High-contrast dispersive readout of a superconducting flux qubit using a nonlinear resonator

0.31 MB·English

by  A. Lupascu

High-contrast dispersive readout of a superconducting flux qubit
using a nonlinear resonator
1 1 2 1 1
A. Lupa¸scu, E.F.C. Driessen, L. Roschier, C.J.P.M. Harmans, and J.E. Mooij
1Kavli Institute of Nanoscience, Delft University of Technology, PO Box 5046, 2600GA Delft, The Netherlands
6 2Low Temperature Laboratory, Helsinki University of Technology, PO Box 2200, FIN-02015 HUT, Finland
0 (Dated: February 4, 2008)
0
Wedemonstrate high-contrast state detection of asuperconducting fluxqubit. Detection is real-
2
izedbyprobingthemicrowavetransmissionofanonlinearresonator,basedonaSQUID.Depending
n on the driving strength of the resonator, the detector can be operated in the monostable or the
a bistable mode. The bistable operation combines high-sensitivity with intrinsic latching. The mea-
J
suredcontrast ofRabioscillations is ashighas87%;of themissing 13%,only3%isunaccounted

7 for. Experiments involving two consecutive detection pulses are consistent with preparation of the
2 qubitstate by thefirst measurement.

] PACSnumbers: 03.67.Lx,85.25.Cp,85.25.Dq
l
l
a
h
mes- ptinurmoSthmuiipsinseffiironecrglomdncaidantunciocdluntiiddnpaegrttoehqsceuefsbroseirotarstlih[z[2e1a]].tiimoRhnpaevlmoeefmacbreokenmaetbnpaltleeieoxasntcsahiobnifelgivlaseeh-mqequdueanbantists- (a)E(Ghz)g,e-055 ge D EEeg (b)<I>(I)qbg,ep-011 ge <<IIqqbb>>eg
at. manipulationschemes[3]andthegenerationofentangled -0.005 0.f000 0.005 -0.005 0.f000 0.005
m two qubit states [4, 5]. Control circuit Readout circuit (c)
- For qubit readout, several detectors have been inves- Icontrol Ls
nd taitgivateelyd epxopoerr,imfoerntraelalsyo.nIsnwgehnicehrala,rtehepirreesffienctileyncnyotiswreell-l Mcontrol Fqb, bias MreadoutFsq, bias C Rin Ir(t)=Ir0(t)cos(2pFt)
o
understood. Most detectors to date rely on irreversible C
c V(t)=V (t)cos(2pFt+c)
[ processesinmesoscopicJosephsoncircuits[6,7,8,9,10]. Qubit r r0 r

Fortheseschemes,energyisdissipatedonthechipwhere
1
thequbitisplaced. Thereforelongwaitingtimesarenec- FIG. 1: Energy (a) and current (b) of the two qubit states
v
4 essarytobringthequbit,readout,andcontrolcircuitsto versus applied magnetic flux. (c) Schematic diagram of the
qubit and its control and readout circuits. Crosses represent
3 their proper initial state. In particular the qubit state is
Josephson junctions. The probe wave is sent through a bias
6 strongly disturbed. More recently, dispersive measure-
1 resistor Rin = 4.7 kΩ. The values of the other circuit com-
0 ment schemes are being investigated, which overcome ponents are Ls ≈0.2 nH, C =67 pF, Mcontrol≈15 fH, and
6 these drawbacks [11, 12, 13, 14, 15]. They are based on Mreadout=15 pH.
themeasurementoftheimpedanceofaresonatorcoupled
0
/ to the qubit. The energy used to probe this resonator
at is mostly dissipated at a place remote from the qubit linearresonator. Inthebistableregimethemeasurement
m chip. Also, the backaction on the qubit is low. However, contrast is very large, 87 %, which is a significant im-
- the qubit relaxation times are typically comparable to provement over previous measurements [6, 7, 8, 9, 10,
d the time necessary to for a reliable measurement of the 13, 14]. We also performed consecutive measurements of
n
impedance. This limits the detection efficiency when a the qubit. We find that the results are consistent with
o
c linear resonator is used [12, 13]. With a nonlinear res- projectionofthequbitstatebyafirstmeasurement,with
: onator,this limitation can be removed,by using a bifur- a probability as high as 90 %.
