Low-frequency measurement of the tunneling amplitude in a flux qubit

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by  M. Grajcar

Low-frequency measurement of the tunneling amplitude in a flux qubit
M. Grajcar,1,∗ A. Izmalkov,1,2 E. Il’ichev,1,† Th. Wagner,1 N. Oukhanski,1 U. Hu¨bner,1
T. May,1 I. Zhilyaev,1,‡ H.E. Hoenig,1 Ya.S. Greenberg,3,§ V.I. Shnyrkov,3,¶
D. Born,1 W. Krech,3 H.-G. Meyer,1 Alec Maassen van den Brink,4 and M.H.S. Amin4
4 1Institute for Physical High Technology, P.O. Box 100239, D-07702 Jena, Germany
0 2Moscow Engineering Physics Institute (State University), Kashirskoe sh. 31, 115409 Moscow, Russia
0 3Friedrich Schiller University, Institute of Solid State Physics, D-07743 Jena, Germany
2 4D-Wave Systems Inc., 320-1985 West Broadway, Vancouver, B.C., V6J 4Y3 Canada
(Dated: February 2, 2008)
n
a
We have observed signatures of resonant tunneling in an Al three-junction qubit, inductively
J
coupledtoaNbLC tankcircuit. Theresonantpropertiesof thetankoscillator aresensitivetothe
2 effectivesusceptibility (orinductance)ofthequbit,whichchangesdrastically asitsfluxstatespass
1
throughdegeneracy. Thetunnelingamplitudeisestimated from thedata. Wefindgood agreement

with thetheoretical predictions in the regime of their validity.
]
n
o PACSnumbers: 85.25.Cp,85.25.Dq,84.37.+q,03.67.Lx
c
- E E
r Several groups, using different devices, have by now +
p
established that superconductors can behave as macro-
u
s scopicquantumobjects.1–3 Thesearenaturalcandidates
. for a qubit, the building block of a quantum computer.
t
a Qubits are effectively two-level systems with time- 2D
m
dependent parameters. One of them is a superconduct-
(a)
- ing loop with low inductance L, including three Joseph-
d
n son junctions (a 3JJ qubit).4 Its potential energy, U = E
o 3j=1EJj(φj), depends on the Josephson phase differ- - Fx
[c Penc3es φj across the junctions. Due to flux quantization F /2
j=1φj =2πΦx/Φ0 (withΦx theexternalmagneticflux 0
3 Pand Φ = h/2e the flux quantum), only two φ ’s are in-
0 j
v dependent. I (t)
7 For suitable parameters, U(φ ,φ ) has two minima M b
1 2
5 corresponding to qubit states Ψl and Ψr, carrying oppo- L J1
6 L C R
sitesupercurrentsaroundtheloop. Thesebecomedegen- T T T A
3 J3
30 weriattheCforthΦexc=ap12aΦci0t.anTcheeoCf ojuunlocmtiobnen1e)rignytrEoCdu(c≡eseq2/u2aCn-, J2 (b)
0 tum uncertainty in the φ . Hence, near degeneracy the
j
/ system can tunnel between the two potential minima. FIG. 1: (a) Quantum energy levels of the qubit vs external
t
a (Since E E E , we deal with a flux qubit; flux. The dashed lines represent the classical potential min-
m E EC ≪yieldsJa≡charJg1e qubit. Coherent tunneling ima. (b) Phase qubit coupled toa tank circuit.
C J
≫
- was demonstrated in both.)
d
In the basis Ψl,Ψr and near Φ = 1Φ , the qubit
n { } x 2 0 matic rf signal at its resonant frequency ω . Then both
o can be described by the Hamiltonian T
amplitude v and phase shift χ (with respect to the bias
c
v: H(t)=−ǫ(t)σz−∆σx ; (1) c(Aur)retnhteIsbh)ifotfitnheretsaonnkanvtolftraegqeuewnicllysdtruoengtolytdheepecnhdanogne
i
X ∆isthe tunneling amplitude. Atbiasǫ=0the twolow- of the effective qubit inductance by the tank flux, and
est energylevels ofthe qubit anticross[Fig.1(a)], with a (B) losses caused by field-induced transitions between
r
a gapof2∆. Increasingǫslowlyenough,thequbitcanadi- the two qubit states. Thus, the tank both applies the
abaticallytransformfromΨl toΨr,stayingintheground probing field to the qubit, and detects its response.
