Table Of ContentBand Engineering in Cooper-Pair Box: Dispersive
6 Measurements of Charge and Phase
0
0
2
Mika Sillanpää1, Leif Roschier, Teijo Lehtinen and Pertti Hakonen
n
a
LowTemperatureLaboratory,HelsinkiUniversityofTechnology,FIN-02015HUT,Finland
J
3
1 Abstract. Low-frequencysusceptibilityofthesplitCooper-pairbox(SCPB)isinvestigatedforuseinsensitivemeasurements
ofexternalphaseorcharge.Dependingonthecouplingscheme,theboxappearsaseitherinductiveorcapacitivereactance
] whichdependsonexternalphaseandcharge.Whilecouplingtothesource-drainphase,wereviewhowtheSCPBlookslike
l atunableinductance,whichpropertyweusedtobuildanovelradio-frequencyelectrometer.Inthedualmodeofoperation,
l
a thatis,whileobservedatthegateinput,theSCPBlookslikeacapacitance.Weconcentrateondiscussingthelatterscheme,
h andweshowhowtodostudiesoffastphasefluctuationsatasensitivityof1mrad/√Hzbymeasuringtheinputcapacitance
- ofthebox.
s
e
Keywords: quantummeasurement,Cooper-pair-box
m
PACS: 67.57.Fg,47.32.-y
.
t
a
INTRODUCTION as a function of the classical fields j = 2p F /F and
m 0
n =C V /e(seeFig.1).
g g g
-
d Josephson junctions (JJ) store energyaccordingto E =
n EJcos(j ), where j is the phase differenceacross the
−
o junction, and the Josephson energy E is related to the
J
c junction critical current I through I =2eE /h¯. Since
C C J
[
JJ’salsotypicallyexhibitnegligibledissipation,theycan
1 be used as reactive circuit components. By combining
v theJosephsonequationsI=I sin(j )andj˙ =2eV(t)/h¯,
C
2 whereV(t)isthevoltageacrossthejunction,wefindthat
9
asingleJJbehavesasanonlinearinductance,
2
1 h¯ L
60 LJ(j )= 2eICcos(j ) = cosJ(0j ), (1) FisIlaGnUdR(tEhic1k.lin(ae))hSacsheamtoattaiclscaopfacthiteanScCePCBS .aTnhdecmhaersgoinscgoepnic-
0 where we defined the linear-regime Josephson induc- ergyEC=e2/(2CS );(b)twolowestenergybandsEk(k=0,1)
/ tanceL =h¯/(2eI ). of the SCPB , for EJ/EC =1.7 (without the parabolic back-
mat QuanJt0um effectsCin mesoscopic JJ’s [1, 2] may mod- sguroscuenpdti−bi(lintgiees)2a/r(e2iCllgu)s,trsaeteedEqb.y(2th)e).aIrnrdouwcstivpearaanlldelctaopajcitainvde
ify Eq. (1) in an important manner. In particular, the ng,respectively.
- Josephson reactance may become capacitive [3, 4]. In
d
this brief communication,we investigate the Josephson
n
o reactance in the split Cooper-pair box (SCPB) geome-
QUANTUMINDUCTANCE
c try, with emphasison detector applications.We first re-
: viewtheinductivesusceptibility,andthenconcentrateon
v With respect to j , the SCPT behaves as an inductance
i discussing the capacitivesusceptibility in the spirit of a
X (Fig.2 (b)),dependent,firstofall, onthe bandindexk,
novelphasedetector.Thediscussionreliesheavilyonthe
aswellasonn andj :
r energy bands [5] E of the SCPB, two lowest of them g
a k
giveninthelimitE /E 1as
E0,1=EC(n2g−2nJg+C2≪)∓ Lkeff(ng,j )= dd2FE2k −1= F2p0 2 dd2jE2k −1. (3)
(2) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(E cos(j /2))2+(2E (1 n ))2 C V2/2
q J C − g − g g TbeheensutrsoendgbnygthdeeppernedseenntceauotfhoLr0esfftowihmenplEemJ/eEnCta≪fa1sthraes-
active electrometer [6], using the scheme of Fig. 2 (a).
