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arXiv:0910.5926v1 [cond-mat.mes-hall] 30 Oct 2009
Transport through open quantum dots: making semiclassics quantitative
Iva Bˇrezinov´a,1, ∗Ludger Wirtz,2Stefan Rotter,1Christoph Stampfer,3and Joachim Burgd¨orfer1
1Institute for Theoretical Physics, Vienna University of Technology,
Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria, EU
2Institute for Electronics, Microelectronics, and Nanotechnology (IEMN), Dept. ISEN,
CNRS-UMR 8520, B.P. 60069, 59652 Vil leneuve d’Ascq Cedex, France, EU
3JARA-FIT and II. Institute of Physics, RWTH Aachen, 52074 Aachen, Germany, EU
(Dated: October 30, 2009)
We investigate electron transport through clean open quantum dots (“quantum billiards”). We
present a semiclassical theory that allows to accurately reproduce quantum transport calculations.
Quantitative agreement is reached for individual energy and magnetic field dependent elements of
the scattering matrix. Two key ingredients are essential: (i) inclusion of pseudo-paths which have
the topology of linked classical paths resulting from diffraction in addition to classical paths and
(ii) a high-level approximation to diffractive scattering. Within this framework of the pseudo-path
semiclassical approximation (PSCA), typical shortcomings of semiclassical theories such as violation
of the anti-correlation between reflection and transmission and the overestimation of conductance
fluctuations are overcome. Beyond its predictive capabilities the PSCA provides deeper insights into
the quantum-to-classical crossover.
PACS numbers:
I. INTRODUCTION
The ability to controllably fabricate, manipulate
and examine structures on the sub-micrometer scale
has let to the observation of quantum phenomena
in electron transport such as, e.g., universal conduc-
tance fluctuations (UCF) in chaotic billiards and weak
localization (WL), which dominate transport at the
nanoscale.1,2By reducing the characteristic system size
below the electronic inelastic mean free path, transport
enters the so-called ballistic regime.3Ballistic electron
transport is a prime candidate for semiclassical descrip-
tions4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26
where the classical trajectories carry an amplitude which
reflects the stability of the classical orbits and a phase
which contains the classical action and accounts for
quantum interference.27 On a more fundamental level,
the semiclassical framework provides a conceptually
powerful bridge between classical and quantum mechan-
ics allowing an intuitive approach to quantum mechanics
and quantum chaos in general, and to transport through
open quantum dots or so-called quantum billiards in
particular.4,5,6
Several semiclassical approximations (SCAs) based
on the approximation of the constant-energy Green’s
function for propagation in a billiard have been proposed
and compared with numerical quantum transport cal-
culations or experiment.8,9,10,11,12,13,14,15,16,17,18,19,28,29
While many qualitative features could be well repro-
duced, quantitative agreement on a system-specific level
has remained a challenge.
One underlying difficulty is the multi-scale nature of the
quantum-to-classical transition for transport through
open quantum dots. For the semiclassical approxi-
mation to hold, the de Broglie wavelength λshould
be vanishingly small compared to all characteristic
dimensions of the device. Such asymptotic theories
have been successfully employed to reproduce, upon
ensemble averaging, random matrix theory (RMT) re-
sults for chaotic cavities (see e.g. Ref. 7,20,21,22,23). A
quantitative comparison on a system-specific level with
full quantum calculations or experiments is, however,
only possible in the non-asymptotic regime where λis
small compared to the linear dimension Dof the dot,
λ≪D, but still comparable to the width of the lead
(or quantum wire) d,λ.d. Moreover, for billiards
with sharp edges the proper asymptotic limit is, rigor-
ously, out of reach. The present theory addresses this
non-asymptotic semiclassical regime, often also referred
to as the “near” semiclassical regime. We show that
the proper inclusion of diffractive contributions allows
to quantitatively reproduce quantum calculations. The
diffractive coupling between classical paths gives rise
to pseudo-paths that are missing in the standard SCA
and are the key to remedy many of the deficiencies of
semiclassical approximations.
We show in the present communication that this pseudo-
path semiclassical approximation (PSCA) can reach
quantitative agreement with full quantum simulations
provided a high-order diffraction theory for the coupling
between classical paths is used. For the scattering at the
leads we develop an approximation involving elements of
both the uniform theory of diffraction (UTD)30,31 and
the geometric theory of diffraction (GTD)32 referred
to in the following as the GTD-UTD approximation.
With these ingredients good agreement with quantum
simulations is found.
One key conceptual insight is the unambiguous identifi-
cation of the paths that contribute to quantum transport.
We apply the present theory to a circle-shaped regular
quantum dot for which the enumeration of paths, more
precisely of path bundles, is easily possible. Unlike for
2
FIG. 1: (Color online) Geometry of the circular billiard of
radius ρwith perpendicular leads of equal width d. In ac-
cordance with previous work33,34,35 on the circular and the
stadium billiard of equal area a= 4 + π, we choose d= 0.25
and ρ=p1 + 4/π.
chaotic dots, for which the exponential proliferation of
contributing paths as a function of the pathlength makes
their unique identification difficult, for regular systems
their enumeration and identification is straightforward
up to large pathlengths.
The circular billiard is depicted in Fig. 1. The leads
are attached at right angle and have equal width d.
In order to probe the local topology of the cavity we
require sufficiently long dwell times such that d/ρ ≪1.
The wavelength of the electron λfulfills λ≪ρfor the
semiclassical limit to hold inside the cavity. However,
as in experimental or numerical studies of quantum
billiards λ.d. Our semiclassical theory can thus
be quantitatively compared with quantum mechanical
numerical calculations for the circular billiard.
This paper is organized as following: In Sec. II we
review both the standard SCA as well as the PSCA.
These approximations differ by the different path sets
entering the corresponding Green’s function. In Sec. III
we introduce the GTD-UTD diffraction approximation
which is a key to the quantitative agreement between the
PSCA and quantum mechanics for transport properties.
The important role of pseudo-paths and a proper diffrac-
tion theory is demonstrated on the level of quantum
mechanical length-area spectra16 in Sec. IV. Finally, we
compare in Sec. Vthe semiclassical predictions for a
variety of quantum transport properties that play a key
role in the understanding of the quantum-to-classical
crossover, in particular conductance fluctuations (CF),
weak localization (WL) and quantum (non-thermal)
shot noise, with quantum calculations.
II. QUANTUM TRANSPORT THROUGH
BILLIARDS
We consider ballistic transport through quantum bil-
liards for which transport properties are determined by
the wave number (k) and magnetic field (B) dependent
quantum mechanical Hamiltonian S-matrix. Dissipative
or dephasing processes are neglected. (We will return to
the effect of decoherence below.) Moreover, we refer to
a “clean” billiard when elastic scattering at a disorder
potential in the interior of the structure is absent. In
this limit, the S-matrix represents elastic scattering at
the boundaries of the billiard only.
The Smatrix elements S(j,i)
n,m (k, B) describe the scatter-
ing from transverse mode min lead ito mode nin lead
j. We denote the transmission amplitudes from lead 1
to lead 2 as tnm (k, B ) = S(2,1)
nm (k, B) and the reflection
amplitudes back into lead 1 as rnm(k, B ) = S(1,1)
nm (k, B).
According to the Landauer formula the conductance gof
a quantum billiard is directly proportional to the total
transmission T(k, B ),
g(k, B) = 2e2
hT(k, B) = 2e2
h
M
X
n=1
M
X
m=1 |tnm(k, B)|2,(1)
where Mis the number of open modes in the leads having
equal width d. The Smatrix elements can be determined
by a projection of the lead modes ϕm(yi) (yiis the trans-
verse coordinate in the lead) onto the Green’s function
G(yj, yi, k, B) for propagation from yito yjat the cav-
ity lead junction of lead i and lead j, respectively. The
S-matrix elements at B= 0 are given by the Fisher-Lee
equations36
tnm(k, B = 0) = −ipkx2,nkx1,m ×
Zdy2Zdy1ϕ∗
n(y2)G(y2, y1, k, B = 0) ϕm(y1),(2)
where kx1,m (kx2,n) is the longitudinal wave number in
lead 1 (lead 2). The prefactor pkx2,nkx1,m is due to flux
normalization. We use atomic units (~=|e|=meff = 1).
At zero magnetic field the mode wavefunctions take the
form
ϕm(y) = r2
dsin mπy
d.(3)
For non-zero magnetic field B6= 0, Eqs. (2,3) have to
be modified (see, e.g., Ref. 33,34 and references therein).
We use the modular recursive Green’s function method
to calculate the exact quantum mechanical S-matrix el-
ements for a given kand B(for details see Ref. 33,34).
A. Semiclassical approximations to transport
The starting point of semiclassical approximations are
the S-matrix elements [Eq. (2)]. In a first step, the quan-
tum mechanical Green’s function Gis replaced by a cor-
responding semiclassical expression which represents the
Fourier-Laplace transform in stationary phase approxi-
mation of the semiclassical limit of the Feynman propa-
gator. It describes propagation along classical paths with
fixed energy. Depending on the class of paths included, a
hierarchy of different semiclassical approximations to the
3
Green’s function results. These are to be distinguished
from the different level of additional approximations em-
ployed in the evaluation of the integral [Eq. (2)] which
projects the Green’s function onto the lead function. The
latter gives rise to another set of semiclassical approxi-
mations to the S-matrix elements.
We focus first on the replacement of Gby a semiclassical
approximation. For the latter we consider two different
levels of approximation, the standard semiclassical ap-
proximations (SCA) and the pseudo-path semiclassical
approximation (PSCA). Both result from the stationary
phase approximation (SPA) to the full Feynman propaga-
tor reducing the continuous set of paths entering Feyn-
man’s path integral to a discrete subset of paths. As-
suming that the classical action Sis much larger than ~
and that well localized and separated stationary points
with δSi= 0 exist, the standard SCA contains exclu-
sively classical paths. However, near sharp edges of the
cavity or near the cavity-lead junctions, the de Broglie
wavelength is not negligibly small and the SPA will fail.
This leads to diffractive corrections which can be taken
into account within the framework of the pseudo-path
semiclassical approximation (PSCA). One of its salient
features is that the basic notion of a propagator consist-
ing of a sum over a discrete set of paths is preserved.
Diffraction effects simply appear as additional contribu-
tions to the path sum [see Sec. II (c)].
The classical action for an electron moving along a path
qis given by Sq=kLq+Baq/c, where Lqis the length
and aqis the directed enclosed area of the path. aqcan
have both positive and negative values depending on the
rotational direction of the path. In all our semiclassi-
cal calculations (standard SCA as well as the PSCA) the
magnetic field enters only via the Aharonov-Bohm phase
Baq/c. The curvature of the paths as well as the effect
of non-zero magnetic field on the diffraction at the lead
(introduced in Sec. III) is neglected since we consider the
regime of weak magnetic fields (ρ≪ck/B with ρbeing
the radius of the circular cavity).
