆ h IQ i (Fig. 1). Figure 3: Detection chain model, taking into account all the added noise modes (in red).
In the end, we record measurements of ˆ S ( t ) and ˆ S ( t ), with ˆ S i ∝ ˆ a i +ˆ h † i . Here ˆ a i ( t ) ∝ ˆ b out ( t − τ i ),where τ i is the time delay on channel i . ˆ h i is a thermal noise with an occupation number of about 65photons, which summarizes the noises added by all the detection steps. In practice, the dominantnoise contribution stems from the amplifiers closest to the sample. Note also that we do not considerhere the effect of the finite bandpass of the filters, which complicates the link between ˆ a i and ˆ b out . C. Computing correlations
From each chunk of signal recorded we compute a chunk of S i ( t ) of the same length 1024. Wethen compute the correlation functions we need as: C X,Y ( τ ) = h X ∗ ( t ) Y ( t + τ ) i = F − ( F ( X ) ∗ F ( Y ))where h ... i stands for the average over the length of the chunk and F is the discrete Fourier transform.Finally, we average the correlation functions from all the chunks and stock this result for furtherpost-processing.To illustrate how we reconstruct the information on ˆ a from S , S , let’s consider the first ordercoherence function g (1) ( τ ) = h ˆ a † ( t )ˆ a ( t + τ ) ih ˆ a † ˆ a i . We start with the product: S ( t ) ∗ S ( t + τ ) ∝ ˆ a † ( t )ˆ a ( t + τ ) + ˆ h ( t )ˆ h † ( t + τ ) + ˆ a † ( t )ˆ h † ( t + τ ) + ˆ h ( t )ˆ a ( t + τ )We then make the hypothesis that ˆ a and ˆ h are independent and hence uncorrelated, which shouldobvioulsy be the case as the noise in the amplifier cannot be affected by the state of the resonator.Then when averaging: h ˆ a ( τ )ˆ h ( t + τ ) i = h ˆ a ( τ ) ih ˆ h ( t + τ ) i = 0as there is no phase coherence in the thermal noise, i.e. h ˆ h i = 0. We then have: h S ( t ) ∗ S ( t + τ ) i ∝ h ˆ a † ( t )ˆ a ( t + τ ) i + h ˆ h ( t )ˆ h † ( t + τ ) i off position: h S ( t ) ∗ S ( t + τ ) i off ∝ h ˆ h ( t )ˆ h † ( t + τ ) i and in the on position: h S ( t ) ∗ S ( t + τ ) i on ∝ h ˆ a † ( t )ˆ a ( t + τ ) i + h S ( t ) ∗ S ( t + τ ) i off such that: g (1) ( τ ) = h S ( t ) ∗ S ( t + τ ) i on − h S ( t ) ∗ S ( t + τ ) i off h S ∗ S i on − h S ∗ S i off Now as we are considering states of the resonator with at most 1 photon, we typically have: h S ∗ S i off ’ h S ∗ S i on (cid:29) h S ∗ S i on − h S ∗ S i off Then any small fluctuation of the gain of the detection chain or of the noise temperature duringthe experiment reduces greatly the contrast on g (1) ( τ ). We hence rely on the cross-correlation X ( τ ) = h S ∗ ( t ) S ( t + τ ) i . Due to a small cross-talk between the two channels this cross-correlationaverages to a finite value even in the off position, but which is 60 dB lower than the autocorrelationof each channel. We hence use: g (1) ( τ ) = X ( τ ) on − X ( τ ) off X (0) on − X (0) off The same treatment allows to compute g ( τ ) with slightly more complex calculations. Theclassical Hanburry Brown-Twiss experiment correlates the signal power over the two channels, i.e.extracts g (2) ( τ ) from h S ∗ S ( t ) S ∗ S ( t + τ ) i . The off value of this correlator is once again muchbigger than the relevant information of the on-off part, and any drift of the amplifiers blurs theaveraged value of g (2) ( τ ).To circumvent this difficulty, we instead use C ( t ) = S ∗ ( t ) S ( t ) as a measure of the instantaneouspower emitted by the sample, provided that the time delay between the two detection lines iscalibrated and compensed for. We then have: g (2) ( τ ) = h C ( t ) C ( t + τ ) i on − h C ( t ) C ( t + τ ) i off ( h C i on − h C i off ) − h C i off h C i on − h C i off − ( X ( τ ) on − X ( τ ) off ) X ( − τ ) off ( h C i on − h C i off ) − ( X ( − τ ) on − X ( − τ ) off ) X ( τ ) off ( h C i on − h C i off ) (12) VI. ELECTROMAGNETIC SIMULATIONS
