Semiclassical theory for transmission through open billiards: Convergence towards
quantum transport
Ludger Wirtz, Christoph Stampfer, Stefan Rotter, and Joachim Burgdo¨rfer Institute for Theoretical Physics, Vienna University of Technology, A-1040 Vienna, Austria
共Received 6 June 2002; published 13 January 2003兲
We present a semiclassical theory for transmission through open quantum billiards which converges towards quantum transport. The transmission amplitude can be expressed as a sum over all classical paths and pseudo-paths which consist of classical path segments joined by ‘‘kinks,’’ i.e., diffractive scattering at lead mouths. For a rectangular billiard we show numerically that the sum over all such paths with a given number of kinks K converges to the quantum transmission amplitude as K→⬁. Unitarity of the semiclassical theory is restored as K approaches infinity. Moreover, we find excellent agreement with the quantum path-length power spectrum up to very long path length.
DOI: 10.1103/PhysRevE.67.016206 PACS number共s兲: 05.45.Mt, 73.23.Ad, 73.50.Bk
I. INTRODUCTION
The aim of semiclassical theory is to bridge the gap be-tween quantum mechanics and its classical limit. Generi-cally, probability amplitudes are calculated by summing over classical paths, each of which carries an amplitude and a phase 关1,2兴. Such an approach facilitates an intuitive under-standing of basic features of quantum mechanics such as ‘‘quantum interference’’ and allows quantitative calculations in the regime of high energies, i.e., short wavelength→0, where full quantum calculations may become impractical. Moreover, semiclassical theory plays an important role in elucidating the signatures of classical chaos in quantum sys-tems whose classical counterpart is chaotic 关1兴. Ballistic transport through billiards has become a popular prototype example关3–13兴: All paths that connect the entrance lead 共or injection quantum wire兲 with the exit lead 共or emission quan-tum wire兲 contribute to the transmission amplitude Tm⬘,m from the mthmode in the entrance lead to the (m
⬘
)thmode in the exit lead. Inside the billiard, i.e., a two-dimensional cav-ity at constant potential, the trajectories are straight lines which are specularly reflected at hard walls. Despite the con-ceptual simplicity of the semiclassical description of ballistic transport, recent applications have revealed fundamental dif-ficulties of the semiclassical theory关4,7–9,14,15,26兴: among many others, unitarity is badly violated with discrepancies in some cases as large as the conductance fluctuations the theory attempts to describe 关8,9兴. Consequently the correla-tion ␦兩T兩2⫽⫺␦兩R兩2 between transmission 共or conductance兲 fluctuations,␦兩T兩2, and the corresponding fluctuations in the reflection 共or resistance兲, ␦兩R兩2, as a function of the wave number k is broken. Also, the ‘‘weak localization’’ effect is considerably underestimated关14,15兴.At first glance, the shortcomings of the semiclassical ap-proximation are not surprising. Hard-walled billiards possess ‘‘sharp edges’’ at the entrance and exit leads. At such points, the length scale aP of spatial variations of the potential ap-proaches zero. Consequently the semiclassical limit /aP
Ⰶ1 cannot be reached no matter how small 共or large k) is.
An obvious improvement of the semiclassical description can be achieved by including into the coupling between the
quantum wire and the billiard effects of diffraction at sharp edges within the framework of the Kirchhoff diffraction关6兴 or Fraunhofer diffraction关7兴. However, despite considerable improvement achieved, the fundamental shortcomings are not accounted for. Remarkably, the discrepancies between the semiclassical path-length power spectrum Pmsc⬘,m(ᐉ) and the corresponding quantum path-length spectrum,
Pmqm⬘,m共ᐉ兲⫽
冏
冕
dkeikᐉTm⬘,m共k兲冏
, 共1.1兲 are more pronounced for classically regular billiards such as the rectangular 关7兴 or the circular 关5,6,26兴 billiard than for chaotic structures such as the Bunimovich stadium 关16– 18,26兴. In other words, the path-length spectrum of a regular system where the number of paths grows linearly as a func-tion of length, Ncᐉ(ᐉ)⬀ᐉ, is much more sensitive to the approximation of the Feynman path integral by the sum over classical paths than the exponentially proliferating path-length spectrum, Ncᐉ(ᐉ)⬀exp(ᐉ), of a chaotic cavity. This observation strongly hints at the lack of missing 共non兲 classical paths as the culprit for the failure. Another hint is provided by the breakdown of the one-to-one correlation be-tween transmission and reflection fluctuations. As classical trajectories that are either ejected through the exit lead con-tributing to T or return back to the entrance lead concon-tributing to R are disjunct subsets, the inequality (␦兩T兩2)sc⫽⫺(␦兩R兩2)
scis anything but surprising and indicates that ad-ditional paths, pseudopaths referred to in the following, are required to couple these disjunct subsets and thereby restore the correlation between transmission and reflection.
pseudo-paths for scattering at open billiards. To this end the quantum transmission amplitude is expanded as a multiple scattering series. In its semiclassical limit, propagation between subse-quent scattering events can be identified as proceeding along classical paths while each scattering corresponds to a non-classical diffractive deflection共‘‘kink’’兲 at the sharp edges of the lead walls. We perform our analysis for a rectangular billiard with exit and entrance leads at opposite sides of the structure共Fig. 1兲 for which all paths and pseudopaths can be easily enumerated 关34兴 and the path sum can be performed until convergence is approximately reached. We find excel-lent agreement with the full quantum calculation for the path-length spectrum and convergence towards the unitarity limit.
The plan of the paper is as follows: In Sec. II we briefly review the standard semiclassical approximation to this prob-lem. In Sec. III we motivate the present multiple scattering approach employed in the enumeration of pseudopaths by revisiting the one-dimensional square well problem. Tran-scription of this problem to the rectangular billiard allows the enumeration of pseudopaths as discussed in Sec. IV. Explicit expressions for the diffractive amplitudes entering the present semiclassical theory at the level of Fraunhofer dif-fraction approximation共FDA兲 are given in Sec. V. Numerical results and comparison with the full quantum results are dis-cussed in Sec. VI followed by a short summary and an out-look to future applications of this approach.
II. STANDARD SEMICLASSICAL APPROXIMATION
The conductance g for ballistic transport as a function of the wave number k through an open billiard is given by the Landauer formula关22兴, g共k兲⫽2e 2 h
冉
m兺
⫽1 N兺
m⬘⫽1 N 兩Tm⬘,m共k兲兩2冊
, 共2.1兲 where N is the number of open modes in the leads共quantum wires兲 of width d. Generically, semiclassical approximations to the transmission amplitudes Tm⬘memploy three steps each of which is connected with a stationary phase approximation共SPA兲 关23兴:
共1兲 The quantum mechanical Feynman propagator KF(rជ
⬘
,rជ,t) 关24兴 leads, after application of the SPA, to the semiclassical Van Vleck propagator KV(rជ⬘
,rជ,t), which con-tains the sum over all classical paths connecting rជ and rជ⬘
in time t关2,25兴.共2兲 The Fourier-Laplace transform of KV(rជ
⬘
,rជ,t) to the semiclassical Green’s propagator Gsc(rជ⬘
,rជ,E), which de-scribes the probability amplitude for propagation from rជ tor
ជ
⬘
at a fixed energy E, is performed by SPA leading to a sum over all classical paths of energy E connecting these two points: Gsc共y 2, y1,k兲⫽ 1 共2i兲1/2y1兺
→y2 兩Dp共y2 , y1,k兲兩1/2 ⫻exp冋
i冉
lp共y2,y1兲⫺ 2p册
. 共2.2兲 Here, lp(y2, y1) denotes the length and p denotes the Maslov index of the path p. We denote the transverse coor-dinates in the entrance/exit lead by y1,2 and suppress the corresponding x coordinates in Eq. 共2.2兲. Dp denotes the weighting factor共deflection factor兲 of the path.共3兲 The transmission amplitudes 共and, equally, reflection
amplitudes兲 from the mode m to the mode m
⬘
are customar-ily expressed as the projection of the Green’s function关evaluated at the energy E⫽ប2k2/(2m)] onto the transverse wave functions m(y1) and m⬘( y2) of the incoming and outgoing modes 关4兴,
Tm⬘,m共k兲⫽⫺
冑
vx2,m⬘vx1,m冕
dy2冕
d y1m*⬘共y2兲⫻GSC共y
2, y1,k兲m共y1兲. 共2.3兲 This double integral is frequently calculated in the SPA as well. This selects those classical trajectories p that enter the billiard with the quantized angle
m⫽sin⫺1
m
dk 共2.4a兲
and exit the billiard at the quantized angle m⬘⫽sin⫺1
m
⬘
dk . 共2.4b兲
In an earlier paper关7兴 we have demonstrated that the SPA in the third step can be avoided by linear expansion of the ex-ponent in Eq. 共2.3兲 allowing for an analytical evaluation of the double integral. Using lead wave functions with longitu-dinal momentum
km⫽
冑
k2⫺兩m/d兩2 共2.5兲 and transverse wave functions⌽m共y兲⫽
冑
2 d再
cos冉
m d y冊
, m odd sin冉
m d y冊
m even, 共2.6兲each integral corresponds to a Fraunhofer diffraction inte-gral. This automatically includes diffractive effects on the
level of the Fraunhofer approximation and yields results comparable to the explicit inclusion of diffractive effects us-ing Kirchhoff theory关6兴. The physical picture that emerges is that classical trajectories representing incoming 共outgoing兲 flux no longer enter共exit兲 the billiard at quantized anglesm
关see Eq. 共2.4兲兴 but with a continuous distribution of angles given by the corresponding diffraction integral. The imple-mentation of this class of diffractive effects for the trajecto-ries leads to a considerable improvement in the transmission and reflection coefficients for low modes and in the path-length spectrum where the position and height of many peaks could be identified with classical paths. Nevertheless the fun-damental difficulties of the semiclassical approximation per-sist. In particular:
共a兲 The unitarity condition
兺
m⬘⫽1 N 兩Tm⬘,m共k兲兩2⫹兺
m⬘⫽1 N 兩Rm⬘,m共k兲兩2⫽1 共2.7兲 is violated 关7,8,26兴. The fluctuations of the semiclassical conductance g around the exact value remains approximately constant and does not decrease with increasing k 共or →0).共b兲 The path-length power spectrum displays a dramatic
overestimate of contributions for long paths 关7–9,26兴 共see Fig. 8 below兲. The position of the peaks is reproduced re-markably well by the semiclassical approximation. However, the approximately exponential decay of the quantum spec-trum contrasts with the inverse linear (l⫺1) decay of the semiclassical spectrum.
It is instructive to classify the standard semiclassical theory in terms of the number of the SPA’s employed. De-pending on the starting point of the description in either time or energy domain one or two SPA’s are involved 关see Eq.
共2.2兲兴 in the propagation. If one neglects diffraction during
the injection and emission, two more SPA’s are needed. Stan-dard semiclassical approximations共SCA兲 are therefore char-acterized by a fixed number of SPA’s with the minimum of at least one, that is, when one starts from a time-independent constant energy description and employs a diffraction ap-proximation for the coupling in and out of the billiard struc-ture.
Going beyond the standard approximation requires taking into account nonclassical paths during the propagation inside the billiards in line with the original Feynman propagator. Identifying and enumerating the relevant nonclassical paths to be included can be performed by casting the quantum problem in a multiple scattering problem. The result will be characterized by an increasing number of SPA’s with an in-creasing number of nonclassical paths.
III. MULTIPLE SCATTERING THEORY: THE ONE-DIMENSIONAL SQUARE WELL
POTENTIAL REVISITED
Our point of departure for the development of the present semiclassical description of quantum transport is the time-independent quantum scattering wave function for multiple scattering. In order to motivate our strategy for enumerating
all classical and nonclassical paths included in the scattering wave function, we revisit the well-known one-dimensional square well problem. Scattering at the square well共SW兲 is a standard problem treated in most quantum mechanics text books关27兴 and easily solved by matching the wave function and its derivative at the two edges of a square well potential of width L and depth⫺V0shown in Fig. 2. The transmission amplitude T(SW) is proportional to the amplitude of the wave function on the right hand side of a square well. In a standard semiclassical approach which is based exclusively on classi-cally allowed paths, there is only one classical path transmit-ted through the well, and consequently
兩TSC
(SW)兩⫽1 共3.1兲
at variance with quantum results. This discrepancy is not surprising as at the edges of a well, the semiclassical crite-rion /ap→0 is violated. We reformulate now the quantum scattering problem in terms of multiple scattering at the two edges. To this end, we consider the square well as a structure composed of three separate substructures, the left edge, the interior of the well, and the right edge 共Fig. 2兲, for each of which we determine separate amplitudes. The transmission amplitude for an incoming wave from the external region from the left with k(e)⫽
冑
2E into the interior of the well with k(i)⫽冑
2(E⫹V0), i.e., forward scattering amplitude at the left edge is given byt(L)⫽2
冑
k(i)k(e)
k(i)⫹k(e). 共3.2兲
Correspondingly, the backscattering共or reflection amplitude兲 at the left edge from the exterior region back into the exterior region is given by
r(L)⫽k
(e)⫺k(i)
k(e)⫹k(i). 共3.3兲
The propagation through the interior of the well from the left to the right (LR) 关or from the right to the left (RL)] is given by the Green’s function
G(LR)共xR,xL兲⫽eik (i)(x
R⫺xL)⫽eik(i)L⫽G(RL)共x
L,xR兲.
