A superconductor free of quasiparticles for seconds
E. T. Mannila, P. Samuelsson, S. Simbierowicz, J. T. Peltonen, V. Vesterinen, L. Grönberg, J. Hassel, V. F. Maisi, J. P. Pekola
AA superconductor free of quasiparticles for seconds
E. T. Mannila, ∗ P. Samuelsson, S. Simbierowicz, † J. T. Peltonen, V. Vesterinen, L. Gr¨onberg, J. Hassel, V. F. Maisi, and J. P. Pekola QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden VTT Technical Research Centre of Finland Ltd,QTF Centre of Excellence, P.O. Box 1000, FI-02044 VTT, Finland (Dated: February 2, 2021)
Superconducting devices, based on the Cooperpairing of electrons, are of outstanding impor-tance in existing and emergent technologies, rang-ing from radiation detectors [1, 2] to quantumcomputers [3]. Their performance is limitedby spurious broken Cooper pairs also known asquasiparticle excitations [4–12]. In state-of-the-art devices, the time-averaged number of quasi-particles can be on the order of one [2, 12–15].However, realizing a superconductor with no ex-citations remains an outstanding challenge. Here,we experimentally demonstrate a superconduc-tor completely free of quasiparticles up to sec-onds. The quasiparticle number on a mesoscopicsuperconductor is monitored in real time by mea-suring the charge tunneling to a normal metalcontact. Quiet excitation-free periods are inter-rupted by random-in-time events, where one orseveral Cooper pairs break, followed by a burstof charge tunneling within a millisecond. Our re-sults vindicate the opportunity to operate deviceswithout quasiparticles with potentially improvedperformance. In addition, our present experi-ment probes the origins of nonequilibrium quasi-particles in it; the decay of the Cooper pair break-ing rate over several weeks following the initialcooldown rules out processes arising from cosmicor long-lived radioactive sources [16–19].
The Bardeen, Cooper and Schrieffer theory of super-conductivity predicts that the number of quasiparticleexcitations should be exponentially small at tempera-tures low compared to the superconducting gap ∆ (di-vided by Boltzmann’s constant k B ). Nevertheless, it isexperimentally well established that a residual popula-tion persists down to the lowest temperatures [4–6], at-tributed to Cooper pair breaking in the superconductordue to non-thermal processes. Residual quasiparticles aredetrimental to the performance of many superconductingdevices, ranging from setting the fundamental limit forthe sensitivity of kinetic inductance detectors [1, 6] to ar-guably limiting the coherence times of superconductingqubits to the millisecond range [7, 8, 16]. Correspond-ingly, their origins and dynamics as well as mitigationschemes have been studied extensively [9–13].In several state-of-the art devices, the observed ultra- low quasiparticle densities correspond to only a few ex-citations within the entire device [2, 12–15]. In this few-particle regime, the number of quasiparticles fluctuatesstrongly in time [2, 6, 12, 15, 20], potentially leaving thedevice free of them for certain periods. Identifying suchperiods would open up possibilities for operation of su-perconducting devices in the absence of quasiparticles.However, measuring their absolute number in real timeis challenging, mainly because breaking of Cooper pairsdoes not change the total charge. In fact, there are only afew time-resolved experimental investigations of the dy-namics in this few-particle regime [2, 5, 13, 15, 21, 22],and none where quasiparticle-free periods have been iden-tified.Here, we show experimentally a mesoscopic supercon-ducting island that is free of quasiparticles for time pe-riods reaching seconds, five orders of magnitude longerthan typical operation times of superconducting qubits[3]. The instantaneous number of quasiparticles is recon-structed from real-time measurements of single-electrontunneling from the island to a normal metal contact, asillustrated in Fig. 1. We observe quiet periods, with notunneling events and no quasiparticles on the island, for99.97% of the time, interrupted by Cooper pair break-ing events, followed by bursts of tunneling within a mil-lisecond. The pair breaking events occur randomly intime, with an average rate 1.5 Hz and a probability ∝ exp( − . N pair ) to break N pair Cooper pairs. Thatis, in 40% of the events, more than one Cooper pair isbroken.Due to the three orders of magnitude faster probingvia single-electron tunneling as compared to the Cooperpair breaking, we are able to measure the decay dynam-ics after individual pair-breaking events, while the time-averaged density n QP ≈ .
01 quasiparticles/ µ m is closeto the lowest reported values [22]. This goes well beyondprevious experiments on time-resolved quasiparticle tun-neling in superconducting structures [2, 5, 21–24] or ex-periments with intentional injection of a large numberof quasiparticles [10, 11]. The very good agreement be-tween the measured dynamics and a rate equation modelfor the quasiparticle number allows for an unambiguousidentification of the excitation-free periods. Our findingscreate opportunities for operating superconducting de-vices while simultaneously monitoring the instantaneous a r X i v : . [ c ond - m a t . s up r- c on ] J a n charge detector (a)
200 nm quasiparticle-free intervals(c) de t e c t o r ou t pu t ( a . u . ) de t e c t o r ou t pu t ( a . u . ) time (ms) time (ms)1000 3088.6 3089.0 3089.4 e xc e ss e l e c t r on s N q u a s i - p a r t i c l e s N Q P superconductor normalmetal tunneljunction (b) e xc e ss e l e c t r on s N FIG. 1.
Real-time monitoring of the number of quasiparticles on a superconducting island via charge tunneling. (a) Sketch of experiment. Pair-breaking radiation (pink arrow) is absorbed in a micron-scale superconducting aluminum island(blue), breaking one or more Cooper pairs and producing quasiparticle excitations (light blue circles). The quasiparticles tunnelout to the normal metallic copper leads (orange) via a tunnel junction as either electrons or holes. The tunneling events aremeasured in real time with a capacitively coupled charge detector (see Methods). (b) False-color scanning electron micrographof the device, with colors as in (a). As no voltage bias is applied across the device, the two tunnel junctions act as a singlejunction. (c) Time trace of charge detector output (red line) and the corresponding charge state (black line), N = 0 , ± N changing by ± N QP on the island as a function of time,inferred from (d). The initial Cooper pair breaking event, occuring on average within 30 µ s before the first tunneling event, isnot directly observed (here indicated by a dashed line). quasiparticle number, allowing for identifying periods forquasiparticle-free operation. This would enable testingpredictions of improved device performance in the ab-sence of quasiparticles [7, 8, 25]. In addition, the statis-tics of number of quasiparticles created might provideadditional information on the origin of the pair-breakingprocesses. Here, we observe that the rate of Cooper pairbreaking events decreased by a factor of 4 over a periodof weeks. This rules out commonly suggested sources ofnonequilibrium quasiparticles, such as insufficient shield-ing against stray light [22, 26] or ionizing radiation [16–19].We monitor the charge on an aluminum superconduct-ing island, sketched in Fig. 1(a) and shown in Fig. 1(b),with volume V = 2 µ m × . µ m ×
35 nm, charg-ing energy E C ≈ µ eV and superconducting gap ∆ ≈ µ eV. The island is coupled to normal metal copperleads via an insulating aluminum oxide tunneling barrier.Measurements were done in a dilution refrigerator witha base temperature of 20 mK at a normal metal elec-tron temperature T N ≈
100 mK, obtained by fitting the temperature dependence of two-electron Andreev tunnel-ing rates (supplement). Coulomb blockade limits the ob-served charge states to N = 0 , ± µ s. Details on thesample fabrication, measurements and device characteri-zation are given in Methods and Supplementary Material.We measure 5 s time traces of the detector output. Arepresentative trace is shown in Fig. 1(c). Quiet periodsup to seconds long, with no tunneling events and zero ex-cess electrons, are interrupted by bursts of two or moreone- or two-electron tunneling events between the chargestates N = 0 , ± × bursts of tunneling. In Fig. 2(a) we show thatthe waiting times between the bursts of tunneling fol-low an exponential distribution, characterized by a rate Waiting time between bursts of tunneling (s) Broken Cooper pairs per burst c oun t s (a) (b)0 1 2 3 0 5 10 experimentexponential fit c oun t s experimentexponential fit FIG. 2.
Statistics of Cooper pair breaking events. (a) Lengths of the quiet periods between bursts of tunnel-ing. The experimental data (circles), binned into intervals ofequal length 0.2 s, is fitted with an exponential curve (solidline). The corresponding Poissonian waiting time distributiondescribes random-in-time bursts with an average burst rateΓ burst ≈ . ≈ . burst overestimates the true burst rate, 1 . N pair ( N QP = 2 N pair ) gen-erated per event. Experimental data (circles) is obtained bycounting the number of single electron tunneling events withineach burst, see Fig. 1(c,d). The solid line is an exponential fit ∼ exp( − λN pair ), weighted by the counts, with λ = 0 . ± . Γ burst ≈ . →± ≈
11 kHz and Γ ± → ≈
20 kHz. Hererate subscripts denote the initial and final charge states.The tunneling rates are thus several orders of magnitudehigher than the rate at which the bursts occur.To infer the instantaneous number of quasiparticles N QP from the time traces we make a number of observa-tions (see supplement for details): (i) The single-electrontunneling rates Γ →± and Γ ± → are independent of thegate offset n g , as expected for quasiparticle excitations[27]. (ii) The number of single-electron tunneling eventswithin a burst is even. (iii) The two-electron events aredue to Andreev tunneling which does not change N QP .(iv) Based on the measured values of T N , ∆ and E C andfurther analysis we conclude that the rate for quasipar-ticles tunneling back into the superconductor from thenormal metal leads is much smaller than both single-and two-electron tunneling rates. (v) Moreover, basedon earlier experiments [10, 28], the recombination of twoquasiparticles into a Cooper pair is more than an or-der of magnitude slower than single-electron tunneling.Taken together, these observations imply the followingcompelling, simple physical picture of the quasiparticledynamics, illustrated in Fig. 1(e): A burst of tunnelingis initiated by the breaking of N pair Cooper pairs, cre-ating N QP = 2 N pair quasiparticles on the island. Theytunnel out to the normal leads as electrons or holes viasingle-electron tunneling events, which decrease the num- ber of quasiparticles one by one, N QP → N QP − N changes by ±
1. After the last quasiparti-cle has tunneled out, within a millisecond from the startof the burst, the superconducting island is completely freeof quasiparticles until the next burst occurs, on averagefor 0.4 seconds. Therefore counting the number of eventschanging N yields directly N QP , which also allows us toinfer N pair for each burst.The statistical distribution of N pair , shown in Fig.2(b), is well described by an exponential ∝ exp( − λN pair )with λ ≈ . ± .
