Tunneling spectroscopy using a probe qubit A. J. Berkley,∗ A. J. Przybysz, T. Lanting, R. Harris, N. Dickson,† F. Altomare, M. H. Amin, P. Bunyk, C. Enderud, E. Hoskinson, M. W. Johnson, E. Ladizinsky, R. Neufeld, C. Rich, A. Yu. Smirnov, E. Tolkacheva, S. Uchaikin, and A. B. Wilson D-Wave Systems Inc. 100-4401 Still Creek Dr., Burnaby BC Canada V5C 6G9 We describe a quantumtunneling spectroscopy techniquethat requires only low bandwidth con- trol. The method involves coupling a probe qubit to the system under study to create a localized probestate. The energyof theprobestateis thenscanned with respect to theunperturbedenergy 3 1 levels of the probed system. Incoherent tunneling transitions that flip the state of the probe qubit 0 occurwhentheenergybiasoftheprobeisclosetoaneigenenergyoftheprobedsystem. Monitoring 2 these transitions allows the reconstruction of the probed system eigenspectrum. We demonstrate thismethod on an rf SQUIDflux qubit. n a PACSnumbers: 85.25.Am,85.25.Cp,03.67.Lx,03.65.Xp J Keywords: spectroscopy; tunnelingspectroscopy; qubit; SQUID;rfSQUID;fluxqubit; quantum annealing; 3 adiabaticquantum computing; quantum computing;Isingspin ] n Recent technological advances have allowed the con- by a generic two-level system Hamiltonian: o struction of mesoscale systems of individual quantum c - elements, including hundreds of trapped ions1, 14 en- Hˆ =−1∆ σˆ − 1ǫ σˆ , (1) pr tangled ions2, nanomagnetic systems assembled out of P 2 P x,P 2 P z,P u magnetic atoms on metallic surfaces3, ultracold 87Rb where σˆ and σˆ are Pauli matrices operating on P, x,P z,P s atomsin opticallattices4, andarraysofsuperconducting and both parameters ǫ and ∆ should be controllable. t. devices5,6. Whilethestudyofsmallnumbersofatomsor The eigenstates of σˆ P with eiPgenvalues +1 and -1 are a z,P devices often involves direct manipulation and full state m |↑iP and |↓iP, respectively. The σˆz,P eigenstates should tomography, these techniques become impractical in the bedistinguishablebyareadoutmechanism. Lettherebe - d mesoscale regime. As a result, there is a need for tools asystemS,governedbysomeHamiltonianHˆS,thatone n thatareapplicablewhenonehasamesoscalesystemwith would like to study. o limited control over its individual elements. To perform QTS we require a coupling between the c Tunneling spectroscopy is a powerful tool for study- probe qubit and a parameter of the system (described [ ing condensed matter systems. It has been used to byanoperatorCˆ)aswellasacontrollablecompensation 2 push the limits of our understanding of many-body bias ǫ coupled to that same parameter. In this case, comp v physics, as in recent studies of two dimensional elec- the system plus probe Hamiltonian can be expressed as: 0 tron systems in high magnetic field using time do- 31 main capacitance spectroscopy7 and scanning tunneling HˆS+P =HˆS +HˆP +Jσˆz,PCˆ+ 1ǫcompCˆ, (2) 6 spectroscopy8. Tunneling spectroscopy can also be used 2 0. to directly validate numerical or analytical models of with J the strength of the probe qubit-system interac- 1 complex systems, such as the single particle states of tion. 2 CdSe quantum dots9 or the electronic wavefunctions of Forgeneralǫ the eigenstatesof S+P arenotrep- comp 1 carbon nanotubes10. Motivated by the ability of tun- resentative of those of S. However, in the special case v: neling spectroscopy to probe the quantum behaviour ǫcomp = −2J, the spectrum of HˆS+P splits into two of mesoscale systems, we have developed an analogous i qualitatively different manifolds, M and M , wherein X method that is applicable when one has limited control the probe qubit P is in state |↑i an↑d |↓i , re↓spectively. P P r over a large system. The large system is probed using H can then be rewritten as: a S+P a dedicated probe qubit with its own readout and low bandwidth control of its Hamiltonian. We have termed Hˆ = ǫ Iˆ −2JCˆ+Hˆ ⊗|↓i h↓| (3) S+P (cid:16) P S S(cid:17) P P this new technique qubit tunneling spectroscopy (QTS). +Hˆ ⊗|↑i h↑| A related method has been proposed in Ref. 11 where S P P the probe qubit must be