The experiment can be schematically represented by figure 4, where the high impedance mi-crowave mode we will use is represented in the green box as a LC resonant circuit. Its resonantpulsation ω and characteristic impedance Z C are given by Z C = r LC ; ω = 1 √ LC . Aiming at a coupling strength r ’ Z C ∼ k Ω, L ∼
60 nHand C ∼
15 fF.1 V bias-tee samplevoltagebiasing radiation collection Josephsonjunctionresonator
Figure 4: Schematic diagram of the experiment. The sample, represented by the green box consists ina Josephson junction galvanically coupled to a high impedance resonator consisting in a on-chip spiralinductor. It is connected to a DC biasing circuit, represented in red, and to a 50 Ω detection line, representedin blue, through a bias Tee.
In order to reduce the capacitance we fabricated the resonator on a quartz wafer, with a smalleffective permittivity ε r = 4 . µ m, which increases thecapacitance to ground, where as a bridge forms a capacitor with every turn of the coil that mustbe taken into account in the microwave simulations.We chose to use an Alumninum bridge, supported by a > µ m BCB layer. BCB is a low lossdielectric which has been developed for such applications by the microwave industries which alsohas a relatively low permittivity.A last parameter of the resonator that can be tuned is its quality factor Q. We consider a simpleparallel LC oscillator with Q = f ∆ f = Z C Z Det , where Z Det is the impedance of the detection line as seen from the resonator and ∆ f the resonancebandwidth at -3dB. To tune Q , we can insert an impedance transformer between the 50Ω measure-ment line and the resonator and thus increase the effective input impedance, to decrease the qualityfactor.In order to simulate our resonators, we use a high frequency electromagnetic software tool forplanar circuits analysis : Sonnet. The system simulated by this software consists in several metalliclayers separated by dielectrics as shown in Fig. 5.Each metallic sheet layer contains a metallic pattern for the circuit, with strip-lines or resonatorsand can be connected to the other layers through vias. Dielectric layers properties and thicknesscan also be chosen.2 Figure 5: Sonnet schematic of the dielectric stack.
This stack is enclosed in a box with perfect metallic walls. The simulated device sees the outerworld through ports that sit at the surface of the box, or are added inside the box, as probes shownin Fig. 6. Sonnet also allows us to insert lumped electric component in the circuit, between twopoints of the pattern. Z S =50 Ω Z env C J DUT
Figure 6: Sonnet port configuration.
As we are interested in the behavior of the environment seen by the junction, we will replaceit by a port, which will act like a probe. We assume that the Josephson energy is small enough3for the admittance of the junction associated to the flow of Cooper pairs to be negligible ; we canthus model the junction as an open port. Furthermore, in order to take into account the junction’sgeometric capacitance, we add a discrete capacitor in parallel to ground as presented in Fig. 6.The other port of the resonator is model by a 50 Ω resistor, modeling the detection line.Using the microwave simulation results, we predict the resonant frequency f , the impedanceseen by the junction Z out 2 , the quality factor Q and the environment characteristic impedance Z C . Coil nb of turns line width line space bridge23.5 1 µ m 2 µ m BCB / 1 . µ mResults f ∆ f Z C Re( Z env ) MAX ( C J = 2 fF) 5,1 GHz 60 MHz 2,05 kΩ 188kΩTable I: Geometric parameters of the resonator and associated characteristics. The corresponding schematic and result of simulations are shown in Fig. 7. Now, the lumpedcapacitor C J represents the capacitance of the Josephson junction alone, the rest of the capacitiancebeing implemented by the surrounding ground. R e ( Z ) ( k Ω ) Frequency (GHz) J = 2fFGround plane Figure 7: Final design drawing (left) and associated simulation result(right). a. Computing current densities
In order to understand the full resonator behavior, we havesimulated current and charge densities at resonance, as shown in Fig. 8.
