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Pulse and hold strategy for switching current measurements Jochen Walter, Erik Tholén, and David B. Haviland Nanostructure Physics, Royal Institute of Technology, AlbaNova University Center, SE-10691 Stockholm, Sweden 7 Joachim Sjo¨strand 0 Department of Physics, Stockholm University, AlbaNova University Center, SE-10691 Stockholm, Sweden 0 (Dated: February 4, 2008) 2 n Weinvestigatebytheoryandexperiment,theJosephsonjunctionswitchingcurrentdetectorinan a environment with frequency dependentdamping. Analysis of thecircuit’s phase space show that a J favorabletopologyforswitchingcanbeobtainedwithoverdampeddynamicsathighfrequencies. A 8 pulse-and-holdmethodisdescribed,whereafastswitchpulsebringsthecircuitclosetoanunstable 1 point in thephasespace when biased at thehold level. Experimentsare performed on Cooper pair transistors and Quantronium circuits, which are overdamped at high frequencies with an on-chip ] RC shunt. For 20 s switch pulses the switching process is well described by thermal equilibrium n µ escape, based on ageneralization of Kramers formula to thecase of frequencydependentdamping. o Acapacitor biasmethod isused tocreate veryrapid,25nsswitch pulses, whereit isobserved that c theswitching process is not governed by thermal equilibrium noise. - r p u I. INTRODUCTION at considerably higher speeds than the static switching s methods, allowing individual quantum measurements to . at A classical non-linear dynamical system, when driven be made with much higher duty cycle. In particular, m the dispersive methods have shown that it is possible toapointofinstability,willundergoabifurcation,where to continuously monitor the qubit.13 However, for both - thesystemevolvestowarddistinctlydifferentfinalstates. d static switching and dispersive methods, the sensitivity At bifurcation the system becomes very sensitive and n ofthe techniqueis improvedbyexploitingthe non-linear the smallest fluctuation can determine the evolution of o propertiesofthereadoutcircuitinapulseandholdmea- c a massivesystemwithhuge potentialenergy. This prop- surementstrategy. Thisimprovedsensitivityofthepulse [ erty of infinite sensitivity at the point of instability can and hold method is not surprising, because when prop- be used to amplify very weak signals, and has recently 2 erly designed, the pulse and hold technique will exploit been the focus of investigation in the design of quantum v theinfinitesensitivityofanon-linearsystematthepoint 4 detectors to readout the state of quantum bits (qubits) of instability. 0 built from Josephson junction (JJ) circuits. Here we ex- 7 amine in experiment and theory a pulse and hold strat- The general idea of exploiting the infinite sensitivity 0 egy for rapid switching of a JJ circuit which is quickly at an instable point is a recurrent theme in applications 1 brought near a point of instability, pointing out several of non-linear dynamics. The basic idea has been used 6 important properties for an ideal detector. We focus on sincetheearlydaysofmicrowaveengineeringinthewell- 0 switching in a circuit with overdamped phase dynamics known parametric amplifier14 which has infinite gain at / t athighfrequencies,andunderdampedatlowfrequencies. the point of dynamical instability. The unstable point a m This HF-overdamped case is relevant to experiments on canbe convenientlyrepresentedasa saddlepointforthe small capacitance JJs biased with typical measurement phase space trajectoriesof the non-lineardynamicalsys- - d leads. tem. In the pulse and hold measurement method an ini- n ClassicalJJshavestronglynon-linearelectrodynamics tial fast pulse is used to quickly bring the system to the o and they have served as a model system in non-linear saddle point for a particular hold bias level. The hold c physics for the last 40 years. More recently it has been level is chosen so that the phase space topology favors a : v shown that JJ circuits with small capacitance can also rapidseparationin to the two basins of attractionin the Xi exhibit quantum dynamics when properly measured at phase space. The initial pulse should be not so fast that low enough temperatures. Experimental demonstration it will cause excessive back action on the qubit, but not r a of the macroscopic quantum dynamics in these circuits so slow that it’s duration exceeds the relaxation time of has relied on efficient quantum measurement strategies, the qubit. The length of the hold pulse is that which is characterized by high single shot sensitivity with rapid required to achieve a signal to noise ratio necessary for reset time and low back action. Some of these measure- unambiguous determination of the resulting basin of at- mentordetectionmethods arebasedonthe switchingof traction. In practice this length is set by the filters and aJJcircuitfromthezerovoltagestatetoafinitevoltage amplifiers in the second stage of the quantum measure- state.1,2,3,4 Other detection methods are based on a dis- ment system. persive technique, where a high frequency signal probes In this paper we discuss pulse and hold detection in the phase dynamics of a qubit.5,6,7,8,9,10,11,12 These dis- the context of switching from the zero voltage state to persive methods have achieved the desired sensitivity the finite voltage state of a JJ. We give an overview of 2 such switching in JJs, focusing on the HF-overdamped (a) case. Switching detectors with overdamped high fre- quencyphase dynamicsaredifferentfromallotherqubit measurementstrategiesimplementedthus far,whereun- I derdampedphasedynamicshasbeenused. However,the C HF-overdamped case is quite relevant to a large number J R I 0 of experiments which measure switching in low capacitance JJs with small critical currents.15,16,17,18 We show that by making the damping at high frequencies large (b) enough, a favorable phase space topology for switching can be achieved.19 In this