Non-linearity in the system of quasiparticles of a superconducting resonator
M. Vignati, C. Bellenghi, L. Cardani, N. Casali, I. Colantoni, A. Cruciani
NNon-linearity in the system of quasiparticles of a superconducting resonator
M. Vignati,
1, 2, ∗ C. Bellenghi, † L. Cardani, N. Casali, I. Colantoni,
3, 2 and A. Cruciani Sapienza Universit`a di Roma – Dipartimento di Fisica, I-00185, Roma, Italy Istituto Nazionale di Fisica Nucleare – Sezione di Roma, I-00185, Roma, Italy Consiglio Nazionale delle Ricerche – Istituto di Nanotecnologia, I-00185, Roma, Italy (Dated: February 19, 2021)We observed a strong non-linearity in the system of quasiparticles of a superconducting aluminumresonator, due to the Cooper-pair breaking from the absorbed readout power. We observed bothnegative and positive feedback effects, controlled by the detuning of the readout frequency, whichare able to alter the relaxation time of quasiparticles by a factor greater than 10. We estimate thatthe (70 ±
5) % of the total non-linearity of the device is due to quasiparticles.
Superconducting resonators are used to build sensitivedetectors, amplifiers and quantum circuits. These de-vices base their working principle on non-linear induc-tances, which can be engineered via Josephson Junc-tions [1–3] or can rely on the intrinsic kinetic inductanceof the superconductor [4–6].In these applications, the superconductor is cooled wellbelow its critical temperature, almost all electrons arebound in Cooper pairs, and the circuit is in principle loss-less. Photon or phonon interactions, however, can breakthe pairs and create quasiparticles, which then recombineon timescales that decrease with their density [7]. Thepresence of quasiparticles, along with two-level systems(TLS) [8, 9], is one of the main source of losses and canlimit the quality factor of the resonator.The readout power absorbed in the circuit can alsobreak the pairs and increase the density of quasiparticles,in a way similar to a temperature increase [10]. Thisleads to the establishment of an electro-thermal feedbackdue to the variation of the absorbed power with thedensity of quasiparticles [11–13]. In this work we reporton the first observation of a non-linear behavior dueto the electro-thermal feedback and its effect on therelaxation time of quasiparticles.The resonator under study consists of a lumped-elementLC circuit coupled to a coplanar wave guide, realized witha 60 nm aluminum lithography on a 2x2 cm wide, 0.3mm thick silicon substrate. The inductor L is 6 cm longand 60 µ m wide, and it is winded in a meander with 5 µ mspacing and closed on a 2-finger capacitor C at distanceof 60 µ m. The resonator is analogous to that presented inRef. [14], and is operated as phonon-mediated [15] kineticinductance detector [4] in a cryogen-free dilution refrig-erator with base temperature of 20 mK. The transmis-sion S of the circuit is measured by means of a hetero-dyne electronics [16]. The low-power, low-temperatureresonant frequency and quality factor are found to be f r, = 2 . Q = 98100, respectively, andthe fraction of kinetic over total inductance is α = 2 . f g across f r, atthree power levels, here quoted in terms of density over the inductor volume ( P g = 5 . µm , 3.3 and 4.7fW/ µm ), are shown in Fig. 1 (top). In the left panelof the figure the magnitude of the transmission S past the chip, corrected for line effects [17], is shownas a function of the detuning in line-widths units, y = Q ( f g − f r, ) /f r, , while the right panel shows the realand imaginary parts of S . The resonance moves signifi-cantly to lower frequencies and becomes asymmetric withincreasing power, a known effect due to the kinetic induc-tance non-linearity. At even higher powers, not shown inthe figure, the resonance enters in a hysteretic regime,implying that a fraction of the transmission is no longeraccessible [11, 18]. The size of the circle in the real andimaginary plane of S , which is directly proportional tothe quality factor of the resonator, changes less evidently.It increases at medium power, as expected with the sat-uration of TLS, and decreases back at high power, aneffect that is attributed to the increase of the number