Bidirectional Single-Electron Counting and the Fluctuation Theorem

Abstract

We investigate theoretically and experimentally the full counting statistics of bidirectional single-electron tunneling through a double quantum dot in a GaAs/GaAlAs heterostructure and compare with predictions of the fluctuation theorem (FT) for Markovian stochastic processes. We observe that the quantum point contact electrometer used to study the transport induces nonequilibrium shot noise and dot-level fluctuations and strongly modifies the tunneling statistics. As a result, the FT appears to be violated. We show that it is satisfied if the back-action of the electrometer is taken into account, and we provide a quantitative estimate of this effect. Comment: 4pages, 2figures

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arXiv:0908.0229v1 [cond-mat.mes-hall] 3 Aug 2009
Bidirectional Single-Electron Counting and the Fluctuation Theorem
Y. Utsumi1, D. S. Golubev2, M. Marthaler3, K. Saito4, T. Fujisawa5,6, and Gerd Sch¨ on2,3
1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
2Forschungszentrum Karlsruhe, Institut f¨ ur Nanotechnologie, 76021 Karlsruhe, Germany
3Institut f¨ ur Theoretische Festk¨ orperphysik and DFG Center for Functional Nanostructures (CFN),
Universit¨ at Karlsruhe, 76128 Karlsruhe, Germany
4Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan
5NTT Basic Research Laboratories, NTT Corporation, Morinosato-Wakamiya, Atsugi 243-0198, Japan
6Research Center for Low Temperature Physics, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan
(Dated: August 3, 2009)
We investigate theoretically and experimentally the full counting statistics of bidirectional single-
electron tunneling through a double quantum dot in a GaAs/GaAlAs heterostructure and compare
with predictions of the fluctuation theorem (FT) for Markovian stochastic processes. We observe
that the quantum point contact electrometer used to study the transport induces nonequilibrium
shot noise and dot-level fluctuations and strongly modifies the tunneling statistics. As a result, the
FT appears to be violated. We show that it is satisfied if the back-action of the electrometer is
taken into account, and we provide a quantitative estimate of this effect.
PACS numbers: 73.23.-b,73.23.Hk,72.70.+m,05.70.Ln
According to the second law of thermodynamics, the
entropy of a macroscopic system driven out of equilib-
rium increases with time until equilibrium is reached.
Thus the dynamics of such a system is irreversible. In
contrast, for a mesoscopic system performing a random
trajectory in phase space and measured during a suffi-
ciently short time, the entropy may either increase or
decrease. The ‘Fluctuation Theorem’ (FT), which re-
lies only on the microreversiblity of the underlying equa-
tion of motion, states that the probability distribution
Pτ(∆S) for processes increasing or decreasing the en-
tropy by ±∆S during a time interval τ obeys the relation
Pτ(∆S)/Pτ(−∆S) = exp(∆S). (1)
Remarkably, the FT remains valid even far from equilib-
rium. It has been proven for thermostated Hamiltonian
systems [1], Markovian stochastic processes [2, 3], and
mesoscopic conductors [4, 5, 6, 7, 8]. The FT is funda-
mentally important for transport theory, one of its con-
sequences being the Jarzynski equality [9, 10], which, in
turn, leads to the 2nd law of thermodynamics. It also
leads to the fluctuation-dissipation theorem and Onsager
symmetry relations [11], as well as to their extensions to
nonlinear transport [4, 5, 6, 7, 8].
In electron transport experiments the entropy produc-
tion is related to Joule heating, ∆S = qeVS/T, where
q is the number of electrons (with charge e) transfered
through the conductor during time τ, and T is the tem-
perature. Hence the FT can be formulated in terms of
the distribution of transfered charge Pτ(q) at sufficiently
long times, τ ? e/I, as follows
Pτ(q)/Pτ(−q) = exp(qeVS/T).(2)
The FT has been experimentally verified in chemical
physics at room temperature [12], while tests of the FT in
mesoscopic transport experiments in milli-Kelvin regime
have been lacking so far. On the other hand, recent ad-
vances in time-resolved charge detection by a quantum
point contact (QPC) [13, 14, 15, 16] made it possible to
measure the distribution Pτ(q) for single-electron tunnel-
ing through quantum dots. This opens the possibility of
testing the FT in mesoscopic transport.
In this article we report on experiments on the
direction-resolved full counting statistics (FCS) of single-
electron tunneling in a double quantum dot (DQD) sys-
tem, which is probed by an asymmetrically coupled
QPC [14]. We analyze the experimental data in the frame
of the FT and find that the form (2) appears to be vi-
olated. However, it is still satisfied, if we replace the
temperature by an enhanced value T∗. We attribute the
apparent overheating to the back action of the QPC de-
tector and provide quantitative estimates of the effect.
Moreover, we note that the form (1) of the FT, valid
for Markovian stochastic processes defined by general
rates [2, 3], is satisfied, and we relate the entropy change
to a ratio of the relevant nonequilibrium tunneling rates.
Experimental test of the FT – The setup of our exper-
iment [14] is shown in Fig. 1. It consists of the DQD
coupled to the QPC detector (Fig. 1a). The left and
right gate voltages VGL,VGRapplied to the quantum dots
are tuned in such a way that only three charge states of
the DQD are allowed: |L?, |R?, and |D?, denoting states
where the left or right dot is occupied with a single elec-
tron, or where both dots are occupied, respectively. Ac-
cordingly, the current through the QPC, which is coupled
asymmetrically to the DQD, switches between three dif-
ferent values (Fig. 1b). This setup allows distinguishing
electron tunneling in different directions. For example,
the switching |L? → |R? corresponds to the transfer of
one electron from the left dot to the right one, while
|R? → |L? signals a transfer in opposite direction. The

