PRL 104, 186803 (2010)

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PHYSICAL REVIEW LETTERS

Inelastic Backaction due to Quantum Point Contact Charge Fluctuations
C. E. Young and A. A. Clerk
Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8(Received 28 October 2009; published 4 May 2010)
We study theoretically transitions of a double quantum-dot qubit caused by nonequilibrium charge
fluctuations in a nearby quantum point contact (QPC) used as a detector. We show that these transitions are
related to the fundamental Heisenberg backaction associated with the measurement, and use theuncertainty principle to derive a lower bound on the transition rates. We also derive simple expressions
for the transition rates for the usual model of a QPC as a mesoscopic conductor, with screening treated at
the RPA level. Finally, numerical results are presented which demonstrate that the charge noise and shotnoise backaction mechanisms can be distinguished in QPCs having nonadiabatic potentials. The enhanced
sensitivity of the charge noise to the QPC potential is explained in terms of interference contributions
similar to those which cause Friedel oscillations.
DOI: 10.1103/PhysRevLett.104.186803

PACS numbers: 73.23.b

A fundamental consequence of the Heisenberg uncertainty principle is that the act of measurement necessarily
perturbs the system being measured; these measurementinduced disturbances are termed ‘‘backaction.’’ Researchon quantum electronic systems has progressed to the point
where backaction effects, often near the limits imposed by
the uncertainty principle, are of key relevance to experiment [1]. They are particularly important in experiments
involving solid state qubits, as they set a limit on how well
one can read out the qubits. In qubit-plus-detector systems,
detector backaction can both dephase the qubit (i.e., T2
effects), as well as drive transitions between the qubit
energy eigenstates (i.e., T1 effects, or ‘‘inelastic’’ backaction). The study of backaction is also of purely fundamental interest, as it can shed light on the physics of thedetector, which is often an interesting nonequilibrium
quantum system in its own right.
In this Letter, we focus on the inelastic backaction
associated with quantum point contact (QPC) charge detectors [2]. QPCs are among the most versatile and widelyused charge detectors in quantum electronics; having acomplete understanding of their backaction is therefore
highly desirable. They are the standard readout for bothcharge and spin quantum-dot qubits [3,4], and have also
been used to make sensitive measurements of nanomechanical oscillators [5]. The backaction of QPC detectors
has been the subject of considerable study. Theoretically,
QPC-induced dephasing has been studied in Refs. [6–8],
and measured experimentally in seminal experiments on
mesoscopic interferometers [9]. Theory has also shownthat the backaction dephasing induced by a QPC can reachthe fundamental quantum limit set by the Heisenberg
uncertainty principle [10–12]. Inelastic QPC backaction
has also received attention. Aguado and Kouwenhoven[13] discuss such effects in a system where a double
quantum-dot is measured by a QPC. They focus exclusively on a mechanism whereby QPC current flucutations(shot noise) lead to potential fluctuations across the double
0031-9007=10=104(18)=186803(4)dot, which in turn drives transitions in the qubit. Several
recent experiments have attempted to investigate this inelastic backaction mechanism [14–17].
Here we focus on an alternate mechanism for inelastic
QPC backaction which involves its nonequilibrium charge
fluctuations, as opposed to its current fluctuations. Recall
that the physics behind these two kinds of noise is very different. Current fluctuations are related to the partitioning ofelectrons into transmitted and reflected streams by the QPC
scattering potential; this results in a binomial distributionof the time-integrated current through the QPC [18]. In
contrast, charge noise results from fluctuations in how longelectrons spend in the scattering region (i.e., dwell-time
fluctuations) [19]. Importantly, the current noise mechanism proposed in Ref. [13] can in principle be completely
suppressed by appropriate circuit design. In contrast, theinelastic backaction arising from QPC charge fluctuations
is the fundamental Heisenberg backaction: for a fixed
measurement strength, it cannot be made arbitrarily small.
Using the uncertainty principle, we derive a rigorous
lower bound on the charge noise inelastic backaction that
exclusively involves directly measurable quantities. We
also derive simple expressions for transition rates valid
for the usual model of a QPC as a coherent mesoscopic
conductor. Our results suggest that the charge noise mechanism may already play a significant role in experiments.
Further, we show that charge fluctuations are more sensitive to the nature of the QPC potential than current noise:
while the current noise is simply determined by the QPCtransmission T , the charge noise also depends on the
adiabaticity of the QPC potential. As a result, it should
be experimentally possible to distinguish the current andcharge noise backaction mechanisms by their dependence
on T . We provide an explanation for this effect in terms ofinterference contributions similar to those that give rise toFriedel oscillations.
Note that other aspects of QPC charge noise backaction
on qubits have been studied before. Büttiker and collabo-

