Spectroscopy of a Cooper-Pair Box Coupled to a Two-Level System Via Charge and Critical Current
V. Zaretskey, B. Suri, S. Novikov, F. C. Wellstood, B. S. Palmer
aa r X i v : . [ qu a n t - ph ] M a y Spectroscopy of a Cooper-Pair Box Coupled to a Two-Level System Via Charge andCritical Current
V. Zaretskey,
1, 2
B. Suri,
1, 2
S. Novikov,
1, 2
F. C. Wellstood,
1, 3 and B. S. Palmer Department of Physics, University of Maryland, College Park, Maryland, 20742 Laboratory for Physical Sciences, College Park, Maryland, 20740 Joint Quantum Institute and Center for Nanophysics and Advanced Materials, College Park, Maryland, 20742 (Dated: June 25, 2018)We report on the quadrupling of the transition spectrum of an Al / AlO x / Al Cooper-pair box(CPB) charge qubit in the . − . frequency range. The qubit was coupled to a quasi-lumpedelement Al superconducting resonator and measured at a temperature of
25 mK . We obtainedgood matches between the observed spectrum and the spectra calculated from a model Hamiltoniancontaining two distinct low excitation energy two-level systems (TLS) coupled to the CPB. In ourmodel, each TLS has a charge that tunnels between two sites in a local potential and induces a changein the CPB critical current. By fitting the model to the spectrum, we have extracted microscopicparameters of the fluctuators including the well asymmetry, tunneling rate, and a surprisingly largefractional change ( − ) in the critical current (
12 nA ). This large change is consistent witha Josephson junction with a non-uniform tunnel barrier containing a few dominant conductionchannels and a TLS that modulates one of them.
PACS numbers: 03.67.Lx, 74.25.Sv, 42.50.Pq, 85.25.Cp
INTRODUCTION
Dissipation and dephasing from two-level systems(TLS) are a serious problem in many superconductingqubits. The aggregate effect of many weakly coupledfluctuators causes /f charge noise, broadband dielec-tric loss, and magnetic flux noise, as well as inhomoge-neous broadening and decreased measurement fidelity inqubits. An individual TLS quantum-coherently cou-pled to a qubit can typically be identified when it leads toa resolvable avoided crossing in the qubit spectrum. Suchavoided level crossings have been observed in phase, flux, charge, quantronium, and transmon qubits.While qubit performance is typically severely degradednear such an avoided crossing, strong qubit-TLSinteractions allow the microscopic details of the TLS tobe determined. Coherent coupling to a long-livedTLS also makes it possible to observe coherent oscilla-tions between a qubit and a TLS or use the TLS as aquantum memory. Two-level fluctuators in superconducting devices canbe classified into three types—charge, flux, or criticalcurrent—depending on the nature of the interaction withthe qubit. The microscopic origin of charge and criticalcurrent fluctuators is believed to be impurity ions suchas H or low coordination bonds in the amorphousdielectric used to build the devices. In phase and fluxqubits, it appears to be possible in principle but difficultin practice to identify the exact nature of the qubit-TLSinteraction. In contrast, detailed spectroscopy on chargequbits or Cooper-pair boxes (CPB) has enabled the iden-tification of discrete charge fluctuators. For example,Kim et al . found TLS’s that behaved as pure chargefluctuators. A moving charge could also modulate thecritical current if it was located in the tunnel barrier. Critical current fluctuations have been frequently seen in Josephson junction devices, but apparently notin Josephson based qubits. This may be due to the diffi-culty of conclusively distinguishing a critical current fluc-tuator from a charge fluctuator. Alternatively, the rela-tively small area of qubit junctions compared to that ofconventional junctions leads to far fewer total fluctua-tors and a corresponding decrease in the probability ofobserving one. Also, qubit measurements are typicallymade at less than
100 mK , where critical current fluctu-ators appear to be frozen out. Josephson junctions are afundamental building block of all superconducting qubitsand an understanding of the origin of critical current fluc-tuations is important for continued improvement of qubitperformance.In this paper we report on a CPB with an unusualspectrum that has multiple spectroscopic features dis-placed in both frequency and in gate charge instead ofan avoided level crossing. We find that the spectrum,including the curvature of the spectral features, can bemodeled well with a critical current fluctuator coupledto a CPB with an excitation energy for the fluctuatormuch less than the qubit energy. By fitting our model tothe spectrum we extract microscopic parameters for thefluctuators.
