Measurement of Energy Decay in Superconducting Qubits from Nonequilibrium Quasiparticles

PHYSICAL REVIEW B 84, 024501 (2011)

Measurement of energy decay in superconducting qubits from nonequilibrium quasiparticles

M. Lenander,1 H. Wang,1,2 Radoslaw C. Bialczak,1 Erik Lucero,1 Matteo Mariantoni,1 M. Neeley,1 A. D. O’Connell,1 D. Sank,1 M. Weides,1 J. Wenner,1 T. Yamamoto,1,3 Y. Yin, 1 J. Zhao,1 A. N. Cleland,1 and John M. Martinis1 1Department of Physics, University of California, Santa Barbara, California 93106, USA 2Department of Physics, Zhejiang University, Hangzhou 310027, China 3Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan (Received 3 January 2011; revised manuscript received 17 May 2011; published 1 July 2011) Quasiparticles are an important decoherence mechanism in superconducting qubits, and can be described with a complex admittance that is a generalization of the Mattis-Bardeen theory. By injecting nonequilibrium quasiparticles with a tunnel junction, we verify qualitatively the expected change of the decay rate and transition frequency in a phase qubit. With their relative change in agreement to within 4 % of prediction, the theory can be reliably used to infer quasiparticle density. We describe how settling of the decay rate may allow determination of whether qubit energy relaxation is limited by nonequilibrium quasiparticles.

DOI: 10.1103/PhysRevB.84.024501 PACS number(s): 74.81.Fa, 03.65.Yz, 74.25.N−,74.50.+r

Superconducting resonators and Josephson qubits are Josephson inductance LJ and admittance YJ from Cooper pair remarkable systems for quantum information processing,1 tunneling has frequency dependence with recent experiments demonstrating three-qubit 2,3 4,5 1 1 2πI0 cos φ entanglement, Bell and Leggett-Garg inequalities, YJ (ω) = = [1 − 2f ()], (1) resonance fluorescence,6 photon jumps,7 and transducers,8 iωLJ iω 0 and entangled-photon NOON states.9 For maximum where φ is the junction phase difference, 0 = h/2e is the coherence, devices are operated at temperatures T well 13 flux quantum, and I0 is the critical current. Nonequilibrium below the superconducting transition Tc to suppress thermal quasiparticles effectively lower the critical current by a factor excitations and quasiparticles. Nevertheless, experiments 1 − 2f () by blocking pair channels. Note that this reduction using resonators and qubits can sense the presence of is not a function of nqp, but depends on the occupation quasiparticles, which arise from nonequilibrium sources such probability at the gap f (). For an exact calculation, the as stray infrared radiation, cosmic rays, or energy relaxation in 10,11 occupation would be for Andreev bound states in the junction materials. It is vitally important to completely understand at energies slightly below the gap;14 for small tunneling the theory of quasiparticle dissipation, determine whether it probability, we assume this equals the occupation of free limits qubit coherence, and identify generation mechanisms. quasiparticles at the gap f (). Thermal quasiparticles have In this paper, we experimentally test the theory of = + 12 an occupation fT (E) 1/[1 exp(E/kT )], which yields the quasiparticle dissipation by measuring the response of a expected temperature dependence tanh(/2kT ). superconducting phase qubit to the injection of nonequilib- = Defining√ a through the relation f () anqp/ncp,we rium quasiparticles. The decay rate of the qubit is shown find a /2πkT for thermal quasiparticles at low qualitatively to depend on the injection and recombination temperature.15 For the case of nonequilibrium quasiparticles rate of quasiparticles. Because the density of quasiparticles generated by an external pair-breaking source, we numerically nqp is not known from injection, we compare the decay rate −0.173 compute a 0.12 (nqp/ncp) ,givinga 1.2 for typical and frequency shift of the qubit with changing nqp to test parameters.13 quantitatively the correct dependence with theory. We also The total junction admittance Yj is also affected by show that the characteristic settling time to steady state is quasiparticle tunneling,12 and is given by consistent with theory, and may allow an additional measure of quasiparticle density. Yj (ω) 1 + i nqp In superconductivity, nonequilibrium quasiparticles at = (1 + cos φ) √ − iπf() 1/R hω¯ hω¯ n energy E can be described using an occupation proba- n 2 cp bility f (E). Quasiparticle effects can be expressed in a − iπ cos φ [1 − 2f ()], (2) two-fluid manner by considering their total density nqp = ∞ hω¯ 2D(E ) ρ(E)f (E)dE, where D(E )/2 is the single-spin f f √ where the last term, from Cooper pair tunneling, has the density of states, is the gap, and ρ(E) = E/ E2 − 2 critical current I0 = π/2eRn re-expressed using the gap and is the normalized density of quasiparticle states. For low junction resistance Rn. The first term gives a different phase dissipation appropriate for qubits, we need only consider small dependence 1 + cos φ, and has contribution from the tunneling occupations n n , where we define n ≡ D(E ) as the qp cp cp f of free quasiparticles [nq] and bound Andreev states [f ()]. density of Cooper pairs. Using BCS theory, nonequilibrium For a junction phase difference φ = 0 the factors from f () = − quasiparticles lower the superconducting gap as 0(1 cancel out, and the admittance is 13 nqp/ncp), where 0 is the gap without quasiparticles. We first describe the effect of nonequilibrium quasiparticles √ 3/2 n Yj (ω) = + qp − on Josephson tunneling. For arbitrary occupation f (E), the (1 i) 2 iπ . (3) 1/Rn hω¯ ncp hω¯

