Transport properties of a periodically driven superconducting single electron transistor

0.77 MB·English

by  Alessandro Romito

Transport properties of a periodically driven superconducting single electron
transistor
Alessandro Romito
Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel and
NEST-CNR-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
9
0
0 Simone Montangero
2 NEST-CNR-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy

n
a Rosario Fazio
J International School for Advanced Studies (SISSA) via Beirut 2-4, I-34014, Trieste, Italy and
0 NEST-CNR-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
2 (Dated: January 20, 2009)

We discuss coherent transport of Cooper pairs through a Cooper pair shuttle. We analyze both
]
l the DC and AC Josephson effect in the two limiting cases where the charging energy EC is either
l
a muchlargerormuchsmallerthantheJosephsoncouplingEJ. InthelimitEJ ≪EC wepresentthe
h detailedbehaviorofthecriticalcurrentasafunctionofthedampingratesandthedynamicalphases.
- The AC effect in this regime is very sensitive to all dynamical scales present in the problem. The
s
effectoffluctuationsoftheexternalperiodicdrivingisdiscussedaswell. Intheoppositeregimethe
e
m systemcanbemappedontothequantumkickedrotator,aclassicallychaoticsystem. Weinvestigate
thetransport properties also in this regime showing that the underlyingclassical chaotic dynamics
t. emergesasanincoherenttransferofCooperpairsthroughtheshuttle. Foranappropriatechoiceof
a theparameters theCooper pair shuttlecan exhibit thephenomenonof dynamical localization. We
m
discussindetailsthepropertiesofthelocalizedregimeasafunctionofthephasedifferencebetween
- the superconducting electrodes and the decoherence due to gate voltage fluctuations. Finally we
d point how dynamical localization is reflected in thenoise properties of the shuttle.
n
o PACSnumbers:
c
[

I. INTRODUCTION ofsingleelectroneffectswiththeintensivelystudiedarea
3
v of Nano-ElectroMechanical systems7. Among the most
7 interestingdevicesinthis areathereis the electronshut-
Soon after the appearance of the microscopic theory
5 tle (for a review see Ref. 8). In its essential realization,
ofsuperconductivity1, Josephsonpredictedaremarkable
6 a shuttle system consists of a small conducting grain,
9 manifestation of macroscopic quantum coherence2 by
in Coulomb blockade regime, oscillating periodically be-
0 showing that two superconducting electrodes connected
tween two electrodes (source and drain). The essential
6 byaninsulatingbarriercansustainadissipationlesscur-
conditiontocharacterizetheshuttlingmechanismisthat
0 rent. Since its discovery, the Josephson effect has had a
t/ tremendousimpactbothinpure3,4 andappliedphysics4. the grain must be in contact with a single electrode at
a any time. Following the original proposal of a normal
One ofthe mostrecentandexciting developmentsin the
m shuttle Gorelik et al.9 introduce the Cooper pair shuttle
researchbasedonthe Josephsoneffectisprobablyinthe
where all the device (electrodes and central island) is in
- implementationofsuperconductingnanocircuitsforsolid
d thesuperconductingstate. Despitethefactthatthecen-
state quantum computation5.
n tralislandisneverconnectedtothetwosuperconductors
o In nanodevices a new energy scale appears,the charg- simultaneously, the Chalmers group has shown that the
c ing energy, and new interesting effects show up due to system is still capable to establish a global phase coher-
:
v the interplay between Josephson coupling and the pres- ence and hence support a finite Josephson current9,10.
Xi ence of charging. The Josephson coupling, leading to The shuttle does not only carry charge, as in the nor-
phase coherence between the two superconducting elec-
mal metal case, but it also establishes phase coherence
r
a trodescanbeunderstoodintermsofthecoherentsuper- between the superconductors. Differently from the nor-
positionofdifferentchargestates. Coulombblockade6on
mal metal case, the Cooper pair shuttle does not need
theothersidetendstolocalizethechargethusdestroying a moving island, it is just a SSET with time dependent
phase coherence. The simplest example of this interplay
Josephson couplings and therefore it can be realized in
is provided by behavior of the supercurrent through a the standard SSET with time-dependent fluxes11. The
Superconducting Single Electron Transistor (SSET)3. It
propertiesoftheCooperpairshuttlecruciallydependon
consists in a small superconducting island connected, by the decoherence mechanism which is also responsible in
tunnel junctions, to two superconducting electrodes.
driving the system toward a steady state. The presence
Additional features emerge if the SSET is coupled to of dissipation modifies the current phase relation, but
mechanical degrees of freedom thus combining the field does not (in general) destroy the Josephson current9,11.
2
The effect ofgatevoltagefluctuations has beenanalyzed
inRef. 11where it hasbeen shownthat decoherencecan
even enhance the Josephson current. Additional work
on the Cooper pair shuttle considered the full counting
statistics of Cooper pair shuttling12, and the possibility
of observing quantum chaotic dynamics13.
The present paper extends our previous works on the
subject11,13. In addition to our previous results we pro-
vide details ofthe derivationofDC Josephsoncurrentin
the limit E /E 1, and consider several extensions
J C
≪
which are important for a connection with possible ex-
periments. We also analyze the effect of fluctuations in
theexternaldrivingandtheeffectofanexternalvoltage.
Moreover we discuss the AC effect and study the inter-
play of various times scales on the spectrum of the AC
current. IntheoppositelimitE /E 1,whichhasnot
J C
≫
beendiscussedintheliteraturesofar,wepresentanalyt-
ical and numerical results on the dynamical localization
and discuss its signatures on the Josephsoncurrent fluc-
Figure 1: (Color online). Upper left panel. Schematic repre-
tuations.
sentationofthesystemdescribedbytheHamiltonianEq.(1).