COOPER-PAIR BOX QUBIT AND READOUT
Our Cooper-pair box (CPB) consists of a supercon-ducting island connected to a superconducting reser-voir (ground) through two ultrasmall Josephson tunneljunctions (critical current I / and junction capacitance C j / ) [see Fig. 1(d)]. We can apply gate voltage V g to a capacitively coupled gate (capacitance C g to the is-land) to control the system’s electrostatic energy. Ap-plying flux Φ to the loop formed by the two junctions CPB T = 25 mKResonator P in P out C c Isolatorsdc BlockBiasTee Dir. Coupler V g f probe f pump Atten.LPFilters T = 4 K LNALNAAtten.Mixer f LO ADCRFIFAmp.Comp.IQ MixerDAC Φ ( d )( a ) ( c )(b) μ m 5 μ m μ m Figure 1. (a) Optical image of the lumped element resonator coupled to a CPW transmission line and surrounded by a perforatedground plane. Light regions are aluminum metalization and dark are sapphire substrate. (b) Optical image of the CPB locatedbetween the capacitor and ground plane. (c) Scanning electron image of the CPB. The twinned features are a consequence of thedouble-angle evaporation and the Josephson junctions are located at the overlap of the two patterns. (d) Simplified schematicof the experimental setup. The CPB is coupled through capacitor C c to a quasi-lumped element LC resonator. Its state isreadout via a coherent heterodyne measurement of a microwave power at frequency f probe transmitted through the device,amplified, mixed with a local oscillator at frequency f LO and finally digitized. The CPB transition frequency is controlled bythe gate voltage V g and an external magnetic flux Φ and its state is coherently manipulated using shaped microwave pulses atfrequency f pump . tunes the effective total critical current I via the re-lation I = I max cos ( π Φ / Φ ) where Φ = h/ e is themagnetic flux quantum.Neglecting quasiparticle states, the Hamiltonian de-scribing a CPB in the charge basis is given by ˆ H CP B = E c X n (2 n − n g ) | n i h n |− E J X n ( | n + 1 i h n | + | n i h n + 1 | ) (1)where E c = e / C Σ is the charging energy, E J = I Φ / π is the Josephson energy, C Σ = C j + C g is thetotal island capacitance to ground, n g = C g V g /e is thereduced gate voltage and | n i is the excess number ofCooper-pairs on the island. For E c > E J the systemis highly anharmonic and only a few charge states areneeded to accurately describe the lowest energy states.For charge qubits with E c ≫ E J and . < n g < . , theHamiltonian can be reduced to H CP B = (cid:18) E c (0 − n g ) − E J / − E J / E c (2 − n g ) (cid:19) (2)which yields the excited state transition energy ~ ω CP B ( n g ) = q (4 E c (1 − n g )) + E J . Near the chargedegeneracy point n g = 1 the transition energy variesparabolically as ~ ω CP B ( n g ) ≈ E J + 8 E c (1 − n g ) /E J .To measure the state of the qubit, we coupled our qubitto a thin-film quasi-lumped element LC resonator [seeFig. 1(d)] that was in turn weakly coupled to a microwave transmission line patterned on the sample chip. To readout the state of the qubit, we apply microwave power atthe resonance frequency of the resonator and record thetransmitted microwave signal S . This is a dispersivereadout in which the qubit state modulates the resonancefrequency of the resonator. In the case of weak qubit-resonator coupling g and large detuning ∆ = ω CP B − ω r the combined CPB-resonator system Hamiltonian isapproximately ˆ H = ~ (cid:18) ω r + g ∆ σ z (cid:19) (cid:18) a † a + 12 (cid:19) + ~ ω CP B σ z (3)where ~ g = (2 E c C c /e ) p ~ ω r / C is the strength of thequbit-resonator coupling energy, C c is the coupling ca-pacitance between the resonator and the island of theCPB, C is the capacitance of the LC resonator, ω r isthe resonance frequency, a † a is the number operator forexcitations in the resonator, and σ z is the Pauli