1. In 40% of the bursts, more than oneCooper pair was thus initially broken. Further insightinto the dynamics of quasiparticle number relaxation isobtained from the probability P ( N QP , t ) of having N QP quasiparticles at a time t after the first tunneling eventin the burst. The probabilities for N QP = 0 to 3 as wellas N QP ≥ t = 0 . P ( N QP , t ) increases(decreases) for N QP even (odd) as a function of t , a con-sequence of having an odd number of quasiparticles onthe island directly after the first tunneling event at t = 0.Figure 3(a) also shows that P ( N QP , t ) decays faster forincreasing N QP . This is consistent with the predictionthat the single electron tunneling rate for N QP quasipar-ticles on the island is proportional to N QP [27]. It isfurther reflected in the time-dependent rates Γ →± ( t )and Γ ± → ( t ) for single electron tunneling shown in Fig.3(b), evaluated in a short interval around time t withinthe bursts. In particular, we see that the long time ( t > µ s) rate from the state N = 0, Γ →± ( t ) ≈
15 kHz,is twice the rate from N = ±
1, Γ ± → ( t ) ≈ µ s. The measured single-quasiparticle rateΓ QP = Γ ± → ( t ) = 8 kHz is close to the estimate 17 kHzobtained based on other experiments (see supplement).The dynamics of N QP within a burst, based on the sim-ple physical picture, are modelled by a rate equation for P ( N QP , t ), see Methods. As the initial condition, we havethe experimentally determined distribution P ( N QP ) ∝ exp( − . N pair ) = exp( − . N QP ) at the start of theburst, see Fig. 2. Only tunneling out of quasiparticlesis accounted for, making Γ QP the only free parameterof the model. From Fig. 3(a) we see that the modelreproduces the experimental results well for all timeswithin the burst and over three orders of magnitude of P ( N QP , t ), for all N QP shown. The best fit is obtained forΓ QP = 8 . →± ( t ) and Γ ± → ( t ) are Γ N → N ± ( k H z ) ≥ N QP (a)(b) P ( N Q P , t ) -3 -2 -1 t (µs from first tunneling event of burst)0102030 -1 → → +1 0 → -1+1 → N FIG. 3.
Dynamics of quasiparticle relaxation within a burst. (a) The probability P ( N QP , t ) to have N QP quasiparticlespresent on the island at time t , for N QP = 0 , , , N QP ≥
4. Already at t = 150 µ s the probability that the burst hasended, P ( N QP = 0), is above one half. The theoretical model (solid lines), see text and Methods, assumes quasiparticlerelaxation by tunneling only, and uses the measured distribution of broken Cooper pairs, Fig. 2(b), as an initial condition. (b)Effective single-electron tunneling rates Γ N → N ± ( t ) as a function of time t from the first tunneling event of a burst. After aninitial decay, the rates saturate to minimum values corresponding to tunneling rates for one (two) quasiparticles on the islandfor N odd (even). The dip in the rates at short times ( t < µ s) is due to finite bandwidth of the experiment, as discussed inMethods. In both panels (a) and (b), experimental data (symbols) are obtained from averaging over 2 × bursts. For thetheoretical model (solid lines) Γ QP = 8 . plotted together with the experimental data. Both theobserved initial decay and the long time saturation arewell captured by the model. The model also allows us(see Methods) to obtain an accurate estimate of the time-averaged number of quasiparticles on the island h N QP i =2 . burst / Γ QP = 4 . × − , measured 130 days after thestart of the cooldown. The corresponding quasiparticledensity is n QP = h N QP i /V = 0.013 quasiparticles/ µ m or x QP = h N QP i / ( D ( E F )∆ V ) ≈ × − , if normal-ized by the Cooper pair density ∆ D ( E F ), with D ( E F ) =2 . × J − m − the density of states. This is compa-rable to the lowest measured values x QP ∼ − reportedin the literature [22] and below the bound x QP ≈ × − estimated due to the ionizing radiation background inRef. [16].In our experiment, the rate of the bursts Γ burst de-creased over time from the start of a cooldown ontimescales of weeks following a power law Γ burst ∝ t − . ,where t is the time after the sample reached tempera-tures below 77 K. This was observed reproducibly in twosubsequent cooldowns of the same sample, as shown inFig. 4. The observed long time decay of Γ burst rules outa number of observed or suggested sources of quasiparti-cles being dominant in our system. In particular, exter-nal infrared or microwave photons [22, 26] and environ-mental radioactivity and cosmic rays [16–19] are all ex- pected to be stationary sources of quasiparticles. More-over, the distribution of Cooper pairs broken per burstdid not change during the cooldown (supplement), sug-gesting that the same source of quasiparticles was dom-inant throughout the experiment. Although heat leaksdecreasing over timescales of weeks are well known inexperiments at low temperatures [29], we cannot certifywhat in our experiment is the mechanism of Cooper pairbreaking.In conclusion, we have demonstrated that a mesoscopicsuperconducting island can remain completely free fromquasiparticles for time periods up to seconds. These find-ings open a new route to identifying periods of quasipar-ticle free operation of superconducting devices, of keyinterest for e.g. superconducting quantum computation.Our approach, giving access to the number of Cooperpairs broken in a given event, also provides novel in-formation on the properties of the source of quasipar-ticles, which is presently an actively investigated and de-bated topic [16–19]. Furthermore, our device could po-tentially be adapted to operate as an energy-resolvingsingle-photon detector in the terahertz range similar tothe quantum capacitance detector [2]. t (days since beginning of cooldown)10 -1 -2 Γ bu r s t ( H z ) n Q P ( qua s i pa r t i c l e s / µ m ) first cooldownsecond cooldowndata of Figs. 1-3power law t -0.9 FIG. 4.
Decay of burst rate over time.
The ob-served rate of Cooper pair breaking events leading to burstsof tunneling Γ burst is shown as a function of time t after thestart of the first (filled diamonds) and second (open triangles)cooldown. A power law fit Γ burst ∝ t − . is shown by a solidline. The right y-axis shows the corresponding time-averagedquasiparticle density n QP , proportional to Γ burst (see text)since the distribution of quasiparticles generated per burst aswell as the single electron tunneling rate were found to beconstant over the entire cooldown period (see supplement).A circle indicates the data used in Figs. 1-3. ACKNOWLEDGEMENTS
We acknowledge useful discussions with O. Maillet andsupport in shielding solutions from J. Ala-Heikkil¨a. Wethank Janne Lehtinen and Mika Prunnila from VTTTechnical Research Center of Finland Ltd also involved inthe JPA development. This work was performed as partof the Academy of Finland Centre of Excellence program(projects 312057, 312059 and 312294). We acknowledgethe provision of facilities and technical support by AaltoUniversity at OtaNano - Micronova Nanofabrication Cen-tre and OtaNano - Low Temperature Laboratory. E.T.M.and J.P.P. acknowledge financial support from Microsoft.V.V. acknowledges financial support from the Academyof Finland through grant no. 321700. P.S. and V.F.M.acknowledge financial support from the Swedish NationalScience Foundation and V.F.M. acknowledges financialsupport from the QuantERA project “2D hybrid mate-rials as a platform for topological quantum computing”and NanoLund.
AUTHOR CONTRIBUTIONS
E.T.M., P.S., V.F.M. and J.P.P. conceived the experi-ment and model and interpreted the results. E.T.M. fab-ricated the sample and performed the experiment withsupport from J.T.P., and E.T.M. analysed the data. P.S.performed the theoretical modeling. S.S., V.V., L.G.and J.H. provided the Josephson parametric amplifier.E.T.M. and J.T.P. integrated the Josephson parametric amplifier into the setup with assistance from S.S. andV.V. The manuscript was written by E.T.M. and P.S.with input from all coauthors.