A. Junction’s capacitance influence
The Josephson junction’s geometric capacitance C J is of the order of few fF (70 fF .µm − ) [18]and is also part of the environment seen by the pure Josephson element according to Z env ( ω ) = Z circuit ( ω )1 + jC J ωZ circuit ( ω ) , Resonator : impedance transformer d) Current Density (A.m -1 ) a) Circuit
04 00003.10 -5 A.m-1C.m-2 charge current
10 21 D e n s i t y ( a . u . ) Port b) Targetted behavior Ω external port internal port= 1M Ω c) Charge Densitiy (C.m -2 ) Figure 8: a) Our circuit has two ports. One external port “1” to be connected to the measurement chaincan be modeled as a 50Ω load. The second port “2” is internal to the circuit and parametrized to mimicthe Josephson junction open between the resonator and the ground plane probes the impedance seen by thefuture junction. These boundary conditions make us expect the resonator to behave like a λ/ λ/
4, the low impedance port 1 corresponds to a node in charge and an anti-node in current,while on the “open” side port 2, there is an accumulation of charges and no current. c) As expected, thereis a charge accumulation on the high impedance side of the resonator. As the coil is used as an inductancebut is also the capacitance of the circuit, there is an accumulation of charge at the periphery, i.e. in thefirst turn of the coil. d) There is indeed no current flowing through the high impedance side and we see anincrease toward the low impedance port. in c) and d) the ground plane shown in fig. 7 is not representedhere as it does not present peculiar current/charge density. Z circuit ( ω ) is the impedance of the resonant circuit connected to the measurement line withoutjunction.By adding a discrete capacitor to ground C = C J at the junction’s position in simulations andtuning its value, one can then observe in Fig. 9 that it is not negligible and must be taken intoaccount. frequency (GHz) R e ( Z e n v ) ( k Ω )
64 50200100 C J : 4 2 fF Figure 9: Real part of the impedance seen by the junction Re[ Z env ( ω )] for different junction capacitances As we aim at building a resonator with a capacitance around 15 fF, C J will account for 10 to20% of the total capacitance of the circuit. As a consequence, both characteristic impedance andresonant frequency will be decreased by 5 to 10%. B. Tuning the bandwidth using quarter wavelength resonator
According to table I and Fig. 7, our resonator is expected to have a bandwidth of ∆ f ∼
60 MHz, which is not much larger than the 3 MHz FHWM of the Josephson radiation due to lowfrequency voltage polarisation noise. It is thus useful to broaden this resonance while preservingthe characteristic impedance and resonant frequency.Keeping the same resonator geometry, one can enlarge its bandwidth by inserting a second res-onator between it and the source to play the role of an impedance transformer (quarter wavelength).Doing so, we can increase the input impedance seen by the coil and broaden the resonance.We have built this second stage of impedance transformer “on chip” between the measurementline (modeled by Z ) and the coil, using a lossless coplanar waveguide (CPW) of length l = λ/ Z Det = Z C Z , Figure 10: Circuit with an additional impedance transformer.∆ f (MHz) Z Det Z c , λ/ µm ) Gap ( µm )60 50 - - -100 100 70 25 10300 400 140 10 50500 600 173 5 67Table II: Influence of an additionnal impedance transformer on the resonator bandwidth. where Z C is the characteristic impedance of the line, Z Det the transformed detection impedance ofthe resonator and Z the 50Ω characteristic impedance of the detection line.One can then choose the impedance seen by the coil (the impedance Z of the transformer) bytuning the characteristic impedance Z C . To do so, textbook calculations allow to choose the goodratio between the width of the central conductor and the distance to ground plane on a particularsubstrate [19]. The bandwidth of the resonator, corresponding to an input impedance of 50Ω, is60 MHz. By adding quarter wavelength transformers, we increase ∆ f as listed in table VI B :Using the quarter wavelength transformer simulations as a first block and the previous coilresults as a second one, the full circuit was simulated, using the Sonnet “netlist” feature. Such acombination of previous simulations assumes no geometric “crosstalk” between the two resonators,which makes sense given that they are shielded from each other by ground planes. We obtainedthe results of Fig. 11.We were then able to check that the λ/ VII. FABRICATION
As mentioned above, we built the circuit of Fig. 12 on a 3 ×
10 mm low permittivity quartz chipwith a single input/output port adapted to a 50Ω measurement line.7 frequency (GHz) R e ( Z e n v ) ( k Ω )
64 5060180120 no λ /470 Ω Ω Ω Z C = 2,2 k Ω Figure 11: : initial simulation result and Netlist simulation results for the 3 λ/ Our fabrication process consists in 3 mains steps. First, we fabricate Niobium based coil andquarterwave impedance transfromers. Then, we connect the center of the coil to its periphery witha bridge and finally, as it is the most fragile element, we fabricate the Josephson junctions.
