Measurement of the effect of quantum phase-slips in a Josephson Junction chain

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by  I. M. Pop

Measurement of the effect of quantum phase-slips in a Josephson junction chain
I. M. Pop,1 I. Protopopov,2,3 F. Lecocq,1 Z. Peng,1 B. Pannetier,1 O. Buisson,1 and W. Guichard1
1Institut N´eel, C.N.R.S. and Universit´e Joseph Fourier, BP 166, 38042 Grenoble-cedex 9, France
2L. D. Landau Institute for Theoretical Physics, Kosygin str. 2, Moscow 119334, Russia
3Institut fuer Nanotechnologie, Karlsruher Institut fuer Technologie, 76021 Karlsruhe, Germany
(Dated: January 26, 2010)
The interplay between superconductivity and Matveev et al.8. Let us consider the Josephson junction
Coulomb interactions has been studied for more chain depicted in Fig. 1(a). The chain contains N junc-
0
than twenty years now1–13. In low-dimensional tions and is biased with a phase γ. We denote E the
1 J
0 systems, superconductivity degrades in the pres- Josephson energy of a single junction and EC = 2eC2 its
2 ence of Coulomb repulsion: interactions tend to charging energy. Here we consider E (cid:29)E . Let Q be
J C i
suppress fluctuations of charge, thereby increas- the charge on each junction and θ the phase difference.
n i
a ing fluctuations of phase. This can lead to the In the nearest-neighbor-capacitance limit the Hamilto-
J occurrence of a superconducting-insulator transi- nian can be written as:
tion, as has been observed in thin superconduct-
6
2 ing films5,6, wires7 and also in Josephson junc- (cid:88)N (cid:88)N
tion arrays9,11–13. The latter are very attractive H = [4EC(Qi/2e)2+EJ(1−cosθi)] ; θi =γ. (1)

] systems as they enable a relatively easy control i=1 i=1
n
of the relevant energies involved in the competi-
o Ignoring the charging energy for the moment, we find
c tion between superconductivity and Coulomb in- the classical ground state, that satisfies the constraint
pr- tseurcaccetsisofnusll.yJuosseedphtosocnrejautnecptiaornticcuhlaairnselhecatvreombeaegn- on the phase (cid:80)Ni=1θi, by minimizing the Josephson cou-
pling energy. The corresponding phase configuration
u netic environments for the reduction of charge
can be easily found for sufficiently large N (N (cid:29) 1)
s fluctuations14–16. Recently, they have attracted
t. interestastheycouldprovidethebasisforthere- and it is given by θi = γ/N. The resulting Joseph-
a son energy hence reads E = E γ2/2N and the chain
alization of a new type of topologically protected 0 J
m is equivalent to a large inductance. If a phase-slip oc-
qubit17,18 or for the implementation of a new cur-
- curs on one of the junctions, say the jth junction, then
d rent standard19. Here we present measurements θ → θ +2π. Since the Josephson energy is periodic in
n that show clearly the effect of quantum phase j j
θ , a phase-slip of 2π does not change the Josephson en-
o slips on the ground state of a Josephson junction j (cid:80)
ergyofthejunctionj. However,theconstraint θ =γ
c chain. We tune in situ the strength of quantum i i
[ is violated after such a phase-slip event. Therefore the
phase fluctuations and obtain for the first time an
2 excellent agreement with the tight-binding model phase difference θi over all other junctions changes a lit-
tle from γ/N to (γ −2π)/N in order to accommodate
v initially proposed by Matveev et al.8.
the bias constraint. A phase-slip on a single junction
7 The Hamiltonian for the theoretical description of su-
1 leads to a collective response of all junctions. Conse-
perconducting circuits can be conveniently obtained by
4 quentlytheJosephsonenergyoftheentirechainchanges
applyingDevoret’scircuittheory20. Here, eachelectrical
2 from E = E γ2/2N to E = E (γ −2π)2/2N. One
elementsuchasaninductance,acapacitorortheJoseph- 0 J 1 J
2. can show similarly that the classical energy needed to
son element can add a degree of freedom. In the case
1 accommodate m phase-slips without violating the con-
of circuits with a small number of electrical elements, a
9 straint is given by E =E (γ−2πm)2/2N. Therefore,
complete analytical description that takes into account m J
0 theclassicalgroundstateenergyofthechain,min{E },
v: all degrees of freedom can be obtained. However, when consistsofshiftedparabolasthatcorrespondrespectivmely
thecircuitscontainanincreasingnumberofelements, as
i to a fixed number m of phase-slips (see Fig. 1(b)). For
X for example Josephson junction chains, even numerical
the special values γ = π(2m+1) the energies E and
solutions of the problem become difficult to obtain when m
r E are degenerate.
