RESONANT COOPER-PAIR TUNNELING: COUNTING STATISTICS
AND FREQUENCY-DEPENDENT CURRENT NOISE

arXiv:cond-mat/0301277 v1 15 Jan 2003

A.A. CLERK
Departments of Applied Physics and Physics
Yale University, PO Box 208284, New Haven, CT 06520-8284

Abstract
We discuss the counting statistics and current noise associated with the
double Josephson quasiparticle resonance point in a superconducting single
electron transistor. The counting statistics are in general phase-dependent,
despite the fact that the average current has no dependence on phase.
Focusing on parameter regimes where the counting statistics have no phasedependence, we use a general relation first derived by MacDonald in 1948
to obtain the full frequency-dependent shot noise directly from the counting
statistics, without any further approximations. We comment on problems
posed by the phase-dependence of the counting statistics for the finitefrequency noise.
1. Introduction
Resonant Cooper-pair tunneling, also known as Josephson quasiparticle
tunneling, refers to transport cycles in superconducting single-electron transistors (SSET’s) which involve the transfer of both Cooper pairs and quasiparticles [1, 2, 3]. They have recently been the subject of renewed attention,
both because of their unusual noise properties [4, 5, 6] and because of theirutility in measuring the state of a charge superconducting qubit [7, 8]. In
terms of noise properties, it has been shown that charge fluctuations associated with these processes can induce a population inversion in a coupled
two-level system (i.e in terms of its charge noise, the transistor effectively
has a negative temperature) [5, 6]. The shot-noise in the current through thetransistor was also found to have remarkable properties [4, 5]. By tuning thestrength of the Cooper pair tunneling relative to the quasiparticle tunneling,
one could effectively tune the Fano factor determining the zero-frequency

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2
shot noise. It was possible to achieve a Fano factor greater than one, which
was interpreted as a consequence of the effective charge associated with thetransport cycle being greater than one. Perhaps more surprisingly, it waspossible to reduce the Fano factor below 1/2, behaviour that was not fully
explained. The finite frequency current noise also showed interesting behaviour [4]– in the regime where the Cooper-pair tunneling dominated thequasiparticle tunneling, a coherent peak in the current noise was predicted
at the Josephson energy.
In this paper, we look specifically at the statistics of transferred charge
and current noise associated with the double Josephson quasiparticle (DJQP)
process in a SSET, a process which allows for near quantum-limited measurement [5], and which has been used in a recent experiment [8]. We pay
careful consideration to the fact that the breaking of gauge invariance bysuperconductivity can make the interpretation of counting statistics more
subtle than in the normal-metal case [10, 12, 13]. One obtains phasedependent counting statistics here despite the fact that transport is insensitive to the overall phase difference between the two superconducting
reservoirs. We identify two regimes in which there is no ambiguity in defining counting statistics, and proceed to use them to directly calculate thefrequency-dependent shot noise, without any further approximations. This
is in contrast to previous treatments both of normal and superconducting
single electron transistors [14, 15, 4], where an additional Markov approximation (beyond what is needed for the counting statistics) is made toobtain the frequency-dependent current noise. The direct connection between counting statistics and frequency-dependent current noise we employ
here allows one to give a simple interpretation of the latter. In particular,
coherent peaks in the shot noise appearing at the Josephson energy canbe directly tied to the time dependence of the counting statistics. We alsosuggest that the suppression of the Fano factor below 1/2 is related to thecoherence of Cooper-pair tunneling.

2. Description and Model for Josephson Quasiparticle Tunneling
A SSET consists of a superconducting, Coulomb-blockaded island which is
attached via tunnel junctions to two superconducting electrodes (Fig. 1).
The SSET Hamiltonian HS = HK + HC + HV + HT has a term HK describing the kinetic energy of source, drain and central island electrons (i.e.
three bulk BCS Hamiltonians), a term HV which describes the work doneby the voltage sources, and a tunneling term HT . The charging interactionis HC = EC (n − N )2 , where EC is the SSET capacitive charging energy, n
is the number of electrons on the central island, and N is the dimensionless

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3

Figure 1. Schematic showing the four steps of the double Josephson quasiparticle process
which can occur in a superconducting single-electron transistor. Circles represent thecentral island of the SSET, while the rectangles are the electrodes. Numbers indicate thecharge of the SSET island.

