Nonequilibrium fluctuations, fluctuation theorems, and counting
statistics in quantum systems
Massimiliano Esposito
Department of Chemistry, University of California, San Diego, La Jolla, California
920930340, USA
and Center for Nonlinear Phenomena and Complex Systems, Université Libre
de Bruxelles, C.P. 231, Campus Plaine, Brussels, Belgium
Upendra Harbola and Shaul Mukamel
Department of Chemistry, University of California, Irvine, California 92697-2025, USA
共Published 2 December 2009兲
Fluctuation theorems 共FTs兲, which describe some universal properties of nonequilibrium fluctuations,
are examined from a quantum perspective and derived by introducing a two-point measurement on
the system. FTs for closed and open systems driven out of equilibrium by an external time-dependent
force, and for open systems maintained in a nonequilibrium steady state by nonequilibrium boundary
conditions, are derived from a unified approach. Applications to fermion and boson transport in
quantum junctions are discussed. Quantum master equations and Green’s functions techniques for
computing the energy and particle statistics are presented.
DOI: 10.1103/RevModPhys.81.1665 PACS number共s兲: 05.30.⫺d, 05.60.Gg, 73.40.⫺c
CONTENTS
I. Introduction 1665
II. Two-Point Measurement Statistics 1668
A. The forward probability 1668
B. The time-reversed probability 1670
III. The Fluctuation Theorem 1670
A. General derivation and connection to entropy 1670
B. Transient fluctuation theorems 1672
1. Work fluctuation theorem for isolated
driven systems 1672
2. Work fluctuation theorem for closed driven
systems 1673
3. Fluctuation theorem for direct heat and
matter exchange between two systems 1673
C. Steady-state fluctuation theorems 1674
IV. Heat and Matter Transfer Statistics in Weakly
Coupled Open Systems 1675
A. Generalized quantum master equation 1675
1. Generalized reservoir correlation functions 1675
2. The Markovian and the rotating wave
approximation 1676
B. Applications to particle counting statistics 1677
1. Fermion transport 1677
2. Boson transport 1678
3. Modulated tunneling 1679
4. Direct-tunneling limit 1680
V. Many-Body Approach to Particle Counting Statistics 1680
A. Liouville space formulation of particle counting
statistics 1681
B. Electron counting statistics for direct tunneling
between two systems 1681
1. Effects of initial correlations 1682
2. The thermodynamic limit 1683
C. Electron counting statistics for transport through a
quantum junction 1683
1. Long-time statistics 1684
2. Recovering the generalized quantum master
equation 1685
3. The Levitov-Lesovik formula 1686
VI. Nonlinear Coefficients 1686
A. Single nonequilibrium constraint 1687
B. Multiple nonequilibrium constraints 1687
VII. Conclusions and Perspectives 1688
Acknowledgments 1689
Appendix A: Time-Reversed Evolution 1689
Appendix B: Fluctuation Theorem for Coarse-Grained
Dynamics 1690
Appendix C: Large Deviation and Fluctuation Theorem 1691
Appendix D: Derivation of the Generalized Quantum Master
Equation 1692
Appendix E: Bidirectional Poisson Statistics 1693
Appendix F: Liouville Space and Superoperator Algebra 1693
Appendix G: Probability Distribution for Electron Transfers 1694
Appendix H: Path-Integral Evaluation of the Generating
Function for Fermion Transport 1695
Appendix I: Grassmann Algebra 1697
References 1698
I. INTRODUCTION
Small fluctuations of systems at equilibrium or weakly
driven near equilibrium satisfy a universal relation
known as the fluctuation-dissipation 共FD兲 theorem
共Callen and Welton, 1951; Kubo, 1957; de Groot and
Mazur, 1984; Stratonovich, 1992; Kubo et al., 1998;
Zwanzig, 2001兲. This relation that connects spontaneous
fluctuations to the linear response holds for classical and
quantum systems alike. The search for similar relations
for systems driven far from equilibrium has been an ac-
REVIEWS OF MODERN PHYSICS, VOLUME 81, OCTOBER–DECEMBER 2009
0034-6861/2009/81共4兲/1665共38兲 ©2009 The American Physical Society1665
tive area of research for many decades. A major break-
through in this regard had taken place over the past 15
years with the discovery of exact fluctuation relations,
which hold for classical systems far from equilibrium.