overdamped situation the ex- C2 VC ternal phase can be treated classically and contributions ofmacroscopicquantumtunneling(MQT)totheswitch- I ing probability can be neglected, in contrast to the un- C J derdampedcase.20Experimentalresultsareshownwhere R1 R I0 2 on-chip RC damping circuits are used to create an HF- overdamped environment. We observe that for longer FIG. 1: (a) RCSJ model of a Josephson junction. (b) Sim- pulses of duration 20 s, the switching process is initi- µ ple model of a junction embedded in an environment with ated by thermal fluctuations in the overdamped system frequency dependentimpedance. and thermal equilibrium is achieved at the base temperature of the cryostat (25 mK). For short pulses of duration < 25 ns, the switching is unaffected by thermal fluctuations up to a temperature of 500 mK, and the washboard potential (marked A in fig 2(a)) correspond- widthoftheswitchingdistributionatlowtemperaturesis ing to the superconducting state where V = hδ˙i = 0. rather determined by random variations in the repeated Increasing the tilt of the potential to i > 1, where local switch pulse. Although the detector apparently had the minima no longer exist, the particle will start to accel- speed and sensitivity required for making a quantum erate to a finite velocity V = hδ˙i> 0 determined by the measurement, we were unfortunately unable to demon- damping. If the tilt is then decreased below i = 1, the strate quantum dynamics of the qubit due to problems particle for an underdamped JJ (Q > 1) will keep on with fluctuating backgroundcharges. moving due to inertia. Further decreasing the tilt below the level i < i , where loss per cycle from damping ex- r ceeds gain due to inertia, the particle will be retrapped II. PHASE SPACE PORTRAITS inalocalminimum. Intermsofthecurrent-voltagechar- acteristic,thiscorrespondstohysteresis,oracoexistence Thenon-lineardynamicsofaDC-drivenJJcanbepic- of two stable states, V =0 and V >0 for a bias fixed in torially represented in a phase space portrait. We begin the region i < i < 1. For the overdamped RCSJ model r by examining the phase space portraits of the resistive the particle will always be trapped in a local minimum and capacitively shunted junction (RCSJ), which is the fori≤1andfreelyevolvingdownthe potentialfori>1 simplest model from which we can gain intuitive under- and there is no coexistence of two stable states. standing of the non-linear dynamics. The RCSJ model A phase space portrait19,21,22 of the RCSJ model is consists of an ideal Josephson element of critical current shown in fig. 2(b). This portrait shows trajectories that I0 biased at the current level I, which is shunted by the the particle would follow in the space of coordinate (δ) parallelplate capacitanceofthe tunneljunction, CJ and versus velocity (δ˙) for a few chosen initial conditions. a resistor R, which models the damping at all frequen- Thetopologyofthephasespaceportraitischaracterized cies (see fig. 1(a)). The circuit parameters define the by severaldistinct features. Fix point attractorsmarked two quantities ωp = I0/ϕ0CJ called the plasma fre- “A” in fig. 2 correspond to the particle resting in a lo- quency,andthequalityfactorQ=ωpRC. Thedynamics calminimum of the washboardpotential, andthe saddle p isclassifiedasoverdampedorunderdampedforQ<1or points marked “S” corresponds to the particle resting in Q > 1, respectively. Here ϕ0 = h¯/2e is the reduced flux anunstablestateatthetopofthepotentialbarrier(com- quantum. parewithfig.2(a)). TwotrajectoriessurroundingAand The circuit dynamics can be visualized by the motion ending at S are the unstable trajectories which define of a particle of mass ϕ2C in a tilted washboard poten- the boundary of a basin of attraction: All initial condi- 0 tial U(δ) = −E (iδ +cosδ) subjected to the damping tionswithinthisboundarywillfollowatrajectoryleading J force ϕ2/R, where the particle position corresponds to to A. We call this the 0-basin of attraction. The thick 0 the phase difference δ across the junction and the tilt line B is a stable limiting cycle, corresponding to a free- i is the applied current normalized by the critical cur- running state of the phase δ, where the circuit is under- rent, i=I/I (see fig 2(a)). Below the critical tilt i=1 going Josephson oscillations with frequency ω = V/ϕ . 0 0 the fictitious particle will stay in a localminimum of the All trajectories leading to the limiting cycle B start in 3 the 1-basin of attraction, which is the regionoutside the (a) S 0-basins. The existenceoftwobasinsofattractioninthe phase space topology, and in particular the clear separa- U tionofall0-basinsbythe1-basin,maketheunderdamped A RCSJcircuit(Q>1)appropriateforaswitchingcurrent detector,aswediscussbelow. FortheoverdampedRCSJ circuit(Q<1)attractorsAandBdonotcoexistforany 0 2 4 6 δ/π fixed bias condition, and it therefore can not be used for 4 a switching current detector. However, the RCSJ model (b) is not always the most realistic model for the dynam- B ics of JJ circuits, as the damping in real experiments is 2 usuallyfrequencydependent,andinthecaseofsmallca- pacitance JJs, this frequency dependence can very much chaAntgehitghhe cfrheaqruaecntecrieosfotfhethdeamorpdinerg.of the plasma fre- δd/dt 0 A S A S A S quencyofthejunction(20-100GHzforAl/AlOx/Altun- nel junctions) losses are typically due to radiation phe- −2 nomena, where the leads to the junction act as a wave i=0.6 guide for the microwaveradiation. If we model the leads Q=4 as a transmission line, the high frequency impedance −4 would correspond to a damping resistance of the order 0 2 4 6 of free space impedance Z ≈ Z /2π = 60 Ω. With the δ/π 0 small capacitance of a typical JJ as used in present ex- (c) 0.01 periments, this damping inevitably leads to overdamped dynamics Q < 1. It should be noted that for small ca- C 0.005 pacitance JJs, underdamped phase dynamics is hard to achieve