1 presentaddress:NationalInstituteofStandardsandTechnology,325 Themeasurementsareperformedbystudyingthephase
Broadway,Boulder,CO80305,USA
shift Q = arg(V /V ) of the "carrier" microwave re- (n = 1,j =0).UsingEq.(2),wehaveL =4L .
out in g eff,0 J0
±
flectedfromaresonantcircuitcontainingtheSCPB.De- For the best electrometer performance, n should be
g
noting by Z the lumped-elementimpedance seen when biasedatthepointsmarkedbycirclesinFig.2(b).From
lookingtowardstheresonancecircuitfromthetransmis- Eq.(2)wealsofindthemaximumgaing whichgrows
m
sion line of impedanceZ =50W , the reflection coeffi- rapidly when E /E 1: g 2(E /E ) 1. Another
0 J C m J C −
≪ ≃
cientofavoltagewaveis important figure is the value of L at the optimal gate
eff
bias which yields g , denoted here as L [9]. To
V Z Z m eff,m
G = Vout = Z−+Z0 =G 0eiQ . (4) someextent,therapidlygrowingLeff,mtowardslowering
in 0 E /E (see Fig. 3) cancels the benefit of growing g
J C m
Sincethewholesetupconsistsinprincipleonlyofreac- fromthepointofviewofchargesensitivity.
tances,theinductivelyreadschemeshouldbesuperiorin
termsofnoiseandback-action[7]overthepreviousfast
electrometer,therf-SET[8],whichreliesonthecontrol
ofdissipation.
FIGURE 3. Numerical values of the SCPB ground band
JosephsoninductanceLeff,0(thatatng= 1,j =0),andLeff,m
(atmaximum gain, atj =0) for atypic±al aluminiumdevice.
Also shown is the "classical" Josephson inductance L in
J0
Eq.(1).
Without going into details, optimal charge sensitiv-
ity limited by zero-point fluctuations in the loaded LC-
oscillatorinFig.2(a)is[10]:
16√2e(L )2√2k T
sQL= eff,m B N, (6)
q g p √h¯F L √Q
m 0 J0 i
where T is the noise temperature of the rf-amplifier,
N
and Q is the internal quality factor of the resonator.
i
Evaluating the values in Eq. (6) numerically, we find
that s 10 7e/√Hz, order of magnitude better than
q −
∼
the shot-noise limit of rf-SET, is intrinsically possible
for the L-SET if Q 103 and T 200 mK. So far,
i N
∼ ∼
FIGURE 2. (a) Schematics of the capacitively coupled the sensitivity in experiment [10, 11] has been limited
"L-SET" inductive rf-electrometer. The resonance frequency by Qi.20 downto sq 2 10−5e/√Hz. The limit of
≃ ×
f 1 =2p (L Lk )C depends on the SCPB Josephson in- Eq.(6) isreachedwhenparametervaluesare chosenso
p− k eff that
ddeurcitvaantciveeL(qikenffv;er(sbe)Lc0alc,uselaeteEdq.m(3o)d)ualtatthioenSCofPBthegroseucnodnednenrggy- w p= F 20(Leff,m+L). (7)
eff 64h¯L L
band,fordifferentEJ/EC,andj =0.Thecirclesmarkoptimal eff,m
biaspointsfortheelectrometeroperation. Equation (7) yields values typically f = w /(2p )
p p
≃
The crucial number for electrometer operation is the 1 2GHz,thoughdependenceof fpisratherweak.
−
differential modulation of L (at the ground band), or
eff
dimensionless"gain":
QUANTUMCAPACITANCE
¶ L
eff
g , (5)
≡ ¶ ng(cid:18)Leff,0(cid:19) The band energies of an SCPB depend on the (gate)
which we have presented as normalizedby Leff,0 which charge ng, see Fig. 1 (b), and the SCPB should then
denotes the Josephson inductance at the special point behavelike a capacitancewith respectto changesof ng
[2,3],whichmeansthatthepointofobservationisatthe Themaximumof f w.r.t.j atn =0isplottedinFig.5.
g
gateelectrode: We consider the experimental setup of Fig. 4 (a),
where the quantum capacitance C is in parallel with
eff
Ck = ¶ 2Ek(j ,ng) = Cg2¶ 2Ek(j ,ng). (8) a(generallymuchlarger)straycapacitanceC,andforms
eff − ¶ V2 −e2 ¶ n2 a resonator with an inductance L. In this scheme, it is
g g
typical to operate in the limit of vanishing internal dis-
PhasemodulationoftheinputcapacitanceC (n ,j )of sipationwhichcorrespondstochangeofphaseQ ofthe
eff g
theSCPBobservedinthismannerisplottedinFig.4(b). reflectedcarrierchangingby2p aroundtheresonantfre-
Asseeninthefigure,C hasastrongphasedependence quency f .
