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Decoherence processes in a current biased dc SQUID J. Claudon, A. Fay, L. P. Lévy and O. Buisson CRTBT-LCMI, C.N.R.S.- Université Joseph Fourier, BP 166, 38042 Grenoble-cedex 9, France (Dated: February 6, 2008) 6 AcurrentbiasdcSQUIDbehavesasananharmonicquantumoscillatorcontrolledbyabiascurrent 0 andanappliedmagneticflux. Weconsiderhereitstwolevellimitconsistingofthetwolowerenergy 0 states|0iand|1i. Wehavemeasuredenergyrelaxationtimesandmicrowaveabsorptionfordifferent 2 bias currents and fluxes in the low microwave power limit. Decoherence times are extracted. The lowfrequencyfluxandcurrentnoisehavebeenmeasuredindependentlybyanalyzingtheprobability n of current switching from the superconducting to the finite voltage state, as a function of applied a J flux. Thehigh frequencypart of thecurrent noiseis derivedfrom theelectromagnetic environment ofthecircuit. Thedecoherenceofthisquantumcircuitcanbefullyaccountedbythesecurrentand 3 fluxnoise sources. 2 PACSnumbers: ValidPACSappear here ] n o c In the past years, coherent manipulation of two and the well-known MQT formula for underdamped JJ[12]: - multi-level quantum systems, efficient quantum read- Γ (I ,Φ ) = aω exp( 36∆U/5~ω ), where a is of order r 0 b b p − p p outs, entanglement between quantum bits have been unity. u achieved[1, 2, 3, 4, 5, 6] demonstrating the full potential TheenvironmentofthedcSQUIDinducesfluctuations s ofquantumlogicinsolidstatephysics. Atpresent,future of the bias current and the bias flux. In this work, we . t developments require longer coherence times [8, 9, 10]. show how the current and flux noise sources can be sep- a In contrast with atomic system, the huge number of arately quantified. This is achieved by escape measure- m degree of freedom makes its optimization a challenging ments of the SQUID at specific working points where it - d problem. Up to now, the most successful strategy has is mostly sensitive to current or flux noise. Using these n been to manipulate the quantum system at particular identified noise sources,the measured decoherence times o working points where its coupling to external noise is are fit precisely as a function of bias current. c minimal[2]. Nevertheless, experimental analysis of deco- Experimental results are analyzed assuming a linear [ herence phenomena in superconducting circuits remains coupling between the SQUID and the environment de- 1 apriorityforafull-controlofquantumexperiments. Dif- grees of freedom. We suppose that current δI and v ferent models for the noise sources have been proposed flux δΦ noises are generated by independent gcaussian 2 to describe the decoherence processes acting on various sourcces. Here, δx (x = I or Φ) is an operator acting 2 qubits[7,8,9,10]. Howeveracompleteandconsistentun- 5 on the environmcent. Their fluctuations are specified by derstanding of decoherence remains a current and open 1 the quantum spectral densities S (ν)[14]. In presence x problem. In this paper, we study decoherence processes 0 of flux microwave (MW) excitation, the total Hamilto- 6 of a phase qubit: the current biased dc SQUID. nian H in the SQUID eigenstates basis 0 , 1 reads: at/0 sonThjuisncstuiopnersc(oJnJd)u,cetainchg wciirtchuiatccroitnisciasltscuorfretnwtoIJ0oasnepdha- Habre=P−abu12lihmν0a1tσbrzic−eshaνnRdcνo0s1(2=πν(Et)1σbx+E0Nb)/hw{.h|Teirhee|σbfiixr}satntderσbmz capacitance C . The junctions are embedded in a super- − m 0 is the qubit Hamiltonian and the second term describes conductingloopofinductanceL ,threadedbyafluxΦ . s b theMWexcitationofreducedamplitudeν atfrequency - R d In the limit where LsI0 ≈Φ0/2π, the phase dynamics of ν. In this notation, νR is also the Rabi precession fre- n thetwojunctionscanbemappedontoafictitiousparticle quency for a tuned excitation (ν = ν ) . The last term 01 o