spin op-erator. This Jaynes-Cummings Hamiltonian yields tran-sitions in which the bare resonator frequency ω r is dis-persively shifted by χ = ± g / ∆ depending on the stateof the qubit. If χ < Γ , where Γ is the resonator linewidth,the average phase of the transmitted signal at ω = ω r islinearly dependent on the excited state occupation prob-ability. On the other hand if χ > Γ , then the in-phaseor quadrature transmitted voltage is proportional to theexcited state occupation probability. Additional complications can arise when the qubit andresonator are coupled to another quantum system, suchas a TLS. If multiple energy levels in the combined system super-conductorsuper-conductor dielectricx L x R E L E R TLS potentialenergy U(x)electric fieldT LR Figure 2. Simplified diagram of the potential energy of acharged TLS in an insulating tunnel barrier. The fluctuatorcan be localized at positions x R or x L with correspondingenergies E R or E L and can tunnel between them with en-ergy T LR . Additionally the Josephson energy E J of the CPBdepends on whether the fluctuator is at x R or x L . have similar detunings from the resonator, the effectivedispersive shift χ eff will have a contribution from eachlevel. Qubit state readout can still be performed as de-scribed for the two level case, but the sensitivity to aparticular state depends on the choice of resonator probefrequency. As we see below, this is our situation.
CHARGE AND CRITICAL CURRENT TLSMODEL
To include the effects on a CPB produced by a com-bined charge and critical current fluctuator, we expandon the charge defect model previously reported by Z.Kim, et al . We assume the fluctuator acts as a two-level system in which a point charge in the tunnel barriercan tunnel between two potential well minima. In theTLS position basis the fluctuator Hamiltonian is givenby H T LS = (cid:18) E L T LR T LR E R (cid:19) (4) where E L and E R are energies of the charge in theleft and right position states and T LR is the tunnel-ing matrix element [see Fig. 2]. For an isolated fluc-tuator the excited state transition energy is given by ~ ω T LS = q ( E R − E L ) + 4 T LR .The charge coupling between the CPB and TLS orig-inates from changes in the electrostatic potential whenthe defect tunnels between its two sites. Using Green’sreciprocation theorem the change in the induced po-larization charge on the island of the CPB when thefluctuator tunnels from the left to the right well is ∆ Q pi = Q T LS ( x R − x L ) cos ( η ) /d where Q T LS is theTLS charge, η is the angle the TLS displacement vectormakes relative to the electric field in the junction and d ≈ is the thickness of the tunnel junction. Forfixed net charge on the island this in turn results in achange in the electrostatic potential of the island givenby ∆ V i = Q T LS C Σ ( x R − x L ) cos ( η ) d . (5)Accounting for the electrostatic charging energy and thework done by the gate voltage source when the pointcharge moves, the coupling Hamiltonian is given by ˆ H CP B − T LS = 2 E c (cid:16) N − n g (cid:17) Q T LS e ˆ x cos ( η ) d (6)where ˆ N is the CPB charge operator that counts thenumber of excess Cooper-pairs on the island and ˆ x is theTLS position operator. Combining Eqs. (2), (4) and (6) we can write the totalHamiltonian for a CPB coupled to a single charge fluc-tuator as ˆ H = ˆ H CP B + ˆ H T LS + ˆ H CP B − T LS . In blockmatrix form this becomes H = (cid:18) H L TT H R (cid:19) (7)where T = T LR I , I is the × identity matrix, and H L and H R are the CPB Hamiltonian with the TLS ineither the left or right well. If we assume E L = 0 then H L = H CP B as given by Eq. (2) and H R = (cid:18) E c (0 − n g ) + E int (0 − n g ) + E R − E J / − E J / E c (2 − n g ) + E int (2 − n g ) + E R (cid:19) (8)where E int = 2 E c Q T LS ( x R − x L ) cos ( η ) /ed sets the en-ergy scale for the charge coupled interaction between thefluctuator and the CPB.If the TLS is in the junction tunnel barrier, it can alsomodulate the critical current depending on its position. This coupling can be accounted for by making the substi-tution E J → E J + ∆ E J / in H L and E J → E J − ∆ E J / in H R .Numerically diagonalizing the resulting × Hamil-tonian H , we find the energy levels and the transitionfrequencies from the ground state to the excited statesof the system. An avoided crossing occurs if the excitedstate of the TLS is resonant with the first excited stateof the CPB at some value of the gate voltage n g . However if the TLS excited state energy lies below theCPB transition minimum the CPB spectrum is twinned,with one parabola corresponding primarily to the excitedstate of the CPB and the other to a joint excitationof the CPB and the TLS. Considered individually, eachparabola bears a strong resemblance to the spectrum of aTLS-free CPB. When the tunneling energy T LR is smallwe can identify the qualitative effects of each parameteron the twinned parabolas. ∆ E J creates an offset alongthe frequency axis and a change in the effective curva-ture while E int creates an offset along the n g axis and“tilts” the parabolas. E R also creates an offset along thefrequency axis that adds to or subtracts from the effectof ∆ E J . Finally T LR determines the size of any avoidedcrossings that are present in the spectrum and determinesthe transition rate induced by a gate perturbation be- tween the ground state and excited states involving theTLS.We can further extend the model by considering the ef-fect of two critical current fluctuators. This is motivatedby the observation of quadrupling of the spectral lines inour data which can’t be explained by the presence of asingle TLS. The total Hamiltonian for a CPB coupled totwo fluctuators in block matrix form is H = H LL T T T T H RL T T T T H LR T T T T H RR (9)where T = T LR, I , T = T LR, I , and T = T I where T accounts for any possible TLS-TLS coupling and theindices refer to the first or second TLS. H ij with i, j ∈{ L, R } is the CPB Hamiltonian with the respective TLSin either the left or right well. For example, H RL is givenby H RL = (cid:18) E c (0 − n g ) + E int, (0 − n g ) + E R, − ( E J − ∆ E J, / E J, / / − ( E J − ∆ E J, / E J, / / E c (2 − n g ) + E int, (2 − n g ) + E R, (cid:19) (10)and H LR has the respective indices swapped. H RR in-cludes the contribution of both TLS and in additionpresent on the diagonal is a CPB mediated TLS-TLSinteraction term of the form E int, E int, / E c . EXPERIMENTAL DETAILS
We fabricated a thin-film lumped-element supercon-ducting microwave resonator using standard photolithog-raphy and lift-off techniques. It was made from a
100 nm thick film of thermally evaporated Al on a c-plane sap-phire wafer that was patterned into a meander inductor( L ≈ ) and interdigital capacitor ( C ≈
400 fF ) cou-pled to a coplanar waveguide transmission line [see Fig.1(a,b)]. The resonance frequency was ω r / π = 5 .
47 GHz with loaded quality factor Q L = 35 , , external qual-ity factor Q e = 47 , , and internal quality factor Q i = 147 , .The CPB was subsequently defined by e-beam lithog-raphy and deposited using double-angle evaporation andthermal oxidation of aluminum to create the
350 nm ×
150 nm
Josephson tunnel junctions [see Fig. 1(c)]. For the e-beam lithography we used a bilayer stack ofMMA(8.5)MAA copolymer and ZEP520A e-beam resistto facilitate lift-off and reduce proximity exposure dur-ing writing. A
30 nm thick Al island and
50 nm thick Alleads were deposited in an e-beam evaporator. As dis-cussed below, measurements of the CPB yielded E c /h inthe . − . range and we tuned E J /h from to E maxJ /h = 7 .
