Full Counting Statistics of Cooper Pair Shuttling

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by  Alessandro Romito

Full Counting Statistics of Cooper Pair Shuttling
Alessandro Romito
NEST-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
Yu. V. Nazarov
9
Department of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Nederlands
0
(Dated: January 19, 2009)
0
2 TheCooperpairshuttleisasimplemodelsystemthatcombinesfeaturesofcoherentandincoher-
n enttransport. Weevaluatethefullcountingstatistics(FCS)ofchargetransferviatheshuttleinthe
a incoherentregime. Wedescribetwolimiting cases whentheFCS allows for classical interpretation.
J Generally, the classical interpretation fails yielding negative and imaginary ”probabilities”. This
9 signalsthatsuperconductingcoherencesurviveseveninincoherentregime. Weevaluatethecurrent
1 noise in some detail.

PACSnumbers:
]
l
l
a
h TheJosephsoneffect1 consistsincoherentCooperpair Thus motivated, we have analyzed the FCS of the
- transfer through a tunnel junction between two bulk Cooper pair shuttle in the most interesting incoherent
s
superconductors. It results in a dissipationless current regime, where no net supercurrent is shuttled. The
e
m presentasagroundstatepropertyofthesystem. This is regime is achieved in the limit when the voltage fluctua-
incontrasttodissipativeelectrontransferinnormalcon- tions are classical. If the fluctuations come from an en-
.
t ductors. In latter case, it is possible to evaluate the full vironment,thisimpliesthatthetemperatureoftheenvi-
a
m countingstatistics(FCS)ofchargetransfers2 intermsof ronmentexceedstherelevantenergyscalesoftheshuttle.
classical probabilities. Thereby, the transport properties Naively, from the absence of the net current one would
-
d are fully understood and characterized: one can predict conclude that no charge transfer occurs in the system.
n not only the average current, but also the current noise However, transfers do occur, the current is zero only in
o and all higher moments of the current distribution func- average,andtheFCSpresentsaconvenientwaytoreveal
c tion3. this circumstance. This stipulates the understanding of
[
One can also access the FCS of a superconducting transportpropertiesofthe shuttle and,generally,the in-

2 Josephson junction4. Due to gauge symmetry breaking terplay between the coherent and incoherent transport.
v
in the superconducting state, this FCS can not be in- Theresultsareasfollows. Iftheperiodoftheshuttling
2
terpreted in classical terms yielding negative ”probabili- is sufficiently long for decoherence to be accomplished,
1
4 ties”. The use and the interpretation of the FCS in this the FCS can be interpreted in terms of classical elemen-
2 case is that it determines the quantum evolution of the tary events: Cooper pair transfers. During the shuttling
0 detector that measures the current5. cycle, either no transfer takes place or a pair is trans-
4 A recent proposal puts forward an interesting way to ferredineitherdirection. Thereisanapparentsimilarity
0
transferCooper pairs: to shuttle them (controllably)be- with the FCS of the pumping in normal systems studied
/
t tween the superconducting electrodes6. Although the in9,10. The FCS in the opposite limit of short cycles al-
a
m originalproposalputsemphasisonmechanicaldegreesof lows for alternative classical interpretation in terms of
freedom, a Cooper pair shuttle is essentially a supercon- a superconducting current randomly switching between
-
d ducting single electron transistor (SSET) with variable two opposite values.
n time-dependentJosephsoncouplingtothesuperconduct- Inthe generalintermediatesituation,the FCScannot
o ing leads7. The Coulomb island of this SSET is brought be interpreted in classical terms. An attempt to evalu-
c
in contact with one electrode at a time, the electrode ate the probabilities per cycle yields negative and even
:
v contacted being periodically altered8. imaginaryvalues. Thisisaclearsignatureofthefactthat
Xi It is interesting that the supercurrent between the thesuperconductingcoherencesurvivesstrongdephasing
leads is achieved as a result of a non-equilibrium driven althoughthiscoherencedoesnotmanifestitselfinnetsu-
r
a process, generally accompanied by dissipation. The co- perconducting current. It was recently explained11 that
herence ofCooper pairstransferredis determined by the any FCS can be characterized directly so one would not
coherenceofdifferentchargestatesintheisland,thisbe- have to measure higher cumulants of the current noise
ing mostly affected by fluctuations of the gate voltage7. onebyone. However,theimmediate physicalvaluemea-
The transportresults fromthe interplaybetweenthe co- suredwouldalsodependonthepropertiesoftheconcrete
herent Josephsoncoupling and decoherence effects. This detector. Thisisalsoawaytoexperimentallyobservethe
iswhytheCooperpairshuttlepresentsamodeltobridge FCS of the Cooper pair shuttle.