in practice as high impedance all the way up to the plasma frequency is desired, and this requires an en- S S S 0 gineering effort where the high impedance leads need to v A A A be constructed very close to the junction.23 However, at lower frequencies (typically below ≈ 10 MHz) the junc- −0.005 i=0.8 tion will see an impedance corresponding to the bias re- Q =1.85 sistor R at the top of the cryostat, which can be chosen 0 Q =0.036 large enough to give Q > 1. The simplest circuit which 1 −0.01 captures the frequency dependence described above, is 0 2 4 6 a JJ shunted by a series combination of a resistor R2 δ/π and a capacitor C2 in parallel with the resistor R1 as (d) 0.01 shown in fig. 1(b). At high frequencies where C is es- 2 sentially a short, the circuit is described by the high- frequency quality factor Q = ω R C , where R is 0.005 1 p || J || the parallel combination of R and R . At low frequen- 1 2 cies where C effectively blocks, the quality factor reads 2 Q = ω R C . This model has been studied previously 0 S S S 0 p 1 J v byseveralauthors.15,19,24,25,26 Castingsuchacircuitina A A A more mathematical language,it can be described by the coupled differential equations19,25 −0.005 i=0.8 Q =1.85 0 δ˙ = Q1 − 1 dU(δ) +v Q0 −1 +i +i (1) Q1=0.0065 Q E dδ Q n1 n2 −0.01 0 (cid:20) J (cid:18) 1 (cid:19) (cid:21) 0 2 4 6 v˙ = ρQ1 − 1 dU(δ) −v+i +i Q21 , δ/π Q3 E dδ n1 n2Q (Q −Q ) 0 (cid:20) J 0 0 1 (cid:21) FIG. 2: (Color online) (a) Tilted washboard model. (b) Un- where v = VC/R1I0 is the reduced voltage across C2 derdamped circuit biased at ir <i<1. (c) Circuit with fre- and ρ = R1CJ/R2C2 reflects the value of the transition quencydependentdampingandQ1>Q1candwithQ1<Q1c frequency, being ω ≈ 1/R C , between high- and low- (d). 2 2 impedance regimes. Phase space portraits for such a circuit are shown in figs. 2(c) and (d). Here the y-axis shows the voltage v 4 whichisdirectlyrelatedtoδ˙. Thetopologyofthis phase for the actual switching process to occur and must be portrait is also characterized by the coexistence of fix- shorter than the relaxation time of the qubit. In the pointsAandthelimiting cycleB(notshown). However, ideal case the measurement time would be the same as for the parameters of fig. 2(c) (Q = 1.85, Q = 0.036 thedurationoftheswitchpulse. Amuchlongertimemay 0 1 and i = 0.8. ), the 0-basins and the 1-basin are now berequiredtoactuallydeterminewhichbasinthesystem separated by an unstable limiting cycle C which does has chosen. This longer detection time is the durationof not intersect a saddle point. An initial condition which theholdlevelneededtoreachasignaltonoiseratiolarger is infinitesimally below or above C will eventually end than 1, which is in practice determine by the bandwidth up either in an attractor A, or on B respectively. In ofthelow-noiseamplifierandfiltersinthesecondstageof fig. 2(c) we also see that the boundaries of the 0-basins thecircuit. Inourexperimentsdescribedinthefollowing aredirectlytouchingoneanotherasaconsequenceofthe sections, we used a low noise amplifier mounted at the existence of C. Thus, it is possible to have a trajectory top of the cryostat which has a very limited bandwidth fromone0-basintoanother0-basin,withoutcrossingthe andhighinput impedance. While thisamplifierhasvery 1- basin. low back action on the qubit circuit (very low current ThissameHF-overdampedmodelcanhoweverproduce noise), it’s low bandwidth increases the detection time a new topology by simply lowering the high-frequency such that individual measurements can be acquired only quality factor Q . As we increase the high frequency at < 10 kHz repetition rate. Since many measurements 1 damping,theunstablelimitingcycleCslowlyapproaches (104) are requiredto get goodstatistics when measuring the saddle points S. For a critical value of Q = Q , C probabilities, the acquisition time window is some 0.5 1 1c andSwilltouchandthephase-portraitsuddenlychanges seconds and the low frequency noise (drift or 1/f noise) itstopology. Fig.2(d)showsthephasespaceportraitfor in the biasing circuitwill thus playa rolein the detector Q = 1.85, Q = 0.0065 < Q and i = 0.8 where we accuracy. 0 1 1c can see that C disappears and adjacent 0-state basins are again separated by the 1-basin – a topology of the same form as the underdamped RCSJ model. IV. FLUCTUATIONS Themeasurementtime ofaswitchingdetectorwillde- III. THE SWITCHING CURRENT DETECTOR pend on fluctuations or noise in the circuit. The phase space portraits display the dissipative trajectories of a The transition from the 0-basin to the 1-basin, called dynamicalsystem, but they do not containany informa- switching, can be used as a very sensitive detector. The tionaboutthe fluctuations whichnecessarilyaccompany idea here is to choose a ”hold” bias level and circuit pa- dissipation. For a switching current detector, we desire rameters where the phase space portrait has a favorable thatthesefluctuationsbeassmallaspossible,andthere- topology such as that shown in figs. 2(b) and (d). A fore the dissipative elements should be kept at as low a rapid ”switch pulse” is applied to the circuit bringing temperatureaspossible. Analyzingtheswitchingcurrent the system from A to a point as close as possible to the detectorcircuitwithathermalequilibriummodel,wecan unstable point S. Balanced at this unstable point, the calculate the rate of escape from the attractor A. This circuit will be very sensitive to any external noise, or to equilibriumescape rate howeveronly sets anupper limit the state of a qubit coupled to the circuit. The qubit on the measurement time. When we apply the switch state at the end of the switch pulse can be thought of as pulse, the goal is to bring the circuit out of equilibrium, determining the initial condition, placing the fictitious and we desire that the sensitivity at the unstable point phase particle oneither side of the basinboundary,from be largeenoughso that