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FIG. 1:
level in each dot, with the energies εL and εR respectively,
contributes to the transport. (b) The QPC current switches
between three values corresponding to the three charge states.
(c) Test of FT (2) at several times. Lines with symbols: loga-
rithm of lhs of Eq. (2) at several times; dashed line: q eVS/T∗
with T∗= 1.37 K; dot-dashed line: q eVS/T. Inset: the dis-
tribution Pτ(q) at τ = 4 ms.
(a) Schematics of the system. Only one energy
time trace of the current taken over a sufficiently long
time τ allows one to determine the distribution of trans-
fered charges Pτ(q). An example of such a distribution
is shown in the inset of Fig. 1c.
First, we perform a direct test of the FT, see Fig. 1c.
The combination ln[Pτ(q)/Pτ(−q)] indeed is a linear
function of the transfered charge q. However, the slope
approaches at long time τ the value eVS/T∗where VS=
300 µV is the value of the applied DQD bias voltage, but
the effective temperature T∗= 1.37 K is a fit parameter
(dashed line), which strongly exceeds the bath tempera-
ture of the leads of T = 130 mK (dot-dashed line).
To further test the time dependence contained in Eq.
(2), we check the integrated form of the FT (Fig. 2a),
?0
q=−∞Pτ(q)
?∞
q=0Pτ(q)
=
?∞
q=0Pτ(q)e−qeVS/T
?∞
q=0Pτ(q)
. (3)
Plotting both sides of Eq. (3) with the value of the elec-
tron temperature, T = 130 mK, we observe a clear dis-
FIG. 2:
terval 0 < τ < 8 ms. The time τ is multiplied by the fre-
quency of electron tunneling I/e ≈ 370 Hz. Squares: lhs of
Eq.(3) (denoted as Pτ(q ≤ 0)/Pτ(q ≥ 0)); dashed line:
rhs of Eq. (3) (denoted as ?exp(−qeVS/T)?q≥0); solid line:
rhs of Eq. (3) with T replaced by T∗= 1.37 K. All three
curves are multiplied by exp(Iτ/e) for clarity. (b) Normal-
ized second, C2 = ?(q − ?q?)2?/?q?, and third cumulants,
C3 = ?(q − ?q?)3?/?q?, of the charge distribution Pτ(q). (c)
The six transitions with Γij between three charge states.
(a) Test of the integrated FT (3) in the time in-
crepancy between both (open squires and dashed line).
However, by adjusting the temperature in the right hand
side of Eq. (3) to the effective temperature (solid line),
we get a good fit for τ ? e/I. As we will discuss below,
the apparent heating is caused by the back-action of the
QPC electrometer, which operates in highly nonequilib-
rium conditions.
The experimental distribution of the transfered charge
(inset of Fig. 1c) deviates strongly from a Gaussian
shape. For illustration the second and the third cumu-
lants of the charge distribution are plotted in Fig. 2b.
The normalized third cumulant remains close to 0.1 at
all times. Since the FT (2) is satisfied for any Gaussian
distribution Pτ(q), the existence of higher cumulants is a
further indication of non-trivial behavior.
Violation and recovery of the FT– Why does the FT
appear to be violated? Quite generally, the FT for the
system with four leads, depicted in Fig. 1a, should be