perturbatively coupled to the − ∆PIˆ ⊗ |↓i h↑| +|↑i h↓| 2 S (cid:16) P P P P (cid:17) system under study. In QTS, the requirement for this weak coupling has been removed through the use of a where Iˆ is the identity operator on system S. In the S compensation bias (as explained below). Further, the casewherethethirdlineofEqn.3isperturbativelysmall, algorithm of Ref. 11 is designed to operate on a gate the first line is the Hamiltonian of M and the second ↓ modelquantumcomputerwhilewedemonstrateQTSon line that of M . Thus, the energy spectrum of M is ↑ ↑ a system with much more limited control. identical to that of Hˆ . Further, the first term of the S QTS requires a probe qubit P that can be described first line of Eqn. 3 shows that the energy of all states 2 ǫ =0, we show how the theoretical energy spectrum of Q the coupled two qubit system splits into two manifolds inFig.1. NotethattheeigenstatesofM showntherein ↑ aresuperpositions ofthe |↑i and|↓i states. Changing Q Q ǫ allowstheloweststateinM ,|↓i |↓i ,tobebrought P ↓ Q P into resonance with the states in M . The system plus ↑ probecantunnelfromM toM throughtheincoherent ↓ ↑ processes labelled as Γ and Γ . 01 02 The twoqubits, probeP andtargetQ, usedinthe ex- periment were rf SQUID flux qubits on a D-Wave quan- FIG. 1. Energy level diagram for the Q+P qubit system tum annealing processor thermalized to a temperature describedbytheHamiltonianinEqn.4atǫQ=0andǫcomp = T = 12mK. A description of a chip similar to that used −2J. The system separates into two distinct manifolds, M ↓ in this study can be found in Ref. 13. The low energy rf and M↑, that differ in the orientation of the probe qubit SQUID flux qubit Hamiltonian14 has a direct mapping P. Spinor notation indicates state of qubits Q and P on onto Eqn. 1: the left and right, respectively. Allowing for weak tunneling isΓny0s2qteubmbeittwinPeetnhfaetchlioleiwtametsaetsneiifnnoeclrdogshy.ersQetnaTttSetpuofrnonMceeleindagnsdpbryfionicndeisitnsieagslivzΓain0lu1geastnhodef HˆQ =−21∆Q(ΦcQcjj)σˆx,Q−ΦxQ(cid:12)(cid:12)IQp(ΦcQcjj)(cid:12)(cid:12)σˆz,Q (5) ↓ (cid:12) (cid:12) ǫP forwhichtransitionrateΓpeaksduetoresonantprocesses where we have performed the substitution ∆Q → Γ01 and Γ02. ∆ (Φccjj) and ǫ → 2 Ip(Φccjj) Φx, with Φccjj and Q Q Q (cid:12) Q Q (cid:12) Q Q (cid:12) (cid:12) Φx beingexternallycontr(cid:12)olledfluxb(cid:12)iasesand Ip(Φccjj) Q (cid:12) Q Q (cid:12) in M↓ can be shifted with respect to those of M↑ by beingthemagnitudeofthequbitpersistentcur(cid:12)(cid:12)rent. Note(cid:12)(cid:12) adjusting the probe energy bias ǫ . For small enough ∆P,thethirdlineofEqn.3givesriPsetoincoherentinter- that both ∆Q and ǫQ (through (cid:12)IQp(cid:12)) are functions of manifold tunneling12 between any state |k↓′i|↓iP of M↓ Φccjj. The functional forms of th(cid:12)(cid:12)ese(cid:12)(cid:12) dependencies are Q and any state |ki|↑iP of M↑ with a rate proportionalto determined by the physicalparameters of the rf SQUID, |∆P hk↓′|ki|2. asdescribedindetailinRef.14. Ifoneconsidersthequbit The QTS method begins by initializing the system Q as an Ising spin, then ∆ corresponds to a transverse Q into the lowest energy state of M↓. The tunneling rate magnetic field, Ip is the magnitude of the spin, and Φx Q Q between manifolds peaks when an eigenstate of M↑ is is an applied longitudinal magnetic field. The physical brought into resonance with the initial system state in Hamiltonian for the probe qubit Hˆ is found by replac- P M↓ by adjusting ǫP. This resonant tunneling transi- ing Q by P in Eqn. 5. The probe qubit had a persistent tionbetweenmanifoldsflipsthestateoftheprobequbit, current Ip = 1.0µA and ∆ /h ∼ 1 MHz. The small which can be easily detected. Thus, to perform QTS, ∆P was(cid:12)(cid:12)chPo(cid:12)(cid:12)sensothatthe traPnsitionrateΓ ofthe probe one measures the initial transition rate Γ ≡ |dP/dt|t=0, qubit was contained within the dc to 3 MHz bandwidth where P is the probability of observing the probe qubit of our slow control lines and to satisfy the incoherent in its initial state, as a function of ǫP, the probe