the measurementis made before which the particle will evolve to the respective attrac- equilibrium is achieved (i.e. before thermal fluctuations tor. The speed and accuracy of the measurement will drive the switching process). depend on how rapidly the particle evolves away from EquilibriumfluctuationscancauseaJJcircuittojump the unstable point S, far enough in to the 0-basin or 1- out of it’s basin of attraction in a process know as ther- basin such that external noise can not drive the system malescape. The randomforcewhichgivesrise tothe es- to the other basin. Fromthis discussionit is clearthat a cape trajectory will most likely take the system through phasespaceportraitwiththe topologyshowninfig.2(c) the saddle point S, because such a trajectory would re- is not favorable for a switching current detector. Here quire a minimum of energy from the noise source.21 For the switching corresponds to crossing the unstable cycle the topology of phase space portraits shown in figs. 2(b) C. Initial conditions which are infinitesimally close to C and (d), thermal escape will result in a switching from willremainclosetoCovermanycyclesofthephase,and a 0-basin to the 1-basin, with negligible probability of a thus a small amount of noise can kick the system back ”retrapping”event bringing the systemback from the 1- and forth between the 0-basins and the 1-basin, leading basintoa0-basin. However,forthetopologyoffig.2(c), to a longer measurement time and increased number of thermalescapethroughthesaddlepointleadstoanother errors. 0-basin, and thus the particle is immediately retrapped The measurement time is that time which is required in the next minimum of the washboard potential. This 5 process of successive escape and retrapping is know as group15,26 whogeneralizedKramersresult. Thetheoreti- phase diffusion, and it’s signature is a non-zeroDC volt- calanalysiswassubjecttotheconstraintthatthedynam- age across the JJ circuit when biased below the critical ics of the voltage across the shunt capacitor v is under- current, i<1. damped (i.e. the quality factor α=R R C I /ϕ (R + 2 || 2 0 0 1 Phase diffusion can occur in the overdamped RCSJ R2) ≫ 1 where R|| is the parallel resistance of R1 and model, or in the HF-overdamped model when parame- R2) so that the dynamics of v is subject to the fast-time ters result in a phase space topology of fig. 2(c). In the average effects of the fluctuating phase δ. Separating latter case, a switching process can be identified which timescales in this way, the switching of v could then be correspondsto the escape froma phase diffusive state to regardedasanescapeoutofameta-potential,B,formed the free running state, or to crossing the unstable lim- bytheaveragedfluctuatingforceinthetiltedwashboard iting cycle C in fig. 2(c) which marks the boundary be- potential F = i−hsinδi−v. Assuming non-absorbing tween the phase diffusive region and the 1-basin. This boundaryconditions,this”escapeoveradissipationbar- basin boundary C is formed by the convergence of many rier” can be written as a generalization of Kramers’ for- trajectories leading to different S, and the escape pro- mula cess of crossing this boundary is fundamentally differ- ′ ′ D(v ) −F F ent than escape from a 0-basin to the 1-basin. Numer- Γ= t exp(B). (3) ical simulations19,27,28 of switching in JJs with such a 2π s(cid:18)λD(cid:19)vb(cid:18)λD(cid:19)vt phase space topology show that escape over the unsta- Here D(v) is the position-dependent diffusion constant, bleboundaryCischaracterizedbylateswitchingevents, and B = vt(F/λD)dv, where v and v stand for the which arise because even a small amount of noise near vb b t bottom and the top of the effective barrier, respectively. this boundary can kick the system back and forth be- R Detailed expressions can be found in refs.15,32 tween the 1-basin and the many 0-basins for a long time In section VI, we use these escape rate formulas to before there is an actual escape leading to the limiting analyze pulse and hold switching measurements. We cycle B. demonstrate that long switching pulses lead to thermal The rate of thermal escape from a 0-basin can be cal- equilibrium switching, whereas short pulses switch the culated using Kramers’ formula22,29,30 circuit in a way that is independent of temperature at ω low temperatures, with the switching distribution deter- 0 Γ=κ exp(−∆E/kBT), (2) minedbynoiseintheswitchpulseratherthannoisefrom 2π the cooled damping circuit. with ∆E being the depthofthe potentialwellfromA to S,k Boltzmann’sconstantandT thetemperature. The B prefactor κω /2π is called the attempt frequency, where V. EXPERIMENTS 0 κ < 1 is a factor which depends on the damping. An- alytical results for κ were found by Kramers in the two Experiments investigating junction current-voltage limiting cases of underdamped (Q>1) and overdamped characteristics (IVC) as well as pulsed switching behav- (Q < 1) dynamics. For the application of Kramers’ es- iorwerecarriedoutinadilutionrefrigeratorwith25mK cape theory we require that ∆E ≫ kBT, i.e. thermal base temperature. A block diagram of the measurement escape is rare, so that each escape event is from a ther- setup is shown in fig. 3(a). A low noise instrumentation mal equilibrium situation. The fluctuations in thermal amplifier (Burr-BrownINA110,noise temperature 1.3 K equilibriumarecompletelyuncorrelatedintime,whichis at 10 kHz) is measuring the voltage across the sample tosaythatthestrengthofthe fluctuationsarefrequency while the sample is biased by a room temperature volt- independent (white noise). Furthermore, the Kramers age source either via the bias capacitor C , or in series b formula assumes absorbing boundary conditions, where with a bias resistor R . The capacitor bias method was b the escape process which leads to a change of the basin usedforexperimentswithfastcurrentpulsesofduration ofattractionhaszeroprobabilityofreturn. Thesecondi- τ = 25 ns, while the conventional resistor bias method p tionsrestrictthedirectapplicationofKramersformulain was used for long pulse experiments with τ = 20 s, as p describingswitchinginJJcircuits31tothecaseoftheun- well as for IVC measurements. µ derdampedRCSJmodelsuchasthatdepictedinfig.2(b). Three different samples are discussed in this paper InprincipleonecouldapplyKramersformulatotheover- which differ primarily in the range of the measured damped RCSJ model, where the escape is from one well switching current (3 nA to 120 nA), and in the type to the next well (switching between adjacent attractors of circuit used for the damping of the phase dynamics. A), but experiments thus far are unable to measure a These different damping circuits are labeled in the order single 2π jump of the phase, as this corresponds to an in which they were implemented, and are represented in extremely small change in circuit energy. fig.3(a)astheblocksF ,F ,F . Theseenvironmentscan 1 2 3 Thermal induced switching of small capacitance be modeled as RC filters with different cut-off frequen- Josephson junctions which experience frequency depen- cies, as schematically be represented in fig. 3(b). dentdampingasmodeledbythe circuitofFig.1(b),was ThekeyparametersforeachsamplearegiveninTable analyzed in experiment and theory by the Quantronics I. Sample I consisted of a Cooper pair transistor (CPT) 6 CPT ShuntJJ F1 F2 F3 Sample Type EJ/EC I0 Isw EJ/EC I0 Isw R1 C1 R2 C2 R3 C3 I CPT 32.9 58.5 3 - - - 60 1 - - - - II CPT 29 51.6 4.2 - - - 60 1 7.2 0.24 - - III Quantronium 2.2 21 12 30.3 158 120 1000 3 7.2 0.24 600 1.4 Currentsarein[nA],resistancesin[Ω]andcapacitances in[nF] TABLE I: An overview over the parameters for the three different samples. Filter F1 resembles the cryostat leads or a cold SMD filter. Filter F2 is theon-chip damping circuit and F3 is an on-chip RC-filter. (a) F used for samples II and III and the bias capacitor 3 were fabricated with a two-step optical lithography pro- C cess. The capacitors were actually two capacitors in se- b ries, formed by a plasma-oxidized Al ground plane cov- R C ered with a Au top plate. The top plates are connected b J F F F to the rest of the circuit via resistors which are formed 1 3 2 V from the same Au film as the top plate, having a typ- I0 ical sheet resistance of 1.2 Ω/2. The capacitors of F 3 could be measured quite accurately, from which we ob- tainaspecific capacitanceof13.6fF/ m2 thatisusedto (b) (c) determine all on-chip capacitors. Figµure 3(c) shows the essential parts of the chip and the components defining the high-frequency environment. The bright rectangular R /2 Cx x F3 F2 area on the left side is the top plate of the capacitor, R /2 andthethinleadsleadingtotherightaretheresistorsof x filter F . Figure 3(d) shows in detail the biasing capaci- 3 500µm tor C The bright trapezoidal area on the left is the top b plate of the capacitor C and the areas surrounded by 2 (d) (e) dashed lines are damping resistors R /2. Figure 3(e) is 2 anelectronmicroscopepictureshowingtheAl/Al O /Al C 2 3 50µm b tunneljunctions,whichwerefabricatedinathirdlayerof electron beam lithography, with the standard two-angle evaporation through a shadow mask. Figure 3(e) shows the quantronium circuit of sample III. F 2 2 µm 5 FIG. 3: (a) Block diagram of the experimental setup. (b) Schematic diagram of the model used to describes the three 2.5 different filters F1,F2,F3 which define the damping circuit. (c) Micrograph of the sample showing the two on-chip RC filters F1 and F2. (d) Magnified view of the center part of A] the chip, F2 and the bias capacitor. (e) Electron microscope [n 0 pictureof theQuantronium (sample III). I −2.5 embeddedinanenvironmentdefinedsolelybythetwisted pair leads of the cryostat which is modeled as F . Sam- −5 1 −75 −50 −25 0 25 50 75 ple II was a CPT fabricated in parallel with Sample I, V [µV] havingnearlyidenticalparameters,differingonlyinthat sample II was embedded in a micro-fabricated on-chip HF-damping circuit F . Sample III is a Quantronium1 FIG. 4: IV curves of sample I without designed RC environ- 2 ment(solidline)andofsampleIIwithspeciallydesignedRC embedded in the same HF-damping F used with sam- 2 environment (dashed line). ple II, but with an additional micro-fabricated on-chip low-passfilter F . The on-chip RC-environmentsF and 3 2 7 The effect of the on-chip HF damping from F on the (a) (b) (c) 2 phase dynamics as can be seen in fig. 4 where the IVC of samples I (solid line) and sample II (dashed line) are a.u.] shown. These two CPT samples differ essentially by the I [ 0 0.5 1 1.5 2 presenceofF insampleII.Weseethatthetypicalphase 100 diffusion sha2pe of the IVC33 of sample I is absent in µ V] 5705 sample II which shows a sharp supercurrent and hys- V [ 25 0 teretic switching. The presence of the on-chip environ- 0 20 40 60 800 20 40 60 800 25 50 75 100 ment in sample II effectively reduces phase diffusion as t [µs] t [µs] t [µs] can be explained by a phase-space topology as shown in figure 2(d). However, the very low value of I is a di- sw FIG. 5: (a) Square pulse and response. (b) Switch pulse rectindicationofexcessivenoiseinthecircuit. Therefore withholdlevelandresponse. (c)Switchpulsewithholdlevel theon-chiplow-passfilterF3 wasimplementedinsample and response, generated by the capacitive bias method. The III, improvingthe switching currentto a value of 75%of finitevoltage in thenon-switch case is duetotheseries filter the critical current. In the remainder of this paper we resistance and thetwo-point measurement setup. concentrate on investigating the switching behavior of sample III. The ability to suppress phase diffusion opens up the possibility to study fast switching with HF-overdamped andthevalueofCb =1.4pF,wecalculateapulseampli- phasedynamicsforthefirsttime. Weusedthepulseand tude of 360 nA throughCb. Due to the symmetry of the holdmethodtomeasureswitchingprobabilitiesofsample