This result is equivalent to the normalized conductivity 16 σ (ω)/σn given by the Mattis-Bardeen theory for thermal quasiparticles17 in the limit kT hω¯ . This shows that resistance in a superconductor can be modeled as a series of Josephson junctions. Quasiparticle damping increases the energy relaxation rate in Josephson qubits. For the phase qubit, where matrix elements are well approximated by harmonic oscillator values, the relaxation rate is given by18 3/2 Re{Yj (E10/h¯ )} 1 + cos φ nqp 1 = √ , (4) C 2RnC E10 ncp where C is the junction capacitance and E10 is the qubit energy. This result is identical to that found from environmental P (E) FIG. 1. (a) Schematic of phase qubit, with critical current theory in the low impedance limit,11 and now includes a I0 2.0 μA and capacitance C 1.0 pF, shunted by admittance cos φ term coming from the interference of electron- and Yj coming from nonequilibrium quasiparticles. The measurement 19 holelike tunneling events. SQUID, shunted by an off-chip resistor Rsq = 30 ,isonthesame Quasiparticles also change the imaginary part of Yj , chip as the qubit (Ref. 21) and generates quasiparticles when switched which then shifts the qubit frequency by δE10/h¯ into the voltage state. The separation between the qubit and SQUID 18 −Im{Yj (E10/h¯ )}/2C using perturbation theory. For the is about 1 mm. The SQUID I-V shows a typical resistively shunted nqp/ncp term in quasiparticle tunneling, the real and imaginary Josephson junction, from which the SQUID critical current and RSQ parts of Yj are equal, which yields a change in angular can be measured. (b) Time sequence of experiment, showing initial frequency − 1/2. Quasiparticles reduce the admittance from pulse for quasiparticle injection at the SQUID, settling time τset,flux Cooper pair tunneling by the factor 1 − (1 + 2a)nqp/ncp due pulses applied to the qubit for frequency or decay time measurement, to a change in and the pair-blocking terms. Including the and a final flux measurement by the SQUID. final f () term from quasiparticle tunneling, the frequency shift is and decay rate. Qubit measurement produces a state-dependent δE 1 1 a − (1 + a) cos φ change in the loop flux that is measured with a SQUID readout 10 =− 1 − , (5) circuit.20 The SQUID is also used to generate quasiparticles h 1 4π b 1 + cos φ when driven into the voltage state Vsq > 0. With the SQUID √ shunted by a resistor Rsq = 30 , the SQUID current Isq can where b = /2E10/π 0.6 for typical parameters. Here, be adjusted to produce a voltage Vsq from ∼0.6 /e to above the shift is normalized to the damping since both are propor- 2/e, greatly changing the generation rate of quasiparticles. tional to nqp. The experimental time sequence is illustrated in Fig. 1(b). The above calculation assumes constant φ, which is not Quasiparticles are initially generated by applying SQUID valid for a phase qubit where we impose the constraint of current to produce Vsq 2/e for a time τinj: quasiparticles constant current bias I. Assuming the standard Josephson then diffuse to the qubit via the ground plane connection. current-phase relationship, the critical current changes both The injection is turned off for a settling time τset,afterwhich from the gap and quasiparticle excitations in the junction, giv- we apply a flux wave form to perform a measurement of the ing δI0/I0 =−(1 + 2a) nqp/ncp. Because the qubit frequency 20 1/4 qubit resonance frequency or decay rate. We then ramp scales as E10 ∝ (I0 − I) , a change in qubit critical current − Isq to measure the flux state for qubit readout, but at a low changes the bias I0 I and results in an additional shift in = − voltage Vsq 0.6 /eso that few quasiparticles are generated. frequency given by δE10/E10 (1/4)δI0/(I0 I). The experiment is typically repeated ∼1000 times to produce Since the equations for both dissipation and frequency shift probabilities. are proportional to nqp, their ratio can be related using qubit 13 We plot in Fig. 2(a) the decay rate and frequency shift of the parameters. Including both frequency shifts, we find qubit versus injected SQUID current for a fixed settling time. When the current Isq produces a SQUID voltage Vsq < 2/e δE δE 1 1 + 2a I E 1/2 10 = 10 − 0 10 . (6) (left of dashed vertical line), we observed almost no frequency h 1 h 1 4 1 + cos φ I0 − I 0 shift δE10/hor change in the decay rate 1. However, when the voltage exceeds the gap and greatly increases the quasiparticle The latter term dominates because the current bias is typically generation rate from pairbreaking, we observe a large increase set close to the critical current I0 − I I0. in qubit decay rate and decrease in frequency. As shown in Fig. 1(a), we experimentally test this theory by The recombination of quasiparticles is tested by keeping the using a phase qubit20 to measure the effect of nonequilibrium injection current and time constant, while varying the settling quasiparticles injected via a separate tunnel junction in the time, as shown in Fig. 2(b). For short times corresponding to superconducting quantum interference device (SQUID). The the highest quasiparticle densities, we see more rapid decay of effect of quasiparticles are modeled by a parallel admittance the qubit. For settling times greater than ∼500 μs the decay Yj that changes the qubit |1 to |0 state transition frequency rate approaches that without injection of quasiparticles.

