Dephasing in quantum chaotic transport: a semiclassical approach
Robert S. Whitney,1 Philippe Jacquod,2 and Cyril Petitjean3, 4
1
Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble, France
Physics Department, University of Arizona, 1118 E. 4th Street, Tucson, AZ 85721, USA
3
Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland
4
Institut I – Theoretische Physik, Universität Regensburg,
Universitätsstrasse 31, D-93040 Regensburg, Germany
(Dated: October 26, 2007)
arXiv:0710.5137v2 [cond-mat.mes-hall] 17 Feb 2008
2
We investigate the effect of dephasing/decoherence on quantum transport through open chaotic
ballistic conductors in the semiclassical limit of small Fermi wavelength to system size ratio,
λF /L ≪ 1. We use the trajectory-based semiclassical theory to study a two-terminal chaotic dot
with decoherence originating from: (i) an external closed quantum chaotic environment, (ii) a classical source of noise, (iii) a voltage probe, i.e. an additional current-conserving terminal. We focus
on the pure dephasing regime, where the coupling to the external source of dephasing is so weak
that it does not induce energy relaxation. In addition to the universal algebraic suppression of weak
localization, we find an exponential suppression of weak-localization ∝ exp[−τ̃ /τφ ], with the dephasing rate τφ−1 . The parameter τ̃ depends strongly on the source of dephasing. For a voltage probe,
τ̃ is of order the Ehrenfest time ∝ ln[L/λF ]. In contrast, for a chaotic environment or a classical
source of noise, it has the correlation length ξ of the coupling/noise potential replacing the Fermi
wavelength λF . We explicitly show that the Fano factor for shot noise is unaffected by decoherence.
We connect these results to earlier works on dephasing due to electron-electron interactions, and
numerically confirm our findings.
PACS numbers: 05.45.Mt,74.40.+k,73.23.-b,03.65.Yz
I.
A.
INTRODUCTION
Dephasing in the universal regime
Electronic systems in the mesoscopic regime are
ideal testing-grounds for investigating the quantum-toclassical transition from a microscopic coherent world
(where quantum interference effects prevail) to a macroscopic classical world1 . On one hand, their size is intermediate between macroscopic and microscopic (atomic)
systems, on the other hand, today’s experimental control
over their design and precision of measurement allows one
to investigate them in regimes ranging from almost fully
coherent to purely classical2,3,4 . The extent to which
quantum coherence is preserved in these systems is usually determined by the ratio τφ /τcl of the dephasing time
τφ to some relevant classical time scale τcl . For instance,
τcl can be the traversal time through one arm of a twopath interferometer5,6,7 , or the average dwell time spent
inside a quantum dot8,9,10,11,12 . In a given experimental
set-up, τφ can often be tuned from τφ > τcl (quantum
coherent regime) to τφ ≪ τcl (purely classical regime) by
varying externally applied voltages or the temperature of
the sample.
Coherent effects abound in mesoscopic physics, the
most important of them being weak-localization, universal conductance fluctuations and Aharonov-Bohm
interferences in transport, as well as persistent currents2,3,4 . The disappearance of these effects as dephasing processes are turned on has raised lots of theoretical8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 and experimental26,27,28,29,30,31,32,33 interest. Focusing on transport
through ballistic systems, dephasing is usually investigated using mostly phenomenological models of dephasing16,17,18,19,20,21 , the most successful of which are the
voltage-probe and dephasing-lead models16,17 . In these
models, a cavity is connected to two external, L (left) and
R (right) transport leads, carrying NR and NL transport
channels respectively. Dephasing is modeled by connecting a third “fictitious” lead to the system, with a voltage set such that no current flows through it on average. Electrons leaving the system through this third lead
are thus re-injected at some later time, with a randomized phase (and randomized momentum). These models
of dephasing present the significant advantage that the
standard scattering approach to transport can be applied
as in fully coherent systems, once it is properly extended
to account for the presence of the third lead.
Using random matrix theory (RMT), the voltage/dephasing probe models34 predict an algebraic suppression of the weak-localization contribution to the conductance (in units of 2e2 /h)35 ,
wl
gRMT
= −
NR NL
[1 + τD /τφ ]−1 ,
[NR + NL ]2
(1)
where −NR NL /[NR + NL ]2 is the weak-localization correction in the absence of dephasing. Similarly, universal
conductance fluctuations become11
2
δgRMT
=
2 NR2 NL2
[1 + τD /τφ ]−2 ,
β [NR + NL ]4
(2)
and
are
thus
damped
below
their
value
(2/β)NR2 NL2 /[NR + NL ]2 (in units of 4e4 /h2 ) for
fully coherent systems with (β = 1) or without (β = 2)
2
time reversal symmetry. For a fictitious lead connected
to a two-dimensional cavity (a lateral quantum dot)
via a point contact of transparency ρ and carrying N3
channels one has τφ = mA/~ρN3 , with the electron mass
m and the area A of the cavity. Similarly, the dwell time
through the cavity is given by τD = mA/~(NL + NR ).
The dephasing and voltage probe models account for
dephasing at the phenomenological level only, without
reference to the microscopic processes leading to dephasing. At sufficiently low temperature, it is accepted that
the dephasing arises dominantly from electronic interactions, which, in diffusive systems, can be well modeled by
a classical noise potential8,9 . Remarkably enough, this
approach reproduces the RMT results of Eqs. (1) and
(2) with τφ set by the noise power. These results are
moreover quite robust in diffusive systems. They are essentially insensitive to most noise-spectrum details, and
hold for various sources of noise such as electron-electron
and electron-phonon interactions, or external microwave
fields. For this reason it is often assumed that dephasing
is system-independent, and exhibits a character of universality well described by the RMT of transport applied
to the dephasing/voltage probe models35 .
B.
Departure from RMT universality
According to the Bohigas-Giannoni-Schmit surmise36 ,
closed chaotic systems exhibit statistical properties of
hermitian RMT37 in the short wavelength limit. Opening up the system, transport properties derive from the
corresponding scattering matrix, which is determined by
both the Hamiltonian of the closed system and its coupling to external leads38 . It has been shown that for not
too strong coupling, and when the Hamiltonian matrix
of the closed system belongs to one of the Gaussian ensembles of random hermitian matrices, the corresponding
scattering matrix is an element of one of the circular ensembles of unitary random matrices39 . One thus expects
that, in the semiclassical limit of large ratio L/λF of the
system size to Fermi wavelength, transport properties of
quantum chaotic ballistic systems are well described by
the RMT of transport. This surmise has recently been
verified semiclassically40.
The regime of validity of RMT is generally bounded
by the existence of finite time scales, however, and it
was noticed by Aleiner and Larkin that, while the dephasing time τφ gives the long time cut-off for quantum
interferences, a new Ehrenfest time scale appears in quantum chaotic system in the deep semiclassical limit, which
determines the short-time onset of these interferences14 .
The Ehrenfest time τE corresponds to the time it takes
for the underlying chaotic classical dynamics to stretch
an initially localized wavepacket to a macroscopic, classical length scale. In open cavities, the latter can be
either the system size L or the width W of the opening to the leads. Accordingly, one can define the closed
cavity, τEcl = λ−1 ln[L/λF ], and the open cavity Ehren-
fest time, τEop = λ−1 ln[W 2 /λF L]41,42 . The emergence
of a finite τE strongly affects quantum effects in transport, and recent analytical and numerical investigations
of quantum chaotic systems have shown that weak localization14,43,44,45,46 and shot noise47,48,49,50 are exponentially suppressed ∝ exp[−τE /τD ] in the absence of dephasing (τφ → ∞). Interestingly enough, the deep semiclassical limit of finite τE sees the emergence of a quantitatively dissimilar behavior of weak-localization and
quantum parametric conductance fluctuations, the latter
exhibiting no τE -dependence in the absence of dephasing51,52,53,54 . These results are not captured by RMT,
instead one has to rely on quasiclassical approaches14,44
or semiclassical methods43,45,46,55,56 to derive them.
C.
Dephasing in the deep semiclassical limit
The behavior of quantum corrections to transport at
finite τE in the presence of dephasing was briefly investigated analytically in Ref. [14], for a model of classical noise with large angle scattering, and numerically in
Ref. [23], for the dephasing lead model with a tunnel barrier. Intriguingly enough, the two approaches delivered
the same result, that quantum effects are exponentially
suppressed ∝ exp[−τE /τφ ]. This suggested that dephasing retains a character of universality even in the deep
semiclassical limit. More recent investigations have however showed that at finite Ehrenfest time, Eq. (1) becomes25 (see also Ref. [24])
g wl = −
NR NL
exp[−(τEcl /τD + τ̃ /τφ )]
,
2
[NR + NL ]
1 + τD /τφ
(3)
with a strongly system-dependent time scale τ̃ . Ref. [25]
showed that, for the dephasing lead model, τ̃ = τEcl + (1 −
ρ)τEop in terms of the transparency ρ of the contacts to
the leads, which provides theoretical understanding for
the numerical findings of Ref. [23]. If however one considers a system-environment model, where the environment is mimicked by electrons in a nearby closed quantum chaotic dot, one has τ̃ = τξ , where
τξ = λ−1 ln[(L/ξ)2 ],
(4)
in terms of the correlation length ξ of the inter-dot interaction potential.
On the experimental front, an exponential suppression
∝ exp[−T /Tc] of weak-localization with temperature has
been reported in Ref. [29]. Taking τφ ∼ T −1 as for dephasing by electronic interactions in two-dimensional diffusive systems, this result was interpreted as the first experimental confirmation of Eq. (3). There is no other theory for such an exponential behavior of weak-localization,
however, the temperature range over which this experiment has been performed makes it unclear whether the
ballistic13,15 , τφ ∼ T −2 , or the diffusive dephasing time
determines the Ehrenfest time dependence of dephasing
(see the discussion in Ref. [24]).
3
Weak localization
Conductance fluctuations
System with environment
τ̃ = τξ
—
Classical noise (microwave, etc)
τ̃ = τξ
τEcl
τ̃ =
+
τ̃ = τξ
e-e interactions within system
System-gate e-e interactions
Dephasing lead:
no tunnel barrier
low transparency barrier
τ̃ ∼ 0, Ref. [24]
1
τ
2 ξ
τ̃ = τEcl
τ̃ = τEop + τEcl ∼ 2τEcl
Shot noise
no dephasing
1 cl
τ ,
2 E
τ̃ ∼
Ref. [24]
τ̃ ∼ 12 τξ , follows from Ref. [24]
τ̃ ∼
2τEcl
τ̃ = 0
(numerics), Ref. [23]
no dephasing, Ref. [57]
Table I: Summary of the known results to date on the nature of the exponential term exp[−τ̃ /τφ ] in the dephasing [cf. Eq. (3)].
Results that are not referenced are obtained in the present article and in Ref. [25]. Here we list the value of τ̃ for different
transport quantities and different sources of dephasing, all in the pure dephasing regime (the phase-breaking regime of Ref. [58]).
The parameter ξ differ slightly from system to system (see text for details) however it is always related to the correlation length
of the interaction with the environment. The results of Ref. [24] neglect τξ -contributions (so “0” could indicate a τ̃ of order
τξ ), they are also only valid for τEcl ≫ τφ .
D.
Outline of this article
In the present article, we amplify on Ref. [25] and extend the analytical derivation of Eq. (3) briefly presented
there. We investigate three different models of dephasing
and show that the suppression of weak-localization corrections to the conductance is strongly model-dependent.
First, we consider an external environment modeled by
a capacitively coupled, closed quantum dot. We restrict
ourselves to the regime of pure dephasing, where the environment does not alter the classical dynamics of the
system. Second, we discuss dephasing by a classical noise
field. Third, following Ref. [56], we provide a semiclassical treatment of transport in the dephasing lead model.
For these three models, we reproduce Eq. (3) and derive
the exact dependence of τ̃ on microscopic details of the
models considered. All our results are summarized in
Table I.
The outline of this article goes as follows. In Section
II, we present the treatment of the system-environment
model, focusing in particular on the construction of a
new scattering approach to transport that incorporates
the coupling to external degrees of freedom. We apply this formalism to a model of an open quantum
dot coupled to a second, closed quantum dot. We
present a detailed calculation of the Drude conductance
and the weak-localization correction, including coherentbackscattering, which explicitly preserves the unitarity
of the S-matrix, and hence current conservation. This
calculation is completed by a derivation of the Fano factor, showing that, in the pure dephasing limit, shot noise
is insensitive to dephasing to leading order. In Section III, we present a model of dephasing via a classical
noise field (such as microwave noise). We consider classical Johnson-Nyquist noise models of dephasing due to
electron-electron interactions within the system, and dephasing due to charge fluctuations on nearby gates. In
Section IV we present a trajectory based semiclassical
calculation of conductance in the dephasing lead model,
both for fully transparent barriers and tunnel barriers.