These are collectively referred to as fluctuation theo-
rems 共FTs兲. In order to introduce these theorems we
adopt the following terminology. A system that follows a
Hamiltonian dynamics is called isolated. By default, we
assume that the Hamiltonian is time independent. Oth-
erwise, it means that some work is performed on the
system and we denote it driven isolated system. A sys-
tem that can only exchange energy with a reservoir will
be denoted closed. If particles are exchanged as well, we
say that the system is open.
The first class of FTs, and the earliest discovered, deal
with irreversible work fluctuations in isolated driven sys-
tems described by a Hamiltonian dynamics where the
Hamiltonian is time dependent 共Bochkov and Kuzovlev,
1977, 1979, 1981a, 1981b; Stratonovich, 1994; Jarzynski,
1997a, 1997b; Cohen and Mauzerall, 2004; Jarzynski,
2004; Cleuren et al., 2006; Horowitz and Jarzynski, 2007;
Jarzynski, 2007; Kawai et al., 2007; Gomez-Marin et al.,
2008兲. An example is the Crooks relation which states
that the nonequilibrium probability p共W兲, that a certain
work w=W is performed by an external time-dependent
driving force acting on a system initially at equilibrium
with temperature

−1
, divided by the probability p
˜
共−W兲,
that a work w=−W is performed by the time-reversed
external driving force acting on the system which is
again initially at equilibrium, satisfies p共W兲/p
˜
共−W兲
=exp关

共W − ⌬F兲兴, where ⌬F is the free-energy difference
between the initial 共no driving force兲 and final 共finite
driving force兲 equilibrium state. The Jarzynski relation
具exp关−

W兴典=exp关−

⌬F兴 follows immediately from
兰dWp
˜
共−W兲= 1. A second class of FTs is concerned with
entropy fluctuations in closed systems described by de-
terministic thermostatted equations of motions 共Evans et
al., 1993; Evans and Searles, 1994, 1995, 1996; Gallavotti
and Cohen, 1995a, 1995b; Cohen and Gallavotti, 1999;
Schöll-Paschinger and Dellago, 2006兲 and a third class
treats the fluctuations of entropy 共or related quantities
such as irreversible work, heat, and matter currents兲 in
closed or open systems described by a stochastic dynam-
ics 共Ross et al., 1988; Crooks, 1998, 1999, 2000; Kurchan,
1998; Lebowitz and Spohn, 1999; Searles and Evans,
1999; Hatano and Sasa, 2001; Seifert, 2005; Chernyak et
al., 2006; Andrieux and Gaspard, 2007b; Esposito et al.,
2007a; Taniguchi and Cohen, 2007; Chetrite and
Gawe¸dzki, 2008兲. As an example for the last two classes,
we give the steady-state FT for the entropy production.
We consider a trajectory quantity s whose ensemble av-
erage 具s典 can be associated with an entropy production
共the specific form of s depends on the underlying dynam-
ics兲.Ifp
共S兲 denotes the probability that s=S when the
system is in a nonequilibrium steady state, then for long
times the FT reads p共S兲/ p共−S兲= exp关S兴. FTs valid at any
time such as the work FTs are called transient FTs while
those who require a long-time limit are called steady-
state FTs.
The FTs are all intimately connected to time-reversal
symmetry and the relations between probabilities of for-
ward and backward classical trajectories. Close to equi-
librium the FTs reduce to the known fluctuation-
dissipation relations such as the Green-Kubo relation
for transport coefficients 共Gallavotti, 1996a, 1996b; Leb-
owitz and Spohn, 1999; Andrieux and Gaspard, 2004,
2007a兲. These classical fluctuation relations have been
reviewed by Maes 共2003兲; Gaspard 共2006兲; Gallavotti
共2007, 2008兲; and Harris and Schutz 共2007兲. Some of
these relations were verified experimentally in mesos-
copic systems where fluctuations are sufficiently large to
be measurable. Work fluctuations have been studied in
macromolecule pulling experiments 共Liphardt et al.,
2002; Collin et al., 2005兲 and in optically driven micro-
spheres 共Trepagnier et al., 2004兲, entropy fluctuations
have also been measured in a similar system 共Wang et
al., 2005兲 and in spectroscopic experiments on a defect
center in diamond 共Schuler et al., 2005; Tietz et al., 2006兲
.