state E−. Since dE−/dΦx is the persistent loop current, The output signal depends on the tank’s quality fac-
the curvature d2E−/dΦ2x is related to the qubit’s suscep- tor Q. Using superconducting coil, values as high as
tibility. Hence, near degeneracy the latter will have a Q 103 can be obtained, leading to high readout sen-
peak, with a width given by ǫ < ∆.5 We present data siti∼vity, e.g., in rf-SQUID magnetometers.8 Such a tank
demonstrating such behavior|in|a∼n Al 3JJ qubit. canthereforebeusedtoprobephasequbits.9ForsmallL,
Our technique is similar to rf-SQUID readout.6,7 The the results are summarized by5
qubit loop is inductively coupled to a parallel resonant
tank circuit [Fig. 1(b)]. The tank is fed a monochro- v =I ω L Q/ 1+(2Qξ)2 , (2)
0 T T
p
2

0.0
0.0
-0.2
-0.2
d) -0.4 0.0
a
(r -0.2 ad)-0.4
c -0.6 -0.4 r
-0.6 (
c -0.6
-0.8 -0.8
-1.0
-0.010-0.0050.000 0.005 0.010
-1.0 -0.8
-0.010 -0.005 0.000 0.005 0.010
(a)
f
x -1.0
-0.010 -0.005 0.000 0.005 0.010
FIG. 2: Tank phase shift vs flux bias near degeneracy. From f
x
the lower to the upper curve (at fx = 0) the driving-voltage
amplitudeVdr ≡I0ωTLTQ takesvalues0.5, 1.0, 1.5, 1.9, 2.9, 1.0 (b) (c)
3.5, 3.9 µV. Inset: theoretical curves for ∆/h = 650 MHz, 0.01
and I0 =0.07, 0.13, 0.20, 0.26, 0.39, 0.47, 0.53 nA. 0.8
)0
0.6
ξ(v,f )= k2Ltanχ2π=dφ2Qcoξs,2φd2E−(f) , ((43)) kc tan 0.4 F width (
x 2Φ2 Z π df2 0.2
0 0
Mv
0.0
f =fx+ sinφ, (5) 0 50 100 150 200 10 100
ω L Φ
T T 0 T (m K) T (mK)
where f =Φ /Φ 1, I is the bias-currentamplitude,
x x 0− 2 0
andk=M/√LLT isthetank–qubitcouplingcoefficient, FIG. 3: (a) Tank phase shift vs flux bias near degeneracy
withM (LT)themutual(tank)inductance. Theground- and for Vdr = 0.5 µV. From the lower to the upper curve
state curvature is10 (at fx = 0) the temperature is 10, 20, 30, 50, 75, 100, 125,
150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude
d2E E2∆2λ2
− = J , (6) oftanχ(circles)andtanh(∆/kBT)(line),forthe∆following
df2 −(E2λ2f2+∆2)3/2 from Fig. 2; the overall scale κ is a fitting parameter. The
J
data indicate a saturation of the effective qubit temperature
where λ(α,g) (with g = EJ/EC) is the conversion fac- at30mK.(c)Fulldipwidthathalfthemaximumamplitude
tor in ǫ = E λf.11 If I vanishes, ξ = 1k2Ld2E /dΦ2 vstemperature. Thehorizontallinefitsthelow-T (<200mK)
becomes an eJxternal par0ameter accountin2g for the−qubixt parttoaconstant;theslopedlinerepresentstheT3 behavior
susceptibility coupled to the tank. For finite I , this has observed empirically for higher T.