METHODSSample fabrication
The sample is fabricated by standard electron-beamlithography and multiple angle deposition on a siliconsubstrate. The normal metal leads of the superconduct-ing island are capacitively shunted by a 2 nm / 30 nm / 2nm thick Ti/Au/Ti ground plane, which helps suppressphoton-assisted tunneling events [30]. The charge de-tector is a normal-metallic single-electron transistor fab-ricated with the laterally proximitized tunnel junctiontechnique [31], expected to minimize quasiparticle poi-soning due to nonequilibrium phonons [11]. A 10 µ mlong chromium capacitor creates the capacitive couplingbetween the detector and superconductor island. Theground plane and coupler are insulated from the actualdevices by a 40 nm thick aluminum oxide layer grownby atomic layer deposition. Further details on the sam-ple design and fabrication are given in the SupplementalMaterial. Measurements
The sample is attached to a copper sample holderweighing approximately 0.5 kg with vacuum grease. Alu-minum wire bonds connect the sample chip and the res-onator chip (see below) to the printed circuit board ofthe sample stage. The bond wires to one of the biasleads and the gate electrode of the superconducting is-land had disconnected during the cooldown, but we wereable to tune n g by applying a voltage to the survivingbias lead instead. The sample holder is closed with asingle indium-sealed cap, which is coated by Eccosorb.The sample holder is thermally anchored to the mixingchamber stage of a cryogen-free dilution refrigerator witha base temperature of 20 mK, with the innermost radia-tion shield attached to the still flange of the refrigerator(temperature between 0.5 K and 1 K).The low-frequency measurement lines to the sampleare filtered with approximately 2 m of Thermocoax [32].The radio-frequency lines are filtered and attenuatedwith commercial attenuators and filters at different tem-perature stages of the refrigerator. We have measuredthe same sample in two subsequent cooldowns. For thesecond cooldown, we added home-made Eccosorb filtersto the input and output RF lines at the mixing chamberstage, as well as disconnecting a RF gate line not used inthis experiment. A detailed wiring schematic is providedin the supplement.We operate the charge detector as a radio frequencysingle-electron transistor [33] at 580 MHz, with detailson the detector given in the Supplemental Material. Wehave verified by changing the probe power that the detec-tor does not cause measurable backaction on the super-conducting island in this experiment. The output signalfrom the detector is amplified by a Josephson paramet-ric amplifier (JPA) with added noise on the order of 100mK [34] at the mixing chamber, followed by semiconduc-tor amplifiers at the 2 K stage and at room temperature.The parametric amplifier is similar to the devices pre-sented in Ref. [34], but has a higher bandwidth. Operat-ing the JPA in the phase-sensitive mode enables chargesensitivities of 2 × − e/ √ Hz and charge detection witha signal-to-noise ratio of 6 in 3 µ s. We note that the mea-sured time trace appears noisier in Fig. 1(c) than in 1(d)although the same data is shown on both panels, but thisis purely a visual effect (see supplement for characterisa-tion of the signal-to-noise ratio). The JPA improves thetime resolution of the experiment by roughly a factor of20.The amplified signal is down-converted and digitizedat room temperature with a Aeroflex 3035C vector dig-itizer. We save both quadratures of the signal in 5 straces with a sampling rate of 4 MHz, and rotate thesignal such that the charge response is along one quadra-ture only. We then low-pass filter the signal digitally witha cutoff frequency of 150 kHz to increase the signal-to-noise. This post-processing filter sets the bandwidth ofour charge detector. We then identify the charge states,bursts of tunneling, and single- and two-electron tunnel-ing events from the filtered time traces as described inthe Supplementary Material. Model
The quasiparticle dynamics is governed by a rate equa-tion for the probability distribution P ( N QP , t ) of having N QP quasiparticles on the island at a time t after thefirst tunneling event in a burst. From the general model,see Supplemental information, considering only single-electron tunneling out from the island we have dP ( N QP , t ) dt = − σ Γ QP N QP P ( N QP , t )+ ¯ σ Γ QP ( N QP + 1) P ( N QP + 1 , t ) (1)where σ = 1 , ¯ σ = 2 ( σ = 2 , ¯ σ = 1) for N QP = 1 , , , .. ( N QP = 0 , , , .. ). The factors σ arise from the factthat starting from the state N = 0, where N QP is even,the quasiparticles can tunnel out as both electrons andholes with two possible final states N = ±
1. On theother hand, starting from N = ±
1, where N QP is odd,strong Coulomb interactions restrict the possible finalstates to N = 0. With the initial condition given from the observed exponential distribution of number of bro-ken Cooper pairs, see Fig. 2(b), the probability distribu-tion plotted in Fig. 3(a) is evaluated numerically. Theeffective, time dependent single-electron tunneling ratesΓ ± → ( t ) and Γ ± → ( t ), plotted in Fig. 3(b), are ob-tained from (see Supplemental information for details)Γ N → N ± ( t ) = Γ QP P N QP σP ( N QP , t ) N QP P N QP P ( N QP , t ) , (2)where the sum runs over N QP = 2 , , , ... for N = 0 and N QP = 1 , , , ... for N = ± h N QP i on the island is discussed. From the probabilitydistribution in Eq. (1) we can evaluate h N QP i along thefollowing lines: From the observed distribution of wait-ing times between Cooper pair breaking events in Fig.2(a) we have that the average burst rate is Γ burst . Dur-ing a burst, the time dependent number of quasiparticles N QP ( t ) on the island is given by N QP ( t ) = X N QP P ( N QP , t ) N QP , N QP = 0 , , , .. (3)The average number h N QP i is then directly given by h N QP i = Γ burst Z ∞ N QP ( t ) dt = Γ burst Γ QP c ( λ ) (4)where a numerical evaluation gives the coefficient c ( λ =0 .
45) = 2 .
5, which depends on the initial condition. Wenote that the 1 / Γ QP dependence arises since P ( N QP , t ),from Eq. (1), depends only on the renormalized time t Γ QP . Moreover, the upper limit ∞ of the integral canbe taken due to the orders of magnitude difference be-tween the burst duration ∼ / Γ QP and the average timebetween bursts 1 / Γ burst . We can also estimate the prob-ability of having undetected quasiparticles on the islandat any given moment of time within the quiet periods,which is less than 10 − based on the probabilities of miss-ing tunneling events due to the finite detector bandwidthand quasiparticles decaying by recombination instead oftunneling (see supplement for details). ∗ elsa.mannila@aalto.fi † Present address: Bluefors Oy, Arinatie 10, 00370Helsinki, Finland[1] P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, andJ. Zmuidzinas, A broadband superconducting detectorsuitable for use in large arrays, Nature , 817 (2003).[2] P. M. Echternach, B. J. Pepper, T. Reck, and C. M. Brad-ford, Single photon detection of 1.5 THz radiation withthe quantum capacitance detector, Nature Astronomy ,90 (2018).[3] M. Kjaergaard, M. E. Schwartz, J. Braum¨uller,P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D.Oliver, Superconducting qubits: Current state of play, Annual Review of Condensed Matter Physics , 369(2020).[4] J. Aumentado, M. W. Keller, J. M. Martinis, and M. H.Devoret, Nonequilibrium quasiparticles and 2 e periodic-ity in single-Cooper-pair transistors, Phys. Rev. Lett. ,066802 (2004).[5] M. D. Shaw, R. M. Lutchyn, P. Delsing, and P. M.Echternach, Kinetics of nonequilibrium quasiparticle tun-neling in superconducting charge qubits, Phys. Rev. B , 024503 (2008).[6] P. J. de Visser, J. J. A. Baselmans, P. Diener, S. J. C.Yates, A. Endo, and T. M. Klapwijk, Number fluctua-tions of sparse quasiparticles in a superconductor, Phys.Rev. Lett. , 167004 (2011).[7] J. M. Martinis, M. Ansmann, and J. Aumentado, En-ergy decay in superconducting Josephson-junction qubitsfrom nonequilibrium quasiparticle excitations, Phys.Rev. Lett. , 097002 (2009).[8] G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H.Devoret, and L. I. Glazman, Quasiparticle relaxation ofsuperconducting qubits in the presence of flux, Phys.Rev. Lett. , 077002 (2011).[9] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf,L. I. Glazman, and M. H. Devoret, Coherent suppres-sion of electromagnetic dissipation due to superconduct-ing quasiparticles, Nature , 369 (2014).[10] C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Ax-line, T. Brecht, R. W. Heeres, L. Frunzio, M. H. De-voret, G. Catelani, L. I. Glazman, and R. J. Schoelkopf,Measurement and control of quasiparticle dynamics ina superconducting qubit, Nature Communications , 1(2014).[11] U. Patel, I. V. Pechenezhskiy, B. L. T. Plourde, M. G.Vavilov, and R. McDermott, Phonon-mediated quasipar-ticle poisoning of superconducting microwave resonators,Phys. Rev. B , 220501(R) (2017).[12] S. Gustavsson, F. Yan, G. Catelani, J. Bylander, A. Ka-mal, J. Birenbaum, D. Hover, D. Rosenberg, G. Samach,A. P. Sears, S. J. Weber, J. L. Yoder, J. Clarke, A. J. Ker-man, F. Yoshihara, Y. Nakamura, T. P. Orlando, andW. D. Oliver, Suppressing relaxation in superconduct-ing qubits by quasiparticle pumping, Science , 1573(2016).[13] A. J. Ferguson, Quasiparticle cooling of a single Cooperpair transistor, Applied Physics Letters , 052501(2008).[14] A. P. Higginbotham, S. M. Albrecht, G. Kirˇsanskas,W. Chang, F. Kuemmeth, P. Krogstrup, T. S. Jespersen,J. Nyg˚ard, K. Flensberg, and C. M. Marcus, Parity life-time of bound states in a proximitized semiconductornanowire, Nature Physics , 1017 (2015).[15] U. Vool, I. M. Pop, K. Sliwa, B. Abdo, C. Wang,T. Brecht, Y. Y. Gao, S. Shankar, M. Hatridge, G. Cate-lani, M. Mirrahimi, L. Frunzio, R. J. Schoelkopf, L. I.Glazman, and M. H. Devoret, Non-Poissonian quantumjumps of a fluxonium qubit due to quasiparticle excita-tions, Phys. Rev. Lett. , 247001 (2014).[16] A. Veps¨al¨ainen, A. H. Karamlou, J. L. Orrell, A. S. Do-gra, B. Loer, F. Vasconcelos, D. K. Kim, A. J. Melville,B. M. Niedzielski, J. L. Yoder, S. Gustavsson, J. A. For-maggio, B. A. VanDevender, and W. D. Oliver, Impactof ionizing radiation on superconducting qubit coherence,Nature , 551 (2020).