PRL 96, 056801 (2006)

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PHYSICAL REVIEW LETTERS

Backaction Noise in Strongly Interacting Systems: The dc SQUID
and the Interacting Quantum Point Contact
A. A. Clerk
Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada(Received 13 July 2005; published 7 February 2006)
We study the backaction noise and measurement the efficiency (i.e., noise temperature) of a dc SQUID
amplifier and, equivalently, a quantum point-contact detector formed in a Luttinger liquid. Using amapping to a dissipative tight-binding model, we show that these systems are able to reach the quantum
limit even in regimes where several independent transport processes contribute to the current. We suggest
how this is related to the underlying integrability of these systems.
DOI: 10.1103/PhysRevLett.96.056801

PACS numbers: 73.23.b, 05.40.a, 74.50.+r

There has been considerable recent interest in studying
detectors and amplifiers which add the minimal possible
noise allowed by quantum mechanics to an input signal [1–7]. Such detectors are necessary for single spin and gravity
wave detection, as well as for quantum control and quantum computation. In the important case where the detector
is a mesoscopic conductor, understanding whether one canachieve the quantum limit (i.e., have a minimal backaction
effect) requires an understanding of both the output current
noise of the system as well as its backaction charge noise.
Theoretically, these quantities have been studied for avariety of mesoscopic detectors, including noninteracting
tunneling point-contact detectors [1,3] and more general
coherent scattering detectors [6,7].
Here, we examine the ideality of mesoscopic detectors
where nontrivial correlation effects are important. One
such system is a quantum point-contact (QPC) detector
formed in an interacting Luttinger liquid; this could be
realized in a quantum Hall edge state or by embedding anoninteracting QPC in a resistive electromagnetic environment [8]. QPCs are in widespread use as readouts ofquantum-dot qubits; in the absence of interactions, they
are known to be able to reach the quantum limit [1,5–7].
What happens now when interparticle interactions areturned on? Note that while the current noise of an interacting QPC has received considerable attention [9,10], its
backaction noise has not been studied.
A second correlated detector of obvious practical importance is the dc SQUID amplifier. While experimentally
it is known that the dc SQUID can approach near quantumlimited operation, the theoretical limit for this system has
not been fully studied. Previous studies either neglected theeffect of the SQUID inductance [4] or were based on thequantum Langevin equation [11,12], an equation which isformally valid only in the limit of high temperatures orextreme dissipation [13].
In this Letter, we calculate the backaction noise andmeasurement efficiency (i.e., noise temperature) of boththe interacting QPC detector and the dc SQUID using
controlled perturbative approaches. We discuss how ineach case the principle of detection is the same: the input
0031-9007=06=96(5)=056801(4)$23.00

signal modulates the tunneling of excitations across thedetector. In the QPC, these excitations are electrons orquasiparticles; in the SQUID, they are Cooper pairs. We
address both the weak-tunneling and strong-tunneling regimes of these systems. In the latter regime a multitude oftunneling processes, each transferring a different numberof particles, will be significant. As the output of the detector essentially averages over these processes, one would
expect there to be lost information and hence excess backaction, much in the same way that multiple channels in anoninteracting QPC lead to a departure from the quantum
limit (QL) [6,7]. We show that this is not the case: one canremain at the QL even when multiple tunnel processes
contribute. We suggest how this result is related to theintegrability of the underlying field theory.
Models.—We start with the dc SQUID. Figure 1 shows atypical setup: two identical Josephson junctions, eachshunted by a resistance R and capacitance C, placed in aring of inductance L threading an external flux x . The
junctions are symmetrically biased by a current bias IB . We
consider the typical case of nonhysteretic operation, andwhere IB > 2IC (IC  2eEJ =@ is the critical current ofeach junction). The dc SQUID is a flux-to-voltage amplifier. The input signal is a small additional flux in whichadds to x ; by varying its value, one changes the voltage V
across the SQUID. The backaction here is created by thecirculating current J around the SQUID loop [11]. J directly couples to the input flux; its fluctuations act as anoisy backaction force on the input system.
We consider the dc SQUID as a linear detector, which
requires only that the signal flux in be small enough thatthe corresponding change in V is / in . A measure of theideality of any linear amplifier or detector is its noise

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C
R
EJ

Φx

IB

V

FIG. 1. Schematic of a dc SQUID amplifier.