We also comment on dephasing in multiprobe configurations. Finally, Section V is devoted to numerical simulations confirming our analytical results. Summary and
conclusions are presented in Section VI, while technical
details are presented in the Appendix.
II.
TRANSPORT THEORY FOR A SYSTEM
WITH ENVIRONMENT
In the scattering approach to transport, the system is
assumed fully coherent and all dissipative processes occur in the leads59 . Apart from its coupling to the leads,
the system is isolated. Here we extend this formalism
to include coupling to external degrees of freedom in the
spirit of the standard theory of decoherence. The coupling to an environment can induce dephasing and relaxation. Here, we restrict ourselves to pure dephasing,
where the system-environment coupling does not induce
energy, nor momentum relaxation in the system. In semiclassical language, we assume that classical trajectories
supporting the electron dynamics are not modified by
this coupling.
The starting point of the standard theory of decoherence is the total density matrix ηtot that includes both
system and environmental degrees of freedom1 . The observed properties of the system alone are contained in
the reduced density matrix ηsys , obtained from ηtot by
tracing over the environmental degrees of freedom. This
procedure is probability conserving, Tr [ηsys ] = 1, but
it renders the time-evolution of ηsys non-unitary, and in
particular, the off-diagonal elements of ηsys decay with
time. This can be
by the basis independent
2quantified
≤ 1, which remains equal to one
purity60 , 0 ≤ Tr ηsys
only in the absence of environment. We generalize this
standard approach to the transport problem.
4
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be tuned by applying external backgate voltages on a
given system.
In the standard scattering approach, the transport
properties of the system derive from its (NL + NR ) ×
(NL + NR ) scattering matrix16
!
sLL sRL
,
(6)
Ŝ =
sLR sRR
sys
lead R
lead L
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env
which we write in terms of transmission (t = sLR ) and
reflection (r = sα,α , α ∈ {L, R}) matrices. From Ŝ,
the system’s dimensionless conductance (conductance in
units of 2e2 /h) is given by
Figure 1: Schematic of the system-environment model. The
system is an open quantum dot that is coupled to an environment in the shape of a second, closed dot.
A.
The scattering formalism in the presence of an
environment
We consider two capacitively coupled chaotic cavities
as sketched in Fig. 1. The first one is the system (sys), an
open, two-dimensional quantum dot, ideally connected to
two external leads. The second one is a closed quantum
dot, which plays the role of an environment (env). The
two dots are capacitively coupled, and in particular, they
do not exchange particles. Thus current through the system is conserved. We require that the size of the contacts
between the open system and the leads is much smaller
than the perimeter of the system cavity but is still semiclassically large, so that the number of transport channels satisfies 1 ≪ NL, R ≪ L/λF . This ensures that the
chaotic dynamics inside the dot has enough time to develop, λτD ≫ 1, with the classical Lyapunov exponent
λ. Electrons in the leads do not interact with the second dot. Few-electron double-dot systems have recently
been the focus of intense experimental efforts61 . Parallel
geometries, of direct relevance to the present work, have
been investigated in Refs. [62,63].
The total system is described by the following Hamiltonian
H = Hsys + Henv + U.
(5)
Inside each cavity the chaotic dynamics is generated by
the corresponding one-particle Hamiltonian Hsys, env . We
only specify that the capacitive coupling potential U is
a smooth function of the distance between the particles.
It is characterized by its magnitude U and its correlation
length ξ such that its typical gradient is U/ξ. Physically, ξ is determined by the electrostatic environment of
the system, such as electric charges on the gates defining
the dots and the amount of depletion of the electrostatic
confinement potential between the gates and the inversion layer in semiconductor heterostructures. Generally
speaking, U and ξ are independent parameters and can
have different values in different systems, and might even
g = Tr(t† t).
(7)
To include coupling to an environment in the scattering
approach, we need to define an extended scattering matrix S that includes the external degrees of freedom. This
is formally done in Appendix A, and our starting point is
Eq. (A11) for the case of an initial product density matrix
(n)
(n)
ηtot = ηsys ⊗ ηenv , with ηsys = |nihn|, n ∈ {1, · · · , NL },
and ηenv , the initial density matrix of the environment.
We define the conductance matrix
D
h
i E
(r)
(n)
(8)
= m Trenv S ηsys
gnm
⊗ ηenv S† m ,
where Trenv stands for the trace over the environmental
degrees of freedom. From this matrix, the dimensionless
conductance is then given by,
g=
NR X
NL
X
(r)
gnm
,
(9)
n=0 m=0
and Eq. (A11) reads
X
g =
n∈R;m∈L
Z
dqdq0 dq′0 hq0 |ηenv |q′0 i
∗
×Smn (q, q0 ) Snm (q, q′0 ) .
(10)
Eq. (10) is the generalization of the Laudauer-Büttiker
formula in the presence of an external environment. It
constitutes the backbone of our trajectory-based semiclassical theory of dephasing.
B.
Drude conductance
The semiclassical derivation of the one particle scattering matrix has become standard64,65,66 . Once we introduce the environment we deal with a bipartite problem,
here we use the two-particle semiclassical propagator developed in the framework of entanglement and decoherence67,68 . The extended scattering matrix elements can
be written as,
Z ∞ Z
Z
hm|yi hy0 |ni
dt dy0 dy
Smn (q, q0 ) = −i
(dsys −1)/2
(2π~)
0
L
R
X Aγ AΓ
ei(Sγ +SΓ +Sγ,Γ )/~ , (11)
×
N d/2
(2π~)
γ,Γ
5
where we take a dsys –dimensional, one-particle system
(throughout what follows we take dsys = 2), and a d–
dimensional, N -particle environment. At this point, S
depends on the coordinates of the environment and is
given by a sum over pairs of classical trajectories, labeled
γ for the system and Γ for the environment. The classical
paths γ and Γ connect y0 (on a cross-section of lead L)
and q0 (anywhere in the volume occupied by the environment) to y (on a cross-section of lead R) and q (anywhere
in the volume occupied by the environment) in the time
t = tγ = tΓ . For an environment of N particles in d
dimensions, q is a N d component vector. In the regime
of pure dephasing, these paths are solely determined by
Hsys and Henv . Each pair of paths gives a contribution
weighted by the square root Aγ AΓ of the inverse determinant of the stability matrix69,70 , and oscillating with
one-particle (Sγ and SΓ , which include Maslov indices)
Rt
and two-particle (Sγ,Γ = 0 dτ U[yγ (τ ), qΓ (τ )]) action integrals accumulated along γ and Γ.
We insert Eq. (11) in Eq. (10), sum over channel indices with the semiclassical approximation45,56
PNL
′
′
n hy0 |nihn|y0 i ≈ δ(y0 − y0 ). For the environment
we make the random matrix ansatz that hq0 |ηenv |q′0 i ≈
′
(2π~)N d Ξ−1
env δ(q0 − q0 ), where Ξenv is the environment phase-space volume. The dimensionless conductance then reads
Z
Z
Z
Z
(2π~)−1 ∞
′
g =
dtdt
dq0 dq dy0 dy
Ξenv
0
env
L
R
X
×
Aγ AΓ Aγ ′ AΓ′ ei(Φsys +Φenv +ΦU ) . (12)
γ,Γ;γ ′,Γ′
This is a quadruple sum over classical paths of the system
(γ and γ ′ , going from y0 to y) and the environment (Γ
and Γ′ , going from q0 to q) with action phases,
Φsys = Sγ (y, y0 ; t) − Sγ ′ (y, y0 ; t′ ) /~,
(13a)
′
(13b)
Φenv = SΓ (q, q0 ; t) − SΓ′ (q, q0 ; t ) /~,
′
ΦU = Sγ,Γ (y, y0 ; q, q0 ; t) − Sγ ′ ,Γ′ (y, y0 ; q, q0 ; t ) /~.
(13c)
We are interested in quantities averaged over variations
in the energy or cavity shapes. For most sets of paths the
phase of a given contribution will oscillate wildly with
these variations, so the contribution averages to zero. In
the semiclassical limit, Eq. (12) is thus dominated by
terms which satisfy a stationary phase condition (SPC),
i.e. where the variation of Φsys + Φenv + ΦU has to be
minimized. In the regime of pure dephasing, individual
variations of Φsys , Φenv and ΦU are uncorrelated. They
are moreover dominated by variations of Φsys and Φenv ,
on which we therefore enforce two independent SPC’s.
The dominant contributions that survive averaging are
the diagonal ones. They give the Drude conductance.
Indeed setting γ = γ ′ and Γ = Γ′ straightforwardly satisfies SPC’s over Φsys and Φenv . These two SPC’s require
t = t′ and lead to an exact cancellation of all the phases
Φsys = Φenv = ΦU = 0. The dimensionless Drude conductance is given by
gD =
Z
∞
dt
Z
dq0 dq
dy0
L
env
0
Z
Z
dy
R
X
γ,Γ
A2γ A2Γ
. (14)
(2π~) Ξenv
From here on, the calculation proceeds along the lines
of Ref. [45]. The main idea is to relate semiclassical amplitudes with classical probabilities. This is done by the
introduction of two sum rules that express the ergodic
properties of open cavities, Eq. (15a), and of closed ones,
Eq. (15b),
X
A2γ [· · · ]γ =
Z
X
A2Γ [· · · ]Γ =
Z
γ
Γ
π
2
−π
2
dθ0 dθ Psys (Y, Y0 ; t) [· · · ]Y0 , (15a)
dp0 dp P̃env (Q, Q0 ; t) [· · · ]Q0 . (15b)
Here, Psys (Y, Y0 ; t) = pF cos θ0 × P̃sys (Y; Y0 ; t), and
P̃sys (Y, Y0 ; t) and P̃env (Q, Q0 ; t) are the classical probability densities. For the system, we need to take into
account the fact that particles are injected, which is why
the classical probability density must be multiplied with
the initial system momentum pF cos θ0 along the injection
lead65 . The phase points Y0 = (y0 , θ0 ) and Y = (y, θ)
are at the boundary between the system and the leads.
In contrast Q0 = (q0 , p0 ) and Q = (q, p) are inside
the closed environment cavity. The momenta are integrated over the entire environment phase-space, while
P̃env (Q, Q0 ; t) will always contain a δ-function which restricts the final energy to equal the initial one, (i.e. |p| =
|p0 | if all N environment particles have the same mass).
The average of Psys over an ensemble of systems or over
energy gives a smooth function. For a chaotic system we
write
D
E
P̃sys (Y; Y0 ; t) =
cos θ
e−t/τD . (16a)
2 (WL + WR ) τD
Likewise, the average of P̃env gives
D
E δ(|p| − |p |)
0
,
P̃env (Q; Q0 ; t′ ) =
|p |
Ξenv0
(16b)
|p |
where Ξenv0 is the size of the hypersurface in the environment’s phase-space defined by |p| = |p0 | (for d =
|p |
N
2, Ξenv0 = (2πpenv
where Ωenv is the area of
F Ωenv )
the environment). Inserting Eqs. (15a), (15b), (16a)
and R(16b) into Eq.
R integrals usR (14), we can perform all
ing env dQ0 ≡ env dq0 dp0 = Ξenv and env dQδ(|p| −
|p |
|p0 |) = Ξenv0 . Then, since NL,R = kF WL,R /π, we recover
the classical Drude conductance,
gD =
NL NR
.
NL + NR
(17)
6
Overview of the effect of environment on
weak-localization
γ’
ε γ’
γ
encounter
11111111111111
00000000000000
00000000000000
11111111111111
ξ
00000000000000
W
’
γ=γ 11111111111111
00000000000000
11111111111111
T
00000000000000
11111111111111
00000000000000 L
W11111111111111
00000000000000
11111111111111
ph
de
g
1
2 ξ
y0
γ’
Lead R
θ0
in
as
The leading-order weak-localization corrections to the
conductance were identified in Refs. [14,55,71] as those
arising from trajectories that are paired almost everywhere except in the vicinity of an encounter. An example of such a trajectory is shown in Fig. 2. At the
encounter, one of the trajectories (γ ′ ) intersects itself,
while the other one (γ) avoids the crossing. Thus, they
travel along the loop they form in opposite directions.
For chaotic ballistic systems in the semiclassical limit,
only pairs of trajectories with small crossing angle ǫ contribute significantly to weak-localization. In this case,
trajectories remain correlated for some time on both sides
of the encounter, the correlated region indicated in pink
in Fig. 2. In other words, the smallness of ǫ requires
two minimal times, TL (ǫ) to form a loop, and TW (ǫ) in
order for the legs to separate before escaping into different leads. In the case of an hyperbolic dynamics one
estimates72 ,
γ
Lε
Lead L
C.