33 GHz .The chip was enclosed in a rf-tight Cu box that was an-chored to the mixing chamber of an Oxford Instrumentsmodel 100 dilution refrigerator at
25 mK . Connectionsto the chip were made with Al wirebonds. We used coldattenuators on the input microwave line and isolators onthe output line to filter thermal noise from higher tem-peratures [see Fig. 1(d)]. A filtered dc bias voltage linewas coupled to the input line using a bias tee before thedevice and a dc block was placed after the sample box.For spectroscopic measurements the resonator wasprobed with a weak continuous microwave signal whilea second pump tone was applied to excite the qubit. Thetransmitted microwave signal at the probe frequency wasamplified with a HEMT amplifier sitting in the He bath[see Fig. 1(d)]. We implemented a coherent heterodynesetup to record the phase and amplitude of the transmit-ted probe signal at
500 ns time steps. After the HEMT,the signal was further amplified at room temperature,mixed with a local oscillator tone to an intermediate fre-quency of and then digitally sampled at a typicalsampling rate of
20 MSa / s . A reference tone split offfrom the probe signal was directly mixed and digitallysampled. Both signals passed though a second stage ofdigital demodulation on a computer to extract the ampli-tude and phase. All components were locked to a
10 MHz
Rb atomic clock. Both the probe and pump tone pow-ers were optimized for ease of data acquisition while alsominimally disturbing the qubit. The probe tone powerwas calibrated via the ac Stark shift. During measure-ment of the qubit state, the probe tone power was set topopulate the resonator with an average ¯ n = 25 photonswhile the concurrent pump tone power was slightly abovethat needed to saturate the CPB transitions. SPECTRUM CHARACTERIZATION
We measured the transition spectrum of the qubit byrecording the transmitted microwave probe signal whilesweeping the dc gate voltage and stepping the frequencyof the second microwave pump signal. Fig. 4(d) shows aplot of the transmitted probe signal amplitude as func-tions of n g and pump frequency f pump with E J tunednear E maxJ . Several unexpected and anomalous featuresare evident. Rather than a single parabola, we ob-served two parabolas with varying curvatures offset by ≈ .
25 GHz in frequency and ≈ . e in charge. Thisspectral structure was stable over the course of fourmonths and persisted as we tuned the transition fre-quency from . − . . Close examination of thefigure reveals sections of two more quite weak parabo-las. A notable change in the spectrum occurred when wetuned E J to bring the transition frequency below thatof the resonator. As seen in Fig. 4(a), four parabolasare clearly visible with the stronger new pair displaced ≈ .
40 GHz below the original two. We note two addi-tional anomalies we observed. First, a “dead zone” waspresent between ≈ . − . where no spectrum wasvisible. Second, only half of the spectral parabolas—one from each pair—were visible when measured with apulsed probe readout. For instance, in Fig. 4(d) bothparabolas were present when we used a continuous mea-surement but only the bottom parabola was visible whenwe used a pulsed measurement at a fixed gate voltage n g = 1 . Some clues about the nature of the fluctuator are ev-ident from an examination of the spectrum. The fre-quency offset between the two parabolas in Fig. 4(d)could be caused by a flux fluctuator that modulates E J .However such a fluctuator’s effect on E J (Φ) would beminimal when the applied flux is near zero and increaseas E J is reduced by an external flux bias. As discussedbelow, this is the opposite of the behavior we observed.Another argument against a simple flux fluctuator (suchas a vortex) or a simple charge fluctuator is that thereare correlated shifts in n g and frequency between theparabolas. In contrast, the observed offsets and curva-ture changes are consistent with a two-level system thatis coupled to the CPB via both charge and critical cur-rent.Several factors indicate that the fluctuator is coher-ently coupled to the CPB. An incoherently coupled lowfrequency critical current fluctuator would be expectedto produce twinning in the resonator dispersive shift χ inaddition to twinning of the spectral lines. This twinningof the dispersive shift would be manifest either as twin-ning of the ground state resonator frequency or broaden- | S 21 | ( d B ) fr (GHz) Figure 3. Plot of the ratio of the transmitted output volt-age to input voltage ( | S | ) versus frequency for