© 2006 The American Physical Society

PRL 96, 056801 (2006)

PHYSICAL REVIEW LETTERS

temperature kB TN . It quantifies the total noise added to the
input by the amplifier. Quantum mechanics requires acertain minimal amount of backaction; as a result, there
is a quantum limit on TN : kB TN  @!=2, where ! is thesignal frequency [14]. For an optimal coupling between thesignal source and the SQUID, TN is set by the noise of theuncoupled detector [15]:skB TN
SV !SJ !  ReSVJ !2 1 (1)

@!=2
@!=22
Here, SV (SJ ) is the symmetrized noise in the output (backaction force) of the detector and SVJ is the cross-correlationnoise.   dhVi=dx is the gain. For a dc SQUID withidentical junctions, SVJ  0. We will focus on the low-!
limit of TN ; it describes the typical situation where ! ismuch smaller than detector frequencies. If our detector isused for qubit detection, the same limit of 1=2 yields themeasurement efficiency ratio [6,7].
We now analyze the SQUID. Heuristically, the bias
current IB will partially flow through the shunt resistors,
and partially through the junctions. The junction current
will be due to the incoherent tunneling of Cooper pairs[16]: Cooper pairs tunnel through the junction, simultaneously dissipating energy in the electromagnetic environment formed by the shunts. The voltage across the SQUID
will then be set by the current flowing through the shunts:V  R=2IB  ICPT;1 ICPT;2 , where ICPT;j , theCooper-pair tunneling current through junction j (j 1; 2), depends both on x and IB . Both the gain  andoutput noise SV can be related to ICPT ; the backaction
circulating current will be given by J  ICPT;1 
ICPT;2 =2.
The above picture is made rigorous by using a CaldeiraLegget representation of the impedances and a pathintegral representation of the Keldysh partition function
Z for the system[16]. TracingR
R out the environment, one
has Z  DD exp@i dtL; . Setting  
1 2 =2 and   1  2 =2  x =0 , and using
the indices c (q) to denote classical (quantum) Keldysh
fields [17] (  c q ,   c q ), we have



X
L
x

EJ cos  tcos t vt
2
0

!
!
X Z
c t
c t0 
1
0
0
^
dt
G t  t :(2)
q t
q t0 ;
The effective voltage bias VB  RIB sets v  2eVB =@. The
matrix Green functions G^  (  ; ) are given in termsof the corresponding impedances Z via (@  1)!
^ !2iReZ ! coth2k!B T  iZ !
!G


; (3)
iZ !
0
4e2
with Z !  2=R
2i!C 2=i!L.

2i!C1 and Z !  2=R

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Introducing sources to calculate V, J, and their fluctuations from Z, one can identify operators for the total
Cooper-pair current ICPT and the backaction current J.
They take the expected form
ICPT  2IC cosx =0  sin

vt;

(4)

Idiff  IC sinx =0  cos

vt;

(5)

One now finds rigorously that hVi  R=2IB  hICPT i;
moreover, at T; !  0, one can directly relate the noisecorrelators of interest to the Cooper-pair shot noise: SV 
R=22 SICPT and SJ  SIdiff .
At this point, we pause to note the analogy between thedc SQUID and the interacting QPC detector. The latter is apotential-to-current amplifier: the input signal modulates
the tunneling amplitude  between two semi-infinite, spinless Luttinger liquid leads; the result is a corresponding
modulation of the QPC current [18]. After bosonizing, thetunnel Hamiltonian is H t   cos, and the Keldysh
partition function for this system is identical to that for thedc SQUID if the phase  is pinned. To make this correspondence [19], we associate the dimensionless conductance 1=  h=2e2 R with the Luttinger interaction
parameter g, 2EJ cosx =0 with the QPC tunneling
amplitude , and 2VB with the QPC bias voltage. The
current through the QPC will then correspond to ICPT ,and the backaction force corresponds to Idiff =IC sinx =0 . This mapping also requires Z be constant over frequencies of interest, implying that the cutoff
frequency !C  1=RC be much larger than v; eRIC =@.
We can thus view the dc SQUID as an interacting QPC
detector where, due to fluctuations in , the magnitude of
the tunnel matrix element is fluctuating.
We now return to the SQUID and address its backaction.
By expanding Z in powers of EJ , we can describe processes involving multiple transfers of Cooper pairs. Suchan expansion is well controlled in the large-IB limit; for theexperimentally relevant limit
1, the expansion converges for all IB > 2IC [16]. The simplest limit is when
only single Cooper-pair tunneling events play a role; for< 1, this requires IC
IB . In this regime, it is sufficient
to calculate to lowest nonvanishing order in EJ [4].
Introducing
8e2 Z 1 d!
ReZ !ei!t  1; (6)J  t 
h 0 !
we find that the measurement efficiency (or reduced noise
temperature) for an optimal x  0 =4 is given byR1

dtsinvtImfexpJ  t J  tg 2
2i
2
2h
R01
  e
; (7)
0 dtsinvtImfexpJ  t  J  tgR
with h2 i  8e2 =h 1
0 d! ReZ !=!. We see that fluctuations in the phase  prevent one from reaching the QL(i.e.,   1). These fluctuations represent an extraneous
source of noise: they reduce the gain of the detector andincrease the backaction noise. Suppressing these fluctua-

