VOLUME 89, N UMBER 17

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21 OCTOBER 2002

Resonant Cooper-Pair Tunneling: Quantum Noise and Measurement Characteristics
A. A. Clerk, S. M. Girvin, A. K. Nguyen, and A. D. Stone
Department of Applied Physics and Physics, Yale University, New Haven, Connecticut, 06511(Received 15 March 2002; published 8 October 2002)
We study the quantum charge noise and measurement properties of the double Cooper-pair resonancepoint in a superconducting single-electron transistor (SSET) coupled to a Josephson charge qubit. Usinga density-matrix approach for the coupled system, we obtain a full description of the measurement
backaction; for weak coupling, this is used to extract the quantum charge noise. Unlike the case of anonsuperconducting SET, the backaction here can induce population inversion in the qubit. We find that
the Cooper-pair resonance process allows for a much better measurement than a similar nonsuperconducting SET, and can approach the quantum limit of efficiency.
DOI: 10.1103/PhysRevLett.89.176804

PACS numbers: 73.21.-b, 73.40.Gk, 73.23.Hk

Among the many open issues related to solid state
quantum computation, the question of how best to measure a solid state qubit remains a particularly interestingone. In the case where the qubit is a Cooper-pair box (i.e.,a Josephson-junction single charge box), the standard
choice for a readout device is the single-electron transistor (SET) [1–6]. An alternate and potentially more
powerful approach is to use a superconducting singleelectron transistor (SSET) biased at a point where thecyclic resonant tunneling of Cooper pairs dominatestransport [7–12]. Such processes, known as Josephson
quasiparticle (JQP) resonances, would appear to be an
attractive choice for use in a measurement as their resonance structure implies an extremely high sensitivity.
However, precisely because of their large gain, theseprocesses may be expected to strongly alter the state of
the qubit in a measurement. To assess the balance between
these two opposing tendencies, a close examination of thephysics of JQP tunneling is required.
In this paper, we focus on a double JQP process (DJQP)(see Fig. 1), which occurs at a lower SSET source-drain
voltage than single JQP processes, and which has beenused in a recent experiment [13]. We assess the potential
of DJQP to act as a one-shot measurement of the state of a
Cooper-pair box qubit. This involves characterizing bothmeas , the time needed to discriminate the two qubit statesin the measurement, and the backaction of the measurement on the qubit, which is described by a mixing rate
mix and a dephasing rate 1=’ . These quantities are
intimately related to the noise properties of the SSET,
which are of interest in themselves, given the novel natureof the DJQP process. meas is determined by the shot noiseof the process, while mix and ’ are related to the charge
noise on the SSET island. While the shot noise of a single
JQP process has been analyzed recently [14], the quantum
charge noise has not been addressed.
To describe the measurement process in our system, we
employ a density-matrix description of the fully coupled
SSET plus qubit system; this is similar to the approachtaken by Makhlin et al. [4] for a SET, but extended to deal
with Josephson tunneling. This approach is not limited bya requirement of weak coupling, as are standard approaches which perturbatively link mix to the transistor
charge noise [5,6]; nonetheless, in the limit of weakcoupling it can be used to calculate the quantum charge
noise of the SSET. We find that the quantum (i.e., asymmetric in frequency) nature of the noise is particularly
pronounced for the DJQP feature, leading to regimes
where the SSET can strongly relax the qubit. Moreover,due to the resonant nature of Cooper-pair tunneling, there
exist regimes where the SSET can cause a pronounced
population inversion in the Cooper-pair box. For typicaldevice parameters, we find that a far better single-shotmeasurement is possible using the DJQP process thanwith a comparable SET. Significantly, one can also approach the quantum limit of measurement efficiency[3,4], where ’ =meas " 1, in a regime which is boththeoretically tractable and experimentally relevant.
Model.—The Hamiltonian of the coupled qubit plus
SSET system is written as H  H S  H Q  H int .
The qubit itself (or ‘‘box’’), described by H Q , consistsof a superconducting metal island in the Coulomb blockade regime where only two charge states are relevant.
These can be regarded as the z eigenstates of a fictitious
spin 1=2. The island is attached via a tunnel junction to abulk superconducting electrode, leading to the form