followingaonedimensionalpathina2D-potential[5]. If is the coupling to the noise sources. For our circuit it is, c the biasing current I is smaller than the SQUID criti- b within linear response, : v cal current Ic, the particle is trapped in a cubic poten- Xi tialwellcharacterizedbyitsbottomfrequencyωp(Ib,Φb) N = hσ rI(θ) δI + rΦ(θ) δΦ andabarrierheight∆U(Ib,Φb)(Fig.1.a). Thequantum −2 xh2π√C hν πL √C hν i r statesinthisanharmonicpotentialaredenoted n ,with b b 0 01c s 0 01c (1) a corresponding energies E , n = 0,1,... In the fo|lliowing, hσ ∂ν01 δI+ ∂ν01 δΦ . n −2 zh(cid:16) ∂I (cid:17) (cid:16)∂Φ (cid:17) i only the loweststates |0iand|1iwill be involved. ForIb b b c b c well below I , these two levels are stable and constitute c where η is the asymmetry inductance parameter (see a phase qubit. below), r (θ) = cosθ + ηsinθ, r (θ) = sinθ and θ I Φ When the bias current I is close to I , ∆U de- is the angle between the escape and the mean slope b c creases and becomes of the order of a few ~ω . The directions in the 2D potential[11, 15]. To first order, p ground state can tunnel through the potential barrier the transverse noise proportional to σ only induces x and the SQUID switches to a voltage state[11]. The depolarization. The longitudinal term proportional to b tunnelling rate Γ of the ground state 0 is given by σ induces ”pure” dephasing. The qubit sensitivity to 0 z | i b 2 (a) 105 2.5 D U ) 104 (c) A) 2 F (a) 10 nWR() (eff 111000123 n 01 mI (esc 1.15 F b2 F b1 b0 U w 100 30 MHz 275 MHz 18 p (b) 106 107 108 109 1010 A) phase n (Hz) n 16 ( I 14 (b) R M L R (n ) L SMC D s oc g f 12 L L -0.4 -0.2 0 0.2 0.4 VMW W50 1 2 Cg CSMC V Reduced bias flux : F /F I b b 0 30 mK FIG. 2: (a) Measured escape current (dots) versus external applied flux fitted to MQT theory (solid line) at 30mK. (b) FIG. 1: (a) Squid cubic-quadratic potential. (b) Electri- cal environment of the SQUID. (c) Calculated effective real ThewidthoftheprobabilitydistributionPesc(Ib)(dots)fitted to the 2D MQT predictions. The solid curve takes the low impedance R versus frequency. eff frequency flux noise into account while the dashed line does not. At bias flux Φb0 (resp. Φb1) the sensitivity to flux noise is zero (resp. small) while it is maximum at Φb2. longitudinal noise is given by the partial derivatives (∂ν /∂I ) and (∂ν /∂Φ ). They depend strongly on 01 b 01 b the experimental working point and increase near the 30mK (ν k T/h = 600MHz ν ). The quantum T B 01 ≡ ≪ critical current. spectral density of the current noise, S (ν) in this en- I vironment is set by the fluctuation-dissipation theorem: The measured SQUID consists of two large aluminum SI(ν) = hν(cid:2)coth(cid:0)2khBνT(cid:1)+1(cid:3)Reff(ν)−1 where Reff(ν)−1 JJs of 15µm2 area (I = 1.242µA and C = 0.56pF) is the real part of the environment circuit admittance. 0 0 enclosinga350µm2-areasuperconductingloop. Thetwo Reff(ν) is calculated using the electrical circuit shown in SQUIDbranchesofinductances L andL contribute to Fig.1.bandisplottedinFig.1.c. Toagoodapproxima- 1 2 thetotalloopinductanceL =280pHwiththeasymme- tion, the root mean square (RMS) current fluctuations s terleycptraormamagenteertiηce=nv(Lir1on−mLe2n)t/Lofst=he0S.4Q1U4.IDThisediemsimgnedediattoe pareeakoefdoarrdoeurnpdk3B0TM/HLzo,ca=fr6eqnuAen.cyMmosutchofstmhaellneroitsheains decouple the circuit from the external world. It consists νT. A simple estimate of the flux noise produced by the of two cascaded LC filters (see Fig. 1.b). A large on- inductive coupling to the 50Ω coaxial line shows it can chipinductanceL =9nHismade oftwolongandthin be neglected in the following. oc superconductingwireswhichvalue,derivedfromthenor- The escape probability P (I ) out of the supercon- esc b mal state resistance, is dominated by the kinetic induc- ducting states is measured at fixed flux using dc current tance. Thegoldthinfilmparallelcapacitor,C 