When decreasing system sizes, quantum effects may be-
come significant. Applying the standard trajectory-based
derivations of FTs to quantum regime is complicated by
the lack of a classical trajectory picture when coherences
are taken into account and by the essential role of mea-
surements, which can be safely ignored in ideal classical
systems. We show that the FTs follow from fundamental
dynamical symmetries that apply equally to classical and
quantum systems.
Earlier derivations of the Jarzynski relation for quan-
tum systems defined a work operator 共Bochkov and Ku-
zovlev, 1977, 1979, 1981a, 1981b; Stratonovich, 1994;
Yukawa, 2000; Monnai and Tasaki, 2003; Chernyak and
Mukamel, 2004; Allahverdyan and Nieuwenhuizen,
2005; Engel and Nolte, 2006; Gelin and Kosov, 2008兲.
Since work is not in general an ordinary quantum “ob-
servable” 共the final Hamiltonian does not commute with
the initial Hamiltonian兲共Talkner et al., 2007兲, attempts
to define such an operator had led to quantum correc-
tions to the classical Jarzynski result. However, the
Jarzynski relation in a closed driven quantum system
may be derived without quantum corrections by intro-
ducing an initial and a final projective measurment of
the system energy in accordance with the quantum me-
chanical measurement postulate. This has been done,
not always in a explicit way by Kurchan 共2000兲, Tasaki
共2000兲, Mukamel 共2003b兲, Monnai 共2005兲, Talkner and
Hänggi 共2007兲, Talkner et al. 共2007兲, and Talkner,
Hänggi, and Morillo 共2008兲. The work is then a two-
point quantity obtained by calculating the difference be-
tween the initial and final energy of the system. When
the reservoir is explicitly taken into account, the Jarzyn-
ski relation has often been derived using a master equa-
tion approach 共De Roeck and Mass, 2004; Esposito and
Mukamel, 2006; Crooks, 2008a, 2008b兲. Alternative deri-
vations can be found in Monnai 共2005兲 and Talkner et al.
共
2009兲.
The derivation of a steady-state FT for quantum sys-
tems has been considered as well 共Jarzynski and Wojcik,
2004; Tobiska and Nazarov, 2005; Andrieux and Gas-
pard, 2006; Esposito and Mukamel, 2006; De Roeck and
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Maes, 2006; Cleuren and den Broeck, 2007; Esposito et
al., 2007b; Harbola et al., 2007; De Roeck, 2007; Saito
and Dhar, 2007; Derezinski et al., 2008; Saito and Ut-
sumi, 2008; Andrieux et al., 2009兲. Because of the need
to describe nonequilibrium fluctuations in closed or
open quantum systems exchanging energy or matter
with their reservoir, many similarities exist with the rap-
idly developing field of electron counting statistics 共Levi-
tov and Lesovik, 1993; Levitov et al., 1996; Gurvitz,
1997; Nazarov, 1999; Belzig and Nazarov, 2001a, 2001b;
Bagrets and Nazarov, 2003; Belzig, 2003; Kindermann
and Nazarov, 2003; Nazarov and Kindermann, 2003; Pil-
gram et al., 2003, 2006; Shelankov and Rammer, 2003;
Kindermann and Pilgram, 2004; Levitov and Reznikov,
2004; Rammer et al., 2004; Flindt et al., 2005, 2008; Wab-
nig et al., 2005; Braggio et al., 2006; Kiesslich et al., 2006;
Utsumi et al., 2006; Emary et al., 2007
; Nazarov, 2007;
Schönhammer, 2007; Bednorz and Belzig, 2008; Snyman
and Nazarov, 2008; Welack et al., 2008兲, where small
nanoscale electronic devices exchange electrons. Fluc-
tuations in such systems can nowadays be experimen-
tally resolved at the single electron level 共Lu et al., 2003;
Fujisawa et al., 2004, 2006; Bylander et al., 2005;
Gustavsson et al., 2006兲. Similarities also exist with the
more established field of photon counting statistics,
where photons emitted by a molecule or an atom driven
out of equilibrium by a laser are individually detected
共Glauber, 1963; Kelley and Kleiner, 1964; Mandel, 1982;
Mandel and Wolf, 1995; Gardiner and Zoller, 2000;
Mukamel, 2003a; Zheng and Brown, 2003a, 2003b; Bar-
kai et al., 2004; Kulzer and Orrit, 2004; Sanda and Muka-
mel, 2005兲.