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 2002 The American Physical Society

0031-9007=02=89(17)=176804(4)$20.00

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

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H Q  12 f4ECQ 1  N Q  z  EJQ x g;(1)

where ECQ is the charging energy of the box, EJQ is the
Josephson coupling energy of the box, and N Q is thedimensionless gate voltage applied to the box. The SSET
consists of a superconducting, Coulomb-blockaded island
which is attached via tunnel junctions to two superconducting electrodes (Fig. 1). The SSET Hamiltonian
H S  HK  HC  HV  HT has a term HK describing
the kinetic energy of source, drain, and central island
electrons, a term HV which describes the work done by thevoltage sources, and a tunneling term HT . The charging
term is HC  ECS nS  N S 2 , where ECS is the SSET
charging energy, nS is the number of electrons on thecentral island, and N S is the dimensionless gate voltage
applied to the island. Finally, the qubit is capacitively
coupled to the SSET: Hint  2ECQ CCC z nS Eint z nS .
Here CC is the cross capacitance between the box andthe central island of the SSET, and C is the total capacitance of the SSET island. Note that we neglect the coupling of the qubit to its environment, as we are interested
here in the intrinsic effect of the SSET on the qubit[15]. We also assume a SSET with identical tunnel junctions, whose dimensionless conductance g satisfies
g=2
1.
The DJQP process occurs when the SSET gate voltage
N S and drain-source voltage 2VDS are such that two
Cooper-pair tunneling transitions (one in each junction)
are resonant. We label these transitions as ns  0 ! 2(left junction) and ns  1 ! 1 (right junction) (seeFig. 1). Resonance thus requires eVDS  ECS and N S 
1=2. In addition, ECS =S (where S is the superconducting gap of the SSET) must be chosen so that the quasiparticle transitions linking the two Cooper-pairresonances are energetically allowed (i.e., nS  2 ! 1
and nS  1 ! 0), whereas transitions which end thecycle (i.e., nS  0 ! 1) are not. We take ECS  S tosatisfy these conditions; this corresponds to the experiment of Ref. [13]. The two quasiparticle transitions which
occur in the DJQP are characterized by a rate , which is
given by the usual expression for quasiparticle tunnelingbetween two superconductors [16]. The effective Cooperpair tunneling rate J emerging from our description [i.e.,
Eq. (3) below] is given by [8]J  

E2JS 
:4  =222(2)Here  is the energy difference between the two charge
states involved in tunneling, EJS is the Josephson energyof the SSET, and we set h  1.
Calculation approach.—We consider the reduced density matrix  of the qubit plus SSET system obtained bytracing out the SSET fermionic degrees of freedom. The
evolution of  is calculated perturbatively in the tunneling Hamiltonian HT , keeping only the lowest orderterms; this corresponds to the neglect of cotunneling
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processes, which is valid for small g and near the DJQP
resonance. Using an interaction representation where only
HT (and not Hint ) is viewed as a perturbation, the equationof motion of  takes the standard form:
Ztd
dt0 hH T t; H T t0 ; t0   F i:
t  dt
1(3)
The angular brackets denote the trace over SSET fermion
degrees of freedom; as we work at zero temperature, F isthe density matrix corresponding to the ground state ofthese degrees of freedom in the absence of tunneling.
To make further progress, we treat the Josephson coupling emerging from Eq. (3) as energy independent andgiven by the Ambegaokar-Baratoff value EJS  gS =8.
We also use the smallness of g to neglect logarithmic
renormalization terms, as was done in Ref. [4]. One canthen solve for the time-independent solution of Eq. (3),which describes the state achieved by the system after all
mixing and dephasing of the qubit by the SSET has
occurred. To describe the dynamics of mixing (i.e., therelaxation of the qubit state populations to their stationary value), we also calculate the corresponding eigenmode of Eq. (3). A Markov approximation is made which
involves replacing t0  by t on the right-hand side ofEq. (3). This approximation is justified as long as the timedependence of  in the mixing mode is weak compared totypical frequencies appearing in the correlators of Eq. (3),requiring here that mix
ECS and EJSECS [17].
Backaction.—We focus here primarily on the mixing
effect of the measurement backaction; dephasing will be
discussed more extensively in Ref. [17]. The mixing rate
mix  rel  exc is set by the rates at which the measurement relaxes and excites the qubit. Let  denote the
N Q -dependent energy difference between the two qubit
states. For weak coupling (Eint), Fermi’s golden rule
relates rel and exc toRthe quantum charge noise of the
SSET island SQ !  dtei!t hnS tnS 0i:


EJQ 2
2
rel=exc  Eint
SQ :(4)
In our approach, these rates may be directly obtained byusing the stationary solution (which gives the postmixing
occupancies of the box eigenstates) and the mixing eigenvalue of Eq. (3). In the limit of weak coupling, one canthen use Eq. (4) to extract SQ . Our method for calculating the quantum noise, which uses the qubit as aspectrum analyzer, is physically intuitive and no moredifficult to implement than standard approaches [6]; in
addition, we are able to calculate rel and exc when the
coupling is not weak, and Eq. (4) fails.
Figure 2 displays the quantum charge noise obtained atzero temperature, using SSET parameters which correspond to Ref. [13]. The solid curve in Fig. 2 is for thecenter of the DJQP resonance —N S  1=2, eVDS  ECS .
Note the sudden asymmetry that develops between
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1

SQ(ω) (seconds)

10

Qubit Charge

2

−10

10

0
0

−15

−1

1
Qubit Gate Voltage

On Resonance
Off Resonanceω / ECS (ECS = ∆S)

2

1

FIG. 2. Quantum charge noise associated with the DJQP
process. The solid curve corresponds to N S , VDS tuned tothe center of the DJQP resonance; the dashed curve corresponds to moving eVDS away from resonance by =4. We takeg  0:5 and S ’ ECS  0:25 meV in the SSET, corresponding
to the device of Ref. [13]; this gives EJS =h ’ 0:04. Inset:
average qubit charge after mixing has occurred for weakcoupling (Eint =EJQ  0:01), as a function of qubit gate voltage
N Q ; see text for details. We take ECQ ’ 77 )eV and EJQ ’27 )eV. The frequency range probed by tuning N Q matches
the range of the main plot; the sharp steps in the average charge
occur at N Q  ’ ECS .

absorption [i.e., SQ j!j] and emission [i.e., SQ j!j]
when j!j increases beyond ECS . These jumps correspondto the opening and closing of transport channels in the
SSET, and their sharpness is a result of the singularity inthe quasiparticle density of states. For example, as ! rises
past ECS , transitions which are normally forbidden in the
DJQP cycle (i.e., nS  0 ! 1) suddenly become energetically allowed if they absorb energy from the qubit, causing a sudden increase in SQ !.
The effect of the SSET quantum charge noise on thequbit is shown in the inset of Fig. 2, where the averagequbit charge hNB i 1  hz i for t  mix is shown as afunction of N Q . Changing N Q tunes the qubit splitting
frequency , allowing one to probe the frequency dependence of the noise. The solid black curve corresponds tobeing at the center of the DJQP feature, and the grey
curve corresponds to the unperturbed qubit ground state.
The features in the quantum noise manifest themselves in
hNB i, a quantity which is accessible in experiment.
Even more interesting is the situation when one tunes
N S or VDS slightly off the DJQP resonance center.
Unlike the case of a SET, where noise asymmetries are
weak for j!j
ECS [6], there are strong features herethat result from the resonant nature of Cooper-pair tunneling. By treating the mixing terms in Eq. (3) perturbatively, analytic expressions can be obtained for thequantum noise in this regime when EJS <  [inRef. [13], EJS =h ’ 0:04]. If one moves away from the
DJQP center by tuning only VDS (i.e., N S  1=2,
eVDS  ECS  V =2), we find j!j < ECS SQ !  J V 
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J V  !=J V  !: (5)4J V  !J V  !  !2