150pF, pulses with ∆t=50µs duration and I amplitude. Each g b ≈ introduces a finite resistor. Its dc value at 30 mK is measurement involves 5000 identical current pulses and R = 0.1Ω giving the gold resistivity ρ = 1.210−8Ωm. the total acquisition time is T = 10s. The escape cur- g g m The secondfilter consists of the bounding wires, with an rent I is defined as the current I where the escape esc b estimated inductance L =3nH, anda surface mounted probabilityP (I )=0.5andthewidthoftheswitching f esc b (SMC)capacitorC =2nFandfour500ΩSMCresis- curve∆I =I I asthedifferencebetweenthecurrents SMC h l − tors. The nominal room temperature microwave signal where P (I ) = 0.9 and P (I ) = 0.1. In Fig. 2, the esc h esc l is guided by 50Ω coaxial lines, attenuated at low tem- dependence of I and ∆I on Φ are plotted. By fit- esc b perature before reaching the SQUID through a mutual ting the escape current curve I (Φ ), the experimental esc b inductance M =1.3pH. Specialcare wastakeninmag- parameters of the SQUID (I ,C ,L ,η) are determined. s 0 0 s neticshieldingandbiaslinesfiltering. Alltheseelectrical Moreover,escape measurements are a sensitive toolto parametersweredeterminedindependently [11,13]. Our characterize noise (frequency range and amplitude). If environment model predicts two resonances (resistance noise frequencies exceed the inverse of a current pulse dips in Fig. 1.c). They were observedin a similar set-up duration ∆t−1, the tunnel rate fluctuates during each and the associated resonance frequencies were in precise current pulse. The escape probability is controlled by agreement with the model. the average Γ escape rate in the frequency window 0 (cid:10) (cid:11) The current noise through the SQUID comes mostly [∆t−1, ν ]: P =1 exp Γ I +δI,Φ +δΦ ∆t T esc 0 b b − (cid:2)−(cid:10) (cid:0) (cid:1)(cid:11) (cid:3) from its immediate environment thermalized at T = [11, 16]. The current noise produced by the electrical 3 6 3 environment lies in this frequency interval. Its effect is P (%) (a) P (%) (b) esc5 esc to decrease I (Φ ) by about 6 nA, the RMS current 2 esc b 4 fluctuations (unobservable in Fig. 2.a). Similarly, the 3 width of the switching curve is not affected. 1 2 Ontheotherhand,ifnoisefrequenciesareslowerthan 1 0 ∆t−1, the tunnel rate is constant during a pulse, but 7.3 7.5 7.7 9.5 9.6 9.7 fluctuates from pulse to pulse. In this limit, the escape n (GHz) n (GHz) probability becomes P = 1 exp Γ I +δI,Φ + 8 esc 0 b b (cid:10) − (cid:2)− (cid:0) T T δΦ ∆t , where the statistical average is now in fre- (c) MW delay (cid:1) (cid:3)(cid:11) (cid:10)(cid:11) 6 quency range from T−1 to ∆t−1. To first order, low fre- m %) ∆quIe.ncTyhnuosis∆edIoiesstnhoetbaffesetctqIueascn,tibtuyttioncprreoabseestthhee woriidgtihn (esc 4 n P 2 01 and the magnitude of the low frequency fluctuations: if the flux Φ is set at the value Φ which maximizes I , b b0 c 0 the SQUID is only sensitive to current fluctuations since 0 200 400 600 ∂ν01 =0. Inthevicinityofthisflux,themeasuredwidth T (ns) ∂Φb delay is explained by the usual MQT theory. Hence the measured RMS current fluctuations in the [T−1, ∆t−1] in- m FIG. 3: (a) and (b) Escape probability versus applied mi- terval (low frequency current noise) is below 0.5nA, the crowave frequency with amplitude νR < 5 MHz at two dif- error bar in ∆I measurements. This is consistent with ferent working points (Ib = 2.288µA,Φb1 = −0.117Φ0) and the 0.1nA RMS value derived from the spectral density (Ib = 0.946µA,Φb2 = −0.368Φ0), respectively. The points of noise at frequencies below ∆t−1. For other applied areexperimentaldataandthecontinuouslinesaretheFourier fluxes, the width is slightly largerthan MQT prediction, transformsoffcoh(t)(seetext). (c)Measuredescapeprobabil- indicating a residual low frequency flux noise. The de- ityversusdelay time(dots) fittedtoan