θ
y
γ
γ=γ ’
Chaotic system
(environment not shown)
(18a)
(18b)
Figure 2: (color online) A semiclassical contribution to weaklocalization for the system-environment model. The paths are
paired everywhere except at the encounter, where one path
crosses itself at angle ǫ, while the other one does not (going
the opposite way around the loop). Here we show ξ > ǫL,
so the dephasing (dotted path segment) starts in the loop
(Tξ > 0).
As long as the system-environment coupling does not
generate energy and/or momentum relaxation, the presence of an environment does not significantly change this
picture. However it does lead to dephasing via the accumulation of uncorrelated action phases, mostly along the
loop, when γ and γ ′ are more than a distance ξ apart.
This is illustrated in Fig. 2 for the case when ξ is less
than W . We define a new timescale Tξ as twice the time
between the encounter and the start of the dephasing,
to paths with an encounter, we can still write this sum
in the form given in Eq. (15a), provided the probability
Psys (Y, Y0 ; t) is restricted to paths which cross themselves. To ensure this we write
Z
Psys (Y, Y0 ; t) = pF cos θ0
dR2 dR1 P̃sys (Y, R2 ; t − t2 )
TL (ǫ) ≃ λ−1 ln[ǫ−2 ],
TW (ǫ) ≃ λ−1 ln[ǫ−2 (W/L)2 ].
Tξ (ǫ) ≈ λ−1 ln[ǫ−2 (ξ/L)2 ].
(19)
Dephasing occurs mostly in the loop part. However if
ξ < ǫL and Tξ < 0, dephasing starts before the paths
reach the encounter. We discuss this point in more detail
below in Section II G.
D.
C
× P̃sys (R2 , R1 ; t2 − t1 )P̃sys (R1 , Y0 ; t1 ) . (20)
Calculating the effect of the environment on
weak-localization
In the absence of dephasing each weak-localization
contribution accumulates a phase difference δΦsys =
EF ǫ2 /(λ~)55,71 . In the presence of an environment, an
additional action phase difference δΦU is accumulated.
Incorporating this additional phase into the calculation
of weak-localization does not require significant departure from the theory at U = 0. We extend the theory of
Ref. [45] to account for this additional phase.
We follow the same route as for the Drude conductance, but now consider the pairs of paths described in
Section II C above, and shown in Fig. 2, while the environment is still treated within the diagonal approximation, Γ′ = Γ. The sum rule of Eq. (15b) still applies. Though the sum over system paths is restricted
Here, we use R = (r, φ), φ ∈ [−π, π] for phase-space
points inside the cavity, while Y lies on the lead as before. We then restrict the probabilities inside the integral to trajectories which cross themselves at phasespace positions R1,2 with the first (second) visit to the
crossing occurring at time t1 (t2 ). We can write dR2 =
vF2 sin ǫdt1 dt2 dǫ and set R2 = (r1 , φ1 ± ǫ). Then, the
weak-localization correction to the dimensionless conductance in the presence of an environment is given by,
Z
Z
wl
−1
g = (π~)
dY0 dǫ Re eiδΦsys F (Y0 , ǫ) , (21a)
L
with,
F (Y0 , ǫ) =
2vF2 sin ǫ
× pF cos θ0
Z
∞
dt
TL +TW
Z
R
dY
Z
t−
TW
2
TL +
Z
C
TW
2
dt2
Z
t2 −TL
TW
2
dt1
dR1 P̃sys (Y, R2 ; t − t2 )
× P̃sys (R2 , R1 ; t2 − t1 )P̃sys (R1 , Y0 ; t1 )
Z
dQdQ0
P̃env (Q, Q0 ; t) exp[iδΦU ]. (21b)
×
Ξenv
7
Comparison with Eq. (34) of Ref.[45] shows that the effect of the environment is entirely contained in the last
line of Eq. (21b). At the level of the diagonal approximation for the environment, Γ′ = Γ, one has
Z
1 t
δΦU =
dτ U rγ (τ ), qΓ (τ )) − U rγ ′ (τ ), qΓ (τ ) ,
~ 0
(21c)
where rγ (τ ) and qΓ (τ ) parametrize the trajectories of
the system and of the environment respectively. We note
that in the absence of coupling, δΦU = 0, the integral
over the environment is one, and we recover the weaklocalization correction of an isolated system (cf. Eq. (35)
of Ref. [45]).
To evaluate Eqs. (21), we need the average effect of the
environment on the system, after one has traced out the
environment. For a single measurement, this average is
an integral over all classical paths followed by the environment, starting from its initial state at the beginning
of the measurement. We therefore also average over an
ensemble of initial environment states, or an ensemble
of environment Hamiltonians, which corresponds to performing many measurements. For compactness we define
h· · · ienv as this integral over environment paths and the
ensemble averaging over the environment,
Z
dQ dQ0
h· · · ienv =
P̃env (Q; Q0 ; t) [· · · ]Q0 . (22)
Ξenv
Without loss of generality we assume that for all r, the
interaction U(r, q) is zero upon averaging over all q. We
can ensure an arbitrary interaction fulfills this condition
by moving any constant term in U into the system Hamiltonian (these terms do not lead to dephasing). Since the
environment is ergodic, we have
U rγ (t), qΓ (t) env = 0 .
(23)
Now we use the chaotic nature of the environment
to give the properties
of the
correlation function
U rγ (t), qΓ (t) U rγ ′ (t′ ), qΓ (t′ ) env .
(i) Correlation functions typically decay exponentially
fast with time in chaotic systems, with a typical
decay time related to the Lyapunov exponent73 .
The precise functional form J(λenv |t′ − t|) of the
temporal decay of the coupling correlator depends
on details of Henv and U, however for all practical purposes, it is sufficient to know that it decays fast, and we approximate it by a δ-function,
J(λenv |t′ − t|) ≃ λ−1
env δ(t).
(ii) We argue that the spatial correlations of U for two
different system paths at the same t also decay,
because the averaging over many paths, and many
initial environment states act like an average over q.
We define K(|r′ −r|/ξ) as the functional form of this
decay of spatial correlations, with K(0) = 1. The
precise form of K(x) depends on details of Hsys ,
Henv and U. In particular, the typical length ξ
of this decay is of order the scale on which U(r, q)
changes between its maximum and minimum value.
Given these arguments, we have
U rγ (t), qΓ (t) U rγ ′ (t′ ), qΓ (t′ )
env
hU2 i
=
K (|rγ ′ (t) − rγ (t)|/ξ) δ(t − t′ ) .
λenv
(24)
Then,
hδΦ2U ienv
Z t
1
dτ2 dτ1
= 2
~ 0
D
× U rγ (τ2 ), qΓ (τ2 ) − U rγ ′ (τ2 ), qΓ (τ2 )
E
× U rγ (τ1 ), qΓ (τ1 ) − U rγ ′ (τ1 ), qΓ (τ1 )
env
2 Z t
hU i
= 2
(25)
dτ 1 − K(|rγ ′ (τ ) − rγ (τ )|/ξ) .
λenv 0
We further make the following step-function approximation for K,
K(x) = Θ(1 − x),
(26)
where the Euler Θ-function is one (zero) for positive (negative) arguments. In principle, this is unjustifiable for
x ∼ 1, however since the paths diverge exponentially
from each other, the time during which x ∼ 1 is of order
λ−1 , while dephasing happens on a timescale τφ which
is typically of order the dwell time, τD . Thus the stepfunction approximation of K(x) will have corrections of
order (λτD )−1 ≪ 1, which we therefore neglect. Once
we have made the approximation in Eq. (26), we see that
non-zero contributions to hδΦ2U i come from regions where
the distance between γ and γ ′ is larger than ξ.
We are now ready to calculate dephasing for those system paths shown in Fig. 2. As defined above, t1 and t2
are the two times at which the path γ ′ crosses itself. Dephasing acts on the loop formed by γ ′ and, as just argued,
it acts once the distance between γ and γ ′ is greater than
ξ, i.e. in the time window from (t1 + Tξ /2) to (t2 − Tξ /2),
where Tξ (ǫ) is given in Eq. (19). We average Eq. (21b)
over the environment and use the central limit theorem
to evaluate the action phase due to the coupling between
system and environment,
h
i
exp[iδΦU ] env = exp − 21 hδΦ2U ienv
= exp [−(t2 − t1 − Tξ )/τφ ] , (27)
where the dephasing rate is
2
τφ−1 ∼ ~−2 λ−1
env hU i.
(28)
Given that h· · · ienv is defined in Eq. (22), we can substitute Eq. (27) directly into Eqs. (21). We thereby reduce the problem to an integral over system paths which
8
cos θ e−(t−t2 −TW /2)/τD
, (29c)
hP̃sys (Y, R2 ; t − t2 )i =
2(WL + WR )τD
with Ωsys being the real space volume occupied by the
system (the area of the cavity). At this point the integral
is the same as without dephasing, except that during the
time (t2 − t1 − Tξ ), the inverse dwell time is replaced by
(τD−1 + τφ−1 ). Thus when evaluating the (t2 − t1 )-integral,
we get the extra prefactor exp [−TL (ǫ)/τφ ] /(1 + τD /τφ )
compared with the equivalent integral result without dephasing. Thus we have
hF (Y0 , ǫ)i ∝ sin ǫ
e−TL (ǫ)/τD −(TL (ǫ)−Tξ (ǫ))/τφ
. (30)
1 + τD /τφ
Since (TL (ǫ) − Tξ (ǫ)) = τξ with τξ defined in Eq. (4),
the ǫ dependence of the τφ−1 -term drops out. This means
that hF (Y0 , ǫ)i simply differs from its value without dephasing by a constant factor, e−τξ /τφ /(1 + τD /τφ ). Thus
the integral over ǫ in Eq. (21a) is identical to the one
in the
of the environment, and takes the form45
R ∞ absence
1+2/(λτD )
Re 0 dǫ ǫ
exp[iEF ǫ2 /(λ~)], where we have assumed ǫ ≪ 1. The substitution z = EF ǫ2 /(λ~) immediately yields a dimensionless integral and an exponencl
tial term, e−τE /τD (neglecting as usual O[1]-terms in the
logarithm in τEcl ). From this analysis, we find that the
weak-localization correction is given by
g wl =
g0wl
exp[−τξ /τφ ],
1 + τD /τφ
(31)
where g0wl is the weak-localization correction at finite-τEcl
in the absence of dephasing,
g0wl = −
NL NR
exp[−τEcl /τD ].
(NL + NR )2
Weak-localization for reflection and coherent
backscattering
We show explicitly that our semiclassical method is
probability-conserving, and thus current-conserving, also
Lead R
Lead L
γ’
γ
Chaotic system
(environment not shown)
Figure 3: (color online) A semiclassical contribution to coherent backscattering for the system-environment model. It
involves paths which return to close, but anti-parallel to themselves at lead L. The two solid paths are paired (within W
of each other) in the cross-hatched region. Here we show
ξ > ǫL, so the dephasing (dotted path segment) starts in the
loop (Tξ > 0). In the basis parallel and perpendicular to
γ at injection the initial position and momentum of path γ
at exit are r0⊥ = (y0 − y) cos θ0 , r0k = (y0 − y) sin θ0 and
p0⊥ = pF (θ − θ0 ).
in the presence of dephasing. We do this by calculating the leading-order quantum corrections to reflection,
showing that they enhance reflection by exactly the same
amount that transmission is reduced. There are two
leading-order off-diagonal corrections to reflection. The
first one reduces the probability of reflection to arbitrary
momenta (weak-localization for reflection), while the second one enhances the probability of reflection to the timereversed of the injection path (coherent-backscattering).
The distinction between these two contributions is related to the correlation between the path segments when
they hit the leads. For coherent-backscattering contributions, these segments are correlated (see Fig. 3), but for
weak-localization contributions, they are not.
The derivation of the weak-localization for reflection
rwl is straightforward and proceeds in the same way as
the derivation for g wl given above, replacing the factor
NR /(NR + NL ) by NL /(NR + NL ). We thus get,
(32)
We see that the dephasing of weak localization is not
exponential with the Ehrenfest time, instead it is exponential with the λF -independent scale τξ given in Eq. (4).
In all cases where ξ is a classical scale (i.e. of similar magnitude to W, L rather than λF ) we see that τξ is much
less than the Ehrenfest time, τξ ≪ τφ . In such cases the
exponential term in Eq. (31) is much less noticeable that
the universal power-law suppression of weak-localization.
E.
y0
g
(29b)
0
in
e−(t2 −t1 −TW /2)/τD
,
hP̃sys (R2 , R1 ; t2 − t1 )i =
2πΩsys
(29a)
as
e−t1 /τD
,
2πΩsys
ξ
ph
hP̃sys (R1 , Y0 ; t1 )i =
11111111111111
00000000000000
T’
00000000000000
11111111111111
ξ
00000000000000
11111111111111
00000000000000W
θ 11111111111111
00000000000000
11111111111111
θ
L
00000000000000
11111111111111
y
de
is almost identical to the equivalent integral for U = 0.