v
i cation transition [16]. In this case, the result of a qubit In our experiments we use a persistent current qubit
X measurement is either of two possible oscillation states (PCQ) [17]. It is formed of a superconducting ring in-
r of the driven resonator. These oscillation states can be terruptedby three Josephsonjunctions. For suitable pa-
a
latched and reliably discriminated irrespective of subse- rameters of the Josephson junctions and for an applied
quentrelaxationofthequbit. Adispersivemethodusing magnetic flux in the qubit loop Φqb close to (n+1/2)Φ0
latching was successfully demonstrated in [14], where it (nisintegerandΦ0 =h/2e),thissystemcanbeoperated
was used for the readout of a quantronium. as a two level system. The qubit Hamiltonian is
In this letter we present experimental results on the 1
H = (ǫσ +∆σ ), (1)
qb 2 z x
readout of a superconducting flux qubit using a disper-
sive method. We observed coherent oscillations in two where ǫ = 2IpΦ0f. ∆ and Ip are fixed parameters, de-
distinct operationmodesofthe detector: monostablefor termined by the charging and Josephson energies of the
weak driving and bistable for strong driving of our non- Josephsonjunctions,andthe frustrationf =(Φ (n+
qb
−
2
1/2)Φ0)/Φ0. Figure 1a shows a plot of the ground (g) (a) fpf P M R (b))0.9
and excited (e) energy levels of the qubit used in our fr Tm df mV
mcisueprFarliseoguntutrtreeiedmn1eitcnnhsteFhsq.oiguwT.bs1hiatber.seicnxhgpemeccoatratrietcisoprneopnvrdaeilsnuegenttoaofteitoahncehocfeiorigcueurnlarsettaiandtge- Ir0,mfam0xIr0 Trm AamCcop rnleittaruoddlo epuut tl ispmeuelse <V> (r000..784.4 4F.5m(G4.6Hz)4.7
time
out and control circuit. The readout circuit is based on (c) (d) 1.2
a DC-SQUID, coupled to the qubit through their mu- Hz)89 V)
mtsuoanaglinniendtduicuctcflatuanxncecien[1M8it]srLealJod(ooΦupts.qΦ),Twh=heicSΦhQdUepIDen+dhsaMsonathJeosteoIptha-;l F(Gpeak567 <V> (mr001..80
sq sq,bias readout qb -4 -2 0 2 4 0.6
Φ andM I aregeneratedbyanexternalcoil f (10-3) 0.0 0.1 0.2 0.3
sq,bias readout qb m T (ms)
and by the qubit, respectively. The contribution of the m
self-generatedflux is negligible compared to Φ and
sq,bias FIG. 2: Measurements in the monostable regime. (a)
nearly constant over the range of variation of Φsq,bias Schematicrepresentationofthequbitcontrol(top)andread-
relevant in our experiments. The two qubit states have out (bottom) sequence; explanations are given in the text.
different currents Iqb (see Fig. 1b), resulting in values of (b)Spectroscopymeasurementfor fm =0,fp =fr =0.0021,
the Josephson inductance different by 2 % in our ex- Tm = 2.4 µs, and Tr = 6 µs. (c) Plot of the qubit transition
≈ frequency, measured spectroscopically, versus the frustration
periment. We form a resonant circuit by including L
in a loop closed by the capacitors C and the small unJ- fm. The solid line is a fit that yields the qubit parameters,
∆ and Ip, given in the text. (d) Measurement of Rabi oscil-
intended stray inductance L (see Fig. 1c). We probe
s lations with the same parameters as in (c) and Fm = 4.602
the transmission of this resonant circuit by sending a GHz, resonant to the qubit energy level splitting. The solid
wave with a frequency F and current amplitude at the line is a fit with an exponentially damped sinusoidal.
sample Ir,0 and measuring the amplitude Vr,0 and rel-
ative phase χ of the voltage of the transmitted wave.