ofquasiparticles.In order to excite the resonator, the light from a room-temperature pulsed LED is driven to one side of the sil-icon substrate by an optical fiber passing through thecryostat. The absorbed light is converted to phonons inthe silicon, which scatter through the lattice until theyare absorbed in the superconductor and break the Cooperpairs. The pair-breaking alters the resonator frequencyand quality factor, which are in turn measured throughthe phase ( δφ ) and amplitude ( δA ) variations of the wavetransmitted past the resonator relative to the center ofthe S circle (see e.g. [19]). Pulses following the opticalexcitation are acquired with a software trigger and aver-aged over around 500 samples to reduce noise in the esti-mation of their shape. The rise time of the pulses is dom-inated by the ring time of the resonator ( Q/πf r = 12 µs )and by the phonon life-time in the substrate ( ∼ µs ),while the decay time is attributed to the relaxation ofquasiparticles [20]. The duration of the LED excitationis 2 µ s, and does not contribute significantly to the shapeof the pulses.The decay time τ d of the pulses, measured as thetime difference between the 90% and the 30% of theirtrailing edge, is shown in the bottom panel of Fig. 1 a r X i v : . [ c ond - m a t . s up r- c on ] F e b y2 − − | | S ] m µ [fW/ g P = 5.2e-03 L P = 3.3 M P = 4.7 H P [kHz] r,0 - f g f50 − } Re{S0 0.2 0.4 0.6 0.8 1 } I m { S − − y2 − − z − − y − − a ± ± ± y1.5 - - - [ m s ] d t Time [ms]0 2 4 6 8 10 12 [ a . u . ] fd Figure 1. Frequency scan of the resonator at different gen-erator powers P g . Magnitude of the transmission | S | as afunction of the detuning in line-widths ( y ) (top, left), realand imaginary part of S (top, right), and decay time of thepulses τ d as a function of y (bottom); Inset: δφ average pulsesat low and medium powers with y ∼
0, and at high powerwith y corresponding to the maximum of the τ d overshoot. as a function of the detuning y . At low power τ d does not depend on y and averages to 0 .
64 ms. Thedependency is instead sizeable at medium and highpowers, and in the latter case it varies from a minimumof 0 .
21 to a maximum of 2 . . N qp quasiparticles are created via pair-breaking,before recombining they store an amount of energy E qp = N qp ∆ (1)where 2∆ is the binding energy of a Cooper pair. Insuperconducting resonators the presence of quasiparti-cles, to a first-order approximation, modifies resonantfrequency f r and quality factor Q with respect to their low-power and low-temperature values ( f r, , Q ) as: x r = − α E qp E (2)1 Q = 1 Q + 2 αβ E qp E (3)where x r = ( f r − f r, ) /f r, , β/ E isexpected to be of the order of the pairing energy of thesuperconductor. The measurable quantity is the trans-mission S which, for the resonator under study, can beexpressed as [19]: S = 1 − QQ c
11 + 2 jQx (4)where Q c is the coupling quality factor and x = ( f g − f r ) /f r is the detuning of the generator frequency withrespect to f r . In turn x can be expressed in terms of thedetuning x with respect to f r, , x = ( f g − f r, ) /f r, ,as: x ’ x + α E qp E (5)where in the calculation we approximated 1 + x r ’ E Pqp = η g P qp τ qp (6) P qp = P g Q Q c Q qp
11 + 4 Q x (7)where η g is the efficiency in the creation of quasiparti-cles [22] [23], τ qp and Q qp are the recombination timeand the internal quality factor of quasiparticles, respec-tively. We finally define the non-linearity parameter as: a qp = αQ E Pqp ( x = 0) E (8)The energy E Pqp , may represent only a fraction of thetotal absorbed energy E P and thus of the non-linearity.In our device the total non-linearity ( a ) manifests itselfin the non-linearity of the kinetic inductance, which hasbeen already extensively studied [15, 19]: a = αQ E P E = α Q Q c P g πf r, γE (9)where γ is dimensionless parameter of order 1 to accountfor a possible difference in the energy scale. Defining y = Qx , we can rewrite Eqns. 5 and 3 in terms of a and a qp as: y = y z + a z y (10) z = QQ = 1 − β a qp, z y (11) y2 − − | | S ] m µ [fW/ g P = 5.2e-03 L P = 3.3 M P = 4.7 H P [kHz] r,0 - f g f50 − } Re{S0 0.2 0.4 0.6 0.8 1 } I m { S − − y2 − − z − − y − − a ± ± ± Figure 2. Left: z = Q/Q as