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Ó 2010 The American Physical Society

PHYSICAL REVIEW LETTERS

PRL 104, 186803 (2010)

rators have studied inelastic backaction due to equilibrium
charge fluctuations, as well as dephasing due to nonequilibrium charge noise [10,20]. While the general form of thenonequilibrium charge noise has been derived [19], its
direct role in generating qubit transitions at weak couplinghas not to our knowledge been previously elucidated.
Related work has examined Fermi-edge physics in aqubit-QPC system [21], as well as an indirect mechanism
where QPC charge noise, in consort with spin-orbit coupling, yields transitions in spin qubits [22].
Model.—While our results are general, we focus on theexperimentally relevant case of a double quantum-dot(DQD) charge qubit electrostatically coupled to a QPC
charge detector (see Fig. 1). The qubit Hamiltonian is
HDQD ¼ 2 ^ z þ tc ^ x , where  is the detuning and tc is
the interdot tunneling amplitude. The eigenstates of ^ z
correspond to an electron occupying either the left or rightdot. We start by considering a completely general model ofa spin-degenerate, single channel QPC with applied bias
2
Vqpc . The QPC conductance is G ¼ 2eh T , where T is theeffective transmission.
The QPC acts as a detector by virtue of the fact that thepotential felt by QPC electrons depends on the qubit’s
charge state. The qubit-QPC interaction Hamiltonian
therefore necessarily includes a term of the form
U
^
^ Q;
H^ int ¼
2 z

Q^ 

Z

d~r

Uð~rÞð^ r~Þ:
U(1)Here, Uðr~Þ is the change in the QPC scattering potential brought about by changing the qubit charge state,and ^ is the electron density operator of the QPC. For
convenience, we introduce the parameter U represent~ in the scattering region of
ing the average value of UðrÞthe QPC. For the electrostatic model in Fig. 1, and in thec EClimit of weak coupling Cc  C0 , we have U ¼ 2C
C0 e
where EC is the charging energy of the coupled rightquantum dot. Via H^ int , the QPC potential and hence theconductance both depend on the qubit state; the QPC
current I^ therefore yields information regarding the qubit
state. We assume throughout the realistic limit of weakcoupling in which linear response theory applies. Thus the

FIG. 1 (color online). Electrostatic model of the coupledcharge qubit-QPC system. Inset: Schematic showing the detuning  of the DQD.

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coupling H^ int changes the average QPC current by an
R 0^
dt ðt  t0 Þh^ z ðt0 Þi, where ½! ¼
amount hIðtÞi
¼ U
2
Ri! h½IðÞ;^^
 @i 1Qð0Þiis the finite-frequency gain of0 de
the detector. At ! ¼ 0, the gain ½0 is simply related toT , the change in the transmission of the QPC brought
about by changing the qubit state:^ ¼ h^ z i
hIi

2e2 Vqpc TU½0h^ z i:
2
2
h

(2)

Equation (2) concisely expresses the fact that, via H^ int , theQPC acts as a charge detector. This interaction is thefundamental qubit-detector coupling: without it one cannot
use the QPC to measure ^ z . Consequently, the backaction
due to H^ int is the Heisenberg backaction associated withthe measurement. Fluctuations in Q^ both dephase the qubit
and induce transitions between its eigenstates: it is theinelastic backaction that we wish to describe. As Q^ is aweighted average of the QPC charge density, we refer to it
as the QPC ‘‘charge’’ in what follows.
Before proceeding, note that one may also have a second
qubit-QPC interaction of the form Hint;2 ¼ e^ z U^ 2 ðtÞ,^where U^ 2 ½!  ZT ½!I½!,and ZT is a transimpedance
that converts QPC current fluctuations into qubit potential
fluctuations. Reference [13] provides a comprehensive
description of the qubit transitions induced by this coupling; Ref. [23] furthers this analysis by considering theeffects of a spatially dependent ZT . Note however that
Hint;2 does not contribute to qubit detection: linear response^ at low frequency. The
tells us that Hint;2 cannot change hIi
backaction mechanism of Ref. [13] is therefore not the
Heisenberg backaction of the QPC. Moreover, the backaction due to Hint;2 can be made arbitrarily small by designing the QPC-DQD circuit to suppress ZT . The backaction
due to fluctuations in I^ can therefore be eliminated without
affecting the sensitivity of the detector.
Heisenberg bound.—Fluctuations in Q^ can drive transitions between the qubit eigenstates; for weak coupling, thegolden rule rates for excitation (Q" ) and relaxation (Q# )tc 2
Þ SQQ ½, whereare given by Q"=# ½ ¼ ðU=@Þ2 ð@R^ Qð0Þi^SQQ ½! ¼ 2 dtei!t hQðtÞ
is the quantum noise specpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
tral density of Q^ and @ ¼ 2 þ 4t2c is the qubit level
splitting. As this backaction is due to the coupling used tomake the measurement, one can use the uncertainty principle to rigorously derive lower bounds on the transition
rates without having to specify a QPC model or the coupling potential Uðr~Þ. Using the Heisenberg inequality
which bounds the symmetric-in-frequency noise of a linear
detector [24,25], as well the relation between theasymmetric-in-frequency noise and dissipative response
coefficients, we find 2 
tjU½j2 ðeUÞ2
RK
Q"=#  c
Re
: (3)