共3.4兲
The corresponding transmission amplitude for the right edge is
t(R)⫽2
冑
k(i)k(e)
k(e)⫹k(i), 共3.5兲
i.e., t(R)⫽t(L)⫽t. The backscattering amplitude for a wave approaching either edge from the interior is given by
r⫽k
(i)⫺k(e)
k(i)⫹k(e), 共3.6兲
i.e., r⫽⫺r(L). The transmission amplitude through the com-posite structure, T, can now be written as a multiple scatter-ing series of repeated traversals through the structure 关see Fig. 2共b兲兴 each calculated with the help of the elementary amplitudes for transmission, reflection, and propagation
关Eqs. 共3.2兲–共3.4兲兴, T⫽t(R)关G(LR)共L,0兲⫹G(LR)共L,0兲rG(RL)共0,L兲rG(LR) ⫻共L,0兲⫹•••兴t(L) ⫽t(R)G(LR)共L,0兲
冉
兺
j⫽0 ⬁ 关rG(RL)共0,L兲rG(LR)共L,0兲兴j冊
t(L). 共3.7兲Inserting explicit expressions 关Eqs. 共3.2兲–共3.6兲兴 gives the geometric series
T⫽t2eik(i)L
兺
j⫽0 ⬁
共r2e2ik(i)L兲j 共3.8兲 with the result
T⫽ 1 cos共k(i)L兲⫺i⑀ 2sin共k (i)L兲 , 共3.9兲 where ⑀⫽k (e) k(i)⫹ k(i) k(e). 共3.10兲
Analogously, the reflection amplitude for the composite structure is given by R⫽r(L)⫹t(L)G(LR)共L,0兲
兺
j⫽0 ⬁ 关rG(RL)共0,L兲rG(LR)共L,0兲兴j ⫻rG(RL)共0,L兲t(L). 共3.11兲Inserting explicit expressions for the substructures yields
R⫽⫺r
冉
1⫺t2e2iLk(i)兺
j⫽0 ⬁ 共r2e2ik(i)L兲j冊
⫽⫺r冉
1⫺ t 2e2ik(i)L 1⫺r2e2ik(i)L冊
. 共3.12兲Obviously, these series of multiple scattering at the discon-tinuous edges converges towards the exact quantum result. Unitarity is trivially satisfied (兩R兩2⫹兩T兩2⫽1). Remarkably, we are not aware of a discussion of this intuitive derivation of the square well transmission problem in any standard quantum mechanics textbook. The key point in the present context is now that the formulation of the exact quantum scattering in terms of multiple traversals can be rephrased in terms of a sum over paths, in the following referred to as pseudopaths, which consist of segments of classical paths connected by amplitudes for nonclassical scattering at edges. In one dimension, the semiclassical propagator coincides with the quantum propagator 关Eq. 共3.4兲兴. Accordingly, the geometric series关Eqs. 共3.8兲 and 共3.12兲兴 can be interpreted as a sum over pseudopaths characterized by an increasing num-ber of traversals through the structure before exiting on ei-ther side. Each traversal corresponds to a classical path seg-ment described by a semiclassical propagator. Edge scattering must be described by a quantum scattering ampli-tude which is an obvious consequence of the fact that at the edge /ap→⬁ no matter how large k is and, therefore, the semiclassical limit is never reached. In one dimension, the semiclassical description in terms of a complete set of pseudopaths is naturally equivalent to the full quantum scat-tering amplitude. The nontrivial generalization of this ap-proach to two 共or higher兲 dimensions is at the core of the present semiclassical approach. In such a case, the sum over pseudopaths is no longer equivalent to the full quantum scat-tering process but provides a systematic approximation tech-nique to include nonclassical effects共or equivalently, contri-butions in increasing orders of ប) into the semiclassical description.
IV. TRANSMISSION THROUGH A RECTANGULAR BILLIARD: FROM QUANTUM
TO SEMICLASSICAL DESCRIPTION
The quantum transport problem through a rectangular bil-liard with opposite leads共Fig. 1兲 with width D and length L can be formulated in terms of a multiple scattering series in direct analogy to the 1D square well. Accordingly, we de-compose the transmission problem into three pieces共Fig. 3兲: the injection 共or transmission兲 from the entrance lead 共or
FIG. 3. Decomposition of a rectangular billiard into three sepa-rate substructures: a junction from a narrow to a wide constriction, a wide constriction of length L, and a junction from a wide to a narrow constriction. Transmission through the junction⫽t, reflec-tion at the juncreflec-tion⫽r, and propagation in between for left to right ⫽G(LR)共or right to left ⫽G(RL)
quantum wire兲 to the left into the billiard t(L)the propagation from the left to the right G(LR)or from the right to the left,
G(RL), of the cavity, and the emission共or transmission兲 from the interior into the exit lead to the right t(R). Likewise, the electron approaching the billiard can be reflected at the en-trance lead with amplitudes r(L) or can be reflected at each junction from the wide to the narrow constriction if the wave approaches the lead mouth from the interior with amplitude
r. The scattering amplitudes at each lead mouth representing
a discontinuity in the potential become now matrices with indices referring to the transverse mode number, e.g., tn,m(L) where m refers to the mode number in the lead and n to the mode number in the rectangular billiard. These scattering amplitudes at the lead mouth共or junctions between constric-tions of different widths兲 cannot be satisfactorily described by a true semiclassical description at /ap→⬁ for any en-ergy of the scattered particle. One can, instead, employ either a full numerical solution of the quantum problem for each junction or, alternatively, an approximate analytic approxi-mation in terms of ‘‘diffraction integrals’’ which are a large k approximation and hence close in spirit to a semiclassical approximation. An explicit evaluation of t and r in terms of Fraunhofer diffraction integrals will be given below.