The paper is organized as follows. In Section II we It consists of a Cooper pair box coupled through externally
present the model of the Cooper pair shuttle. In Sec- switched Josephson junctions to phase biased superconduc-
tionIIIweanalyzethetransportpropertiesoftheshuttle tors. Upper right panel. Time dependence of the left and
in the Coulomb blockade regime. We present the details rightJosephson energieswithinasingleperiod. Lowerpanel.
of the formalism to determine the steady state density SketchoftheCooperpairshuttle’scycle. Thethreeintervals
matrix,andtoderivethesteadystateJosephsoncurrent, L, C and R, within the period T = tL+t→+tR+t←, cor-
whosephysicalfeaturesarediscussedinSubsectionIIIC. respond to the situations: (L) EJ(L)(t) = EL, EJ(R)(t) = 0
Subsections IIID and IIIE are dedicated respectively to (interactiontimeatleftlead);(C)EJ(L)(t)=0,EJ(R)(t)=0
(freeevolutiontimeinforwardandbackwarddirections);(R)
the effect of fluctuations of the external driving on the
Josephsoncurrentandto the effect ofanappliedvoltage EJ(L)(t)=0, EJ(R)(t)=ER (interaction timeat right lead).
bias,the AC Josephsoneffect. The chaotic regimeofthe
Cooper pair shuttle is the subjectof SectionIV. We dis-
makingtheislandtomoveorbytuningintimemagnetic
cuss the dynamics of the charge in the central island, in
fluxesandgates. Thishastobe contrastedwiththecase
Subsection IVA. A new feature which appear as com-
ofsingleelectronshuttlewherefortheimplementationof
paredtothe kickedrotatorisanextraphaseshiftduring
the shuttle a mechanical moving island is necessary14,15.
the kicks which is due to the superconducting phase dif-
The system is described by the following Hamiltonian
ferenceoftheelectrodes. Thisphaseshiftplaysakeyrole
since it is responsible for time reversal symmetry break- H =E (t)[nˆ n (t)]2 E(b)(t)cos(ϕˆ φ ). (1)
ingwhoseconsequencesforthedynamicsareinvestigated 0 C − g − J − b
b=L,R
inSubsectionIVB. InSectionsIVCwedevelopthe nec- X
essary formalism to calculate the full counting statistics In Eq.(1) nˆ is the number of excess Cooper pairs in the
in the chaotic regime. The concluding remarks are pre- central island and ϕˆ is its conjugate phase, [nˆ,ϕˆ] = i.
sented in Section V. Several technical details are given The charging energy is given by E (t) = (2e)2/2C −(t)
C Σ
in the Appendices A, B and C. Throughout the paper with C (t) = C (t) + C (t) + C (t) the total capaci-
Σ g L R
kB =1. tance of the SSET (CL/R/g are the various capacitances
indicated in Fig. 1), E(L,R)(t) are the Josephson cou-
J
plings to the left or right lead respectively, and n (t) =
g
II. THE MODEL C (t)V (t)/2e is the gate charge which can be tuned via
g g
the gate voltage V .
g
The Cooper pair shuttle is schematically shown in TheHamiltonianinEq.(1)isnothingelsethanaSSET
Fig. 1. It consists of a central island connected to two with an externaldrive contained in the time dependence
superconductingelectrodesandcapacitivelycoupledtoa of the Josephson energies and of the gate voltage. If the
gatevoltage. Thesuperconductingislandissmallenough time dependence of the coefficients is neglected the sys-
such that charging effects have to be included. The tem is a SSET whose physics is known both in the case
two leads are macroscopic and their phases φ can be of macroscopic junctions (E E ) and in the pres-
L,R J C
treatedasclassicalvariables. Thecouplingsofthe island ence of charging effects (E ≫E )3. By introducing a
J C
≪
to the leads are time-dependent. This time dependence time dependence of the coefficients, it is possible to ex-
isgivenbyexternalmeansandcanbeachievedeither by plore different regimes. A case of adiabatic change of
3
E(b)(t) is that of a Cooper pair sluice which has been not require any mechanically moving part. Here the
J
experimentally and theoretically discussed in the litera- time dependence of the Josephson couplings and n is
g
ture16,17. The shuttling mechanism we are interested in obtained by a time dependent magnetic field and gate
isessentiallycharacterizedbythesequenceoftimelapses voltage, respectively. The setup consists of a more com-
during which the grainexchanges charges with the leads plicated superconducting nanocircuit in a uniform mag-
and time intervals during which it is isolated from the netic field as sketched in Fig.2. By substituting each
leads. The island is said to be in contact with one of Josephson junction by SQUIDs, it is possible to control
the leads when the corresponding Josephson coupling is the E(b)(t) by tuning the applied magnetic field piercing
J
non-zero (with value EL, ER) (configurations L and R the loop. The presence of three type of loops with dif-
in Fig.1). In the intermediate region (configuration C), ferent area, A ,A ,A allows to achieve independently
L R C
EJ(L)(t) = EJ(R)(t) = 0. Note that both Josephson cou- thethreecases,whereoneofthetwoEJ’siszero(regions
plingareneveronatthesametime. AsinRef.10weem- L,R) or both of them are zero (region C), by means of a
ployasuddenapproximation(whichrequiresaswitching uniform magnetic field. If the applied field is such that
time ∆t ~/E ) and suppose E(L,R)(t) to be step a half-flux quantum pierces the areas AL,AR or AC, the
≪ L(R) J Josephson couplings will be those of regions R,L and C,
functions in each region (see Fig. 1). In case of a me-
respectivelyandtheHamiltonianofthesystemcanbeex-
chanical realization of the Cooper pair shuttle such an
actlymappedontothatofEq.(1). Moreover,bychoosing
approximation is well justified due to the rapid decay of
the ratios A /A = 0.146, and A /A =0.292 the two
the Josephson coupling with the distance between the C R C L
Josephson coupling are equal, E =E =E . This im-
grain and the lead. For later convenience we define the L R J
plementationhasseveraladvantages. Itallowstocontrol
functions
thecouplingwiththeenvironmentbysimplyvaryingthe
Θ (t) = θ(t)θ(t t), (2) time dependence ofthe appliedmagneticfield. The time
L L
−
Θ (t) = θ(t (t +t ))θ(t +t +t t), (3) scaleforthevariationofthemagneticfieldshouldbecon-
R L → L → R
− − trolledat the same levelas it is done in the implementa-
in order to write the time dependent Josephson energies tionofJosephsonnanocircuitsforquantumcomputation
as (see Ref. 5 for an extensive discussion).