Assuming phase space ergodicity for the system, we get
the probabilities
rwl =
r0wl
exp[−τξ /τφ ],
1 + τD /τφ
(33)
where r0wl = − exp[−τEcl /τD ] NL2 /(NL + NR )2 is the finiteτEcl correction in the absence of dephasing.
We next calculate the contributions to coherentbackscattering, extending the treatment of Ref. [45] to
account for the presence of dephasing. As before, the environment is treated in the diagonal approximation. The
coherent-backscattering contributions correspond to trajectories where legs escape together within TW /2 of the
encounter. Such a contribution is shown in Fig. 3. The
correlation between the system paths at injection and
exit induces an action difference δΦsys = δScbs not given
by the Richter-Sieber expression. It is convenient to write
this action difference in terms of relative coordinates at
9
the lead (rather than at the encounter). The system action difference is then δScbs = −(p0⊥ + mλr0⊥ )r0⊥ where
the perpendicular difference in position and momentum
are r0⊥ = (y0 − y) cos θ0 and p0⊥ = pF (θ − θ0 ). As
with weak-localization, we can identify three timescales,
′
TL′ , TW
, Tξ′ , associated with the time for paths to spread
to each of three length scales, L, W, ξ. However unlike
for weak-localization we define these timescales as a time
measured from the lead rather than from the encounter.
Thus we have
coupling, the environment part of Eq. (12) would then
be
Z
dqdq0 X
∆env t
,
(38)
AΓ AΓ′ eiΦenv = 1 + µ
Ξenv
~
′
Γ,Γ
where µ is a number of order one, and ∆env is the environment level-spacing (for a two-dimensional environment containing a single particle, ∆env ∼ ~2 /mL2env ).
The first term above comes from the diagonal approximation used throughout this article, while the second
Tℓ′ (r0⊥ , p0⊥ ) ≃ λ−1 ln[(mλℓ)2 /|p0⊥ + mλr0⊥ |2 ] (34)
term is a weak-localization correction. This correction
becomes of order the system’s weak-localization correcwith ℓ = {L, W, ξ}. Writing the integral over Y0 as
tion on the timescale t ∼ τD , so there is a priori no reason
an integral over (r0⊥ , p0⊥ ) and using pF sin θ0 dY0 =
to neglect it.
dp0⊥ dr0⊥ , the coherent-backscattering contribution is
The environment part of our calculation differs howZ
ever from the form-factor in that it corresponds to the
dp
dr
0⊥ 0⊥
Re eiδScbs F cbs (Y0 ) .
(35)
rcbs =
time-evolution of the environment during the time it
L pF sin θ0
takes for a particle to be transported across the system.
Therefore, the sum in Eq. (38) is not restricted to periAfter integrating out the environment in the same manodic orbits, and the unitarity of the environment’s timener as for weak-localization, we get
evolution imposes that µ = 0. Furthermore, unitarity
must be preserved even in the presence of a finite-U, as
F cbs (Y0 )
(36)
Z
Z ∞
long as there is no exchange of particles between system
=
dY
dthPsys (Y, Y0 , t)i exp[−(t − Tξ′ )/τφ ]
and environment. We thus conclude that we do not need
L
TL′
to consider the weak-localization type corrections to the
′
′
′
′
environment evolution because they cancel.
exp
−
(T
−
T
/2)/τ
−
(T
−
T
)/τ
NL pF sin θ0
D
φ
L
W
L
ξ
=
.
π(NL + NR )
1 + τD /τφ
Now we can proceed as for U = 0, pushing the momentum integral’s limits to infinity, and evaluating the
r0⊥ −integral over the range W , with the help of an Euler
Γ-function. We finally obtain
rcbs =
r0cbs
exp[−τξ /τφ ],
1 + τD /τφ
(37)
in terms of r0cbs = exp[−τEcl /τD ] NL /(NL +NR ), the finiteτEcl coherent backscattering contribution in the absence
of dephasing. Hence rcbs + rwl = −g wl for all values
of τξ and τφ , and our approach is probability- and thus
current-conserving.
F.
Weak-localization corrections in the
environment
So far, we have only considered cases where we make
a diagonal approximation for the environment. On the
face of it this seems a little unreasonable. For instance, if
the system and environment are of similar sizes then one
would expect that diagonal contributions for the system
and weak-localization for the environment would be as
important as the contributions calculated above.
Since the environment is a closed cavity, one would
naively think that the weak-localization contribution for
the environment should be calculated in a similar manner
to the form-factor in Refs. [71,74]. In the absence of
G.
Regime of validity of the semiclassical
calculation
Throughout this article we assumed that the systemenvironment coupling is weak enough not to modify the
classical paths in the system. Formally, this assumption can be rigorously justified by invoking theorems on
structural stability75 . However, care should be taken in
extrapolating our results to the limit ξ → 0, since the
force on the particle is the gradient of the interaction potential, ∼ U/ξ. We therefore estimate the minimum ξ for
which we can legitimately assume that classical system
paths are left unchanged by the system-environment coupling. This will give the bound on the regime of validity
of our approach.
To see significant dephasing we need τφ ∼ τD , so we
cannot take the interaction strength to zero, instead we
require that hU2 i ∼ λenv ~2 /τD , see Eq. (28). This induces a typical force ∼ hU2 i1/2 /ξ ∼ (~/ξ)(λenv /τD )1/2 on
the particle. To see if this noisy force significantly modifies the paths in the vicinity of the encounter, we compare
it with the relative force of the chaotic system Hamiltonian on the particle at the encounter. Since the perpendicular extension of the encounter is δr⊥ ∼ (LλF )1/2 ,
and the duration of the encounter is of order the Lyapunov time ∼ λ−1 , the system force goes like mλ2 δr⊥ ∼
mλ2 (LλF )1/2 . Estimating λ−1 ∼ vF /L as is typical of
chaotic billiards, the ratio of the noisy force to the system force becomes [λenv LλF /(ξ 2 λ2 τD )]−1/2 . Thus one
10
ξ
ξ
dephasing
γ’
γ
Lε γ ε γ ’
dephasing
1
2 ξ
γ
γ’
θ
y
Lead R
y0
g
θ0
in
as
Lead L
ph
de
11111111111111
00000000000000
00000000000000
11111111111111
00000000000000
W
γ=γ 11111111111111
’
00000000000000
11111111111111
00000000000000
11111111111111
T
L
00000000000000
W11111111111111
00000000000000
11111111111111
γ=γ ’
Chaotic system
(environment not shown)
Figure 4: (color online) Dephasing of weak-localization when
ξ ≪ Lǫ ∼ (LλF )1/2 , and hence Tξ (ǫ) is negative, see Eq. (19).
The dephasing starts and ends in the “legs” rather than the
loop. In the language of disordered systems this means the
dephasing affects the diffusons as well as the cooperon.
can ignore the modifications of the classical paths due to
the coupling to the environment, as long as
1/2
λenv /λ
1/2
.
(39)
ξ ≫ (λF L) ×
λτD
We see that ξ can easily be less than the typical encounter
size δr⊥ ∼ (LλF )1/2 (remember that λτD ≫ 1 is always
assumed). Thus our method is not only applicable for
ξ up to the system size, where dephasing happens only
in the loop. It is also applicable for ξ smaller than the
encounter size, in which case the time Tξ (ǫ) is negative,
and dephasing occurs in part of the legs as well as the
whole of the loop, see Fig. 4.
Finally we caution the reader that the whole semiclassical method used in this article relies on the lead
width being greater than the encounter size, this requires
that λτD ≪ (L/λF )1/2 , thus we cannot access the regime
ξ ∼ λF , which is dominated by stochastic diffraction at
the leads.
H.
Shot noise in the presence of an environment
When the temperature of the electrons in the leads
coupled to a chaotic system is taken to zero, there is no
thermal noise in the current through the device. However
there is still the intrinsically quantum noise which originates from the wave-like nature of the electrons. This
zero-temperature noise is known as shot noise84 . In the
absence of dephasing, shot noise has been well-studied
using RMT35 , quasi-classical field theory47 , and semiclassical methods49,50,76 .
It is generally argued that the shot noise is unaffected
by the presence of an environment which causes dephasing but not heating of the electrons – the regime of
phase-breaking of Ref. [57]. This belief is founded on the
fact that (i) the dephasing-lead model gives a dephasingindependent shot noise, (ii) kinetic equations – in which
interference effects are ignored – give the same shot noise
as full quantum calculations. Here we show explicitly
that, under the assumption that the system-environment
coupling does not heat up the current-carrying electrons,
indeed, the coupling to the environment does not affect
shot noise.
We start with the formula for the zero-frequency shot
noise power through a system coupled to an environment.
This formula is derived in Appendix A 2, and is given as
SRR (0) in Eq. (A17). We use Eq. (11) to write each matrix element as sums over classical paths. This gives us a
sum over eight paths – 4 system paths and 4 environment
paths – as sketched in Fig. 5. The system paths are as
follows
•
•
•
•
γ1
γ2
γ3
γ4
from
from
from
from
y01
y03
y03
y01
on
on
on
on
lead
lead
lead
lead
L to y1 on lead R,
R to y1 on lead R,
R to y3 on lead R,
L to y3 on lead R.
The sums over lead modes and the trace over the environment density matrix are performed in the same manner
as for the conductance, [see above Eq. (12)], which results
in
S =
Z
Z
X
e3 V
dy
dy03 dy1 dy3
Asys eiΦsys
01
3
(2π~) L
R
γ1,···γ4
Z
X
× dq01 dq03 dq1 dq3
Aenv ei(Φenv +ΦU ) . (40)
Γ1,···Γ4
Here, Asys = Aγ1 Aγ2 Aγ3 Aγ4 , Φsys = (Sγ1 − Sγ2 +
Sγ3 − Sγ4 )/~ and we absorbed all Maslov indices into
the actions. Similarly, Aenv = AΓ1 AΓ2 AΓ3 AΓ4 , Φenv =
(SΓ1 − SΓ2 + SΓ3 − SΓ4 )/~, and ΦU = (Sγ1,Γ1 − Sγ2,Γ2 +
Sγ3,Γ3 − Sγ4,Γ4 )/~. As argued above, in the regime of
pure dephasing, Φsys , Φenv and ΦU are uncorrelated. We
thus first pair up the system paths to minimize Φsys , this
pairing is the same as it would be in the absence of the
environment (compare Fig. 5 to Fig. 1 of Ref. [50]). We
see from the construction of Eq. (A17), that all paths
reach the encounter at the same time77 , t′1 .
Now we make the crucial observation that, for any
given set of system paths, we have γ1 ≃ γ2 for times
greater than t′1 . Thus for times greater than t′1 we can
write the sum over Γ1, Γ2 in the second line of Eq. (40)
11
(a)
y 03
System
y 01
0 t 1 −t 2
q03
Environ.
q’01
q01
encounter
γ3
γ1
y1
Γ2
t’1
Γ3
Γ4
(b)
y 03
System
y 01
γ3
γ1
t 1 −t 2
0
y3
Γ1
time
t 1 t 1 −t 2 +t3
q3
q1
encounter
y3
y1
t’1
time
t 1 t 1 −t 2 +t3
Environ.
Figure 5: (color online) Sketch of typical trajectories which
contribute to shot noise in the presence of an environment. In
both (a) and (b) the system (environment) paths are sketched
above (below) the time axis. In (a) we show a contribution
which will survive system averaging, because the system paths
are paired almost everywhere with an encounter at time t′1 .
There is no constraint on the environment paths, as yet. For
simplicity we show only system paths γ1 and γ3, with path
γ2 and γ4 being the same as γ1 and γ3 except that they cross
at the encounter. Depending on the choice of t′1 with respect
to the other timescales, t1 , t2 , t3 , this pair of system paths
could represent any of the contributions in Fig. 1 of Ref. [50].
In (b) additional constraints are imposed on the Γi ’s, after
one has integrated over all initial and final positions of the
environment. This integration removes all contributions from
environment propagation when the system paths are paired.
as
Z
Γ2 must be the same at time t′1 . This is sketched in
Fig. 5; the paths Γ1 and Γ2 to the right of the encounter
in Fig. 5a are replaced in Fig. 5b by the constraint that
paths Γ1 and Γ2 meet at time t′1 . Note that we cannot use
Eq. (42) to integrate out paths Γ1 and Γ2 for arbitrary
times before t′1 because the system paths γ1 and γ2 are
then different enough that the potential V ′ (q, t) will be
different for the two propagators in Eq. (41). However
we can use the same argument to integrate out the pair
Γ3-Γ4 after time t′1 , and to integrate out the pairs Γ1-Γ4
and Γ2-Γ3 before the time t′1 . After all these pairs are
replaced by δ-functions, we get the situation shown in
Fig. 5b.