21 OCTOBER 2002

In the limit where !
=2, Eq. (5) simply corresponds
to classical telegraph noise (the SSET spends only appreciable time in the states nS  0 and nS  1). However, forfinite V and !, Eq. (5) indicates that the noise develops apronounced asymmetry, even though j!jEC . In particular, if V > 0, one has SQ j!j > SQ j!j, implying that emission by the SSET exceeds absorption. Thisbehavior is shown by the dashed curves in Fig. 2, whichcorrespond to N S  1=2, V  =4. This effect is adirect consequence of the resonant nature of Cooper-pairtunneling—by emitting energy, both Cooper-pair tunneling processes in the DJQP cycle become more resonant,while absorbing energy pushes them even farther fromresonance. The result is a population inversion in the qubit
at zero temperature, which in turn leads to a striking,nonmonotonic dependence of qubit charge on N Q(dashed curve in the inset of Fig. 2) [15]. Note that if
one moves away from the center of the DJQP resonanceby changing only the N S , no asymmetry in the noise
results, as now emission (or absorption) moves one of the
Cooper-pair transitions in the DJQP process farther towards resonance, while it moves the other transition
farther away from resonance. Letting V  0 and N 
4ECS N S  1=2, we have for EJS < :2

SQ !  J N 

N !
1  8   !J N  !
2 E4 =2 J N
JS:4J N  !J N  !  !2
(6)

Measurement rate.—To determine the measurement
time meas , we extend our density-matrix description toalso include m, the number of electrons that have tunneledthrough the left SSET junction [4,14]. We are thus able tocalculate the distribution of tunneled electrons Pm; tji,where i "; # denotes the initial state of the qubit. meas isdefined as the minimum time needed before the two
distributions Pm; tj " and Pm; tj # are statistically distinguishable [4]:
2I"  I#1 p p :
(7)
meas
2f" I"  2f # I#
Here I" and I# are the average SSET currents associatedwith the two qubit states, and f" and f# are the associatedFano factors which govern the zero-frequency shot noisein the current. In the absence of the qubit, the densitymatrix equations for the SSET yield the following for the
single Fano factor f:
3
1 E2JS 3=22  2f  1 ;(8)
2
2 =2 2  2  E2JS =22
where we take eVDS  ECS ,   N  4ECS N S 
1=2. Equation (8) indicates that the effective charge ofthe carriers in the DJQP process is 3e=2 in the limit where  EJS . In this limit, Cooper-pair tunneling is the ratelimiting step in the cycle; electrons effectively tunnel inclumps of e or 2e, leading to an average charge of 3e=2.
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1/ τ
meas
Γmix

−5

Rate / Γ

10

χ

0.5

−10

10

0

0
0

1

3

Qubit Gate Voltage

E /Γ
4 JS

2

FIG. 3. 1=meas , rel , and exc vs qubit gate voltage N Q for astrongly coupled system, where Eint =EJQ ’ 0:3 (i.e., CC =C 0:05). A good measurement is possible for a wide range of gate
voltages. Inset: Heisenberg efficiency ,  ’ =meas at weakcoupling, as a function of EJS =.

We consider meas in the limit of weak coupling (Eint) and weak mixing (EJQ). Taking V  0 andN  =2 for near optimal gain, and using Eqs. (6) –(8), we find that the intrinsic signal-to-noise ratiomeas mix 1=2 of the measurement, in the relevant regime EJS < , is given by
r
4

:
(9)
S=NDJQP  j cot*j3=2Here cot* 4ECQ 1  N Q =EJQ , and we take J 0 < ECS . If a SET in the sequential-tunneling regime isused for the qubit measurement, it was found in Refs. [3,4]that the optimal S=N is given by ( < ECS )s
 2 g2
S=NSET  +j cot*j
(10)
 2;
eVDS