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formulated in terms of the joint probability distribution
Pτ(q,q′) of two charges eq and eq′transmitted through
the DQD and the QPC, respectively [7],
Pτ(q,q′) = exp??qeVS+ q′eVQPC
where T is the temperature of the leads.
the charge eq is measured, Eq. (4) should be integrated
over eq′and afterwards, the right hand side reduces to
exp[eVS/T]Pτ(−q) only if VQPC=0, which leads to the
apparent violation of the FT.
Then why is the FT recovered by introducing the ef-
fective temperature T∗? In the experiment, the DQD is
in the sequential tunneling regime. Then the probability
distribution Pτ(q) can be derived from a master equa-
tion approach of the FCS [3, 17, 18]. It begins with the
‘modified’ master equation
?/T?Pτ(−q,−q′), (4)
Since only
∂tp(t) = Γ(λ)p(t),
pT= (pL,pR,pD), (5)
where ps is the occupation probabilities of the charge
states (s = L,R,D). The transition matrix has a mod-
ified form Γ(λ) depending on the counting field λ mea-
suring the electron transfer through the barrier between
the two quantum dots
Γ(λ)=


−ΓRL− ΓDL
ΓRLeiλ
ΓDL
ΓLRe−iλ
−ΓLR− ΓDR
ΓDR
ΓLD
ΓRD
−ΓLD− ΓRD

. (6)
The characteristic function (CF) takes the form [17]
Zτ(λ) =?
qPτ(q)eiqλ= eTeτΓ(λ)pst, (7)
where eT= (1,1,1) and pstis the stationary state, which
is found from the equation Γ(0)pst= 0. In the long-
time limit, τ ≫e/I, the CF acquires an exponential form
Zτ(λ) ≈ eτF(λ), where F(λ) is the eigenvalue of the ma-
trix Γ(λ) with the largest real part. It satisfies
0 = det[Γ(λ) − F I] = F3+ KF2+ K′F
−ΓDRΓRLΓLD(eiλ− 1) − ΓDLΓLRΓRD(e−iλ− 1), (8)
where I is the identity matrix. Since the parameters K
and K′are independent of λ, we observe without solving
Eq. (8), that the CF satisfies the identity,
Z(λ) = Z(−λ + ieVS/T∗),
T∗= eVS/ln[(ΓDRΓRLΓLD)/(ΓDLΓLRΓRD)].(10)
(9)
Performing the inverse Fourier transform of Eq. (9), we
arrive at the FT in the form (2) with T replaced by T∗.
The argument of the logarithm in Eq. (10) is the ratio
of products of the tunneling rates corresponding to for-
ward and backward cycles (i.e., the counterclockwise and
clockwise cycles shown in Fig. 2c). The logarithm of this
ratio gives the entropy production associated with the
transfer of one electron through the system [2, 3]. While
in the present example we find this transparent result we
note that in more general systems with many transport
cycles, it is in general not possible to define a unique
effective temperature [3].
In the experiments the tunneling rates are estimated as
ΓDR=4kHz, ΓRD=0.3kHz, ΓDL=1kHz, ΓLD=1.5kHz,
ΓLR= 1.7kHz, and ΓRL= 1.8kHz.
a regime where they do not suffer from the finite band
width of the QPC detector (∼ 10 kHz). The effective
temperature derived from Eq. (10), T∗=1.14 K, agrees
well with the value 1.37 K obtained directly from the FT.
QPC back-action – The tunnel rates are influenced by
environmental effects. In particular, the DQD is neces-
sarily coupled to the QPC, as modeled by capacitors CL
and CRin Fig. 1a. The nonequilibrium current noise of
the QPC [19] produces fluctuations of the potentials of
the quantum dots eδVL,R, which in turn influence the
tunnel rates [15, 20] as known from the so-called P(E)-
theory [21]. Introducing three such functions Pj(E) with
j = L,R,dd we find
The values lie in
ΓRL = 2π|tdd|2Pdd(EL− ER),
?
ΓDR = ΓL
dE f(ED− ER− µL+ E)PL(E). (13)
(11)
ΓDL = ΓR
dE f(ED− EL− µR− E)PR(E), (12)
?
Here tddis the matrix element describing tunneling be-
tween the two dots, the rates ΓL and ΓR characterize
the coupling between the dots and the leads. f(E) is
the Fermi function, where we fix the chemical potentials
of the leads as µL= −µR= eVS/2. The energies of the
charge states EL,R,D include the electrostatic energy of
the electric field stored in the capacitors. The rates ΓLR,
ΓLD and ΓRD are given by the same expressions where
the Fermi function f should be replaced by 1−f and
the argument of the functions Pj should be taken with
opposite sign.
The spectral functions PL/R/ddare expressed in terms
of the phase operators ˆ ϕL/R(t) =
ˆ ϕdd= ˆ ϕR− ˆ ϕLas follows
?tdt′δˆVL/R(t′) and
Pj(E) =
?
dt
2πeiEt?
dω
2π
eiˆ ϕj(t)e−iˆ ϕj(0)?
e2[S0,j(ω) + SQPC
≈
?
dt
2πeiEt
×exp
??
V,j(ω)](e−iωt− 1)
ω2
?
. (14)
Here S0,j(ω) is the spectral density of thermal fluc-
tuations of equilibrium environments, including the
impedance of the external circuit Zext (Fig. 1a),
etc. The part SQPC
V,j
within the Gaussian approxima-
tion is proportional to the non-equilibrium and non-
symmetrized current noise of the QPC as SQPC
V,j(ω) =