We use alow-noisecryogenicamplifier,describedin[19]. switches to state 1. Thermal fluctuations will cause the
When F is close to the resonance frequency Fres, the transition 0 1 to occur randomly at Ir0 <IBh, with a
→
transmission depends strongly on small changes of LJ rate that increases exponentially as Ir0 approaches IBh.
and thus on the qubit state. In our experiment F The bistable measurement regime relies on the fact that
res
≈
800 MHz and the quality factor Q 100. The qubit the bifurcation current I depends on the value of the
Bh
≈
state iscontrolledbyapplyinga magneticflux inits ring magneticfluxintheSQUIDloopΦ . Forafixeddriving
sq
Φ (t) = Φ +M I (t). Φ is gener- condition of the oscillator,the probability for the transi-
qb qb,bias control control qb,bias
atedusinganexternalcoilandsettoavalueclosetoone tion 0 1 to take place is very sensitive to variations in
→
of the operation points (n+1/2)Φ0. McontrolIcontrol(t) Φsq, andthus to changesin the qubit state. The monos-
is generated by an on-chip line and used to change the table measurement regime does not exploit the 0 1
→
qubit state either adiabatically or by resonant pulses. transition, but relies instead on the measurement of the
The qubit, resonant circuit, and control line are fabri- voltageandphaseofthedrivenresonator,whichalsode-
cated on an oxidized silicon substrate by using standard pend on Φ .
sq
electronbeamlithography. Measurementsareperformed Wepresentourexperimentalresultsstartingwithmea-
in a dilution refrigerator,at temperatures of 30 mK. surements in the monostable regime. We control the
≈
The Josephson inductance of the SQUID depends on qubit state by using a pulse as shown in Fig. 2a (top).
thecurrentinanonlinearway. Ourresonantcircuitisde- We start with a preparation part (P), by fixing the frus-
scribed,to agoodlevelofapproximation,by the Duffing tration f to a value f , for a time much longer than the
p
oscillator model [20]. This model predicts that a certain qubit energy relaxation time. This results in a Boltz-
driving condition of the resonator (given by the driving mann distribution for the qubit initial state, set by the
frequency F and amplitude Ir0) results in one unique effective temperature Tqb,eff. The qubit is prepared in
forced oscillation state when F > F (1 √3/(2Q)). the ground state if T E E . After qubit state
res qb,eff e g
− ≪ −
When F < F (1 √3/(2Q)), two oscillations states preparation,thefrustrationf ischangedadiabaticallyto
res
−
are possible for IBl < Ir0 < IBh, and only one state the value fm, where qubit manipulation (M) is done by
otherwise. I and I are the values of the driving cur- using a microwave burst of frequency F and duration
Bl Bh m
rent where bifurcation occurs, dependent on F and the T . Finally,thefrustrationf ischangedadiabaticallyto
m
parameters of the resonant circuit. The oscillation state the value f (for readout - R), preserving the weight of
r
denotedby0existswhenIr0 <IBh,andthestate1when theenergyeigenstatescreatedatM.Readoutispossible,
Ir0 >IBl. Without fluctuations the resonator, driven at provided ǫ(fr) > ∆, because the two qubit states have
frequency F and amplitude Ir0 slowly increasing from significantly diff∼erent currents (see Fig. 1b). To measure
Ir0 =0, resides in state 0 until Ir0 reaches IBh, when it the state of the qubit, a microwave burst is sent to the
3
rraestoionnatTorr (wsietehFfrigeq.u2ean-bcyotFto,ma)m.pTlihtuedseeqIur0e,nmcaex,shaonwdnduin- (aI)r0,sw Ir0 Tsw T (b%)) 1680000Fsq, bias1 98 %
Fig. 2a is repeated typically 105 times, and the average I latch (w40
of the voltage of the transmitted wave hVr0i, which is r0,latch Ps20 F
a measure of qubit population, is calculated. Figure 2b 0 0 sq, bias2
4 6 8 10 12
shows the result of a typical spectroscopy measurement. time I (nA)
(c) (d) r0,sw
The position of the qubit spectroscopy peaks, F , is 100 100
peak