(b)
E (t)=E Θ (t nT) , b L,R (4) For a quantitative comparison with the results de-
J b b − ∈{ }
n∈N scribed here, the magnetic field should vary on a time
X
scale shorter than ~/E , typically a few nanoseconds
ThetotalcapacitanceC (t)isweaklydependentontime J
Σ with the parameters of Ref. 18. This is possible with
at the contact regions50 and therefore we assume it to
present day technology19. At a qualitative level the fea-
be constant during the intervals L and R (obviously the
tures of the Josephson current presented in this paper
same hold for E (t) = E ). In the intermediate region
C C do not rely on the step-change approximation of the
(C) it is not necessary to specify the exact time depen-
Josephsoncouplings. Theseeffects areobservableevenif
dence of E (t), as it will be clear in Section III.
C the magnetic field changes on time-scales comparable or
In the rest of the paper we study the transport prop-
slowerthan E . The only strictrequirementis that only
erties of the Cooper pair shuttle. The transfer of charge J
one Josephson coupling at the time is switched on.
isexpressedbythepresenceofacurrentatleftandright
contacts. The corresponding current operators are, in
the Schr¨odinger picture,
E
Iˆ (t) = 2e L sin(ϕˆ ϕ ) Θ (t), (5)
L ~ − L L
E III. COULOMB BLOCKADE REGIME
Iˆ (t) = 2e R sin(ϕˆ ϕ ) Θ (t), (6)
R ~ − R R
corresponding to the coherent exchange of Cooper pairs We first consider the system when E ,E E , i.e.
L R C
≪
between the grain and the left or right lead respectively. in the Coulomb blockade regime. In addition, the gate
Due to the periodical external driving, any interaction voltage is chosen so that 0 < n (t) 1/2. Namely
g
≥
with the external environment leads to a steady state, n (t) = 1/2 when the system is in contact with one of
g
where every observable is periodic. We will essentially the lead and n (t) = const. (0,1/2) in the remaining
g
∈
ignoretransienteffectsandconcentrateonthestationary time ofthe cycle. Ourchoice(the sameofRef.9)results
values of physical observables. in having exact charge degeneracy during the Joseph-
son contacts, then enhancing charge transfer. A differ-
entcondition,ofeasierexperimentalrealization,inwhich
A. Cooper pair shuttle with time-dependent fluxes n (t) = const. is discussed in Appendix A. In this limit
g
one can restrict the Hilbert space of the system to the
Beforeanalyzingindetailthe transportproperties,we one spannedby the twochargestates n=0 , n=1 .
{| i | i}
discussawaytorealizeaCooperpairshuttlewhichdoes The Hamiltonian of the system restricted to the two di-
4
be fully coherent. By including fluctuations, the shuttle
will be described by 2 2 density matrix that obeys the
×
following Bloch equation:
∂ρˆ ı
= Hˆ (t)ρˆ ρˆHˆ (t) 2γ(ρˆ σ ρˆσ ) . (9)
∂t −~ 0 − 0 − − z z
(cid:16) (cid:17)
The only stationary solution of this equation is trivial:
ρˆ = ˆ1/2, this corresponds to the absence of any aver-
age superconducting current. This is a combined effect
of the decoherence term and Josephson coupling. In the
absence of Josephson coupling, voltage fluctuations can
not cause transitions between the chargestates so no re-
laxation takes place. With Josephson coupling switched
on,thevoltagefluctuationscausetransitionsbetweenthe
stationary states separated by energy E(L,R). Classical
J
voltage fluctuations result in equal transition rates with
increasing and decreasing energy. The vanishing of the
critical current has a simple explanation, the classical
Figure 2: (Color online). The setup for the implementation
noise mimics a bath at high temperature51. No coher-
oftheshuttleprocessbymeansofatime-dependentmagnetic
ence can be established at temperatures much higher
field. Theinsetshowsthetimedependenceoftheappliedfield
than the Josephson coupling energy. Nevertheless this
(inunityofB0 =Φ0/(2AC),Φ0 isthefluxquantum)inorder
torealize Cooper pairshuttling. Thedifferentloop areascan model is worth to be considered because, as shown in
be chosen in order to obtain EL=ER. Ref. 12 there is an high temperature regime where the
average current is zero but still coherence manifests in
the higher moments of currentfluctuations. At low tem-
mensional vector space, reads peratures, T . E(L,R) the interactions with the bath
b J
can lead to a density matrix ρˆ = ˆ1 and then to a non-
1 E(b)(t) 6
Hˆ0 =EC(t)[ ng(t)]σz J e−iφbσ++h.c. vanishing supercurrent.
2− − 2
b=L,R
X (cid:0) (cid:1)
(7)
B. Quantum noise
where we used the 2 2 Pauli matrices σ (i = x,y,z)
i
×
with the standard notation σ =(σ σ )/2.