filter stages F1 to F3, only half of this 25 ns pulse cur- III as a function of the amplitude of the switch pulse for rentflowsthroughthe junction, withthe otherhalfflow- twocases: Alongpulseof20 swherethe switchingwas ing through the filter to ground. Thus the peak current µ found to be controlled by equilibrium thermal escape, throughthejunctionduringthe25nspulseIp =180nA, and a short pulse of 25ns, where the switching is clearly which is larger than I0. Exceeding I0 for this very short a non-equilibrium process. timeisnotunreasonable,bearinginmindthatthecircuit Thelongpulseswereformedbyapplyingasquarevolt- is heavily overdamped at high frequencies, and a strong kick will be needed to overcome damping and bring the age pulse through the bias resistor. The response to a simple square pulse is shown in fig. 5(a), where the ap- phase particle close to the saddle point. plied voltage pulse is shown, and several scope traces of The hold level for these fast pulses is 40 s,very much µ the measured voltage over the CPT are overlayed. Here longer than the switch pulse, and its duration is set by we see that the switching causes an increase in the volt- the time needed for the response voltage to rise above age over the sample which can occur at any time during the noise level. The rate of this voltage rise depends on the applied pulse. In order to do statistics we want to the holdcurrentlevelbecauseafter the switchwe arees- unambiguously count all switching events. Late switch- sentiallychargingupthesecondstagefilterandleads,F3 ing events are difficult to distinguish fromnon-switching and F1, with the hold current, Ihold = Cb(dV/dt)hold = eventsasthevoltagedoesnothavetimetoriseabovethe 56nA.Forthe lowlevelofholdcurrentusedinthese ex- noise level. We can add a trailing hold level as shown in periments, we can follow the voltage rise at the junction fig. 5(b). This hold level and duration must be chosen with the 100 kHz bandwidth low noise amplifier at the so that there is zero probability of switching on the hold top of the cryostat. Typically we turn off the hold cur- part of the pulse. The response to such a pulse shows rentandresetthedetectorwhenthesamplevoltageis 30 thatitis noweasyto distinguishswitchfromnon-switch V, so thatthe junction voltageis alwayswell below the µ events. Inthiscasetheholdlevelisusedsimplytoquan- gap voltage V2∆ = 400 V, and therefore quasi-particle µ tize the output, and the switching which occurs during dissipation during the hold can be neglected. the initial switch pulse is found to be a thermal equilib- Pulsed switching measurements were performed were rium escape process as discussed below. a sequence of 103 to 104 identical pulse-hold-reset cycles The fast pulses were formed with a new technique wasappliedtothesamplewhilerecordingthevoltagere- where a voltage waveform consisting of a sharp step fol- sponse of the sample. A threshold level was used to dis- lowedby linearvoltagerise isprogrammedinto anarbi- tinguish switching events (1) from non-switching events trary waveform generator. The slope on the sharp step (0)asdepictedinfig.6(a). The maximumresponsevolt- (dV/dt) is typically 6–7 times larger than the linear age achieved during each cycle is found and a histogram pulse rise during the hold, (dV/dt) . The voltage waveform of these values is plotted as seen in fig. 6(b). The hold hold is propagated to the chip through a coax cable having levelanddurationareadjustedsoastoachieveabimodal negligibledispersionforthesharp25nsvoltagestepused. distribution in the histogram, with zero events near the Anon-chipbiascapacitorC willdifferentiatethevoltage threshold level, meaning that there is zero ambiguity in b waveformtogiveasharpcurrentpulsefollowedbyahold determining a switcheventfrom a non-switchevent. We level,I =C dV/dt,whichisshowninfig.5(c). Fromthe further check that the hold level itself, without the lead- b measured step amplitude needed to switch the junction ing switch pulse, gives no switches of the sample. The 8 sequence of switching events is stored as a binary se- (a) (b) 60 60 quence Y in temporal order. From this sequence we can i 0 1 1 0 1 0 0 1 0 1 calculate the switching probability, 40 40 ] V N V] µ P = N1 Yi (4) µV [ 20 20 [eak p i=1 V X 0 0 and the auto-correlationcoefficients, −20 −20 0 0.5 1 0 0.1 0.2 r = Ni=−1k(Yi−Y)(Yi+k−Y). (5) t [ms] percent k P Ni=1(Yi−Y)2 (c) (d) 0.1 wherekisthe”lag”betwPeenpulses. Theauto-correlation 0.3 is a particularly important check for statistical independence of each switching event. A plot of r for 0.2 k k = 1...1000 is shown in figure 6(c) and the randomness k0 1 r r andlowlevelofr indicatesthatallswitching eventsare 0.1 k notinfluencedbyanyexternalperiodicsignal. Whenthe circuit is not working properly, pick up of spurious sig- 0 −0.1 nals up to the repetition frequency of the measurement, 0 0.5 1 0 200 400 clearly shows up as a periodic modulation in the auto- k [1000] τ [µs] correlation r . Of particular importance is the correla- w k tioncoefficientforlagoner whichtellshowneighboring 1 switching events influence one another. Fig. 6(d) shows FIG.6: (a)Responseofthesampletoapulsesequenceresult- inginswitches(1)andno-switches(0)ofthesample. (b)Peak r as a function of the wait time τ between the end 1 w voltageobtained duringacurrentpulse,indicatinggood sep- of the hold level and the start of the next switch pulse. aration between the switch and non-switch signal. (c) Auto- Forlargevaluesofτ ,r fluctuatesaround0notexceed- w 1 correlation function rk. (d) Correlation coefficient r1 vs. the ing 0.05, which shows that any influence of a switching wait time, with fittedexponential decay function. or non-switching event on the following measurement, is statistically insignificant. As τ is decreased however, a w positive correlationis observed,with r increasing expo- 1 nentially with shorter τw. Positive correlation indicates