B. Standard semiclassical approximation
The Green’s function within the standard SCA enter-
ing Eq. (2) is given by:
GSCA(yj, yi, k, B) = 2π
(2πi)3/2X
q:yi→yjq|Dq(yj, yi, k)|
×exp iSq(yj, yi, k, B)−iπ
2µq,(4)
where Dq(yj, yi, k) is the deflection factor (a measure
for the divergence of nearby trajectories) and µqis the
Maslov index. The deflection factor is defined as
|Dq(yj, yi, k)|=1
|kxj||kxi|
∂2Sq
∂yj∂yi(5)
where xi(yi) is the longitudinal (transverse) component
of the trajectory’s starting point (i) and end point (j),
FIG. 2: (Color online) Two-dimensional length vs. enclosed
area distribution of classical paths within the open (black
dots) and closed (orange dots) circular billiard for (a) reflec-
tion and (b) transmission. Each point represents one classical
path which connects the centers of each lead. The inset of (a)
shows typical paths for the path topology of the three first
branches.
respectively. The Maslov index increases by two for ev-
ery reflection at the hard wall boundary of the billiard
and by one when passing a focal point along the trajec-
tory. Eq. (4) contains a sum over all classical paths q
connecting the entrance lead iwith the exit lead j(see
Sec. II B 1).
Evaluation of Eq. (2) with GSCA,
tSCA
nm (k, B) = −ipkx2,nkx1,m ×
Zdy1Zdy2ϕ∗
n(y2)GSCA(y1, y2, k, B)ϕm(y1),(6)
proceeds either numerically11 or analytically by invoking
another set of SPAs. It was recognized from very early
on that the SPA as applied to Eq. (6) is poorly justified
in the non-asymptotic regime when λ.d. Therefore,
various diffraction integral approximations have been
proposed.10,12 The transmission amplitudes then take the
4
form:
tSCA
nm (k, B) = −1
√2πi pkx2,n kx1,m X
q:y0
i→y0
j
cn(θ2, k, d)
×q|Dq(k)|exp iSq(k, B)−iπ
2µq
×cm(θ1, k, d),(7)
expressed in terms of diffraction coefficients cm(θ, k, d)
describing the diffractive coupling from the entrance lead
mode minto the dot and from the dot into the exit lead
mode n. Deviating from previous calculations, we intro-
duce for cm,n(θ , k, d) a combination of Keller’s geometric
theory of diffraction (GTD)32 and the “uniform theory
of diffraction” (UTD).30,31 The derivation of cm(θ, k, d)
within the GTD-UTD is given in App. A 2. The inclusion
of diffraction effects in terms of diffraction coefficients in
Eq. (7) preserves the structure of the semiclassical trans-
mission amplitude in terms of a discrete sum over paths
contributing to transport. The diffraction coefficients
provide θ- and k-dependent weighting factors for each
path contributing to the transmission from mode mto
mode n. Within the framework of the diffractive cou-
plings into and out of the leads, the entrance and exit
leads are treated as point scatterers.15 Each path bundle
which connects the entrance and the exit lead is replaced
by an appropriately weighted representative path q con-
necting the center of the entrance lead with the center of
the exit lead. Consequently one can replace the deflec-
tion factor as given in Eq. (5) by its value in the closed
circular billiard Dq= 1/pkLq, where Lqis the length of
the path.
The diffractive lead-dot couplings in Eq. (7) should be
distinguished from diffractive corrections included in the
propagation in the interior of the billiard. We refer to
Eq. (7) as the standard SCA while inclusion of diffrac-
tive corrections in the billiard corresponds to the PSCA.
1. Paths entering the standard semiclassical Green’s
function
Classical paths in a regular billiard (such as the cir-
cle) feature a highly ordered structure of their length
and enclosed area distribution (see Fig. 2). The branch
structure of the “length-area” distribution of paths con-
necting the entrance with the exit point is a specialty
of the circular cavity. Along each branch the number of
bounces off the wall increases by one from one path to
its next higher neighbor. The points of convergence of
each branch mark those paths that bounce off the wall
infinitely many times and thus run exactly along the cav-
ity boundary. In the limit θ→π/2 (where θis the en-
trance angle as given in Fig. 1) each branch contains an
infinite number of paths. In our numerical calculations
(see Sec. V) paths near this cluster point effectively do
not contribute as they are cut off by vanishing diffraction
coefficients cn(θ→π/2, k)→0.
The distribution of paths eventually reflected back to the
entrance point [Fig. 2(a)] is symmetrically distributed
relative to the a= 0 axis due to time-reversal symmetry.
Every path has a counterpart of equal length but oppo-
site sign of the enclosed area. The lowest left and right
branches consist of polygons with a number of revolutions
nR= 1 in the cavity. The polygons can be characterized
by an angle φ=2π
nCwhere nCis the number of corners.
Along each branch nCincreases by one from one path to
the next. The paths of the next higher branches revolve
twice (nR= 2) around the circle and φ=2πnR
nC=4π
nC.
All higher branches can be described analogously.
The branches of transmitted paths are not symmetric rel-
ative to the a= 0 axis but show a clear off-set (Fig. 2(b)).
Path pairs with similar length do not have, in general,
the same topology but differ in the number of bounces
off the hard wall boundary. As a consequence, these path
pairs have different Maslov indices and thus do not inter-
fere constructively. Fig. 2also illustrates the difference
between the open and closed billiard, i.e., the effect of
“path shadowing” or suppression of longer paths due to
their prior exit from the structure. All paths that would
be geometrically reflected off the closed lead in the closed
system are missing in the open billiard. The difference
between the closed and open billiard is particularly evi-
dent in reflection since all paths with four-fold symmetry
leave the cavity via lead 2 before being reflected back to
lead 1. This is system-specific for the circular billiard
with perpendicular leads (see Fig. 1).
The distinctly different path distributions for transmis-
sion and reflection point already to a clear structural de-
ficiency of the SCA. Many quantum properties of trans-
port are a consequence of the intrinsic coupling between
transmission and reflection. The standard SCA does not
incorporate this quantum aspect of non-locality. Classi-
cally, the path sets of transmission and reflection are dis-
joint. The distribution of transmitted paths is markedly
different from the one of reflected paths. In quantum
transport transmitted and reflected paths are intertwined
and must share the information on the relative phases.
A semiclassical theory that reproduces quantum features
must therefore allow for coupling between the path sets
associated with transmission and reflection. This is the
key property of pseudo-paths discussed in the following.
C. Pseudo-path semiclassical approximation
The pseudo-path semiclassical approximation14,15 goes
beyond the standard SCA by systematically including
diffractive corrections into the propagation in the inte-
rior of the billiard. In the present case, diffractive cor-
rections arise from multiple back scattering, i.e., internal
reflections at the leads. We point to the conceptual sim-
ilarity to “pseudo-orbits” introduced in Ref. 37 for the
study of the density of states in a closed wedge billiard
and to “Hikami boxes”2,38 introduced for elastic scatter-
ing at short-ranged potentials in the interior of a diffusive
5
quantum dot.
In line with multiple scattering theory the pseudo-path
semiclassical Green’s function can be expressed in terms
of a (semiclassical) Dyson equation,15
GPSCA =GSCA +GSCAV GPSCA
=GSCA ∞
X
i=0
(V GSCA)i.(8)
In the present case, GSCA plays the role of the unper-
turbed Green’s function and the perturbation “poten-
tial” Vaccounts for the (internal) diffractive scatterings
at the lead opening. The unperturbed propagation inside
the cavity GSCA is equivalent to the free propagation in
the “closed” system.
For a two-terminal system the perturbation potential V
is given by
V=V10
0V2,(9)
where V1and V2describe the diffractive scattering off
lead 1 and lead 2, respectively. The semiclassical Dyson
equation, Eq. (8), reads
GPSCA
1,1GPSCA
1,2
GPSCA
2,1GPSCA
2,2=GSCA
1,1GSCA
1,2
GSCA
2,1GSCA
2,2
+GSCA
1,1GSCA
1,2
GSCA
2,1GSCA
2,2V10
0V2GPSCA
1,1GPSCA
1,2
GPSCA
2,1GPSCA
2,2.(10)
To first order, the PSCA to the Green’s function (denoted
by GPSCA(1)) connecting lead iwith lead jincludes terms
in Vof the form,
GPSCA(1) =X
l=1,2
GSCA
j,l VlGSCA
l,i
=X
l=1,2X
q′
j,l X
ql,i
GSCA
q′
j,l v(θq′
l, θql, k, d)GSCA
ql,i .
(11)
Eq. (11) may serve as example to illustrate the physics
entering Eq. (10). It describes propagation from lead ito
lead jvia one intermediate visit to lead lwhere diffractive
internal back scattering with amplitude v(θq′
l, θql, k, d)
takes place. The path from lead ito lead l,ql,i as well
as from lead lto lead j,q′
j,l, are classical paths described
by GSCA. Diffractive scattering couples the incident path
ql,i with angle θqlto the exiting path q′
j,l with angle θq′
l
thereby coupling two disjoint subsets of classical paths
and generating a first-order pseudo-path. The determi-
nation of the diffraction coefficient v(θq′
l, θql, k, d) will be
discussed in more detail in Sec. III. From a conceptual
point of view, the pseudo-path semiclassical approxima-
tion [Eq. (10)] is closely related to the diagrammatic per-
turbation theory.1Both leads 1 and 2 (i.e., V1and V2)
act much like “Hikami boxes”,2,38 where electrons can-
not be described semiclassically since the characteristic
FIG. 3: (Color online) Length-area distribution of first-order
pseudo-paths (blue dots) for reflection (a) and transmission
(b). Zeroth-order pseudo-paths (i.e., the classical paths) are
denoted by orange dots (same as Fig. 2). The number of first-
order pseudo-paths up to length L= 40 is larger by more than
a factor of 60 compared to the number of classical paths.
potential length scale (in our case the sharp edges of the
leads) is smaller than the electron wavelength λ. Thus,
the wave nature of electrons has to be taken into account
as diffraction at the point scatterers allowing classically
distinct path sets to mix. This is crucial for the quantum
corrections to the transmission and reflection amplitudes
and repairs some of the deficiencies of the standard SCA
(see Sec. V).
1. Paths entering the pseudo-path semiclassical Green’s
function
Within the PSCA pseudo-paths are formed by joining
classical paths together via diffraction. With each in-
creasing order of the PSCA the length-area distribution
gets more and more densely filled with paths, or equiva-
lently, the number of paths increases with the order of the
PSCA (Fig. 3). Pseudo-paths form product sets of classi-
cal paths, e.g., joining a given classical path for transmis-
sion with a classical path contributing to reflection (or in
reverse order) forms the set of transmitted pseudo-paths
to first order. Reflected first-order pseudo-paths result
from joining two classical transmitted paths or two clas-
6
sical reflected paths. This coupling allows to recover the
non-locality of quantum transport. Higher-order pseudo-
paths are constructed analogously. With increasing or-
der and increasing (combined) pathlength the total num-
ber of pseudo-paths exponentially proliferates. This is in
sharp contrast to the power-law growth of purely classical
paths for regular systems and explains why the effect of
diffractive scattering is more likely visible in regular than
in chaotic systems where the exponential proliferation of
classical orbits may mask the diffractive contributions.