± x y
±
In order to evaluate the current, Eqs.(5,6), or the av- In order to analyze the low temperature regime, we
erage value of any observable, we need to compute the need to take into account the quantum features of the
reduced density matrix of the central island ρ(t). The bath. As the most important source of fluctuations in
steady state density matrix depends on the specific de- the charge regime are gate voltage fluctuation we couple
coherence mechanism. The main source of decoherence the shuttle via the charge operator nˆ to an environment
in the Cooper pair shuttle is due to gate voltage fluctua- described by the Caldeira–Leggett model20,
tions,eitherinducedbytheelectromagneticenvironment
or by background charges. Hint =nˆOˆ+Hbath =nˆ λi(ai+a†i)+Hbath . (10)
i
X
In Eq.(10), H is the bath Hamiltonian, with boson
A. Classical noise bath
annihilation/creationoperators of the i th mode a , a†,
− i i
H = ω (a†a +1/2). Due tothe periodicity ofthe
Ataclassicallevelvoltagefluctuationscanbeincluded bath i i i i
externaldriving the time evolutionofthe systematlong
by adding a classical stochastic term to n (t). The
g time, t PT, can be determined by iterating the evo-
Hamiltonian in Eq. (7) is modified by the presence of ≫
lution of the density matrix ρ(t) over one single period.
the extra term,
This evolution can be computed through a linear map
Hˆ =Hˆ0+ξ(t)σz, (8) Mt→t+T defined by
ρ(t+T)= [ρ(t)]. (11)
where ξ(t) has a white noise spectrum Mt→t+T
With the following choice of parametrizing ρ(t) =
ξ(t) = 0
h istoc 1/2[1I +σ r(t)], where i = x,y,z and ri(t) = σi , the
ξ(t)ξ(t′) = ~2γδ(t t′) map in Eq·. (11) assumes the form of of a generhaliaffine
h istoc −
map for the vector r(t):
where γ is the inverse decoherence time. If we neglected
the fluctuations, the time evolution of the system would r(t+T)=M r(t)+v , r (0) R3 (12)
t t 1
∈B ⊂
5
where is the Ballof unitary radiusin R3. The matrix approximation. The solution of the master equation in
1
B
M fulfills the property the contact region can be obtained in the form
t
M v v v (0), (13) r(t )=exp(G t )r(0) 2γ G−1 [1I exp(G t )]w .
| t |≤| | ∀ ∈B1 L L L − L L · − L L L
(18)
as we will see from its explicit form determined below. TheparametersoftheHamiltonianenterthefinalresults
In the long time limit, the system reaches a periodic only through the combinations θ = E t /~
L(R) L(R) L(R)
steady state, and γ t . Due to the condition γ E /~
L(R) L(R) L(R) L(R)
the parameter ~γ /E does not enter≪the results
L(R) L(R)
r (t)=(1I M )v , (14)
∞ t t at lowest order.
−
During that part of the cycle when the island is dis-
if, and only if, det(1I M )=0. When this condition is
− t 6 connected from both electrodes, the situation is simpler.
not satisfied the external bath introduced in Eq. (10) is
Since nˆ is conserved, the evolution can be determined
not effective and the system never loses memory of the
exactly
initial conditions.
The stationary limit is the fixed point of 21.
The expression of r (t), and therefore ofMρt→(t+t)T = r(t→+tL)=exp(G→t→)·R(χ→)r(tL) . (19)
∞ ∞
1/2[1I +σ r (t)], uniquely determines the steady state
∞ In the previous equation we defined
·
of the system.
The periodic time dependence of any physical observ- γ 0 0
→
able A is given by G = −0 γ 0 (20)
→ →
 − 
0 0 1
Aˆ(t) =Tr ρ (t)Aˆ , (15)
h i { ∞ }  
and
where the operator Aˆ is in the Schr¨odinger representa-
tion. cos(χ ) sin(χ ) 0
→ →
−
The assumption of a stepwise varying Hamiltonian R(χ )= sin(χ ) cos(χ ) 0 . (21)
→ → →
 
considerably simplifies the form of the map t→t+T, 0 0 1
M
whichnowcanbeexpressedasacompositionofthetime
 
evolutions of ρ in the intervals L, C, R (see Fig.1). In where χ = tL+t→E (t)(1 2n )/~. The rate γ
each time interval it is straightforward to solve the cor- → tL C − g →
depends only on the properties of the bath. Its explicit
responding master equation for the reduced density ma- R
time-dependence varies when the time scale is compared
trix22. In the portion of the cycle corresponding to the
withthe inverseultravioletbathmode cut-off,1/ω ,and
c
islandbeingincontactwiththe leftelectrodethemaster theinversebathtemperature,~/T 23,24. Anexpressionof
b
equation reads
γ in terms of bath parameters can be obtained within
→
the same Born-Markov approximation discussed above
r˙(t)=G (t)r(t)+2γ w (16)
L L L in the case of a weakly coupling between the bath and
with w† =tanh(E /T ) cosφ , sinφ , 0 and the system. It gives γ→ = 2παTb/~ in which case γ→ is
L L b L L independentontimeandthedecayispurelyexponential.
2γ (cid:0) 0 EL sinφ(cid:1) The same equation holds in the backward free evolution
− L − ~ L time
GL = 0 −2γL −E~L cosφL . (17)
E~L sinφL E~L cosφL 0 r(T)=exp(G←t←) R(χ←)r(T t←), (22)
  · −
Here, γ is the dephasing rate (for this portion of the
L whereG← is definedasG→ inEq.(19)with the replace-
cycle), depending onthe temperature of the bath, which
ment γ γ . In addition to the dynamical phases
→ ←
is taken in thermal equilibrium at temperature T . The →
b χ and θ , also the phase difference φ=φ φ
→(←) L(R) L R
master equation when the island is in contact with the −
entersindeterminingthephysicalobservables. Theeffect
rightsuperconductingleadgoesalongthesamelineswith
ofdamping is characterizedby the dimensionless quanti-
the substitution L R (thus introducing a dephasing
ties γ t , and γ t .