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FIG. 3. Parametric plot of qubit frequency shift δE10/hand decay rate 1 for various injection currents. A linear relation is observed, consistent with theoretical predictions that both should scale as nqp. Gray line indicates ﬁt to the data. Qubit number is 2 as shown in Table I.

times the predicted value.13 The difference is probably due to two-level states.23 With confidence that we can accurately extract nqp from our experiment, we now analyze the time dependence of the data FIG. 2. (Color online) Effect of quasiparticles on the phase qubit. plotted in Fig. 2(b). The decay of quasiparticles can readily be (a) Plot of qubit decay rate 1 (circles) and frequency shift δE10/h calculated for the case of small density when they mostly have (triangles) as a function of SQUID injection current Iinj,forafixed energy near the gap.13 In this limit, the integral of Eq. (7) in time of injection τinj = 200 μs and settling τset = 300 μs. Dashed Ref. 11 gives a recombination rate r = (21.8/τ0)(nqp/ncp), vertical line indicates SQUID voltage Vsq = 2/e, above which where τ0 400 ns is the characteristic electron-phonon cou- quasiparticle generation increases rapidly. (b) Plot of decay rate 22 = pling for aluminum. If rqp is the injection rate, the differential versus settling time (circles) for fixed injection current Iinj 20 μA r equation dnqp/dt =−2 nqp + rqp can be solved to give and time τinj = 200 μs. Quasiparticles are observed to recombine on ∼ a time scale 300 μs. Bottom blue line is theory of Eq. (9) with nqp τ0/43.6 22 = = electron-phonon coupling τ0 = 400 ns. Theories with additional (case rqp 0), (7) ncp t − t0 qubit decay (top red, τ0 = 200 ns) and nonequilibrium quasiparticles n = n eq r eq t − t , of Eq. (8)(gray,τ0 = 400 ns) are also shown, each fit with τ0 qp ( qp) coth[ ( ) ( 0)] (8) and a parameter matching 1 at 10 ms. Qubit number is 2, with r eq 1/2 with equilibrium values ( ) = [(43.6/τ0)(rqp/ncp)] and I0 = 2.03 μAandC = 1.063 pF. eq r eq (nqp) /ncp = (τ0/43.6)( ) , where t0 is the integration constant. After quasiparticle injection, Eq. (8) describes a decay that initially follows Eq. (9), but levels off to a steady-state eq r eq Although these data show qualitative agreement with value (nqp) after time 1/( ) . expectations, direct quantitative analysis is difficult because We plot in Fig. 2(b) the prediction for the case of no of uncertainties in predicting nqp from modeling the gen- equilibrium quasiparticles rqp = 0[Eqs.(4) and (D9)], which eration, recombination, and diffusion of the quasiparticles. depends only on τ0. To account for the constant rate at times However, an accurate quantitative test of the theory can be 1 ms, we consider two hypotheses. One is the qubit has obtained by comparing the change in decay rate with the an additional dissipation mechanism, so that the net decay change in qubit frequency, which are both expected to scale as nqp. TABLE I. Qubit parameters showing average cos φ, bias parame- This comparison is shown in Fig. 3 for the range of injection ter (I0 − I)/I0, operating frequency, slope extracted from experiment currents plotted in Fig. 2. With both quantities proportional data as plotted in Fig. 3, slope predicted by theory of Eq. (6), and their to nqp, a linear relation is expected and observed in the ratio for three separate devices. We observe experimental agreement data. The slope δE10/h 1 is thus a quantitative test of the within experimental uncertainty, as denoted in parentheses. Qubits 3, theoretical prediction given in Eq. (6). We extracted the slope 4, and 5 are the same device tested at different times. Calculation − from this type of plot for three devices and five experiments, of cos φ and (I0 I)/I0 accounts for the inductive shunt in the as summarized in Table I. We find that the experimentally flux-biased phase qubit. measured slope is on average only slightly larger (3.6%) than I0−I E10 δE10 δE10 expt cos φ (GHz) [ ]expt [ ]theor predicted by theory and well within experimental uncertainty I0 h h 1 h 1 theor (14%). 1 0.12(3) 0.042(5) 6.523 −1.42(9) −1.34(19) 1.05(16) We have also compared the dissipation and frequency shift 2 0.17(4) 0.066(6) 6.743 −1.30(3) −1.03(13) 1.22(15) in superconducting coplanar resonators made from aluminum, 3 0.19(4) 0.071(3) 6.413 −1.13(6) −1.03(09) 1.09(12) where quasiparticles are generated by raising the temperature. 4 0.33(5) 0.130(3) 7.375 −0.67(2) −0.80(14) 0.83(14) We find that the shift in resonance frequency and inverse 5 0.19(4) 0.071(4) 6.413 −1.04(6) −1.04(10) 0.99(11) quality factor also scale together, with a slope that is 0.77