[17] L. Cardani, F. Valenti, N. Casali, G. Catelani, T. Charp- entier, M. Clemenza, I. Colantoni, A. Cruciani, L. Gironi,L. Gr¨unhaupt, D. Gusenkova, F. Henriques, M. Lagoin,M. Martinez, G. Pettinari, C. Rusconi, O. Sander,A. V. Ustinov, M. Weber, W. Wernsdorfer, M. Vignati,S. Pirro, and I. M. Pop, Reducing the impact of radioac-tivity on quantum circuits in a deep-underground facility,arXiv preprint arXiv:2005.02286 (2020).[18] C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford,L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe,C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois,and R. McDermott, Correlated charge noise and relax-ation errors in superconducting qubits, arXiv preprintarXiv:2012.06029 (2020).[19] J. M. Martinis, Saving superconducting quantum proces-sors from qubit decay and correlated errors generated bygamma and cosmic rays, arXiv preprint arXiv:2012.06137(2020).[20] C. M. Wilson, L. Frunzio, and D. E. Prober, Time-resolved measurements of thermodynamic fluctuations ofthe particle number in a nondegenerate Fermi gas, Phys.Rev. Lett. , 067004 (2001).[21] N. J. Lambert, M. Edwards, A. A. Esmail, F. A. Pollock,S. D. Barrett, B. W. Lovett, and A. J. Ferguson, Experi-mental observation of the breaking and recombination ofsingle Cooper pairs, Phys. Rev. B , 140503 (2014).[22] K. Serniak, S. Diamond, M. Hays, V. Fatemi, S. Shankar,L. Frunzio, R. Schoelkopf, and M. Devoret, Direct disper-sive monitoring of charge parity in offset-charge-sensitivetransmons, Phys. Rev. Applied , 014052 (2019).[23] D. J. Van Woerkom, A. Geresdi, and L. P. Kouwenhoven,One minute parity lifetime of a NbTiN Cooper-pair tran-sistor, Nature Physics , 547 (2015).[24] M. Hays, G. de Lange, K. Serniak, D. J. van Wo-erkom, D. Bouman, P. Krogstrup, J. Nyg˚ard, A. Geresdi,and M. H. Devoret, Direct microwave measurementof Andreev-bound-state dynamics in a semiconductor-nanowire Josephson junction, Phys. Rev. Lett. ,047001 (2018).[25] T. Karzig, W. S. Cole, and D. I. Pikulin, Quasi-particle poisoning of Majorana qubits, arXiv preprintarXiv:2004.01264 (2020).[26] R. Barends, J. Wenner, M. Lenander, Y. Chen,R. C. Bialczak, J. Kelly, E. Lucero, P. O’Malley,M. Mariantoni, D. Sank, H. Wang, T. C. White, Y. Yin,J. Zhao, A. N. Cleland, J. M. Martinis, and J. J. A. Basel-mans, Minimizing quasiparticle generation from stray in-frared light in superconducting quantum circuits, Appl.Phys. Lett. , 113507 (2011).[27] O.-P. Saira, A. Kemppinen, V. F. Maisi, and J. P. Pekola,Vanishing quasiparticle density in a hybrid Al/Cu/Alsingle-electron transistor, Phys. Rev. B , 012504(2012).[28] V. F. Maisi, S. V. Lotkhov, A. Kemppinen, A. Heimes,J. T. Muhonen, and J. P. Pekola, Excitation of singlequasiparticles in a small superconducting Al island con-nected to normal-metal leads by tunnel junctions, Phys.Rev. Lett. , 147001 (2013).[29] F. Pobell, Matter and Methods at Low Temperatures , 3rded. (Springer-Verlag, 2007).[30] J. P. Pekola, V. F. Maisi, S. Kafanov, N. Chekurov,A. Kemppinen, Y. A. Pashkin, O.-P. Saira, M. M¨ott¨onen,and J. S. Tsai, Environment-assisted tunneling as an ori-gin of the Dynes density of states, Phys. Rev. Lett. ,026803 (2010). [31] J. V. Koski, J. T. Peltonen, M. Meschke, and J. P.Pekola, Laterally proximized aluminum tunnel junctions,Applied Physics Letters , 203501 (2011).[32] A. B. Zorin, The thermocoax cable as the microwave fre-quency filter for single electron circuits, Review of Scien-tific Instruments , 4296 (1995).[33] R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov,P. Delsing, and D. E. Prober, The radio-frequency single- electron transistor (RF-SET): A fast and ultrasensitiveelectrometer, Science , 1238 (1998).[34] S. Simbierowicz, V. Vesterinen, L. Gr¨onberg, J. Lehti-nen, M. Prunnila, and J. Hassel, A flux-driven Josephsonparametric amplifier for sub-GHz frequencies fabricatedwith side-wall passivated spacer junction technology, Su-perconductor Science and Technology , 105001 (2018). superconductor free of quasiparticles for secondsSupplementary Material E. T. Mannila, ∗ P. Samuelsson, S. Simbierowicz, † J. T. Peltonen, V. Vesterinen, L. Gr¨onberg, J. Hassel, V. F. Maisi, and J. P. Pekola QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden VTT Technical Research Centre of Finland Ltd,QTF Centre of Excellence, P.O. Box 1000, FI-02044 VTT, Finland (Dated: February 2, 2021)The supplementary material includes details on sample fabrication, measurements, data analysis,and modeling, and additional data.
SAMPLE FABRICATION
We start the sample fabrication by depositing Ti/Au/Ti ground planes (yellow in Supplementary Fig. S1(a)) andalignment markers 2 nm/30 nm/2 nm thick on a 525 µ m thick silicon substrate covered by 300 nm of thermal siliconoxide. In a second lithography step, we deposit a 10 nm thick and 10 µ m long chromium wire (diagonal red line inSupplemental Fig. S1(a)), which couples the superconducting island and charge detector capacitively. The layers arecovered by an insulating layer of 40 nm thick aluminum oxide grown by atomic layer deposition.The superconducting island and charge detector were fabricated simultaneously in a third electron-beam lithographystep using a Ge-based hard mask and four-angle evaporation. We evaporate first 35 nm of aluminum, forming thesuperconducting island. The tunnel junctions are formed by oxidizing in 2 mbar of O for 90 s, followed by depositionof 35 nm of copper which forms the normal metal leads of the superconducting island. Next, we evaporate a further20 nm of aluminum, oxidize for 30 s in 1 mbar of O2 and evaporate 60 nm of copper to form the charge detector.We use the laterally proximized junction technique [1] to create a normal-state charge detector with aluminum oxidetunnel junctions: the 20 nm thick aluminum film is inversely proximized by the clean contact to the 35 nm copperfilm. We expect this choice to minimize detector backaction due to nonequilibrium phonons [2].The design of the sample is such that the superconducting island does not overlap with any of its shadow copies
200 nm2 µm (a) (b)
FIG. S1. (a) Scanning electron micrograph of a device lithographically identical to the measured device. The ground planes (2nm Ti/30 nm Au/2 nm Ti, colored yellow), and the capacitive coupler electrode (10 nm Cr, colored red) are covered by a 40nm thick insulating aluminum oxide layer grown by atomic layer deposition. The other coupler electrode visible on the rightside of the sample is not used in this experiment. (b) Close up of charge detector. a r X i v : . [ c ond - m a t . s up r- c on ] J a n magnetic shield
60 K CPL = 20 dB
Cu sample boxindium-sealed
LP80 MHz T h e r m o c o a x IR IR radiation shield(attached to still flange)
Eccosorbsamplechip x2 θ
12S 12S pulses for JPA bandwidthmeasurement21S2S θ
12 S VNA / spectrumanalyzer RF digitizer(Aeroflex 3035C)probemonitor(VNA)Probe
JPA DCflux biasProbe JPA flux pumpCancellationtone output
FIG. S2. Schematic of the measurement circuitry, filtering and shielding used in the experiment. The components highlightedwith pink circles were added to the setup for the second cooldown (see text). produced by the multi-angle evaporation, and its normal leads overlap with their copies only at a distance 1.5 µ maway, far enough to suppress any potential proximity effect. We note that the normal metal layers of the device, aswell as the gold ground plane, may serve as phonon traps [3, 4]. These layers cover approximately 40% of the chipsurface within the mm-scale immediate vicinity of the device and roughly 20% of the entire chip with an area ofroughly 1 cm . MEASUREMENT SETUP
Supplemental Figure S2 shows a schematic of the measurement setup used. As described in Methods, the DC linesused to tune the gates of the superconducting island and the charge detector are filtered with 1 to 2 m of Thermocoax[5]. The RF input lines (probe tone (highlighted in green), cancellation tone (yellow), JPA pump (orange)) areattenuated at the different temperature stages, and circulators provide approximately 40 dB isolation between the -110 -105 -100 power at input capacitor (dBm) power at input capacitor (dBm) -2 -1 Γ bu r s t ( H z ) (a) o cc upa t i on p r obab ili t y o f N = a t n g = (b) FIG. S3. (a) Burst rates at n g = 0 as a function of the probe power applied, measured 54 (circles) or 123 (triangles) daysafter the start of the first cooldown. The cross indicates the data in Figs. 1-3 of the main text. (b) At higher input powersthan used in the main text, the backaction increases the population of the charge state N = 1 at n g = 1, although the burstrate at n g = 0 is unaffected (see Supplemental Fig. S8 for 2 e -periodic occupation probabilities of the charge states versus n g ). sample and JPA. The probe and cancellation are low-pass filtered at the mixing chamber stage with commercialfilters (Mini-Circuits VLFX), while for the second cooldown we added home-made Eccosorb filters closest to thesample holder (boxes labeled ”IR”). For the second cooldown, we also disconnected the RF drive line, not used inthe present experiments, at the RF input of the bias tee at the mixing chamber stage and terminated the port witha 50 Ω cryotermination (indicated in pink). As shown in Fig. 4 of the main text, in both cooldowns the main sourceof quasiparticles was a time-dependent process not affected by these changes.We use a room-temperature setup similar to that used in Ref. [6] to create the probe, interferometric cancellationand JPA pump tones from a single RF generator. An additional pump signal generator visible in SupplementalFig. S2 was used for initial JPA tuning in the phase-preserving mode, but it was switched off for the measurementspresented here. The output signal (purple) is amplified at 2 K and at room temperature and measured with eithera vector network analyser (initial detector characterization), spectrum analyser (sensitivity measurements) or RFdigitizer (time-domain traces). CHARGE DETECTOR AND JOSEPHSON PARAMETRIC AMPLIFIER CHARACTERIZATIONCharge detector