→ L(R) L(R) →(←) →(←)
rate γ ). Both dephasing rate can be obtained in the
R From Eqs. (16-19) it is easy to check that M ful-
t
Born-Markov approximation22, which requires that the
fills the property in Eq. 13 except for the following val-
bathautocorrelationtimeisthesmallesttimescaleinthe
ues (γ ,γ ,γ ,γ ) = (0,0,0,0) or (γ ,γ ,θ ,θ ) =
L R → ← L R L R
problem. This treatment is valid provided that γ
L(R) (0,0,kπ/2,hπ/2)),k, h integers, when det(1I M )=0.
T /~,E /~, and that the time interval t is mu≪ch − t
b L(R) L(R) Inthesecases,thesystemkeepsmemoryofitsinitialcon-
longer than both ~T−1 and ~E−1 . As an example, for ditions and it never approaches the steady state. This is
b L(R)
anOhmic bathwithcouplingto the environmentα 1, however an artificial situation, because other sources of
one has γ = (π/2)αE coth(E /2T )/~20.≪G dissipation are present and will drive the system to a
L(R) L(R) L(R) b L
istimeindependentasaconsequenceoftheBorn-Markov steady state.
6
Let us comment on the assumption of Heaviside func- the period T is much larger than the inverse dephasing
tions for E (t) we used to determine the steady state rates,the shuttle mechanismis expectedto be inefficient
L(R)
densitymatrix. Itdefinesthesimplestmodeltocatchthe and the critical current is strongly suppressed. In the
features of the shuttling mechanism, i.e. the existence of followingwewilldescribeaquiterichscenario,depending
different regimes during a single period time evolution. ontherelativevalueofthe varioustimescalesandphase
In fact the precise shape of the Josephson energy pulses shifts.
is not relevant, changing it will change the definition of As charge is conserved by the coupling to the envi-
the dynamics phases χ, θ, -eqs. (18), (21)- which en- ronment, [nˆ,Hˆ ] = 0, current can flow only trough
int
ter as parameters in the results for the density matrix. the electrodes. Therefore, in the Heisenberg picture,
What is indeed neglected in our model is the effect of Iˆ (t)+Iˆ (t)+nˆ˙ = 0. By integrating over a period the
L R
excitinghigherenergymodesandtheeffectsofgatevolt- average current reads (I = I I)
R L
− ≡
age fluctuations during the switching time. These are
goodapproximationsforswitchingtimes∆t.~/EJ and I =Tr{nˆ(ρ(tL)−ρ(0))}=Tr{nˆ(ρ(T/2)−ρ(0))} .
∆tγ 1 for any dephasing rate i , ,L,R .
i
≪ ∈{→ ← } We set the initial time within a period at the beginning
of contact with the left lead.
C. DC Josephson current Using the steady state density matrix Eq.(14) we can
derive a formal expressionfor the DC Josephsoncurrent
in the system:
The asymmetry between emission and absorption of
quanta from the bath leads to a nontrivial fixed point e
I = z (1I M )−1v z (1I M )−1v (23)
(Eq.(14))forthemap ,thusleadingtoanonvanishing T · − 0 0− · − T/2 T/2
M
Josephsoncurrentthroughthe Cooper pair shuttle. The
(cid:2) (cid:3)
current depends on the quantum dynamical evolution of where z is the unitary vector (0,0,1)T, stands for the
thechargeontheislandandontheinterplaybetweenthe usual scalar product in R3, and M , v· , M , v are
T/2 T/2 0 0
decoherence and the periodic driving. If, for example, defined in Eq. (12). Their explicit form is
M = exp(G t ) R(χ ) exp(G t ) exp(G t ) R(χ ) exp(G t ) (24)
0 ← ← ← R R → → → L L
· · · · ·
v = 2exp(G t ) R(χ ) γ G−1 (1I exp(G t ))w +γ exp(G t ) exp(G t )
0 − ← ← · ← · R R · − R R R L R R · → →
·R(χ→)·G−L1·(1I −exp(G(cid:2)LtL))wL , (25)
(cid:3)
and M , v are obtained from M , v by the ex- case of θ = θ = θ, χ = χ = χ, γ = γ = γ ,
T/2 T/2 0 0 L R → ← L R J
change of right and left Josephson contacts and of for- γ = γ = γ , t = t = t , t = t = t , Fig. 3
→ ← C L R J → ← C
wardandbackwardfreeevolutiontime. Thismeansthat shows a typical plot of I as a function of θ and φ. De-
M = M , v = v with acting on the param- pending onthe value ofθ (a similarbehavioris observed
T/2 0 T/2 0
P P P
eters θ ,χ ,γ t ,γ t ,φ as: as a function of χ), the critical current can be negative,
L(R) →(←) L(R) L(R) →(←) →(←) L(R)
i.e. the system can behave as a π-junction. The current
: (L, ,R, ) (R, ,L, ). (26) dependence on the various phases is the result of the in-
P → ← ⇒ ← ←
terferencebetweendifferentpathcorrespondingtodiffer-
The expression of the current, which depends on all the enttimeevolutionsforthechargestatesinthegrain. By
previous parameters can be obtained analytically from changing γ t and γ t , certain interference paths are
J J C C
Eq.(23) by explicitly writing M0 and v0 in terms of the suppressed,resultinginashiftoftheinterferencepattern
various parameters. The current depends only on the and ultimately in a change of the sign of the current, as
phase difference between the two superconductors φR shown in Fig.3.
