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Flux Qubits and Readout Device with Two Independent Flux Lines B.L.T. Plourde, T.L. Robertson, P.A. Reichardt, T. Hime, S. Linzen, C.-E. Wu, and John Clarke Department of Physics, University of California, Berkeley, CA 94720-7300 (Dated: February 2, 2008) 5 We report measurements on two superconducting flux qubits coupled to a readout Supercon- 0 ductingQUantumInterferenceDevice(SQUID).Twoon-chipfluxbiaslinesallow independentflux 0 control of any two of the three elements, as illustrated by a two-dimensional qubit flux map. The 2 application of microwaves yields a frequency-flux dispersion curve for 1- and 2-photon driving of the single-qubit excited state, and coherent manipulation of the single-qubit state results in Rabi n a oscillations and Ramsey fringes. This architecture should be scalable to many qubits and SQUIDs J on a single chip. 9 PACSnumbers: 03.67.Lx,85.25.Cp,85.25.Dq 2 ] n Superconducting quantum bits (qubits) based on that are substantially larger than values used previously o charge [1, 2], magnetic flux [3, 4], and phase difference in 3-junction qubits, which have relied on external coils c across a Josephson junction [5, 6] are attractive candi- to generate large magnetic fields [4, 9, 10]. At the same - r dates for the basis of a quantum computer because of time, the mutual inductance between the on-chip flux p their inherent scalability using established thin-film fab- linesandthequbitmustbesufficientlysmallforthenoise u s ricationtechniques. Advantagesofthefluxqubitinclude generatedbythecircuitrysupplyingthefluxbiascurrent . its immunity to the ubiquitous charge noise in the sub- not to be the limiting source of decoherence [11]. t a strate and that it can be configured with no direct elec- In this Letter, we report measurements on two qubits m trical connections. One type of flux qubit consists of anda readoutSQUID thatmeet these criteria. We illus- d- a superconducting loop interrupted by three Josephson tratetheorthogonalizationoftheappliedfluxesbymeans n junctionswithcriticalcurrentsI0,I0,andαI0(α<1)[7]. of a two-dimensional flux map. We report spectroscopy o When the applied flux bias ΦQ is at a degeneracy point on one of the two qubits that matches the expected dis- c (n+1/2)Φ0(nisanintegersuchthat|ΦQ−nΦ0|≤Φ0/2; persionforafluxqubit. Inaddition,weobservespurious [ Φ0 ≡ h/2e is the flux quantum), a screening supercur- resonances, some of which may be related to defects in 2 rent JQ can flow in either direction around the loop. the tunnel barriers of the qubit. We perform coherent v The ground and first excited states of the qubit corre- manipulation of the single-qubit state resulting in Rabi 9 spond to symmetric and antisymmetric superpositions oscillations and Ramsey fringes. In prior measurements 7 6 of the two current states and are separated by an en- of coherent oscillations [9, 10], the readout SQUID was 1 ergy ∆. Here, ∆/h is the tunnel frequency between the connected directly to the flux qubit and thus detected a 0 current states, typically a few GHz. When ΦQ is away combination of flux and phase changes. In contrast, our 5 from a degeneracy point, the energy difference between SQUIDiselectricallyisolatedfromthequbitanddetects 0 the two superposed states is ν = (∆2 + ǫ2)1/2, where only flux changes. / at ǫ = 2JQ[ΦQ −(n+1/2)Φ0]. The state of the qubit is Figure1showsthedevicelayout. ThereadoutSQUID m measuredbycouplingthescreeningfluxgeneratedbyJQ has a calculated inductance L = 358 pH, and each of S to a hysteretic dc superconducting quantum interference - the two qubits it encloses has a calculated inductance d device(SQUID).Thisfluxdeterminesthebiascurrentat L =143pH.Thecalculatedmutualinductancebetween n which the SQUID switches out of the zero-voltagestate. Q each qubit and the SQUID is 61 pH. One pair of series- o c In addition to the development of scalable interqubit connected flux bias lines is arranged near the top of the v: couplings [8], a prerequisite for scaling to a system of SQUIDandasecondnearthebottom;thus,fluxinqubit