two differentpreparations of the qubit state. The filled black squares andblack curve show the transmission with the qubit biased at n g = 1 and driven to a mixed ground and excited state. Thetwo dips are resonances at ω ′ r ± χ eff. The filled red circlesand red curve show the transmission with the qubit in theground state and far detuned from the resonator at n g = 0 .The single dip is the bare resonator frequency at ω r . ing of the resonator linewidth. We didn’t observe eitherof these effects. Instead we observed an effective disper-sive shift χ eff consistent with contributions from multiplelevels [see Fig. 3]. We determined the effective dispersiveshift χ eff and effective resonator frequency ω ′ r by record-ing the resonator response with the qubit in the groundand excited states. We also measured the bare resonatorfrequency ω r by far detuning the qubit from the resonatorby biasing at n g = 0 [see Fig. 3]. As expected ω ′ r = ω r and the effective dispersive shift χ eff differed between theexcited states corresponding to the various parabolas. Fi-nally, in previous cases of incoherent fluctuator couplingwe found that the qubit was rendered inoperable. Yet in this case we were able to measure qubit excitedstate lifetimes T in the − μ s range and record Rabioscillations for all of the parabolas.The strength of the qubit-TLS coupling indicates thatthe TLS was located close to the CPB Josephson junc-tions, either in the tunnel barrier itself or on the surfaceof the CPB island. Furthermore, we note that the spectrawere e periodic in n g . This is the expected periodicityfor a charge fluctuator that is in the tunnel barrier , andsuch a fluctuator would need to be in the AlO x tunnelbarrier to produce a critical current change. FITTING AND DISCUSSION
We first fit the single TLS model to the measured spec-trum at several different external flux bias values. In our n g f pu m p ( GH z ) (dB) −22−21.5−21−20.5−20−19.5−19 n g f pu m p ( GH z ) (°)−160−155−150−145 n g f pu m p ( GH z ) (dB)−23−22−21−20−19 n g f pu m p ( GH z ) (dB)−21.5−21−20.5−20−19.5−19 (a) (b)(c) (d) Figure 4. Measured transition spectrum of the CPB at four different external magnetic flux Φ bias values. The red lines arethe theoretical spectrum using a Hamiltonian consisting of a single charged two-level fluctuator coupled to a CPB. In (a) and(b) (data sets .
39 GHz is a charge noiseartifact.Table I. Fit parameters for the model of a single two-level fluc-tuator coupled to a CPB. The corresponding spectra are plot-ted in Fig. 4. E c and E J are the CPB charging and Joseph-son energies. E R is the TLS potential energy well asymmetry, E int is the charge coupling between the TLS and the CPB, ∆ E J is the change in the CPB Josephson energy when theTLS tunnels between wells and T LR is the TLS tunneling rate.Data set E c /h (GHz) . . . . E J /h (GHz) .
64 4 .
16 5 .
93 6 . E J /h (GHz) .
50 1 .
54 1 .
84 2 . E R /h (GHz) .
62 0 .
62 0 .
62 0 . E int /h (GHz) .
35 0 .
35 0 .
35 0 . T LR /h (GHz) .
01 0 .
01 0 .
06 0 . device E c is comparable to E J , so we needed to include charge states in the CPB Hamiltonian block matrices tobetter approximate the CPB behavior. We initially fo-cused only on the top two parabolas to better understandthe effects of the model parameters and the relation be-tween the one and two TLS models. The solid red curvesin Fig. 4 show the predicted spectrum for those parabo-las and the fits look reasonable.The optimal fit parameters are summarized in Table Iand give reasonable results for all values of the flux bias.Individual fit parameters could typically be varied by ap-proximately 20% while maintaining a reasonable lookingfit. The large uncertainty is partly due to the fact thatthe frequency offset between the twinned parabolas arisesfrom both ∆ E J and E R . Additionally the model pre-dicts avoided crossings which were too small to resolve, Table II. Fit parameters for the model of two two-level fluctua-tors coupled to a CPB. The corresponding spectra are plottedin Fig. 5. E c and E J are the CPB charging and Josephsonenergies. E R, and E R, are the potential energy well asym-metries for the first and second TLS. Similarly E int, and E int, are the charge coupling between the first and the sec-ond TLS and the CPB and ∆ E J, and ∆ E J, are the changesin the CPB Josephson energy when the TLS tunnel betweentheir respective wells. T LR, and T LR, are the TLS tunnelingrates while T is the TLS-TLS coupling strength.Data set E c /h (GHz) . . E J /h (GHz) .
79 3 . E J, /h (GHz) .