−
φ . It is an odd function with respect to the actionof
L
P
defined by Eq. (26). From this observation follows that, Another interesting aspect of the Josephsoncurrentis
even for φ = 0, there can be a supercurrent between that it is a non-monotonous function of γ t , i.e. by
J J
the leads as long as the evolution over a cycle is not increasing the damping, the Josephson current can in-
-invariant. In this sense the systembehaveslike a non- crease. The behavior as a function of the dephasing
P
adiabatic Cooper pair pump. rates is presented in Fig.3. The presence of a maximum
The main features of the Josephson current in the Josephson current at a finite value of γ t can be un-
J J
Cooper pair shuttle have been discussed in Ref. 11 and derstood by noting that the current is vanishing in the
we recall them here for completeness however providing strong and weak damping limits. In both the limits sim-
a number of new results and additional details. In the ple analytic expressions are available.
7
0.67
2 p
Ι
3 p
2 T/e
0.2
q
p 0 0
-0.2
p
1
2 0.5 γ

-0.67 - CtC
0 0.5γ e
0 p f 2 p 3 p
- JtJ
e
1
0
Figure 3: (Color online). Left panel. Supercurrent (in units of e/T) as a function of the superconductor phase difference φ
and of the phase accumulated during the contact to one of the electrodes θ. The other parameters are fixed as: χ = 5π/6,
e−γJtJ = 3/4, e−γCtC = 4/5. The plot is obtained for Tb ≪ EJ. Right panel. Average current (Tb ≪ EJ) as a function of
the dephasing rates, with φ = −3π/4, θ = 7π/10, χ = 5π/6. As a function of γJtJ, the supercurrent has a not-monotonous
behavior. Notethe changeof sign in thecurrent obtained byvarying decoherence rates in each time interval separately.
If the dephasing is strong, I can be expanded in powers of e−γL(R)tL(R) and e−γ→(←)t→(←) and, to leading order,
2e E E
I tanh L e−(γLtL+γ←t←)sin(2θL)sin(φ χ←)+tanh R e−(γRtR+γ→t→)sin(2θR)sin(φ+χ→) .
∼ T T − T
(cid:20) (cid:18) b (cid:19) (cid:18) b (cid:19) (cid:21)
(27)
For simplicity we assume that the Josephson energies at left and right contacts are equal, as well as the contact and
free evolution time. In this case the previous expression is simplified to
2e E
Istrong tanh J e−(γJtJ+γCtC)cos(χ→+χ←)sin(2θ) sin(φ+(χ→ χ←)). (28)
∼ T T −
(cid:18) b (cid:19)
ItisworthtonoticethepresenceofanetDCcurrenteven evolution of the density matrix in a period. This is not
in the case of φ = 0 as argued from general argument a serious limitation for the experimental setups.
presented before. The role of breaking the -invariance In the opposite limit of weak damping defined by
P
is then played by the difference of the dynamical phases γ t γ t 1
J J C C
≪ ≪
accumulated in the forwardand backwardfree evolution
2e E γ t (cosφ+cos2χ)tanθsinφ
time intervals, χ→ χ←. If instead we assume a perfect I tanh J J J .
PDC-incvuarrriaenntceo,fwReefr.−e1c1over the known expression for the weak ∼ T (cid:18)Tb (cid:19)γCtC 1+cosφcos2χ (30)
Thecurrenttendstozeroifthecouplingwiththebathis
I 2etanh EJ e−(γJtJ+γCtC) negligible during the contact time. In this case the time
strong
∼ T T evolution in the intervals L,R is almost unitary, while,
(cid:18) b (cid:19)
cos(2χ)sin(2θ) sinφ. (29) in the region C, pure dephasing leads to a suppression
of the off-diagonal terms of the reduced density matrix
Strong dephasing is reflected in the simple (i.e. sinφ) ρ(t). As a result, in the stationary limit the system is
∝
current-phase relationship and in the exponential sup- described by a complete mixture with equal weights. At
pression of the current itself. Strong dephasing, in fact, the point (γ t ,γ t ) = (0,0) our model is not defined
J J C C
suppresses coherent transport over multiple cycles, thus asdiscussedatthe endofSectionIII. Thelimiting value
giving a corresponding suppression of higher harmonics of the current in approaching such point depends on the
in the current-phase relationship, i.e. a suppression of relative strength γ t ≶γ t between this two parame-
J J C C
terms sin2m+1(φ), m N. ters.
∝ ∈
For the sake of simplicity, from now on, we present all Thecurrenttendstozeroinbothlimitingcasesoflarge
the result in the case of perfect -invariance of the time and small γ t . Therefore one should expect an optimal
J J
P
8
coupling to the environment where the Josephson cur- simple terms is the limit γ 0, for a fixed value of θ.
C
→
rentis maximum. Aregimewherethe crossoverbetween For example, at θ =π/4 the current reads
the strong and weak damping cases can be described in
2e EJ 2e−γJtJ[2e−2γJtJ cosφ+(1+e−4γJtJ)cos2χ]sinφ
I = tanh (31)
T (cid:18)Tb (cid:19) (1+e−2γJtJ)(1+e−2γJtJ cosφcos2χ+e−4γJtJ)
Inthelimitofvanishingγ t ,Eq.(31)correspondstothe fluctuations of the switching times. This means that the
J J
situation discussed in Ref. 9. Indeed, both expressions contacttimes t , t and the free evolutiontimes t , t
L R → ←
are independent of the dephasing rates. The difference take,atanycyclei,arandomvaluet (i),b=L,R, , .