The multiple scattering expansion of the transmission am-plitude T and the reflection amam-plitude R are given in analogy to Eqs.共3.7兲 and 共3.11兲 by matrix equations,
T共E兲⫽t(R)G(LR)
冉
兺
j⫽0 ⬁ 共rG(RL)rG(LR)兲j冊
t(L), 共4.1兲 and R共E兲⫽r(L)⫹t(L)G(LR)冉
兺
j⫽0 ⬁ 共rG(RL)rG(LR)兲j冊
rG(RL)t(L). 共4.2兲Here and in the following we suppress the energy E or wave number k dependence for notational simplicity. For the propagator we choose a mixed representation which is local in x, and employs a spectral sum over transverse modes,
G(LR)共xR,xL兲⫽G(RL)共xL,xR兲⫽
兺
n 兩n典
eikn(xR⫺xL)
具
n兩. 共4.3兲The transverse modes in the billiard are described by the wave functions n共y兲⫽
冦
冑
2 D cos冉
n D y冊
, n odd冑
2 D sin冉
n D y ,冊
n even, 共4.4兲and xR,L are the x coordinates of the right 共left兲 lead with
xR⫺xL⫽L. Equations 共4.1兲 and 共4.2兲 represent a full
quan-tum description of the transport amplitudes T and R in terms of a multiple scattering series. In the following we will in-vestigate its semiclassical limit.
1. Single and double traversals
In this subsection we begin to develop an improved semi-classical approximation to the quantum mechanical expres-sions Eqs. 共4.1兲 and 共4.2兲 for certain classes of short paths, featuring few traversals through the structure. The key fea-ture of this derivation is the transition from discrete mode numbers to continuous angles of incident trajectories.
The amplitude for transmission from mode mLin the left lead to mR in the right lead关Eq. 共4.1兲兴 reads explicitly for up to three successive traversals
TmR,mL⬇
兺
n tmR,n (R) eiknLt n,mL (L) ⫹兺
n,n⬘,n⬙ tm R,n⬙ (R) eikn⬙L ⫻rn⬙,n⬘e ikn⬘Lr n⬘,ne iknLt n,mL (L) ⫹••• 共4.5兲 with kn⫽冑
k2⫺共n/D兲2.The crucial step towards a semiclassical approximation to Eqs. 共4.1兲 and 共4.5兲 is now to associate the traversals with classical paths. For clarity we present first the derivation for a path that traverses the cavity just once corresponding to the first term in Eq.共4.5兲. We then extend our derivation to mul-tiple traversals through the cavity and show that the sum over multiple traversals can be performed by mapping the problem on combinatorics of lattice vectors in an extended zone scheme of a rectangular lattice. The method of the ex-tended zone scheme has also been used to determine a semi-classical expansion of the transmission amplitudes in terms of a finite number of continued fractions 关28兴.
In order to associate the first term in Eq. 共4.5兲 with clas-sical paths we have to map the transverse quantum number n onto an injection angle . For this task we employ the Pois-son sum formula 关29兴. Without loss of generality we take here and in the following mL and mR to be odd integers. Because of inversion symmetry, the subspaces of even and odd mode number completely decouple. Analogous expres-sions can be derived for even mL,R. The single traversal共t1兲 contribution becomes Tm R,mL (t1) 共k兲⫽1 4 ␣⫽⫺⬁
兺
⬁冕
⫺⬁ ⬁ tmR共兲tmL共兲 ⫻eiL冑
k2⫺冉
D冊
2⫹i␣(⫺1)d. 共4.6兲 The sum over integers n can be replaced by an integral over the continuous variable . Since the 共discrete兲 mode num-bers n are associated with quantized angles sin⫽n/(kD), the integration over the variable can be associated with an angle integration via the substitution⫽kD sin共兲→d⫽kD
waves, we neglect the latter which would make only an ex-ponentially small contribution to the transport for large L. With the substitution 关Eq. 共4.7兲兴 we arrive at
Tm R,mL (1) ⫽kD 4 ␣⫽⫺⬁
兺
⬁冕
⫺/2 /2 tm R d 共兲t mL d 共兲⫻eik[L cos()⫹␣D sin()]⫺i␣cos共兲d, 共4.8兲 where tm
L
d (),t mR
d () are the injection共ejection兲 amplitudes and the transverse mode index n is converted to a continuous angle variable. For these amplitudes we will later employ the FDA in line with previous semiclassical treatments 关7兴. The action in the exponent of Eq. 共4.8兲
S共兲⫽k关L cos共兲⫹␣D sin共兲兴, 共4.9兲
has a simple geometric interpretation in the extended zone scheme of the rectangular lattice 关Fig. 4共a兲兴. The action is that of a path connecting the center of the left side of the unit cell, the position of the entrance lead, with the center of the right side of the␣th replica of the unit cell in the transverse direction. The path possesses at most one discontinuity, i.e., a discontinuous displacement of the trajectory⌬rជ perpendicu-lar to the direction of propagation such that kជ⌬rជ⫽0 and no contribution to the classical action is accumulated along the displacement⌬rជ. Note that Eq.共4.8兲 is, apart from neglect-ing evanescent modes, equivalent to the original quantum expression for single traversals. This set of straight-line paths connecting the entrance lead with any one of the replicas of the exit lead featuring no more than one lateral displacement amounts to the representation of a full path sum of the Feyn-man propagator for this process.