@
Zin ½
2@2 
4SII ½Here, RK  eh2 and SII ½! ¼ 12 ðSII ½! þ SII ½!Þ is the186803-2

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PHYSICAL REVIEW LETTERS

symmetrized QPC current noise. The input impedanceof the QPC as seen by the qubit is Zin ½! 
R
i! ^
^
@=ð! 1h½QðÞ; Qð0ÞiÞ.
As Zin is directly related
0 de
to the QPC charge-charge susceptibility, it can be measured by driving Rabi oscillations in the DQD and measuring the induced oscillations in the QPC charge Q^ using asecond QPC, similar to the experiment of Ref. [26]. The
magnitude and phase of the resulting oscillations in Q^
provide a measure of Zin ½!.
The bound in Eq. (3) is completely general, and does notrely on any assumptions regarding the QPC. Most importantly, all of the parameters on the right-hand side are
intrinsic properties of the QPC and directly measurable.
2
Note also that RefZ1
in g vanishes as ! when ! ! 0,reflecting the capacitive coupling between qubit and QPC.
Backaction transition rates.—We now specialize andtreat the QPC as a coherent scatterer, described by ascattering matrix s with both time-reversal and inversion
symmetries. The effects of screening in the QPC are described using the method developed by Büttiker and collaborators [10,20,27], which combines the circuit model inFig. 1 with the random-phase approximation (RPA).
Within this model, the spatial variations in the QPC charge
and internal potential are taken into account by using acapacitive network to describe them [28,29].
In the presence of screening, the total (screened) change
in QPC potential Utot ðr~Þ, resulting from a change in thequbit state, differs from the bare (i.e., unscreened) potentialchange Uð~rÞ appearing in Hint . It is Utot ðr~Þ, not Uð~rÞ,which determines the change in the QPC transmission
T . Screening also modifies SQQ . Importantly, within
the RPA, one finds that T and SQQ are both determinedby the same parametricchange of the scattering matrix,Rgiven by s ¼ d~rUtot ð~rÞ½s=Uðr~Þ, where =Uð~rÞ
is a functional derivative with respect to the local QPC
potential [30]. Using this approach [29], we find that, evenin the presence of screening, one can relate the qubit
transition rates to T without needing to know the precise
form of Utot ðr~Þ. In the low-temperature limit kB T 
feVqpc ; @g, and for @  eVqpc , we find 2tðT Þ2 eVqpc  @Q" ½ ¼ c;(4)@ T ð1  T Þ
4@
 