024501-3 M. LENANDER et al. PHYSICAL REVIEW B 84, 024501 (2011) rate is the sum of the above prediction with a constant offset. The second assumes constant injection of nonequilibrium quasiparticles, as given by Eq. (8). We plot these two additional predictions, which show somewhat different dependencies with time. For settling times 300 μs, we believe that the departure from theory is due to diffusion effects that are not included in this simple model.11 Numerical simulations13 show that such deviations are reasonable given that the injection and qubit junctions are well separated. Note the small dip in 1 at time ∼1 ms is similar in magnitude to prior temperature measurements.11 Although both hypotheses can roughly ex- FIG. 4. Parametric plot of fractional frequency shift δf /f versus plain the data for times 300 μs, the scenario that ﬁts a bit dissipation 1/Q, for various quasiparticle occupations nqp changed by varying the the sample temperature. The device is an aluminum better is when nonequilibrium quasiparticles are the dominant coplanar resonator fabricated on sapphire. The slope of the data is in decay mechanism. good agreement with predictions (gray line) at temperatures above Fluctuations in the quasiparticle density cause the qubit 24 250 mK. Deviations at low temperature are believed to come from frequency to jitter, producing dephasing. Using a Ramsey the two-level states. fringe protocol, measurements were made of the decay time

T2 at different quasiparticle injection currents, where we The factor of 2 in the definition of nqp comes from saw a decrease in T2 at high quasiparticle densities. After integration only over positive energy, whereas excitations arise correcting for the decrease in T2 from a change in T1 using from both electron states above and below the Fermi energy. = + 1/T2 1/2T1 1/Tφ, we found that Tφ was unchanged to Note that this integral already contains the two possible spin within experimental error. We conclude that other dephasing states in the definition of D(Ef ). mechanisms are dominant in the phase qubit. In conclusion, we have used the theory of quasiparticle APPENDIX B: QUASIPARTICLES IN COPLANAR admittance to predict the qubit frequency shift and energy RESONATORS decay from nonequilibrium quasiparticles. When injecting quasiparticles from a nearby junction, we find good qualitative The relationship between quasiparticle damping and fre- agreement with theory. Good quantitative agreement was quency shift may also be tested in superconducting res- observed between the relative change of qubit decay and onators. Here microwave transmission is measured to extract frequency with changing quasiparticle density. Monitoring the the resonance frequency f and the quality factor Q,with time dependence of qubit decay after injection should allow the quasiparticle density changed by simply increasing the future experiments to positively identify if nonequilibrium temperature.25 As shown in Fig. 4, we find that with increasing quasiparticles are the limiting decay mechanism in supercon- quasiparticle density, dissipation increases and the resonance ducting qubits. frequency decreases, in a similar manner as for junctions. To compare with theory, we use solutions to the Mattis- ACKNOWLEDGMENTS Bardeen conductivity that is valid for the regime kT ∼ 16,17 We thank F. K. Wilhelm and M. H. Devoret for invaluable hω¯ . These results can be expressed in terms of an discussions. This work was supported by IARPA under ARO admittance function discussed in the main paper, but with a modification of the the 1 + i term, Award No. W911NF-09-1-0375. M.M. acknowledges support from an Elings Postdoctoral Fellowship. H.W. acknowledges √ 3/2 n Yj (ω) = + qp − partial support by the Fundamental Research Funds for the (dR idI ) 2 iπ , (B1) 1/R hω¯ n hω¯ Central Universities in China (Program No. 2010QNA3036). n cp Devices were made at the UC Santa Barbara Nanofabrication d = 2 2x/π sinh(x)K (x), (B2) R √ 0 Facility, a part of the NSF-funded National Nanotechnology d = 2πx exp(−x)I (x), (B3) Infrastructure Network. I 0 where x = hω/¯ 2kT . The theoretical prediction, indicated by APPENDIX A: NON-EQUILIBRIUM QUASIPARTICLES the gray line in Fig. 4, is in reasonable agreement with the data at high temperatures. The experimental slope at temperatures We are interested in how nonequilibrium quasiparticles above 200 mK is 0.77 times that given by theory. The deviation affect the properties of a Josephson junction in qubit devices. is largest below about 120 mK, and believed to arise from Since we are concerned with very-low-temperature operation, two-level states that are not included in this model. The we consider phonon temperatures sufficiently low that no response from nonequilibrium quasiparticles has recently be quasiparticles exist due to thermal generation. The nonequi- calculated.26 librium quasiparticles are generated with an unknown mechanism, and then relax their energy via the emission of phonons. APPENDIX C: NUMERICAL SOLUTION OF The quasiparticles typically have energy E close to the gap, QUASIPARTICLE RECOMBINATION from which they eventually decay via recombination. From electron-phonon physics, we know that the qusiaparticles relax For numerical computations of nonequilibrium quasiparti- to energies very near the gap, as calculated in Ref. 11. cle density from relaxation and recombination,11 the integrals