We derive now the semiclassical limit of Eq. 共4.8兲 by applying the SPA. The SPA is valid if the action S() is rapidly varying on a scale of 2, i.e., if kLⰇ or kDⰇ. Since for a given mode n, k⯝n/d, these conditions are only fulfilled for all n provided that D/dⰇ1 or L/dⰇ1, i.e., if at least one of the linear dimensions of the rectangular billiard is large compared to the lead width. With this restric-tion in mind, we obtain from the starestric-tionary phase condirestric-tion 0⫽S
⬘
共兲⫽k关⫺L sin共兲⫹␣D cos共兲兴 共4.10兲the ‘‘stationary’’ angles
␣⫽arctan␣
D
L , 共4.11兲
which coincide with the angles of the classical straight-line paths of length l␣connecting the lattice point of the entrance lead with the lattice point of the␣th replica of the exit lead
关Fig. 4共b兲兴. The stationarity condition eliminates the
discon-tinuous displacement and renders the paths fully classical. The number␣ corresponds to the number of horizontal zone boundaries the trajectory has crossed. The second derivative yields the deflection factor for each path
S
⬙
共兲⫽⫺k关L cos共兲⫹␣D sin共兲兴 ⫽⫺kL共L
2⫹␣2D2兲cos共
␣兲⫽⫺kl␣. 共4.12兲 We thus obtain the semiclassical transmission coefficient for single traversals Tm R,mL (t1) ⫽kD 2
冑
1 2i␣⫽⫺⬁兺
⬁冑
1 kl␣e i(kl␣⫺␣)t mR d 共 ␣兲 ⫻cos␣tmL d 共 ␣兲. 共4.13兲The phase ␣ in Eq. 共4.13兲 is associated with the Maslov index for ␣ crossings of the ‘‘hard-wall’’ boundaries 共i.e.,
horizontal zone boundaries兲 the trajectory has crossed on its way to the␣thexit lattice point关see Fig. 4共b兲兴. The classical path sum in Eq.共4.13兲 corresponds to a sum over all lattice vectors connecting the lattice point of the entrance lead with exit points in the␣thunit cell.
The next term in the multiple scattering expansion关Eqs.
共4.1兲, 共4.2兲兴 corresponds to the double traversal t2. A t2
contribution does not appear in the expansion of the trans-mission amplitude关Eq. 共4.5兲兴. Note that odd 共even兲 number of traversal correspond to transmission共reflection兲. Accord-ingly, the lowest-order reflection amplitude is
Rm L⬘,mL (t2) ⫽
兺
n,n⬘ tm L⬘,n⬘ (L) eikn⬘Lr n⬘,neiknLtn,mL (L) . 共4.14兲Before again applying the Poisson formula, we decompose the internal reflection amplitude at the vertical wall into the geometric hard-wall reflection amplitude of the closed bil-liard and the diffractive amplitude for the lead opening
rn⬘,n⫽⫺␦n⬘,n⫹rn⬘,n d
. 共4.15兲
The term ⫺␦n⬘,n corresponds to geometric scattering at the hard vertical wall of the closed billiard including its phase jump共Maslov index兲 while the amplitude rd corresponds to the diffractive scattering at the mouth of the lead. Accord-ingly, Eq. 共4.14兲 can be decomposed into terms involving different numbers of diffractive scatterings inside the cavity,
Rm L⬘,mL (t2) ⫽
兺
n tm L⬘,n (L) ei(2knL⫺)t n,mL (L) ⫹兺
n,n⬘ tm L⬘,n⬘ (L) eikn⬘Lr n⬘,n d 共k兲eiknLt n,mL (L) . 共4.16兲The first term with zero internal diffractive scattering is for-mally completely equivalent to the first term for the trans-mission amplitude in Eq. 共4.5兲, while the second term with one internal diffractive scattering contains one additional sum over transverse mode numbers. The semiclassical ap-proximation for the zero-diffraction term proceeds as above and yields in analogy to Eq. 共4.13兲
Rm L⬘,mL (t2) 共zero diffraction兲 ⫽kD2
冑
21i兺
␣⫽⫺⬁ ⬁冑
1 kl␣, ⫻ei[kl␣,⫺␣⫺(⫺1)]t mL⬘ d 共␣,k兲cos␣tmL d 共 ␣,k兲, 共4.17兲with l␣,⫽
冑
(L)2⫹(␣D)2 and⫽2. The index in Eq.共4.17兲 corresponds in the extended zone scheme to the lattice
coordinate in the horizontal direction 关Fig. 4共c兲兴. The zero-diffraction paths correspond to straight-line rays emanating from the entrance point (␣⫽0,⫽0) reaching the lattice
points (␣,⫽2). More generally,⫽even correspond to re-flected paths, while⫽odd correspond to transmitted paths. The one-diffraction contribution in Eq.共4.16兲 can now be converted into contributions from pseudopaths, more pre-cisely, of two segments of classical paths by applying the Poisson formula twice to eliminate the summation over n and
n
⬘
. Each Poisson sum, in the semiclassical limit, requires an additional SPA. A straightforward evaluation leads toRm L⬘,mL (t2) 共one diffraction兲 ⫽
冉
kD 2冊
2冉
冑
1 2i冊
2兺
␣2⫽⫺⬁ ⬁兺
␣1⫽⫺⬁ ⬁冑
1 kl␣ 2冑
1 kl␣ 1 ⫻ei[k(l␣ 1⫹l␣2)⫺(␣1⫹␣2)]tm L⬘ d 共␣2,k兲 ⫻cos␣2r d共 ␣2,␣1,k兲cos␣1tmL d 共 ␣1,k兲, 共4.18兲 where ␣i⫽arctan ⫺1␣iD L 共4.19兲are the continuous angle variables replacing the mode num-bers ni in rd. The path described by Eq.共4.18兲 corresponds to two classical path segments 共Fig. 5兲, one connecting the origin (0,0) with the lattice point (␣1,⫽1) followed by one that connects (␣1,⫽1) with (␣2,⫽2). The two seg-ments are joined through a diffractive scattering amplitude rd which changes directions of the path, thereby introducing nonclassical ‘‘kinks’’ into an otherwise classical path. For convenience we absorb prefactors in Eqs. 共4.13兲 and 共4.18兲 by rescaling the diffractive scattering amplitude rd and the injection and emission amplitudes td as
tm共兲⫽
冑
kD cos/2tmd共兲, 共4.20a兲
r共
⬘
,兲⫽冑
cos⬘
cos kD 2 rd共
⬘
,兲. 共4.20b兲Explicit expressions will be given in the following section within the framework of the Fraunhofer diffraction approxi-mation 共FDA兲. It will be shown that tm
d
() contains an in-verse factor of
冑
D and rd(⬘
,) contains an inverse factor of D. After rescaling through Eqs. 共4.20a兲 and 共4.20b兲 this leads to an independence of the scattering amplitudes on the cavity dimension D. This important feature is, however, not restricted to the FDA, but can be proven generally for dif-fraction amplitudes in the far-field region. Of course, the far-field approximation is only valid if the two leads are suf-ficiently far apart. This requires that kDⰇ or kLⰇ, acondition that was already encountered in the derivation of Eq. 共4.10兲.
