Measurement crosstalk between two phase qubits coupled by a coplanar waveguide
Fabio Altomare, Katarina Cicak, Mika A. Sillanpää, Michael S. Allman, Adam J. Sirois, Dale Li, Jae I. Park, Joshua A. Strong, John D. Teufel, Jed D. Whittaker, Raymond W. Simmonds
MMeasurement crosstalk between two phase qubits coupled by a coplanar waveguide
Fabio Altomare , Katarina Cicak , Mika A. Sillanp¨a¨a , Michael S. Allman , , Adam J. Sirois , , Dale Li , JaeI. Park , Joshua A. Strong , , John D. Teufel , Jed D. Whittaker , , and Raymond W. Simmonds National Institute of Standards and Technology, 325 Broadway, Boulder CO 80305, USA Helsinki University of Technology, Espoo P.O. Box 2200 FIN-02015 HUT Finland University of Colorado, 2000 Colorado Ave, Boulder, CO 80309-0390, USA
We analyze the measurement crosstalk between two flux-biased phase qubits coupled by a resonantcoplanar waveguide cavity. After the first qubit is measured, the superconducting phase can undergodamped oscillations resulting in an a.c. voltage that produces a frequency chirped noise signal whosefrequency crosses that of the cavity. We show experimentally that the coplanar waveguide cavity actsas a bandpass filter that can significantly reduce the crosstalk signal seen by the second qubit whenits frequency is far from the cavity’s resonant frequency. We present a simple classical descriptionof the qubit behavior that agrees well with the experimental data. These results suggest thatmeasurement crosstalk between superconducting phase qubits can be reduced by use of linear orpossibly nonlinear resonant cavities as coupling elements.
In recent years, much effort has been spent fabricatingsuperconducting circuits with embedded Josephson junc-tions (JJs) as a promising platform for developing a quan-tum computer. In particular, superconducting qubits,broadly classified as charge, flux, and phase , have re-cently achieved coherence time longer than 7 µ s , andsingle shot visibility close to 90% . Various schemes havebeen devised to couple several qubits in a more complexcircuit: coupling through JJs , inductive coupling , ca-pacitive coupling and coupling through a resonant copla-nar waveguide (CPW) cavity have all been achieved.The quantum mechanical nature of CPW cavities hasalso been demonstrated by generating arbitrary Fockstates through the use of a coupled phase qubit . Ad-ditionally, a protocol for the preparation of arbitrary en-tangled states of two phase qubits and a CPW cavity hasbeen developed .Here, we will focus on two flux-biased phase qubits coupled through a CPW cavity and will show howthe CPW cavity plays a crucial role in the reduction ofmeasurement crosstalk .Measurement crosstalk in coupled flux-biased phasequbits results from their unique formation in ametastable well of a double-well potential and the mea-surement scheme used for determining the qubit state.The schematic of a typical phase qubit circuit is shownin Fig. 1(a). The phase qubit is essentially a reso-nant LC circuit in parallel with a Josephson inductance( L J = h/ e/ (2 πI (Φ)) = Φ / (2 πI cos(2 π Φ / Φ ))), whereΦ is th flux quantum, Φ is the flux applied to the circuitloop and I the JJ critical current. The potential energyas a function of the superconducting phase difference ( ϕ )across the JJ is presented in Fig. 1(b) for a particular fluxbias. With a relatively strong anharmonicity, the quan-tized energy levels in the left well can be individuallyaddressed with transition frequencies between the lowestquantized state ( | (cid:105) ) and the first excited state ( | (cid:105) ) inthe microwave region. The occupation probability of thequbit’s first excited state is measured by applying a fastflux pulse or measure pulse (MP) that tilts the well for afew nanoseconds so that only the | (cid:105) state can tunnel out to the right well, as shown in Fig. 1(c). Because thephase qubit is formed in a metastable region of the po-tential, its ’ground state’ energy is naturally higher thanthe global ground state of the system. Upon tunneling,this additional energy is released so that the phase ofthe qubit (classically) undergoes large oscillations in thedeeper right well. Following the MP the flux is adjustedto form a symmetric double-well potential. After tens ofmicroseconds, when the system has relaxed due to dissi-pation, a DC SQUID detects the