a function of the detuning withrespect to the power-shifted resonant frequency ( y ) at low(black), medium (blue) and high (red) powers; Right: y as afunction of y (dots) along with fits for a of Eq. 10 (lines). where a = a ( Q = Q ) and a qp, = a qp ( Q = Q ). Theseequations describe the detuning in line-widths with re-spect to the power-shifted resonant frequency ( y ), andthe fractional change of the quality factor ( z ) as a func-tion of y , respectively. It has to be underlined that thedetuning is affected by the total non-linearity ( a ), whilethe quality factor is affected only by the portion of non-linearity due to quasiparticles ( a qp ).The values of ( z, y ) in Eq. 10 are calculated from the S data (cf. Suppl. Mat.) and shown in Fig. 2 (left).At low-power the value of z decreases with | y | , which re-veals that the quality factor is dominated by TLS as itlowers with decreasing absorbed power. At higher pow-ers the same behavior is observed at high | y | , while for | y | around zero it decreases because of the power-generatedquasiparticles. It has to be noticed that, at least for theresonator under study, the z variation with y at fixedpower is only at few % level, presumably because of acompensation between the quality factors of quasiparti-cles and TLS. If quasiparticles had dominated the qualityfactor, we could have used Eq. 11 to extract a qp, fromfits to the ( z, y ) data, provided that β is measured inde-pendently. Nevertheless, as it will be shown next, it ispossible to estimate a qp, directly from the decay-time ofthe pulses, without introducing the effect of TLS in themodel, and thus other free parameters.The points in the ( y, y ) plane are shown in Fig. 2(right) along with fits of Eq. 10 for a and usingvalues of z from the ( z, y ) graph. The results are a = 0 . ± . , . ± .
01 and 0 . ± .
02 for thelow, medium and high powers, respectively, with a 6%systematic error added from the model (cf. Suppl. Mat.).The presence of a population of quasiparticles at equi-librium, along the frequency and quality factor of theresonator in Eqns. 2 and 3, modifies the recombinationtime as τ qp ∝ /N qp [7]. One can therefore map the re-combination time and the shift of the resonant frequency as: 1 τ qp = 1 τ qp, − y r τ k (12)where τ k embeds the physics governing the dependency offrequency and recombination time with N qp , τ qp, is thesaturation value of the recombination time at low-powerand low-temperature [24], and y r = − a qp, z y (13)is the shift of the resonant frequency due to the powerabsorbed in quasiparticles.When there are M quasiparticles out of equilibrium,as in the case of excitation with the light pulses, theirtime evolution can be described as dMdt = − Mτ qp − δN Pqp M ! (14)where the first term of the product accounts for the re-combination, while the second for the extra quasiparticles δN Pqp injected or removed by the variation of the absorbedpower. With a first order approximation [25] we obtainfrom Eq. 6: δN Pqp M = δE Pqp M ∆ = − δxδx qp a qp F ( y ) (15) F ( y ) = 8 y (1 + 4 y ) + δQ − δx y ) (16)where δx and δQ − are the variations in detuning and in-verse quality factor following the excitation, respectively, δx qp = αM ∆ /E is the contribution to δx due only toquasiparticles, F ( y ) accounts for the feedback sign anddependency on y , and a qp accounts for its intensity.Putting together Eqns. 14 and 15 we obtain the ex-pression for the relaxation time: τ rel = τ qp − δN Pqp M = τ qp δxδx qp a qp F ( y ) (17)In presence of quasiparticles’ non-linearity only, δx/δx qp = 1 and δQ − /δx = 2 /β . Other non-linearities,however, alter these parameters by adding a dependencyon y . Including these effects would add free parametersto the model, therefore we chose to estimate δx and δQ − /δx directly from the δφ and δA pulses acquired ateach y (cf. Suppl. Mat.). The value of δx qp is estimatedfrom the pulses at high | y | , where the non-linearities aresuppressed.We fit the measured decay time of signals as a func-tion of y with τ qp, , τ k and a qp, as free parameters(Fig. 3). From the figure one can see that the first-order model we proposed reproduces well the data with a qp, = 0 . ± .
01 and a qp, = 0 . ± .