A. Resonator: coil and λ/ In order to be able to test the samples at 4K, we chose to built niobium based resonator. AsNiobium is of a much better quality when sputtered than evaporated, we used a top down approachfor this step.A 100 nm a layer of Niobium was first deposited on a 430 µm thick Quartz wafer at 2nm/s usinga dc-magnetron sputtering machine and then patterned by optical lithography and reactive ionetching (RIE).In order to pattern the resonators, we used an optical lithography process. The classical opticallithography process used a resist thick enough so that all the niobium between the lines can beremoved before all the resist is etched, the S1813 from Shipley.It was spinned according to the following recipe:1. 110 (cid:176) C prebake of the substrate on hot plate2. Resist spinning : S1813, 4000 rpm 45” / 8000 rpm 15”3. 2 min rebake on hot plateUsing these parameters, and performing interferometric measurements, we measured a resist layerof 1450 nm. The sample was then exposed with a Karl-S¨uss MJB4 optical aligner, with a dose of150 mJ/cm − (15 secs). Finally it was developed using microposit MF319 during 90 seconds andrinsed in deionized water for at least 1 min.The next step of the process is the reactive ion etching of the niobium film : we used a mixtureof CF and Ar (20/10 cc) at a pressure of 50 µ bar (plasma off) and a power of 50 W (209V) for 48 Figure 12: Photograph of the chip used for the experiments described in the main text.Figure 13: Fabrication of the coil. Left : photograph after optical lithography. Right : photograph afterniobium etching. minutes 45 seconds (150 nm). After this process, the sample was cleaned in 40 (cid:176)
C acetone for 10minutes to remove any resist residues and rinsed in IPA.The quarter wave resonator was fabricated at the same time as the coil by Niobium etching.9
Figure 14: 70 Ω quarter wavelength impedance transformer and coil. The whole chip is 3 ×
10 mm . B. Bridge
As we decided to use a dielectric spacer to support the bridge, we added two additional steps tothe fabrication process. One for the dielectric spacer, the second one for the brdige itself. One ofthe main difficulties of these steps is that, as the pads to connect the bridge is small, they requirevery precise alignment. a. Dielectric support
We chose to work with polymers derived from B-staged bisbenzocy-clobutene, sold as Cyclotene 4000 by Dow Chemicals and choose the lower viscosity, in order toobtain a spacer between 0.8 and 1.8 µ m thick: XU35133. The process was performed according tothe following recipe:1. 2 minutes prebake at 110 (cid:176) C2. Primer AP 3000 rpm 30 secs3. BCB XU : 3000rpm, 45secs/ 8000 rpm 15 secs4. 3 minutes rebake @80 (cid:176)
CUsing this technique, we obtained 1650 nm thick layers. The sample was then exposed with theMJB4 optical aligner, with during 3 seconds. The development of this resist is quite difficult as itis not dissolved by acetone:1. 30 secs on hot plate (70 (cid:176)
C) : to avoid that the bridge flows2. DS 3000 rinsing for 1 minute3. TS 1100 rinsing for 30 seconds4. 1 min rinsing in deionized water5. the sample was then dried while spinningIn order to obtain a flat surface and remove all resist residues, an RIE SF / O etching was performedfor 30 seconds (20/2 cc, 10 µ bar, 50W) as shown in Fig 15. Finally, the sample was rebaked during30 minutes at 190 (cid:176) C to stabilize the resist.0 µm00.10.20.30.40.50.60.70.80.9 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 µmL R µm-0.100.10.20.30.40.50.60.70.8 0 5 10 15 20 25 30 35 40 45 µmRL
Width
Height
TIR