flux in the qubit loop,allowing us to discriminate between the two circulatingcurrent states where the phase either relaxed in the leftwell or the right well. These correspond to the qubitstates | (cid:105) if the qubit did not tunnel and | (cid:105) if the qubitdid tunnel.The oscillations of the qubit phase in the right wellproduce an oscillating voltage across the JJ with a rel-atively large size, representing roughly hundreds of mi-crowave quanta. This voltage signal can excite any de-vices coupled to the qubit when their resonant frequencymatches that of the oscillation. Because the right well po-tential is weakly anharmonic, as the amplitude decreasesdue to dissipation the frequency rises, producing a chirpcrosstalk signal spanning over 10 GHz. With direct ca-pacitive coupling between two phase qubits , this pro-cess results in the second qubit being excited over itsmetastable barrier whenever the first qubit is measuredin the | (cid:105) state. Due to the nonlinear dynamics of thesystem, there is a finite amount of time required (severalnanoseconds) for the second qubit to be excited after thetunneling of the first qubit. It was found that the lossof qubit information due to measurement crosstalk couldbe avoided by measuring the two qubits simultaneously (within ∼ . Unfortunately, theexcess energy released by a metastable phase qubit dur-ing a tunnelling event is unavoidable, so that measure-ment crosstalk can be a serious problem for systems withmany coupled qubits.The situation is quite different if two phase qubits are a r X i v : . [ c ond - m a t . s up r- c on ] F e b Qubit 1 Qubit 2CPW a) b) c) FIG. 1. (a) Equivalent electrical circuit for two flux-biasedphase qubits coupled to a CPW cavity (modelled as a lumpedelement harmonic oscillator). C i is the total i − qubit (or CPWcavity) capacitance, L i the geometrical inductance, L j,i theJosephson inductance of the JJ, R i models the dissipation inthe system. (b) U ( ϕ, ϕ e ) is the potential energy of the phasequbit as function of superconducting phase difference ϕ acrossthe JJ and the dimensionless external flux bias ϕ e = Φ2 π/ Φ .∆ U ( ϕ e ) is the difference between the local potential maxi-mum and the local potential minimum in the left well at theflux bias ϕ e . (c) During the MP, the potential barrier ∆ U ( ϕ e )between the two wells is lowered for a few nanoseconds allow-ing the | (cid:105) state to tunnel into the right well where it will(classically) oscillate and lose energy due to the dissipation. coupled through a CPW cavity. Here we show both ex-perimentally and using the classical description of Ref. 13that the CPW cavity acts as a bandpass filter, and thesecond device is excited only if it is in resonance with theCPW cavity. This suggests a simple and effective way toreduce the measurement crosstalk between coupled de-vices and has recently been implemented .The device we use has already been described else-where , and consists of two flux-biased phase qubits ca-pacitively coupled through a 7 mm open-ended copla-nar waveguide whose half-wave resonant mode frequencyis ω r / π ≈ . . The CPW cavity, with characteris-tic impedance Z r ≈
50 Ω close to resonance, is equiva-lent to a lumped element resonator with L r = 2 Z r /πω r ≈ pH , C r = π/ ω r Z r ≈ . pF . The lifetime of an ex-citation in the CPW cavity is T r ≈ µ s, which yields R r = T r /C r = 1 . M Ω. The qubit’s parameters are: L = 690 pH, C = 0 . I = 0 . µ A, T = 170 ns, R = 240 k Ω, and L = 690 pH, C = 0 . I =0 . µ A, T = 70 ns, R = 100 k Ω, with C x = 6 . ≈ . = 0 . c , and for the second at a fluxΦ = 0 . c . For each qubit, Φ ci is the critical flux atwhich the left well of Fig. 1(b) disappears.For our experiment, we initially determine theoptimal ’simultaneous’ timing between the two MPs thattakes into account the different cabling and instrumentaldelays from the room-temperature equipment to thecold devices. Then, as a function of the flux applied tothe two qubits, we measure the tunneling probabilityfor the second (first) qubit after we purposely inducea tunneling event in the first (second) qubit. Theresults are shown in Fig. 2(a,c). The probability offinding the second (first) qubit in the excited state asa result of measurement crosstalk is significant only ina region around ϕ /ϕ c = Φ / Φ c ≈ Φ / Φ c ∼ . ϕ /ϕ c = Φ / Φ c ≈ Φ / Φ c ∼ .