i many qubits is that the attendantreadout,filtering, and 1(2)issuppliedpredominantlybythelower(upper)flux X bias circuitry also scale. A particular challenge is that lines. Themutualinductancebetweeneachqubitandits r the flux bias must be settable for each element individ- associatedflux lines wasdesignedto be 4-5pH, enabling a ually. This mandates the use of on-chip flux-bias lines ustoapply∼2Φ0 withacurrentof1mA.Thiscriterion in an arrangement that enables one to apply a combi- dictated the relatively large qubit self-inductances com- nation of currents to address any given qubit or SQUID pared with those in previous experiments [4, 9, 12, 13]. while maintaining all other flux biases at constant val- We fabricated the device on an oxidized Si substrate ues. Furthermore, the bias currents required to change using electron-beam lithography and double-angle evap- thefluxover(say)±1Φ0shouldnotbesolargethatitbe- oration to form the Al-AlOx-Al tunnel junctions. The comesimpracticaltodeliverthemtoachipcooledtomil- Al lines for the qubit and SQUID loops were 1 µm wide, likelvintemperatures. This requirementestablishesmin- and those for the flux bias 10 µm wide. Each SQUID imum self-inductances of the qubit and readout SQUID junction was 175× 200 nm2 with a critical current of 2 of the system: the SQUID flux Φ , and the qubit fluxes S ΦQ1 and ΦQ2. Given the two flux lines, we can set any twofluxesarbitrarily;thethirdisfullyconstrained. This can be expressed succinctly by the matrix equation (cid:2)ΦS ΦQ1 ΦQ2(cid:3)=MMMFFF111QQS21 MMMFFF222QQS21 ΦΦΦ0Q0Q0S21II112, (1) where the M are mutual inductances between the vari- ij ous flux lines and loops, the Φ0 account for static back- j ground fields, and the I are currents in the flux lines. i We turn now to our experimental results. Figure 2(a) showstheswitchingprobabilityofthe SQUIDversusthe amplitude of the current pulses applied to it for two dif- FIG. 1: (a) Chip layout. Dark gray represents Al traces, ferent values of ΦQ2 at constant ΦS. Because we hold lightgrayAuCutraces. Padsnearupperedgeofchipprovide Φ constant, we can measure the displacement of the S two independent flux lines; wirebonded Al jumpers couple two curves for constant sensitivity in the SQUID. The left and right halves. Pads near lower edge of chip supply measurementfidelity,whichisthedifferencebetweenthe currentpulsestothereadoutSQUIDandsenseanyresulting switching probabilities, has a maximum of about 60%. voltage. (b)Photographofcenterregionofcompleteddevice. Segments of flux lines are visible to left and right of SQUID, Figure 2(b) shows Is50% vs. ΦS; Is50% is the pulse am- which surroundsthetwo qubits. plitude for which the switching probability is 50%. The effectsofthechangingfluxinthequbitsaresmallonthis scale. Figure 2(c) shows Is50% vs. ΦQ1 for constant ΦS; 220 nA. For qubits 1 and 2, the larger junctions had ar- as ΦQ1 is varied, a flux of approximately the same mag- eas of 250×250 nm2 and 180×200 nm2, approximate nitude and opposite sign is applied to qubit 2. Thus, as critical currents I0 (scaled from SQUID junction areas) ΦQ1 isincreasedatconstantΦS,Is50% abruptlyincreases of 390 nA and 230 nA, and α-values (based on junction when qubit 1 flips state and decreaseswhen qubit 2 flips areas) of 0.49 and 0.68, respectively. The different junc- state. tion parameters were chosen to increase the probability To determine the parameters in Eq. (1), we sampled of obtaining one viable qubit; in fact, qubit 2 displayed Is50% at ∼ 20,000 different settings of I1 and I2. We fit goodcharacteristics. A42nm-thickAuCufilmdeposited these data to a parametric model of the response of the and patternedprior to the Al deposition providedquasi- SQUIDIsfit(Φ)tothe totalSQUIDfluxΦT. Todescribe particle traps near the junctions [14], 100 Ω shunts on the SQUID modulation, we use the ad hoc expression each of the four flux lines, and 500 Ω and 1275 Ω series 15 15 resistors on each end of the pulse lines and sense lines, Φ Φ Ifit(Φ)= a cos2πi + b sin2πi +d, (2) respectively. s X i Φ0 X i Φ0 i=1 i=2 