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PHYSICAL REVIEW LETTERS

tions (i.e., pinning ) requires that the loop inductance L besmall. More precisely, for
1, Eq. (7) becomesPn; t 

1  ~ 2M Z t
X
iEJ

@
2M
X Y

1

Mn

2 



ReZ v ReZ v 2:
ReZ v  ReZ v

(8)X

f tICPT  ttJ tg:

eiv j tj 1

Zt
t2M1

dt2M

j j tj ei tj j~;~ j1

Reaching the QL in this large bias, weak-tunneling regime
thus requires that Z =Z is small at the characteristic
Josephson frequency v set by the bias current. Thus, inthe L ! 0 limit, our system always reaches the QL as longas tunneling is weak, irrespective of further details of the
impedance Z . As the SQUID is equivalent to a interacting
QPC in this limit, we can also conclude that an interacting
QPC in the weak-tunneling regime always reaches the QL,
irrespective of g. For repulsive interactions (g < 1), thisregime corresponds to small voltages [18]. Using the usual
duality between large and small g [19], this conclusion also
holds for a QPC detector near perfect transmission, where
the input signal modulates the strength of a weakly backscattering impurity. If the backscattering is weak enoughthat it can be treated to leading order (which requires large
voltage for g < 1), one will again always be at the QL.
We next consider the L ! 0 limit, but now consider
regimes where higher-order tunneling plays a role.
Integrating out the phase  in each term of Z, we obtaina ‘‘Coloumb gas’’ description of Z [19]. We introduce
auxiliary source fields in the action which couple to the
Cooper-pair current and to the backaction force:
L src 



dt1

(9)



To interpret the resulting expansion of Z, we make ananalogy to the Schmid model, a dissipative tight-binding(TB) model [20]. It describes a particle on a 1D latticewhich feels a force from an oscillator bath having a spectral density A!  8e2 =h!Z !. In this mapping,x
corresponds to the tunnel matrix eleE~J  2EJ cos 0ment of the TB model, and 2eVB to a constant external
force. The expression for Z (at ;   0) may now be castas a sum over tunnel events which take the particle from an
initial density matrix localized at x  0 to one localized atx  n, where n is arbitrary. Each term in the expansion
describes the amplitude of a process involving 2M tunnel
events occurring at times t1 to t2M . Each event can movethe particle either to the left or to the right, and can occur
either on the forward or backwards Keldysh contour. Eachtunnel event is thus labeled by a charge j  1 which
gives the direction of the tunnel event, and a charge jwhich determines the contour of the event: j j  1 (1)for an event on the forward (backwards) contour. Finally,
because of decoherence from the bath, each tunnel process
results in a P
final density matrix state which is diagonal; wethus havej  0 for each term. We thus have Z Plimt!1 n Pn; t, with F;~ ~; ~ :(10)

n is the net displacement of a given process, and the factor
F describes interactions among the charges arising from
integrating out ; its precise form is given in
Refs. [16,19,21]. Sums overP charges in Eq.
P (10) are restricted to those satisfying j j  0 and j j  2n.
We see that the source field
couplesto the timeP
derivative of the net displacement n  j =2. Thus, theCooper-pair current ICPT corresponds to the velocity of theparticle in the TB model. This has been used previously to
calculate the current noise in this system [22]. We also seethat the source  couples to j j . Thus, the backaction
force corresponds
P to the time derivative of the ‘‘quantum
charge’’ z  j j =2; z is the net number of forward
contour minus backward contour tunnel events. For a given
M and n, z may range from M  jnj to M  jnj.
To make further progress, we follow Refs. [21,22] andconsider the Laplace transform representation of Z. As
detailed in Refs. [21,22], one may then discuss Z in termsof ‘‘irreducible clusters.’’ Letting s denote the Laplacetransform variable, these are tunnel processes (i.e., a setof and  charges) which, in the s ! 0 limit, yield a finite
constant contribution to sdZ=dts. Unlike Refs. [21,22],
we track both the classical displacement n and the quantum
charge z; each irreducible process is characterized by aparticular value of n and z. In the long time limit, each suchprocess is statistically independent, with its amplitude n;zacting as an independent rate; one obtains Poissonian
statistics for both n and z. At T  0 and finite bias, one
finds that all rates with n < 0 vanish. One can also shown;z   n;z  . A straightforward calculation then yields
X
 X
1
1
X
hICPT inncl n;
(11)
n;z 
2e
z
n0
n0
X
 X
1
1
X
SICPT
2