Different types of approaches have been used to de-
rive these FTs and describe these counting experiments.
The first is based on the quantum master equation
共QME兲共Gurvitz, 1997; Rammer et al., 2004; Flindt et al.
2005, 2008; Wabnig et al., 2005; Braggio et al., 2006; Es-
posito and Mukamel, 2006; Kiesslich et al., 2006; De Ro-
eck and Maes, 2006; Emary et al., 2007
; Esposito et al.,
2007b; Harbola et al., 2007; Welack et al., 2008兲. Here
one starts with an isolated total system containing the
system and the reservoir in weak interaction. By tracing
the reservoir degrees of freedom, taking the infinite res-
ervoir limit and using perturbation theory, one can de-
rive a closed evolution equation for the reduced density
matrix of the system. The information about the reser-
voir evolution is discarded. However, the evolution of a
quantum system described by a QME can be seen as
resulting from a continuous projective measurement on
the reservoir, leading to a continuous positive operator-
valued measurement on the system. Such interpretation
allows one to construct a trajectory picture of the system
dynamics, where each realization of the continuous mea-
surement leads to a given system trajectory 共Brun, 2000,
2002; Gardiner and Zoller, 2000; Nielsen and Chuang,
2000; Breuer and Petruccione, 2002兲. The QME is recov-
ered by ensemble averaging over all possible trajecto-
ries. This unraveling of the QME into trajectories has
been originally developed in the description of photon
counting statistics 共Wiseman and Milburn, 1993a, 1993b;
Plenio and Knight, 1998; Gardiner and Zoller, 2000;
Breuer and Petruccione, 2002兲. Another approach is
based on a modified propagator defined on a Keldysh
loop which, under certain circumstances, can be inter-
preted as the generating function of the electron count-
ing probability distribution 共Nazarov, 1999, 2007; Belzig
and Nazarov, 2001a, 2001b; Belzig, 2003; Kindermann
and Nazarov, 2003; Nazarov and Kindermann, 2003;
Kindermann and Pilgram, 2004兲. Using a path integral
formalism, the propagator of the density matrix of a
“detector” with Hamiltonian p
2
/2m interacting with a
system can be expressed in terms of the influence func-
tional that only depends on the system degrees of free-
dom 共Feynman and Vernon, 1963兲. The modified propa-
gator is the influence functional when the system is
linearly coupled to the detector 共with coupling term xA,
where x is the position of the detector and A is a system
observable兲 in the limit of very large detector inertia
m→ ⬁. It is only under some specific assumptions 共such
as a classical detector where the detector density matrix
is assumed diagonal兲 that the modified propagator be-
comes the generating function associated with the prob-
ability distribution that the detector momentum changes
from a given amount, which can be interpreted as the
probability to measure the time average of the system
observable A: 兰
0
t
d
A共
兲.IfA is an electric current, then
the integral gives the number of electrons transfered. An
early quantum FT for electronic junctions has been de-
rived in this context by Tobiska and Nazarov 共2005兲
based on the time-reversal invariance of the Hamil-
tonian quantum dynamics. Different derivations of
quantum FTs relying on this approach have been consid-
ered by Saito and Dhar 共2007兲 and Saito and Utsumi
共2008兲. A third semiclassical scattering approach is often
used in electron counting statistics 共Pilgram et al., 2003,
2004; Jordan et al., 2004; Nagaev et al., 2004; Pilgram,
2004兲. This can be recovered from the modified propa-
gator approach as recently shown by Snyman and Naz-
arov 共2008兲, but will not be addressed here.