1 tc 2 ðeUÞ2
RK
Re
: (5)
Q# ½ ¼ Q" ½ þ
 @Zin ½
@2 
Here we have taken s and s to be approximately energy
independent within the energy ranges of interest.
Equations (4) and (5) are the central results of this Letter.
They allow one to make a simple prediction for the chargenoise-induced inelastic backaction using only experimentally accessible QPC quantities. The excitation rate Q" hasa particularly simple form that depends only on the QPC
bias Vqpc , transmission T , and sensitivity T . This is instark contrast to the current noise backaction rates dis-cussed in Ref. [13], where one must also estimate the valueof a transimpedance that is difficult to measure. Simple
estimates using Eq. (4) and typical experimental values(e.g., Ref. [16]) suggest the expected charge noise backaction is at least comparable in magnitude to the shot noisemechanism in recent experiments. We stress that screening
only enters through the measurable quantities T and Zin(see also Ref. [10]).
Charge noise backaction signature.—Next we outline a
way to distinguish the charge and current noise backaction
mechanisms in experiment. Here we note that the shotnoise mechanism of Ref. [13] yields a backaction excitation rate I"  SII  T ð1  T Þ. Thus an observed dependence of " on the QPC transmission of the form
T ð1  T Þ (and hence a maximum at T ¼ 1=2) is often
regarded as a signature of the shot noise mechanism[14,17]. From Eq. (4), we find that, in general, Q" will
not have the same dependence on T : the only case where it
Tdoes is when T ð1T
is a constant that does not change asÞT and T are simultaneously varied by changing gate
voltages. As noted in previous work [10,11], this conditionis satisfied if and only if the QPC potential is adiabatic [31]and so is the qubit-QPC coupling [i.e., the variation of~ is smooth compared to the Fermi wavelength F ]. IfUðrÞthese conditions are violated, Q" ðT Þ will not have its
maximum at T ¼ 1=2. This provides a way to distinguishthe charge and current noise backaction mechanisms.
To test this assertion, we have numerically studied thecharge noise associated with a model two-dimensional
QPC saddle point potential using a recursive Green function approach extended to include the calculation of local
properties [32]. We consider the potential22Uðx; yÞ ¼ We4x =w y2 ;(6)where W is the quantity modified by the coupling to the
DQD. The gating effect of the qubit is therefore modeled asa shift Uðx; yÞ þ Uðx; yÞ ¼ ½1 þ U=ða2 WÞUðx; yÞ,
where a is the lattice spacing. Note that our goal is not toadvocate a particular coupling model, but rather to study
the dependence of the charge noise on the QPC adiabaticity. In Eq. (6), the adiabaticity of both the potential and thecoupling is set by the barrier width w.
Figure 2 shows Q" as a function of transmission. In thelimit of an adiabatic potential and coupling (i.e., wF ),Q" is to a good approximation proportional to T ð1  T Þ,
and therefore to SII . In contrast, in the nonadiabatic casew  F , the T dependence of Q" is markedly different: it
is strongly skewed towards T > 1=2. Observation of thisasymmetry offers a way to distinguish the charge noise andcurrent noise backaction.
Role of Friedel oscillations.—The increased charge
noise backaction for T > 1=2 is due to interference contributions that are the nonequilibrium analogue of Friedel
oscillations [33]. In one dimension, the continuity equation

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PHYSICAL REVIEW LETTERS

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backaction, the charge noise backaction need not be maximal at QPC transmission T ¼ 1=2.
This work was supported by NSERC, FQRNT, and
CIFAR.

FIG. 2 (color online). Numerical results for the zero-frequency
qubit excitation rate Q" as a function of QPC transmission T .
We use a lattice spacing a ¼ 6:2 nm, giving a hopping energy
t ¼ @2 =ð2m a2 Þ ¼ 14:5 meV, where m ¼ 0:068me in GaAs.
For EF ¼ 16 meV, the Fermi wavelength is F ¼ 6a. The
dashed (red) line is for an adiabatic barrier with w ¼ 10F ,while the solid (blue) line is for a nonadiabatic barrier with w ¼
F ; each curve is scaled by its maximum value. In the nonadiabatic case, Q" is asymmetric and has its maximum at T >1=2. Inset: the corresponding transmission curves as a function
of W.



1 Z
d
d ^
^ tÞ
dx
UðxÞ Iðx;
(7)
Q¼
dx
dt
U
relates the charge operator Q^ to a weighted average of thecurrent flowing into the scattering region of the QPC. Thisis distinct from and independent of the transport current(flowing through the QPC) used to define the QPC conductance and shot noise SII . The current operator I^ of acoherent conductor contains spatially varying terms that
oscillate with a wavelength F =2. These terms give rise toFriedel oscillations in the density-density correlation function, but do not contribute to, e.g., the average conductanceor low-frequency shot noise [34]. In contrast, we find thatthey can enhance the nonequilibrium charge noise. Their
contributions to SQQ ½! via Eq. (7) correspond to interferences between incoming and transmitted waves, and thus
scale as T . If Uð~rÞ varies slowly on the scale of F , thenthe spatial averaging described by Eq. (7) strongly suppresses these terms. In the opposite limit, no such averaging occurs; the terms thus enhance the charge noise at
higher values of T . Note that the same interferences yield
the surprising result that the total charge in a perfect (i.e.,
T ¼ 1), zero temperature one-dimensional wire of length
L fluctuates; at ! ¼ 0, the spectral density is SQQ ½0 ¼
ðe3 V@=4Þ½sinðkF LÞ=EF 2 , where V  EF =e is the bias
and EF is the Fermi energy.
Conclusion.—The fundamental Heisenberg backactionof QPC qubit detectors is due to nonequilibrium charge
fluctuations. We have shown that the corresponding inelastic transition rates in the qubit can be expressed exclusively
in terms of quantities that are directly measurable in experiment. We have also shown that, unlike the current noise[1] M. H. Devoret and R. J. Schoelkopf, Nature (London) 406,
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