gate voltage applied to the island. We consider a SSET with identical tunnel
junctions, whose dimensionless conductance g satisfies g/(2π)  1.
The DJQP process occurs when both the transistor gate voltage N anddrain-source voltage 2VDS are tuned such that two Cooper-pair tunnelingtransitions, one in each junction, are resonant. We label these transitions
as n = 0 → 2 in the left junction, and n = 1 → −1 in the right junction(see Fig. 1). The double resonance point for these transitions occurs at
eVDS = ECS and NS = 1/2. In addition, EC /∆ (where ∆ is the superconducting gap of the SSET) must be chosen so that the quasiparticle
transitions linking the two Cooper pair resonances are energetically allowed(i.e. n = 2 → 1 and n = −1 → 0), whereas unwanted transitions which
would interrupt the cycle (i.e. n = 0 → 1) are not. We take EC = ∆ tosatisfy these conditions; this corresponds to the experiment of Ref. [8].
Assuming that all the above conditions are met, and that T  EC ,transport through the SSET will be dominated by the DJQP process sketchedin Fig. 1. The standard theoretical description of this process [2] is obtainedby looking at the dynamics of the reduced density matrix ρ describing
the charge n of the central SSET island. Here, only four charge states areimportant; further, off-diagonal terms need only be retained between states
involved in Josephson tunneling. Letting ρ(i,j) = hn = i|ρ|n = ji, we canrepresent the non-zero elements of this reduced density matrix as an 8component vector:ρ~≡ρ(−1,−1) ρ(0,0) ρ(1,1) ρ(2,2) ρ(−1,1) ρ(1,−1) ρ(0,2) ρ(2,0)

(1)

By starting with the von Neumann equation for the evolution of the fulldensity matrix, treating the tunneling Hamiltonian to lowest order in perturbation theory, and then tracing out the fermion degrees of freedom (see,e.g. Ref. [15]), one obtains the an evolution equation for ρ. Taking both

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4junctions to be on resonance, we have:d ~~ρ(t) = U † M U ρ(t)
dt

(2)


0
0
−Γ
0
0
0
−i E2J i E2J
 Γ
0
0
0
0
0
−i E2J i E2J 



 0
EJ
EJ
0
0
Γ
i 2 −i 2
0
0




0
0
−Γ
0
0
i E2J −i E2J 
 0
,
M =
EJ
EJ

 −i 20i 20 −Γ/2 0
0
0

 EJ
EJ

 i 2
0
−i
0
0
−Γ/2
0
0
2


EJ

 0 −i EJ
0
i
0
0
−Γ/2
0
2
2
EJ
EJ
0 −i 2000 −Γ/2
0
i 2




−iφR

U = diag 1, 1, 1, 1, e

iφR

,e

−iφL

,e

iφL

,e



(3)

(4)

EJ = g∆/8 is the standard Ambegaokar-Baratoff value for the Josephsonenergy emerging from perturbation theory in the tunneling Hamiltonian;φL and φR are the phases of the superconducting reservoirs. Γ is the rate
associated with the two quasiparticle transitions occurring the DJQP cycle,
and is given by the usual expression for quasiparticle tunneling between two
superconductors [16]; for simplicity, we assume the two transitions to haveequal rates. We have also made a Markov approximation to obtain Eq. (2),which is valid as long we do not probe the evolution of ρ on timescales
smaller than ~/EC .
The zero eigenvector ρ~0 of the evolution matrix M in Eq. (3) corresponds
to the stationary value of the reduced density matrix describing the island
charge; note that in general, it will have non-zero off-diagonal elements.
The diagonal elements of ρ0 (which are probabilities) can be obtained froma classical rate equation; one eliminates the off-diagonal elements from Eq.
(2) by expressing them in terms of the diagonal elements. Letting p~ be the
4-vector of these probabilities, we are lead to the rate equation:−Γ − γ 0 γ
0


d
Γ
−γ 0
γ
 ~p(t) = 0(5)p(t) = 
~


γ
0 −γ
Γ
dt
0
γ 0 −Γ − γ
where γ = EJ2 /Γ represents a “rate” for Cooper-pair tunneling. Eq. (5)represents an incoherent model for the DJQP process, and is sufficientfor many purposes (e.g., calculating the average current hIi = 3γΓ/(4γ +2Γ)). The full stationary value of the density matrix (including off-diagonal

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5
terms) is given by:
ρ
~0 =



1
γ, γ + Γ, γ + Γ, γ, iEJ e−iφR , −iEJ eiφR , −iEJ e−iφL , iEJ eiφL
4γ + 2Γ
(6)