The length-area distribution of first order pseudo-paths
contained in Eq. (10) contributing to reflection is (of
course) still symmetric [Fig. 3(a)]. In addition, new
branches appear with classical and pseudo-path part-
ners of approximately equal length. The change in the
branch structure is more dramatic in the spectrum of
transmitted paths [Fig. 3(b)]. The pseudo-paths lead
to a “symmetrization” of the length-area distribution.
The symmetrization results primarily from paths which
change their rotational direction through diffractive scat-
tering [see the pseudo-path in the inset of Fig. 3(b)].
Branches of classical paths are now completed by sym-
metric pseudo-path “partner” branches of approximately
equal length and different enclosed area.
The weight and the phase of interfering paths are strongly
influenced by the diffraction coefficient v(θ′, θ, k, d). As
will be demonstrated in Sec. V, the previously employed
Fraunhofer theory of diffraction12,13,14,15 is not suffi-
ciently accurate as to give quantitatively reliable results
for transport properties. The same holds for the Kirch-
hoff theory of diffraction10 which is closely related to the
Fraunhofer theory of diffraction and gives similar results
for the diffraction coefficients (see Fig. 5in the follow-
ing section). For a quantitative agreement between the
PSCA and quantum mechanics, it is thus necessary to
go beyond low-order diffraction approximations and use
a more sophisticated theory of diffraction. We present
such a theory in the following section.
III. DIFFRACTION AT THE LEAD
The contribution of a given classical path with path-
length Lqto the standard semiclassical Green’s function,
GSCA, is
GSCA
q=1
p2πkLq
eikLq−i3π/4−iπ
2µq.(12)
Eq. (12) is the basic building block entering the
Dyson Eq. (10) together with the diffraction coefficient
v(θ′, θ, k, d). In line with the far-field approximation un-
derlying diffraction theory, v(θ′, θ, k, d) is assumed to be
independent of the length of the path approaching or ex-
iting the diffractive scattering region.
A successful application of the pseudo-path semiclas-
sical approximation outlined in the preceding subsec-
tion requires the determination of accurate diffraction
coefficients v(θ′, θ, k, d). Different approximations have
FIG. 4: (Color online) Sketch of diffraction at an open lead
mouth: a path qreaches the orifice under an angle θand is
backscattered into a path q′that leaves with an angle θ′(the
angles θand θ′as depicted in the figure have opposite signs).
The dashed lines denote that ~r and ~r ′are in the far field
region.
been used in the past for the inclusion of diffraction
effects: Kirchhof diffraction approximation (KDA),10
Fraunhofer diffraction approximation (FDA),12,13,14,15
geometric theory of diffraction (GTD),32 and the “uni-
form theory of diffraction” (UTD).30,31 We have devel-
oped a new theory for the reflection at open lead mouths
by combining the GTD with the UTD (the GTD-UTD)
to take into account paths that scatter multiple times
between the edges of the leads, (see appendix A 1). With
this theory we obtain diffraction coefficients in excellent
agreement to quantum mechanics.
Consider, as a test case, the diffractive scattering (Fig. 4)
at the lead mouth described by the first-order term
[Eq. (11)],
GPSCA(1)
q′,q =GSCA
q′v(θ′, θ, k, d)GSCA
q.(13)
The incoming path qis incident at angle θ(measured
with respect to the surface normal) and is diffractively
scattered into angle θ′under which path q′leaves the
scattering region. We compare (Fig. 5) the present GTD-
UTD theory with the FDA, the KDA and exact quan-
tum mechanical (QM) calculations.10 Even for a typical k
value in the low mode regime (k=2.5π/d), the agreement
between the GTD-UTD and the exact QM calculations
is very good whereas both the FDA and the KDA display
clear deviations from the QM values. While these devi-
ations do not appear dramatic at first glance, they are,
in fact, quantitatively very important as the diffraction
coefficient enters the Dyson series [Eq. (10)] to all orders.
Note, however, that the GTD-UTD would fail in the limit
θ→π/2 [as indicated by the kink at θ′=θ= 3π/8 in
Fig. 5(c)]. This deficiency is of no concern for the present
applications as the probability for diffractive scattering
tends to zero in this limit. The maximum of the diffrac-
tive reflection probability is clearly around the specular
value θ′≈ −θ. However, it is important to note that
7
0
10
20
30
40
50
60
-0.4 -0.2 0 0.2 0.4
|v(θ’,θ,k,d)|2
θ’/π
θ=3π/8
(c) FDA
KDA
GTD-UTD
QM
0
10
20
30
40
50
60
|v(θ’,θ,k,d)|2
θ=π/4
(b)
0
10
20
30
40
50
60
|v(θ’,θ,k,d)|2
θ=π/8
(a) FDA
KDA
GTD-UTD
QM
-4
-2
0
2
4
6
-0.4-0.2 0 0.2 0.4
v(θ’,θ,k,d)
Im Re
FIG. 5: (Color online) Absolute square of the diffraction co-
efficient |v(θ′, θ, k, d)|2for diffractive scattering (see Fig. 4)
within the FDA, the KDA, the GTD-UTD and exact quan-
tum mechanical data for angle of incidence (a) θ=π/8, (b)
θ=π/4, and (c) θ= 3π/8 at k= 2.5π/d. The vertical lines
in each frame mark the angles of specular reflection. The
quantum mechanical (QM) and KDA coefficients are taken
from Ref. 10. Inset of Fig. (b): the real and imaginary part
of v(θ′, θ, k, d) within the GTD-UTD.
the probability distribution possesses a local maximum
at the backscattering angle θ′≈θ. A non-negligible part
of the electron wave is back-scattered into the direction
from where it came from. This diffractive back-reflection
should not be confused with the well-known Andreev
back-reflection in which the back-reflected particle simul-
taneously undergoes a particle-hole exchange.39 Back-
reflected paths are responsible for the symmetrization of
the distribution of transmitted paths [Fig. 3(b)] and are
crucial for the understanding of the weak-localization dip
in the transmission.16
While the KDA and the FDA have been successfully used
in the past to explain certain features of conductance
fluctuations,10,12,13,14,15 only the GTD-UTD is precise
enough to reproduce transport semiclassically on a quan-
titative level (see Sec. V). In Sec. V B we compare the
results for transport properties obtained by implementing
the GTD-UTD and the FDA to full quantum mechanical
calculations.
FIG. 6: (Color online) Two-dimensional distribution of
the absolute square of the quantum S-matrix element
|t22(k, B )|2=T22(k, B) as a function of the wavenumber k
and the magnetic field B.
IV. PATHS IN QUANTUM TRANSPORT
The information on paths governing quantum trans-
port can be reliably extracted from the two-dimensional
Fourier transforms of the quantum mechanical S-matrix
elements,16 Snm(k, B ). The S-matrix elements display
a strongly fluctuating pattern as a function of kand B
(see Fig. 6for T22(k, B)). Since the canonically conjugate
variables to the wavenumber kand the magnetic field B
are the length Land the directed area a, respectively,
the Fourier transform
˜
Sj,i
nm(L, a) = Zdk ZdB e−i(kL+B
ca)Sj,i
nm(k, B) (14)
allows the unambiguous identification of quantum paths
contributing to quantum transport via their length and
enclosed area [Fig. 7(a)-(d)]. No apriori-assumption
as to the existence of classical paths q(L, a) with path-
length Land area aenters Eq. (14). The two-dimensional
pathlength-area spectrum allows to identify both classi-
cal as well as non-classical contributions to the full quan-
tum spectra. Fig. 7displays examples of pathlength-area
spectra |˜
Sij (L, a)|2. [Note that Fig. 7(c) is the absolute
square of the Fourier transform of the transmission am-
plitude whose absolute square is plotted in Fig. 6]. Obvi-
ously the strong fluctuations of conductance in quantum
transport (Fig. 6) are the result of the interference of
clearly identifiable (quantum) paths (Fig. 7).
The quantum mechanical S-matrix elements Snm(k, B )
are determined with the help of the modular recur-
sive Green’s function method33,34 and then numerically
Fourier transformed. Finite discretized intervals must
be used when performing the Fourier transform numer-
ically. The integration intervals are denoted by ∆k=
kmax −kmin (∆B=Bmax −Bmin) and the numerical
grid spacings by δk (δB). Accordingly, the resolvable
length is ∆L= 2π/δk and the resolvable area interval
is ∆a= 2π/(δB/c). The magnitude of the S-matrix
elements decreases with increasing length, which is an
8
FIG. 7: (Color online) The absolute square of the Fourier transform |˜
S(L, a)|2as a function of length Land enclosed directed
area a: (a) ˜
R22(L, a), (b) ˜
R11(L, a), (c) ˜
T22(L, a), and (d) ˜
T11(L, a). The color shading is determined by log (|˜
S(L, a)|2+ 1).
The insets show the probabilities along the entire resolved length. The integration of Eq. (14) is performed numerically over
k∈[2.2,3.45]π/d discretized with 251 points and B/c ∈[−3,3] with 121 points. The spectra are compared to the distribution
of classical paths (black dots) at zero Bfield (the curvature of the trajectories in the present magnetic field and energy regime
is negligible).
obvious consequence of open systems: the probability to
stay within the cavity decreases with increasing length.
The parameter δk must be chosen sufficiently small such
that the maximum resolvable length (∆L) lies already in
the region of strongly reduced amplitudes. Otherwise,
the Fourier spectrum is visibly back-folded onto the fun-
damental interval. The magnetic field interval is further
restricted by the requirement that the curvature of the
paths is negligible, i.e., the cyclotron radius ck/B should
be much larger than the circle radius ρ. We chose the in-
terval ∆kand ∆Bas well as the number of interval points
such that a maximum length of ∆L= 100 is resolved [in
Fig. 7(a)-(d) only the contributions with L≤40 are
shown, the insets contain the entire spectrum]. Except
for R11, the absolute square of the S-matrix elements is
considerably damped at a length of ∆L= 100. Thus, the
graphs represent essentially the entire length-area spec-
trum. Only for R11, contributions with L > 100 are
non-negligible which leads to a back-folding near L= 0
[the lowest branch structure near L= 2 in Fig. 7(b)].
The quantum length-area spectra provide detailed infor-
mation on the paths contributing to transport through
a specific system. They represent the paths entering the
full Feynman path integral. The following general trends
can be observed: Long paths are more prevalent in S-
matrix elements connecting low mode numbers (in the
present case they contribute more strongly to S-matrix
elements ˜
S11 than to ˜
S22). Lower modes favor smaller
entrance and exit angles that are associated with longer
paths with a larger number of bounces off the cavity
boundary. The most important observation is the re-
markably close correspondence of the quantum mechani-
cal length-area spectrum to its classical counterpart. Im-
portant contributions are located near classical paths.