01 for the - - - [ m s ] d t qp,0 t – k t – qp,0 a 0.01 – qp,0 t – k t – qp,0 a 0.01 – y1.5 - - - qp,0 t – k t – qp,0 a 0.01 – qp,0 t – k t – qp,0 a 0.01 – Figure 3. Decay time of the pulses ( τ d ) as a function of thepower-shifted detuning ( y ) for the medium (top) and high(bottom) powers. The solid lines are fits of the model forthe relaxation time including feedback effects (Eq. 17). Thedashed lines indicate the contribution to these fits due to thesteady change of τ qp with absorbed power (Eq. 12) . medium and high powers, respectively. We note that a qp /a = a qp, /a = (68 ±
3) % and (72 ±
3) % for thethe medium and high powers, respectively, pointing tothe fact that quasiparticles account for a large fractionof the total non-linearity. Combining the two measure-ments and including the systematic error from the modelwe obtain a qp /a = (70 ±
5) %.In order to deepen the understanding of the observedphenomena, we also study the variation of the decay timewith temperature when the resonator is biased at lowpower. By doing this, we remove the generator feedbackand, isolate the behaviour of τ qp with a steady populationof quasiparticles. Figure 4 shows the resonance shift withtemperature (top) and the decay time as a function of theresonant frequency shift (bottom). Fitting Eq. 12 to thedata points we find τ qp, = 0 . ± .
01 and τ k = 0 . ± . a qp a = η g γ πf r, τ qp Q qp , (18)allows in principle to derive η g /γ , provided that Q qp isknown at least to some approximation. From the defini-tion of the total quality factor, Q − = Q − c + Q − i , and as-suming that the internal quality factor, Q i , is dominatedby TLS and quasiparticles, Q i − ’ Q T LS − + Q − qp , wecan argue that the maxima of z in the frequency sweepsof Fig. 2 (left) correspond to Q T LS ’ Q qp . The value of y2 − − − − | | S T [mK]
20 100150 200230 250
I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Q − − − − − − − − − z Temperature [K]0 0.1 0.2 r y − − − / Q = Q r / Z [ m s ] qp τ r0 y0.6 − − − β r0 y0.6 − − − / Q = Q r / Z / ndf χ
06 / 1 − TLS Δ ± − β ± χ
06 / 1 − TLS Δ ± − β ± r,0 y0.6 − − − ] - [ m s - τ qp,0 τ ± k τ ± qp,0 τ ± k τ ± i − × i / Q qp τ − × r0 y0.6 − − − / Q − − − i Q (teo) qp Q calc
TLS Q qp − × qp / Q qp τ − × qp,teo /Q qp τ qp,teo /Q -1 ) -1 } qp0 τ -{ -1qp τ ( Figure 4. Temperature and frequency scan of the resonatorat low generator power. Magnitude of the transmission | S | as a function of y (top) and inverse of the decay time of thesignals τ d as a function of the shift of the resonant frequencyin line-widths ( y r, ) at each temperature (bottom). The solidline is a fit for Eq. 12. Q qp can be therefore estimated as:1 Q qp (cid:12)(cid:12)(cid:12) z max ’ (cid:18) z max Q − Q c (cid:19) (19)Using the values of τ qp calculated with Eqns. 12 and13 in the point ( y | z max , z max ) and the measured valueof Q c ( Q c = 115100, cf. Suppl .Mat.), from Eq. 18 weobtain η g /γ = (18 ±
3) % and (16 ±
3) % for the mediumand high power data, respectively. Assuming γ = 1, thevalue of η g is in line with the predictions in Ref. [26].Our results reveal the existence of a population ofquasiparticles generated from the readout power whichundergoes a strong electro-thermal feedback and whichsignificantly modifies the properties of the superconduct-ing circuit. As an example, the positive feedback couldbe exploited to increase the response of superconductingcircuits at signal frequencies below 1 / πτ rel . The neg-ative feedback, instead, could be exploited to make thecircuit more resistant to quasiparticles’ perturbations.Materials with different intrinsic values of τ qp /Q qp couldbe studied to enhance or suppress the non-linearity fromquasiparticles.The authors acknowledge useful discussion with J. Loren-zana, A. Monfardini and I. M. Pop, and thank the per-sonnel of INFN Sezione di Roma for the technical sup-port, in particular A. Girardi and M. Iannone. Thiswork was supported by the European Research Council(FP7/2007-2013) under Contract No. CALDER 335359and by the Italian Ministry of Research under the FIRBContract No. RBFR1269SL. ∗ Corresponding author: [email protected] † Now at Technische Universit¨at M¨unchen, Physik-Department, D-85748, Garching, Germany[1] J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev.Lett., , 1543 (1985).