b
→ ←
inthedetailsofthecurrent-phase(s)relationshiparedue It is reasonable to assume that the fluctuations of any
to the different environment. switchingtimeareindependentontheothers. Intermsof
In all the three cases presented here, Eqs.(29, 30, 31), timeintervalst (i),itfollowsthatthefluctuations∆t (i)
b b
the change of sign of the current as a function of the aroundtheaveragevaluet ,areuncorrelatedatdifferent
b
phase shifts θ or χ is present. periods. Within the same period the fluctuations of any
two distinct time intervals are also independent. Hence
D. Effect of driving fluctuations ∆t (i) =0 (32)
h b it
The expressions for the current, discussed in the pre- and
vious Section, depend on the specific form of the cou-
pling between the system and the reservoir. The deco- h(∆ta(i))n(∆tb(j))mit =h(∆ta(i))nith(∆tb(j))mit (33)
herenceweconsideredsofaroriginatesfromgatevoltage
fluctuations. In addition in the shuttling mechanism an if i =j a = b. The integer valued arguments of ∆tb()
6 ∨ 6 ·
unavoidable coupling to an environment producing fluc- labels the periods and the subscript index runs over the
tuations in the period and shape of the driving is also set L,R, , . The average on the stochastic process
→ ←
present. We are therefore interested also in considering is defined by . We will discuss later the distribution
hit
the effect of fluctuations in the time dependence of the function of ∆tb(i).
external parameters on the Josephson current. Having By using the same notation in Eq. (11), we can write
assumedastep-liketimedependenceoftheparametersof the evolution of the density matrix after a finite number
theHamiltonian,noiseinthe externaldrivingconsistsin of cycles, h>1, as
h h−1 h−2h−2
r(t+ T(k))= M (λ)r(t)+ M (λ+µ+1)v (λ)+v (h 1). (34)
t t t t
−
k=1 λ=0 λ=0µ=λ
X Y X Y
In the previous equationthe expressionsM (i) and v (i) Note that if we had considered only the external driv-
t t
are those defined in Eq. (11). The index in parenthesis ing fluctuations(i.e. neglecting the effect ofgatevoltage
indicates the explicit dependence of both M and v on fluctuations) we would have found, in the steady state,
t t
various t (i). We refer to T(k) = t (k) as the pe- ρ (t) 1I. In this case in fact the evolution of r(t) con-
b b b ∞ ∝
riod although the time evolution is no longer periodic sists in an alternate sequence of rotation on the Bloch
P
(before averaging); T(k) is in fact the time the shuttle sphere around the (1,0,0) and (0,0,1) axes. Due to un-
takes to complete a cycle, k labels the number of cycles. certaintyintherotationanglesitisarandomwalkwhich
Thefirsttermintheright-hand-sideofEq.(34)vanishes leads, at long time, to a uniform distribution over the
in the long-time limit h . Averaging the previous Bloch sphere. The nontrivial result in Eq. (35) arises
→ ∞
expression over the fluctuations of the switching times because of the interplay between the two stochastic pro-
according to Eq. (33) is straightforward,leading to cesses of gate voltage fluctuations and switching time
fluctuations. They are independent because there is a
r (s) =(1I M )−1 v . (35) time scale separation between these two processes: Cor-
h ∞ it −h sit h sit
9
relations in the quantum bath do occur on a time scale We are interested in averaging the current after the
τ T, while the time intervals t (i) do not fluctuate system has reached its steady state
c b
≪
within any single cycle.
N
2e
hIit =Nl→im∞* N T(i)Trnˆ M0→ hj−=11T(h)+tL(j)−M0→ hj−=11T(h) [ρ(0)]+ , (36)
i=1 Xj=1(cid:16) P P (cid:17) t
 
P
In orderto proceedfurther we needto specify the distri- can determine the quantitative details of the Josephson
butionfunctionP(∆t (i)). Letusnotethatthedistribu- current, rather it can grasp the main features of physi-
b
tion is meaningful only if P(∆t (i)) = 0 for ∆t (i) < 0. cal effects due to imprecision in controlling the external
b b
We consider driving. For the sake of concreteness, let us consider the
limit of strong dephasing (Eq. (29)), when the expres-
P(∆tb(i))=θ(∆tb(i)+τ)θ(τ ∆tb(i))/2τ, (37) sion for the current considerably simplifies. The strong
−
dephasing leads to a rapid loss of memory of the initial
with τ < tb b as a toy model: the underlying physical conditions. One can suppose that this occurs after one
∀
idea is that the switching time can be controlled with cycle independently on the averaging process. It follows
a precision 2τ and the switching can happen with equal that, in Eq. (36),
probability in the interval [t τ,t +τ]52. We do not
b b
−
expect that this simple form of the distribution function
N
2e
hIit =Nl→im∞* N T(i)Trnˆ M hj−=11T(h)→ hj−=11T(h)+tL(j)−1I [ρ∞(0)]+ , (38)
i=1 Xj=1(cid:16) P P (cid:17) t
 
P
depends only onthe stochastic parametersofthe last (=jth) cycle and ρ (0) onparametersofthe j 1thcycle.