Moreover the branch structure of classical paths is re-
produced [Fig. 7(a)-(d)]. On the other hand, there are
distinct structures which do not correspond to classical
paths and which can be identified using the distributions
in Fig. 3(a) and (b). The quantum mechanical length-
area spectra confirm the existence and substantial role
of non-classical paths, the pseudo-paths: these are those
paths that are one or several times diffractively reflected
9
-20 -10 0 10 20
Area a
5
10
15
20
25
Length L
0
1
2
3
4
5
6
QM
(a)
-20 -10 0 10 20
Area a
5
10
15
20
25
Length L
0
1
2
3
4
5
6
GTD-UTD
(b)
-20 -10 0 10 20
Area a
5
10
15
20
25
Length L
0
1
2
3
4
5
6
FDA
(c)
FIG. 8: (Color online) ˜
T22(L, a) as a function of length Land enclosed directed area a: (a) full quantum S-matrix, (b) the
PSCA using the GTD-UTD, and (c) the PSCA using the FDA for the diffraction coefficients. Several regions with non-classical
paths are highlighted by red circles.
off the lead before exiting the cavity. Two examples are
given in the following: For ˜
T22 we identify an ensemble
of diffractive paths which, among others, contribute to a
symmetrization of the spectrum [non-classical branches
in Fig. 7(c), two of them together with the classical
partner branches are marked by arrows]. R11 reveals the
importance of paths that are geometrically reflected off
the open lead (e.g., the periodic contributions near a= 0
along the length axis belong to horizontal paths bounc-
ing increasingly many times back and forth).
The importance of a given class of paths to quantum
transport can be delineated by inverting this decomposi-
tion process. Deleting a selected class of paths (classical
or non-classical) from ˜
Sij (L, a) and performing the in-
verse Fourier transform gives rise to truncated S-matrix
elements ¯
Sij (k, B) from which certain path contributions
have been removed in a controlled manner. This is the
key to detailed quantitative tests of semiclassical the-
ories. Since summation of the (P)SCA over arbitrar-
ily long paths is prohibitively complicated we can com-
pare truncated quantum and semiclassical S-matrix el-
ements where both quantum and classical paths only
up to a maximum pathlength L≤Lmax are included.
The length-area spectra also allow sensitive tests for the
proper diffractive weight of a given class of paths in a
semiclassical theory. To this end, we first calculate the
S-matrix elements within the PSCA and then perform
the Fourier transform [Eq. (14)] in analogy to quantum
calculations. To analyze the role of a proper diffraction
coefficient we use either the GTD-UTD (which gives good
agreement with quantum mechanics, see Fig. 5) or the
FDA (with poor agreement with quantum mechanics, see
likewise Fig. 5). The fact that back-reflection into the
cavity is poorly described within the FDA is mirrored in
the semiclassical length-area spectrum where important
non-classical (diffractive) contributions have a far too low
weight [Fig. 8(c)]. In particular, the diffractive change
of the rotational direction is insufficiently described (see
Fig. 5for θ′>0). A clear indication for the essen-
tial role of the corresponding paths is the improvement
within the GTD-UTD. The length-area spectrum within
the GTD-UTD remarkably reproduces even fine details
of the quantum mechanical spectrum [compare Fig. 8(a)
and (b)]. The non-classical path sets in Fig. 8can be
identified using the path distributions within the PSCA
(Fig. 3). The length-area spectra do not leave any am-
biguity as to which paths contribute by which phase and
weight. By taking into account pseudo-paths and weight-
ing them with the appropriate diffraction coefficients the
quantum mechanical path spectrum is reproduced on a
quantitative level. In the following section we demon-
strate how the accurate representation of the length-area
spectra within PSCA directly translates into quantitative
reproduction of transport properties.
V. APPLICATION TO TRANSPORT
THROUGH REGULAR BILLIARDS
Phase coherent ballistic transport governed by quan-
tum interference influences the conductance in several
important ways: the conductance strongly fluctuates
as a function of the Fermi energy, the magnetic field
or the cavity geometry (conductance fluctuations, CF).
The conductance is, on average, suppressed compared
to the classical prediction and increases with external
magnetic field (weak localization, WL). The noise carries
the signatures of quantum mechanical uncertainty (shot
noise).
Path interference has been the key to the
understanding of phase coherent ballistic
transport.7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,40
For chaotic systems and large mode num-
bers quantum transport properties have been
attributed17,18,19,20,21,22,23 to the interference of clas-
sically allowed paths (SCA). We demonstrate in the
following for the circular billiard at low mode numbers
(accessible to experiments) that CF, WL, and shot noise
cannot be explained by the interference of classical paths
alone. In this regime, the SCA overestimates the CF,
does not reproduce the weak-localization dip and also
shows poor quantitative agreement with the quantum
10
mechanical prediction for shot noise. These difficulties
can be overcome with the PSCA when a high-level
diffraction approximation, the GTD-UTD, is employed.
For technical reasons, we perform in both the SCA and
the PSCA the summation over paths only up to a maxi-
mum cut-off length Lmax. Correspondingly, we truncate
the full quantum scattering matrix elements by setting
all elements ˜
Snm(L, a) with lengths exceeding Lmax to
0 and carry out the inverse Fourier transform. This
allows a quantitative comparison between semiclassical
and quantum calculations unaffected by the (inevitable)
unitarity deficiency of a truncated semiclassical path
sum.
A. Conductance fluctuations
Conductance fluctuations (CF) have been identified as
a direct manifestation of phase coherent transport (see
e.g. Refs. 1,2,8,18,41,42,43,44 and references therein).
The strong fluctuations of the conductance as a function
of, e.g., energy originate from path interference and
thus give evidence for the wave nature of electrons in
quantum dots. The CF offer one of the most stringent
testing grounds for a semiclassical theory, when good
agreement on the level of each individual S-matrix
element is required.
In the following we compare the results for CF within the
PSCA with the GTD-UTD, the SCA and the quantum
mechanical calculations as a function of kat vanishing
magnetic fields B= 0. Results for B6= 0 averaged over
kcan be found in Sec. V B.
The semiclassical and quantum mechanical results both
display strong fluctuations of the conductance [i.e., the
total transmission T(k, B = 0)]. Their amplitude is,
however, extremely sensitive to any deficiencies in the
semiclassical approximations (inappropriate weighting
of paths, missing paths). We emphasize that the
comparison of Fig. 9is on a fully differential level. No
energy or ensemble average is involved. Unsurprisingly,
the agreement between the SCA and the quantum
data is poor and on a level of qualitative agreement at
best [Fig. 9(a) and (b)]. The functional dependence
of T(k, B = 0) and R(k , B = 0) seems only weakly
related to the quantum mechanical prediction. By
ensemble averaging (e.g., over a suitable kinterval)
these discrepancies would be (partially) removed (or
masked). To correctly reproduce the fluctuations in
T(k, B = 0) = PM
m,n |tmn(k, B = 0)|2, all individual
mode-to-mode amplitudes tmn have to be accurate.
Obviously, the contribution of pseudo-paths included
in the PSCA but missing in the SCA significantly
improves the agreement with the quantum conductance
fluctuations [Fig. 9(c) and (d)].
Even for averaged quantities, such as the averaged
conductance hTi∆kand the variance of the conductance
fluctuations σ=phT2i∆k− hTi2
∆k, the deficiencies of
the SCA are still visible, in particular at larger cut-off
lengths Lmax. Both hTi∆kand σare overestimated
(Fig. 10) as compared to quantum mechanics which can
be attributed, in part, to the lack of correlation among
purely classical paths.
In contrast, the PSCA shows very good agreement with
quantum mechanics for both the averaged conductance
as well as for the variance. For averaged quantities
(Fig. 10) the inclusion of up to 4th order diffractive
scattering for Lmax = 40 is sufficient while for fully
differential quantities (Fig. 9) 5th order corrections for
Lmax = 40 still improve the agreement.
We point out that for the present system, taking into
account paths up to a length of Lmax = 50 which
corresponds to 16 radial traversals through the billiard,
one reaches 70% of the unitarity level within both the
truncated quantum mechanics and the PSCA. The
essential contributions to CF, WL and shotnoise (see
the following sections) is thus rooted in this subset of
relatively short paths.
B. Weak localization
Weak localization (WL) is a well-known quantum
correction to the classical diagonal approximation (see
e.g. Refs. 1,2,8,16,17,18,19,20,28,45,46,47,48,49 and ref-
erences therein) where the later corresponds to the re-
striction to terms q=q′in the double sum over paths
when calculating |tSCA
nm |2from Eq. (7). Off-diagonal
terms q6=q′give rise to quantum interferences which
manifest themselves as an increase of the averaged total
reflection hR(B)i∆k=Pm,nhRnm (B)i∆kat B= 0 in
form of a pronounced peak. Correspondingly, the aver-
aged total transmission hT(B)i∆k=Pm,n hTnm(B)i∆k
features a dip which is an immediate consequence of uni-
tarity. For the investigation of the weak localization
dip (peak) we employ an average over a small window
∆k= [2.2−2.8]π/d of the kdependence of the probabil-
ities: hTnm(B)i∆k=R∆kdk|tnm (k, B )|2,hRnm(B)i∆k=
R∆kdk|rnm(k, B)|2.
Since semiclassical theories are, by construction, not
necessarily unitary at a given level of approximation,
the anti-correlated peak-dip structure near B= 0 pro-
vides a sensitive test for semiclassical approximations. It
has been shown that the quantum anti-correlation be-
tween reflection and transmission (hR(B)i∆k− hR(B=
0)i∆k) = −(hT(B)i∆k−hT(B= 0)i∆k) requires a corre-
lation of transmitted and reflected paths.16 This correla-
tion is absent in the SCA such that no transmission dip
is reproduced [see Fig. 11 (b)].
The role of diffraction in a quantum billiard manifests
itself by a very good agreement of the PSCA with quan-
tum mechanical results [see Fig. 11 (a)]. Tests for WL
as a function of the cut-off length Lmax of the paths as
well as of the order of diffractive scattering included show
that convergence toward the (truncated) quantum result
11
0
0.5
1
1.5
2
2.5
2.2 2.4 2.6 2.8
R(k,B=0)
(c) Reflection
PSCA GTD-UTD
QM
0
0.5
1
1.5
2
2.5
2.2 2.4 2.6 2.8
T(k,B=0)
k in π/d
(d) Transmission
PSCA GTD-UTD
QM
0
0.5
1
1.5
2
2.5
2.2 2.4 2.6 2.8
R(k,B=0)
(a) Reflection
SCA
QM
0
0.5
1
1.5
2
2.5
2.2 2.4 2.6 2.8
T(k,B=0)
k in π/d
(b) Transmission
SCA
QM
FIG. 9: (Color online) (Conductance) Fluctuations in the transmission T(k, B = 0) and reflection R(k , B = 0) as a function
of k. The maximal included length is Lmax = 40. (a) Reflection and (b) transmission within the SCA (red/green solid line),
and quantum mechanics (black dashed line). For the diffraction coefficients cm(θ , k, d) entering the SCA we have used the
GTD-UTD. (c) Reflection and (d) transmission within the PSCA with the GTD-UTD (red/green solid line), and quantum
mechanics (black dashed line). The PSCA includes diffractive scattering up to 5th order.