a taking into account all degrees of freedom. Nevertheless m+1
TakingnowintoaccountthefinitechargingenergyE ,
ourmeasurementsdemonstratethatthegroundstateofa C
quantumphase-slipscanliftthedegeneracyatthepoints
phase-biased Josephsonjunctionchain(seeFig. 1(a))can
γ =π(2m+1). In the limit E (cid:29)E , the hopping ele-
bedescribedbyasingledegreeoffreedom. Althoughthe J C
ment for the quantum phase-slip can be approximated
chain is a multi-dimensional object, the effect of quan- √
(cid:112)
tum phase-slips can be described by a single variable m, by21,22: v = 16 EJEC/π(EJ/2EC)1/4e− 8EJ/EC.
Since a phase-slip can take place on any of the N junc-
that counts the number of phase-slips in the chain.
tions, the hopping term between the two states |m(cid:105) and
We start by giving a short introduction on the low-
|m+1(cid:105) is given by Nv. Therefore, using a tight-binding
energy properties of a Josephson junction chain which
approximation, the total Hamiltonian for the chain is
have been studied in terms of quantum phase slips by
2
given by: ponentially suppressed with N and E /E 8.
J C
H|m(cid:105)=E |m(cid:105)−Nv[|m−1(cid:105)+|m+1(cid:105)]. (2)
m
γ
Fig. 1(b) shows the numerical calculation of the two I
lowest eigenenergies of this Hamiltonian for three differ- ΦS ΦC
ent ratios EJ =20, 3 and 1.3 in the case of a 6-junction
chain. Fig.EC1(c) shows the corresponding current-phase θ2 δ θ5
relation of the chain in the ground state. The chain’s
Φ
supercurrentisobtainedbythecalculationofthederiva- C
tive of the ground state energy E : i = 2e∂Eg. For
g S h¯ ∂γ
C L
50 mK
I
bias
V
FIG.2. Measurementcircuit. The6-SQUIDchainisinserted
in a superconducting loop. The flux Φ created by on chip
C
coils controls the phase difference γ over the chain. The flux
Φ throughtheSQUIDscanbecontrolledindependentlybya
S
secondcoil. Wedenotethephasedifferenceovertheread-out
junction δ.
Read-out junction SQUID at φ =0
S
SRO =(121±5)103nm2 SSQ =(30±2)103nm2
CRO =5.8±0.2fF C =1.4±0.1fF
RRO =968±5Ω RSQ =3800±450Ω
N N
FIG. 1. (a) Schematic picture of the phase-biased Joseph- IRO =330±2nA ISQ =83±9nA
C C
sonjunctionchain. (b)EnergylevelsofaJosephsonjunction
chain with N = 6 as a function of bias phase γ for differ-
ent ratios EJ/EC. For EJ/EC = 20 (black lines) no split- TABLE I. Parameters of the sample: size, capacitance,
ting is visible at the crossing points. For EJ/EC = 3 (red normal-state resistance and critical current of the read-out
lines) a gap emerges that increases rapidly with decreasing junction and a single SQUID of the chain. The critical cur-
EJ/EC. Blue lines show the energy levels for EJ/EC =1.3. rentvarianceforthejunctionsinthechainisestimatedtobe
For each EJ/EC, the two lowest-lying states have been cal- smaller than 4%.
culated by numerical diagonalization of the hamiltonian (2).
(c)Current-phaserelationforthe ground stateEg(γ)for the To measure the effect of quantum phase-slips on the
same E /E ratios as in (a). The supercurrent is calculated
J C ground-stateofaJosephsonjunctionchainwehavestud-
from the derivative of the energy band: i = 2e∂Eg. The
S h¯ ∂γ ied a chain of 6 junctions. Our measurement setup and
chain current is reported in units of the critical current of a
the junction parameters are presented in Fig. 2 and Ta-
singlechainjunctioni = 2eE . (d)Schematicpictureofthe
0 h¯ J ble I. Each junction in the chain is realized by a SQUID
chain shunted by the read-out junction. (e) Escape potential
inordertoenabletunableJosephsoncouplingE . Inthis
fortheJosephsonjunctionchaininparallelwiththeread-out J
way we can tune in situ the E /E ratio by applying a
junctionforthreedifferentflux-biasesφC intheread-outloop J C
(see Fig. 2). The ground-state of the chain clearly modifies uniform magnetic flux ΦS through all SQUIDs, and con-
the escape potential of the read-out junction. sequently we can control the strength of quantum phase
fluctuations. For our measurements we placed this chain
large values of E /E , quantum phase fluctuations are in a closed superconducting loop, threaded by the flux
J C
very small (v ∼ 0) and the current-phase relation has a ΦC, containing an additional shunt Josephson junction
sawtooth-like dependence with a critical current that is that is used for the read-out of the chain state. The flux
approximatively N/π times smaller than that of a single ΦC enablesthecontrolofthebiasphaseγ =ΦC−δ over
junction of the chain. We call this regime the ”classical” the chain.