where + is of order unity. As the quasiparticle transitiong
rate   2eVDS , we see that the S=N achieved using
DJQP is parametrically larger (in 2=g  1) than thatobtained for the SET. This enhancement results largely
from the narrow width of the DJQP feature —the energy
scale over which the current changes (and thus the gain)
is set by  rather than VDS . The gain and S=N ratio of theSET could be improved by working in the cotunnelingregime; however, this would result in a much larger meas(meas / g2 ), making one more susceptible to unwanted
environmental effects. In contrast, the DJQP feature hasboth a large gain and a short meas (i.e., meas / 1=g).
Shown in Fig. 3 as a function of N Q are meas ; rel , andexc for a strongly coupled device (CC =C  0:05), withall other parameters as listed in the caption of Fig. 2. We
have taken V  0 and N  =2 for optimal gain.
Figure 3 confirms that an excellent measurement is indeed
possible, with S=N2 > 100.
We have also studied the efficiency ,  ’ =meas ofmeasurement using DJQP for a weak coupling (Eint
EJS ; ) and  < ECS , where ’ is the measurementinduced dephasing time [17]. Unlike an SET in the sequential-tunneling regime, where , / g2 is always much
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less than the quantum limit ,  1 [3,4], here , is controlled by the ratio EJS =. As shown in the inset of Fig. 3,
by tuning this ratio, , can be made to approach thequantum limit. Here, for each value of EJS =, we haveset VDS and N S to optimize the gain. Measurement using
DJQP is able to reach a high efficiency when EJS ’  bothbecause of the symmetry of the process and because of
the coherent nature of Josephson tunneling; the large gainof the process is also important [17]. Clearly, the DJQP
process allows for a far superior measurement of a
Cooper-pair box qubit than a SET.
We thank M. Devoret, K. Lehnert, and R. Schoelkopf
for useful discussions. This work was supported by the
NSF under Grants No. DMR-0084501 and No. DMR0196503, and by the Army Research Office under Grant
No. ARO-43387-PH-QC.

[1] R. J. Schoelkopf et al., Science 280, 1238 (1998).
[2] A. N. Korotkov et al., in Single-Electron Tunneling andMesoscopic Physics, edited by H. Koch and H. Lubbig(Springer, Berlin, 1992).
[3] M. H. Devoret and R. J. Schoelkopf, Nature (London)
406, 1039 (2000).
[4] Y. Makhlin et al., Phys. Rev. Lett. 85, 4578 (2000); Rev.
Mod. Phys. 73, 357 (2001).
[5] A. Aassime et al., Phys. Rev. Lett. 86, 3376 (2001).
[6] G. Johansson et al., Phys. Rev. Lett. 88, 046802 (2002).
[7] T. A. Fulton et al., Phys. Rev. Lett. 63, 1307 (1989).
[8] D.V. Averin and V. Ya. Aleshkin, JETP Lett. 50, 367(1989); Physica (Amsterdam) 165-166B, 949 (1990).
[9] A. Maasen van den Brink et al., Phys. Rev. Lett. 67, 3030
(1991).
[10] Y. Nakamura et al., Phys. Rev. B 53, 8234 (1996).
[11] S. Pohlen, Ph.D. thesis, Harvard University, 1999 (unpublished).
[12] The use of only Cooper-pair tunneling in a modified
SSET has been studied in A. B. Zorin, Phys. Rev. Lett.
76, 4408 (1996); cond-mat/0112351. The use of quasiparticle-only processes in a SSET was studied in A. N.
Korotkov, Appl. Phys. Lett. 69, 2593 (1996).
[13] K. Lehnert et al. (to be published).
[14] M-S. Choi et al., Phys. Rev. Lett. 87, 116601 (2001).
[15] To observe the population inversion predicted here, environment-induced relaxation must be weak compared tothe mixing from the measurement. For the devices ofRef. [13], the electromagnetic environment may betreated as a 50  impedance whose voltage fluctuations
cause relaxation; this leads to a relaxation rate of envt 
106 Hz. The effect of 1=f noise is negligible given thelarge qubit splitting frequency. To have the measurementinduced mixing rates rel=exc [Eq. (4)] exceed envt , a
coupling strength CC =C ’ 0:1 is sufficient.
[16] See, e.g., M. Tinkham, Introduction to Superconductivity(McGraw-Hill, New York, 1996).
[17] ’ can be obtained in a manner analogous to that used inRef. [4]. A full derivation and discussion will be presented in A. A. Clerk et al. (to be published).

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