Leveling: 2 zonesZoom: 46% 2 bars 2 zonesParameters... T h i c k n e ss distance from starting pointStepper measurement of BCB resist profile before and after etching Figure 15: The flatness of the sample was measured with a stepper. After 30 seconds of SF / O etching,resist residues have disappeared b. Bridge’s line As the spacer is quite thick, this step requires a thicker resist. We usedAZ5214 and obtained a 1.5 µ m layer of resist according to the recipe:1. 72 (cid:176) C prebake on hot plate2. microposit primer : 6000rpm for 30 seconds3. AZ5214 : 4000 rpm during 60s, 8000rpm during 10s4. 2min rebake at 100 (cid:176)
C with a bescher on top of the sampleThe sample was then aligned and exposed during 7s using the MJB4. As the AZ5214 is a negativeresist which can be reversed, we rebaked the sample for 3min at 120 (cid:176)
C and performed a flood expo-sure for 25 seconds. The development was then performed using diluted AZ 400K with deionizedwater (1:4) for 1 min. Finally, the BCB was covered with a 200 nm layer of aluminum after 12seconds Argon etching to ensure good contacts with the coil.
Figure 16: Left: in a first step, a BCB brick is deposited with optical lithography. Right: in a second stepthe core of the coil is connected with an aluminum bridge. C. Josephson junction
As explained in the main text, for r ’
1, strong anti-bunching effects are expected when theresonator is, in average, almost empty. The maximum photon emission rate is given by˙ n = Re[ Z ( ω )] I ~ ω , from which the mean occupation number n can be deduced by:˙ n = n Γ , with Γ = 2 π HMBW, the leaking rate of the resonator. In order to estimate the targeted resistanceof the junction, one uses the Ambegaokar-Baratoff formula and Josephson relations (taking intoaccount that DCB will renormalizes E J by a factor of exp( − πZ c / R Q )): I = π ∆2 eR N , E J = ϕ I C π . In order to be able to tune E J with a magnetic field, a SQUID geometry is used for the josephsonjunction: two junctions are placed in parallel to form a loop, which behaves as a single effectivejunction tunable with the external magnetic flux applied to the loop.As a small capacitance is required for the resonator, junctions must be as small as possible,but big enough to be reproducible and lead to a good symmetry between the two branches of theSQUID. Assuming a symmetry of 90%, E J can then be reduced by a factor of 10 tuning the fluxwith a little coil on top of the sample.Assuming a bandwidth ∆ ω ∼
100 MHz, a characteristic impedance Z C ∼ k Ω, a critical current I of 1 nA and a symmetry of 90%, one can estimate the minimal amount of photon in resonator : n = 1 / . ˙ n Γ = Z C I h (∆ ω ) . e − πZc/RQ ∼ . ω the half maximum bandwidth of the resonator (FWHM). These parameters require anormal state resistance for the SQUID of R N ∼ k Ω. a. Fabrication principle Samples are made of aluminum based tunnel junctions, fabricated bydouble angle evaporation through a suspended shadow mask, using the standard Dolan technique[20]. By adjusting the angles of evaporation, two adjacent openings in the mask can be projectedonto the same spot, creating an overlay of metallic films as shown in fig. 17. The first film isoxidized before the second evaporation to form the tunnel barrier.In order to have reproducible as well as small junctions, we used a cross shape as shown in Fig.18. b. SQUID fabrication
PMMA/PMGI resist bilayer spining :1. 2 min rebake at 110 (cid:176)
C2. Ti prime 6000 rpm 30 secs3. PMGI SF8 : 3000rpm, 45secs/ 6000 rpm 15 secs ( ≈ ± (cid:176) C with bescher2
Figure 17: Double angle evaporation principle: two metallic layers are evaporated onto the same spot,creating an overlay of metallic films. As the first layer was oxydized, the two electrodes are separated byan insulator and form a Josephson SIS junction.