Focusing on ξ much greater than the encounter size,
the above method can be used to integrate out the environment paths for all t > t′1 and all t < t′1 . This leaves a
single point (the environment state at t = t′1 ) to integrate
over. Doing this we see that Eq. (40) reduces to
S=
e3 V
(2π~)3
Z
dy01
L
Z
dy03 dy1 dy3
R
X
AΓ1 AΓ2 ei(SΓ1 −SΓ2 +Sγ1,Γ1 −Sγ2,Γ2 )/~
Γ1,Γ2
≃
Z
dq1
=
Z
∗
dq1 K′env (q1 , t1 ; q′′1 , t′1 ) K′env (q1 , t1 ; q′1 , t′1 ) (41)
X
AΓ1 AΓ2 ei(SΓ1 −SΓ2 +Sγ1,Γ1 −Sγ1,Γ2 )/~
Γ1,Γ2
where we set γ2 = γ1 to get the second line. We define K′env (q1 , t1 ; q′1 , t′1 ) as the propagator for the environment evolving under an effective time-dependent potential, V ′ (q, t), which is the sum of U(rγ1 (t), q) and the potential term in Henv . Since both propagators in Eq. (41)
evolve under the same potential (because γ1 = γ2), we
can use the basic property of propagators78 that
Z
∗
dq1 K′env (q1 , t1 ; q′ , t′1 ) K′env (q1 , t1 ; q′′ , t′1 )
= δ(q′′ − q′ )
(42)
to integrate out these propagators for times greater than
t′1 . In their place we have a constraint that path Γ1 and
Asys eiΦsys . (43)
γ1,···γ4
This is identical to the shot noise formula in the absence
of an environment. Thus we have completely removed
the environment from the problem without affecting the
shot noise of the system at all. To calculate the shot
noise now, one simply needs to follow the derivation for
a system without an environment in Ref. [50]. The result
is most conveniently written in terms of the Fano-factor,
F , which is the ratio of the shot noise to the Poissonian
noise 2ehIi, where hIi = 2e2 g D V /h is the average current.
One gets
F ≡ S/2ehIi =
dq1
X
NL NR
exp[−τEop /τD ]. (44)
(NL + NR )2
This is of course independent of the coupling to the environment.
As a final comment, we note that above we kept
only the leading O[N ] term in the shot noise. There is
a hierarchy of weak-localization-like corrections O[N a ],
a = 0, −1, . . . to this result76 , which are suppressed by
dephasing in much the same way as the weak-localization
correction to conductance. Thus for τEop ≪ τD , we can
expect the environment to cause a cross-over from the
result in Ref. [76] to the result in Eq. (44) with τEop = 0.
Hence in the classical limit of wide leads (NL,R ≫ 1)
the environment’s effect is negligible, however for narrow leads (NL,R ∼ 1) the environment’s effect may be
significant.
III.
CLASSICAL NOISE
We have shown that, to capture the effect of dephasing on weak localization, it is sufficient to treat the environment at the level of the diagonal approximation.
We thus observe that, in the semiclassical limit of short
12
wavelength, λF /Lenv → 0, a quantum chaotic environment has the same dephasing effect on weak-localization
as the equivalent classical chaotic environment. Because
correlations typically decay exponentially fast in classical
hyperbolic systems, this makes the effect of this classical
environment very similar to a classical noise field with
a suitably chosen spatial and temporal correlation function. In this Section we show that the conclusions that
we draw for a quantum environment can also be drawn
for a classical noise field. One common experimental
example of such a field is microwave radiation, applied
to the chaotic dot either by accident or on purpose27 .
A second example, is the common theoretical treatment
of electron-electron interactions as a source of classical
(Johnson-Nyquist) noise8,9 .
We add a new term to the system Hamiltonian of the
form, Vnoise (t). We assume this term is weak enough that
it does not affect the classical paths, but strong enough to
modify the phase acquired along such paths. The phase
difference for a pair of paths contributing to the conductance is (Φsys + Φnoise ), where Φsys is given in Eq. (13a)
and
Z
Φnoise = dt V (rγ ′ ; t) − V (rγ ; t) /~.
(45)
We now assume that the noise is Gaussian distributed
with
V rγ ′ (t); t V rγ (t′ ); t′
= hV 2 i Knoise (|r2 − r1 |/ξ) Jnoise (λnoise |t2 − t1 |). (46)
Here, Knoise (x) gives the form of the spatial decay of the
correlation function (on a scale ξ), and Jnoise (t) gives the
form of the temporal decay of the correlation function (on
a scale which we call λ−1
noise to make the analogy with the
notation in Section II C). We can now follow the derivation in Section II C) by replacing U(rγ (t), qΓ (t)) with
V (rγ ; t) throughout. We assume that the correlations in
time are short enough to be treated as white-noise-like,
Jnoise (x) ∝ δ(x), and that the spatial correlations decay fast enough that we can justify the approximation
in Eq. (26). This directly leads to the same result for
weak-localization as in Eq. (31), where now
2
τφ−1 ∼ ~−2 λ−1
noise hV i.
A.
(47)
Noise due to electron-electron interactions in a
2-dimensional ballistic system.
Here we consider the noise generated by electronelectron interactions in a ballistic chaotic system. In such
a system, dephasing is caused by noise with momenta (δpvectors) larger than the inverse system size, L−1 . Thus
the dephasing processes are the same as those for ballistic motion in disordered systems (δp-vectors greater than
inverse mean free path). Such processes were first studied in Ref. [13] for spinless electrons, while more recently
Ref. [15] explored the full crossover from ballistic to diffusive motion for electrons with spin. The effect of a
finite Ehrenfest time on such dephasing was considered
in Ref. [24], which found that the dephasing rate in the
vicinity of the encounter has a logarithmic dependence on
the perpendicular distance between paths. This led them
to observe that the electron-electron interaction in a 2dimensional ballistic system dephases weak-localization
exponentially with the Ehrenfest time. We repeat their
derivation here and show that
wl
=
ge−e
g0wl
exp − τEcl + 21 τLT /τφ ,
1 + τD /τφ
(48)
where τLT is given by Eq. (31) with ξ equaling a thermal
length scale LT = ~vF /kB T . Ref. [24] neglected the τLT term in the exponent since it is often small.
In our qualitative derivation of this result, we treat
the electron-electron interaction as classical-noise, rather
than using the perturbative field theory approach in
Refs. [13,15,24]. Our approach is similar in spirit to those
for diffusive systems8,9 . The screened electron-electron
interaction gives a correlation function of the form13
δ ω − m−1 p · δp
2
hVδp,ω i ∝
× J(ω),
(49)
|δp|
where we assume |δp| ≪ |p|, so the energy difference
between a particle with momentum (p + δp) and p is
p · δp/m. The factor of 1/|δp| comes from the imaginary part of the screened Coulomb interaction, is due to
the polarization bubble and corresponds to the fluctuations of the electron sea at momentum and energy (δp, ω).
The δ-function ensures that energy and momentum are
conserved in the interaction between the system and the
environment. The function J(ω) gives the weight of environment modes excited at energy ω. At temperature T
−1
it is typically of the form J(ω) ≈ sinh(ω/kB T ) . It
is convenient to write δp in terms of components parallel
and perpendicular to the relevant classical path, i.e. parallel/perpendicular to p. Then,
δ ω − vF δpk
2
hVδp,ω i ∝
× J(ω),
(50)
[δp2k + δp2⊥ ]1/2
and we have
hV (rγ ′ (t′ ); t′ )V (rγ (t); t)i
Z
Z
′
′
2
=
dd δp dωei[δpδr(t ,t)+ω(t −t)]/~ hVδp,ω
i(51a)
Z
∝
dωJ(ω) exp i2ω(t′ − t)/~
#
"Z
pF
exp[iδp⊥ δr⊥ /~]
. (51b)
×Re
dδp⊥ 2
[δp⊥ + (ω/vF )2 ]1/2
~/L
Here, δr(t′ , t) = (rγ ′ (t′ ) − rγ (t)). To get the second line,
we wrote δr(t′ , t) = (δrk , δr⊥ ) in the basis parallel and
13
perpendicular to p, we then inserted Eq. (50), evaluated the δpk -integral and noted that δrk = vF (t′ − t).
Eq. (51) expresses the correlator of the Coulomb interaction along classical trajectories in the form convenient
for our semiclassical approach, Eq. (46). At first sight
the form of the interaction in Eqs. (49,50) does not appear to correspond to a classical noise-field, since it is
a function of the momentum, pγ (t), of system paths.
The correlator must however be evaluated on weak localization loops, in which case one can use the fact that
rkγ ′ (t) = rkγ (t) = vF t throughout the encounter region
to perform the Fourier transform. Thus the interaction
term in Eq. (51b) is equivalent to a classical-noise field
which is a single function of δr and t for all pγ . This
function must simply be chosen such that the integral
over δpk reduces to Eq. (51b).
The time-dependent part of the correlator is given by
the ω-integral in Eq. (51b). We assume that the temperature is high enough, kB T > ~vF /L, that the correlation time becomes shorter than the time of flight L/vF
through the cavity. Accordingly, we treat the noise as
δ-correlated in time, and set t′ = t from now on.
We next investigate the properties of the real part of
the δp⊥ -integral in Eq. (51b), giving the spatial dependence of the correlator. We write it as
"Z
#
pF δr⊥ /~
dx eix
G(δr⊥ ) = Re
, (52)
[x2 + (δr⊥ /Lω )2 ]1/2
δr⊥ /L
where Lω = ~vF /ω is the distance a system particle will
travel on the timescale that the ω-energy component of
the noise fluctuates. For the ballistic model of the ee interactions to be valid we need that Lω ≪ L, but
we assume that Lω ≫ λF . We can easily evaluate this
integral in the following regimes79 ,
for δr⊥ ≪ λF ,
ln[Lω /λF ],
G(δr⊥ ) ≃ ln[Lω /δr⊥ ],
for λF ≪ δr⊥ ≪ Lω , (53)
0,
for Lω ≪ δr⊥ ∼ L,
where we neglected all O[1]-terms (for example the result for δr⊥ ≫ Lω is actually O[Lω /δr⊥ ] . O[1]). The
crossover between these regimes is smooth. We thus conclude that Knoise (x), as defined by Eq. (46), becomes
Knoise (δr/Lω ) =
ln[Lω /δr⊥ ]
ln[Lω /λF ]
(54)
This function does not decay fast as δr⊥ grows, and thus
it cannot be treated in the manner we do elsewhere in
this article, i.e. we cannot write an equation analogous
to Eq. (26). Instead we note that the dephasing rate in
the vicinity
of the encounter (where λF ≪ δr⊥ ≪ Lω )
goes like G(0) − G(λF ≪ δr⊥ ≪ Lω ) = ln[δr⊥ /λF ].
While the dephasing rate in the loop, τφ−1 , goes like
G(0) − G(δr⊥ ∼ Lω ) ∼ ln[Lω /λF ]. Now we note that
the integral over ω is dominated by ω ≃ kB T ≪ EF , thus
we can define LT = ~vF /kB T and write ln[Lω /λF ] =
ln[LT /λF ] − ln[ω/kB T ]. We can neglect the second term
as it is much smaller than the first, and thereby replace
Lω with LT in the above formulae. In this way we reproduce the result in Ref. [24], that the dephasing rate
in the vicinity of the encounter is
τ̃φ−1 (δr⊥ ≪ LT ) = τφ−1
ln[δr⊥ /λF ]
ln[LT /λF ]
(55)
while τ̃φ−1 (δr⊥ & LT ) = τφ−1 . This is rather different from the systems considered elsewhere in this article where the dephasing rate is approximately zero for
δr⊥ < ξ and approximately constant for δr⊥ > ξ.