0
0.2
0.4
0.6
0.8
1
0 0.02 0.04 0.06 0.08 0.1
<T>∆k
1/Lmax
(a)
SCA
PSCA
QM
0
0.2
0.4
0.6
0 0.02 0.04 0.06 0.08 0.1
σ
1/Lmax
(b)
SCA
PSCA
QM
FIG. 10: (Color online) Comparison between the PSCA, the SCA, and quantum mechanics (QM) for (a) the averaged trans-
mission hTi∆kas a function of the inverse cut-off length 1/Lmax and (b) the averaged variance of the conductance fluctuations
σ. The average is performed in the interval k∈[2.2,2.8]π/d. The data within the PSCA are calculated in 3rd order for lengths
Lmax = 10−30, in 4th order for Lmax = 40 and 5th order for Lmax = 50. The QM results extend to the exact value (Lmax =∞).
12
0.3
0.4
0.5
0.6
0.7
-0.4 -0.2 0 0.2 0.4
B/c
(a) Lmax=40
Lmax=30
Lmax=20
PSCA
-0.4 -0.2 0 0.2 0.4
0.3
0.4
0.5
0.6
0.7
B/c
(b) Lmax=40
Lmax=30
Lmax=20
SCA
FIG. 11: (Color online) Comparison between truncated quan-
tum mechanics, the SCA, and the PSCA for the weak local-
ization dip in transmission for different Lmax. (a) PSCA using
the GTD-UTD for all diffraction coefficients (red solid line).
(b) Standard SCA (red solid line) (using the GTD-UTD for
the diffraction coefficients cm(θ, k, d) for entering and exiting
the circular cavity.) Black dashed lines: quantum results.
is reached for Lmax = 40 when diffraction up to fourth
order is included. We note an improvement compared to
previous third-order calculations16 especially due to the
inclusion of the (small) real part of the diffraction coeffi-
cient v(θ′, θ, k, d).
It is instructive to analyze the build-up of the weak local-
ization peak in reflection and of the dip in transmission
from individual S-matrix elements. In reflection only the
diagonal elements R11(B) and R22(B) show a peak while
the off-diagonal elements R12 (B) and R21(B) exhibit a
dip [Fig. 12 (c), (d)]. This is due to the fact that time-
reversal symmetric paths contributing to Rnm (B) with
different parity of modes mand nacquire an additional
phase-shift of π. As an example consider R21 (B), enter-
ing in mode m= 1 and exiting in mode m= 2 produces a
phase shift of π. Thus the time-reversal symmetric paths
interfere destructively.
In transmission the major contribution to a dip originates
from T22(B) in the chosen energy window of two open
modes [Fig. 12 (e), (f)]. Note that T21 (B) = T12(B) be-
cause of the Onsager relation t(2,1)
nm (k, B) = t(1,2)
mn (k, −B)
and the symmetry of the circular billiard which ensures
that t(1,2)
mn (k, −B) = t(2,1)
mn (k, B). Due to time-reversal
symmetry, R(B) = R(−B) giving rise to a symmetric
peak in Ras a function of B(Fig. 12). For unitary trans-
port, T(B) = 1 −R(B), which implies a symmetric dip
in transmission. For the truncated quantum mechanics
and semiclassics where long paths are omitted unitarity
is not preserved and T(B) is not exactly symmetric but
features a slight shift of the minimum toward B < 0.
Overall, the agreement between the PSCA and the full
quantum calculation (truncated at the same pathlength)
0
0.1
0.2
0.3
-0.4 -0.2 0 0.2 0.4
B/c
(e)
T22
T21 T12
T11
-0.4 -0.2 0 0.2 0.4
0
0.1
0.2
0.3
B/c
(f) T22
T21 T12
T11
0
0.1
0.2
0.3
0.4
(c) R22
R12 R21
R11
0
0.1
0.2
0.3
0.4
(d) R22
R12 R21
R11
0.5
0.6
0.7 (a)
QM
T
R 0.5
0.6
0.7
(b)
PSCA with GTD-UTD
T
R
FIG. 12: (Color online) Weak localization for Lmax = 40 as
produced by quantum mechanics: left column (a),(c),(e), and
PSCA with the GTD-UTD: right column (b), (d), (f ).
is remarkable (Fig. 12). The residual minor deviations
are mainly ascribed to deficiencies in the diffraction co-
efficients for which we use an analytical far-field approxi-
mation (Sec. III and appendix A 1 and A 2). As, e.g., the
diffraction coefficients cm(θ, k, d) enter Rnm,Tnm to the
fourth power, a small deficiency in cm(θ, k , d) can have
a sizeable effect on Rnm ,Tnm. The offset of the total
reflection Rand transmission Twithin PSCA [Fig. 12
(b)] compared to the quantum mechanical result [Fig. 12
(a)] mainly originates from imperfections in R22 and T22
[compare Fig. 12 (c), (d), and (e), (f)]. The imperfec-
tion could be cured by including a correction factor of
≈0.97 −0.99 in c2(θ, k, d).
To demonstrate how sensitively transport properties de-
pend on the weights of the contributing paths we show
the results for weak localization calculated within PSCA
but now using the FDA instead of the GTD-UTD for the
diffraction coefficients (Fig. 13). (We have used 3rd or-
der of the PSCA here, since the deviation between 3rd
and 4th order is much smaller than the errors due to the
simpler diffraction theory.) It is striking that especially
both R22 and T22 seem to be underestimated which is
due to deficiencies in c2(θ, k, d) within the FDA. It is
worthwhile pointing out that the FDA diffraction could
not be “repaired” by a correction factor, as the errors
in R22 and T22 are different. In other words, transmis-
sion is more “sensitive” to a correct implementation of
pseudo-paths than reflection. Of particular importance
are paths which change the rotational direction.16 Since
the FDA does not give sufficiently weight to this class of
13
0
0.1
0.2
0.3
-0.4 -0.2 0 0.2 0.4
B/c
(e)
T22
T21 T12
T11
-0.4 -0.2 0 0.2 0.4
0
0.1
0.2
0.3
B/c
(f)
T22
T21 T12
T11
0
0.1
0.2
0.3
0.4
(c) R22
R12 R21
R11
0
0.1
0.2
0.3
0.4
(d)
R22
R12 R21
R11
0.3
0.4
0.5
0.6
0.7
(a)
QM
T
R
0.3
0.4
0.5
0.6
0.7
(b)
PSCA with FDA
T
R
FIG. 13: (Color online) Weak localization for Lmax = 40.
Left column (a),(c),(e): quantum calculations, right column
(b),(d),(f): PSCA with diffraction coefficients from FDA.
0
0.1
0.2
0.3
0.4
0 0.02 0.04 0.06 0.08 0.1
F
1/Lmax
SCA
PSCA
QM
FIG. 14: (Color online) The k-averaged Fano factor Fas a
function of the inverse cut-off length Lmax. The average is
performed over the interval k∈[2.2,2.8]π/d. Comparison
between PSCA with the GTD-UTD, the SCA, and quantum
mechanics (QM). (Lmax =∞corresponds to the exact result.)
pseudo-paths (see Fig. 5) the transmission dip cannot be
well reproduced.
C. Shot noise
Another quantity characteristic for quantum trans-
port is the quantum shot noise power of the current
(see e.g. Refs. 1,18,21,42,50,51,52,53,54,55,56,57 and ref-
erences therein). At zero temperature (T= 0), the time-
dependent current noise is due to the granularity of the
electron charge and carries information about the wave
vs. particle nature of charge transport. The Fano fac-
tor Fmeasures the amount by which the noise in phase
coherent transport is suppressed relative to the Poisso-
nian value of uncorrelated classical electrons. Within the
Landauer-B¨uttiker picture, Fcan be expressed as50
F=hT r(t†tr†r)i∆k
hT r(t†t)i∆k
=hPnτnηni∆k
hPnτni∆k
,(15)
with τn,ηnbeing the eigenvalues of the Hermitian ma-
trices t†tand r†r, respectively.
Calculating the shot noise power from non-unitary scat-
tering matrices is obviously a delicate matter, as replace-
ment of rtrby 1 −tttleads, unlike for unitary descrip-
tions, to different results. Furthermore, such a replace-
ment may result in negative and thus unphysical values
for the shot noise power, as non-unitary scattering ma-
trices allow for the possibility of having τn>1 such that
(1 −τn)<0 (for very high mode numbers as, e.g., in
Ref. 18 such a situation may, however, be unlikely). By
using Eq. (15), such difficulties can be avoided as both
the transmission and reflection eigenvalues τn, ηnare, by
construction, real and positive not only for the truncated
quantum calculation but also for the PSCA and the stan-
dard SCA. The standard SCA result for Fstrongly devi-
ates from the quantum mechanical data (see Fig. 14). For
small cut-off lengths Lmax , the value of Fis smaller but
increases more rapidly than the quantum mechanical re-
sult with Lmax. The PSCA yields very good agreement
with the quantum mechanical result for the shot noise
Fano factor F. Note that for the largest cut-off length
Lmax = 50 the Fano factor Fis already converged to its
asymptotic value F≈0.28 suggesting that long paths do
not play a significant role for F. This result agrees with
the finding51,52 that the shot noise power is of similar
magnitude for regular and chaotic billiards as differences
in the dynamics are most strongly felt by very long paths.
VI. SUMMARY
We have presented a semiclassical theory which is able
to quantitatively reproduce full quantum results for scat-
tering through microstructures with a specific geometry,
in the present case a circular shaped billiard with leads
oriented 90odegrees relative to each other. The present
approach does not invoke the limit of large mode num-
bers, (where the de Broglie wavelength λis small rela-
tive to the lead width) but requires λto be small only
on the scale of the linear dimension of the microstruc-
ture (the circle). This non-asymptotic semiclassical the-
ory allows a direct comparison with quantum calcula-
tions as well with experiments on a system-specific level
for individual S-matrix elements avoiding any ensemble
14
averaging or fit parameters. This level of agreement al-
lows us to perform detailed semiclassical investigations of
quantum transport quantities such as the conductance
fluctuations, the weak localization, and the shot noise.