5. PMMA A6 : 6000rpm, 60secs ( ≈ ± (cid:176) C (with bescher)As the quartz is very sensitive to charging effects, we placed an additional 7nm layer of aluminumof top of the resist to evacuate charges during EBL. The full wafer was then covered by a thick layerof UVIII resist which can be removed in IPA and sent to IEF for dicing. Actually, as the Quartzsubstate has an hexagonal symmetry, it cannot be cleaved.We then performed EBL on single chips using an FEI XL30 SEM with a dose of 300 µ C.cm − at 30 kV. The focus was tuned a three point on the sample using 20 nm gold colloids.The development process then consisted in :1. 35 secs MIF 726, 15 secs ODI to remove the aluminum layer3 Figure 18: Left: SEM image of the SQUID. Right: zoom on one of the Josephson junction with size95 ×
87 nm
2. 60 secs MIBK + IPA (1:3), 30 secs IPA, 15 secs ODI to open the Josephson junction patterns3. 25 secs MIF 726, 1min ODI, 15 secs ethanol to have a nice undercut
Double oxidation junctions
Finally, we deposited and oxidized aluminum to form highlyresistive Josephson junctions using double angle evaporation technique. In order to fabricate veryresistive Josephson junctions, the group of J.P. Pekola [18] raised the idea of oxidizing not onelayer of aluminum but to do it twice. By evaporating an additional subnanometer thick layer of Alimmediately after oxidizing the first layer, and oxidizing this fresh very thin layer, one thus obtainthicker barriers.The key parameter of this recipe is the thickness of the intermediate thin Al layer. As it willbe completely oxidized we can achieve resistances up to 1 M Ω with 0,4nm. Using this process, thesurfacic capacitance of the junction is estimated to 70 fF /µm i.e. ∼ (cid:176) : 20 nm Al @ 1 nm.s-13. O2/Ar (15/85 %) oxidation 300 mbar during 20 min4. 0.25 nm Al @ 0.1 nm.s-15. O2/Ar (15/85 %) oxidation 667 mbar during 10 min6. 24 (cid:176) : 80 nm Al @ 1 nm.s-1The lift-off of the resist was done by putting the sample in 60 (cid:176) C remover-PG during 40 minutes.In order to get uniform resistance values and limit Josephson junctions aging, they were rebakedon a hot plate at 110 (cid:176)
C during one minute.The chip was then stuck on the PCB with UVIII resist and bonded to the single input/ outputport using aluminum wires as shown in Fig. 12.4 a)b)c)d)
Figure 19: Josephson junction fabrication steps : a) Josephson junction shape : PMMA development b)Undercut : PMGI development c) & d) optical microscope view of the junctions after lift-off
VIII. CALIBRATION PROCEDURE
To calibrate in-situ the gain G of the detection chain and the impedance Z ( ν ) seen by thejunction, we measure the power emitted by the junction in two different regimes. First, we biasthe junction well above the gap voltage V gap = 210 µV and measure the voltage derivative of thequasiparticle shot noise power spectral density P ( ν ): ∂ P ( ν ) ∂V = 2 eR t Re Z ( ν ) | R t + Z ( ν ) | , (13)5with R t = 222 ± V to measure the power at ν = h/ eV resulting from theinelastic tunneling of Cooper pairs emitting single photons, as given by Eq. 2 of the main text: P = 2 e E ∗ J ~ Re Z ( ν = 2 eV /h ) . (14)The different power dependences on Z ( ν ) in these two regimes allows for an absolute determi-nation and Z ( ν ), the latter being shown in red in Fig.3b of the main text. With an absolutedetermination of Z ( ν ) at hand, we can use the normal quasi-particles shot noise as a calibratednoise source to determine in situ the gain and noise temperature of our two detection chains. [1] M. Hofheinz, F. Portier, Q. Baudouin, P. Joyez, D. Vion, P. Bertet, P. Roche, and D. Esteve, Phys.Rev. Lett. , 217005 (2011), URL http://link.aps.org/doi/10.1103/PhysRevLett.106.217005 .