[2] M. J. Feldman, P. T. Parrish, and R. Y. Chiao, Journalof Applied Physics, , 4031 (1975).[3] G. J. Dolan, T. G. Phillips, and D. P. Woody, AppliedPhysics Letters, , 347 (1979).[4] P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, andJ. Zmuidzinas, Nature, , 817 (2003).[5] B. Ho Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas,Nature Physics, , 623 (2012).[6] F. B. Faramarzi et al. , (2021), arXiv:2012.08654.[7] S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang,S. Jafarey, and D. J. Scalapino, Phys. Rev. B, , 4854(1976).[8] J. M. Martinis, K. B. Cooper, R. McDermott, M. Stef-fen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P.Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev.Lett., , 210503 (2005).[9] C. R. H. McRae, H. Wang, J. Gao, M. R. Vissers,T. Brecht, A. Dunsworth, D. P. Pappas, and J. Mutus,Review of Scientific Instruments, , 091101 (2020).[10] P. De Visser, J. Baselmans, J. Bueno, N. Llombart, andT. Klapwijk, Nature Communications, (2014).[11] P. J. de Visser, S. Withington, and D. J. Goldie, Journalof Applied Physics, , 114504 (2010).[12] S. E. Thompson, S. Withington, D. J. Goldie, and C. N.Thomas, Superconductor Science and Technology, ,095009 (2013). [13] C. N. Thomas, S. Withington, and D. J. Goldie, Super-conductor Science and Technology, , 045012 (2015).[14] L. Cardani, N. Casali, I. Colantoni, A. Cruciani,F. Bellini, M. G. Castellano, C. Cosmelli, A. D’Addabbo,S. Di Domizio, M. Martinez, C. Tomei, and M. Vignati,Appl. Phys. Lett., , 033504 (2017).[15] L. J. Swenson, A. Cruciani, A. Benoit, M. Roesch, C. S.Yung, A. Bideaud, and A. Monfardini, Appl. Phys. Lett., , 263511 (2010).[16] O. Bourrion, C. Vescovi, A. Catalano, M. Calvo,A. D’Addabbo, J. Goupy, N. Boudou, J. F. Macias-Perez,and A. Monfardini, JINST, , C12006 (2013).[17] M. S. Khalil, M. J. A. Stoutimore, F. C. Wellstood, andK. D. Osborn, J. Appl. Phys., , 054510 (2012).[18] L. Swenson, P. K. Day, B. H. Eom, H. G. LeDuc,N. Llombart, C. M. McKenney, O. Noroozian, andJ. Zmuidzinas, J. Appl. Phys., , 104501 (2013).[19] J. Zmuidzinas, Annu.Rev.Cond.Mat.Phys., , 169 (2012).[20] M. Martinez, L. Cardani, N. Casali, A. Cruciani, G. Pet-tinari, and M. Vignati, Phys. Rev. Applied, , 064025(2019).[21] P. J. de Visser, D. J. Goldie, P. Diener, S. Withington,J. J. A. Baselmans, and T. M. Klapwijk, Phys. Rev.Lett., , 047004 (2014).[22] P. J. de Visser, J. J. A. Baselmans, S. J. C. Yates, P. Di-ener, A. Endo, and T. M. Klapwijk, Appl. Phys. Lett., , 162601 (2012).[23] η g depends on the number of quasiparticles, and thus on P qp [26]. This dependency can be neglected in our first-order model.[24] R. Barends, J. J. A. Baselmans, S. J. C. Yates, J. R. Gao,J. N. Hovenier, and T. M. Klapwijk, Phys. Rev. Lett., , 257002 (2008).[25] In the calculation of derivative, we approximated d ( τ qp /Q qp ) dM = 0 since the dependency of τ qp and Q qp onthe number of quasiparticles cancels to a good approxi-mation [22].[26] D. J. Goldie and S. Withington, Superconductor Scienceand Technology, , 015004 (2013). on-linearity in the system of quasiparticles of a superconducting resonator:Supplementary material M. Vignati,
1, 2, ∗ C. Bellenghi, † L. Cardani, N. Casali, I. Colantoni,
3, 2 and A. Cruciani Sapienza Universit`a di Roma – Dipartimento di Fisica, I-00185, Roma, Italy Istituto Nazionale di Fisica Nucleare – Sezione di Roma, I-00185, Roma, Italy Consiglio Nazionale delle Ricerche – Istituto di Nanotecnologia, I-00185, Roma, Italy (Dated: February 19, 2021)