energy inter-manifold tunneling condition. An on-chip tunable bias. Scanning ǫP and locating peaks in Γ allows one to coupler13 betweenthetwoqubitswasprogrammedtoat- map out the eigenspectrum of M↑ which is identical to tain an interqubit mutual inductance M = 2.0pH. The that of Hˆ if the compensation bias ǫ is set to −2J. resulting form for J in Eqn. 4 is J = M Ip Ip . With S comp (cid:12) Q(cid:12)(cid:12) P(cid:12) Note that errors in this compensation bias will skew the these parameters, the spectral gap in the(cid:12)M(cid:12)↓(cid:12) m(cid:12)anifold, energyspectrumofthe probedsystem. Theerrorsinthe as depicted in Fig. 1, satisfied 4J =4M Ip Ip ≫k T (cid:12) Q(cid:12)(cid:12) P(cid:12) B extracted energy spacings of the spectrum are bounded over the range of Ip encountered in the(cid:12)se(cid:12)e(cid:12)xpe(cid:12)riments. by the compensation bias error. Consequently,the(cid:12)(cid:12)reQw(cid:12)(cid:12)asnegligiblethermalactivationout To experimentally demonstrate QTS we take a target of the initial state |↓i |↓i to higher levels within M . Q P ↓ systemcomprisingasinglequbitQ,governedbyaHamil- With these parameters, the compensation bias is explic- tqounbiiatnPHˆwQit(hEsqtnre.n1gtwhitJhtPhro→ugQh)a. mWuetucaoluσˆpzleinttheerapcrtoiobne pitlliyed21bǫcyomadpσdˆzin,Qg=an−oJffsσˆezt,QM=I−pM∼(cid:12)(cid:12)I1Qpm(cid:12)(cid:12)(cid:12)(cid:12)IΦPp(cid:12)(cid:12)σtˆoz,tQheanflduxisbaiaps- to qubit Q. In this case, Hˆ → Hˆ and Cˆ → σˆ in (cid:12) P(cid:12) 0 S Q z,Q Φx of qubit Q. This com(cid:12)pen(cid:12)sation bias requires only Eqn. 2, yielding Q careful calibration of probe parameters. The experimental method for initialization and read- 1 HˆQ+P =HˆQ+HˆP +Jσˆz,Pσˆz,Q+ ǫcompσˆz,Q. (4) out is the same as the two-qubit cotunneling technique 2 described in Ref. 15. The experiments described herein ∆ is chosen to be small compared with all other terms differed from the cotunneling experiment in three re- P sotheeigenstatesoftheprobearetogoodapproximation gards: First, in QTS one intentionally sets ∆ ≪ ∆ , P Q |↑i and|↓i . Settingǫ =−2J thenyieldsaHamil- thus exploring an extreme limit of the mismatched tun- P P comp tonianoftheformgiveninEqn.3. Fortheparticularcase nelingenergyconfigurationdescribedinRef.15. Second, 3 (a) 0.8 0.4 0 −4 −2 0 2 4 6 8 10 (b) (c) 1 10 10 8 8 0.8 6 6 0.6 4 4 2 2 0.4 0 0 0.2 −2 −2 −1 0 1 −1 0 1 0 (d)6 FIG. 3. IQp(ΦcQcjj) vs ∆Q(ΦcQcjj) from the Q spectra for a range of settings of the control bias Φccjj. The data points 5 Q from thefitsin Fig. 2(d)are labelled with arrows. Thedata 4 also have horizontal error bars approximately the size of the symbols. Results have been fit to the rf SQUID model from 3 Ref. 14 (solid line) with the rf SQUID capacitance C as the 2 only free parameter. 1 0 pair of Gaussian peaks in order to locate the peak cen- −1.5 −1 −0.5 0 0.5 1 1.5 ters. ExamplemapsoftheinitialtransitionrateΓversus ǫ for a range of qubit Q flux biases around Φx =0 are P Q FIG. 2. (a) QTS tunneling rate data versus ǫp for ΦxQ = shown in Figs. 2(b) and (c) for two values of the control 0.1mΦ0 and ΦcQcjj = 0.637. Peaks in Γ are readily asso- flux bias ΦcQcjj (and therefore ∆Q). For clarity the cen- ciated with the processes Γ01 and Γ02 denoted in Fig. 1. ters of the Gaussian peaks have been indicated by white The centers of the peaks are found by fitting to a sum circles. The peak positions reveal the avoided crossing of two Gaussian peaks. (b) Qubit Q energy spectra ob- betweenlocalizedspinstates|↑i and|↓i . InFig.2(d), tained by QTS at control bias values: ΦcQcjj/Φ0 = 0.640; (c) wesummarizethedifferenceinpQrobeenerQgyδǫP between ΦcQcjj/Φ0 = 0.637. In both plots, the ordinate is the probe the two peaks as a function of ΦxQ. We then fit those re- bΦtrixQaanss∝eitniǫeoQrngyraapǫtpPeliΓe=din2t(cid:12)(cid:12)otIhPpqe(cid:12)(cid:12)uΦ(bΦxPitxQa,QnǫP.d)tTphhleaenaegb.rsacWyissshcaaitleiescitinhrdceliecflsautdexesnbtoihatees suusliItnnsgto∆orQdtheearnetdiog(cid:12)(cid:12)eIncQpsrp(cid:12)(cid:12)oseascscthrfrueecmekpooafurtarhmeQeHtTearSms.ilrteosnuilatsn,inwEeqhna.v5e, thecentersofpeaksinΓfoundbytheGaussianfits. Avoided