VI. ANALYSIS that a switching event (a ”1”) is more likely to be fol- lowed by another switching event. Fig. 6(d) shows a fit The switching probabilities were thus measured and to correlation r to the function 1 the dependence on the amplitude of the switch pulse, P(I ) wasstudied as as a function of temperature. Each p τ w measurementofP(I )beganwithapulsesequenceshav- r =3.345·exp − . (6) p 1 33.3 s ing pulse amplitude resulting in a switching probabil- (cid:18) µ (cid:19) ity P = 0, and the pulse amplitude was successively We can extrapolate the fit to the time τ∗ = 40.25 s increased until P = 1. The measurement produces an µ wheretheauto-correlationbecomesr1 =1,meaningthat ”S-curve” as shown in figure 7, where the experimental oncethecircuitswitchesitwillalwaysstayinthe1-state. dataforthelongpulsedurationτ =20 sisshownwith p In our experience, increasing the capacitance of filter F1 crosses. The S-curves were taken at temµperatures 100, causes τ∗ to increase, from which we infer that the in- 200, 300, 400 and 500 mK (right to left) respectively. crease in the correlationr1 for shortτw is resulting from We compare the measured data to theoretical predic- errorswherethe detectoris notproperlyresetbecauseit tionsbasedonthermalescapeasdiscussedinsectionIV. does not have time to dischargethe environment capaci- The filter F causes a rounding of the applied square 1 tance before a new pulse is applied. For the experiment voltage pulse, which is accounted for by calculating the shown in figure 6 the time constant of the environment escape probability for a time dependent current32, was estimated to be 3 s. These observations indicate that it is necessary to µbring the junction voltage very 1 i P =1−exp − Γ(i′)di′ (7) close to zero before the retrapping will occur, and the di/dt detector will reset. For good statistics many pulses are (cid:18) Z0 (cid:19) required and a short duty cycle is desirable in order to wheretheescaperateΓcanbefoundusingeithereqns.2 avoid effects from low frequency noise as discussed sec- or 3. The simulated S-curves using eqn. 3 are plot- tionIII. By studying the correlationcoefficientr inthis ted in figure 7 as solid lines for the temperatures cor- 1 way, we can choose an optimal duty cycle. responding to the measured data. Sample parameters 9 1 escape is from a saddle point, so that the non-absorbing experiment boundary condition assumed in the theory is valid. For simulation comparison, we use the overdamped Kramers formula 0.75 (equation 2) to simulate the S-curve and calculate I sw and∆I,whichisshownbythedashedlineinfig.8. Here P 0.5 theprefactorκ(Q)isgiveninref.25 andwehaveusedthe high frequency quality factor Q = 0.027 as determined 1 0.25 by the resistor R only. We see that the Kramers for- 2 mula overestimates I by some 25% (fig. 8(a)) and is sw 0 worse than the simulation based on eqn. 3, in reproduc- 0.5 0.6 0.7 0.8 ingthetemperaturedependenceof∆I (fig.8(b)). Infact, I /I the experimental data for the 20 s pulses only shows a p 0 µ weakincreasein∆I overthe temperaturerangestudied, FIG. 7: Switching probability as a function of pulse height whereas both theoretical curves predict a slight increase in the temperature range T = 100, 200, 300, 400, 500 mK in ∆I. Thus an equilibrium thermal escape model ex- (right to left) for a pulse duration τp = 20 s. Crosses show plains the data for long, 20 s pulses reasonably well µ µ measured data and simulated data is shown as solid lines. and the data is better explained by the theory of escape with frequency dependent damping, than by the simpler theory embodied in the overdamped Kramers formula. However the correspondence with the former theory is usedforthiscalculationarethemeasuredbias(including not perfect. We may explain these deviations as being filter) resistance R = 11600 Ω, the measured high fre- 1 due to the fact that the quality factor α=4.49 (see sec- quencydampingresistor,R =7.2Ω,thehighfrequency 2 tion IV) does not really satisfy the condition for validity dampingcapacitanceC =0.207nF,the junctioncapac- 2 of the theory, α≫1. itance C = 30 fF, and the calculated critical current J Experimental data for the short pulses of duration I = 148 nA. The critical current I = 148 nA is not 0 0 τ = 25 ns generated by the capacitive bias method is the barecriticalcurrentI =158nAsincethe quantron- p 0 plotted in fig. 8 as circles. Here we see that the value iumwasbiasedatamagneticfieldsuchthatapersistent of I is constant in the temperature range studied, in- currentof≈10nAwasflowingintheloop. Theseparam- sw dicating that escape is not from a thermal equilibrium etersareallindependently determined,andnotadjusted state. For the ideal phase space topology, as shown to improve the fit. However, the capacitance of filter F 1 in figure 2(d), the initial pulse would bring the phase was uncertain, having a nominal value of 10nF, and un- particle arbitrarily close to the saddle point S for the known temperature dependence below 4 K. Cooling the hold bias level. If the separation in to the basins of at- samecapacitanceto4K,weobservedadecreaseofC by 1 tractionoccurs before thermal equilibrium canbe estab- around10%. This capacitorC determines the rounding 1 lished, we would not expect temperature dependence of of the square voltage pulse, and thus the time depen- I . In this case,the width of the switching distribution dence of the current applied to the junction. We found sw will be determined not by thermal fluctuations, but by that it was necessary to assume C = 3 nF in order for 1 other sources of noise, such as random variation in the thesimulationstoagreewithexperiment. Thislowvalue height of the switch pulse. These variations are signifi- ofC at lowtemperatures is not unreasonable,ascircuit 1 cant because the 1/f noise from the waveform generator simulationswiththenominalvalueof10nFshowedthat mustbetakenintoaccountwhengeneratingthetrainof the