After rescaling, the zero-order diffraction 关or zero-kink contribution, Eq. 共4.17兲兴 transforms to
Rm L⬘,mL (2t) 共zero diffraction兲 ⫽
冉
1 2i冊
1/2兺
␣⫽⫺⬁ ⬁冑
1 kᐉ␣,⫽2e i[kᐉ␣,2⫺(␣⫹1)] ⫻tmL⬘共␣兲tmL共␣兲. 共4.21兲 The one-kink contribution becomesRm L⬘mL (2t) 共one diffraction兲 ⫽
冉
21i冊
1/2冉
1 2i冊
1/2兺
␣1⫽⫺⬁ ⬁兺
␣2⫽⫺⬁ ⬁冑
1 kᐉ␣ 1,0冑
1 kᐉ␣ 2,0 ei[k(ᐉ␣1⫹ᐉ␣2)⫺(␣1⫹␣2)]tm L⬘共␣2兲r共␣2,␣1兲tmL共␣1兲. 共4.22兲With Eqs. 共4.18兲, 共4.21兲, and 共4.22兲 we have all ingredients at our disposal to write down the complete multiple scatter-ing series.
2. Generalization to arbitrary paths
Complete multiple scattering series 关Eqs. 共4.1兲 and 共4.2兲兴 can be formulated in the semiclassical limit as a sum over lattice vectors connected by an increasing number, K, of kinks. We first note that all amplitudes with a different num-ber of traversals关Eqs. 共4.13兲, 共4.21兲兴 and different numbers of kinks 关Eq. 共4.22兲兴 contain as an ingredient the 2D semi-classical Green’s function关Eq. 共2.1兲兴 for propagation along a lattice vector (⌬␣,⌬) of the rectangular lattice,
GSC共⌬␣,⌬兲⫽
冉
1 2ikᐉ⌬␣,⌬冊
1/2
ei[kᐉ⌬␣,⌬⫺(⌬␣⫹⌬⫺1)].
共4.23兲
For an arbitrary S matrix element共which stands for either a reflection amplitude RmL⬘,mL or a transmission amplitude
TmR,mL) we have Sm⬘,m⫽Sm⬘,m (0) ⫹
兺
K⬎0 ⬁ Sm ⬘,m (K) 共4.24兲with a zero-kink (K⫽0) S matrix element
Sm(0)⬘,m⫽
兺
␣⫽⫺⬁ ⬁兺
⭓1 tm⬘关共␣,兲G SC共␣,兲t m关共␣,兲兴, 共4.25兲where(␣,) is given by Eq.共4.28d兲. The sum extends over all lattice points (␣,) of the right half plane with the re-striction that is odd if the matrix element Sm⬘,mstands for a transmission amplitude and is
even if it represents a reflection amplitude. Equation共4.25兲 is equivalent to the standard semiclassical approximation, i.e.,
Sm(0)⬘,m⯝SmSCA⬘,m 共4.26兲
when diffractive injection and emission 关7兴 is included. However, one important difference is worth noting: The splitting of the reflection amplitude for vertical walls into a geometric and a diffractive term关Eq. 共4.15兲兴 corresponds to the treatment of the lead opening as a pointlike scatterer. Therefore, Sm(0)⬘,m does not include the effect of geometric path shadowing, i.e., the elimination of longer paths due to their premature exit through the opening of finite width. This leads to contributions of paths to Sm(0)⬘,mwhich are not present in SmSCA⬘,m. However, the amplitudes of these paths are re-duced by interference with higher-order diffractive paths of the same or very similar length.
Geometric reflections at the open lead are closely related to the so-called ‘‘ghost paths’’ which were introduced by Schwieters et al.关6兴, in order to treat diffractive backscatter-ing at the open lead mouths usbackscatter-ing Kirchhoff diffraction theory. However, in关6兴 only specularly reflected ghost-paths corresponding to straight lines in the extended zone scheme were used. They comprise only a small subset of the com-plete set of kink paths共nonspecularly reflected pseudopaths兲. It will become evident below that the whole set of kink-paths is needed in order to describe properly the path-length power spectrum and to restore unitarity.
With an increasing number of kinks, Eq.共4.24兲 represents corrections to the standard semiclassical approximation of increasing order (kD)⫺K/2, or equivalentlyបK/2. For a given
Sm(K)⬘,m⫽
兺
⌬␣1⫽⫺⬁ ⬁兺
⌬1⭓1 ⬁ •••兺
⌬␣K⫹1⫽⫺⬁ ⬁兺
⌬K⫹1⭓1 ⬁ tm⬘ ⫻关共⌬␣1,⌬1兲兴兿
K⬘⫽1 K 兵GSC共⌬␣K⬘,⌬K⬘兲 ⫻r关共⌬␣K⬘⫹1,⌬K⬘⫹1兲,共⌬␣K⬘,⌬K⬘兲兴其 ⫻GSC共⌬␣ K⫹1,⌬K⫹1兲tm关共⌬␣K⫹1,⌬K⫹1兲兴. 共4.27兲We have introduced the following abbreviations
⌬␣i⫽␣i⫺␣i⫺1 共4.28a兲 with␣0⫽0 and 兺⌬␣i⫽␣, ⌬i⫽i⫺i⫺1⭓0 共4.28b兲 with0 and兺i⌬i⫽, ᐉ⌬␣,⌬⫽
冑
共⌬L兲2⫹共⌬␣D兲2, 共4.28c兲 共⌬␣,⌬兲⫽tan⫺1冉
⌬␣D ⌬L冊
. 共4.28d兲The geometric picture underlying Eq.共4.27兲 is the sum over all pseudopaths consisting of all classical paths of arbitrary length joined by K kinks. In other words: Each lattice point reached by the ray of classical trajectories emanating from the entrance lead into the positive half plane spawns a new bundle of classical trajectories into the positive half plane with positive⌬. This process is repeated K times共Fig. 6兲. Apart from the diffractive amplitudes r and t, Eq. 共4.27兲 contains only quantities calculated from classical dynamics. We refer to this semiclassical approximations in the follow-ing as pseudopath semiclassical approximation共PSCA兲. The present formulation therefore, extends the semiclassical
ap-proximation by including higher-order contributions inប. It is important to point out some of the limitations of validity of the present formulation:
共a兲 While the present approach allows the systematic
in-clusion of certain classes of corrections of increasing order in (kᐉ)⫺K/2and thus of increasing order in the effective បK/2, this expansion does not assure the inclusion of all corrections to a given order. This is due to the fact that an independent approximation of the diffractive amplitudes r and t is re-quired which is not yet specified. The latter may allow for another expansion in (kd)⫺n which would, in turn, provide additional contribution to a given order in ប. Moreover, in general, cross terms of the form (kd)⫺n(kᐉ)⫺m may appear.