phase-slipregime. Whenquantumphasefluctuationsin- We have measured the switching current of the entire
crease, i.e. E /E decreases, the current-phase relation Josephsonjunctioncircuitcontainingboththechainand
J C
becomes sinusoidal and the critical current becomes ex- the read-out junction. The switching current was deter-
minedfromtheswitchingprobabilityat50%. Theescape
3
probability as a function of bias current I shows the ofthemeasuredswitchingcurrent. Letuspointoutthat
bias
usual ”S-form” with a width of ≈20 nA. We apply typi- we have used the nominal values for E and E calcu-
J C
cally10000biascurrentpulsesofamplitudeI anddu- lated from the characteristics of the sample indicated in
bias
ration ∆t <∼ 1µs, and measure the switching probability Table I. The normal-state resistance for a single chain
as the ratio between the number of switching events and junction has been deduced from the measured normal-
the total number of pulses. The results of the switching state resistance of the read-out junction by considering
currentmeasurementsasafunctionoffluxΦ areshown thesizeratiobetweenthetwo. Weevaluatetheprecision
C
inFig. 3. Fromtheseswitchingcurrentmeasurementswe of the determination of E and E to be in the range of
J C
deduce the effect of quantum phase-slips on the ground ±10%. This error bar on E and E yields an uncer-
J C
state of the chain. tainty δN = 1 for the effective number of junctions N
Themeasuredswitchingcurrentcorrespondstothees- contributing to the phase-slip amplitude. This confirms
cape process out of the total potential energy U con- thegoodhomogeneityofourjunctionsandthecollective
tot
tainingthecontributionsoftheread-outjunctionandthe nature of the phase-slip events.
chain:

¯h
Utot(δ,ΦC)=EJROcos(δ)+Eg(ΦC −δ)− 2eIbiasδ. (3) 200 EJ/EC=3
180
Here E is the ground state of the 6-SQUID chain calcu-
g
lated by solving the Hamiltonian (2). As EJRO (cid:29)Eg the ) 160
bmiaaisnedcoremapd-oonuetnjtuinnctUiotontEisJRtOhceops(oδt)e−nti2h¯aelIboifastδh.eFciugr.r1en(et)- (nA 128000 EJ/EC=2
shows the escape potential at constant bias-current for W
three different flux values φC corresponding to three dif- IS 160
ferent biasing phases γ over the chain. Let us point out 200
EJ/EC=1
that the position of the minimum of the potential U is
tot
in good approximation independent of the value of the 180

flux φ . Therefore the bias phase difference γ over the
C 160
chaindependsonlyonthefluxφ . Asaconsequence,the
C -3π -2π -π 0 π 2π 3π
φ dependenceofthemeasuredswitchingcurrentresults
C φ
from the γ dependence of the chain’s ground state.
C
The escape from the potential U occurs via Macro-
tot
scopic Quantum Tunneling (MQT). The MQT rate
for an arbitrary potential can be calculated in the FIG.3. Measuredswitchingcurrent(blackpoints)asafunc-
limit of weak tunneling using the dilute instanton-gas tion of φC over the chain for three different EJ/EC ratios.
approximation26. Within this model, the escape rate Γ The measurement noise for each point is about 0.2 nA. The
out of the washboard potential U (γ) reads27: red lines represent theoretical calculations for the switching
tot
current using formulas (5) and (4).