eff p
in the limit E /E 1 around j = p . Exactly at Similarly as in the inductive readout, there are here
J C
j = p , Cooper-pai≫r tunneling is comp±letely blocked, no internal noise sources except quantum fluctuations
±
and C reduces to classical series capacitance of the intheresonator.Typically,therefore,sensitivityisagain
eff
junctionsandCg,thatis, (C1+C2)−1+Cg−1 −1. limited by noise of the preamplifier:spectral density of
The input capacitance depends sensitively (quadrati- thevoltagenoisereferredtopreamplifierinputissVout=
cally) on the coupling ca(cid:2)pacitance Cg, and(cid:3)even when √2kBTNZ0, which can be regarded as a phase noise of
Cg ismadeunusuallylargesuchthatitpracticallylimits themicrowavecarrier,sQ =sVout/Vout.Whenthecarrier
thechargingenergy,C typicallyremainsverysmall,in amplitudeisoptimallylarge,itcanbeshownthatunder
eff
thefemto-Faradrange,seerighthandscaleofFig.4(b). the conditions mentioned, V = e Z C 1 . When
However,ithasbeensuggestedthattheextremelystrong out 2Cg 0 L2p
referred as an equivalent flux noise at qdetector input
phasedependencecouldbeusedforfast,reactivelyread
usingEq.(9),theresultbecomes
phase detection [4]. This "CSET" mode of operation is
somewhatdualtothe"L-SET"electrometry. sj =¶ Q s/Q ¶j =2√p e CC √kfBTENZ0
(cid:18) g(cid:19) m C (10)
4√p C√k T Z
B N 0
,
≃ fme
where the last form follows from the assumption that
at high E /E , charging energy is limited by the large
J C
gate capacitance. This is the ultimate limit with ad-
vanced junction fabrication (very thin oxide). The pre-
dicted phase sensitivity is plotted in Fig. 5. Evidently,
sensitivityimproveswithdecreasingstraycapacitanceC,
sincethisresultsinlargermodulationoftotalcapacitance
C+Ceff.Weseethatsj <10−6rad/√Hz,farbeyondan
equally fast rf-SQUID, is possible in principle at high
E /E 10andalowstraycapacitanceC C .
J C g
∼ ∼
We investigated the discussed phase detection exper-
imentally in the scheme of Fig. 4 (a), with the param-
eter values E =0.30 K, E =0.83 K, E /E =0.36,
J C J C
C =0.65fF,C=250fF,C/C =380,andL=160nH.
g g
Except C , the sample parameters were determined by
g
microwave spectroscopy [12]. To the input bias coil of
thephasedetector,weappliedlow-frequencymodulation
by0.013F at80Hz.Itsamplitudewascalibratedrely-
0
FIGURE4. (a)Schematicsoftheexperimentusedtostudy ing on F -periodicity of the static response. This way,
0
how the SCPB appears as a tunable capacitanceCeff; (b) left weobtainedasensitivityof1.3mrad/√Hz,seetheblack
scaleisthecalculatedsecondng-derivativeoftheSCPBground curveinFig.6,limitedbythe4Kamplifiernoise,which
band atng=0, andright scaleisthecorresponding effective figureisevenbetterthanexpected(seeFig.5).
capacitanceifCg=1fFandCS =2fF.
We shall now discuss Fig. 6 in more detail. Both the
An important figure of merit for phase sensitivity is curvesweremeasuredatafluxbiasclosetoj p which
thedifferentialgain,analogoustoEq.(5): yieldsthe largestgain f . For the blackcurve∼, f(n ,j )
m g
was further maximized by tuning n close to 1, which
g
¶ C also yielded a high level of low-frequencynoise as can
eff
f . (9)
≡ ¶j Cg2/(2CS )! be seen in the data. Since the low-frequency noise is
This yields the gray line in Fig. 6, with the numbers
around10Hzbeingcomparabletobigjunctions.
FIGURE5. Leftscale(dashedlines):Phasesensitivitypre- FIGURE 6. Measured equivalent flux noise at CSET in-
dicted for the CSET, Eq. (10), first form, if Z0 =50W and put (left scale, black curve) and critical current noise (right
EC =1K,fordifferent ratiosof thegatecapacitancetostray scale, gray curve). Low-frequency flux modulalation by
capacitance. Right scale: the maximum gain fm of the phase 0.013F 0,RMSat80Hzwasusedasamarker.
detector.Experimentalpointisgivenbytherectangle(notethat
ithadalargercapacitanceratioof 380).
∼
ACKNOWLEDGMENTS
significantly reduced when we tuned n =0 where the
g
response is insensitive to charge fluctuations (the gray We thank T. Heikkilä, F. Hekking, R. Lindell, Yu.
curve), we assign the increased noise around n =1 to Makhlin,M.Paalanen,andR.Schoelkopfforcomments
g
theubiquitouslow-frequencybackgroundchargenoise. and useful criticism. This work was supported by the
Since the low-frequencynoise at n =0 is free from Academy of Finland and by the Vaisala Foundation of
g
theeffectofchargenoise,wewereabletodirectlymea- theFinnishAcademyofScienceandLetters.
sureintheschemetheapparentfluxnoise,whichweat-
tribute to critical-current fluctuations. The power spec-
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2e 2e ¶ Ceff −1
sIC = h¯ sEJ = h¯ fmsj C0 ¶ E . (13)
(cid:18) J (cid:19)