024501-4 MEASUREMENT OF ENERGY DECAY IN ... PHYSICAL REVIEW B 84, 024501 (2011) over energy have to be put into discrete form. For a binning wherewehaveusedtheBCSresult/kTc = 1.76. Here, size given by d in energy, we define the number of excitations D(Ef )/2 is the single-spin density of states, and we define the in bin i as Cooper pair density ncp ≡ D(EF ). ≡ The time dependence of the quasiparticle density can be ni d ρ( i )f ( i ). (C1) understood via the rate equation Using this definition, the total quasiparticle density is d r nqp =−2 nqp + rqp, (D5) nqp 2 dt = ni . (C2) ncp d n 43.6 n 2 r i qp =− qp + qp , (D6) The scattering and recombination rates of Eqs. (6) and (7) dt ncp τ0 ncp ncp of Ref. 11 can then be expressed in a discrete form, where a recombination event removes two quasiparticles, − 2 2 s ( i j ) and rqp is the single-particle quasiparticle injection rate. The → = 1 − N ( − )d ρ( ) i j τ (kT )3 p i j j second equation is for the normalized quasiparticle density, j 0 c i j 2 and has a recombination rate that is proportional to nqp because (C3) of the two-body electron-electron interaction. ≡ s The equilibrium quasiparticle density is given by setting Gij , (C4) dnqp/dt = 0, yielding density and recombination rates j 2 2 eq 1/2 ( i + j ) (n ) τ r τ r = 1 + N ( + ) qp = 0 qp = 0 r eq i,j 3 p i j ( ) , (D7) τ0(kTc) i j ncp 43.6 ncp 43.6 j × d ρ( )f ( )(C5)43.6 r 1/2 43.6 (n )eq j j ( r )eq = qp = qp . (D8) ≡ r τ0 ncp τ0 ncp Gij nj , (C6) j The first equation is close to what was found numerically in where the phonon occupation factor is Np(E) = Ref. 11. The second is given by the geometric mean of the 1/| exp(−E/kTp) − 1| and the bracketed terms are the normalized injection and the characteristic electron-electron G factors. We have also assumed small occupation, so that in interaction rates. Eq. (C3)weuse1− f → 1. We compared the results of this simple calculation with The coupled differential equations for the change in the numerical solutions for a range of injection rates and found excitation number are excellent agreement for nqp/ncp 0.001. Even at large density = d nqp/ncp 0.1, Eq. (D7) is a reasonable approximation as n = Gs n − Gs n − (1 + δ )Gr n n , (C7) its prediction is only 40% larger than that obtained via dt i ji j ij i ij ij j i j j numerics. For no injection of quasiparticles r = 0, the differential where δ is the Kronecker δ and accounts for the annihilation qp ij equation can be integrated to give of two quasiparticles when in the same bin (i = j). With the physics expressed in matrix form, a solution can be readily n qp = τ0/43.6 solved numerically. , (D9) ncp t − t0

APPENDIX D: QUASIPARTICLE DECAY where t is the time and t0 is an integration constant, which is approximately the time at which the quasiparticles start to cool. The physics of quasiparticle relaxation and recombination The solution to the differential equation for a ﬁnite injection was discussed in Ref. 11. Although the paper described solu- rate is tions for the nonequilibrium occupation f (E) using numerical eq r eq methods, quasiparticle decay physics can be understood in nqp = (nqp) coth[ ( ) (t − t0)], (D10) the case of low density where they are mostly occupied at the gap. The electron-electron recombination rate of a single where the coth term is replaced by tanh if the quasiparticle quasiparticle, starting from Eq. (7) of Ref. 11, can be well density increases with time. At short times the term coth r t = approximated using 1/ r t, which then gives Eq. (9) and a time dependence that scales only with the electron-phonon coupling time τ0. There is 1 ∞ ( + )2 2 r d 1 + ρ( )f ( )(D1) a relatively sharp crossover to the long-time behavior where the τ (kT )3 quasiparticle density neq is constant with time. The crossover 0 c qp 2 2 ∞ time is given by 1/( r )eq. 1 (2) 1 + d ρ( )f ( )(D2) The inverse of the crossover time thus gives the equilibrium τ (kT )3 2 0 c recombination rate ( r )eq, which is related to the density using 4 nqp = (1.76)3 (D3) Eq. (D7) and the parameter τ0. Comparing this density with τ0 D(EF ) that found from the qubit decay rate allows one to determine n whether quasiparticles are the limiting decay mechanism for = 21.8 qp , (D4) the qubit. τ0 ncp

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FIG. 6. Plot of a = f ()/(nqp/ncp)versusnqp/ncp, found from numerical simulation. Based on the values of nqp/ncp in Fig. 2(b) in the paper, we ﬁnd a 1.2. For a wide range of injection rates, a can −0.173 be well approximated by the power law a 0.12 (nqp/ncp) .

to the simple bulk analysis for the behavior at long times 250 μs.