82) where the resonantfrequency of the second (first) qubit is close to the CPWcavity frequency.To provide a qualitative description of these results,we write the Lagrangian for the two qubits coupledthrough a CPW cavity (Fig. 1 (a)) as L = 12 C V + 12 C V + 12 C r V r + 12 C x ( V − V r ) +12 C x ( V r − V ) − U ( φ ) − U ( φ ) − Φ r L r (1)where Φ r / L r = L r I r / U i = E L,i (cid:20)
12 ( ϕ i − ϕ e,i ) − L i L J,i cos ϕ i (cid:21) (2)with E L,i = (Φ / π ) /L i . The dimensionless flux, ϕ e,i =2 π Φ i / Φ , determines the profile of the potential energyfor the qubit. Using the Josephson relations (to substi-tute V i with ϕ i ) and solving for the equation of motion,after including the damping term, we obtain : C ¨ ϕ + C x ( ¨ ϕ − ¨ ϕ r ) = − ∂U ∂ϕ − R ˙ ϕ C r ¨ ϕ r + C x ( ¨ ϕ − ¨ ϕ r ) + C x ( ¨ ϕ r − ¨ ϕ ) = − ϕ r L r − ˙ ϕ r R r C ¨ ϕ + C x ( ¨ ϕ r − ¨ ϕ ) = − ∂U ∂ϕ − R ˙ ϕ (3)where ϕ r = 2 π Φ r / Φ . As can be seen from Eq. 3, thetreatment of the qubits and the CPW cavity is fully clas-sical. The initial conditions for the solution of this sys-tem of differential equations are described below. Thesecond qubit and the oscillator begin with zero kineticenergy ( ˙ ϕ = ˙ ϕ r = 0) and have zero potential energy;zero energy is defined at the bottom of the left well. To a)b) c)d) xx FIG. 2. (Color) Measurement crosstalk: (a) Experimentaltunneling probability for qubit 2, after qubit 1 has alreadytunneled as function of the (dimensionless) flux applied tothe qubits. The left ordinate displays the resonant frequencyas measured from the qubit spectroscopy. The right ordinatedisplays the ratio between the applied flux and the critical fluxfor qubit 2. (b) Simulation: ratio between the maximum en-ergy acquired by the second qubit and the resonant frequencyin the left well ( N l ) as a function of the flux applied to thequbits. The left ordinate displays the oscillation frequency asdetermined from the Fast Fourier Transform of the energy ofqubit 2. The right ordinate displays the ratio between theapplied flux and the critical flux for qubit 1. Temporal tracescorresponding to the two x ’s are displayed in Fig. 3. (c-d)Same as (a-b) after reversing the roles of the two qubits. understand the initial conditions for the first qubit it isuseful to recall the physics of the measurement. Whenthe MP is applied, the flux approaches the critical value(approximately 0 . ϕ c ) over a short period of time, sothat the first excited state tunnels out with unit proba-bility. Once tunneled, this qubit can be assumed to havezero kinetic energy ( ˙ ϕ = 0), to have a phase value justto the right side of the residual local maximum betweenthe two wells (Fig. 1(b)), giving it an initial potentialenergy ∼ . U ( ϕ e ) below the local maximum value .In addition, we assume that the decay rate in the rightwell is comparable to that in the left well, and the simu-lation is run for times ∼ T , after which the qubit phasehas relaxed to rest. We have checked that small varia-tions in these assumptions do not meaningfully affect theresults of our simulations. From these initial conditionsthe phase of the first qubit(classically) undergoes dampedoscillations in the anharmonic right well. Because of theanharmonicity of the potential, when the amplitude ofthe oscillation is large, the frequency of the oscillationsis lower than the unmeasured qubit frequency. As thesystem loses energy due to the damping, the oscillationfrequency increases