EXPERIMENTAL SETUP
The device reported in the present work was fabricated on a high-quality (FZ method) intrinsic silicon (100)substrate, with high resistivity and double side polished. The resonator and the CPW were patterned by electronbeam lithography on a single 60 nm thick aluminum film, deposited using electron-gun evaporation (see Ref. [S1] formore details). The chip was mounted in a copper holder, fixed using PTFE supports, connected to SMA read-outusing wedge bonding (see Fig. S1 left), and then installed at the coldest point of a He/ He dilution refrigerator.The readout scheme is shown in Fig. S1 (right). The microwave generated at room temperature by the electron-ics [S2] enters the cryostat with power P txcryo , is reduced by 56 dB down to P txg with attenuators placed at the differenttemperature stages of the cryostat and is sent to the CPW on the chip. The microwave at the output of the chip, withpower P rxg , is driven to a CITLF-3 amplifier [S3] placed at the 4 K temperature stage and featuring an amplification of35 dB at 2.5 GHz. Past the amplifier the signal exits the cryostat with power P rxcryo and is sent back to the electronics.When the resonator is driven at off-resonance frequencies, in principle P txg = P rxg . However by measuring P txcryo and P rxcryo and by tracing back the powers at the resonator including the attenuation of the cables, we find P rxg /P txg = − P g quoted in the main text are the mean values between P txg and P rxg , which are not used to derive the results but are reported for comparison with other devices. M. Vignati
300 K 40 K 4 K 0.8 K 20 mKP g +35 dB-20 dB-10 dB -6 dB -20 dB h e t e r odyn e e l ec t r on i c s a n a l og a nd d i g it a l P cryo resonator tx P cryorx tx P grx Figure S1.
Left: photo of the aluminum resonator on the silicon substrate mounted in the copper holder; right: readoutscheme of the resonator placed in the dilution refrigerator. a r X i v : . [ c ond - m a t . s up r- c on ] F e b ANALYSIS OF S DATA
The values of
Q/Q c and of y are extracted directly from the real and imaginary parts of the S data. From Eq. 4we have: < (1 − S ) = QQ c
11 + 4 y (S1) = (1 − S ) = − QQ c y y (S2)so that: y = − = (1 − S ) < (1 − S ) (S3) v = QQ c = < (1 − S ) + = (1 − S ) < (1 − S ) (S4)The advantage of this method is that any dependency of the quality factor on the absorbed power is directly estimatedpoint by point from the S data without any interpolation.The characteristics of the resonator Q c , Q and f r, are evaluated from the low power data. By setting a = 0 inEq. 10, we have that: y = yz → f g = f r, (cid:18) yv Q c (cid:19) (S5)The above equation is used to extract f r, and Q c with a linear fit of f g vs y/v (Fig. S2 a and b). The value of Q isthen calculated as: Q = v ( y = 0) Q c (S6)and from the definition of the quality factor ( Q − = Q − c + Q − i ), one can derive the internal quality factor. It hasto be stressed that Q has to be considered only as reference value, i.e. the value we obtain when biasing the circuiton-resonance with P g = 1 . Q , giventhe dependency of the internal quality factor on the absorbed power.Once f r, , Q and Q c are fixed to the low-power values, the values of y and z = vQ c /Q are calculated for thedata points at all powers and then Eq. 10 is used to fit the data. Since the equation has a 3rd degree dependency on y , the solution features discontinuities when the resonator is in the bifurcation regime (see Ref. [S4]). Therefore wechose to invert the equation and fit y as a function of y to ease the minimization procedure: y = yz − a z y (S7)As it can be seen from Fig. 2 (right) the fits do not perfectly match the data. One of the reasons could be that theimpedance of the resonator and of the coplanar-wave-guide, which is also made of aluminum, may change with powerand thus Q c could slightly