FIG. 5. (Color online) Plot of quasiparticle density versus settling time for a superconducting disk of radius 5 mm with quasiparticle APPENDIX F: DEPENDENCE OF GAP ON injection at the center. Points are for the simulation at four radii, r = 0 QUASIPARTICLES (black), 0.5 mm (red), 1.25 mm (green), and 5 mm (blue). Large The change in the superconducting gap with quasiparti- changes are observed for small radii, which show a time dependence cles can be calculated starting from the BCS gap equation, but close to that predicted by the full theory of Eq. (9) of main paper assuming a small nonequilibrium populationf (E), (black line) and for the zero background of Eq. (8) (cyan line). At large radii much greater than the characteristic diffusion length of θD ∼1 mm, no change in quasiparticle density is seen. The simulation = D(E )V dE ρ (1 − 2f ), (F1) f E for radius 1.25 mm is in reasonable agreement with experimental data. θD dE θD 1 1 = D(Ef )V √ − dE ρ 2f (F2) 2 − 2 E E APPENDIX E: QUASIPARTICLE DECAY WITH DIFFUSION 2θD nqp D(Ef )V ln − , (F3) The analysis in the last section assumes a bulk (uniform) D(Ef ) model where there is no diffusion of quasiparticles. Here, we describe a numerical solution for quasiparticle decay including where V is the attraction potential and θD is the Debye energy. relaxation, recombination, and diffusion using the simple Solving for the gap, one finds geometry of a thin superconducting disk of radius 5 mm. 1 n We use constant quasiparticle injection throughout the disk = 2θ exp − − qp (F4) D D(E )V D(E ) and a large injection pulse into the center of the disk F F at time t =−200 μstot = 0. Because diffusion depends n = − qp on the quasiparticle energy, the calculation keeps track of 0 exp (F5) D(EF ) the occupation probability for both the radius and energy nqp variables. 0 1 − , (F6) In Fig. 5 we plot quasiparticle density versus settling time ncp in a manner similar to that in the main paper, but for four radii. We find differing behavior depending on the ratio of the radius where 0 is the normal expression for the BCS gap with no with the diffusion length ∼1 mm, as computed for e-e diffusion quasiparticles. in Fig. 3 of Ref. 11. For small radii, we see a dependence on time that matches closely with the bulk theory, as described APPENDIX G: NUMERICAL DETERMINATION OF in the main paper. For a radius much larger than the diffusion OCCUPATION PARAMETER length, the quasiparticle density does not change. For the radius close to the diffusion length, we observed behavior between To determine the effect of nonequilibrium quasiparticles, the two limits—a reduced peak density but a relaxation to the both the quasiparticle density nqp and the occupation proba- steady-state value that has a similar time scale as for a small bility at the gap f () must be calculated. We plot in Fig. 6 = radius. the quantity a f ()/(nqp/ncp) versus nqp/ncp obtained We note that the actual qubit device has interruptions in from numerical computations for a wide range of injection the ground plane due to the device geometry, so that this rates. We find that the results are well approximated by computation will not exactly match the experimental data. a line on the log-log plot, implying that the dependence However, the model mimics the time dependence of the can be well approximated by the power-law formula a −0.173 data fairly well above 10 μs, so it is reasonable to compare 0.12 (nqp/ncp) .

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APPENDIX H: CURRENT-PHASE RELATIONSHIP WITH inductance LJ 0 = 0/2πI0, the qubit resonance frequency QUASIPARTICLES is given by E 1 1 The current-phase relationship from Josephson tunneling is 10 √ − 1/4 given by [2(I0 I)/I0] . (I4) h 2π LJ 0C n We thus calculate the decay rate of the qubit, = − qp − I(φ) I0 sin φ 1 [1 2f ()], (H1) ncp 3/2 1 + cos φ 2eI0 nqp 1 √ (I5) where I = π /2eR and the dependence of on quasi- 2 πC E10 ncp 0 0 n particles is now explicitly shown. To first order, the fractional 1 + cos φ h 1 3/2 n = √ qp (I6) change in the critical current is 2 2 2π LJ 0C E10 ncp δI nqp 2 3/2 0 =− + 1 + cos φ h (2πE10/h) nqp (1 2a) . (H2) √ √ (I7) I0 ncp 2 2 2π 2(I0 − I)/I0 E10 ncp An interesting question is whether the quasiparticle tun- E I 1/2 1/2 n = (1 + cos φ) 10 0 qp . (I8) neling terms should also be included in the current-phase h I0 − I E10 ncp relation. For the junction current to only be a function of phase, it must arise for a purely inductive component of By taking the ratio of Eqs. (I3) and (I8), the quasiparticle the junction admittance, which corresponds to terms with a densities cancel out, and we can relate the dissipation to the frequency dependence that scales as 1/iω. Tunneling of free frequency shift, quasiparticles should not be included√ since it has an additional 1/2 1/2 δ(E10/h) 1 1 + 2a I0 E10 frequency dependence (nqp/ncp) /hω¯ . The Andreev bound =− . + − (I9) states of the quasiparticles have an inductance, so the ac 1 4 1 cos φ I0 I Josephson relation can then be used to find the current φ APPENDIX J: JOSEPHSON EFFECT FOR ARBITRARY 0 IABS (φ) =− Im{ωYABS (φ)} dφ (H3) QUASIPARTICLE OCCUPATION 2π 0 = I0[φ + sin φ]f (). (H4) Here we calculate the effect of a nonequilibrium population of quasiparticle states on Josephson tunneling, as appropriate Note that this term increases the critical current, and if one for qubit devices. The current proportional to cos δ is also replaces φ → sin φ it cancels out the decrease coming from evaluated for the case of gaps that are not equal. The Josephson tunneling. Since many experiments have shown that results are readily obtained using standard second-order the temperature dependence of the current-phase relation is perturbation theory and simple integration of intermediate given by only Josephson tunneling, we do not use this Andreev formulas. bound state term in our calculations. In addition, our data are The work expands on Ref. 27, which calculated the not consistent with including this term since it has the effect Josephson effect at zero temperature. In the paper, the section of reducing 2a in Eq. (H2) to a value below a. on Josephson tunneling is the starting point of this calculation. The Josephson effect is derived by calculating the second- order change in energy to a superconducting state from a APPENDIX I: RELATING DISSIPATION AND THE tunnel junction. The tunneling Hamiltonian in second-order FREQUENCY CHANGE perturbation theory is given by Since both dissipation and the fractional critical-current (2) 1 change are proportional to n , the magnitude of these effects H = HT HT , (J1) qp T are related. The fractional change in the qubit resonance i i frequency E10/h can be calculated knowing that its dominant 1/4 where i is the energy of the intermediate state i. Because the scaling is E10/h ∝ (I0 − I) , where I is the qubit bias † terms in HT have both γ and γ operators, the second-order current, giving Hamiltonian has terms that transfer charge across the junction − but do not change the superconducting state, thus giving a δ(E10/h) = 1 δ(I0 I) − (I1) change in the energy of the state. This differs from first-order E10/h 4 I0 I tunneling theory, which produces current only through the real 1 I δI = 0 0 (I2) creation of quasiparticles. 4 I0 − I I0 Because HT has terms that transfer charge in both direc- + n tions, H H will produce terms that transfer two electrons =−1 2a I0 qp T T . (I3) to the right, two to the left, and with no net transfer. With 4 I0 − I ncp no transfer, a calculation of the second-order energy gives a Quasiparticle dissipation can be likewise written in terms constant value, which has no physical effect. We first calculate terms for the transfer of two electrons to the right from of the quasiparticle density, provided we first re-express the −→ −→ −→ −→ capacitance C into qubit parameters. Using the Josephson (H T + + H T −)(H T + + H T −). Nonzero expectation values