Γ=Aexp[−B],
Fig. 4 shows the measured switching current ampli-
where A and B are given by: tude ∆ISW and the corresponding theoretical calcula-
tions as a function of E /E . For each measurement,
J C
A=(cid:113)h¯ω03σeI with I =2(cid:82)σ(cid:113)h¯2Utot(x)dx (4) EJ has been calculated using the flux dependence of the
8πh¯ 0 4ERO SQUID’s Josephson coupling: E (Φ ) = h¯ i (Φ ) with
C J S 2e 0 S
(cid:16) (cid:17)
B =(cid:82)σ(cid:104)(cid:113) h¯2ω02 − 1(cid:105)dx. i0(ΦS) = ICSQcos πΦΦS0 . In order to distinguish be-
0 16ECROUtot(x) x tween the suppression of the switching current that is
due to quantum phase fluctuations and the one that is
We have denoted by σ the width of the barrier and by
simplyduetothewell-knowncancellationoftheSQUID’s
x the phase coordinate measured from the minimum of
critical current as a function of flux, we plot the switch-
the washboard potential. The plasma frequency is ω =
0 ing current amplitude divided by the critical current of
(cid:113)
8ECROh¯Ut(cid:48)o(cid:48)t(0), where ECRO is the charging energy of the a single SQUID i0. We see that the measured switching
read-out junction. current amplitude follows very well the predicted theo-
Knowing the escape rate Γ, we can calculate the retical suppression of the switching current oscillations
switching probability: inthepresenceofquantumphasefluctuations. Fromour
measurements we can also deduce the strength of the
P(I )=1−exp[−Γ(I )∆t]. (5) quantum phase-slip amplitude. With decreasing E /E
bias bias J C
ratio from 3 to 1 the quantum phase-slip amplitude in-
The results are shown as red lines in Fig. 3. The theory creases from 0.8 GHz to 2.7 GHz. In addition, in Fig.
fitsverywellbothinamplitudeandshapetheoscillations 4 we have plotted for comparison the calculation for the
4
EJ/ k T These results open the way for the use of quantum
B
phase-slips in JJ networks for the implementation
0 7 13 19 26 33 40
0.30 of a new current standard, the observation of Bloch
oscillations19, the fabrication of topologically protected
classical chain (v ~ 0)
0.25 qubits25 and the design of new superconducting circuit
elements.
0 0.20
i

/ Methods

W 0.15
quantum
IS chain The circuit was fabricated on a Si/SiO2 substrate and
∆ 0.10
the Al/AlO /Al junctions were obtained using standard
x
shadow evaporation techniques. The aluminum oxide
0.05
was obtained by natural oxidation in a controlled O at-
2
mosphere. The sample was mounted in a closed copper
0.00
blockwhichwasthermallyconnectedtothecoldplateof
0.0 0.5 1.0 1.5 2.0 2.5 3.0 a dilution refrigerator at 50mK. All lines were strongly
E /E filteredbylow-passfiltersatthecryostatentranceandby
J C
thermocoaxial cables and π filters at low temperatures.
The switching current I of the circuit is obtained
FIG. 4. Comparison between the measured (black points) SW
performing the following sequence. We use a series of M
and the calculated (red empty circles) switching current am-
current steps of equal amplitude I to bias the junc-
plitude as a function of the EJ/EC ratio. Note that the bias
switchingcurrentamplitudeisdividedbythefluxdependent tion. We count the number of transitions to the voltage
critical current of a single SQUID i0, in order to reveal the state MSW and thus obtain the value of the switching
effect of quantum-phase fluctuations. The top curve (blue probability PSW = MMSW corresponding to the applied
emptycircles)showsthetheoreticalcalculationoftheswitch- I . BysweepingtheI amplitudeandrepeatingthe
bias bias
ing current amplitude in the absence of quantum phase fluc- above sequence, we measure a complete switching his-
tuations. The lines are guides for the eye. togram, P vs I . The P = 50% bias current is
SW bias SW
called the switching current of the circuit, I .
SW
Thechoiceoftheread-outjunctioncriticalcurrentIRO
switching current amplitude in the case when quantum C
for an optimal measurement of i is not straightforward.
S
phase fluctuations would be negligibly small: v ∼ 0. As
On the one hand one would like to have IRO (cid:29) i , but
expected,wegetapracticallyflatdependenceonE /E . C S
J C on the other hand the width of the switching histograms
Further on, the upper x-axis of Fig. 4 shows the ra-
w increases with IRO and so does the probabilistic noise
tio E /k T of the Josephson energy with respect to the C √
J B in the measurement w/ M. For reasonable measuring
thermal energy at T = 50mK. Since E (cid:29) k T, ther-
J B time scales, the number of current steps M is limited to
mal fluctuations are excluded to explain the suppression
valuesofabout104. Ifwewanttomeasuresupercurrents
of the switching current with decreasing E /E . Ad-
J C fortheSQUIDchainintherangeof1nA,IRO needstobe
ditional measurements (not shown here) reveal a con- C
intherangeof100nA. Wehaveusedaread-outjunction
stant switching current amplitude up to a temperature
with a critical current IRO =330nA which offers a good
of T =100mK. C
trade off.