vary. Also, because of the presence of TLS, f r, could slightly increase with power. Inorder to take into account these effects, the above equation can be therefore modified as it follows: y = ∆ y r, + yv Q Q c − (cid:18) vQ c Q (cid:19) a y (S8)where ∆ y r, accounts for the resonant frequency shift and the free parameters are ∆ y r, , Q c /Q and a . The fits withthis 3-parameter model better reproduce the behavior of the ( y , y ) data (see Fig. S2 c and d), with a values around6% lower than the results from the fits with the model quoted in the main text (Fig. 2). We consider the discrepancybetween models as systematic error on the estimation of a .In order to reduce the systematic error, however, one should take into account of the Q c correlation with theprocedure applied to deconvolve the transmission line effects [S5]. Extra calibrations of the transmission-line withpower should be performed, but they were outside the scope of this work since the results presented are consideredrobust. I [ADC]1000 − − Q [ AD C ] Graph = -16.8 C I = -168.7 C Q = 1319.1 A I = 1200.2 A Q = 0.95 Φ Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × M a gn it ud e [ d B ] Graph abs(Big0) = 1128.5arg(Big0) = 1.354abs(Big0-c) = 1297.0arg(Big0-c) = 1.369 Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × P h a s e [r a d ] Graph
Tau = 36.6 ns = 0.45 Ψ I0 0.2 0.4 0.6 0.8 1 Q − − = 0.3341 θ Frequency [kHz]2556.25 2556.3 2556.35 2556.4 2556.45 2556.5 × d B − − − − Graph
Frequency [kHz]2556.25 2556.3 2556.35 2556.4 2556.45 2556.5 × P h a s e − − − − Graph y2 − − v vs y c v = Q/Q y/v2 − f - [ k H z ] f 31 ± c Q 121 ± offset f 0 ± f 31 ± c Q 121 ± offset f 0 ± * (y/v)) c (1+1/Q f = f (y=0) = 664001 i QQ(y=0) = 98098 y2 − − − − a × a ) I [ADC]1000 − − Q [ AD C ] Graph = -16.8 C I = -168.7 C Q = 1319.1 A I = 1200.2 A Q = 0.95 Φ Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × M a gn it ud e [ d B ] Graph abs(Big0) = 1128.5arg(Big0) = 1.354abs(Big0-c) = 1297.0arg(Big0-c) = 1.369 Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × P h a s e [r a d ] Graph
Tau = 36.6 ns = 0.45 Ψ I0 0.2 0.4 0.6 0.8 1 Q − − = 0.3341 θ Frequency [kHz]2556.25 2556.3 2556.35 2556.4 2556.45 2556.5 × d B − − − − Graph
Frequency [kHz]2556.25 2556.3 2556.35 2556.4 2556.45 2556.5 × P h a s e − − − − Graph y2 − − v vs y c v = Q/Q y/v2 − f - [ k H z ] f 31 ± c Q 121 ± offset f 0 ± f 31 ± c Q 121 ± offset f 0 ± * (y/v)) c (1+1/Q f = f (y=0) = 664001 i QQ(y=0) = 98098 y2 − − − − a × b ) I [ADC]30000 − − − − − − Q [ AD C ] − − − Graph = 4034.6 C I = 289.6 C Q = 33455.3 A I = 30192.1 A Q = 2.61 Φ Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × M a gn it ud e [ d B ] Graph abs(Big0) = 28922.7arg(Big0) = 2.987abs(Big0-c) = 32876.9arg(Big0-c) = 3.014
Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × P h a s e [r a d ] − − − Graph
Tau = 36.5 ns = 0.44 Ψ I0 0.2 0.4 0.6 0.8 1 Q − − = 0.3520 θ Frequency - 2556363.024869 [kHz]100 − − d B − − − − Graph
Frequency - 2556363.024869 [kHz]100 − − P h a s e − − Graph y2 − − v vs y c v = Q/Q y2 − − y − − r,0 y Δ ± a 0.005 ± /Q c Q 0.002 ± γ ± r,0 y Δ ± a 0.005 ± /Q c Q 0.002 ± γ ± c Q (y=0) = 913504 i QQ(y=0) = 104677 = 1.0671 Q(y=0)/Qa(y=0) = 0.4564 y2 − − − − a c ) I [ADC]5000 − Q [ AD C ] − Graph = -4187.0 C I = -2528.2 C Q = 40217.5 A I = 36177.4 A Q = -0.05 Φ Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × M a gn it ud e [ d B ] Graph abs(Big0) = 34738.7arg(Big0) = 0.318abs(Big0-c) = 39521.7arg(Big0-c) = 0.346
Frequency [kHz]2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 × P h a s e [r a d ] − Graph
Tau = 36.5 ns = 0.44 Ψ I0 0.2 0.4 0.6 0.8 1 Q − − = 0.3520 θ Frequency - 2556363.024869 [kHz]150 − − − d B − − − − Graph