Our studies show unambiguously that for reproducing
these quantities correctly, two major ingredients are, in-
deed, crucial: (1) the inclusion of “pseudo-paths” in the
semiclassical propagator which are diffractively backscat-
tered from the interior side of the cavity openings, (2) a
sufficiently accurate description of the diffraction coef-
ficients for the injection, ejection and back-reflection of
particle flux at the cavity openings. We meet the lat-
ter requirement by developing a combined geometric and
uniform theory of diffraction (GTD-UTD). Pseudo-paths
are crucial for reproducing the conductance fluctuations
in transport and lead to a reduction of its variance. Also
for the weak-localization effect we find that pseudo-paths
are crucial, as no signature of weak localization appears
in the transmission through the circular cavity without
their contribution (even when the advanced diffraction
theory is employed for all truly classical paths). For the
shot noise power we showed that a standard semiclassical
calculation (without pseudo-paths) gives sizeable discrep-
ancies. The inclusion of pseudo-paths leads to agreement
with the quantum mechanical result. We emphasize that
the parameter regime in which we have identified the
above effects of pseudo-paths coincides with the typical
situation in quantum transport experiments. The latter
usually feature only a few open lead modes M.
The present results raise several interesting questions for
future investigations: the comparison between the semi-
classical approximations (PSCA and SCA) and full quan-
tum calculations were performed for truncated path sums
up to a finite path length L≤Lmax. The primary rea-
son for the truncation was technical, as the number of
diffractive pseudo-paths exponentially proliferates with
L→ ∞ also for classically regular structures and ex-
act path sums become prohibitively difficult to perform.
There is, however, a second conceptual motivation. In
the experiment, decoherence due to inelastic scattering
limits phase-coherent transport to pathlengths L≤lφ,
where the phase-decoherence mean free path lφtypically
allows only a moderate number of traversals across the
cavity. The latter restriction rules out that very long
paths with L > lφcontribute to quantum interference in
the experiment, a feature which is naturally incorporated
by way of the cut-off length Lmax in our semiclassical the-
ory. Clearly, such long paths can still provide incoherent
contributions. The present approach may thus contribute
to a semiclassical unterstanding of decoherence effects in
regular cavities.28
In the present treatment of diffractive scattering, both
internal diffraction at the open lead mouth giving rise
to pseudo-paths as well as the coupling between leads
and cavity was performed for sharp edges. The weight
of diffractive contributions can be changed by “round-
ing off” the lead opening. An investigation of the de-
pendence of the weak localization on the smoothness of
the edges is currently underway.64 The introduction of
rounded corners has, however, another profound effect,
apart from changing the weight of diffractive scatter-
ing: an open circle with rounded edges of the leads is
no longer regular but features a mixed phase space. This
raises the question as to the interplay between diffractive
scatterings at the lead opening and chaotic scattering in
the interior of the billiard. For generic chaotic systems
a number of alternative semiclassical theories has been
proposed to explain universal features of quantum trans-
port (see e.g. Refs. 17,18,19,20,21,22,23 and references
therein). These theories typically employ an ensemble
average and rely on a ~→0 limit which makes them
complementary to the present system-specific approach
for finite ~. Bridging the gap between these two frame-
works would be of great interest. One key ingredient
would be to clarify the interplay between the diffraction-
based pseudo-paths and the chaos-based correlated clas-
sical path pairs (Richter-Sieber orbits20). The relative
weight for characteristic quantum transport effects car-
ried by these two classes of paths when both are present
(as in a chaotic billiard with sharp edges) remains an
open question.
Acknowledgments
We thank Piet Brouwer, Tobias Dollinger and Klaus
Richter for helpful discussions. This work was supported
by the Austrian FWF under Grants No. FWF-SFB016
“ADLIS” and No. 17359, and the FWF doctoral program
“Complex Quantum Systems”. L.W. acknowledges sup-
port by the PHC “Amadeus” of the French Ministry of
Foreign and European Affairs.
APPENDIX A: GEOMETRIC AND UNIFORM
THEORY OF DIFFRACTION FOR THE
COUPLING OF QUANTUM LEADS TO A
BILLIARD CAVITY
In a quantum billiard, the electron propagation in
the leads is determined by a quantum mechanical wave,
while the propagation inside the ballistic cavity can
be described semiclassically, i.e., by propagation along
classical trajectories with a quantum phase. The strong
scattering effects that occur especially in the low mode
regime at the orifices have been described in past works
on transport through open billiards by the Kirchhoff
diffraction approximation (KDA)10 or on the level of the
Fraunhofer diffraction approximation (FDA).12,13,14,15
In the high-mode regime diffraction has been neglected
altogether.8,17,18,19,20,21,22,23,24,26 Both the KDA and
the FDA perform about equally well (see, e.g., Fig. 5).
For the identification of pseudo-paths in the Fourier
spectra of the conductance fluctuations, the KDA and
the FDA have been sufficiently precise. However, in
order to recover unitarity of the semiclassical S-matrix
15
FIG. 15: (Color online) Propagation between two points ~r and
~r ′in a semi-infinite plane in the presence of an open (lead).
Contributions from the direct path (black dotted line), from
specular reflection (blue solid line) , and from scattering at
the orifice (red dashed line) are depicted.
and weak-localization in transmission, a more precise
diffraction theory must be implemented. For multiple-
scattering paths, higher order products of the diffraction
weights occur and even small errors are rapidly am-
plified. We have therefore implemented a combination
of the geometric theory of diffraction (GTD)32 and
the uniform theory of diffraction (UTD)30,31 for the
diffraction coefficients in open billiards referred to in
the following as the GTD-UTD. Both theories have
been previously applied separately to the calculation of
higher-order scattering corrections to Gutzwiller’s trace
formula in closed quantum-billiards.31 We discuss in
section A 1 diffractive backscattering at a semi-infinite
half-plane with an orifice and in section A 2 diffraction
during propagation from a lead into a semi-infinite plane.
1. The GTD-UTD for backscattering into the
cavity
The propagation between two points in a semi-infinite
plane with a connected lead is depicted in Fig. 15. In the
absence of a lead, the propagation between two points
~r and ~r ′in a semi-infinite plane is described by a sum
of two Green’s functions, G(~r ′, ~r, k) = Gdir(~r ′, ~r, k ) +
Grefl(~r ′, ~r, k). The first contribution corresponds to the
direct path from ~r to ~r ′(black dotted line) and is in the
semiclassical approximation [Eq. (12)] given by
Gdir(~r ′, ~r , k) = eikL(~r ′,~r)−i3π/4−iµπ/2
p2πkL(~r ′, ~r )
=: GSCA(~r ′, ~r, k ),(A1)
where L(~r ′, ~r) is the distance between the two points
and µis the Maslov index (µ= 0 in this case). The
second term corresponds to a propagation via a classical,
specularly reflected path (blue solid line):
Grefl(~r ′, ~r , k) = eik[L(~r ′,~a)+L(~a,~r )]−i3π/4−iπ
p2πk [L(~r ′, ~a) + L(~a, ~r )] .(A2)
The orifice gives rise to an additional scattered wave for
which we assume in the far-field limit (|~
k~r | ≫ 1 and
|~
k~r ′| ≫ 1) a cylindrical wave emanating from the center
of the orifice (point ~
b). Invoking far-field approximations
is at the heart of semiclassical diffraction theories, the
validity of which need testing on a case-by-case basis (see
below).
The diffraction contribution to the Green’s function (red
dashed line in Fig. 15) is
GPSCA(~r ′, ~r, k ) = eikL(~r ′,~
b)−i3π/4
q2πkL(~r ′,~
b)
v(θ′, θ, k, d)×
eikL(~
b,~r)−i3π/4
q2πkL(~
b, ~r )
=GSCA(~r ′,~
b, k)v(θ′, θ, k, d)×
GSCA(~
b, ~r , k).(A3)
We refer to this term as the (first-order) pseudo-path
semiclassical contribution, because it corresponds to a
classically forbidden path. We note the different scal-
ing of GPSCA and Grefl with k. The amplitude of Grefl
scales as 1/pk(L1+L2) while the diffractive contribu-
tion scales as 1/√k2L1L2. At large distances (far-field)
and/or large k, the geometric reflection amplitude dom-
inates over the diffraction amplitude, as expected.
The diffraction coefficient v(θ′, θ, k, d) as a function of the
incoming and outgoing angles has been calculated in the
past in the Kirchhoff (KDA) and Fraunhofer diffraction
approximations (FDA). Both approaches have in com-
mon that the amplitude of the cylindrical diffractive wave
emanating from the orifice is determined from an inte-
gration over the orifice using as a source the amplitude
and the phase of the unperturbed incoming wave in the
absence of the boundary. In other words, they are im-
plementations of Huygens’ principle according to which
each point of the lead opening is the source of an out-
going circular wave. For details, we refer the reader to
Ref. 10 (for the KDA) and to Ref. 12,14 (for the FDA).
Keller’s geometric theory of diffraction (GTD)32 has a
different point of departure: it originates from the far-
field approximation of the exact Sommerfeld’s solution
for the wave scattering at a wedge58 [see Fig. 16 (a)].
In the following we discuss the application of the GTD
and its refinement within the framework of the uniform
theory of diffraction UTD30 to the problem of the diffrac-
tive scattering at the lead mouth [Figs. 16 (b) and (c)].
Starting point of our determination of the diffraction co-
efficient v(θ′, θ, k, d) is the decomposition of the orifice
into two wedges with an inner angle of π/2 and outer
angle of 3π/2 [see Fig. 16 (a)]. In the far-field limit, the
incident wave can be regarded asymptotically as a plane-
wave. The GTD describes the scattering of a plane wave
at an infinitely sharp wedge32 [Fig. 16 (a)]. The total
wavefunction is a sum of the incoming plane wave, the re-
flected plane wave and an outgoing cylindrical diffracted
16
FIG. 16: (Color online) (a) Diffraction at a wedge. (b)
Diffraction at a lead described as diffraction at two wedges.
The reference path is marked by a red dashed line. The angle
is counted positive for a path lying on the left of the verti-
cal axis, for a path on the right the angle is negative. (In
the present example θis negative and θ′is positive). The
points ~r and ~r ′are assumed to be in the far field region. (c)
Higher-order corrections enter via paths scattered between
the wedges.
wave, emanating from the edge. There are two discon-
tinuities: at the shadow boundary [φ′−φ=π, for the
definition of the angles see Fig. 16 (a)] and the bound-
ary of geometric reflection (φ′+φ=π). Outside a small
region around these two angles, the diffracted wave can
be described by an outgoing cylindrical wave modulated
by a smooth diffraction coefficient. The diffractive part
of the Green’s function between the two points ~r and ~r ′
in Fig. 16 (a) can thus be approximated as:65
GPSCA(~r ′, ~r , k) = GSCA(~r ′, ~r0, k )1
2D(φ′, φ), GSCA(~r0, ~r, k),
(A4)
with
D(φ′, φ) = −2sin π/N
N"1
cos π
N−cos φ′−φ
N
−1
cos π
N−cos φ′+φ
N#,(A5)
where N= 3/2 is the exterior angle (in units of π) of a
perpendicular wedge and ~r0is the position of the corner
of the wedge [Fig. 16 (a)].