exponentiallaw with pendence of ∆I on Φ shown in Fig. 2.b is perfectly ex- T1 = 95ns (continuous line). The inset specifies the timing b plained by a gaussian low frequency flux noise. Its RMS of the measurement pulse which follows the MW excitation amplitude, δΦ2 1/2 =5.5 10−4Φ ,isextractedfrom pulse. (cid:10) LF(cid:11) × 0 the fit shown in Fig. 2.b and is attributed to the flux noise in the [100mHz,20kHz] frequency interval. The during a time T =300ns, and measuring its popula- MW originoffluxnoisemaybeduetovorticestrappedinthe tionwithincreasingtimedelayT aftertheendofthe delay fouraluminumcontactpadslocatedata0.5mmdistance MW pulse. As shown in Fig. 3.c, the escape probability from the SQUID. follows an exponential relaxation with a characteristic Hereafterwediscussdephasingandrelaxationinduced time T1. In Fig. 4, measured resonant frequency ν01, bythenoisesourcespreviouslyidentified. Theseincoher- relaxation time T1 and the inverse of microwave width ent processes are experimentaly studied with low power ∆ν−1 are plotted versus current bias for the two differ- spectroscopyandenergyrelaxationmeasurement. Asde- ent applied fluxes Φb1 (close to Φb0) and Φb2 shown in scribedinRef.[5],aMWfluxpulseisappliedfollowedby Fig. 2. ν01, T1 and ∆ν−1 decreases as Ib gets closer to a 2ns duration dc flux pulse to perform a fast but adia- Ic. For these two applied fluxes, the ν01 dependence fits batic measurement of the quantum state of the SQUID perfectly the semiclassical formulas for a cubic potential (Fig. 3.c inset). The duration T = 300ns of MW [17]usingthesameSQUIDelectricalparametersasthose MW pulses is sufficient to reach the stationnary state where extracted from escape measurements. the population p of the level 1 only depends on ν and ThedepolarizationrateT−1isgivenbythesumT−1 = 1 1 1 the amplitude νR. The microw| aive amplitude νR is cal- ΓR+ΓE oftherelaxationΓR andtheexcitationΓE rates. ibrated using Rabi like oscillations[5]. In the two level These two rates are calculated using Fermi golden rule. experiments discussed in this paper, the measured es- At low temperature, excitation can be neglected and ΓR cape probability P induced by the dc flux pulse can reads: esc be interpreted as Pesc =Pe|s0ci+(Pe|s1ci Pe|s0ci) p1(ν,νR). r2(θ) r2(θ) Pe|snci denotes the escapeprobabilityo−ut ofthe×pure state ΓR = 4CIhν SI(ν01)+ L2CΦ hν SΦ(ν01). 0 01 s 0 01 n . In Fig. 3.a and 3.b, the escape probability versus | i microwave frequency ν are plotted at two different bias- Neglecting the high frequency part of the flux noise, ing points. The experimental curves present a resonant one obtains T = 2R (ν )C /r2(θ) where R (ν ) = 1 eff 01 0 I eff 01 peak whichposition and full width at half maximum de- (2πL ν )2/R (ν ). R (ν)=αR isthehighfrequency oc 01 g 01 g s fine the resonant frequency ν01 and ∆ν. Spectroscopy resistance of the gold capacitor where Rs = √πµ0ρgν is experiments are performed in the linear regime and ∆ν the surface resistanceand α is a dimensionless geometri- is experimentally checked to be independent of the MW calparameter. TheT versusI dependenceiswellfitted 1 b amplitude. Relaxationmeasurementswereperformedby withα=200astheonlyadjustableparameter(Fig.4.b). populatingthe 1 statewithlowpowerMWtunedatν This value is the right order of magnitude for the geom- 01 | i 4 12 2π∂ν01 2 +∞dνS (ν)sinc(πνt) , where (∂ν /∂I ) is z) F (cid:0)extra∂cItbe(cid:1)d Rd−ir∞ectly Ifrom the slop(cid:3)e of the exp0e1rimebntal H G 10 b1 curve of Fig. 4.a. We neglect flux noise contributions ( F withfrequencieshigherthan20kHz. Sincetheacquisition 1 0 b2 time of absorptionspectra andescape measurementsare n 8 similar,theSQUIDundergoesthesameRMSfluxfluctu- ationsinthe twoexperiments. Inthese conditions,f (t) 120 Φ takes the simple gaussian form: f (t) = exp 1t2 ) Φ (cid:2)− 2 × (ns 80 (cid:0)2π∂∂νΦ0b1(cid:1)2(cid:10)δΦ2LF(cid:11)(cid:3). (∂ν01/∂Φb) was