We consider fluctuations in the output of a two-point
projective measurement 共of energy, particle, charge,
etc.兲. This allows us to avoid the detailed modeling of
detectors and their dynamics. The projective measure-
ment can be viewed as an effective modeling of the ef-
fect of the system-detector interaction on the system or
as resulting in a fundamental way from the quantum
measurement postulate. The three other approaches
共unraveling of the QME, modified propagator on
Keldysh loop, and the scattering approach兲 can be re-
covered in some limits of the two-point measurement
approach. This provides a unified framework from
which the different types of FTs previously derived for
quantum systems can be obtained.
In Sec. II, we give the general expression for the prob-
ability of the output of a two-point measurement at dif-
ferent times on a quantum system described by the
quantum Liouville equation. The calculation is repeated
for a system described by the time-reversed dynamics. In
Sec. III, we start by discussing the basic ingredients re-
quired for FTs to hold. We use these results to derive
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Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009
three transient FTs, the Jarzynski and Crooks relation in
isolated and closed driven systems and a FT for matter
and heat exchange between two systems in direct con-
tact. We also show that a steady-state FT can be derived
for matter and heat exchange between two reservoirs
through an embedded system. In Sec. IV, we consider a
small quantum system weakly interacting with multiple
reservoirs. We develop a projection superoperator for-
malism to derive equations of motion for the generating
function associated with the system reduced density ma-
trix conditional of the output of a two-point measure-
ment of the energy or number of particles in the reser-
voirs. We apply this generalized quantum master
equation 共GQME兲 formalism to calculate the statistics
of particles or heat transfer in different models of gen-
eral interest in nanosciences in order to verify the valid-
ity of the steady-state FT. In Sec. V, we present a non-
equilibrium Green’s function formalism in Liouville
space, which provides a powerful tool to calculate the
particle statistics of many- body quantum systems. In
Sec. VI, we show that the FTs can be used to derive
generalized fluctuation-dissipation relations. Conclu-
sions and perspectives are drawn in Sec. VII.
II. TWO-POINT MEASUREMENT STATISTICS
We consider an isolated, possibly driven, quantum sys-
tem described by a density matrix
ˆ
共t兲, which obeys the
von Neumann 共quantum Liouville兲 equation
d
dt
ˆ
共t兲 =−
i
ប
关H
ˆ
共t兲,
ˆ
共t兲兴. 共1兲
Its formal solution reads
ˆ
共t兲 = U
ˆ
共t,0兲
ˆ
0
U
ˆ
†
共t,0兲. 共2兲
The propagator
U
ˆ
共t,0兲 = exp
+
再
−
i
ប
冕
0
t
d
H
ˆ
共
兲
冎
⬅ 1+
兺
n=1
⬁
冉
−
i
ប
冊
n
冕
0
t
dt
1
冕
0
t
1
dt
2
¯
⫻
冕
0
t
n−1
dt
n
H
ˆ
共t
1
兲H
ˆ
共t
2
兲 ¯ H
ˆ
共t
n
兲共3兲
is unitary U
ˆ
†
共t ,0兲=U
ˆ
−1
共t ,0兲 and satisfies U
ˆ
†
共t ,0兲=U
ˆ
共0,t兲
and U
ˆ
共t , t
1
兲U
ˆ
共t
1
,0兲=U
ˆ
共t ,0兲. We use the subscript ⫹ 共⫺兲
to denote an antichronological 共chronological兲 time or-
dering from left to right. We call Eq. 共2兲 the forward
evolution to distinguish it from the the time-reversed
evolution that will be defined below.
A. The forward probability
We consider an observable A
ˆ
共t兲 in the Schrödinger
picture whose explicit time dependence solely comes
from an external driving. For nondriven systems A
ˆ
共t兲
=A
ˆ
. In the applications considered A
ˆ
共t兲 will be either an
energy operator H
ˆ
or a particle number operator N
ˆ
. The
eigenvalues 共eigenvectors兲 of A
ˆ
共t兲 are denoted by a
t
共兩a
t
典兲: A
ˆ
共t兲=兺
a
t
兩a
t
典a
t
具a
t
兩.