We now calculate the effect of such a δr⊥ -dependent
dephasing rate. We note that close to the encounter,
δr⊥ = 21 ǫLeλτ where τ is the time measured from the
encounter. We split the dephasing into two contributions. The first contribution is where δr⊥ ≥ LT (here
dephasing is time-independent, at the rate τφ−1 ), the second is where the paths have λF < δr⊥ < LT (here
dephasing is time-dependent). The boundary between
the two contributions is at τ = TT (ǫ)/2, where we define TT (ǫ) = λ−1 ln[L2T /(Lǫ)2 ]. The lower bound on
the second contribution (δr⊥ = λF ) is at time τ =
TT (ǫ)/2 − λ−1 ln[LT /λF ]. Then the exponent induced
by the dephasing is
t2 − t1 − TT (ǫ)
−2
−
τφ
= −
Z
t2 − t1 − TT (ǫ)
−
τφ
TT /2
TT /2−λ−1 ln[LT /λF ]
λ−1 ln[LT /λF ]
τφ
dτ
τ̃φ δr⊥
(56)
where to evaluate the integral we defined τ ′ = τ −
TT (ǫ)/2. The first term (which comes from the dephasing in the loop) alone would give no exponential term
in the dephasing. The integral of that term over t2 − t1
gives it a form TL (ǫ) − TT (ǫ) = τLT giving an exponent
like in Eq. (31) with ξ = LT . However we also have
the second term which gives dephasing in the vicinity of
the encounter, we can write it in terms of an Ehrenfest
time using λ−1 ln[LT /λF ] = τEcl − 21 τLT . Summing the
two terms we find that the exponential term in the dephasing goes like τ̃ /τφ , with τ̃ = τEcl + 12 τLT . This is
the result which we gave in Eq. (48) and was found in
Ref. [24] (neglecting the τLT -term).
Because λF is the scale of Friedel oscillations, one
might have expected that electron-electron interactions
give a noise with a correlation length ξ ∼ λF , which
would lead to a suppression ∝ exp[−2τEcl /τφ ] of weak localization, instead of exp[−τEcl /τφ ]. The factor of 2 difference between the correct result and this naive argument,
is due to the fact that all scales (i.e. all values of δp)
contribute to the noise induced by the electron-electron
interactions.
14
3 dimensional metal gates
00
11
11
00
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00 D
11
λF
0000000000
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00
11
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0000000000
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00000000
11111111
00000000
11111111
00
11
1111111111
0000000000
L
000000000
111111111
D
000000000
111111111
2 dimensional sample
000000000
111111111
Figure 6: Sketch of a typical chaotic quantum dot in a twodimensional electron gas (2deg). The dot is defined by metallic gates which are biased to deplete the 2deg everywhere except in the regions defining the leads and the dot. These
gates are a distance D above the 2deg (with D ≫ λgate
). At
F
finite temperature, the electrons at the surface of these metallic gates will fluctuate, leading to noise which will be felt by
the electrons in the chaotic quantum dot.
B.
Noise due to the coupling to the electrostatic
environment
Our aim here is to show that electron-electron interactions do not automatically lead to dephasing which is an
exponential function of the Ehrenfest time. It only happens when the q-integral is divergent at its upper limit,
with an upper cut-off of order pF . If the q-integral is
cut-off by some other length scale, then the dephasing
of weak-localization will be independent of the Ehrenfest
time.
The example we consider is noise in a two-dimensional
system due to electron-electron interactions between the
system and the gates. A typical experimental set-up is
sketched in Fig. 6. The gates are bulk metal, the electrons in the system will feel fluctuations of electrons at
the surface of the gates. One can expect that these fluctuations at the surface of the gate are sufficiently wellconfined to two-dimensions (by screening in the bulk of
the gate), that they will cause a noise field in the system
of a form similar to Eq. (49). However the fact that the
distance between the gates and the chaotic system is D
means that the natural upper cut-off on the q-integral will
be ~/D and not pF . Assuming D ≪ L, Lω , we replace pF
by ~/D throughout the derivation in Section III A, and
find that
wl
ggate
e−e
g0wl
=
exp[−τξ /τφ ],
1 + τD /τφ
1/2
−1
(57)
2
with ξ = (LT D)
and hence τξ = λ ln[L /(LT D)].
The dephasing rate here is τφ−1 ∝ D−1 . In systems in
which dephasing is dominated by thermal system-gate
interactions, we therefore expect a dephasing that is τE independent and algebraic in τφ for D ∼ L.
In real experiments the gates are typically much more
disordered than the chaotic system, thus we can easily
have a situation in which the thermally excited modes
in the gates are diffusive – or even localized by a charge
trap at the edge of a gate – in which case their noise
field will be similar to that in Ref. [8,9]. This modifies
the integrand of the q-integral, however the upper cutoff will still be ~/D, therefore dephasing will again be
τE -independent.
In general, dephasing is due to a combination of e-e interactions within the system and the Coulomb coupling
between the system and external charge distributions in
gates and other reservoirs of charges. Which of these
sources of dephasing dominates is determined by microscopic details which we do not discuss here, in particular the temperatures and mean-free-paths of both system
and gates13 .
IV.
DEPHASING LEAD MODEL
In its simplest formulation the dephasing lead model
consists of adding a fictitious lead 3 to the cavity. This
is illustrated in Fig. 7. Contrary to the two real leads
L,R, the potential voltage on lead 3 is tuned such that
the net current through it is zero. Every electron that
leaves through lead 3 is replaced by one with an unrelated
phase, leading to a loss of phase information without loss
of current.
In this situation the conductance from L to R is given
by16
g = TRL +
TR3 T3L
,
T3L + T3R
(58)
where Tnm is the conductance from lead m to lead n
in the absence of a voltage on lead 3. We separate the
Drude and weak-localization parts of Tnm ,
D
+ δTnm + O[NT−1 ],
Tnm = Tnm
(59)
D
where the Drude contribution, Tnm
, is O[NT ] and the
weak-localization contribution, δTnm , is O[NT0 ] and NT =
NL + NR + N3 is the total number of channels in this
three terminal geometry. We expand g for large NT
and collect all O[NT ]-terms (Drude contributions) and
all O[NT0 ]-terms (weak-localization contributions) to get
g = g D + g wl with
D
+
g D = TRL
D D
TR3
T3L
,
D
D
T3L + T3R
g wl = δTRL +
(60a)
D 2
D 2
(TR3
) δT3L + (T3L
) δT3R
. (60b)
D
D
(T3R + T3L )2
These equations form the basis of our semiclassical
derivation of weak localization in the dephasing lead
model. We first consider the case of a dephasing lead
perfectly coupled to the cavity, and then move on to consider a dephasing lead with a tunnel barrier of transparency ρ < 1. We finally discuss multiple dephasing
leads.
15
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Fermi wavelength appears in place of ξ, so the time scale
in the additional exponential suppression is the Ehrenfest
time, τEcl .
lead R
ei φ
lead L
2.
lead 3 (dephasing)
Figure 7: Schematic of the dephasing-lead model. The system
is a open quantum dot with an extra lead (lead 3), whose
voltage is chosen to make zero current flow (on average) in
that lead. This extra lead thereby causes dephasing without
loss of particles.
A.
In the absence of tunnel barrier, we can go further
and calculate conductance fluctuations at almost no extra cost. From Eqs. (58) and (59), we get the following
expression for the variance of the conductance to order
O[NT0 ]
var g = var TRL
D 4
D 4
(T3R
) var T3L + (T3L
) var T3R
+
D + T D )4
(T3R
3L
Dephasing lead without tunnel-barrier
1.
Nn Nm
,
NT
Nn Nm
= −
exp[−τEcl /τ̃D ]
NT2
D
Tnm
=
D
D 2
(T3R
T3L
) covar (T3L , T3R )
D + T D )4
(T3R
3L
+ 2
D 2
(T3L
) covar(TRL , T3R )
D + T D )2
(T3R
3L
+ 2
D 2
(T3R
) covar(TRL , T3L )
.
D + T D )2
(T3R
3L
(63)
τφ−1
(64)
−1
= (τ0 L)
W3 .
We have τ̃D−1 = τD−1 + τφ−1 , and from this we find the
Drude conductance and weak-localization correction,
g D = g0D ,
(65)
g0wl
1 + τD /τφ
exp[−τEcl /τφ ].
varTij =
(62)
τD−1 = (τ0 L)−1 (WL + WR ),
(66)
Here g0D and g0wl are the results for a two lead cavity in
the absence of dephasing, Eqs. (17) and (32)
The weak-localization correction with a dephasing lead
has a similar structure to that with a real environment.
However here the time scale involved in the additional
exponential suppression contains no independent parameter analogous to ξ. We could have expected that the
width of the dephasing lead would play a role similar to
ξ. However this turns out not to be the case, instead the
(67)
In the universal regime, Ref. [10] gives, to order O[NT0 ]
in the presence of time-reversal symmetry,
(61)
where τ̃D−1 = (τ0 L)−1 (WL + WR + W3 ), in terms of τ0 ,
the time of flight across the system. We substitute these
results into Eq. (60a) and Eq. (60b), and write the answer
in terms of the dwell time in the two lead (L and R)
geometry, τD , and the dephasing rate, τφ−1 , which we
define as the decay rate to lead 3,
g wl =
+ 2
Weak-localization
The Drude conductance and weak-localization correction from lead m to lead n in a three-lead cavity are
δTnm
Universal conductance fluctuations
covar (Tij , Tik ) =
Ni2 Nj2
,
NT4
(68)
Ni2 Nj Nk
.
NT4
(69)
Ref.[53] showed that Eq. (68) remains valid even at finite
τE /τD . Inspection of their calculation for varTij shows
that the same conclusion also applies to covar (Tij , Tik ),
and thus Eq. (69) still holds independently of τE /τD . Together with Eq. (63), this straightforwardly leads to
var g =
1
NR2 NL2
.
(NR + NL )4 (1 + τD /τφ )2
(70)
In the universal regime, this result was previously derived
in Ref. [11]. We thus conclude that, in the dephasing lead
model without tunnel-barrier, conductance fluctuations
exhibit the universal behavior of Eq. (70). Below, we
confirm this result numerically.
B.
Dephasing lead with tunnel-barrier
Putting a tunnel-barrier on the dephasing lead 3 is attractive because one can avoid the local character of the
dephasing lead model by considering a wide third lead
with an almost opaque barrier11. Additionally, this is
the model studied numerically in Ref. [23] in the context of conductance fluctuations. Weak-localization and
shot noise in this model have been considered within the
16
trajectory-based semiclassical approach in Ref. [56], and
here we only mention the main results.
According to Ref. [56], when all leads are connected to
the cavity via tunnel-barriers, with the barrier on lead m
having transparency ρm ∈ [0, 1], the Drude conductance
D
between lead m and n, Tnm
, and the weak-localization
correction, δTnm , are
D
Tnm
= ρn ρm Nn Nm /N,
(71)
!
Ñ
Nn Nm
ρn + ρm −
2
N
N
op
cl
× exp −τE /τD2 − (τE − τEop )/τD1 . (72)
P
−1
−1
Here, τP
=
(τ0 L)−1 n ρn Wn and τD2
=
D1
−1
ρ
(2
−
ρ
)W
are
the
single
path
and
(τ0 L)
n
n
n
n
the paired paths survival
P Wn is the
P times respectively,
width of lead n, N = k ρk Nk and Ñ = k ρ2k Nk .
Now we assume that ρL = ρR = 1 so only the dephasing lead has a tunnel barrier. Substituting the Drude
and weak-localization contributions into Eq. (60b) we
find that
δTnm = ρn ρm
g
wl
g0wl
=
exp −(1 − ρ)τEop /τφ − τEcl /τφ . (73)
1 + τD1 /τφ
The argument in the exponent in Eq. (73) has a simple
physical meaning. It is the probability that a path survives throughout the paired-region (τEop /2 on either side
of the encounter) without escaping into lead 3, multiplied
by the probability to survive the extra time (τEcl − τEop )
unpaired without escaping into lead 3 (to close a loop of
length τEcl ). The first probability is exp[−(2 − ρ)τEop /τφ ]
while the second is exp[−(τEcl − τEop )/τφ ].
We note that if we consider a nearly opaque barrier,
the relevant time scale involved in the exponent is τEcl +
τEop ≃ 2τEcl . Thus by tuning the opacity of the barrier, we
can vary the exponential contribution to dephasing from
exp[−τEcl /τφ ] to exp[−2τEcl /τφ ], but we cannot remove the
exponent. In particular, we cannot mimic dephasing due
to a real environment with ξ ∼ L, Eq. (31), since it has
only a power-law dephasing.
There is to date no theory for conductance fluctuations at finite Ehrenfest time in the presence of tunnel
barriers, and constructing such a theory would require a
formidable theoretical endeavor. Numerically, Ref. [23]
observed that, in contrast to the dephasing lead model
without tunnel-barrier, conductance fluctuations in the
presence of a dephasing lead with a tunnel barrier are
exponentially damped ∝ exp[−τE /τφ ].
C.