repeated the measurements and analysis that led to crossings between two localized states, explicitly labelled in Fig. 2(d) for several values of Φccjj. By doing so, we (b) as |↑iQ and |↓iQ, are visible. (d) The difference in probe Q energy δǫP between thetwo peak centersas a function of Φxq generated maps of the qubit parameters ∆Q and (cid:12)IQp(cid:12) as for the datasets in (a) and (b). Results have been fit to the (cid:12) (cid:12) dispersion of the Hamiltonian in Eqn. 5 using ∆Q and (cid:12)IQp(cid:12) a function of ΦcQcjj. A plot of the relationship b(cid:12)etw(cid:12)een as free parameters. (cid:12) (cid:12) Ip and∆ isshowninFig.3. Thiscurveiscompletely (cid:12) Q(cid:12) Q (cid:12) (cid:12) d(cid:12)ete(cid:12)rmined by the rf SQUID inductance L and capaci- tance C of qubit Q. We have fit these results (solid line in QTSwe use relatively largeoffset biasesǫcomp/(cid:12)IQp(cid:12)= in Fig. 3) to a physical rf SQUID Hamiltonian (Eqn. 4 1mΦ0 inordertosatisfythecompensationconditio(cid:12)ne(cid:12)m- in Ref. 14) taking L= 355.5pH, as determined by inde- bodied in Eqn. 3. Third, whereas the dynamics studied pendentmeasurements,andusingC asafreeparameter. in Ref. 15 involved incoherent tunneling of the pair of The best fit returned C = 118±2 fF, which is a physi- qubits between localized initial and final spin states, in callyreasonablevalue,givenJosephsonjunctionsizesand QTS the final state can place qubit Q in a delocalized qubit wiring geometry. The single parameter fit models (superposition) state, as depicted in Fig. 1. thedatawell,implyingthatQTShascorrectlyextracted A scan of the initial transition rate Γ versus ǫ at the low energy spectrum of rf SQUID qubit Q. P Φx ∼ 0 and Φccjj = 0.637 is shown in Fig. 2(a). The QTScould,inprinciple,yieldmorethanjusttheeigen- Q Q data clearly show two distinct peaks. These peaks are spectrum of system S. In particular, there is signifi- readily identified as the processes Γ and Γ indicated cant information contained in the spectral weight of the 01 02 in Fig. 1. We fit such scans to a model composed of a peaks in Γ. For example, the spectral lines inferred 4 from Fig. 2(b) are less pronounced for the upper level ifolds of qualitatively different states. Transitions be- atΦx <0 andforthe lowerlevelatΦx >0. This is due tween the two manifolds are monitored as a function of Q Q to the proportionality of the initial transition rate Γ to the energy bias of one of the manifolds. Transition rate thesmalloverlapoftheinitialstateofQ(∼|↓i )withits peaks correspond to the presence of eigenstates in the Q final state (∼ |↑i ). Choosing ǫ = +2J, instead of target manifold under study at that energy bias. We Q comp −2J,yieldsasysteminwhichthestatesinM exchange validated this method by verifying that rf SQUID flux ↓ roles,thus yielding a new initial state |↑i |↓i . Repeat- qubit energy spectra measured in this manner are con- Q P ing the QTS experiment with this configuration should sistent with an rf SQUID Hamiltonian. QTS provided a thenswaptheregionsofhighandlowpeakvisibilityseen direct measurement of the first qubit’s energy splitting in Fig. 2(b). Thus the spectral weight contains informa- ∆Q that was three orders of magnitude larger than the tion about the wavefunction of the probed system. measurement bandwidth ∆P. While the demonstration in this paper was limited to Further information could be gleaned from the line- a single qubit, QTS is extensible to larger numbers of shapes of inter-manifold tunneling processes. We chose qubits and to other physical systems, provided one has to fit tunneling rate peaks to Gaussians as we had an- good control and readout of the probe and a method of ticipated that their lineshapes would be dominated by applying a compensation bias. We anticipate that QTS the incoherenttunneling ofthe slowprobequbitP,asin will be a valuable tool for studying mesoscale systems. Ref. 16. A detailed analysis of the physical mechanisms The authors would like to acknowledge: F. 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