[2] G.-L. Ingold and H. Grabert, Europhys. Lett. , 371 (1991).[3] B. Kubala, V. Gramich, and J. Ankerhold, Phys. Scr. T165 , 014029 (2015), URL http://stacks.iop.org/1402-4896/2015/i=T165/a=014029 .[4] A. D. Armour, B. Kubala, and J. Ankerhold, Phys. Rev. B , 214509 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.96.214509 .[5] V. Gramich, B. Kubala, S. Rohrer, and J. Ankerhold, Phys. Rev. Lett. , 247002 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.111.247002 .[6] D. Walls and G. Milburn, Quantum Optics (Springer Berlin Heidelberg, 2009), ISBN 9783540814887.[7] M. P. da Silva, D. Bozyigit, A. Wallraff, and A. Blais, Phys. Rev. A , 043804 (2010), URL https://link.aps.org/doi/10.1103/PhysRevA.82.043804 .[8] S. Boutin, D. M. Toyli, A. V. Venkatramani, A. W. Eddins, I. Siddiqi, and A. Blais, Phys. Rev. Applied , 054030 (2017), URL https://link.aps.org/doi/10.1103/PhysRevApplied.8.054030 .[9] E. del Valle, A. Gonzalez-Tudela, F. P. Laussy, C. Tejedor, and M. J. Hartmann, Phys. Rev. Lett. ,183601 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.109.183601 .[10] E. del Valle, New Journal of Physics , 025019 (2013), URL http://stacks.iop.org/1367-2630/15/i=2/a=025019 .[11] C. Dory, K. A. Fischer, K. M¨uller, K. G. Lagoudakis, T. Sarmiento, A. Rundquist, J. L. Zhang,Y. Kelaita, N. V. Sapra, and J. Vuˇckovi´c, Phys. Rev. A , 023804 (2017), URL https://link.aps.org/doi/10.1103/PhysRevA.95.023804 .[12] C. M. Caves, Phys. Rev. D , 1817 (1982).[13] D. Bozyigit, C. Lang, L. Steffen, J. M. Fink, C. Eichler, M. Baur, R. Bianchetti, P. J. Leek, S. Filipp,M. P. da Silva, et al., Nat Phys , 154 (2011).[14] J. M. Fink, M. Kalaee, A. Pitanti, R. Norte, L. Heinzle, M. Davan¸co, K. Srinivasan, and O. Painter,Nature Communications , 12396 EP (2016), article, URL http://dx.doi.org/10.1038/ncomms12396 .[15] M.-C. Harabula, T. Hasler, G. F¨ul¨op, M. Jung, V. Ranjan, and C. Sch¨onenberger, Phys. Rev. Applied , 054006 (2017), URL https://link.aps.org/doi/10.1103/PhysRevApplied.8.054006 .[16] T. Hasler, M. Jung, V. Ranjan, G. Puebla-Hellmann, A. Wallraff, and C. Sch¨onenberger, Phys. Rev.Applied , 054002 (2015), URL https://link.aps.org/doi/10.1103/PhysRevApplied.4.054002 .[17] C. Altimiras, O. Parlavecchio, P. Joyez, D. Vion, P. Roche, D. Esteve, and F. Portier, Applied PhysicsLetters , 212601 (2013), https://doi.org/10.1063/1.4832074, URL https://doi.org/10.1063/1.4832074 .[18] T. Holmqvist, M. Meschke, and J. P. Pekola, Journal of Vacuum Science & Technology B:Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena , 28(2008), https://avs.scitation.org/doi/pdf/10.1116/1.2817629, URL https://avs.scitation.org/doi/abs/10.1116/1.2817629 . [19] I. Wolff, Coplanar Microwave Integrated Circuits (Wiley, 2006).[20] G. J. Dolan, Applied Physics Letters , 337 (1977), https://doi.org/10.1063/1.89690, URL https://doi.org/10.1063/1.89690https://doi.org/10.1063/1.89690