0
to be averaged over a bias cycle 0 < φ < 2π. The re-
sulting integral (4) turns out to involve a weight cos2φ,
since the effective time-dependent coupling is (kf˙)2 [cf. as 190 650 nm2 while one is smaller, so that α
× ≡
theφ-derivativeofEq.(5)],proportionaltothesquareof EJ3/EJ1,2 0.8. The critical current was determined
the voltage the tank induces in the qubit. The resulting by measuri≈ng an rf-SQUID prepared on the same chip7
equations are coupled and nonlinear, but readily solved as Ic = 2eEJ/¯h 380 nA. With EC/h 3 GHz, one
numerically. finds g 60 and≈λ 4.4. The loop are≈a was 90 µm2,
≈ ≈ −
For the tank, we prepareda square-shapeNb pancake with L = 38 pH. We measured v by a three-stage cryo-
coil on an oxidized Si substrate. The line width of the genicamplifier,placedat 2Kandbasedoncommercial
≈
20 windings was 2 µm, with a 2 µm spacing. Predefined pseudomorphichighelectronmobilitytransistors. Itwas
alignmentmarksallowplacing a qubitin the center. For slightly modified from the version in Ref. 12 to decrease
flexibility, only the coil was made lithographically; an itsback-actiononthequbit. Theinput-voltagenoisewas
externalcapacitanceC isusedtochangeω intherange < 0.6 nV/√Hz in the range 1–35 MHz. The noise tem-
T T
5–35 MHz. For the selected tank (L 50 nH, C perature was 300 mK at 32 MHz. The effective qubit
470pF),weobtainedω /2π=32.675TM≈HzandQ T72≈5 temperatured∼ue tothe amplifier’sback-actionshouldbe
T
from the voltage–frequency characteristic. ≈ considerably lower because of the small k 2 10−2.
≈ ·
The 3JJ qubit structure was manufactured out of Al Theχ(f )curvesmeasuredatvariousI andamixing-
x 0
by conventional shadow evaporation. The area of two of chamber temperature T = 10 mK are shown in Fig. 2.
the junctions was estimated using electron microscopy The narrow dip at f = 0 directly corresponds to the
x
3
one in Eq. (6), in line with the qualitative picture below for ∆/h = 650 MHz independently obtained above from
Eq. (1). With device parameters as above, all quanti- the low-T width. The good agreement strongly sup-
ties in Eqs. (2)–(6) are known, but ∆ only in principle: ports our interpretation, and is consistent with ∆ be-
its exponential sensitivity to α and especially g makes ingT-independentinthe relevantrange.16 Ofcourse,for
it notoriously hard to calculate a priori. Hence, it is higherT the dip willwashout; we observeawidth T3
∝
treated as a free parameter; calculated curves for the above a crossover temperature 225 mK. For T of this
≈
best fit ∆/h=650 MHz are shown in the inset. For the order, deviations from the two-state model can be ex-
largest I the experimental and theoretical curves dis- pected, especially for f = 0. This behavior outside the
0 x
6
agree, for the rapid change of Φ then leads to Landau– qubit regime has not been pursued.
x
Zener transitions13,14 suppressing the dip. In conclusion, we have observed resonant tunneling in
TheT-dependenceofχisshowninFig.3. Forincreas- amacroscopicsuperconductingsystem,containinganAl
ing T the dip’s amplitude decreases while, strikingly, its flux qubit and a Nb tank circuit. The latter played dual
width is unchanged [Fig. 3(c)]. Both are a simple mani- control and readout roles. The impedance readout tech-
festation of the Hamiltonian (1) yielding σ = (ǫ/Ω) niqueallowsdirectcharacterizationofsomeofthequbit’s
z
h i ×
tanh(Ω/k T),15 Ω = √ǫ2+∆2. This result of equilib- quantum properties, without using spectroscopy.2,3 In a
B
rium statistics of course assumes that the t-dependence range 50 200 mK, the effective qubit temperature has
∼
of ǫ(t) is adiabatic. However, it does remain valid if been verified [Fig. 3(b)] to be the same as the mixing
the full (Liouville) evolutionoperatorof the qubit would chamber’s(after∆hasbeendeterminedatlowT),which
contain standard Bloch-type relaxation and dephasing is often difficult to confirm independently.
terms (which indeed are not probed5) in addition to MHSAandAMvdBaregratefultoA.Yu.Smirnovand
the Hamiltonian dynamics (1), since the fluctuation– A.M. Zagoskin for fruitful discussions, and to P.C.E.
dissipation theorem guarantees that such terms do not Stamp for the remark on effective thermometry. MG
affectequilibriumproperties. Normalizeddipamplitudes wants to acknowledge partial support by the Slovak
areshownvs T inFig. 3(b)togetherwith tanh(∆/k T), Grant Agency VEGA (Grant No. 1/9177/02).