共b兲 The present analysis assumes that the leads are the
only interior points where diffractive scattering occurs. This is correct not only for the rectangular billiard explicitly treated here but also a large class of other structures such as the circle or the Bunimovich stadium. However, for other structures containing diffractive edges 共interior angles ⭓) or concave curvatures additional classes of pseudopathes must be included.
共c兲 On a more technical level, the present definition
as-sumes that the SPA evaluations 关e.g., Eq. 共4.8兲兴 can be per-formed under the assumption that the diffractive amplitudes t and r are slowly varying functions. Depending on the chosen approximation for the latter and on the size of (kd), this may lead to additional corrections in the SPA integrals and in Eqs.
共4.24兲–共4.27兲. Clearly, the latter restriction can be removed
once the analytic approximation to r and t is specified. In the path sum Eq.共4.27兲 organized in terms of the num-ber of kinks the total path length
ᐉp⫽
兺
i ᐉ⌬␣i,⌬i 共4.29兲 and hence, the classical action Sp⫽kᐉp is not fixed. In a numerical implementation of the semiclassical theory, it is advantageous to include only terms up to a given maximum action Sp⭐Sp
max
, or equivalently
ᐉp⭐ᐉmax. 共4.30兲
The pseudo-path sum Eq. 共4.24兲 can be reformulated in terms of the fixed maximum total length and variable K,
Sm⬘,m共ᐉp⭐ᐉmax兲⫽
兺
K⫽0 KmaxSm(K)⬘,m共ᐉp⭐ᐉmax兲. 共4.31兲
With the restriction共4.30兲 also the number of included kinks is restricted to K⭐Kmax. Furthermore, the sum over all lat-tice vectors⌬␣i,⌬i appearing in Eqs.共4.25兲 and 共4.27兲 is restricted as well. Geometrically, Eq. 共4.31兲 can be easily visualized 共Fig. 6兲. The length of all rays of classical paths emanating from the entrance lead into the positive half plane is limited to ᐉp⭐ᐉmax. At all lattice points inside the semi-circle of radiusᐉmax each classical trajectory spawns a new generation of rays of trajectories into a new semicircle of radius ᐉmax centered about a respective lattice point. How-ever, only the subset of those is included in the sum 关Eq.
共4.31兲兴 for which the target lattice point lies again in the
original semicircle 关i.e., which is subject to the constraint
共4.29兲兴. This process can be repeated up to a maximum
num-ber given by the maximum numnum-ber of lateral displacements max⫽
冋
ᐉmax
L
册
, 共4.32兲where the bracket stands for the largest integer less than the argument with the additional constraint that max is odd
共even兲 for transmission 共reflection兲. Equation 共4.32兲 follows
from the fact that for each daughter generation of spawned trajectories ⌬⭓1. Consequently, Kmax⭐max⫺1.
The important point to be noted is that the number of pseudopaths included in Eq.共4.31兲 proliferates exponentially withᐉmax while the number of classical paths, i.e., the num-ber of paths included in Sm(0)⬘,m grows only linearly. The loss of information on an exponentially growing number of dif-fractive pseudopaths is at the core of the failure of the stan-dard semiclassical approximation, in particular for the path-length power spectrum and the unitarity.
V. FRAUNHOFER DIFFRACTION APPROXIMATION
The expression for the scattering matrix in terms of the semiclassical sum of pseudopaths contains amplitudes t and r for injection, emission, and diffractive scattering at the lead mouth. Because of the sharp edges with /ap→⬁ for all k, calculation of r and t is not feasible within standard semi-classical theory. An applicable approximation close in spirit to a semiclassical approximation is a diffraction approxima-tion valid for short wavelength, i.e., kdⰇ1 关30兴. We use in the following the Fraunhofer diffraction approximation which we have previously used within the framework of the standard semiclassical approximation. Accordingly, both td and rd can be expressed in terms of the fundamental diffrac-tion integral共for odd n),
IiFDA共,n兲⫽ 1
冑
2wi⫺d/2冕
d/2ei(kni⫹k sin)yd y 共5.1兲
with
w1⫽d,
w2⫽D,
kn1⫽n/d, 共5.2兲
kn2⫽n/D.
The index i⫽1 refers to the wave incident from the quantum wire while the index i⫽2 refers to the wave approaching the mouth of the lead from the inside. Evaluation of the integral for odd integer yields
IiFDA共,n兲⫽
冑
2wi
冉
sin关共kni⫹k sin兲d/2兴
kn
i⫹k sin
冊
. 共5.3兲The amplitude for injection 共emission兲 can be expressed for quantized angles in terms of Eq.共5.3兲 as
tn,mFDA⫽
冑
2 D再
I1 FDA冋
sin⫺1冉
n kD冊
,m册
⫹I1 FDA冋
sin⫺1冉
n kD冊
,⫺m册冎
. 共5.4兲For continuous angles the amplitude for injection or emis-sion appearing in Eq.共4.8兲 is accordingly given by
tmd共兲⫽tmFDA共兲⫽
冑
2 D关I1 FDA共,m兲⫹I 1 FDA共,⫺m兲兴. 共5.5兲After rescaling according to Eq.共4.20兲 the injection or emis-sion amplitude appearing in the PSCA关Eq. 共4.27兲兴 becomes
tm共兲⫽
冑
kD cos/2tm d共兲 ⫽冑
2 cos kd冋
sin关共km(1)⫹k sin兲d/2兴 sin⫹km(1)/k ⫹sin关共k sin⫺km (1)兲d/2兴 sin⫺km(1)/k册
. 共5.6兲 The diffractive part of the reflection amplitude 关Eq. 共4.15兲兴 back into the billiard is given in Fraunhofer approximation by rnd⬘,n⫽rnFDA⬘,n⫽冑
2 D再
I2 FDA冋
⬘
⫽sin⫺1冉
n⬘
kD冊
,n册
⫹I2 FDA共⬘
,n兲冎
. 共5.7兲In the semiclassical limit discussed in the previous section, Eq. 共5.7兲 simplifies further. When the transformation from the discrete quantum numbers (n
⬘
,n) to continuous angle variables (⬘
,) via the Poisson formula is employed, kn⬘
is mapped onto the continuous function k sin⬘
. If we extend the range of angles to (⫺/2⭐⬘
⭐/2), the two terms in Eq. 共5.7兲 become equivalent and reduce tord共
⬘
,兲⫽2冑
2DI2
FDA共
⬘
,兲⫽4D
sin关kd/2共sin
⬘
⫹sin兲兴k共sin
⬘
⫹sin兲 . 共5.8兲After rescaling关Eq. 共4.20兲兴, the diffraction or kink amplitude entering the PSCA关Eq. 共4.27兲兴 is given by
r共
⬘
,兲⫽kD2
冑
cos⬘
cosr d共⬘
,兲⫽2
冑
cos⬘
cos sin关kd/2共sin⬘
⫹sin兲兴 sin⬘
⫹sin .The amplitudes for injection or emission tm() 关Eq. 共5.6兲兴 and for internal diffractive reflection r(
⬘
,) 关Eq. 共5.9兲兴 en-tering our semiclassical theory possess the remarkable fea-ture that they are independent of the geometric parameter of the cavity共i.e., D). This observation indicates that the semi-classical multiple scattering expansion 关Eqs. 共4.24兲 and共4.27兲兴 should be applicable irrespective of the geometry of
the cavity.