The basic quantity in the following will be the joint
probability to measure a
0
at time 0 and a
t
at time t,
P关a
t
,a
0
兴⬅Tr兵P
ˆ
a
t
U
ˆ
共t,0兲 P
ˆ
a
0
ˆ
0
P
ˆ
a
0
U
ˆ
†
共t,0兲 P
ˆ
a
t
其
= P
*
关a
t
,a
0
兴, 共4兲
where the projection operators are given by
P
ˆ
a
t
= 兩a
t
典具a
t
兩. 共5兲
Using the properties P
ˆ
a
t
=P
ˆ
a
t
2
and 兺
a
t
P
ˆ
a
t
=1
ˆ
, we can verify
the normalization 兺
a
t
a
0
P关a
t
,a
0
兴=1. Consider two com-
plete Hilbert space basis sets 兵兩i , a
0
典其 and 兵兩j ,a
t
典其, where i
共j兲 are used to differentiate between the states with same
a
0
共a
t
兲. The basis 兵兩i , a
0
典其 is chosen such that it diagonal-
izes
ˆ
0
共this is always possible since
ˆ
0
is Hermitian兲.We
can also write Eq. 共4兲 as
P关a
t
,a
0
兴 =
兺
i,j
P关j,a
t
;i,a
0
兴, 共6兲
where
P关j,a
t
;i,a
0
兴⬅兩具j,a
t
兩U
ˆ
共t,0兲兩i,a
0
典兩
2
具i,a
0
兩
ˆ
0
兩i,a
0
典. 共7兲
The probability distribution for the difference ⌬a=a
t
−a
0
between the output of the two measurements is
given by
p共⌬a兲 =
兺
a
t
a
0
␦
„⌬a − 共a
t
− a
0
兲…P关a
t
,a
0
兴, 共8兲
where
␦
共a兲 denotes the Dirac distribution. It is often
more convenient to calculate the generating function
共GF兲 associated with this probability
G共兲⬅
冕
−⬁
⬁
d⌬ae
i⌬a
p共⌬a兲 = G
*
共− 兲
=
兺
a
t
a
0
e
i共a
t
−a
0
兲
P关a
t
,a
0
兴. 共9兲
The nth moment 具⌬a
n
典 of p共⌬a兲 is obtained by taking
nth derivative of the GF with respect to evaluated at
=0:
具⌬a
n
典 = 共− i兲
n
冏
n
n
G共兲
冏
=0
. 共10兲
We further define the cumulant GF
Z共兲 =lnG共兲. 共11兲
The nth cumulant K
n
of p共⌬a兲 is obtained by taking nth
derivative of the cumulant GF with respect to evalu-
ated at =0:
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Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009
K
n
= 共− i兲
n
冏
n
n
Z共兲
冏
=0
. 共12兲
The first cumulant coincides with the first moment which
gives the average K
1
=具⌬a典. Higher order cumulants can
be expressed in term of the moments. The variance K
2
=具⌬a
2
典−具⌬a典
2
gives the fluctuations around the average,
and the skewness K
3
=Š共⌬a −具⌬a典兲
3
‹ gives the leading or-
der deviation of p共⌬a兲 from a Gaussian. When measur-
ing the statistics of quantities associated to nonequilib-
rium fluxes, in most cases 关but not always 共Esposito and
Lindenberg, 2008兲兴 the cumulants grow linearly with
time and it becomes convenient to define the long-time
limit of the cumulant GF
S共兲 = lim
t→⬁
1
t
Z共兲, 共13兲
which measures the deviations from the central limit
theorem 共Sornette, 2006兲.