Frequency - 2556363.024869 [kHz]150 − − − P h a s e − − − − Graph y2 − − v vs y c v = Q/Q y2 − − y − − r,0 y Δ ± a 0.004 ± /Q c Q 0.001 ± γ ± r,0 y Δ ± a 0.004 ± /Q c Q 0.001 ± γ ± c Q (y=0) = 685325 i QQ(y=0) = 101018 = 1.0298 Q(y=0)/Qa(y=0) = 0.7348 y2 − − − − a d )Figure S2. a ) v vs y values for the data at low power; b ) generator frequency f g as a function of y for data at low power. Theline is a fit for f r, and Q c with Eq. S5, the parameters Q i ( y = 0) and Q(y=0) are calculated as described in the text; c ) fit ofdata at medium power with the 3-parameter model (Eq. S8); d ) fit of data at high power with the 3-parameter model. ESTIMATION OF δx AND δQ − /δx The pulses are measured as phase ( δφ ) and amplitude ( δA ) variations of S with respect to the center of theresonance “circle”. It has to be noticed that, given the variation of Q with y , the S data do not lye on a perfectgeometrical circle. However, given that this variation amounts at maximum to 5% at medium at high powers (seeFig.2 left), we still fit the data with a circle function to evaluate its center and radius.From Eq. 4 the variations are: δφ = 4 Qδx y (S9) δA = 2 QδQ − y (S10)where δA is normalized to the radius of the circle. Therefore the quantities δx/δx qp and δQ − /δx can be calculatedas: δxδx qp = z H z (1 + 4 y )(1 + 4 y H ) δφδφ H (S11) δQ − dx = 2 δAδφ (S12)were δφ H is the phase variation at the maximum detuning y H , where feedback effects are suppressed (see Eq. 16).The transfer function of the resonator, which at y = 0 and at low-power is a single-pole low-pass filter with cutoffequal to 1 / (2 πτ r ), becomes asymmetric between positive and negative signal frequencies when y = 0 at low power and Time [samples]0 500 1000 1500 2000 2500 3000 [r a d ] y = -0.55 φδ A δ Time [samples]500 600 700 800 900 1000 A δ / φ δ − − − − Time [samples]0 500 1000 1500 2000 2500 3000 [r a d ] y = 0.43 φδ A δ Time [samples]500 600 700 800 900 1000 A δ / φ δ − − − − Figure S3.
Left : Average δφ (red) and δA (blue) waveforms acquired with 2 µ s sampling time. Data taken at high power for y = − .
55 (top) and y = 0 .
43 (bottom); right : Ratio between waveforms ( δφ/δA ) zoomed around the pulse region. The linesindicate the position of the maximum of the δφ pulses. The ratio in correspondence of the raising edge is not constant becauseof the different shape with detuning. The trailing edge has the same shape and the average is used as estimate of δφ/δA . at any y in the non-linear regime. This implies that the δ -response of the resonator is no longer real, but contains animaginary part. This effect mixes the δφ and δA responses at frequencies around and above 1 / πτ r . Since δφ (cid:29) δA the mixing affects evidently only δA , and in particular the raising edge of the pulse which is of the order of τ r .Figure S3 (left) shows the δφ and δA waveforms at high power and for relatively high detunings. As it can be seenthe δA pulse exhibit an overshoot ( y <
0) or an undershoot ( y >
0) due to this “asymmetric cutoff”. Therefore while δφ is estimated directly from the pulse amplitude, the value of δQ − /dx is evaluated from the ratio of the trailingedges of the pulses, i.e. at signal frequencies well below the cutoff of the resonator since τ rel (cid:29) τ r (Fig. S3 right). ∗ Corresponding author: [email protected] † Now at Technische Universit¨at M¨unchen, Physik-Department, D-85748, Garching, Germany[S1] I. Colantoni, F. Bellini, L. Cardani, N. Casali, M. G. Castellano, A. Coppolecchia, C. Cosmelli, A. Cruciani, A. D’Addabbo,S. Di Domizio, et al. , Journal of Low Temperature Physics, 1 (2016), ISSN 1573-7357.[S2] O. Bourrion, C. Vescovi, A. Catalano, M. Calvo, A. D’Addabbo, J. Goupy, N. Boudou, J. F. Macias-Perez, and A. Mon-fardini, JINST, , C12006 (2013).[S3] “Cosmic Microwave Technology, Inc – CITLF3,” .[S4] L. Swenson, P. K. Day, B. H. Eom, H. G. LeDuc, N. Llombart, C. M. McKenney, O. Noroozian, and J. Zmuidzinas, J.Appl. Phys., , 104501 (2013).[S5] M. S. Khalil, M. J. A. Stoutimore, F. C. Wellstood, and K. D. Osborn, J. Appl. Phys.,111