We now consider the lead opening as being composed of
two wedges [Fig. 16 (b)]. The obvious conceptual diffi-
culty lies in the fact that the two wedges are, in general,
not in the far-field limit (kd ≫1) of each other. With
this caveat in mind, diffraction at the lead can be con-
sidered within the GTD to result from the interference
of two paths that are diffracted at the corners of the
two wedges limiting the lead. Summing up the diffrac-
tion weights and phases from the two paths, we obtain
the Green’s functions of the reference pseudo-path as in
Eq. (A3) but with the GTD reflection coefficient
vGTD(θ′, θ, k , d) = 1
2DL(θ′, θ)e−ik d
2(sin θ′+sin θ)
+1
2DR(θ′, θ)e+ik d
2(sin θ′+sin θ).
(A6)
In Eq. (A6) we have neglected the difference of the incom-
ing angles at the left and at the right wedge, respectively,
i.e., we set θ1=θ2=θ[Fig. 16 (c)]. Likewise, for the out-
going angles, we set θ′
1=θ′
2=θ′. The phase differences
of the right/left pseudo-path with respect to the reference
path emanating from the center of the orifice can then be
written in linear approximation with respect to the trans-
verse lead coordinate as k∆L=±ik d
2(sin θ′+ sin θ). The
coefficients in Eq. (A6) are defined as
DL(θ′, θ) = D(π/2−θ′, π /2−θ)
DR(θ′, θ) = D(π/2 + θ′, π /2 + θ).(A7)
We note that both DLand DRhave a singularity at the
reflection boundary for θ′=−θ. However, in the sum
of the two terms, the singularities cancel out and the re-
sulting backscattering amplitude is perfectly smooth (see
Fig. 17, red chain dotted line).
The comparison of the GTD with the exact quantum
calculation for the diffraction coefficient, |v(θ, θ′, k, d)|2
(Fig. 17), reveals sizeable deviations. Two deficiencies
are noteworthy: the almost complete missing of the
back-reflection peak and the failure at grazing angles
θ′, θ →6=π/2. The diffraction coefficient should approach
zero in this limit but, instead, converges toward a finite
value (Fig. 17). This behavior originates from treating
the diffraction at the lead as two independent local phe-
nomena of diffraction at two separate wedges.
The diffraction at a lead can be treated within the
UTD by using a double-wedge diffraction coefficient (see
Ref. 59,60 and a more recent paper for arbitrary configu-
rations of the wedges, Ref. 61). The double-wedge diffrac-
tion coefficient cannot be separated into a sequence of
single-wedge diffraction coefficients and contains rather
involved mathematical expressions such that the beauty
and structural simplicity of a semiclassical approach is
lost. We present in the following an ansatz for double-
wedge diffraction which assumes a separation of the
diffraction process into a sequence of diffraction events
and we verify its validity by comparing with quantum
mechanical results from Ref. 10. We show that the short-
comings of Eq. A6 can be (to a large extent) remedied by
taking diffractive paths of higher order into account, i.e.
paths that pass between the two wedges once or several
times. This drastically improves the agreement with the
quantum mechanical result. (We point to the conceptual
similarity of our approach to the treatment of double-
wedge diffraction illuminated by transition region fields
by a sum over higher-order diffracted fields.62 ) In a first
step, we include paths that scatter once between the two
edges [see green line Fig. 16 (c)]. There are two such
17
0
10
20
30
40
-0.4 -0.2 0 0.2 0.4
|v(θ’,θ,k,d)|2
θ’/π
θ=π/4
GTD: jmax=0
GTD-UTD: jmax=1
jmax=5
jmax=10
QM
FIG. 17: (Color online) Absolute square of the diffraction
coefficient |vGTD(θ′, θ, k , d)|2(or vGTD−UTD with jmax = 0)
compared with |vGTD−UTD(θ′, θ, k , d)|2for jmax = 1, jmax = 5
and jmax = 10 (for the definition of jmax see text), k= 2.5π/d.
The quantum mechanical (QM) result is taken from Ref. 10.
paths. One approaches the right wedge at angle θ. It
is scattered into the angle π/2 (with respect to the sur-
face normal), and at the left wedge, it is scattered into
the angle θ′. The other path is scattered from the left
wedge to the right one with the same entrance and exit
angles. The weights of this pair of paths cannot be deter-
mined from the GTD diffraction coefficients [Eq. (A5)]:
The GTD diffraction coefficient fails in the limit of φ→0
and φ′→π[definition of angles as in Fig. 16 (a)], because
this is in proximity of the shadow boundary, into which
the horizontal paths are scattered. This problem can be
overcome by invoking the uniform theory of diffraction
(UTD).
Contrary to the GTD, the UTD is also valid on the zone
boundaries. The outgoing cylindrical wave is multiplied
by a diffraction coefficient which depends not only on the
two angles φ′and φbut also on the distances r′and r,
and on the wavenumber k,
DUTD(φ′, φ, r′, r, k) = −eiπ
4
N×
X
σ,η=±1
σcot π+η(φ′−σφ)
2N×
Fkrr′
r+r′aη(φ′−σφ),(A8)
where a±(β) = 2 cos22πN n±−β
2and n±is the integer
which most closely satisfies 2πN n±−β=±π. The func-
tion Fis defined as a generalized Fresnel integral:
F(x) = −2i√xe−ix Z∞
√x
dτeiτ 2(A9)
and has the asymptotic form
F(x) = 1 −i1
2x−3
4
1
x2+.... (A10)
For x→ ∞, i.e., for large distances and outside the tran-
sition zones, F(x) = 1, and DUTD(φ′, φ, r′, r, k) reduces
to DGTD(φ′, φ). Within the transition zone, the distance
dependence of DUTD leads to a deviation of the scattered
wave from a purely cylindrical wave. This is necessary
to ensure the continuity of the total wave-function at
the zone boundaries. Furthermore, by using the UTD
diffraction coefficient we partially take into account the
fact that the two wedges are not in the far-field region
with respect to each other.
Since the GTD fails for large (near grazing) angles, we
opt for a piecewise construction: We combine GTD and
UTD referred to in the following as GTD-UTD such
that the diffraction of the path with the smaller (abso-
lute value) of the angle is treated by the GTD and the
path with the larger angle on the level of the UTD. Ac-
cordingly, to first order the GTD-UTD correction to the
diffraction coefficient can be written as
vGTD−UTD,1(θ′, θ, k, d) =
1
2UL(θ′,−π/2, d, k)eikd(−sin θ′+sin θ)/2
√2πkd
1
2DR(+π/2, θ)
+1
2UR(θ′,+π/2, d, k)eikd(+ sin θ′−sin θ)/2
√2πkd
1
2DL(−π/2, θ),
(A11)
where [in analogy to Eq. (A7)] we have defined the UTD
diffraction coefficients at the left (L) and right (R) wedge
as
UL(θ′, θ, r, k) = DUTD(π/2−θ′, π/2−θ, r′→ ∞, r, k)
UR(θ′, θ, r, k) = DUTD(π/2 + θ′, π/2 + θ, r′→ ∞, r, k).
(A12)
In Eq. (A11), the diffraction of the incoming path (with
angle θ), is treated on the GTD level and the diffrac-
tion of the outgoing path (angle θ′) on the UTD level.
This construction introduces a first-order discontinuity
(“kink”) at |θ|=|θ′|. This kink is, however, negligible
for small angles and visible only at large angles close to
π/2, in other words the formula breaks down in the limit
θ, θ′→π/2, see, e.g., Fig. 5(c). (One could alterna-
tively ignore the fact that the UTD is not multiplicative
in the near-field and employ for both scattering events
UTD, thus avoiding the kink. This ansatz, however,
breaks down similarly for grazing incidence: the diffrac-
tion coefficient does not approach zero for the outgoing
angle θ′→π/2. For small and medium angles this ap-
proach behaves equally well as the presented GTD-UTD
approach.) At large incident and/or outgoing angles the
diffraction coefficient is already strongly suppressed such
that the resulting error is small.
The inclusion of the first order GTD-UTD correction
(jmax = 1, see Fig. 17) already considerably improves the
agreement with the quantum diffraction pattern. Higher
order diffraction corrections include paths that are scat-
tered several times between the wedges. This includes
18
paths that are incident and backscattered at an angle
±π/2 at the wedge. This is exactly on the reflection
boundary. In this limit, Reiche has shown63 that the
diffraction pattern of a plane wave with unit amplitude
incident on the wedge with an angle of ±π/2 reduces to
a reflected plane wave with amplitude 1/2 and a cylin-
drical wave. The Green’s function of a higher order path
which scatters jtimes between the wedges is, therefore,
a product of the GTD diffraction coefficient, the UTD
diffraction coefficient, and the Green’s function for free
propagation along the distance jd, acquiring a factor 1/2
and a phase of πfor each reflection at a wedge. Summing
the diffraction corrections up to order jmax we obtain
vGTD−UTD(θ′, θ, k, d) = vGTD (θ′, θ, k, d) +
1
4
jmax
X
odd: j=1
UL(θ′,−π/2, jd, k)gj(k)eiΦ−+DR(+π/2, θ) + UR(θ′,+π/2, j d, k)gj(k)eiΦ+−DL(−π/2, θ) +
1
4
jmax
X
even: j=1
UR(θ′,+π/2, jd, k)gj(k)eiΦ++ DR(+π/2, θ) + UL(θ′,−π/2, j d, k)gj(k)eiΦ−−DL(−π/2, θ),
(A13)
where
gj(k) = 1
√2πkjd
1
2j−1ei(kjd+(j−1)π)(A14)
and
Φ±± =kd
2(±sin θ′±sin θ).(A15)
Fig. 17 demonstrates that the diffraction coefficient
v(θ′, θ, k, d)GTD−UTD is converged for jmax = 5. Fur-
thermore, the condition v(θ′, θ, k, d)→0 for θ, θ′→π/2
is fulfilled and the agreement with the fully quantum
mechanical backscattering weight is excellent. The
agreement deteriorates for increasing entrance angles
θbut is still satisfying compared to the simple Fraun-
hofer diffraction approximation (FDA), (see Fig. 5).
Fortunately, large angles do not play an important role
because the overall diffraction weight is very low. The
GTD-UTD provides a remarkable compromise between
simplicity and accurate representation of quantum
mechanical results for diffraction at a lead attached to a
semi-infinite half-plane.
FIG. 18: (Color online) A lead of width dcoupled to a half-
infinite plane. An incoming wave in mode mcan be separated
into two rays θ=±θm=±arcsin mπ/dk which diffractively
scatter at the lead wedges. The reference path is denoted by
a red dashed line.