filpetlvoeotltfestdphleaitsdtiaantgfaufnwoclilttoihowntihnoegfeftrxhopemefcrtEuasqtt.iroa1nt,ifyoonirelftdhmsetiqhnueFbqiitgu.be2intce.rpgaAy- P(%)sw468000 g 87 % e P(%)sw468000
20
rameters ∆ = 4.58 GHz and I = 360 nA. Using strong 20
p 0
microwave pulses, with a frequency F equal to the en- 4 6 8 10 12 0
ergy levels splitting of the qubit, weminduce Rabi oscil- Ir0,sw(nA) 0 10 20 Tm3(0ns) 40 50
lations between the qubit energy eigenstates. A typical
result is shown in Fig. 2d, together with a fit with a ex- FIG. 3: Measurements in the bistable regime. (a) Envelope
oftheACreadoutpulse. (b)Switchingprobabilitycurvesfor
ponentially damped sinusoidal signal. We observe Rabi
oscillationswithfrequenciesashighas100MHz,varying two values of Φsq,bias: Φsq,bias1 = 2.335 Φ0 and Φsq,bias2 =
2.340 Φ0. (c) Switchingprobability curves,for qubitin theg
linearly with the amplitude δfm of the microwaveburst. and e states, for fp = 0.0161, fm = 0.00, fr = 0.0126, Tsw ≈
We now turn to a presentation of the experiments us- 100ns,andT ∼10µs. (d)Rabioscillationwiththesame
latch
ing the bistable operation mode. To characterize experi- setting as for (c), and Isw optimized for maximum contrast.
mentallythe transitionbetweenthe states0and1 ofthe
driven oscillator, we use the methods developed in [16].
We drive the resonator with an AC pulse of fixed fre- for 7 %. The qubit relaxation during the measure-
≈
quency F < F (1 √3/(2Q)) and modulation of the ment(butnot duetothemeasurementprocess)accounts
res
−
amplitude as shown in Fig. 3a. During the first part of for 2%. The remainder of 3% occurs during the
≈ ≈
thepulse,ofdurationT ,theamplitudeissettoavalue switchingonofthedetector. InFig.3cweshowthemea-
sw
Ir0,sw closetoIBh,suchthatthereisasignificantproba- surement of Rabi oscillations with measurementsettings
bility for the 0 1 transition of the oscillator. The sec- optimized for maximum contrast. Each measurement is
ond part of the→pulse is used to discriminate the states repeated 104 timesinordertoreducethespreadofthe
∼
0 and 1 by measuring the amplitude and phase of the average when measuring superpositions.
voltage of the resonator. In this part Ir0 is set such that We finally address the question of how the qubit state
both transitions 0 1 and 1 0 have negligible rates is changed by the measurement [21]. This requires the
→ →
anda long time Tlatch allowsfor amplifier noise suppres- use of two consecutive measurements; the second mea-
sion. We first measure the probability Psw of switching surement is used to characterize the effect that the first
between the states 0 and 1, with the qubit in its ground measurement had on the qubit. Using two consecutive
state, for two values of the magnetic flux in the SQUID readout pulses, of the type shown in Fig. 3a, has the
loop Φsq,bias1 and Φsq,bias2. Their difference equals the disadvantage that the minimum duration of the latch-
change in magnetic flux when the qubit state changes ing plateau, due to the noise of our amplifiers, is com-
from g to e. The switching probability Psw is shown in parable to the qubit relaxation time. The qubit would
Fig. 3b as a function of the amplitude Ir0,sw. The figure undergo significant energy relaxation before the second
showsthatit is possible,inprinciple, to detectthe qubit measurement takes place. We choose instead a readout
state with an efficiency as high as 98 %. pulse as shown in Fig. 4a, consisting of two intervals
Using the bistable operation mode, we next measure of large amplitude, realizing two consecutive measure-
changes in the state of the qubit. We use a controlpulse ments R1 and R2. We use the labels r1 and r2 to de-
as shown in Fig. 2a (top) and a readout pulse as shown note the outcome of the measurements R1 and R2; they
in Fig. 3a. The shape of the control pulse is optimized take the values 0 (1) if switching did not (did) occur.