Toeliminateexternalmagneticfieldfluctuationsween- closedthechipina6×16×22mm3cavitymachinedintoa where the ai, bi, and dare fit parameters. The totalflux copper block and plated with Pb. This superconducting coupled to the SQUID is well approximated by cavitystabilizes the field, enabling us to acquiredata for periodsupto 48hours. A1mm-diametersuperconduct- ΦT =ΦS +∆ΦQ1jQ1(ΦQ1)+∆ΦQ2jQ2(ΦQ2), (3) ingloop∼3mmabovethechipsuppliedmicrowaveflux. whichneglectstheself-screeningoftheSQUID.Here,the The sample holder was attached to the mixing chamber ∆Φ = J M are the amplitudes of qubit screening of a dilution refrigerator at 50 mK. All electrical leads Qi Qi QiS flux changes near the degeneracy point, referred to the to the experiment were heavily filtered at several differ- SQUID, and the qubit circulating currents are described ent temperatures with a combination of lumped circuit by dimensionless periodic functions of unit amplitude and copper powder low-pass filters [15]. Measurements of the state of a qubit as a function of flux, microwave jQi(Φ)={sin[2πgQi(Φ−nΦ0)/Φ0]}/sinπgQi, (4) power and frequency were made by pulsing the current in the SQUID and detecting whether or not it switched with fitting parameters g . The j (Φ) are discontinu- Qi Qi out of the zero-voltage state by means of a low-noise, ous when n increments. room-temperatureamplifier. Theflux bias currentswere Fitting the Is50%(I1,I2) data, we obtain the following suppliedbyhighlystablepotentiometers,controlledbya parameters for Isfit(ΦT): MF1Q1 = 3.96 pH, MF2Q1 = computer over a fiber-optic link [16]. −0.77 pH, MF1Q2 = 1.37 pH, MF2Q2 = −3.31 pH, Therearethreeappliedfluxesthatdeterminethestate MF1S = 8.30 pH, MF2S = −6.30 pH, ∆ΦQ1 = 4.39 3 FIG. 3: Spectroscopy of qubit 2. Enhancement and suppres- sion of Is50% is shown as a function of ΦQ2 and fm relative to measurements in the absence of microwaves. Dashed lines indicatefittohyperbolicdispersionfor1-and2-photonqubit excitations. The 2-photon fit is one-half the frequency of the 1-photon fit. Inset containing ∼ 23,000 points is at higher resolution. point are indicated by the circle. The remaining discussion of the experiments is con- FIG. 2: (a) SQUID switching probability vs. amplitude of cerned with the effects of an applied microwave flux on bias current pulse near qubit 2 transition. The two curves qubit2. Photonsofenergyhf =ν drivetransitionsbe- m represent the states corresponding to ΦQ2 = 0.48Φ0 (red) tweenthe two qubit states, producing peaks and dips on and ΦQ2 = 0.52Φ0 (blue); ΦS is held constant. Each curve the qubit transitions. By measuring I50% as a function Ecoancthaipnesri1o0d0opfooinstcsillaavteiornagceodn8ta,i0n0s0∼tim5,e0s0.0(bfl)uxIs5v0a%luvess., aΦnSd. of ΦQ2 and fm, we determined the dsispersion relation shown in Fig. 3. This measurement contains a total each switching current is averaged 8,000 times. (c) Depen- dence of Is50% on ΦQ1 for constant ΦS. (d) Qubit flux map. of ∼ 75,000 points and took 48 hours to acquire, thus demonstrating the excellent flux stability in our system. The dispersion is well described by the hyperbolic relation ν = (ǫ2 +∆2)1/2, with ∆/h = (3.99±0.05) GHz mΦ0, ∆ΦQ2 = 4.04 mΦ0, gQ1 = 0.64, gQ2 = 0.51. This and (1/h)dǫ/dΦQ2 = (896±5) MHz/mΦ0. Part of the fit captures all the essential features of the data, having spectrum corresponding to 2-photon transitions is also a root-mean-square residual of 1.8 nA, that is, less than shown. 0.5% of the maximum value of I50%. Figure 3 shows two types of deviation from ideal be- s In Fig. 2(d), we plot I50%−Ifit(Φ ) versus the cur- havior, more evident in the inset which was measured 4 s s S rents I1 and I2; lines of constant ΦS, ΦQ1, and ΦQ2 weeks before the full spectrum. First, there are sharp are indicated. This qubit flux map displays only the suppressions of the critical current which are indepen- flux contributions of the two qubits. For example, in- dent of ΦQ2 and hence occur at particular constant val- side the square, where ΦQ1 ≈ Φ0 and ΦQ2 ≈ −Φ0/2, ues of fm, for example, at 7.4 GHz, that we believe arise Is50% exhibits an abrupt