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† † are obtained for only two out of the four terms, as given by → uveiφ(−γ γ + γ γ ), (J6) −→ −→ −→ −→ 0 0 1 1 −−→ + † † (2) H T + H T − H T − H T + → iφ − + H = (J2) c−kck uve ( γ0 γ0 γ1γ1 ), (J7) T i i † † † −iφ † −iφ c c− = (uγ + ve γ )(uγ − ve γ ) (J8) † † † † k k 0 1 1 0 (c c )(c c ) + (c c )(c c ) = | |2 L R −L −R −L −R L R → −iφ − † + † t (J3) uve ( γ0 γ0 γ1γ1 ), (J9) i i † † † † c c → uve−iφ(−γ γ + γ γ ), (J10) † † + † † −k k 0 0 1 1 (cLc−L)(cRc−R) (c−LcL)(c−RcR) = |t|2 . (J4) where we have only included pairs of quasiparticle operators γ

i i that leave the superconducting state unchanged, as needed for The pairs of electron creation and annihilation operators a calculation of the energy change from tunneling. Inserting can be computed, giving these operators into Eq. (J4) and deﬁning the phase difference = − = + iφ † − iφ † δ φL φR, we ﬁnd ckc−k (uγ0 ve γ1 )(uγ1 ve γ0 ) (J5)

† † † † † † † † −−→ (−γ γ + γ γ )(−γ γ + γ γ ) + (−γ γ + γ γ )(−γ γ + γ γ ) (2) = | |2 iδ L0 L0 L1 L1 R0 R0 R1 R1 L0 L0 L1 L1 R0 R0 R1 R1 HT t e (uLvL)(uRvR) (J11) i i − † † − † † − † † − † † 2 iδ γL0γL0γR1γR1 γL1γL1γR0γR0 γL1γL1γR0γR0 γL0γL0γR1γR1 = |t| e (uLvL)(uRvR) + EL + ER −EL − ER i † † † † † † † † γ γ γ γ + γ γ γ γ γ γ γ γ + γ γ γ γ + L0 L0 R0 R0 L1 L1 R1 R1 + L1 L1 R1 R1 L0 L0 R0 R0 , (J12) EL − ER −EL + ER

− − + − where we have computed the intermediate energy i using =−1 fL fR fLfR + fLfR + fR fLfR a positive (negative) energy E for the creation (annihilation) EL + ER EL + ER EL − ER + i of a quasiparticle. The quantity uv = /2E describes the fL − fLfR amplitude for the virtual quasiparticle to be both electron- and + (J16) −E + E + i holelike, which allows a net transfer of charge by two electrons. L R 1 − f − f f − f We now change the sum to an integral over electron states =− L R + P R L according to EL + ER EL − ER ∞ ∞ + iπ(fL + fR − 2fLfR)δ(EL − ER) (J17) → N0L dξL N0R dξR 1 2fREL − 2fLER −∞ −∞ =− + P i + 2 − 2 ∞ ∞ EL ER EL ER + + − − = 2N0L ρL dEL 2N0R ρR dER, (J13) iπ(fL fR 2fLfR)δ(EL ER), (J18) L R where G comes from quasiparticle operators [bracket terms in = where√ N0 is the normal density of states, and ρ Eq. (J12)] after removing a factor of 2 because of the pair of E/ E2 − 2 is the (normalized) superconducting density of states 0 and 1. We take → 0+, and the integration over the states. zero of energy in the denominator is performed using 1/(x + By describing the superconducting state with an occupation i ) = P(1/x) + iπδ(x), where P is the principle part and δ(x) probability of quasiparticles f = f (E), the quasiparticle is the Dirac δ function. operators for the creation then destruction of a quasiparticle The total second-order Hamiltonian for the tunneling of is weighted by 1 − f , while the process of destruction then two electrons in both directions is creation is weighted by f . The tunneling Hamiltonian is then −−→ ←−− given by H (2) = H (2) + H (2) (J19) T T T −−→ ∞ −−→ (2) 2 iδ = (2) + H =|t| e N0LN0R 2 ρLdEL HT H.c. (J20) T −−→ L ∞ = 2Re H (2) , (J21) L R T × 2 ρRdER 2G, (J14) R 2EL 2ER where H.c. is the Hermitian conjugate. This result can be expressed in more physical terms by (1 − fL)(1 − fR) fLfR (1 − fL)fR G =− − + noting that the junction resistance can be written as EL + ER −EL − ER EL − ER + i − 2 fL(1 fR) 1 4πe 2 + (J15) = |t| N0RN0L. (J22) −EL + ER + i Rn h¯