as seen by the CPW cavity. Whenthe crosstalk voltage has a frequency close to the CPWcavity frequency, it can transfer energy to the CPW cav-ity. If the second qubit’s frequency matches that of theCPW cavity then the cavity’s excitation can be trans-ferred to the second qubit. In Fig. 2(b) we plot, for thesecond qubit, the ratio ( N l ) between the maximum en- E ne r g y Q b2 ( N l ) Time (ns) E ne r g y C P W ( A . U . ) c)b) E ne r g y Q b1 ( A . U . ) a) FIG. 3. (Color) Simulated energy for (a) the first qubit, (b)the CPW cavity, (c) the second qubit. (Red): ϕ = 0 . ϕ c and ϕ = 0 . ϕ c . The first qubit decays exponentially upto t ≈ ns . At this time the frequency of the oscillation inright well matches the CPW cavity resonant frequency andthe qubit transfers part of its energy to the CPW cavity. Thesecond qubit is resonating at a different frequency and it isminimally exited by the incoming microwave voltage. Thiscorresponds to the red x of Fig. 2(b) (Black): ϕ = 0 . ϕ c and ϕ = 0 . ϕ c . In this case the first qubit transfers partof its energy to the CPW cavity at t ≈ ns because itstarts at a lower energy in the deep well. At this flux thesecond qubit is in resonance with the cavity and it is excitedup to the sixth quantized level. This corresponds to the whitex of Fig. 2(b) ergy acquired and (cid:126) ω p , where ω p is the plasma frequencyof the qubit in the left well, as a function of the fluxesin the two qubits. The crosstalk, measured as the maxi-mum energy transferred to the second qubit, is maximumat a flux ϕ /ϕ c ∼ . ≈ . GHz , determined by taking the FastFourier Transform of the oscillations in energy over time(see Fig. 3 (a-c)). Reversing the roles of the two qubits,we find that for the first qubit the crosstalk is maximumat a flux ∼ . ϕ c , corresponding to an excitation fre-quency of ≈ . GHz (Fig. 2(d)). These values weredetermined for qubit 2 (qubit 1) by performing a Gaus-sian fit of N l versus flux (or frequency) after averagingover the span of flux (or frequency) values for qubit 1(qubit 2). Notice that the crosstalk transferred to qubit2 (qubit 1) is flux independent of qubit 1 (qubit 2) andsubstantial only when the cavity frequency matches thefrequency of qubit 2 (qubit 1). The results of the sim-ulations are in good agreement with the experimentaldata. To gain additional insight into the dynamics of thesystem, we plot the time evolution of the energy for thequbits and the CPW cavity (Fig. 3 (a-c)) for two differentsets of fluxes in the two qubits. At ϕ = 0 . ϕ c and ϕ = 0 . ϕ c (red x in Fig. 2(b)) the first qubit decaysexponentially for a time t < ∼ ns (Fig. 3 (a-c)-Red). At t = 123 ns there is a downward jump in the energy of thefirst qubit while the energy of the CPW cavity exhibitsan upward jump. At this time, the frequency of oscilla-tion in the right well matches the CPW cavity resonantfrequency, so part of the qubit energy is transferred tothe CPW cavity. However, since the second qubit is noton resonance with the CPW cavity, it does not get signif-icantly excited by the microwave current passing throughthe capacitor C x .At ϕ = 0 . ϕ c and ϕ = 0 . ϕ c (white x inFig. 2(b)), the dynamics of the first qubit and the CPWcavity are essentially unchanged, except that the CPWcavity frequency is matched at a different time ( t =103 ns ) because the first qubit starts at a lower energy inthe deep well (Fig. 3 (a-c)-Black). 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