36 1 . E R, /h (GHz) .
62 0 . E int, /h (GHz) − . − . T LR, /h (GHz) .
00 0 . E J, /h (GHz) − . − . E R, /h (GHz) − . − . E int, /h (GHz) .
13 0 . T LR, /h (GHz) .
04 0 . T /h (GHz) .
04 0 . and this meant we could place an upper bound on theTLS tunneling strength T LR . We note that the data setswith different applied flux only require E J and ∆ E J tobe adjusted, which is consistent with changing flux bias,except for a change in T LR when the qubit is tuned frombelow to above the resonator ω r . The model also predictsa nearly flat TLS spectral line in the − range,roughly equal to the transition frequency of the isolatedfluctuator. We didn’t observe such a feature, perhaps be-cause our resonator perturbative measurement techniquewas insensitive to a low frequency TLS-only transition.It is important to consider if other models can explainour observations. We can eliminate a coherently coupledflux fluctuator using the same reasoning used to excludethe incoherently coupled case. In particular this suggeststhat the unusual spectrum isn’t due to coupling to a mov-ing vortex. Another possibility is that the data could befit by a charged fluctuator with ∆ E J = 0 . Such modelwould predict a large “tilt” of the parabolas that disagreeswith data covering a wider n g and frequency span.We also fit the entire spectrum of four parabolas tothe two TLS model [see Eqs. 9 and 10]. The solid redcurves in Fig. 5 show the best fit spectrum superposedon the data. The optimal fit parameters are summa-rized in Table II. The vertical lines at n g ≈ ± . and n g ≈ ± . are due to the resonant crossing betweenthe qubit parabolas and the resonator line at ω r . Al-though the fits are reasonable and capture all of the ma-jor features, the fit parameters contain one surprise. Ifwe assume two independent fluctuators, the simplest as-sumption in light of the strong shielding of electric fieldsin the dielectric of the Josephson junction by the super-conducting electrodes, then we would expect T LR, = 0 and T LR, = 0 while T = 0 . However our fit yields T LR, = 0 while T LR, = 0 and T = 0 which suggestscoupled TLS’s or more complicated microscopic behav-ior. Furthermore, we note that several of the TLS pa-rameters, such as the charge coupling E int, , change val-ues when switching from the single to the double TLSmodel. This indicates that the two TLS model is neededto explain the full quadrupled spectrum and suggests thatthere is significant interaction between the TLS’s.There are some noteworthy implications from themagnitude of the fit parameters. First, we note that ∆ E J /E J ≈ . The large relative size of ∆ E J to E J suggests that the junction tunnel barrier is non-uniform with a few dominant conduction channels andthat the TLS is located near and modulates one of thesechannels. Second, the TLS tunneling matrix element T LR . .
06 GHz is small compared to the other ener-gies in the system, indicating that the TLS is tunnelingbetween fairly well isolated sites. We can also place alower bound on T LR by noting that for T LR < .
01 GHz the spectra would be too faint to observe. If the excitedstate of such a TLS were resonant with the first excitedstate of the CPB, the resulting avoided crossing would bevery small and difficult to resolve. Our extracted tunnel-ing matrix element values are also significantly smallerthan those reported by Z. Kim, et al ., which were inthe −
13 GHz range. There is a similar relation be-tween the range of well asymmetry values extracted byus, E R = 0 . − . , and those reported by Z. Kim, et al ., E R = 7 −
39 GHz . Assuming a TLS charge of Q T LS = e and a tunnel barrier thickness of d = 1 nm ,we estimate the maximum hopping distance of the defectat . − . Å. This is in agreement with the bounds of . − . Å found by Z. Kim, et al .Discrete critical current fluctuators have been re-ported in current biased Josephson junctions, identi-fied via either a random telegraph signal in the volt-age time trace or a signature Lorentzian bump in thenoise spectrum.