step acrossthe ΦQ2 line, as JQ2 from electromagnetic modes which couple to the qubit. changes direction discontinuously. However, there is no Second, we see disruptions of the dispersion curve that step across the ΦQ1 line, as JQ1 is continuous. Further- are suggestive of coupling between the qubit and other more, one can scan the switching current along a line two-state systems. One instance of this second anomaly, where Φ is held constant at an arbitrary value, to ob- originally centered near 7 GHz in the inset and shifted S serve the effects of flux only on the qubits. Finally, spe- to 6.5 GHz in the full spectrum, is remarkablysimilarto cialpointswherethetwoqubitsareeachatadegeneracy those reported for phase qubits, andmay be of the same 4 tainedthe relaxationtimeτ =281nsat10.0GHz from R an exponential fit. We measured the dephasing time τ φ fromRamseyfringes[20]. Wefirstappliedaπ/2pulseto tipthequbitstatevectorintotheequatorialplane,where it dephased. To calibrate the π/2 pulses, we chose a mi- crowavepulsewidthbasedonaRabioscillationmeasure- mentmadeatthesameoperatingpoint. Afteravariable time delay τ, we applied a second π/2 pulse followed by ameasurementofthe SQUIDswitchingprobabilityfora fixedbiascurrentpulseamplitude. Whenwechoseami- crowavefrequencyoffresonanceweobserveddampedos- cillations of the SQUID switching probability [Fig. 4(c)] with a fit decay time of the oscillations τ = 6.6 ns. φ Figure 4(d) shows the Ramsey fringe frequency for 100 values of the microwave frequency, together with the fit FIG. 4: Coherent manipulation of qubit state. (a) Rabi to two lines with slopes of magnitude unity, as expected oscillations, scaled tomeasured SQUIDfidelity,asafunction for coherently driven Ramsey fringes. ofwidthof10.0GHzmicrowavepulses. (b)Rabifrequencyvs. In conclusion, we have demonstrated on-chip flux bias 10.0GHzpulseamplitude;lineisleastsquaresfittothedata. lines which allow us to vary the flux applied to two of (c)Ramseyfringesforqubitsplittingof9.95GHz,microwave the three devices independently. The mutual inductance frequency of 10.095 GHz. (d) Ramsey fringe frequency vs. microwave frequency. Lines with slopes ±1are fitsto data. betweenthefluxlinesandthequbitsisweakenoughthat the characteristic impedance of the flux lines (say 50 Ω) should not limit our coherence times [11]. Although in this Letter we have concentrated on the quantum coher- origin [17]. ent properties of a single flux qubit, we note that the Weperformedcoherentmanipulationofthequbitstate two flux qubits in this design in principle could be cou- by varying the duration of a resonant microwave pulse pled controllably using the circulating currents in the dc for fixed frequency andamplitude. Upon applying fixed- SQUID [8]. This systemof qubits, SQUID andflux lines amplitude bias current pulses, we observed Rabi oscilla- should be readily scalable. tionsintheswitchingprobabilityoftheSQUIDasafunc- We thank M.H. Devoret, D. Esteve, C.J.P.M. Har- tion of the microwave pulse width at approximately 100 mans, J.M. Martinis, R. McDermott, J.E. Mooij, R.J. frequencies ranging from 4.37 to 13.86 GHz. Generally Schoelkopf,D.Vion,andF.K.Wilhelmforhelpfuldiscus- speaking, the amplitude of the oscillations is higher and sions. ThisworkwassupportedbytheAirForceOfficeof their decaytime is longerfor qubit flux bias points away Scientific Research under Grant F49-620-02-1-0295, the from the spurious splittings shown in the spectroscopy. Army ResearchOffice under GrantDAAD-19-02-1-0187, An example is shown in Fig. 4(a), where each of the the National Science Foundation under Grant EIA-020- 195 points was averaged 20,000 times. We reference the 5641, and the Advanced Research and Development Ac- amplitude of these oscillations to the qubit by scaling tivity. to the SQUID measurement fidelity. We observed Rabi oscillations with the longest decay time when we oper- ated at a bias current of 296 nA, where the fidelity was 14.4% for this particular value of Φ and the SQUID S switched infrequently [Fig. 2(a)]. 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