2. The GTD-UTD for coupling from lead modes
into the cavity
The flux-normalized wave-function for mode min a
lead of width doriented parallel to the x-axis is
ψm(x, y) = s2
dkx,m
eikx,mxsin (ky,m y)
=−is1
2dkx,m
[ei(kx,mx+ky,m y)
−ei(kx,mx−ky,m y)],(A16)
where ky,m =mπ/d is the transverse momentum compo-
nent and kx,m =pk2−(mπ/d)2the longitudinal com-
ponent. Eq. (A16) can be viewed as two rays emanating
19
0
1
2
3
4
5
6
7
-0.4 -0.2 0 0.2 0.4
kx,2|c2(θ,k,d)|2
θ/π
(a) GTD: k=2.1π/d
GTD-UTD: k=2.1π/d
0
1
2
3
4
5
6
-0.4 -0.2 0 0.2 0.4
kx,2|c2(θ,k,d)|2
θ/π
(b) GTD: k=2.5π/d
GTD-UTD: k=2.5π/d
FIG. 19: (Color online) The absolute square of the coupling coefficient |cm(θ, k, d)|2multiplied by kx,m for m= 2 at two
different wavenumbers (a) k= 2.1π/d and (b) k= 2.5π/d within the GTD and the GTD-UTD. Note the breakdown of the
GTD for θ→ ±π/2 near k≈2π/d.
with angles ±θm=±arcsin(mπ/dk) (Fig. 18). In the
GTD approximation, each ray hits an edge of the lead
mouth and the two cylindrical waves emanating from the
edges cause, in turn, an interference pattern at large dis-
tances. The GTD diffraction for scattering from the lead
mode minto the half-space (in our application the cav-
ity) can be written in direct analogy with Eqs. A6 and
A7 as
cGTD
m(θ, k, d) = −ieimπ
2
√2dkx,m h1
2DL(θ, θm)eimπ
2e−ik d
2sin θ
−1
2DR(θ, θm)e−imπ
2eik d
2sin θi,(A17)
with the scattering coefficients at left and right wedge
DL(θ, θm) = D(π
2−θ, 3π
2−θm),
DR(θ, θm) = D(π
2+θ, 3π
2−θm).(A18)
The two paths have a phase difference of mπ at the lead
mouth and thus a phase difference ±mπ/2 relative to the
reference path that starts at the center of the orifice (see
Fig. 18). With the flux normalization factor pkx,m from
Eq. (2)pkx,mcGTD
m(θ, k, d) is dimensionless .
We include now higher-order scattering events on the
level of the UTD in order to improve Eq. (A17),
cGTD−UTD
m(θ, k, d) = cGTD
m(θ, k, d)−ieimπ
2
√2dkx,m ×
1
4"jmax
X
odd:j=1
UR(θ, +π/2, j d, k)gj(k)eiφ++DL(−π/2, θm)−UL(θ, −π/2, jd, k)gj(k)eiΦ−− DR(+π/2, θm)−
jmax
X
even:j=1
UR(θ, +π/2, j d, k)gj(k)eiΦ+−DR(+π/2, θm) + UL(θ, −π/2, j d, k)gj(k)eiΦ−+DL(−π/2, θm)#,
(A19)
where
Φ±± =±kd
2sin θ±mπ
2,(A20)
gj(k) is given in Eq. (A14) and UL,URare given by
Eq. (A12). Fig. 19 illustrates that the UTD corrections
20
become most important when the value of kis close to a threshold, i.e., when the angle θmis close to π/2.
∗Electronic address: iva.brezinova@tuwien.ac.at
1S. Datta, Electronic Transport in Mesoscopic Systems
(Cambridge University Press, 1995).
2E. Akkermans and G. Montambaux, Mesoscopic Physics
of Electrons and Photons (Cambridge University Press,
2006).
3C. W. J. Beenakker and H. van Houten, Solid State Physics
4, 1 (1991).
4M. C. Gutzwiller, Chaos in Classical and Quantum Me-
chanics (Springer Verlag, New York, 1991).
5N. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315
(1972).
6M. Brack and R. K. Bhaduri, Semiclassical Physics (Fron-
tiers in Physics, Westview Press, 2003).
7R. Bl¨umel and U. Smilansky, Phys. Rev. Lett. 64, 241
(1990).
8H. U. Baranger, R. A. Jalabert, and A. D. Stone, Chaos
3, 665 (1993).
9T. Blomquist, Phys. Rev. B 66, 155316 (2002).
10 C. D. Schwieters, J. A. Alford, and J. B. Delos, Phys. Rev.
B54, 10652 (1996).
11 T. Blomquist and I. V. Zozoulenko, Phys. Rev. B 64,
195301 (2001).
12 L. Wirtz, J.-Z. Tang, and J. Burgd¨orfer, Phys. Rev. B 56,
7589 (1997).
13 L. Wirtz, J.-Z. Tang, and J. Burgd¨orfer, Phys. Rev. B 59,
2956 (1999).
14 L. Wirtz, C. Stampfer, S. Rotter, and J. Burgd¨orfer, Phys.
Rev. E 67, 016206 (2003).
15 C. Stampfer, S. Rotter, J. Burgd¨orfer, and L. Wirtz, Phys.
Rev. E 72, 036223 (2005).
16 I. Bˇrezinov´a, C. Stampfer, L. Wirtz, S. Rotter, and
J. Burgd¨orfer, Phys. Rev. B 77, 165321 (2008).
17 S. Rahav and P. W. Brouwer, Phys. Rev. Lett. 95, 056806
(2005).
18 S. Rahav and P. W. Brouwer, Phys. Rev. B 73, 035324
(2006).
19 P. Jacquod and R. S. Whitney, Phys. Rev. B 73, 195115
(2006).
20 K. Richter and M. Sieber, Phys. Rev. Lett. 89, 206801
(2002).
21 P. Braun, S. Heusler, S. M¨uller, and F. Haake, J. Phys. A.:
Math. Gen. 39, L159 (2006).
22 S. Heusler, S. M¨uller, P. Braun, and F. Haake, Phys. Rev.
Lett. 96, 066804 (2006).
23 P. W. Brouwer and S. Rahav, Phys. Rev. B 74, 085313
(2006).
24 P. Pichaureau and R. A. Jalabert, European Physical Jour-
nal B 9, 299 (1999).
25 E. Bogomolny, Nonlinearity 13, 947 (2000).
26 N. Argaman, Phys. Rev. B 53, 7035 (1996).
27 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and
Path Integrals (MacGraw-Hill, New York, 1965).
28 A. M. Chang, H. U. Baranger, L. N. Pfeiffer, and K. W.
West, Phys. Rev. Lett. 73, 2111 (1994).
29 C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F.
Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506
(1992).
30 R. Kouyoumjian and P. H. Pathak, Proc. IEEE 62, 1448
(1974).
31 M. Sieber, N. Pavloff, and C. Schmit, Phys. Rev. E 55,
2279 (1997).
32 J. B. Keller, J. Opt. Soc. Am 52, 116 (1962).
33 S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgd¨orfer,
Phys. Rev. B 62, 1950 (2000).
34 S. Rotter, B. Weingartner, N. Rohringer, and
J. Burgd¨orfer, Phys. Rev. B 68, 165302 (2003).
35 H. Ishio and J. Burgd¨orfer, Phys. Rev. B 51, 2013 (1995).
36 D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).
37 T. Szeredi and D. A. Goodings, Phys. Rev. E 48, 3529
(1993).
38 S. Hikami, Phys. Rev. B 24, 2671 (1981).
39 A. Andreev, Sov. Phys. JETP 19, 1228 (1964).
40 X. Yang, H. Ishio, and J. Burgd¨orfer, Phys. Rev. B 52,
8219 (1995).
41 J. Tworzyd lo, A. Tajic, and C. W. J. Beenakker, Phys.
Rev. B 69, 165318 (2004).
42 P. Jacquod and E. V. Sukhorukov, Phys. Rev. Lett. 92,
116801 (2004).
43 M. Y. Kharitonov and K. B. Efetov, Phys. Rev. B 78,
033404 (2008).
44 B. Weingartner, S. Rotter, and J. Burgd¨orfer, Phys. Rev.
B72, 115342 (2005).
45 M. Hartung, T. Wellens, C. A. M¨uller, K. Richter, and
P. Schlagheck, Phys. Rev. Lett. 101, 020603 (2008).
46 E. Larose, L. Margerin, B. A. van Tiggelen, and
M. Campillo, Phys. Rev. Lett. 93, 048501 (2004).
47 J. P. Robinson, H. Schomerus, L. Oroszl´any, and V. I.
Fal’ko, Phys. Rev. Lett. 101, 196803 (2008).
48 M. Kopp, H. Schomerus, and S. Rotter, Phys. Rev. B 78,
075312 (2008).
49 V. A. Gopar, S. Rotter, and H. Schomerus, Phys. Rev. B
73, 165308 (2006).
50 Y. M. Blanter and M. B¨uttiker, Phys. Rep. 336, 1 (2000).
51 F. Aigner, S. Rotter, and J. Burgd¨orfer, Phys. Rev. Lett.
94, 216801 (2005).
52 S. Rotter, F. Aigner, and J. Burgd¨orfer, Phys. Rev. B 75,
125312 (2007).
53 B. A. Khoruzhenko, D. V. Savin, and H.-J. Sommers, Phys.
Rev. B 80, 125301 (2009).
54 M. Novaes, Phys. Rev. B 75, 073304 (2007).
55 J. Tworzyd lo, A. Tajic, H. Schomerus, and C. W. J.
Beenakker, Phys. Rev. B 68, 115313 (2003).
56 C. H. Lewenkopf, E. R. Mucciolo, and A. H. CastroNeto,
Phys. Rev. B 77, 081410(R) (2008).
57 J. Tworzyd lo, B. Trauzettel, M. Titov, A. Rycerz, and
C. W. J. Beenakker, Phys. Rev. Lett. 96, 246802 (2006).
58 A. Sommerfeld, Vorlesungen ¨uber Theoretische Physik,
Optik (Dieterich’sche Verlagsbuchhandlung, 1950).
59 M. Schneider and R. J. Luebbers, IEEE Trans. Antennas
Propag. 39, 8 (1991).
60 M. Schneider, A uniform solution of double wedge-edge
diffraction (Ph.D. thesis, Pennsylvania State University,
1988).
61 M. Albani, IEEE Trans. Antennas Propag. 53, 702 (2005).
62 P. D. Holm, IEEE Trans. Antennas Propag. 44, 879 (1996).
21
63 F. Reiche, Annalen der Physik 37, 131 (1911).
64 T. Dollinger et al., (unpublished).
65 The factor 1/2 (atomic units) in Eq. (A4) originates from
~2/2m(SI units) and assures that the Green’s function is
correctly normalized [such that ( ~2k2
2m−ˆ
H~r ′)G(~r ′, ~r, k) =
δ(~r ′−~r)]. We use the same normalization as in Ref. 6,
section 7.5.4.