In conclusion, we present for the first time a detailed
experimental characterization of the effect of quantum Acknowledgements
phase-slips on the ground-state of a Josephson junction
chain. These phase-slips are the result of fluctuations We thank B. Dou¸cot, D. Est`eve, F. Hekking and L.
induced by the finite charging energy of each Josephson Ioffe for fruitful discussions. We are grateful to the team
junction in the chain. The experimental results can be of the Nanofab facility in Grenoble for their technical
fitted in very good agreement by considering a simple support in the sample fabrication. Our research is sup-
tight-binding model for the phase-slips8. Our measure- ported by the European STREP MIDAS and the French
ments also show that a JJ chain under phase bias con- ANR ”QUANTJO”.
straint can behave in a collective way very similar to a
single macroscopic quantum object.
1 R.M. Bradley, S. Doniach, Phys. Rev. B,37 , 1138 (1984). 4 R. Fazio, H.S.J. van der Zant, Phys. Reports, 355, 235
2 L.I.Glazman,A.I.Larkin,Phys.Rev.Lett.,79,0031-9007 (2001).
(1997). 5 D.B. Haviland, Y. Liu, A.M. Goldman, Phys. Rev. Lett,
3 A. Finkel’stein, Physica B, 197, 636 (1994). 56, 378 (1986).
5
6 H.M.Jaeger,D.B.Haviland,A.M.GoldmanandB.G.Orr, (2008).
Phys. Rev. Lett, 56, 378 (1986). 18 S.Gladchenko,D.Olaya,E.DupontFerrier,B.Douc¸ot,L.
7 F. Sharifi, A.V. Herzog and R.C. Dynes, Phys. Rev. Lett, Ioffe and M.E. Gershenson, Nature Physics,5,48,(2009).
71, 428 (1993). 19 W. Guichard and F. Hekking, cond-mat 0909.3749.
8 K.A. Matveev, A.I. Larkin, and L.I. Glazman, Phys. Rev. 20 M. H. Devoret, Quantum Fluctutations in Electrical Cir-
Lett. 89, 096802 (2002). cuits, Les Houches,Session LXIII (Elsevier, Amsterdam,
9 K.AndersonandD.B.Haviland,Phys.Rev.B67,092507 1995), Chap. 10.
(2003). 21 K.K. Likharev and A.B. Zorin, J. Low Temp. Phys. 59,
10 B. Abeles, Phys. Rev. B 15, 2828 (1977). 347 (1985).
11 E.Chow,P.Delsing,andD.B.Haviland,Phys.Rev.Lett. 22 D.V. Averin, A.B. Zorin and K.K. Likharev, Sov. Phys.
81, 204 (1998). JETP 61, 407 (1985).
12 Y. Takahide, H. Miyazaki, Y. Ootuka, Phys. Rev. B 73, 23 M.L. Della Rocca, M. Chauvin, B. Huard, H. Pothier, D.
224503 (2006). Esteve, and C. Urbina, Phys. Rev. Let. 99,127005, (2007)
13 HisaoMiyazaki,Masterthesis,QuantumFluctuationsand 24 A.O. Caldeira and A.J. Leggett. Ann. Phys., 149:374,
Dissipative PhasetransitioninJosephson Chains, Univer- (1981).
sity of Tsukuba. 25 B. Douc¸ot and L. B. Ioffe, AIP Conf. Proc., 1134, 128,
14 S.Corlevi,W.Guichard,F.W.J.Hekking,andD.B.Havi- (2009).
land, Phys. Rev. Lett. 97, 096802 (2006). 26 S. Coleman, The Uses of Instantons, from: The whys of
15 V.E. Manucharyan, J. Koch, L.I. Glazman and M.H. De- subnuclearphysics,(PlenumPress,NewYork,1977),page
voret, Science 326, 113-116 (2009). 805.
16 J. Koch, V. Manucharyan, M.H. Devoret and L.I. Glaz- 27 N. Didier, PhD-Thesis, Universit´e Joseph Fourier, Greno-
man, Phys. Rev. Lett. 103, 217004 (2009). ble, (2009).
17 I.M. Pop, O. Buisson, K. Hasselbach, I. Protopopov, W.
Guichard and B. Pannetier, Phys. Rev. B, 78, 104504,