The charge detector, shown in Supplemental Fig. S1(b), is a normal-metallic single-electron transistor with room-temperature resistance 55 kΩ. It is connected in parallel to a resonant circuit on a separate silicon chip, formed ofan evaporated aluminum spiral inductor and mostly parasitic capacitance, with the input and output signal coupledthrough finger capacitors, similarly to the RF thermometer described in Ref. [7] and the RF-SET used in Ref. [8].The resonant frequency of the circuit was 580.35 MHz in the first cooldown and 581.28 MHz in the second. Weattribute this change to slight shifting of the chips or bond wires and corresponding changes in parasitic capacitancedue to the thermal cycle. We apply a probe signal at the resonant frequency f probe and adjust the probe power andgate offset of the detector for maximum sensitivity, but do not apply a DC voltage bias. The probe power used in thedata presented in the main text was -104 dBm at the input capacitor. JPA operation
The output of the charge detector is amplified by the Josephson parametric amplifier, which is otherwise similarto the devices presented in Ref. [6] but has a larger gain-bandwidth product. We operate the amplifier in the phase-sensitive mode, where the pump frequency f pump = 2 f probe , and tune to between 18 and 30 dB gain in the amplifiedquadrature (18 dB for the data shown in Figs. 1-3). We cancel the steady-state probe tone interferometrically toovercome the dynamic range limitations of the JPA, which manifest as excess noise apparent in the time domainsignal even at powers smaller than the 1-dB saturation point. The cancellation tone is combined with the probe toneby a directional coupler between the sample and JPA, as shown in Supplemental Fig. S2. In practice, we first fix thepump signal and DC flux bias to the JPA as well as the magnitude of the probe signal. We then tune the phase of the -5 0 5 detector output (a.u.) c oun t s frequency (MHz) frequency (MHz) -110-100-90-80-70 po w e r ( d B m ) ×10 -4 probe onprobe offJPA off and detunedprobe onprobe offJPA on (a) (b) (c) N =0 N =+1 N =-1 FIG. S4. Sensitivity measurement with JPA on (a) and with JPA off and detuned (b). (c) Histogram of the signal levels ofthe filtered trace shown in Fig. S5 (circles), while solid lines are Gaussian fits corresponding to the three charge states. Thearrows indicate the thresholds used for determining the charge states (see below). probe signal with the cancellation tone off such that the charge response is in the amplified quadrature and adjustthe gate offset of the charge detector to a sensitive operating point. Then, we adjust the phase and amplitude of thecancellation tone with the JPA pump off such that the output amplitude corresponding to the N = 0 charge state isclose to zero. Charge sensitivity and signal-to-noise ratio of charge readout
In Supplemental Fig. S4(a,b), we show the response of our charge detector to an 0 . e RMS excitation at 20 kHzapplied to the detector through the DC gate, measured with a spectrum analyser bandwidth of 1 kHz. The linelabeled ”RF drive” in Supplemental Fig. S2 was not used in this experiment due to uncertainty of its transmissionat low frequencies. Here, the JPA gain is 20.5 dB and the input probe tone to the sample is -101 dBm (comparedto 18 dB and -104 dBm for the data shown in the main text). The signal-to-noise ratios are 31 dB and 18 dB withthe JPA on and off, corresponding to sensitivities of 2 . × − e/ √ Hz and 1 . × − e/ √ Hz, respectively. Forthe JPA off measurement, we have also detuned the JPA flux bias so that the JPA resonance is moved outside thefrequency range of interest. The broad noise peak visible around the carrier tone arises from the noise of the detectorbut also from true tunneling events, as the detector is coupled to the superconducting island during this measurementas well. The peaks in the spectrum at 130 kHz from the carrier arise from electronic pickup that couples to the gateof the superconducting island which we were unable to eliminate in the experiment. For comparison, we also showthe measured spectra when the probe and cancellation tones are turned off. With the JPA off, the noise floor isdominated by the amplifier (Caltech CITLF2) at the 2 K stage of the cryostat, which has a noise temperature of 3.5K. Although the measured charge sensitivities are about two orders of magnitude worse than the state of the art [9],they are sufficient for single-shot detection of charge jumps of roughly 0 . e , corresponding to single-electron tunnelingevents in the superconducting island, with high fidelity.The operating point of the detector and JPA in the charge sensitivity measurement shown above is not exactly thesame as the one used in the data shown in Figs. 1-3 of the main text, and these results are merely indicative of ourgeneral detector performance. We additionally characterize the signal-to-noise of the charge measurement directlyfrom the traces measured in the time domain. In Supplemental Fig. S4(c), we show a histogram of a filtered time tracefrom the dataset used in the main text, with Gaussian fits corresponding to the three charge states observed. Thefinite count of points between the separate peaks originate from the transitions between the peaks, where the signalspends time in intermediate values due to the filtering. Defining the signal-to-noise ratio like in Ref. [10] as the ratioof the separation of the peaks to the standard deviation of the fitted Gaussian distributions, we obtain signal-to-noiseratios of 6.5 and 6.7 for the transitions N = − ↔ N = +1 ↔
0, respectively. In this experiment, we need ahigh signal-to-noise ratio to be able to distinguish reliably noise from the odd charge states, which are occupied with time (ms) -505 de t e c t o r ou t pu t ( a . u . ) time (ms) -505 de t e c t o r ou t pu t ( a . u . ) ×10- ×10 -4 (a)(b) N =1 N =0 N =-1 N =1 N =0 N =-1 FIG. S5. (a) Portion of the timetrace histogrammed in Supplemental Fig. S4, zoomed in (b) to cover a single burst of tunneling.The shaded areas and horizontal lines indicate the thresholds for the different charge states, which are also indicated as arrowsin Supplemental Fig. S4(c). In panel (b), the triangles indicate two-electron Andreev tunneling events. a probability on the order of 10 − . As the Gaussian fits in Fig. S4(c) do not overlap, the probability of interpretingexperimental noise as spurious transitions is small. Detection bandwidth
The 3-dB bandwidth of the resonant circuit is approximately 1 MHz. We measure the bandwidth of the JPA inthe phase-sensitive mode similarly as in the supplementary material of Ref. [11] by applying square pulses to thecancellation line with similar amplitude as the charge detection signal, while the probe signal is turned off. Even with30 dB gain, the JPA rise time is less than a microsecond. Under these conditions, the detector bandwidth is limitedby the post-processing low-pass filter.
DATA ANALYSISFiltering
For the data shown in Figs. 1-3 of the main text, we record 5 s traces of the detector output with a sampling rateof 4 MHz and filter the signal digitally to achieve a high signal-to-noise ratio. For some of the data shown in thesupplementary material and Fig. 4, we have used also slightly longer or shorter traces and sampling rates between2 MHz and 10 MHz. For the data shown in Figs. 1-3 of the main text, we first subtract a moving median of thesignal over 10 ms to remove low-frequency drifts. Then, we apply a digital low-pass filter with a cutoff of 150 kHz,which sets the detector rise time to 3 µ s. This low-pass filter sets the effective bandwidth of the charge detector, andthis filtered data is the detector output shown in Fig. 1 of the main text. We do not use the moving median filterin cases where more than one charge state is occupied with a probability larger than 1%, such as in obtaining the n g dependence of the tunneling rates shown below. Determining the instantaneous charge state
We rotate the measured IQ signal such that the charge response is only along one quadrature and analyse thetraces one by one. To identify the signal levels corresponding to each charge state, we filter the signal with a cutofffrequency of 50 kHz, histogram the signal values in this filtered trace and identify peaks in the histogram with aminimum separation of 3 × − in the units on the x-axis of Supplemental Fig. S4(c). We exclude the traces wherethe algorithm has not correctly identified 3 peaks corresponding to the three populated charge states at n g = 0. Wethen define threshold signal values halfway between the peaks and the minima between each adjacent pair of peaksin the histogram, which are shown in Supplemental Figs. S4(c) and S5. Then, we use a hysteretic trigger to assignthe charge states based on the trace filtered with a cutoff frequency of 150 kHz: a transition from charge state N to N is considered to have happened only when the signal value crosses the threshold corresponding to state N .For the data acquired during the second cooldown, shown in Fig. 4 in the main text, we additionally identify thestate of an additional fluctuator which causes jumps by 1 e occurring at a rate on the order of 1 Hz in the gate offsetof the superconducting island, as discussed below. Identifying tunneling events and bursts