3. Counting Statistics
In order to obtain information about the number of electrons that havetunneled through one of the SSET junctions, the density matrix approachof the previous section must be embellished. The standard approach forsingle electron transistors [15, 4] is to employ a “counter” scheme where one
explicitly tracks the dynamics of, say, mL , the number of electrons whichhave tunneled through the left junction of the SSET. The Hilbert space of
the SSET is expanded so that each state is now also labelled by a value ofmL , and the tunnel Hamiltonian is modified so that it raises or lowers thisindex as appropriate (e.g., by introducing an auxiliary raising operator F †such that [mL , F † ] = F † ). The resulting equation for the reduced density
matrix describing both n and mL is simplified if one Fourier transforms indρ~(k) = U † M (k)U ρ~(k), where we havethe latter variable. One finds that dt
defined:P
ikm
if n1 = 0, n2 = 2,
 PmL e L hn1 , mL |ρ|n2 , mL + 2i
ikm
L hn , m + 2|ρ|n , m ieif
n1 = 2, n2 = 0, (7)ρ(n1 ,n2 ) (k) =
1
L
2
L
L
 mP
ikm
L
hn1 , mL |ρ|n2 , mL iotherwise.
mL e


0
0
−Γ
0
0
0
−i E2J i E2J Γeik
i E2J
0
0
0
0
0
−i E2J



 0
0
0
Γ
i E2J −i E2J
0
0




0
0
−Γ
0
0
i E2J e2ik −i E2J e2ik  0
M (k) = 
 −i E2J0i E2J0−Γ/2 0
0
0

 EJ
EJ

 i 2
0
−i
0
0
−Γ/2
0
0
2


EJ −2ik 0 −i EJ
0
i 2 e
0
0
−Γ/2
0
2
EJ
EJ −2ik0i 20 −i 2 e
0
0
0
−Γ/2
(8)
The probability distribution for mL is thus easily obtained from the characteristic function p(k, t):Z πZ πdk −ikmL
dk −ikmL X
p(mL , t) =ep(k, t) ≡e
ρ(n,n) (k, t)
(9)
−π 2π
−π 2π
n


ρ
~(k, t) = eM (k)t ρ~(k, t = 0)

(10)

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6
It only remains to specify the initial state of the transistor (i.e. ρ~(k, t = 0)).
The natural choice is to place the transistor n degree of freedom in its
stationary state, and choose mL = 0 with probability one (i.e. ρ~(k, t =0) = ρ
~0 ) [15]. p(mL , t) would then represent the stationary probability thatmL electrons are transferred through the left junction in a time t. While
this procedure is perfectly well defined for non-superconducting systems[15], it is problematic in the superconducting case. The stationary state of
the SSET has coherence between different charge states– thus, one cannot
have both that mL = 0 and the coherence required in the stationary state.
A possible remedy for this problem would be to suppress the off-diagonal
elements of the initial density matrix; of course, this procedure needs tobe justified. Note that given the linearity of Eq. (10) in the initial density
matrix, we can separate incoherent and coherent contributions to p(mL , t)(i.e. those arising, respectively, from the diagonal and off-diagonal elementsof the initial density matrix).
To gain further insight, it is instructive to consider an alternate to
the “counter” scheme discussed above. One can also obtain a tunnelledcharge distribution for our system by following the general prescription
for obtaining full counting statistics discussed in Ref. [9, 10]; we denote
this distribution as p0 (mL , t). The procedure here involves considering theeffects of coupling an ideal measurer of current to the transistor. In practice,a gauge transformation is made to eliminate the interaction with the measurement device, resulting in a phase eiλ(t) being attached to the tunnel
matrix element t [11]. Using the fact that the perturbative calculationof ρ in the tunneling Hamiltonian has a Keldysh structure (see e.g. Ref.
[15]), one finds that λ(t) = δ + k on the forward Keldysh contour, andδ − k on the backwards Keldysh contour. δ is a phase which depends on
the initial state of the detector; for non-superconducting systems, gauge
invariance ensures that it does not play a role. This procedure results inan expression for p0 (k, t) which is identical in form to Eq. (10), except that
now the initial state is uniquely specified as ρ
~0 , and the evolution matrix
M undergoes a δ-dependent unitary transformation which only affects thecoherent contribution to p(k, t):
M → V †M V
V = diag 1, 1, 1, 1, 1, 1, e−i(k+δ) , e−i(k−δ)(11)
Thus, one has in general phase-dependent counting statistics for our system,
meaning that there is a dependence on the phase shift δ introduced by
the measurement. This phase-dependence exists despite the fact that the
average current is not sensitive to the phase difference between the two
superconducting reservoirs. As discussed in Ref. [10, 12], the results of any
charge-counting experiment will now be dependent on the initial state ofthe detector. We thus see that the ambiguity in defining the initial state of