One way we can compare ourTLS’s to others is to calculate the effective defect area A eff given by A eff = (∆ E J /E J ) A j . For our devicefind A eff ≈ ,
000 nm where A j = 350 ×
150 nm is the junction area. This value is much larger thanthe A eff = 1 − reported in similar junctions, the A eff ≈
600 nm seen in larger area junctions, orthe A eff = 72 nm found in similar area high- T c su-perconductor grain boundary junctions. On the otherhand, the absolute value of the critical current fluctua-tion ∆ I ≈ we observed is close to that reportedin both similar area ( ∆ I = 9 . ) and larger junc-tions ( ∆ I ≈ ). One notable difference that mightaccount for some of these discrepancies is that the crit-ical current density of our sample (
23 A / cm ) is smallerby an order of magnitude or more than the referencedsamples. If we assume that the conductance of a tunnel-ing channel is similar between the various devices, this isconsistent with a small number of tunneling hot spots inour junction. n g f pu m p ( GH z ) (dB) −22−21.5−21−20.5−20−19.5−19 n g f pu m p ( GH z ) (°)−160−155−150−145 (a) (b) Figure 5. Measured transition spectrum of the CPB at two different external magnetic flux Φ bias values. The red lines arethe theoretical spectrum using a Hamiltonian consisting of two charged two-level fluctuators coupled to a CPB. Plots (a) and(b) (data sets COMMENTS AND CONCLUSION
The longitudinal relaxation rate of a TLS in anamorphous solid is expected to be limited by /T = α ~ ω T LS T LR coth ( ~ ω T LS / k B T ) where T is temperatureand α is a material dependent constant. From the re-sults of Z. Kim, et al . we estimate /α ≈ μ s · GHz · h for the dielectric AlO x in the tunnel junction barrier.Our fit values then place an upper bound on the TLSexcited state lifetime of T . . This bound is con-sistent with a relatively long TLS lifetime and with ourqubit T ≈ − μ s . The excited states of the systemare mixtures of pure CPB and TLS excited states, so thedecay rate is a weighted average of the pure CPB andTLS decay rates. For example, according to our fits tothe model at n g = 1 the lower parabola in Fig. 4(d)is composed of a CPB excitation and an jointCPB plus TLS excitation while the upper parabola is an
CPB excitation and a joint CPB plus TLS ex-citation. Only when both the qubit and TLS decay ratesare small, as is our case, will the system decay time belong in both parabolas.Finally it’s worthwhile to speculate why this behaviorwas observed in our sample. In order to observe spectraltwinning rather than an avoided crossing, the TLS needsto be coupled to the qubit but have a transition frequencyless than E J /h . That this occurred is a statistical coin-cidence. Observing two such defects in the same sampleis less likely, and the TLS fit parameters suggest they arecorrelated. Furthermore, we are biased in selecting sam-ples for detailed study that have especially conspicuousfeatures, such as large avoided crossings or anomalousspectra, and the parameter values of such samples arelikely to be somewhat unusual. While our simple model provides a good fit to therecorded spectrum, it leaves other questions unanswered.The resonator wasn’t included in the model but some ofour observations suggest that it may produce significanteffects on the spectrum. Inclusion of the resonator in themodel would allow a theoretical calculation of the ex-pected dispersive shift and a comparison with the data.A more complete model may also elucidate the role, ifany, the resonator played in the the large difference inthe visibility of the different parabolas when the qubitwas tuned from below to above the resonator ω r or the“dead zone” we observed between ≈ . − . whereno spectrum was visible. Perhaps the most puzzling fea-ture was that half of the spectral parabolas weren’t vis-ible when measured with a pulsed probe. Unfortunatelyadditional data on this issue wasn’t obtained.In conclusion we have examined the transition spec-trum of a CPB that had an anomalous quadrupling ofthe spectral lines. A microscopic model of one or twocharged critical current fluctuators coupled to a CPB wasused to fit the spectrum. The fits were in good agreementwith the data, reproduced the key features in the spec-trum, and allowed us to extract microscopic parametersfor the TLS’s. Our tunneling terms were much smallerthan those reported by Z. Kim, et al . in their measure-ments of avoided crossings. Finally, the large fractionalchange ∆ E J /E J of − suggests that the tunnelbarrier is non-uniform in thickness with the TLS hop-ping blocking a dominant conduction channel. ACKNOWLEDGMENTS
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