to achieve high fidelity for qubit state preparation. Fig- For this double-measurement pulse the probability that
ure 3c shows the switching probability curves when the switching occurs is P2 (ψ ,I1 ,I2 ,T ) =
sw | iinit r0,sw r0,sw delay
qubit is prepared in the g or e state by using a resonant 1 P(r1 = 0 r2 = 0); here ψ init is the initial
microwave pulse at M (see Fig. 2a). The maximum sep- qu−bit state, Ir10,s∧w and Ir20,sw are th|eiamplitudes for R1
aration between the two curves is 87%. This is smaller and R2, Tdelay is the delay between the two measure-
than the value of 98% corresponding to a measurement ments, and P(r1 = 0 r2 = 0) is the probability that
∧
of a magnetic flux difference equal to the qubit signal switching does not occur during either R1 or R2. We
(seeFig.3b). Detailedexperimentalstudiesindicatethat also read out the qubit state with a single measurement
the most important contribution to the observed loss of pulse by setting the amplitude Ir20,sw = Ir0,latch. The
contrast is the adiabatic shift from M to R, accounting switchingprobabilityforthissingle-measurementpulseis
4
(a) I (b) 100 A no-switch event results in a post-measurement state
r0 80
III12rr00,,ssww T1swTdelay T2sw Tlatch P(%)sw246000 g e nmineeaaFrsliuygr.eeq4mdueanslthtooswtathtaee.feaxTsctihteiendcrresegtaaiostene,coiorvrreerresspwpechotniicvdhestothfoethtcheuerpvrreees--
r0,latc0h R1 R2 0 2 4 1 6 8 10 gionwheretheswitchingprobabilityforthegroundstate,
time I r0,sw(nA) shown in Fig. 4b, increases; this can be interpreted as a
(c) (d) 100 u transition from weak to strong measurement. Note that
%)
rast (%)4600 e|r=0) (1468000 87654ppppp/////88888 uslitmseianitdgaottihfoetnwdooofusobunlcecl-yemsgseiiavvseiunprguemilnseefosnrtamspausthlisooewnsnhooninwtnFhieign.qF3uaibgih.ta4ssattaihntee-
Cont200 P(q= 200 321ppp///888 when r1 =0.
0 2 4 6 8 10 In this paper we presented measurements of a super-
2 4 6 8 10
Tdelay(ms) I1 (nA) conductingqubitusingadispersivereadoutscheme. Op-
r0,sw
eration in the bistable mode results in the observation
FIG. 4: Results of the correlation measurements. (a) En- of Rabioscillations with veryhigh contrast,significantly
velope of the double-measurement pulse. (b) The switching improvedoverother measurements of this type. We also
probabilitycurvesforthequbitinthegroundorexcitesstates, present measurements which indicate that a single mea-
with a single-measurement pulse with Ts1w ≈ 125 ns. (c) The surementpreparesthequbitstatewithafidelityofabout
contrast of Rabioscillations measured with a pulse as shown
90 %. These results establish dispersive readout as very
in (a), with I1 = I2 =7.7 nA and T1 = T2 ≈ 125 ns,
r,sw r,sw sw sw suited for flux qubits and make it promising for readout
as a function of T . The solid line is a fit with an expo-
delay
of entangled two-qubit states.
nentialdecay. (d)Conditional probabilityforthequbittobe
intheexcitedstate, whenthefirstmeasurementdidnotlead We acknowledge the help of P. Bertet and R.N.