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7
the transistor in the “counter” scheme translates here to needing to know
the initial state of the detector. Note that the incoherent part of p(k, t) is
the same in the two schemes, while the coherent part differs. The process ofignoring the initial coherence of the density matrix in the “counter” scheme
corresponds here to assuming an initial detector state which is completely
delocalized in the phase δ.
Despite the above caveats, we can determine the counting statistics unambiguously in two limits where the off-diagonal elements of the stationary
state ρ~0 are suppressed– either Γ  EJ or EJ  Γ (c.f. Eq. (6)). In
these limits the coherent contribution to p(k, t) vanishes, implying that thecounting statistics are identical in both schemes, and have no dependence
on the phase shift δ. Note that in general, the magnitude of the off-diagonal
elements of ρ in the stationary state are driven both by the size of EJ /Γ
and by the population asymmetry of the two charge states involved (i.e.
ρ02 ∝ (ρ00 − ρ22 )). In the limit of large EJ , it is the lack of population
asymmetry which suppresses the off-diagonal elements. In this limit, thesymmetric and anti-symmetric superpositions of charge states are equally
populated, leading to a vanishing of the off-diagonal matrix elements.
3.1. THE LIMIT EJ  Γ

In this limit, the SSET effectively gets stuck in the states n = 0 and n = 1
waiting for the relatively slow Cooper-pair transitions to occur; we canthink of the rate γ (c.f. Eq. (5)) as describing effective transitions between
n = 0 and n = 1 (i.e. γ describes both a Cooper-pair event and thesubsequent quasiparticle event). We find for the probability distributionp(m, t) ≡ p(mL , t) = p(mR , t) (i.e. the junctions are identical) :2l−γt (γt)
if m = 3l e(2l)!
−γt (γt)2l+1e(12)p(m, t) =if m = 3l + 1 or m = 3l + 2 2 (2l+1)!
0otherwise.

We have neglected terms which are small as (EJ /Γ)2 , and chosen m = 0
at t = 0. Again, as the coherent contribution to p(m, t) vanishes to leading
order, there is no ambiguity in defining the counting statistics. The statistics
can be given a simple interpretation. Each Cooper-pair plus quasiparticle
event is described by the rate γ and a Poisson distribution. After an evennumber of γ transitions, a multiple of 3 electrons must have been transferredthrough a junction. However, after an odd number of γ transitions, there
is an equal probability of having had an extra two electrons transferred(i.e. if we are looking at the left junction, n = 0 → 2 → 1) or of having
a single extra electron transferred (i.e. n = 1 → −1 → 0 for the left

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8junction). Note that there is an asymmetry here which favours values ofm which are an integer multiple of 3; this is the analog of the even-odd
asymmetry found for the single JQP process [4]. Also, note that in thislimit the counting statistics are identical if we use the incoherent model of
Cooper-pair tunneling described by Eq. (5); coherence plays no role.
3.2. THE LIMIT EJ  Γ

In this limit, we can think of there being long periods of coherent Josephson
oscillations which are interrupted by infrequent quasiparticle transitions;note that we still assume EJ  EC = ∆, so that it does not effect theenergetics of quasiparticle tunneling. Defining the Poisson distribution Λ(l)as:
Λ(l) = e−Γt/2(Γt/2)ll!

(13)

we find up to terms of order EJ /Γ:
 



cos(EJ t)
1

Λ(2l)
3
+
+
Λ(2l
+
1)
+
Λ(2l
−
1)