initial pulse would not exceed the hold level, which pulses over the time window of the measurement which clearly is not possible because excellent latching of the was about 0.5 sec. In our experiments however, we may circuit was observed. not have achieved a constant hold level since the volt- Fromthe experimentalandsimulatedS-curves,we de- age ramp from the waveform generator is not perfectly fine the switching currentofthe sample asthe pulse am- smooth. Knowingthe bias capacitorwe cancalculate an plitude that gives 50 % switching P(I ) = 0.5 and the sw average hold level of i = 0.35, somewhat lower than resolution is defined from the S-curve by ∆I = I (P = hold p the critical value of i =0.67 necessary to achieve the 0.9)−I (P = 0.1). A comparison of experimental and hold p phase space topology of figure 2(d). Nevertheless, we theoretical I vs. T and ∆I/I vs. T is shown in sw sw observe excellent latching of the circuit for these 25 ns fig. 8. We see that the experimental data for the long switch pulses. We conclude that the observed tempera- pulses (points marked by an X) are in reasonably good ture independence of I , and the fact that I exceeds agreement with the simulated values when the theory of sw sw I by20%isconsistentwithaveryrapidswitchingofthe switching in an environment with frequency dependent 0 junction. dampingisused(escapefromameta-potential,equation 3) which is plotted as a solid line in fig. 8(a). We note Wecanruleoutexcessivethermalnoiseasareasonfor that for the 20 s pulses, escape occurs at bias currents the temperature independent value of I for the short sw µ i ≈ 0.7, where the phase space has a topology as shown pulses. BymeasuringthegatevoltagedependenceofI sw in figure 2(d). Hence we can neglect phase diffusion and as a function of the temperature, a clear transition from 10 1.3 switching current of the two qubit states at the optimal (a) readout point differ by 9.6 nA. Numerous experiments 1.1 20 µ s 25 ns were made with microwave pulses and continuous microwave radiation to try and find the qubit resonance. /Iw0 0.9 eqn. 3 eqn. 2 However, due in part to uncertainty in the qubit circuit Is parameters (level separation) and in part to jumps in background charge, no qubit resonance was detected in 0.7 these experiments. VII. CONCLUSION 0.5 0.1 (b) Fast and sensitive measurement of the switching cur- 0.08 rentcanbeachievedwithapulse-and-holdmeasurement method, where an initial switch pulse brings the JJ cir- I/I 0.06 cuit close to an unstable point in the phase space of ∆ the circuit biased at the hold level. This technique ex- 0.04 ploits the infinite sensitivity of a non-linear dynamical system at a point of bifurcation, a common theme in 0.02 manysuccessfulJJqubitdetectors. We haveshownthat 100 200 300 400 500 with properly designed frequency dependent damping, T [mK] fast switching can be achieved even when the high fre- quencydynamicsoftheJJcircuitareoverdamped. With FIG. 8: Switching current normalized to critical current (a) an on-chip RC damping circuit, we have experimentally andrelativeresolution(b)ofsampleIII.Crossesindicatemea- studiedthethermalescapeprocessinoverdampedJJs. A sured values for τp =20 s, solid and dashed lines are calcu- capacitorbiasmethodwasusedtocreateveryrapid25ns µ lated values using a generalization of Kramers’ large friction switch pulses. We demonstrated fast switching in such result and Kramers’ original result, respectively. Circles are overdamped JJs for the first time, where the switching measured values for τp =25 ns. was not described by thermal equilibrium escape. The methodspresentedhereareasimpleandinexpensiveway to perform sensitive switching current measurements in Josephson junction circuits. While we have shown that 2etoeperiodicitywasobservedinthetemperaturerange the sensitivity can be high, the effect of back-action of 250mKto300mK.Forthesizeofthesuperconductingis- such a detector is still unclear and might be a reason landusedinthisexperiment,wecanestimateacrossover temperature T∗ ≈300 mK,abovewhichthe free energy why no quantum effects were observed. In contrast to difference between even and odd parity goes to zero.34 the readout strategy presented here, all other working qubit-readout strategies, both static switching and dis- Hence, we know that the sample is in equilibrium with the thermometer below T∗, and therefore heating effects persive, are based on underdamped phase dynamics. thatmightoccurinthe shortpulse experiments,cannot explain the fact that the observed I is independent of sw temperature. Acknowledgments Thus we have achieved a very rapid, 25 ns measure- menttimeoftheswitchingcurrent,whichshouldbesuffi- cientformeasurementofthequantumstateofaquantro- This work has been partially supported by the EU nium circuit. For qubit readout, not only the measure- projectSQUBIT II, and the SSF NanoDev Center. Fab- ment time is important, but also the resolution of the ricationandmeasurementequipmentwaspurchasedwith detector. Forthe 25ns pulse,we obtainedthe resolution thegeneroussupportoftheK.A.Wallenbergfoundation. of ∆I/I = 0.055, or ∆I = 9.9 nA. This implies that We acknowledge helpful discussion with D. Vion and M. sw single shot readout is possible for a Quantronium with H.DevoretandtheoreticalsupportfromH.Hanssonand parameters E = 0.5 K and E /E = 2.5 where the A. Karlhede. C J C 1 D. Vion, A. Aassime, A. Cottet, H. Pothier, C. Urbina, 4 B. L. T. Plourde, T. L. Robertson, P. A. Reichardt, D. Esteve, and M. H.Devoret, Science 296, 886 (2002). T. Hime, S. Linzen, C.-E. Wu, and J. Clarke, Phys. Rev. 2 I.Chiorescu,Y.Nakamura,C.J.P.M.Harmans,andJ.E. B 72, 060506 (2005). Mooij, Science 299, 1869 (2003). 5 E. Il’ichev, N. Oukhanski, A. Izmalkov, T. Wagner, 3 J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, M. Grajcar, H.-G. Meyer, A. Y. Smirnov, A. Maassen Phys. Rev.Lett. 89, 117901 (2002). vandenBrink,M.H.S.Amin,andA.M.Zagoskin,Phys.