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κ2
j|Zt(ω)|2SQPC
I
(ω) [20], where
SQPC
I
=e2
π
??
±
T (1 − T )(ω ± eVQPC)
1 − e−(ω±eVQPC)/T
+
2T2ω
1 − e−ω/T
?
.
Here T is the QPC transparency and Ztis the impedance
of the electromagnetic environment seen by the QPC;
Zt(ω) = 1/(−iω¯C + 1/¯R) where the resistance and ca-
pacitance are¯R=1/(R−1
CLCR/(CL+CR). The factors κj characterize the cou-
pling between QPC and quantum dots given by certain
ratios of the capacitances.
When the QPC is in equilibrium VQPC=0, the func-
tions Pj obey the detailed balance Pj(E)/Pj(−E) =
eE/Tand thus ΓRD/ΓDR= e(ED−ER−µL)/T, etc. From
Eqs. (10-13) it follows that in this case T∗=T. However,
in the experiment, the QPC is biased at a rather high
voltage eVQPC≫T and the tunnel rates through the cen-
tral barrier are approximately ΓRL/LR≈Γmax/[1+(ER−
EL)2/Γ2
and Γmax= 2|tdd|2/Γ0. From the experimental values
Γmax≈ 7 kHz, Γ0≈ 30µeV, VQPC=800 µV and T =0.19,
the parameters can be roughly estimated as |tdd| ≈ 30
MHz, κdd¯R ≈ 5 kΩ. The latter value is in agreement
with other experiments, see i.e. [15]. Since the difference
between the tunnel rates in opposite directions is signifi-
cantly reduced, ΓLR≈ΓRL, we observe that the effective
temperature T∗is enhanced. From Eq. (10) we roughly
estimate T∗∼ eVS/ln[πeVS/Γ0] ≈ 1 K, in good agree-
ment with our previous findings, thus further supporting
the QPC back-action model. In order to reduce T∗one
can reduce VQPCor the external impedance seen by the
QPC¯R.
Other environmental effects do not change our find-
ings by much. In GaAs nanostructures acoustic phonons
modify the tunneling properties via piezoelectric and de-
formation coupling. At experimental values of the QPC
current IQPC≈ 12nA the phonons stay almost in equi-
librium [16]. Therefore, the phonon effect is absorbed in
the equilibrium part of the spectral density S0,j, and does
not affect the FT. An intrinsic back-action of the QPC
is often addressed in the context of quantum measure-
ment [22]. However, in the present case we estimate this
effect to be negligible. Detailed discussions of these envi-
ronmental effects and derivations of the theory presented
in this article, based on the real-time diagrammatic tech-
nique [19, 23], will be published elsewhere [24].
Summary – We have investigated experimentally and
theoretically the fluctuation theorem for Markovian
stochastic processes by studying the direction-resolved
full counting statistics in a double quantum dot system
via a nearby quantum point contact electrometer. We
found an apparent violation of the fluctuation theorem,
which we attribute to the non-equilibrium electromag-
netic fluctuations generated by the shot noise of the quan-
tum point-contact electrometer. We also demonstrated
QPC+Z−1
ext) and¯C=(3C0+CG)/2+
0], where Γ0 = T (1 − T )(e2κdd¯R)2eVQPC/2π
that the FT is recovered if we adopt an effective value
for the temperature ten times higher than the electron
temperature. This effective temperature depends on the
entropy production associated with the transfer of one
electron through the double quantum dot, which in turn
can be expressed by a ratio of forward to backward tun-
neling rates.
We would like to thank M. Hettler for helpful discus-
sions. This work has been supported by Strategic In-
ternational Cooperative Program the Japan Science and
Technology Agency (JST) and by the German Science
Foundation (DFG).
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