between the limits of coherentand incoherenttransport. The system of interest is presented schematically in
The model is simple indeed, one can restrict the consid- Fig. 1. It consists of a Cooper pair box, or Coulomb
eration to just two quantum states. island,connectedwithJosephsonjunctions tothe super-
2
conducting leads L and R. Conventionally,6,7,8 we as- externallycontrolled
Josephsonjunctions
sume the separation of energy scales, ∆ ≫ E ≫ E ,
C J
∆ being the superconducting energy gap, E being the
C
charging energy of the box, and E being a typical
J
Josephsonjunctionenergy. Undertheseconditionsquasi-
φ nˆ,ϕˆ φ
particles degrees of freedom are not involved in the sys- L R
tem’s dynamics. In addition, the gate voltage is chosen
to bring two charge states of the box, say, |0> and |1>
tothedegeneracypoint. Undertheseassumption,wecan
restricttheconsiderationtothesetwostatesonlyandthe Vg Cooper
shuttle is described by the Hamiltonian pairbox
E
E(b)(t)
Hˆ(t)=2eV(t)σz −b=XL,R J2 (cid:0)eiφbσ++σ−e−iφb(cid:1) , EJ EJL(t) EJR(t)
(1)
which we write in terms of 2 × 2 Pauli matrices σ ,
z
σ ≡(σ ±σ )/2inthespacespannedby|0>,|1>. V(t) 0 tJ tJ+tC 2tJ+tC T t
± x y
is proportional to the deviation of the gate voltage from
the valuecorrespondingtothe degeneracypointandφ ,
R Figure 1: Upper panel. Cooper pair shuttle consists of a
φL are the phases of the macroscopic superconductors. Cooper pair box coupled to superconducting leads through
Only φ = φR −φL is a physical quantity which we will Josephson junctions. Two charge states are tuned to degen-
assume to be fixed; this can be obtained by closing the eracy point by the gate voltage. Lower panel. The specific
circuitendpointsinalooppiercedbyaconstantmagnetic time dependenceof E(R,L) within a single period T provides
J
flux. E(L,R)(t)aretime-dependentJosephsonenergiesof theshuttling.
J
left and right junction. For the sake of concreteness, we
assumestepwiseperiodicvariationofbothJosephsonen-
absence of Josephson coupling, voltage fluctuations can
ergies with time as shown in the lower panel of Fig. 1.
not cause transitions between the chargestates so no re-
Each lead is contacted during time interval t and the
J
laxationtakes place. With Josephsoncoupling, the volt-
box contactsnoleadduring time intervalt foreachpe-
C
agefluctuationscausetransitionsbetweenthe stationary
riod T = 2(t +t ). We call t and t the “Josephson
J C J C
states separated by energy 2E at V = 0. Classical
contact” and the “free evolution” time respectively. An J 0
voltage fluctuations result in equal transition rates with
idea to realize the required time dependence in Joseph-
increasing and decreasing energy. In fact, the ratio of
son coupling by means of an experimentally realizable
these rates is given by Bolzman factor exp(2E /T ), T
SQUIDs based device has been proposed in Ref. 7. J b b
being the temperature ofthe environmentproducing the
We assume V(t) to be a classical stochastic vari-
fluctuations. As shown in Ref. 7, proper account of this
able with white noise statistics, hV(t)i = V and
hV(t)V(t′)i =γ~2/(2e)2δ(t−t′),whersetohc·.i d0efines factorleadstoanisotropicρˆ6=ˆ1andtothenetsupercur-
stoc. stoc. rent vanishing at T ≫ E . This defines the incoherent
the averageoverthe fluctuations. Thus defined, γ is just b J
regime under consideration.
the inverse decoherence time of the two charge states. If
Let us evaluate the FCS. One starts with the assump-
we neglected the fluctuations, the time evolution of the
tionthatatransportprocesscanbecharacterizedbythe
systemwouldbe fully coherentgovernedby thecoherent
probabilities P (N) of N electrons transferred through
part of the Hamiltonian Hˆ ≡ Hˆ . In this case the τ
c thecontactinatimeintervalτ andattemptstocompute
stoc.