n
n2 cl n;
(12)

n;z
4e2
z
n0
n0
X
 X
1
1
X
SJ
2

z
z2 q z;
(13)
n;z 
e2 B
z1
z
n0
where
B  tan2 x =0 . We can interpret cl n Pz n;z as a ‘‘rate’’ associated with the incoherent tunneling of n Cooper pairs; each such process contributes to thecurrent independently. Moreover, we see that backaction
noise results from an uncertainty in how tunnel events are
distributed between the two Keldysh contours (i.e., z
fluctuates).
As many independent processes contribute to the current, one’s first guess is that the system will no longer be at

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PHYSICAL REVIEW LETTERSPRL 96, 056801 (2006)

the QL (i.e.,  > 1), as generically, there is lost information associated with averaging over the different transport
processes. To address this issue, we first note a remarkable
result found by Saleur and Weiss: the rates cl n P2nz n;z are exactly proportional to EJ and contain nohigher-order terms [22]. Formally, this relation resultsfrom cancellations of terms arising in perturbation theory.
It implies cl n  n;0 , and that irreducible processes
involving both backwards and forward tunneling events
make no net contribution to charge transport.
CPT i
Using the above result, along with   dhId
~i tanx =0  E0J dhIdECPT
,
we
see
that
a
sufficientcondi~Jtion for having   1 is cl n  2 Req z  n. We
have explicitly calculated the rates n;z to order EJ 6 ,which involves calculating 10 partial rates n;z , each ofwhich has contributions from numerous charge sequences.
We find that a much stronger relation is satisfied: if both nand z are nonzero,Ren;z  0;(14)

otherwise,2 Re n0;z 2znz;0 / EJ  :Here  is the strength of the QPC backscattering potential,
and C is an energy-independent constant. Remarkably,
the energy-dependent transmission T " for quasiparticles
in the boundary sine-Gordon model satisfies the exact same
equation [see Eq. (13) of Ref. [10] ]. This suggests a deep
connection between integrability and the QL. Note alsothat by using the results of Refs. [8,10], one can see that ina weakly interacting QPC, the ‘‘physical’’ transmission
coefficient (i.e., for electrons, not for quasiparticles) satisfies Eq. (16); this follows from the fact that in the exact
solution of Ref. [10], dT =d / dT =d logE, where E isenergy. Thus, if one is concerned with the validity ofEq. (16) on the physical transmission coefficient, this isalso known to be satisfied in a weakly interacting QPC.
In conclusion, we have calculated the backaction noiseand measurement efficiency for both the dc SQUID amplifier and the interacting QPC detector. Using a perturbative approach, we have shown that it is still possible toreach the quantum limit in regimes where multiple higherorder tunneling processes play a role.
We thank NSERC, FQRNT, and the Canada Research
Chairs program for support.

(15)

Note the duality between transport in the classical (i.e., n)and quantum (i.e., z) directions. The only irreducible process yielding a classical displacement n has precisely 2ntunnel events all occurring in the same direction. Similarly,
the only irreducible process increasing the quantum chargeby z has exactly 2z tunnel events on the same Keldysh
contour. The cancellations which yield these results, andthus a quantum-limited backaction, require a purely Ohmicbath spectrum. This is always the case for the interacting
QPC. For the dc SQUID, one needs the Josephson frequency v to be much smaller than the cutoff frequency
1=RC in order to reach the quantum limit at strong
tunneling. Note that Eqs. (14) and (15) are not a directconsequence of the result for cl n found in Ref. [22].
Also note that, using duality [16,19], our conclusions also
apply to the case where a perturbative expansion in EJ does
not converge, as one can similarly analyze an alternate
convergent expansion for Z.
The remarkable cancellations that lead to the result of
Eqs. (14) and (15) are, similar to the result cl n / EJ 2n ,a result of the integrability of the model studied here.
Integrability may be used to exactly calculate the current
and current noise in this system [10]. The solution is given
in terms of the scattering of quasiparticles in a boundarysine-Gordon model. One calculates an energy-dependent
quasiparticle transmission coefficient T ", and then uses
this to calculate the current and noise. Note that for anoninteracting QPC, where again the signal of interest
modulates a localized potential, reaching the QL requires
the QPC transmission T " to satisfy [6,7]dT" CT"1  T":d

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