After identifying the charge states from the raw data, we identify the tunneling events as single-electron or two-electron tunneling events. We classify a tunneling event as Andreev tunneling if the signal changes from N = +1to 1 or vice versa within the detector rise time 3 µ s, ascertained from the measured waiting time distribution in thestate N = 0 similarly as in Ref. [12]. The Andreev tunneling rates in the data shown in Figs. 1-3 in the maintext are Γ − → +1 = 5 . +1 →− = 0 . t sep = 250 µ s in charge state N = 0 belong to separate bursts. With this choice, to interpret two successivebursts erraneously as one requires that a second Cooper-pair breaking event happens within time t burst + t sep from thestart of the burst, where t burst = 170 µ s is the mean length of a burst. Given that the Cooper pair breaking events occuraccording to a Poisson process with rate Γ burst = 1 . − exp[ − Γ burst ( t burst + t sep )] = 6 × − .On the other hand, the probability of splitting a long burst into two is given by the probability that the system werein state N = 0 and N QP ≥ t sep without a tunneling event occurring. The tunneling rate out of the state N = 0 and N QP = 2 is 4Γ QP = 32 kHz, so this probability is exp( − QP t sep ) = 3 × − . Hence the total probabilityof misinterpreting bursts is on the order of 10 − . After identifying the bursts and tunneling events, Figs. 2(b) and3(a) of the main text are directly obtained from counting the number of tunneling events.In plotting the waiting times between the bursts of tunneling in Fig. 2(a) of the main text, we leave out the intervalsat the start and beginning of each 5 s long time trace. Due to the finite length of the time traces, long quiet times areunderrepresented in the measured data, which means the fit for the rate parameter is an upper limit. In Fig. 4, weplot instead the burst rate calculated as dividing the number of bursts by the total measurement time, which producesan unbiased estimate of the burst rate. The error bars in Figs. 2 and 4 representing the statistical uncertainty arisingfrom the counts of events in each bin would be roughly the same size as the markers and are thus not shown. Effective tunneling rates within the bursts
We calculate effective tunneling rates within bursts Γ N → N ( t ) from charge states N to N as a function time t fromthe first tunneling event of a burst. For single-electron transitions, N = N ±
1, for Andreev rates, N = N ±
2. Therates are calculated as averages over time bins by dividing the number of transitions from N to N within a bin bythe total time spent in state N within that bin. The time spent in state N = 0 after a burst has ended is not includedin the denominator, so that the rates are calculated only within the bursts. The bin widths are adjusted from 10 µ sat short times to 50 µ s at t > µ s, as there are less long bursts available for calculating statistics. We show onlythose data points where we have statistics calculated from at least 10 bursts with events in the corresponding bin.The error bars in Fig. 3(b) of the main text and Supplemental Figs. S6 and S8 indicate statistical uncertainty basedthe count of events in each bin.The single-electron tunneling rates are shown in Fig. 3(b) of the main text, while the Andreev tunneling rates areshown in Supplemental Fig. S6. The rate Γ − → +1 ( t ) is constant within the burst. This is expected theoretically,as the Andreev tunneling events are insensitive to the number of quasiparticles present and thus should be constant t (µs from first tunneling event of burst) ! N ! N ' ( k H z ) -1 ! +1+1 ! -1 FIG. S6. Andreev tunneling rates within the bursts of tunneling from the same dataset as the single-electron rates shown inFig. 3(b) of the main text. within the bursts. The slight decrease of the rate Γ +1 →− ( t ) at t < µ s is due to misinterpration of single-electrontransitions as Andreev, and the effect is larger because there are less true Andreev events in this direction. InSupplemental Fig. S9, discussed below, we show the measured occupation probabilities of the charge states calculatedas averages over the same time bins as the tunneling rates, again not including the time after a burst has ended.Note that in both panels of Fig. 3 of the main text as well as Supplemental Figs. S6 and S8, time t is measuredfrom the first tunneling event of each burst. This occurs typically within less than 30 µ s after the initial Cooper-pair-breaking event, as the rate out of the state with N QP = 2 is 4Γ QP = 32 kHz. Effect of finite bandwidth on the measurement
Based on the detector performance above, we can estimate the probability that a burst of tunneling would beundetected due to finite bandwidth of the charge detector [13]. The shortest burst is one which starts with only twoquasiparticles and correspondingly has only two tunneling events ( N = 0 → ± → QP = 8 kHz. Such an event will be missed if thesystem stays in the state N = ± µ s, with probability 1 − exp(3 µ s × ≈ →± at t < µ s in Fig.3(b) of the main text. To quantify this effect, we have generated simulated time traces from the model and analysedthis simulated data with the same procedure as the experimental data. The main effect is that the distributionof number of quasiparticles per burst is skewed towards lower numbers, and hence our estimate for the underlyingparameter λ = 0 . →± in Fig. 3(b) at t < µ s is due to the finite risetime. RATE EQUATION MODEL
We describe the combined quasiparticle and charge dynamics within a burst with a rate equation for the jointprobability P ( N QP , N, t ) to have N QP = 0 , , , ... quasiparticles and N = − , , t . Charge states with | N | ≥ dP ( N QP , N, t ) dt = X N QP ,N Γ N QP → N QP ,N → N P ( N QP , N , t ) − X N QP ,N Γ N QP → N QP ,N → N P ( N QP , N, t ) . (1)The charge-quasiparticle states, denoted ( N, N QP ), and the four different transfer processes N QP → N QP , N → N accounted for, with corresponding rates, are shown schematically in Supplemental Fig. S7. First, quasiparticles cantunnel out of the island as either electrons or holes with a rate proportional to the N QP [15] (yellow arrows). As thequasiparticles are generated in pairs, no branch imbalance is created. Starting from N = 0, tunneling can happeninto both of the two states N = ± σ , ¯ σ in Eq. (1) in Methods. Second, quasiparticles mayin principle also tunnel back into the superconductor from the normal leads. We neglect the process N = 0 → ± N QP → N QP + 1 (backtunneling starting from N = 0) due to its high energy cost (see discussion below), but includebacktunneling starting from states with N odd (red arrows). Third, we account for quasiparticle recombination witha rate scaling as N for N QP ≥ N = ± N QP -independent rate. The prefactors of each of these rates are parameters of the model. We point out that the (0,0)(0,4) (1,3)(-1,1) (1,1)(-1,3) (0,2) Quasiparticle out tunneling Quasiparticle back tunneling Andreev tunneling Pair recombination
FIG. S7. Left: Charge-quasiparticle states (
N, N QP ) with processes for transfer between states denoted with arrows. Right:List of state transfer processes with corresponding rates. model is charge symmetric, that is, the rates are unaffected by changing N = ± ∓
1. While the model couldreadily be extended to charge asymmetric rates, this is not needed for the purposes in the present paper, as only theAndreev tunneling rates, which do not contribute to the quasiparticle dynamics, are expexted to be asymmetric inthe experiment.To obtain the time evolution of P ( N QP , N, t ) after a pair breaking event occuring at time t PB , we take as initialcondition the distribution of number of broken pairs N pair = N QP / P ( N QP , N, t PB ) = e − λ QP N QP (cid:0) e λ QP − (cid:1) δ N, , (2)for N QP = 2 , , .. and zero otherwise, with λ QP = λ/ .
45. However, since only the electron tunneling eventsfollowing the pair breaking event itself can be detected in our setup, in a number of cases we consider the distribution P ( N QP , N, t ) after the first tunneling event occuring at t = 0 (not introducing a separate notation, the distributionconsidered is clear from the context). The initial condition for this conditional probability distribution is given by P ( N QP , N,
0) = e − λ QP ( N QP +1) (cid:0) e λ QP − (cid:1) δ N, + δ N, − N QP = 1 , , .. and zero otherwise. This distribution accounts for the time evolution from the pair breaking to thefirst tunnel event. We point out that in obtaining Eq. (3) from (2), quasiparticle recombination is neglected. This ismotivated by the observation that Γ QP > Γ R N QP for all relevant N QP , as discussed below.From the probability distribution P ( N QP , N, t ) after the first tunneling event occuring at t = 0, the differentquantities shown can be obtained. The formally infinite set of equations in (1) is solved by truncating the quasiparticlenumber at some maximum N QP = N maxQP , such that the result is independent of N maxQP . In the main text (includingMethods) the basic model, considering only quasiparticle out-tunneling, is discussed. For the quantities analyzed inthe Supplementary information, the theoretical burst length distribution, shown in Supplemental Fig. S12, is obtainedby a Monte Carlo simulation of Eq. (1). Moreover, the effective time-dependent single-electron rates Γ ± → ( t ) and -2 -1 0 1 2-2 -1 0 1 2 -2 -1 0 1 2 o cc upa t i on p r obab ili t y P ( N ) s i ng l e - e l e c t r on t unne li ng r a t e Γ N → N ± ( H z ) t w o - e l e c t r on t unne li ng r a t e Γ N → N ± ( H z ) (a) (b) (c) n g -1 -2 -3 -4 n g n g N -3-2-10123 FIG. S8. Measured (a) occupation probabilities P ( N ), (b) single-electron tunneling rates Γ N → N ± and (c) two-electron tun-neling rates Γ N → N ± as a function of the gate offset n g . Symbols are experimental data, while lines are simulated results (seetext). In panels (b) and (c), dashed (solid) lines and crosses (circles) indicate transitions where N decreases (increases). Γ →± ( t ) are given by Γ N → N ± ( t ) = P N QP P ( N QP , N, t )Γ N QP → N QP − ,N → N ± P N QP P ( N QP , N, t ) (4)where the sum runs over N QP = 2 , , , ... for N = 0 and N QP = 1 , , , ... for N = ±
1. The probability for the emptystate (0 ,
0) is thus not included in the definition of the effective rates, in agreement with how the rates are obtainedfrom the experimental data. Hence, the rates describe only the tunneling within the bursts.From P ( N QP , N, t ) we can also obtain the time dependent probabilities for having a charge N = 0 , ± N = ±
1) or an even ( N = 0) charge, defined as P ( N = ± , t ) = X N QP =1 , , .. [ P ( N QP , N = 1 , t ) + P ( N QP , N = − , t )] (5)and P ( N = 0 , t ) = X N QP =2 , ,.. P ( N QP , N = 0 , t ) (6)respectively. As for the effective rates, the probabilty for N QP = 0 is not included, in order to only describe theprobabilities within the bursts. SYSTEM CHARACTERIZATION
In this section, we discuss estimates for the various parameters contributing to the rates in our model consideredabove, and show that in our experiment the dominant process after a Cooper-pair breaking event is the tunneling outof quasiparticles from the island, leading to the simple model used in the main text.