We next turn to computing the GF. The initial density
matrix can be expressed as
ˆ
0
=
ˆ
¯
0
+
ˆ
ញ
0
, 共14兲
where
ˆ
¯
0
=
兺
a
0
P
ˆ
a
0
ˆ
0
P
ˆ
a
0
,
ˆ
ញ
0
=
兺
a
0
⫽a
0
⬘
P
ˆ
a
0
ˆ
0
P
ˆ
a
0
⬘
. 共15兲
ˆ
¯
0
commutes with A
ˆ
共0兲. Using the fact that f共A
ˆ
兲
=兺
a
P
ˆ
a
f共a兲, where f is an arbitrary function, and using
also
兺
a
0
e
−ia
0
P
ˆ
a
0
ˆ
0
P
ˆ
a
0
= e
−i共/2兲A
ˆ
共0兲
ˆ
¯
0
e
−i共/2兲A
ˆ
共0兲
, 共16兲
we find, by substituting Eq. 共4兲 into Eq. 共9兲, that
G共兲 =Tr
ˆ
共,t兲, 共17兲
where we have defined
ˆ
共,t兲⬅U
ˆ
/2
共t,0兲
ˆ
¯
0
U
ˆ
−/2
†
共t,0兲共18兲
and the modified evolution operator
U
ˆ
共t,0兲⬅e
iA
ˆ
共t兲
U
ˆ
共t,0兲 e
−iA
ˆ
共0兲
. 共19兲
For =0,
ˆ
共 , t兲 reduces to the system density matrix
and U
ˆ
共t ,0兲 to the standard evolution operator. Defining
the modified Hamiltonian
H
ˆ
共t兲⬅e
iA
ˆ
共t兲
H
ˆ
共t兲e
−iA
ˆ
共t兲
− ប
t
A
ˆ
共t兲, 共20兲
we find that U
ˆ
共t ,0兲 satisfies the equation of motion
d
dt
U
ˆ
共t,0兲 =−
i
ប
H
ˆ
共t兲U
ˆ
共t,0兲. 共21兲
Since U
ˆ
共0,0兲=1
ˆ
,weget
U
ˆ
/2
共t,0兲 = exp
+
再
−
i
ប
冕
0
t
d
H
ˆ
/2
共
兲
冎
, 共22兲
U
ˆ
−/2
†
共t,0兲 = exp
−
再
i
ប
冕
0
t
d
H
ˆ
−/2
共
兲
冎
. 共23兲
Equations 共17兲 and 共18兲 together with Eqs. 共22兲 and 共23兲
provide an exact formal expression for the statistics of
changes in A
ˆ
共t兲 derived from the two-point measure-
ments.
We note that if and only if the eigenvalues of A
ˆ
are
integers 共as in electron counting where one considers the
number operator兲, using the integral representation of
the Kronecker delta
␦
K
共a − a
⬘
兲 =
冕
0
2
d⌳
2
e
−i⌳共a−a
⬘
兲
, 共24兲
Eq. 共18兲 can be written as
ˆ
共,t兲 =
冕
0
2
d⌳
2
ˆ
共,⌳,t兲, 共25兲
where
ˆ
共,⌳,t兲⬅U
ˆ
⌳+共/2兲
共t,0兲
ˆ
0
U
ˆ
⌳−共/2兲
†
共t,0兲. 共26兲
We see that by introducing an additional ⌳ dependence,
we were able to keep the initial density matrix
ˆ
0
in Eq.
共26兲 instead of
ˆ
0
as in Eq. 共18兲.
The current operator associated with A
ˆ
共t兲 is given by
I
ˆ
共t兲⬅
i
ប
关H
ˆ
共t兲,A
ˆ
共t兲兴 +
t
A
ˆ
共t兲. 共27兲
As a result,
I
ˆ
共h兲
共t兲 =
d
dt
A
ˆ
共h兲
共t兲, 共28兲
where the superscript 共h兲 denotes the Heisenberg repre-
sentation A
ˆ
共h兲
共t兲⬅U
ˆ
†
共t ,0兲A
ˆ
共t兲U
ˆ
共t ,0兲. We can write Eq.
共20兲 as
H
ˆ
共t兲 = H
ˆ
共t兲 − បI
ˆ
共t兲 + O共
2
ប
2
兲. 共29兲
In the semiclassical approximation where terms
O共
2
ប
2
兲 are disregarded, the GF 共17兲关with Eqs. 共18兲,
共22兲, and 共23兲兴, after going to the interaction representa-
tion, becomes
G共兲 =Tr
再
exp
−
冋
i
2
冕
0
t
d
I
ˆ
共h兲
共
兲
册
ˆ
¯
0
⫻exp
−
冋
i
2
冕
0
t
d
I
ˆ
共h兲
共
兲
册
冎
. 共30兲
This form is commonly found in the modified propaga-
tor approach, described in the introduction, to counting
statistics 共Kindermann and Nazarov, 2003; Nazarov and
Kindermann, 2003; Kindermann and Pilgram, 2004兲.
Note that in these publications the full initial density
matrix
ˆ
0
is used in Eq. 共30兲 instead of
ˆ
0
.
1669
Esposito, Harbola, and Mukamel: Nonequilibrium fluctuations, fluctuation …
Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009