2l
4
2






cos(EJ t)
1
1
Λ(2l
+
1)
+
Λ(2l
+
2)
1
−
2l+2
4
2
2
p(m, t) =cos(EJ t)
1
1

Λ(2l)
1
−
Λ(2l
+
1)
+

4
2

22l


0

if m = 3l,
if m = 3l + 1,
(14)
if m = 3l + 2,
otherwise

where again, we have assumed an initial state with m = 0 without any
ambiguity, as the coherent contribution to p(m, t) is negligible. We canagain think of the quasiparticle transitions as effectively being describedby a rate γ = Γ/2 and a Poisson distribution. Now, however, after aneven number of transitions (say 2l), it is not certain that 3l electrons will
have been transferred through a junction; rather, due to weakly damped
Josephson oscillations, there is also a probability to find 3l ± 2 electronstransferred. Even more remarkably, there are signatures of coherent oscillations in p(m, t), this despite the fact that the initial coherence of thedensity matrix is irrelevant. As we will see in the next section, the timedependence of the counting statistics has a direct impact on the frequency
dependence of the current noise. Note that the oscillation serves to modulate the asymmetry which favours m being a multiple of three. For Γt  1,the probability of m being a multiple of 3 is given by:p(m = 3l) →5 + cos(EJ t)
8(15)

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9
4. Current Noise and Counting Statistics
In this section, we calculate the frequency dependent current noise SI (ω)
associated with the DJQP process. As is standard, the capacitance of thejunctions CL and CR may be used to connect this quantity (which includes the effects of displacement currents) to the noise associated with thetunneling currents in each contact [14]:SI (ω) =

C2
CL2
CL CR
SLL (ω) + R SRR (ω) +(SLR (ω) + SRL (ω))
CΣ
CΣ
CΣ

where
Sαβ = lim lim

Z T

T →∞ t0 →∞ −Tdteiωt hh{Iα (t + t0 ), Iβ (t0 )}ii.

(16)

(17)

The order of limits here ensures that we are taking averages with respect tothe stationary density matrix of the SSET; note also that we are calculating
the classical part of the current noise, which is a symmetric function of
frequency. Previous calculations of this quantity for both normal and superconducting transistors have effectively made use of the so-called quantum
regression theorem, which involves making an additional Markov approximation beyond that necessary to calculate the reduced density matrix ρ[14, 15, 4]. It turns out that this is not necessary; one can directly connect
the particle-current noise correlators to the counting statistics using a formula first derived by MacDonald [17] . Using the definition Iα (t) = ∂t mα (t),one can show:
Z ∞
d
2
(18)
Sαβ (ω) = 2e ωdt (sin ωt) hhmα (t)mβ (t)ii,
dt
0
where at t = 0, mL = mR = 0 with certainty, and the system is described by
the stationary density matrix ρ0 . The integral in thisR ∞expression should beinterpreted within the theory of distributions, i.e. 0 dt sin ωt = 1/ω. We
see from Eq. (18) that the frequency-dependent current noise is directly tied
to the time-dependence of the second moment of the tunneled charge distribution. At long times, hhm(t)2 ii → f It/e, where f is the Fano factor; thisleads to the usual expression Sαα (ω = 0) = 2ef I. A frequency-dependentS(ω) indicates that hhm(t)2 ii deviates from this linear in time behaviour atshort times. We thus see that in addition to being useful for calculations,Eq. (18) provides a straightforward way to interpret frequency-dependent
current noise in terms of counting statistics. Note that unlike approaches
using the quantum regression theorem [14, 4], one does not need to add an
ad-hoc term to Eq. (18) to obtain the correct noise in the high-frequency
limit.

DJQPNoise2.tex; 15/01/2003; 22:41; p.9

10

80

S (ω) / (e2 Γ)

60

40

P(m)

0.1

0
20

40

60

80

I

m

20

0
0

5

ω / Γ 1015

Figure 2. Frequency dependent current noise for the DJQP process at EJ = 10Γ; thelarge peak at ω = EJ is the result of oscillations in the asymmetry of the tunneledcharge distribution. Inset: Distribution of tunneled charge p(m) for the same parametersat Γt = 67; the asymmetry which favours multiples of 3 is clearly visible.

We apply Eq. (18) to the DJQP process in the limit EJ  Γ, where
the counting statistics are well-defined and phase-independent. As noted in
the previous section, the coherent Josephson oscillations at frequency EJ /~modulates the asymmetry in p(m, t) which favours multiples of 3. This inturn modulates the second moment of the distribution (i.e. the greater theasymmetry, the smaller the width); one finds from Eq. (14):9Γt 9 − e−Γt cos(EJ t)  −Γt/4−3Γt/4e−ehhm (t)ii =
+
−
8
8
2
2

(19)

Eq. (18) then implies that there should be a peak in SLL = SRR at ω =EJ of order e2 EJ2 /Γ, which is much larger than the zero frequency noise(9/4)e2 Γ. Shown in Fig. 2 are results obtained for SI (ω) for EJ = 10Γ,
CL = CR ; a sharp peak is indeed visible. Similar results for the single JQP
process were found in Ref. [4]. We see here that the peak in the current
noise is directly related to the modulation in time of the asymmetry of thecounting statistics by coherent Josephson oscillations.