quantumstateofthecentralgrDainEwouldacquiredynam- thecharacteristicfunctionofthisprobabilitydistribution
ical phases 2θ = EJtJ/~ during the Josephson contact defined as2
timesand2χ=2eV t /~duringthefreeevolutiontimes
0 C
(we assume that V0 is only present during both intervals e−S(λ,τ) = Pτ(N)eıλN. (3)
of the free evolution time). N
X
Withfluctuations,theshuttlewillbedescribedbya2×
The general quantum expression for this function can
2densitymatrixthatobeysthefollowingBlochequation:
be obtained by coupling the system to a detector for a
∂ρˆ ı measuring time τ and interpreting the detector readout
∂t =−~ Hˆc(t)ρˆ−ρˆHˆc(t) −2γ(ρˆ−σzρˆσz) . (2) in terms of charge transfer5. The FCS in this case is
(cid:16) (cid:17) definedasintegralkernelthatrelatestheinitialandfinal
The only stationary solution of this equation is quite density matrices of the detector, and reads
trivial: ρˆ∝ˆ1, this corresponds to the absence of the av-
eraged superconducting current. Therefore, the shuttle e−S(λ,τ) =Tr U (τ,0)ρˆ(0)U† (τ,0) , (4)
operates in incoherent regime. This is a combined effect +λ −λ
h i
of the decoherence term and Josephson coupling. In the ρˆ′(τ)
| {z }
3
where Uλ(τ,0) = −→Te−ıR0τ Hˆλ(s)ds is the unitary evolu- Leading corrections to adiabatic FCS are exponentially
tionoperatorcorrespondingtothemodifiedHamiltonian small, ≃exp(−2t γ).
J
Hˆ =Hˆ +λIˆ/e, Iˆbeing the operatorof the electric cur- Another interesting limit is that of small Joseph-
λ
rent. Two such operators provide non-unitary evolution son couplings E ≪ ~γ and sufficiently long cycles
J
oftheinitialdensitymatrixofthesystemρˆ(0)intoρˆ′(τ). ~/t ,~/t ≪ E . In this case, the charge relaxation
C J J
Forourshuttle model,theimplementationofthisgen- time ≃ γ~2/E2 may be long, exceeding both decoher-
J
eral scheme is especially simple since the density matrix ence time and cycle duration. The relevant parameter
is just 2 × 2 matrix. Following Ref. 4, we also gauge exp(−t E2/(~2γ)) ≡ f determines the efficiency of the
J J
the counting field λ to the left electrode, this yields chargerelaxationduringacycle. TheFCSbecomesmore
Hˆ = Hˆ (φ → φ + λ). We derive an equation for complicated, for f ≃1 it is given by
λ c L L
ρˆ′ that looks very similar to the Bloch equation (2)
1
µ¯(λ)= (1+f2)cos2λ+fsin2λ+(1−f2)×
∂ρˆ′ ı 2
=− Hˆ ρˆ′−ρˆ′Hˆ −2γ(ρˆ′−σ ρˆ′σ ) . (5)
∂t ~ +λ −λ z z 1 1−f 2
(cid:16) (cid:17) × cos2λ 1− sin2λ . (9)
The difference is that the Hamiltonians governingevolu- 2vu (cid:18)1+f(cid:19) !
tion of ”bra”’s and ”ket”’s differ by opposite λ shifts. u
t
This system of 4 linear equations can be solved with and, generally speaking, all pN 6=0. This is because the
initial conditions ρˆ′(t ) to obtain a linear map M that chargecannotcompletely relaxduringa cycle providing
0
givesρˆ′ afteracycle,ρˆ′(t +T)=Mρˆ′(t )Allfoureigen- ”memory effect” so that charge transfers in different cy-
0 0
values µ of M satisfy the condition |µ | ∈ [0,1]. Since cles are not independent and elementary event of charge
i i
we study statistics of low-frequency fluctuations, τ ≫T transfer can encompass several cycles. Still, probabili-
by definition. Therefore, the FCS is determined by the ties remain positively defined owing to the fact that in
eigenvalue with the greatest magnitude, the limit of long cycles the FCS depends neither on the
superconducting phase φ nor on dynamical phases θ,χ.
e−S(λ,τ) =µ¯(λ)τ/T , where 1−µ¯= min (1−µi). (6) Beyondthislimit,theFCSdoesdependonφ. Asmen-
i=1,...,4
tioned in Ref. 4, the classical interpretation in this case
Thisisinaccordancewiththemethodandtheresult12for may fail and one can not assure that pn are positive or
even real. Indeed, we have found this in numerical cal-
FCSofchargetransportdescribedbyamasterequation.
culations at γT ≃ 1 (Fig. 2). This clearly signals that
The most transparent way to present the FCS is to
define the probabilities p to transfer N electrons per the superconducting coherence remain in the system al-
N
cycle, as it has been done in Ref. 9 to characterize the though the decoherence completely destroys the average
supercurrent.