toswitching,fordifferentqubitinitialstates;Tdelay =800ns. SchoutenandusefuldiscussionswithI.Siddiqi,R.Vijay,
and L. Vandersypen. This work was supported by the
SQUBIT project and the Large Scale Installation Pro-
Ps1w(|ψiinit,Ir10,sw) = 1−P(r1 = 0), where P(r1 = 0) is gramULTI-3ofthe EuropeanUnion, the Dutch Organi-
the probability that switching does not occur during the
zation for Fundamental Research on Matter (FOM), the
measurement R1. Figure 4b shows the switching proba-
Academy of Finland, and the NanoNed program.
bility curves, for a single-measurement pulse, as a func-
tion of I1 . (The timing for this single-measurement
r0,sw
pulse is as used in the experiments described below, but
differentfromthosethatledtothedatashowninFig.3c).
In a first experiment we measure the contrast of Rabi [1] M. Devoret, A. Wallraff, and J. Martinis, cond-
oscillations with a double-measurement pulse as a func- mat/0411174.
tionofthedelaytimeTdelay betweenR1andR2,asshown [2] M. A. Nielsen and I. L. Chuang, Quantum Computation
inFig.4c. Theobserveddecreaseisduetotherelaxation andQuantumInformation (CambridgeUniversityPress,
2000).
of the qubit between R1 and R2. A fit with an exponen- [3] E. Collin et al.,Phys. Rev.Lett. 93, 157005 (2004).
tial decay,shownby the solid line in Fig.4c, yields a de- [4] T. Yamamoto et al., Nature425, 941 (2003).
cay time which is the same as the qubit relaxation time. [5] R. McDermott et al.,Science 307, 1299 (2003).
This shows that the correlations between the results of [6] D. Vion et al.,Science 296, 886 (2002).
measurements R1 and R2 are due to the change in the [7] I. Chiorescu et al.,Science 299, 1869 (2003).
qubit state, and not to the detector dynamics. [8] T. Dutyet al.,Phys. Rev.B 69, 140503 (2004).
In a second experiment we set the initial state of the [9] O. Astafiev et al., Phys.Rev.B 69, 180507 (2004).
[10] K.B. Cooper et al.,Phys.Rev.Lett.93,180401 (2004).
qubitto a definedvalue andwe measurethe jointproba-
[11] M. Grajcar et al.,Phys.Rev. B69, 60501 (2004).
bility P(r1 = 0 ∧r2 = 0) with a double-measurement [12] A. Lupa¸scu et al.,Phys.Rev. Lett.93, 177006 (2004).
pulse and the probability P(r1 = 0) with a single- [13] A. Wallraff et al.,Phys.Rev. Lett.95, 060501 (2005).
measurement pulse. This allows calculating the prob- [14] I. Siddiqiet al., cond-mat/0507548.
ability for r2 = 0 when r1 = 0. Therefore the con- [15] M.A.Sillanp¨a¨aetal.,Phys.Rev.Lett.95,206806(2005).
ditional probability P(q = er1 = 0), for the qubit [16] I. Siddiqiet al., Phys.Rev.Lett. 93, 207002 (2004).
| [17] J. E. Mooij et al.,Science 285, 1036 (1999).
to be in state e when the first measurement gave the
[18] A. Barone and G. Paterno, Physics and Applications of
result r1 = 0, can be calculated. This is plotted in
the Josephson effect (John Wiley and Sons, 1982).
Fig. 4d versus Ir10,sw, for initial qubit states given by [19] L. Roschier and P.Hakonen, Cryogenics 44, 783 (2004).
ψ init = cos(θ/2)g + exp(iφ)sin(θ/2)e obtained by [20] M. Dykman and M. Krivoglaz, Physica A 104, 480
| i | i | i
Rabi rotation; θ is indicated in Fig. 4d for each curve, (1980).
and φ is an unimportant phase angle. For large I1 [21] V. Braginsky and F. Khalili, Quantum Measurement
r0,sw
all the curves collapse to a single value of about 90%. (Cambridge University Press, Cambridge, 1992).
≈