Multiple dephasing leads
The n probe dephasing model consists of adding n fictitious leads to the cavity (labeled {3, · · · , n + 2}) in addition to lead L, R. The voltage on each supplementary
lead is tuned so that the current it carries is zero. Without loss of generality we defined VR = 0, then we get the
set of equations
IR = TRL VL + TTR V
0 = I = −Tsub V + TL VL
(74a)
(74b)
where the superscript-T indicates the transpose. The
column-vectors I, V have an ith element given by the
current or voltage (respectively) for the dephasing lead
i ∈ {3, n + 2}. The column-vectors TL and TR have an
ith element given by TLi and TRi , respectively. Finally
Tsub has matrix elements given by
[Tsub ]ij = Ni δij − Tij
hP
i
=
δij − Tij (1 − δij ) ,
T
kj
k6=j
(75)
(76)
g = TLR + TTL T−1
sub TR .
(77)
where again i, j ∈ {3, n + 2}. Substituting V from
Eq. (74b) into Eq. (74a) and using IR = gVL gives us
the conductance from L to R as
Thus finding g requires the inversion of Tsub . This is
cumbersome, so instead we present a simple argument to
extract only the information we are interested in – the
nature of the exponential in the dephasing.
We argue that whatever the formula for conductance
for n dephasing leads is, we can expand it in powers of N
and collect the O[N 0 ]-terms to get a formula for weaklocalization of the form
g wl = δTLR +
n+2
X
j=3
Aj δTLj + Bj δTjR +
n+2
X
Cij δTij .(78)
i,j=3
Here, the sum is over all dephasing leads. To get the prefactors Aj , Bj , Cij we would have solved the full problem
by inverting Tsub , however we can already see that, to
leading order, they are combinations of Drude conductances and thus independent of the Ehrenfest time. In
contrast all the weak-localization contributions contain
an exponential of the form [τD1 and τD2 are defined below Eq. (72)]
exp[−τEop /τD2 + (τEcl − τEop )/τD1 ] .
(79)
Thus defining τφ−1 as the rate of escaping into any of the
Pn
dephasing leads, so τφ−1 = (τ0 L)−1 j=3 ρj Wj , we see
that g wl decays with an exponential
g wl ∝ exp[−(1 − ρ̃)τEop /τφ − τEcl /τφ ],
(80)
P
where we define ρ̃ such that ρ̃τφ−1 = (τ0 L)−1 j ρ2j Wj .
We have just shown that multiple dephasing leads
cause an exponential suppression of the weak-localization
which is qualitatively similar to that caused by a single dephasing lead. The exponent is proportional to the
Ehrenfest time, and contains no independent parameter
analogous to ξ.
17
V.
NUMERICAL SIMULATIONS
A.
Open kicked rotators
Because of the slow, logarithmic increase of τE with
the size M of the Hilbert space, the ergodic semiclassical
regime τE & τD , λτD ≫ 1 is unattainable by standard
numerical methods. We therefore follow Refs. [23,44,48,
51,52,80] and consider the open kicked rotator model.
1.
Dephasing lead model
where |γl |2 = ρl ∈ [0, 1] gives the transparency
of the
S
contact to the external channel l, and i {li } denotes
the ensemble of cavity modes coupled to the external
leads. Below we focus on perfectly transparent contacts,
ρl = γl = 1, ∀l.
The conductance (60a) in the dephasing lead model is
obtained from Eq. (83) with n + 2 = 3 leads. The transport leads carry N = NL = NR channels, which defines
the dwell time through the system as τD /τ0 = M/2N .
The dephasing time τφ /τ0 = M/N3 is defined by the
number N3 of channels carried by the third, dephasing
lead, and the Ehrenfest time is given by
τEcl = λ−1 [ln M + O(1)] .
The system is described by the time-dependent Hamiltonian
H=
X
(p − p0 )2
+ K cos(x − x0 )
δ(t − nτ0 ).
2
n
(81)
The kicking strength K drives the dynamics from integrable (K = 0) to fully chaotic (K & 7). The Lyapunov
exponent in the classical version of the kicked rotator is
given by λτ0 ≈ ln(K/2). In most quantum simulations,
however, one observes an effective Lyapunov exponent
λeff instead that is systematically smaller than λ by as
much as 30%81 . Two parameters, p0 and x0 , are introduced to break two parities and drive the crossover from
the β = 1 to the β = 2 universality class37 , corresponding
to breaking the time reversal symmetry.
We quantize the Hamiltonian of Eq. (81) on the torus
x, p ∈ [0, 2π], by discretizing the momentum coordinates
as pl = 2πl/M , l = 1, . . . M . A quantum representation
of the Hamiltonian of Eq. (81) is provided by the unitary
M × M Floquet operator U , which gives the time evolution for one iteration of the map. For our specific choice
of the kicked rotator, the Floquet operator has matrix
elements
2
′
2
(82)
Ul,l′ = M −1/2 e−(πi/2M)[(l−l0 ) +(l −l0 ) ]
X
2πim(l−l′ )/M −(iMK/2π) cos[2π(m−m0 )/M]
×
e
e
,
m
with l0 = p0 M/2π and m0 = x0 M/2π. The Hilbert space
size M is given by the ratio L/λF of the linear system
size to the Fermi wavelength.
To investigate transport, we open the system by defining n + 2 ≥ 2 contacts to leads via absorbing phase-space
strips P
[li − Ni /2, li + Ni /2 − 1], i = 1, 2 . . . n + 2. With
NT = i Ni , we construct a NT × NT scattering matrix
from the Floquet operator U as82
h
i−1
p
p
S(ε) = I − PP† −P e−iε I−U I − P† P
U P† . (83)
The NT × M projection matrix P, which describes the
coupling to the leads, has matrix elements
(
S
γl δlm if l ∈ i {l(i) },
Pl,m =
(84)
0
otherwise,
2.
(85)
The open kicked rotator coupled to an environment
We extend the kicked rotator model to account for the
coupling to external degrees of freedom. The exponential
increase of memory size with number of particles forces us
to focus on an environment modeled by a single chaotic
particle. Therefore, we follow Ref. [68] and consider two
coupled kicked rotators (with i = sys, env)
H = Hsys + Henv + U,
X
(pi − p0 )2
+ Ki cos(xi − x0 )
δ(t − nτ0 ),
Hi =
2
n
X
U = ε sin(xsys − xenv − 0.33)
δ(t − nτ0 ).
(86)
n
In this model, the interaction potential U acts at the
same time as the kicks, which facilitates the construction
of the S-matrix. For this particular choice of interaction, the correlation length ξ = L so that one expects
a universal behavior of dephasing, as in Eq. (1). The
quantum representation of the coupled Hamiltonian is
a unitary (Msys Menv ) × (Msys Menv ) Floquet operator.
We open the system (and not the environment) to two
external leads by means of extended projectors Ptot =
P (L) ⊗ Ienv + P (R) ⊗ Ienv . A straightforward generalization of Eq. (83) defines a (NT Menv )×(NT Menv ) extended
scattering matrix, from which we evaluate the conductance via Eq. (A11). We focus on the symmetric situation
where the two leads carry the same number N of channels
and average our data over a set of pure but random initial environment density matrices ηenv (q, q′ ; t = 0). We
estimate the dephasing time from the ε-induced broadening of two-particle levels in the corresponding closed
two-particle kicked rotator, τφ−1 = 0.43(ε/~eff )2 , with
~eff = 2π/M , the effective Planck constant68 .
B.
Weak-localization with dephasing
To investigate weak-localization, we follow the same
procedure as in Ref. [83] of taking a constant nonzero p0 ,
δg(2.4) − δg(0)
18
a)
0.2
0.1
∆g
0.1
0.01
0.0
0.001
0.001
0.01
0.1
τD/τφ
1
10
b)
0.2
Figure 9: (Color online) Amplitude δg(m0 /mc = 2.4) −
δg(m0 /mc = 0) of the weak localization correction to the conductance as a function of τD /τφ for the double kicked rotator
model (circles) and the open kicked rotator with transparent
dephasing lead (squares). The black line gives the universal
algebraic behavior (1+τD /τφ )−1 /4, and the red line is a guide
to the eye, including an exponential decay with τEcl = 2.78, on
top of the universal decay.
∆g
0.1
0.0
0.0
1.0
2.0
m0 /mc
Figure 8: a) Magnetoconductance curves ∆g(x0 ) = g(x0 ) −
g(0) for the double kicked rotator model (see text) with
Ksys = Kenv = 34.08 (λeff ≈ 2), τD /τ0 = 8, ξ/L = 1
and Hilbert space sizes Msys = 256, Menv = 16. Different
symbols correspond to different dephasing times τφ /τD = ∞
−1
(~−1
eff ε = 0, circles), τφ /τD = 4.8 (~ eff ε ≃ 0.25, squares),
−1
τφ /τD = 1.2 (~eff ε ≃ 0.5, diamonds), τφ /τD = 0.3 (~−1
eff ε ≃ 1,
downward triangles) and τφ /τD = 0.07 (~−1
eff ε ≃ 2, upward
triangles). Data are averaged over 25 different lead positions,
each with 25 different quasi-energies and 10 different initial
environment states. b) Magnetoconductance curves for the
open kicked rotator with transparent dephasing lead (see text)
with K = 34.08, τD /τ0 = 8 and Hilbert space size M = 256.
Different symbols correspond to different dephasing times
τφ /τD = ∞ (circles), τφ /τD = 5 (squares), τφ /τD = 1.25 (diamonds), τφ /τD = 0.5 (downward triangles) and τφ /τD = 0.25
(upward triangles). Data are averaged over 225 different lead
positions, each with 50 different quasi-energies.
while varying x0 (which hence plays the role of a magnetic
field). The obtained magnetoconductance in the absence
of dephasing is Lorentzian45,83 ,
g wl (Φ) =
g wl (0)
,
1 + (m0 /mc )2
(87)
√
with mc = 4π/K M τD . In Fig. 8 we compare the suppression of weak-localization for the system-environment
kicked rotator [panel a), top] and the dephasing lead
kicked rotator [panel b), bottom]. For both models, we
show five magnetoconductance curves, corresponding to
five different ratios τφ /τD . All curves exhibit the expected Lorentzian behavior vs. m0 /mc , however, the amplitude of the magnetoconductance is reduced as τφ /τD
is reduced. This allows one to extract the τφ -dependence
of g wl . For the system-environment model, we found
no significant departure from our analytical prediction,
Eq. (31) with τξ = 0 (since the interaction in Eq. (86)
has ξ = L). The same behavior is observed for the dephasing lead model, as long as τφ /τD is large, however,
compared to the system-environment kicked rotator, the
damping of magnetoconductance accelerates as τφ /τE becomes comparable to or smaller than one. When this
regime is reached, magnetoconductance curves for the
dephasing lead model lies significantly below those of the
system-environment model, even when the latter corresponds to shorter dephasing times (compare in particular
the upward triangles in both panels of Fig. 8). This behavior is further illustrated in Fig. 9, where we plot the
amplitude δg(m0 /mc = 2.4) − δg(m0/mc ) of the weak localization corrections to the conductance as a function of
τD /τφ . The data for the double kicked rotator nicely line
up on the universal algebraic behavior (1 + τD /τφ )−1 /4
without any fitting parameter. This is clearly not the
case for the dephasing lead model, where an additional,
exponential dependence on τφ emerges. We attribute
this to the exponential damping factor ∝ exp[−τE /τφ ]
of Eq. (66). The data presented in Figs. 8 and 9 for
τφ−1 = 0 exhibit a weak dependence on exp[−τEcl /τD ] only,
with τEcl . 0.8, which we attribute to terms of order one
in Eq. (85).
All collected data (including some that we do not
19
present here) thus confirm qualitatively – if not quantitatively – the validity of Eq. (66) for the dephasing lead
model.
C.
Conductance fluctuations in the dephasing lead
model
Conductance fluctuations were studied numerically in
Ref. [23] for the dephasing lead model with a tunnel barrier of low transparency, and an exponential damping
var(g) ∝ exp[−2τE /τφ ] was reported. Instead, here we
consider a model in which the dephasing lead is transparently coupled to the system. In Fig. 10, we show
data for var(g) against τφ /τD , which is varied only by
varying the width of the third, dephasing lead. There
are four data sets (empty symbols) corresponding to a
fixed classical configuration at different stages in the
quantum-classical crossover, i.e. with increasing ratio
M = L/λF ∈ [128, 8192]. As M increases, so does
τE /τφ , however no change of behavior of var(g) is observed. These data are compared to a fifth set obtained
in the universal regime, τE /τD ≪ 1, and the universal
prediction of Eq. (70) (dashed line). These data confirm
our analytical result, Eq. (70), that var(g) exhibits no
Ehrenfest time dependence for the dephasing lead model
with perfectly transparent contacts.
VI.