computed using the 1 known electrical parameters of the SQUID[19]. T Within linear response, the shape of the resonance 40 curve is proportional to the Fourier transform (FT) of 50 f (t). Resonance curves in Fig. 3.a and 3.b are fit- ns) 30 tecdohusing Pesc Pe|s0ci FT fcoh (ν ν01) (continuous ( − ∝ { } − 1 line). Our model explains perfectly the shape of the ex- - 10 perimental curves. In Fig. 4.c, the theoretical width ∆ν Dn 5 extracted from the curve FT f (ν) is in very good coh { } agreement with experimental data without free param- 0.5 1 1.5 2 2.5 eter. When I gets close to I , the partial derivatives b c Bias current : I (m A) (∂ν /∂I )and(∂ν /∂Φ )increase: thenoisesensitivity b 01 b 01 b increases and ∆ν broadens. For bias points correspond- FIG.4: Resonanttransitionfrequency(a),relaxationtime(b) ing to Φb2, the width is due to current and flux noise. andmicrowavewidth(c)asfunctionofbiascurrentatΦb1 = For a bias flux equal to Φb1, the effect of flux noise is −0.117Φ0 and Φb2 = −0.368Φ0 respectively right and left small and the width is dominated by current noise. At curves. Symbols correspond to experiments and continuous this flux, for I < 1.95µA, our model predicts satellite b line to model predictions. resonances around ν which are not observed. Other 01 noise mechanism may blur the predicted satellite peaks. In conclusion, we have shown how the flux and cur- etry of the gold capacitor. rent noise present in this controlled quantum circuit can Relaxationaloneistooweaktoexplainthevalueof∆ν be separately identified. We measured the decoherence and ”pure” dephasing also contribute to the linewidths. times at low microwave power where the quantum cir- First,we considerthe time evolutionofthe reducedden- cuitcanbereducedtoatwolevelsystem. Analyzingthe sity matrix in the basis 0 , 1 in the absence of MW. {| i | i} coupling of the SQUID to the known noise sources, the The linear coupling to noise sources induces a time de- measured relaxation times and the resonance width can cayoftheamplitudef (t)ofthecoherenceterms. Since coh be fully understood. currentandfluxnoisesareindependent,f (t)isfactor- coh izedasf (t)=f (t)f (t)exp( 2t/T )wheref (t)and We thank E. Colin, V. H. Dao, K. Hasselbach, F.W.J coh I Φ 1 I − f (t)arerespectivelythe”pure”dephasingcontributions Hekking,B.Pannetier,P.E.Roche,J.Schrieff,A.Shnir- Φ due to current and flux noises. manforveryusefuldiscussions. Thisworkwassupported The current contribution is given by the well-known bytwoACIprogramsandbytheInstitutdePhysiquede gaussian noise formula [10, 18]: f (t) = exp 1t2 la Matière Condensée. I (cid:2)− 2 × [1] T. Yamamoto et al.,Nature425, 941 (2003). Grenoble, France (2005). [2] D.Vion et al.,Science 296, 886 (2002). [14] The quantum spectral density is defined as follows : [3] J.M.Martinisetal.,Phys.Rev.Lett.89,117901(2002). Sx(ν)= dt δx(0)δx(τ) e−i2πντ. [4] I.Chiorescu et al., Science299, 1869 (2003). [15] V. LefevrRe-Se(cid:10)gcuin ect al.,(cid:11)Phys. Rev.B 46, 5507 (1992). [5] J. Claudon et al., Phys.Rev.Lett. 93, 187003 (2004). [16] J.M. Martinis and H.Grabert, Phys. Rev. B 38, 2371 [6] A.Wallraff et al.,Nature431, 162 (2004). (1988). [7] K.B. Cooper et al.,Phys. Rev.Lett. 93, 180401 (2004). [17] A.I.LarkinandY.N.Ovchinnikov,Sov.Phys.JETP64, [8] O.Astafiev et al.,Phys.Rev. Lett.93, 267007 (2004). 185 (1986). [9] P.Bertet et al.,cond-mat/0412485. [18] A. Cottet, PhD thesis, Université Paris VI, (2002). [10] G. Ithier et al., Phys.Rev.B 72, 134519 (2005). [19] (∂ν01/∂Φb) could be measured by changing slighlty Φb. [11] F. Balestro et al., Phys.Rev.Lett. 91, 158301 (2003). Infact,WekeptΦ asconstantaspossibleduringexper- b [12] A.O. Caldeira and A.J. Leggett, Ann. Phys. 149, 374 iments,becauseofastaticfluxhysteresisabout10−3Φ0. (1983). [13] J. Claudon, PhD Thesis, Université Joseph Fourier,