Charging energy, superconducting gap and electron temperature
We determine the charging energy E C ≈ µ eV and superconducting gap ∆ ≈ µ eV of the device by fitting themeasured occupation probabilities of the charge states N at elevated bath temperatures T bath between 120 mK and215 mK, where we expect the device properties to be set by thermal excitations. The obtained values are consistentwith the device geometry and typical gap values for thin aluminum films. Due to the charging energy E C ≈ . n g ≈ | N | ≥ > ∆. In contrast, thecharging energy cost for occupying the charge states N = ± N = 0 , ± n g , measured at therefrigerator base temperature T = 20 mK in similar conditions as the data shown in Figs. 1-3 of the main text. Eachexperimental point here consists of averages over 30 s of data, with the tunneling rates calculated as the number ofevents divided by the time spent in the starting charge state. The solid lines are simulations similar to those in Refs.[8, 14]. They incorporate the rates shown in Supplemental Fig. S7, as well as thermally activated backtunneling ofquasiparticles, and Andreev tunneling events with rates given in Ref. [16]. As Andreev rates are insensitive to thepresence of quasiparticles in the superconductor in the few-quasiparticle regime considered here, the n g dependence ofthe Andreev rates allows fitting the electron temperature at the base temperature T N = 100 mK. This temperature alsoreproduces the occupation probability of the state N = 1 at n g = 1. Here, we do not include a Cooper pair breakingrate in the simulations, and thus the occupation probabilities and tunneling rates resulting from these nonequilibriumevents around n g = 0 are not reproduced by our model. The observation of gate-independent single-electron tunnelingrates, like in Ref. [8], is a strong indication that the observed tunneling events are due to nonequilibrium Cooper pairbreaking on the superconducting island. For the data shown in Figs. 1-3 of the main text, n g ≈ .
05. In the datashown in Fig. 4 of the main text, the burst rate is obtained from data where the gate offset is between n g = − . n g = 0 . Single-quasiparticle tunneling rates
Quasiparticle tunneling out
A theoretical value for the single-quasiparticle tunneling rate for removing a quasiparticle from the island canbe estimated as Γ
QP, theory = (2 e R T V D ( E F )) − [15]. The tunnel resistance of the two tunnel junctions of thesuperconducting island in series was measured at room temperature to be 520 kΩ, and we estimate the low-temperatureresistance of the two junctions in parallel, relevant in the experiment where no bias is applied, to be R T = 150 kΩ.This incorporates a typical 15% increase in the resistance when cooling down. Given the literature value for thedensity of states at the Fermi level in the normal state for both spins D ( E F ) = 2 . × J − m − [17], and theknown island volume V = 35 nm × µ m × µ m, we obtain Γ QP, theory ≈
17 kHz. As literature values for thedensity of states differ by up to a factor of 1.5 [17, 18] and there is some uncertainty in the junction resistances aswell, we find the estimated value in fair agreement with the measured Γ QP = 8 . Quasiparticle backtunneling
The data shown in the main text was obtained at n g ≈ .
05. At this operation point, the expected rate for thethermally activated transition N = 0 → ± N QP = 0 → − Hz), and in any case, the rate of such events is bounded by the measured burst rate.The thermally activated rate of the transition N = ± → N QP = 1 → N = ± E C , ∆ and n g , although the independence ofmeasured quantities on n g , shown below, also supports the conclusion that backtunneling from the leads is negligible.We have performed also simulations with a finite rate Γ B . This rate can be expected to be independent of the N QP ,the number of quasiparticles already present, as we always have only very few excitations present.A sensitive probe of this backtunneling process are the time-dependent occupation probabilities P ( N, t ) of thecharge states within bursts of tunneling (in contrast with the probabilities P ( N QP , t ) discussed in the main text).This can be intuitively be understood as follows: If there is no backtunneling, at long times from the initial pair-breaking event, a burst must either end or be in the charge state ±
1. If there is a nonzero backtunneling rate,there would be a non-zero occupation probability for the state N = 0 even at long times. Calculations show thatthe long-time limiting value is independent of the initial conditions (number of Cooper pairs broken) and givenby P ( N = 0) = ( − − q + p (1 + q )(9 + q )) /
2, where q = Γ B / Γ QP is the backtunneling rate normalized by thesingle-quasiparticle tunneling rate.1 t (µs from first tunneling event of burst) -4 -3 -2 -1 o cc upa t i on p r obab ili t y P ( N =+1, t ) + P ( N =-1, t ) P ( N =0, t ) ! B / ! QP = 0.01 ! B / ! QP = 0.005 ! B / ! QP = 0.0025 ! B / ! QP = 0 FIG. S9. Occupation probabilities of the charge states N as a function of time t within the bursts of tunneling. Filled squaresindicate the probability P ( N = +1 , t ) + P ( N = − , t ), while open circles show the probability of P ( N = 0 , t ). Lines aresimulated probabilities P ( N = 0 , t ) with varying backtunneling rates Γ B . In Supplemental Fig. S9, we show the distribution P ( N = 0 , t ) and P ( N = − , t ) + P ( N = +1 , t ) obtainedfrom the same dataset as used in Figs. 1-3 of the main text. Here, we show the total probability of the odd states P ( N = −
1) + P ( N = +1), as unequal Andreev tunneling rates in the experiment change the relative probabilitiesof the states N = +1 and N = −
1, although they do not influence the quasiparticle dynamics. From the measuredupper bound, we calculate Γ B < . QP and are thus justified in neglecting backtunneling. Hence, each observedsingle-electron tunneling event must correspond to an event removing a quasiparticle from the island, which enablesreconstructing the instantaneous quasiparticle number. Recombination rate
We estimate the recombination rates based on the expression Γ rec ( N QP ) = Σ∆ N / [12 ζ (5) D ( E F ) k B V ] ≡ Γ R N given in Ref. [14]. Here, Σ is the electron-phonon coupling constant in the normal state, ζ is the Riemann zeta function,and k B the Boltzmann constant. We use the measured values of the superconducting gap ∆ ≈ µ eV and volume V = 2 × . × . µ m . With Σ = 2 × W K − m − consistent with literature values [19–21] and our recentmeasurements on similar Al films [22], we obtain Γ R = 25 Hz and N QP Γ R < Γ QP up to N QP = 320. Alternatively, themeasured recombination constant r = 1 /
80 ns in Ref. [23], connected to Γ R = r/ ( D ( E F )∆ V ), we obtain Γ R = 50 Hz,although the effect of phonon trapping [24] is likely somewhat weaker in our experiment due to the thinner film anddifferent substrate used. Hence we are justified in ignoring recombination in our basic model, shown in the main text.Given these rates, we estimate the probability of a Cooper-pair-breaking event being undetected because thequasiparticles recombined before a tunneling event took place. An upper limit can be obtained by considering anevent where only a single Cooper pair was broken, which leads to the shortest bursts. The relevant tunneling rate tocompare with is the total tunneling rate out of the state N = 0, N QP = 2, which is 4Γ QP = 32 kHz. With Γ R = 25 Hz,in only 0.3% of the cases the quasiparticles would recombine before tunneling out. Even with the order of magnitudehigher value of Σ = 1 . × W K − m − in Ref. [14], only 3% of Cooper-pair breaking events would be undetecteddue to recombination. A non-zero recombination rate also reduces the count of single-electron tunneling events ina detected burst if some quasiparticles decay before tunneling out, meaning that in the presence of non-negligiblerecombination the true proportion of events which break more than one Cooper pair would be greater than 40%. Probability of undetected quasiparticles within the quiet periods
In the quiet periods between the bursts of tunneling, quasiparticles may be present without leading to detectedtunneling events either if the tunneling events were missed by our detector due to the finite bandwidth. We estimatedabove that the 1.4% of bursts of tunneling that are shorter than 3 µ s will be missed, and the probability that at any2 Γ b u r s t ( H z ) Γ b u r s t ( H z ) (a) (b) -1 -0.5 0 0.5 1 n g
20 40 60 80 100 120 140 T bath (mK) T bath = 20 mK n g ≈ FIG. S10. Burst rate shown as a function of (a) gate offset n g , measured at a bath temperature of 20 mK, and (b) T bath ,measured at n g ≈
0. All data shown here was measured between 125 and 140 days from the start of the first cooldown. Theerror bars in both panels reflect statistical uncertainty, but there may be systematical errors in the burst rate measured athigher temperatures due to changes in the charge detector performance. In panel (a), a new burst is defined to start after t sep = 250 µ s with no tunneling events in any charge state, whereas elsewhere we require 250 µ s in state N = 0. t (days) n g T bath (mK) (a) (b) (c)
1e tunneling events per burst
1e tunneling events per burst -3 -2 -1 p r obab ili t y
1e tunneling events per burst
FIG. S11. Distribution of single-electron tunneling events versus (a) time from start of the cooldown, (b) gate offset, or (c)bath temperature,. The data in panels (b) and (c) was measured 125 to 140 days after the start of the first cooldown. In allpanels, the black circles correspond to the dataset shown in the main text, while the solid line indicates exponential fits. given moment quasiparticles would be present due to such an event is Γ burst × . × µ s = 6 × − . Anotherpossibility is that the quasiparticles recombined before tunneling out. Estimating cautiously that this happens after1% of the Cooper pair breaking events, the probability that quasiparticles were present due to such an event wouldbe Γ burst × . × µ s = 5 × − . QUASIPARTICLE NUMBER DISTRIBUTION AND SINGLE-QUASIPARTICLE RATE VERSUS TIMEWITHIN COOLDOWN, GATE OFFSET AND BATH TEMPERATURE