CONCLUSIONS
We have investigated the dephasing properties of open
quantum chaotic system, focusing on the deep semiclassical limit where the Ehrenfest time is comparable to
or larger than the dwell time through the system. We
treated three models of dephasing. In the first one, the
transport system is capacitively coupled to an external
quantum chaotic system. For that model, we developed
a new scattering formalism, based on an extended scattering matrix S, including the degrees of freedom of the
environment. Transport properties are extracted from S,
once the environment has been traced out properly. In
that model, we find that, in addition to the universal algebraic suppression g wl ∝ (1 + τD /τφ )−1 with the dwell
time τD through the cavity and the dephasing rate τφ−1 ,
weak-localization is exponentially suppressed by a factor
∝ exp[−τξ /τφ ], with a new time scale τξ depending on
the correlation length of the coupling potential between
the system and the environment.
The second model we treated is that of dephasing due
to a classical noise field. We show that the new time scale
τξ plays the same role here as in the system-environment
model. We then consider a classical Johnson-Nyquist
noise model of electron-electron interactions. We show
that ξ ∼ λF (and so τξ equals the Ehrenfest time) when
dephasing is dominated by electron-electron interactions
within the system, but that ξ ∼ D when dephasing is
0.1
δg
2
0.05
0
0
1
τD/τφ
2
3
Figure 10: Variance of the conductance vs. τD /τφ for the open
kicked rotator with K = 14. and τD /τ0 = 5 (empty symbols),
transparently coupled to a dephasing lead. Different symbols
correspond to different Hilbert space sizes (and hence different
τEcl ) M = 128 (squares, τEcl /τD = 0.6), M = 512 (diamonds,
τEcl /τD = 0.75), M = 2048 (upward triangles, τEcl /τD = 0.9)
and M = 8192 (downward triangles, τEcl /τD = 1.1). Additional data for K = 144., τD = 25 and M = 2048 are also
shown (full circles, τEcl /τD = 0.08). The dashed line shows
the universal behavior of Eq. (2). Unlike for weak-localization
(see Fig. 8) and for the dephasing lead model with partial
transparency48 , the behavior of δg 2 remains universal and
shows no noticeable dependence on τEcl /τD . Data are averaged
over 50 different quasi-energies and from 50 (for N = 8192)
to 500 (for N = 128 and 512) different lead positions.
dominated by interactions between electrons in the system and those in a gate, a distance D away.
The third model we treated is the dephasing lead
model. We found a similar exponential suppression of
weak-localization. To our surprise, however, it is the
Fermi wavelength, not the dephasing-lead’s width, which
plays a role similar to ξ in that model. This inequivalence
between the dephasing lead model and dephasing due to
a real environment or classical noise can be most clearly
seen in a situation where the interaction with a real environment has ξ ≃ L. Then the environment induces
only power-law dephasing, which is impossible to mimic
with a dephasing lead. For the dephasing lead model,
we also showed analytically and numerically that conductance fluctuations exhibit only power-law dephasing
if the connection between the cavity and the dephasing
lead is perfectly transparent. This is to be contrasted
with the exponential dephasing observed for the dephasing lead model with tunnel barrier, and reflects the fact
that the presence of tunnel barriers violates a sum rule
otherwise preserving the universality of conductance fluctuations vs. τE /τD 53 .
Related results have been obtained for conductance
fluctuations in Ref. [24], where different behaviors have
been predicted for external sources – which give simi-
20
lar dephasing as our environment model – and internal
source of dephasing – which qualitatively reproduce the
prediction of Ref. [14].
ACKNOWLEDGMENTS
working on this project, C. Petitjean was supported
by the Swiss National Science Foundation, which also
funded the visits to Geneva of P.Jacquod and R.Whitney.
P. Jacquod expresses his gratitude to M. Büttiker and the
Department of Theoretical Physics at the University of
Geneva for their hospitality. This work was initiated in
the summer of 2006 at the Aspen Center for Physics.
We thank I. Aleiner, P. Brouwer, M. Büttiker and M.
Polianski for useful and stimulating discussions. While
Appendix A: SCATTERING APPROACH TO TRANSPORT IN THE PRESENCE OF AN
ENVIRONMENT
Here we extend the scattering approach to transport to account for those environmental degrees of freedom which
couple to the system being studied. We follow the lines of the derivation of the expression for noise presented in
Ref. [84], focusing on the two-terminal configuration. The following derivation is valid in the limit of pure dephasing,
when there is no energy/momentum exchange between system and environment.
1.
Current operator and conductance
In the presence of an environment, the current operator at time t on a cross-section deep inside of lead α = L, R
(where there is no system-environment interaction) reads
Z
i
Xh
′
e
ˆ
dE dE ′ ei(E−E )t/~
â†αn (E ′ )âαn (E) − b̂†αn (E ′ )b̂αn (E) × Ienv .
(A1)
Iα (t) =
h
n∈α
The second quantized operators â(†) and b̂(†) create and destroy incoming and outgoing system particles respectively.
Since the environment particles carry no current, the current operator acts as the identity operator Ienv in the
environment sub-space. We could write Ienv in terms of second quantized operators, however for our present purpose,
it is more convenient to write it as
Z
Ienv =
dq|qihq|.
(A2)
As in Ref. [84] we now back evolve the outgoing-states into incoming-states, this time with an S-matrix which also
depends on the coordinates of the environment,
XZ
b̂αn (E)hq| =
dq0 Sαβ;nj (q, q0 ; E, Ẽ) âβj (Ẽ) hq0 |.
(A3)
β;j
Here, Sαβ;nj (q, q0 ; E, Ẽ) gives the transmission amplitude from channel j in lead β with energy Ẽ to channel n in
lead α with energy E, while simultaneously, the environment evolves from q0 to q. We can set Ẽ = E, because
throughout this article we only consider the regime of pure dephasing. Using (A3) we rewrite the current operator as
"
Z
Z
X
′
e
′
i(E−E
)t/~
Iˆα (t) =
dq dE dE e
â†αn (E ′ ) âαn (E) hΨenv |qihq|Ψenv i
h
n
#
Z
†
XX
†
′
′
′
′
− dq0 dq0
Sαγ;nj (q, q0 ) Sαβ;nk (q, q0 ) âγj (E ) âβk (E) hΨenv |q0 ihq0 |Ψenv i . (A4)
γ,β jk
We next rewrite the first line of Eq. (A4) as
Z
X
dq
â†αn (E ′ )âαn (E)hΨenv |qihq|Ψenv i
n
=
Z
dq0 dq′0
XX
γ,β jk
δ(q′0 − q0 )δγα δβα δjk â†γj (E ′ )âβk (E)hΨenv |q′0 ihq0 |Ψenv i.
(A5)
21
Finally, we write the current operator in terms of the initial environment density-matrix ηenv (q0 , q′0 ) =
hq0 |Ψenv ihΨenv |q′0 i,
Z
(A6)
Iˆα (t) =
dq′0 dq0 Iˆα(red) (q′0 , q0 ; t)ηenv (q0 , q′0 ),
where we defined the reduced current operator as
Z
Z
X X jk
′
e
dq dE dE ′ ei(E−E )t/~
Iˆα(red) (q′0 , q0 ; t) =
Bγβ (α, E, E ′ ; q′0 , q0 )â†γj (E ′ ) âβk (E),
h
γ,β j,k
Z
†
X
jk
Sαγ;nj (q, q′0 ) Sαβ;nk (q, q0 ).
Bγβ
(α, E, E ′ ; q′0 , q0 ) = δ(q′0 − q0 )δγα δβα δjk − dq
(A7a)
(A7b)
n
The current is obtained by taking the expectation value of the current operator over the system, using
hhâ†γj (E ′ ) âβk (E)ii = δγβ δjk δ(E − E ′ ) fβ (E)
(A8)
where fβ is the Fermi function in lead β. The unitarity of S implies,
Z
†
X
dq
Sδγ;nj (q, q′0 ) Sδβ;nk (q, q0 ) = δ(q′0 − q0 )δβγ δjk .
(A9)
δ,n
We use this latter equality to rewrite the Kronecker δ’s in Eq. (A8). Finally the current in the left lead is
Z
Z
†
h
i
eX
dq′0 dq0 dq dE SLR;nk (q, q′0 ) SLR;nk (q, q0 ) fL (E) − fR (E) ηenv (q0 , q′0 ).
hhIL ii =
h
(A10)
n,k
In the limit of zero temperature in the leads, and assuming that the scattering matrix is not too strongly energy
dependent, Eq. (A10) leads to the linear conductance
Z
†
e2 X
dq′0 dq0 dq SLR;nk (q, q′0 ) SLR;kn (q, q0 )ηenv (q0 , q′0 ),
(A11)
G=
h
n,k
with scattering matrices to be evaluated at the Fermi energy. From Eqs (A10) and (A11), we see that both current
and conductance are obtained by tracing over the environmental degrees of freedom of the square of the extended
scattering matrix. Besides this prescription, these two equations are extremely similar to their counterpart in the
standard scattering approach to transport. We also note that conductance fluctuations can be obtained by squaring
Eqs. (A10) and (A11).
It is legitimate to expect that a complex environment – such as the chaotic system considered in this article –
has a complicated initial wavefunction, which, under ensemble averaging, is uncorrelated with itself on all scales
greater than the environment wavelength. This justifies us treating the initial environment state as hηenv (q0 , q′0 )i ∼
(2π~)N d δ(q0 − q′0 )/Ξenv .
2.
Current noise
We follow similar steps as in the previous Section to calculate the zero-frequency current noise. However now we
have two current operators, and hence two creation and two Rannihilation operators for the system84 . The environment
has two Ienv operators, each of which we write in the form dq|qihq| to get the current time-correlator as
*
*Z
+
+
(red)
Iˆβ (t)Iˆα (0) =
dq03 dq01 dq′01 Iˆ
(q′01 , q03 ; t) Iˆα(red) (q03 , q01 ; 0) ηenv (q01 , q′01 ) .
(A12)
β
(red)
The reduced current operator Iˆβ (q′ , q; t) is given in Eq. (A7a). The zero-frequency noise power is obtained from
the product of the deviations from the average current at times 0 and t. Consequently it is proportional to
Z
Z
DD
EE
dt dq03 dq01 dq′01
Iˆ(red) (q′01 , q03 ; t) − hhIˆ(red) (q′01 , q03 ; t)ii Iˆ(red) (q03 , q01 ; 0) − hhIˆ(red) (q03 , q01 ; 0)ii
×ηenv (q01 , q′01 )
(A13)
22
There is only one trace over the environment here because we assume we measure the current as a function of time
in a given experiment (with a given initial ηenv ), and then extract the average current (and the deviations from it)
from that data set.
We need to take the following expectation value of products of creation/annihilation operators over the system84
â†βm (E2 ) âγn (E1 )â†β ′ m′ (E4 ) âγ ′ n′ (E3 ) −
â†βm (E2 ) âγn (E1 )
â†β ′ m′ (E4 ) âγ ′ n′ (E3 )
=
δβγ ′ δγβ ′ δmn′ δnm′ δ(E2 − E3 ) δ(E1 − E4 )fα (E2 )[1 ∓ fβ (E1 )],
(A14)
where the minus (plus) sign stands for fermions (bosons). From this we finally get the zero-frequency noise power
Z
Z
XX
e2
nm
mn
Sαα′ (0) =
dE
dq03 dq01 dq′01
Bβγ
(α, E, E; q′01 , q03 )Bγβ
(α′ , E, E; q03 , q01 )
h
β,γ m,n
× fβ (E)[1 ∓ fγ (E)] + [1 ∓ fα (E)]fβ (E) ηenv (q01 , q′01 ).
(A15)
All the relevant information for shot noise is contained in the diagonal Sαα , α = L, R. Shot noise is obtained by
calculating this latter expression in the limit of low temperature but finite voltage bias V between the two leads. In
that case, it is easily checked that the contribution to B arising from the first term on the right-hand side of Eq. (A7b)
does not contribute, and one gets
Z
†
XZ
X X
e2
dE
dq3 dq1 dq03 dq01 dq′01
Sαβ;m′ m (q3 , q′01 ; E) Sαγ;m′ n (q3 , q03 ; E)
Sαα (0) =
h
m,n m′ ,n′
β6=γ
†
× Sαγ;n′ n (q1 , q03 ; E) Sαβ;n′ m (q1 , q01 ; E) fβ (E)[1 ∓ fγ (E)] + [1 ∓ fα (E)]fβ (E) ηenv (q01 , q′01 ).(A16)
We finally assume a slow dependence of S on E, in which case the integral over the energy is easily performed, giving
a factor eV . For a two-lead device (L,R), we find that
Z
†
X
X
2e3 V
SRR (0) =
dq3 dq1 dq03 dq01 dq′01
SRL;m′ m (q3 , q′01 ; E) SRR;m′ n (q3 , q03 ; E)
h
m∈L n,m′ ,n′ ∈R
†
(A17)
× SRR;n′ n (q1 , q03 ; E) SRL;n′ m (q1 , q01 ; E) ηenv (q01 , q′01 ).
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