pumping of normal electrons,
Thisbecomesevidentwhenanalyzingtheoppositelim-
1 π iting case of very short cycles γT ≪ 1. In this case,
p = dλe−iNλµ¯(λ). (7)
N 2π the cumulants of the current are contributed by FCS at
Z−π
λ≃γT ≪1. In this region, the FCS reads
The FCS in the limit considered is always an even func-
tionofλ,sothatthepairsareshuttledineitherdirection µ¯−1 γ˜ γ˜ 2 λI 2
s
with equal probability, p =p . ≃− + − , (10)
N −N T 2 s 2 e
We analyze the FCS in several limiting cases. First (cid:18) (cid:19) (cid:18) (cid:19)
we consider the limit of long periods 1/t ,1/t ≪ γ ≪ where I is the superconducting current in the absence
C J s
E /~. This is in fact an adiabatic limit since the relax- of decoherence γ =0 and γ˜ is the relaxation time in the
J
ation and decoherence are fully developed within each limit γ →0:
time interval t ,t . In this case,
C J
1 1 2e∂α γ tJ tC
µ¯(λ)=cos2λ → p0 = 2;p±2 = 4, (8) Is = T ∂φ, γ˜ = sin2α T PJ + T PC
(cid:18) (cid:19)
so that, each shuttling between the superconductors where α ≡ arccos cos2θcos2χ−sin2θcosφ and P ,
J
transfers either one Cooper pair or none, this occurs P arealwayspositivepolynomialfunctionsinsin(·)and
C
(cid:0) (cid:1)
with equal probabilities. The pair is transferred with cos(·) of the phases φ,χ,θ:
equalprobabilities ineither direction. Itis interesting to
P (φ,χ,θ) = 2sin2(2θ)(1+cosφcos(2χ))
note that this simple result is quite general and relays C
on neither the periodicity of shuttling nor the concrete PJ(φ,χ,θ) = 2 1+sin2θcosφcos(2χ) +
time dependence of EJ(t) provided the adiabaticity is − co(cid:0)s(2θ) cos2θcos2(2χ)−(cid:1)sin2θcos2φ
preserved. The shuttle may even return (several times)
to the same superconducting terminal before contacting FCS of this type corres(cid:0)ponds4 to a supercurrent ra(cid:1)n-
the opposite one: the charge transfer is in this case as- domly switching at time scale 1/γ˜ ≫T between the op-
sociated with two trips between the opposite terminals. posite values ±I , so that the coherence is preserved at
s
4
thistimescale. Thechargesaretransferredinlongtrains
of ≃1/γ˜T of elementary charge, this leads to significant
(a) low-frequency current noise.
p
N
Low-frequency noise S(ω = 0) is obtained di-
0,8
rectly from the definition of the FCS, S(0) =
0
0.6 −e2/T lim ∂2(µ¯/∂λ2). From the Eq. (9) we obtain
λ→0
0,4 S(0)=2e2/T×(1−f)/(1+f)intheadiabaticlimit. The
2
noise is enhanced in the opposite limit of short cycles,
0.2
4 S(0) = 2I2/γ˜ ≫ e2/T and is sensitive to all dynamical
s
0 0.5 1 2 E /γ phases. Thenumericalresultsintheintermediateregime
J
(Fig. 3) show in addition a quasi-oscillatory dependence
on the dynamical phase θ.
(b) The authors appreciate many highlighting discussions
ℜ(p ) with R. Fazio. This work was supported by the EU
N
(IST-SQUBIT, Grant No. HPRN-CT-2002-00144) and
0,8 0 by Fondazione Silvio Tronchetti Provera.
0,6
0,4
2
0,2
4
−0.02 1 2 4 6 8 E /γ
J S(0)T
4e2
(c) 0.4
ℑ(pN) 2
0
0,08
0.8
0,04
1
1.6
0 1 2 4 6 8 E /γ 0.5 10
4 J
−0.04
0
2 0 1 2 4 6 8 10 12 14 EJ/γ
−0.08
Figure3: Low-frequencynoisein theintermediateregime for
Figure 2: Fourier components, pN, with N = 0,2,4 of µ¯(λ) different values of γtJ, as expressed by each line label. The
plottedversusEJ/~γ. Eachlineislabeledbythecorrespond- noiseanditsoscillations withθ increasewith decreasingγtJ.
ing value of N. Adiabatic regime (a): γtJ = 5; positive Otherparameters are fixed to tJ =tC, χ=π/5, φ=π/3 for
probabilities. General situation (b-c): γtJ = 0.9; pN can be all plots.
negative and even immaginary. Other parameters are fixed
to tJ =tC, χ=π/5, φ=π/4 for all plots.
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