In this section, we study the origin of the quasiparticles in further detail. In Supplemental Fig. S10, we show thatthe burst rate was independent of n g over a wide range, which is expected as the bursts are due to Cooper pair breakingevents occurring independent of the state of the superconducting island. Near n g = 1, the burst rate increases due tothermal activation of single-electron and Andreev tunneling. The burst rate at n g ≈ T bath = 85 mK (Supplemental Fig. S11). The independence ontime within the cooldowns supports the conclusion that a single time-dependent quasiparticle source was dominant3 burst length (ms) burst length (ms) -3 -2 -1 p r obab ili t y burst length (ms) t (days) n g T bath (mK) (a) (b) (c) FIG. S12. Burst length distributions versus (a) time from beginning of the second cooldown, (b) gate offset, or (c) bathtemperature,. The data in panels (b) and (c) was measured 124 to 140 days after the start of the first cooldown. Black circlesin all panels correspond to the dataset shown in Figs. 1-3 in the main text, and the black solid line is the theoretical burstlength distribution predicted by the basic model. In (c), the green solid line is the burst length distribution calculated with ascaled single-quasiparticle tunneling rate 13 kHz, compared to 8 kHz used for the base-temperature data. over the whole experimental period. At temperatures above 100 mK, the mean number of single-electron events perburst increases somewhat, which could be due to increased rates for thermally activated backtunneling (see above).They could also be due to noise interpreted as spurious tunneling events, since the charge detector performancedeteoriates with increasing temperature.To study the dependence of the single-quasiparticle tunneling rate Γ QP on these control parameters, as a complementto the quantities discussed in the main text we show in Supplemental Fig. S12 the measured distribution of the burstlengths. This quantity requires less statistics to extract reliably than the effective rates shown in the main text,and our simulations show that after the initial 200 µ s the distribution decays exponentially with the slope set bythe single-quasiparticle rate. The initial deviation at t < µ s from the exponential behaviour is due to the finiteprobability of more than one Cooper pair breaking and allows for an independent determination of the parameter λ used as the initial condition for the simulations. We find that the single-quasiparticle rate is unchanged by the timeduring cooldown or n g , as expected. At an increased T bath , the single-quasiparticle rate increases by roughly a factorof 50%, but the predictions of our simple model fit the measured burst length distribution exceedingly well when onlythis parameter is changed, as shown in Supplemental Fig. S12(c). e JUMPS OF n g During the first cooldown, we occasionally observed large discrete jumps of n g on the timescales of hours in additionto slow drifts of the background charge, as is typical in experiments on single-electron devices. During the secondcooldown, however, we observed jumps of n g by 1 occurring at a rate on the order of 1 Hz, and several such jumpsare shown in the time trace of Supplemental Fig. S13. We attribute such jumps to electrons tunneling in and out ofa localized state, perhaps in the oxide layer of the tunnel junctions or the surface of the superconductor, similar to[25], and it is plausible that such a localized state might become more strongly coupled after a thermal cycle.For the data shown in Fig. 4 of the main text and Supplemental Fig. S13 measured in the second cooldown, wehave inferred the gate offset from the measured traces by low-pass filtering the data with a cutoff set to 100 Hz, so thatthe fast bursts of tunneling are filtered out. Such a filtered trace is shown also in Supplemental Fig. S13. We thenidentify the instantaneous value of n g by applying similar thresholds as in determining the charge states, describedpreviously, and use only those bursts of tunneling where the gate offset is close to 0.In principle, bursts of tunneling could be triggered by an electron tunneling from such a localized state (or from theleads due to photon-assisted tunneling), instead of Cooper-pair-breaking events. However, in such a case we wouldexpect to have only one excess quasiparticle on the island, which might lead to multiple Andreev tunneling events onthe island, but this cannot explain the observed decay of the single-electron tunneling rates as a function of the timewithin a burst as shown in Fig. 3(b) of the main text. Hence we conclude that the bursts of tunneling are indeed dueto Cooper-pair breaking events on the superconducting island. Even during the first cooldown, roughly 0.3% of the4 time (ms) time (ms)time (ms) de t e c t o r ou t pu t ( a . u . ) de t e c t o r ou t pu t ( a . u . ) ×10 -3 ×10 -3 ×10 -3 (a)(b) (c) de t e c t o r ou t pu t ( a . u . ) b c3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000-2023510 3511 3512 3513 3514 3515-202 3700 3701 3702 3703 3704 3705-202 FIG. S13. Portions of time trace from second cooldown, filtered with 150 kHz cutoff (red) and 100 Hz cutoff (black). The latterenables distinguishing whenever n g has changed by one, which happened four times within the period shown. Panels (b) and(c) show 5-ms portions of the trace where n g ≈ n g ≈
1, respectively. measured bursts of tunneling were longer than 1.5 ms, which should be exceedingly unlikely according to our model.Such long bursts might be explained by quasiparticles becoming trapped in a localized state, from which they escapewith a slower rate, which would also explain the slight decrease in the rate Γ ± → after t > µ s. We emphasizethat at t = 500 µ s, over 90% of bursts have already ended so this effect is only a minor correction to the dynamics. ∗ elsa.mannila@aalto.fi † Present address: Bluefors Oy, Arinatie 10, 00370 Helsinki, Finland[1] J. V. Koski, J. T. Peltonen, M. Meschke, and J. P. Pekola, Laterally proximized aluminum tunnel junctions, AppliedPhysics Letters , 203501 (2011).[2] U. Patel, I. V. Pechenezhskiy, B. L. T. Plourde, M. G. Vavilov, and R. McDermott, Phonon-mediated quasiparticlepoisoning of superconducting microwave resonators, Phys. Rev. B , 220501(R) (2017).[3] F. Henriques, F. Valenti, T. Charpentier, M. Lagoin, C. Gouriou, M. Mart´ınez, L. Cardani, M. Vignati, L. Gr¨unhaupt,D. Gusenkova, J. Ferrero, S. T. Skacel, W. Wernsdorfer, A. V. Ustinov, G. Catelani, O. Sander, and I. M. Pop, Phonontraps reduce the quasiparticle density in superconducting circuits, Applied Physics Letters , 212601 (2019).[4] K. Karatsu, A. Endo, J. Bueno, P. J. de Visser, R. Barends, D. J. Thoen, V. Murugesan, N. Tomita, and J. J. A.Baselmans, Mitigation of cosmic ray effect on microwave kinetic inductance detector arrays, Applied Physics Letters ,032601 (2019).[5] A. B. Zorin, The thermocoax cable as the microwave frequency filter for single electron circuits, Review of ScientificInstruments , 4296 (1995).[6] S. Simbierowicz, V. Vesterinen, L. Gr¨onberg, J. Lehtinen, M. Prunnila, and J. Hassel, A flux-driven Josephson parametricamplifier for sub-GHz frequencies fabricated with side-wall passivated spacer junction technology, Superconductor Scienceand Technology , 105001 (2018).[7] K. L. Viisanen, S. Suomela, S. Gasparinetti, O.-P. Saira, J. Ankerhold, and J. P. Pekola, Incomplete measurement of workin a dissipative two level system, New Journal of Physics , 055014 (2015).[8] E. T. Mannila, V. F. Maisi, H. Q. Nguyen, C. M. Marcus, and J. P. Pekola, Detecting parity effect in a superconductingdevice in the presence of parity switches, Phys. Rev. B , 020502(R) (2019).[9] S. Schaal, I. Ahmed, J. A. Haigh, L. Hutin, B. Bertrand, S. Barraud, M. Vinet, C.-M. Lee, N. Stelmashenko, J. W. A.Robinson, J. Y. Qiu, S. Hacohen-Gourgy, I. Siddiqi, M. F. Gonzalez-Zalba, and J. J. L. Morton, Fast gate-based readoutof silicon quantum dots using Josephson parametric amplification, Phys. Rev. Lett. , 067701 (2020).[10] D. Razmadze, D. Sabonis, F. K. Malinowski, G. C. M´enard, S. Pauka, H. Nguyen, D. M. van Zanten, E. C. O Farrell,J. Suter, P. Krogstrup, F. Kuemmeth, and C. M. Marcus, Radio-frequency methods for Majorana-based quantum devices:Fast charge sensing and phase-diagram mapping, Phys. Rev. Applied , 064011 (2019). [11] D. Rist`e, J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo, Initialization by measurement of a superconductingquantum bit circuit, Phys. Rev. Lett. , 050507 (2012).[12] V. F. Maisi, O.-P. Saira, Y. A. Pashkin, J. S. Tsai, D. V. Averin, and J. P. Pekola, Real-time observation of discreteandreev tunneling events, Phys. Rev. Lett. , 217003 (2011).[13] O. Naaman and J. Aumentado, Poisson transition rates from time-domain measurements with a finite bandwidth, Phys.Rev. Lett. , 100201 (2006).[14] V. F. Maisi, S. V. Lotkhov, A. Kemppinen, A. Heimes, J. T. Muhonen, and J. P. Pekola, Excitation of single quasiparticlesin a small superconducting Al island connected to normal-metal leads by tunnel junctions, Phys. Rev. Lett. , 147001(2013).[15] O.-P. Saira, A. Kemppinen, V. F. Maisi, and J. P. Pekola, Vanishing quasiparticle density in a hybrid Al/Cu/Al single-electron transistor, Phys. Rev. B , 012504 (2012).[16] D. V. Averin and J. P. Pekola, Nonadiabatic charge pumping in a hybrid single-electron transistor, Phys. Rev. Lett. ,066801 (2008).[17] P. Lerch and A. Zehnder, Quantum Giaever detectors: STJ’s, in Cryogenic particle detection , edited by C. Enss (Springer,2005) pp. 217–266.[18] N. W. Ashcroft and N. D. Mermin,
Solid State Physics (Saunders, 1976).[19] R. L. Kautz, G. Zimmerli, and J. M. Martinis, Self-heating in the Coulomb-blockade electrometer, Journal of AppliedPhysics , 2386 (1993).[20] J. P. Kauppinen and J. P. Pekola, Electron-phonon heat transport in arrays of Al islands with submicrometer-sized tunneljunctions, Phys. Rev. B , R8353 (1996).[21] M. Meschke, J. P. Pekola, F. Gay, R. E. Rapp, and H. Godfrin, Electron thermalization in metallic islands probed byCoulomb blockade thermometry, Journal of Low Temperature Physics , 1119 (2004).[22] E. T. Mannila, V. F. Maisi and J. P. Pekola, in preparation.[23] C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Axline, T. Brecht, R. W. Heeres, L. Frunzio, M. H. Devoret, G. Catelani, L. I.Glazman, and R. J. Schoelkopf, Measurement and control of quasiparticle dynamics in a superconducting qubit, NatureCommunications , 1 (2014).[24] A. Rothwarf and B. N. Taylor, Measurement of recombination lifetimes in superconductors, Phys. Rev. Lett. , 27 (1967).[25] T. M. Eiles, J. M. Martinis, and M. H. Devoret, Even-odd asymmetry of a superconductor revealed by the Coulombblockade of Andreev reflection, Phys. Rev. Lett.70