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2 − 2 In addition, The Josephson tunneling current is given by We next compute the ER integral for the ER/(EL ER) term. We define w = E / , and note that the integration 2e ∂ H (2) L R I = T . (J23) over ER depends on whether EL is greater or less than R, j h¯ ∂δ Combining all of these equations, the total Josephson current ∞ E I = P dE R R (J35) is given by the integrals 2R R 2 − 2 R 2 − 2 EL ER ∞ ∞ ER R 2 L R ∞ Ij = sin δ P ρLdEL ρLdER 1 z πeRn L R EL ER = P dz√ (J36) 2 − z2 − w2 1 f E − f E 1 z ⎧1 × − 2 R L L R − π cos δ ∞ + 2 − 2 ⎨0(w>1) EL ER EL ER 1 =−√ × (J37) ∞ ⎩ x2−1 LR 1 − w2 arctan (w<1) × ρ ρ dE (f + f − 2f f ) , 1−w2 L R E2 L R L R 1 max(L,R ) π 1 (J24) =− √ θ(R − EL). (J38) 2 1 − w2 We note that the sin δ term in Eq. (J24) corresponds to Eq. (22) 28 of the Ambegaokar-Baratoff calculation. This result implies that when integrating over EL, no contri- We evaluate these integrals by first considering, without√ loss bution comes from R to infinity, so the full integral is 2 2 of generality, that L <R.Usingρ/E = / E − = and y ER/L > 1, we compute that the temperature- R L independent term 1/(EL + ER) gives for integration over EL I2RL = dEL 2fLI2R (J39) L 2 − 2 ∞ EL L L 1 I1L = dEL (J25) + R L 2 − 2 EL ER L R EL L =−π fL dEL . (J40) ∞ L 2 − 2 2 − 2 1 1 EL L R EL = dx√ (J26) 1 x2 − 1 x + y arccosh y In the limit where L is close to R such that fL is constant = . (J27) over the region of integration, the integral can be evaluated as y2 − 1

29 The remaining integration over ER gives =− − 2 I2RL πfL(L) LEllipticK[1 (L/R) ] (J41) ∞ 2 = R Larccosh( ER/L) π I1LR dER (J28) − 2fL(), (J42) R E2 − 2 E2 − 2 4 R R R L − LR L R where in the last equation we have taken the limit → = = π EllipticK (J29) L R L + R L + R . π 2 For the case of equal gaps , the ﬁrst two integrals give a = (for = = ), (J30) Josephson current with a sin δ dependence, 4 L R = where the last equation uses EllipticK(0) π/2. From nu- 2 merical integration, we have found that Eq. (J29) is only Ijs = (I1LR + I2RL)sinδ (J43) πeRn approximate for L = R. π For the next term in Eq. (J24) that has the principle = [1 − 2f ()] sin δ (J44) 2 − 2 L part of EL/(EL ER), we ﬁrst integrate over EL. With the 2 eRn assumption < , the integral always passes across the L R = π pole at E giving tanh[/2kT ]sinδ (thermal). (J45) R 2 eRn ∞ = L EL I2L P dEL 2 2 (J31) 2 2 E − E The last equation assumes a thermal population of quasipar- L E − L R = + L L ticles given by f (E) 1/[1 exp(E/kT )], which yields the ∞ 1 x Ambegaokar-Baratoff formula28 for the Josephson current. = P dx√ (J32) 2 2 The Josephson current can also be calculated for an 1 x2 − 1 x − y ⎧ ∞ arbitrary quasiparticle occupation under the assumption that ⎨ y2−1 − arctanh (x>y) the difference of the gaps R L is much larger than the 1 x2−1 =− × (J33) typical width of the quasiparticle distribution. The contribution y2 − 1 ⎩ x2−1 arctanh 2− (x

024501-9 M. LENANDER et al. PHYSICAL REVIEW B 84, 024501 (2011) is given by density is constant with f = fl + fR − fLfR, numerical integration gives the approximate formula R 2 cos δ 2sinδ R L I − −0.1 + L − 0.5ln 1 − L f. (J48) Ij2 − fL dEL (J46) jc eR eRn R R n 2 − 2 L E2 − 2 R L L L For the case where the gaps greatly differ, and in the limit n −sin δ RL qpL discussed in the previous paragraph, the current is , (J47) eRn 2 − 2 N0LL δ ∞ R L −2 cos L R Ijc ρL(R) 2 ρR dE fR (J49) eRn R R cos δ 2 n Note that these equations have a contribution from quasi- =− L qpR , (J50) particle occupation only from the left electrode, which has eRn 2 − 2 N0RR lower gap. This makes sense since a more exact theory of R L Andreev bound states has suppression of the critical current which has a form and magnitude similar to the thermal current. from occupied states in the gap, which has to have energy Note that for the cos δ current, quasiparticles contribute below that of the lowest gap. from the higher gap side of the junction. This contrasts the For the cos δ term, we ﬁrst note that the current diverges behavior of the sin δ current, which has a contribution from logarithmically for L → R. Assuming the quaisparticle the superconducting electrode with lower gap.

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