B
∗ On leave from Department of Solid State Physics, Come- W. Krech, and H.-G. Meyer, Phys. Rev. B 66, 224511
nius University,SK-84248 Bratislava, Slovakia. (2002).
† Electronic address: [email protected] 10 A minus sign is missing in Eq.(17) of Ref. 5.
‡ OnleavefromInst.ofMicroelectronicTechnology,Russian 11 Let us rectify the discussion of λ, Eq. (12) in Ref. 5. In
Academy of Science, 142432 Chernogolovka, Russia. Eq. (1) of Ref. 2, ǫ is expressed in terms of the persistent
§ On leave from Novosibirsk State Technical University, current (subsequently given numerically for their sample
20 K.Marx Ave., 630092 Novosibirsk, Russia. parameters)insteadofEJ,whichimplicitlyincorporatesλ.
¶ OnleavefromB.VerkinInst.forLowTemperaturePhysics In T.P. Orlando et al., Phys. Rev. B 60, 15398 (1999),
and Engineering, 310164 Kharkov,Ukraine. the flux–energy conversion is carried out in Eq. (32) and
1 Y. Nakamura, Yu.A. Pashkin, and J.S. Tsai, Nature 398, Appendix B. Our λ differs from their r1/EJ only in an
786 (1999); J.R. Friedman, V. Patel, W. Chen, S.K. O(g−1/2)correction[takenintoaccountbelowourEq.(6)],
Tolpygo,andJ.E.Lukens,Nature406,43(2000);D.Vion, plus an overall sign immaterial for λ2. This correction de-
A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, creasing λ(hereby∼9%)maybesmall,butithasaclear
D. Esteve, and M.H. Devoret, Science 296, 886 (2002); interpretation: as fx increases, the upper well, ultimately
J.M. Martinis, S. Nam, J. Aumentado, and C. Urbina, losing classical stability, becomes softer. Hence, its local
Phys. Rev.Lett. 89, 117901 (2002). zero-pointenergydecreases,slightlycounteractingtherise
2 C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N. of the potential minimum.
Schouten,C.J.P.M.Harmans,T.P.Orlando,S.Lloyd,and 12 N. Oukhanski, M. Grajcar, E. Il’ichev, and H.-G. Meyer,
J.E. Mooij, Science 290, 773 (2000). Rev.Sci. Instrum.74, 1145 (2003).
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M.Grajcar,H.-G.Meyer,A.Yu.Smirnov,A.Maassenvan Proc. R.Soc. London A 137, 696 (1932).
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Lett. 91, 097906 (2003). phys.Lett.
4 J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van 15 Using thissimple tanh law also for thenet qubitresponse
der Wal, and S. Lloyd, Science 285, 1036 (1999). ∝ ∂ hσ i accounts for the following subtlety. While our
ǫ z
5 Ya.S. Greenberg, A. Izmalkov, M. Grajcar, E. Il’ichev, methodisquasi-equilibrium,on the scale of one bias cycle
W. Krech, H.-G. Meyer, M.H.S. Amin, and A. Maassen ∼ω−1 relaxationisslow.Hence,inthefinite-T generaliza-
T
van den Brink, Phys. Rev.B 66, 214525 (2002). tion, flux derivatives such as in Eq. (6) become adiabatic
6 A.H. Silver and J.E. Zimmerman, Phys. Rev. 157, 317 ratherthan isothermal. ForfiniteT and v, theproblem is
(1967). essentially nonequilibirum and outside ourscope.
7 E. Il’ichev et al., Rev.Sci. Instrum.72, 1882 (2001). 16 S.-X.Li, Y. Yu,Y.Zhang, W. Qiu, S.Han, and Z. Wang,
8 V.V. Danilov and K.K. Likharev, Zh. Tekh. Fiz. 45, 1110 Phys.Rev.Lett.89,098301(2002)andrefenercestherein.
(1975) [Sov. Phys.Tech. Phys. 20, 697 (1976)].
9 Ya.S. Greenberg, A. Izmalkov, M. Grajcar, E. Il’ichev,