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11
5. The Intermediate Regime EJ ' Γ
The noise properties in the regime EJ ' Γ are especially interesting, as
it is in this regime where the DJQP process can be used to make a nearquantum-limited measurement [5]. In this regime, the off-diagonal elementsof the stationary density matrix ρ~0 are by no means small, and thus thecoherent, phase-dependent contribution to the counting statistics will playa role. While this in itself is not a problem, the resulting status of SI (ω)
as calculated using Eq. (18) becomes unclear, as the quantity ∂t hhm2 (t)ii
is phase dependent. The conclusion would thus appear to be that for resonant Cooper-pair tunneling, the finite-frequency current noise is phasedependent, and consequently is sensitive to details of the measurement.
We close by pointing out that the value of the zero frequency noise
remains phase independent and unambiguous regardless of the ratio EJ /Γ.
At zero frequency, each of Sαβ are equal; thus, we only need to know∂t hhm2 (t)ii in the large time limit. As discussed in Ref. [15], this is determined completely by the lifting of the zero eigenvalue of the evolution
matrix M (k = 0) in Eq. (3) by non-zero k; the contribution to hhm2 (t)ii
from other eigenmodes of M are damped away in the long time limit.
Restricting ourselves to only this “lifted” zero-mode, we have:p(k, t) → eλ0 (k)t (1 + A(k, δ)),(20)

where A(k, δ) contains all k-dependent terms involving the initial density
matrix, and the δ-dependent eigenvectors (left and right) corresponding to
λ0 (k); A vanishes for k → 0. As:
hhm2 (t)ii = −d2
log p(k, t)
dk2

(21)

it follows from Eq. (18) that2

SI (0) ≡ Sαα = 2ed2
− 2 λ0 (k)
dk
k=0



(22)

The phase-dependent term A is thus explicitly seen to play no role. Analyzing the eigenvalue λ0 (k) of M (k) in Eq. (8), one find for the Fano factor
f [5]:#"
6EJ2 Γ2
3
(23)
f=
1−
2
2
Γ2 + 2E 2J

In both the limits EJ  Γ or EJ  Γ , f → 3/2. This can be understood as
an effective charge– each Cooper-pair plus quasiparticle transition transfers
on average 3/2 electrons per junction. Interestingly, when EJ ∼ Γ, f drops

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12√
below 1/2, reaching a minimum of 3/8 when EJ = Γ/ 2. This behaviour
is reminiscent of double tunnel junction systems, or of a normal SET; inboth cases, there are two rates involved in transport, and the Fano factor
reaches a minimum of 1/2 when these rates are equal. The behaviour in such
systems can be understood completely classically by the self-averaging that
occurs when one sequentially combines two independent Poisson processes.
In contrast, the fact that f drops below 1/2 for the DJQP process is a directconsequence of the coherence of Cooper-pair tunneling. To underscore thispoint, one can calculate f using the incoherent model of Eq. (5). While
this procedure yields the correct value of hIi and the correct value of f for
extreme values of EJ /Γ, it only gives a minimum f of 1/2. One finds:#"
6EJ2 Γ2
2EJ2 Γ2
3=f+
fincoherent =
1−
(24)

2
2
2
Γ2 + 2E 2
Γ2 + 2E 2JJNote that the incoherent calculation of f is always an overestimate; as
expected, coherence between charge states suppresses the zero-frequency
current noise, as it tends to make tunnelling events more regular.
6. Conclusions

We have studied the counting statistics and finite-frequency shot noiseof the DJQP process in a superconducting single electron transistor, attempting to clarify some of the remarkable noise features found in previous
studies. In general, the counting statistics are phase-dependent, despite
the fact that the average current has no phase dependent. An interestingquestion remains how this phase dependence impacts the finite frequency
noise, given Eq. (18) which directly relates the two quantities.
Acknowledgements
I am grateful to S. M. Girvin for useful discussions, and to L. Fedichkin forbringing Ref. [17] and Eq. 18 to my attention. This work was supported by
the NSA and ARDA under ARO contracts ARO-43387-PH-QC, by the NSF
under DMR-0196503 & DMR-0084501, and by the W.M. Keck Foundation.

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