∞
M −
By considering the expansion of the denominator in the previous equation as
N N
1/( T(i))=1/(NT)(1 ∆t (i)/NT +...) (39)
b
−
i=1 i=1 b
X XX
together with Eq. (33), Eq. (36) reduces to
N
2e
hIit =Nl→im∞NT *Trnˆ M hj−=11T(h)→ hj−=11T(h)+tL(j)−1I [ρ∞(0)]+ +O(1/N) . (40)
Xj=1(cid:16) P P (cid:17) t
 
In the N limit, only the first term in the previous equation is non-vanishing. It means that, as a consequence
→ ∞
of the strong dephasing, the effect of fluctuations in the term 1/T in the definition of the currentare ineffective. The
final expression for the averagecurrent is
2e E t E t
I = sinφ e−γCt←cos C ← e−γJtLsin J L (41)
h it T ~ ~
(cid:28) (cid:29)t(cid:28) (cid:29)t
which, in the lowest orders in the small parameter τE /~, reads
C
2e 1 E τ 2 1 E τ 2 ~γ E ~γ
C C C J J
I 1 I + sin(2θ)sin(2χ) cos(2θ)cos(2χ) sinφ ,
h it ≈ T (" − 6 (cid:18) ~ (cid:19) # strong 3 (cid:18) ~ (cid:19) (cid:20)EC × − EC EC (cid:21) )
(42)
with the further condition 1/T γ E /~ E /~ γ 1/T. The current I is defined as
J J C C strong
≪ ≪ ≪ ≫ ≫
10
the currentin absence ofdriving fluctuations and is that 40
obtained in Eq. (29). The leading order correction to
I doesnotmodify the functionaldependence ofthe
strong 30 10
currentonthedynamicalphases: Itisasimplerenormal-
izationofthepre-factor,1 1 (E τ)2/(6~2). Eq.(42) I/I0
C
→ − 0
smhoodwisfictahtaitonhsigohfetrhoerfduenrcctoiornreacltdioenpsencdanenicnestoefatdhelecaudrrteonat I/I020 0 0.4 0.8w T/(2 p)1.2 1.6 2
on dynamical phases.
10
E. AC Josephson effect
0
0 2 4 6 8 10 12 14 16
When a bias voltage is applied to a Josephson junc- w T / (2 p)
tion it results in an alternating current. This is the AC
Josephson effect which can be derived from the expres-
sion for the Josephson current I = 2~eEJsin(φR φL) Figure4: (Coloronline). AbsolutevalueoftheFouriertrans-
−
s2uepVp.leImnetnhteedprbesyenthteprJoobsleepmhstohneraelltaetrinoantidndtg(φcRur−renφtLw)i=ll fpolromtisofobthtaeinJeodsefpohrstoJn=cutrCre=ntTi/n4u,nEiCtsTo/f~I=0 =602,eEEJJT//~~. =Th6e,
~
not be simply sinusoidal, it is therefore convenient to γJT = γCT = 0.001. In the inset we plot the Fourier trans-
form of the Josephson current in a restricted range of fre-
consider its frequency spectrum defined as
quencies for γJT = γCT = 0.001 (black dotted line) and for
γJT =γCT =0.1 (full red line).
I˜(ω)= dte−iωt I (t). (43)
R h i
Z
(inthesimplecaseofasinusoidalcurrentitreadsI˜(ω) where I(t) = Tr{sin(ϕˆ−φL−eVt/~)ρ(t)}, according
δ(ω 2eV/~) for ω >0). ∝ to the definition of current in Eq. (5).
− We computed numerically the time evolution of the
The application of a finite voltage bias in the Cooper
densitymatrixoftheislandandobtainedfromitthefre-
pairshuttlegivesrisetoaquiterichsituation,duetothe
quencyspectrumoftheJosephsoncurrent. Asawarmup
interplay of of the voltage bias effect and the underlying
weconsiderthesimplestcasewhichconsistsinneglecting
periodic motion of the shuttle. The relative magnitude
of the characteristic frequencies, 2eV/~ and 1/T, the effect of voltage bias. This is the same case consid-
∼ ∼ ered in previous Sections, in which, however,we analyze
determine different regimes that we are going to investi-
the instantaneous current rather then the averaged one.
gate.
The results arepresentedin Fig 4. The spectrumclearly
We set the electric voltage of the left and right elec-
signalstheperiodicityofthetimedependenceofthecur-
trode respectively equal to V = V/2 and V = V/2.
L R
− rent signal and of the presence of the Θ function. In
Thereisnodissipativecurrentthroughthesystemaslong L
the steady state the current is repeats periodically and
as eV 2∆. Inthe limit C ,C .C , and (V/2) V
L R g g
≪ ≪ then (s+kT)= (s) inEq.(45). As aconsequencethe
the Hamiltonian of the system is still given by Eq. (1). I I
current spectrum presents peaks at integer multiples of
The effectofthe electricpotential V in the twoleads
L(R)
the frequency ω =2π/T,
canbeincluded,bymeansofagaugetransformation,into n
timedependentphasesofthecondensatewavefunctions,
ψL(R) →e2ieVL(R)t/~ ψL(R) , or equivalently, I˜(ω)= A˜nδ(ω−2πn/T) (46)
n
X
(cid:12) (cid:11) φ φ(cid:12) +(cid:11) 2eV t/~ .
(cid:12) L(R) (cid:12)L(R) L(R)
→ with
The Hamiltonian describing the effect of voltage bias in
tJ
the Cooper pair shuttle becomes A˜ = ds exp( 2πins/T) (s) .
n
− I
Z0
Hˆ = E (t)[nˆ n (t)]2
AC C g
E Θ (t)−cos(ϕˆ φ eVt/~) The form of A˜n is fixed by the expression of the den-
J L L
− − − sity matrix at the fixed point through (s), (s) =
E Θ (t)cos(ϕˆ φ +eVt/~). (44) I I
− J R − R y Rz(φL)r(s),andwithr(s)determinedbyEq.(18). The
·
signal (s) consists of damped oscillations at frequency
We are interested in the frequency dependent current E /~,I(s) exp( γ s)sin(E s/~+α ). The Fourier
J J J 0
I ∝ −
E transform of such a signal determines the characteristic
I˜(ω) = 2e J dte−iωtΘ (t) (t)
~ ZR L I fAe˜atudreisspolfatyhseocsucirlrleantitosnpseacttrfurmeqpureensceinesteTd/itnFmigo.d4u.lNatoewd
= 2eE~J k∈Ze−ikωT Z0tLdse−iωsI(s+kT)(.45) |bTyhnea| npaotwuerre-laowf tdheecaoythfuenrcptieoanks(∝sh1o/wωn, fionrJωthe≫inEsJet/~i)n.
X