VI. NUMERICAL RESULTS
We present in the following numerical results within the framework of the present semiclassical theory. We focus on two properties for which previous semiclassical descriptions faced major problems: unitarity and the path-length power spectrum. We compare our results with the previously pro-posed semiclassical description in which both injection and emission was treated in FDA, but only classical paths were included for propagation inside the rectangular billiard. This corresponds to the approximation Sm⬘,m⬇Sm⬘,m
(0)
关Eq. 共4.24兲兴
with tm() given by Eq. 共5.6兲. We also perform detailed comparison with quantum calculations. For the latter we ex-ploit the correspondence between the semiclassical and the quantum multiple scattering series. Since any semiclassical calculation can only be performed for up to a certain maxi-mum path length ᐉmax corresponding to a maximum lateral lattice displacementmax关Eq. 共4.32兲兴, errors due to the trun-cation of the contributing paths should be disentangled from the errors due to the semiclassical approximation itself. This is easily accomplished within the present multiple scattering theory as the corresponding quantum multiple scattering ex-pansion 关Eqs. 共4.1兲 and 共4.2兲兴 can be truncated at the same max so that semiclassical and quantum calculations can be directly compared at the same level of truncation. Moreover, the quantum multiple scattering series depends also on the same substructure amplitudes t and r for injection, emission, and internal reflections. Calculating the latter quantum me-chanically and truncating the multiple scattering series at max⫽11 yields the result for the unitarity 共Fig. 7兲 labeled as TQM. This truncated quantum result is only approximately unitary (⬇0.95) with a k dependent fluctuation of⬇0.05. If we replace in the quantum multiple scattering series the amplitudes t and r by its FDA approximation关Eqs. 共5.4兲 and
共5.7兲兴, the resulting unitarity limit at the same level of
trun-cation, labeled as TQM-FDA, reaches value of 0.85 with
⬇0.05. Turning now to the semiclassical theory based upon
the FDA approximation including both classical and pseudo-paths, we observe convergence as a function of a number of kinks K included (K⭐10) towards the corresponding quan-tum unitarity limit. The only difference to the quanquan-tum result is the larger fluctuation (⬇0.15). This is in stark contrast to standard semiclassical approximations. The standard semi-classical result 共without geometric path shadowing, K⫽0) exceeding the unitarity limit can be corrected for path shad-owing denoted by SSCA关7,28兴. Longer paths (␣,) are shad-owed by shorter paths (␣/n,/n) when ␣/n, /n are inte-gers for an integer n. Shadowing also removes geometric reflection at the open lead mouth 共the ‘‘ghosts’’兲. This cor-rection, also shown in Fig. 7, reduces the transmission
am-plitude without restoring the unitarity. The present results illustrate that the semiclassical diffraction 共or kink兲 expan-sion关Eq. 共4.24兲兴 converges towards the quantum calculation at the same level of input and truncation. It furthermore in-dicates that the largest residual error is caused by the FDA for t and r rather than from the semiclassical approximation to the paths sum itself, as indicated by the deterioration of the quantum multiple scattering series with FDA amplitudes. Employing diffraction approximations beyond the FDA such as the geometric theory of diffraction关31,32兴 or the uniform theory of diffraction 关33兴 would most likely result in further improvements.
Turning now to the path-length power spectrum for the transmission amplitude, S11⫽T11,
P共ᐉ兲⫽
冏
冕
dkeikᐉT 11共k兲冏
2
, 共6.1兲
the standard approximation T11(k)⬇S11 (0)
(k), 关Eq. 共4.24兲兴 fails badly at large path length共Fig. 8兲. More precisely, while the peak positions which can be associated with classical paths and labeled by lattice vectors ␣, agree well with the full quantum spectrum, the peak heights are overestimated up to one order of magnitude关Fig. 8共a兲兴. The quantum peak heights decrease exponentially while the standard semiclas-sical approximation exhibits only an inverse linear decay proportional to ᐉ␣,⫺1. The present semiclassical theory that includes pseudopaths with a finite number of kinks, by con-trast, reproduces the quantum path-length spectrum very well
关Fig. 8共b兲兴, even fine details at large ᐉ are remarkably well
reproduced. The decay at largeᐉ becomes exponential rather than linear 共Fig. 8c兲. This observation is the key to the un-derstanding of the role pseudopaths play in quantum and semiclassical transport. The exponential suppression of the path-length spectrum at large ᐉ is a consequence of the de-structive interference of an exponentially proliferating set of
FIG. 7. Test of unitarity on different levels of semiclassical ap-proximation to the rectangular billiard. Comparison between: quan-tum mechanical calculation with truncated multiple scattering ex-pansion (max⫽11), TQM; quantum multiple scattering with FDA
pseudopaths. For each long classical zero-kink path with lat-tice vector␣,we find a large number n of few-kink paths␥ with nearly the same length
ᐉ␣,(␥)⬇
兺
K⬘⫽1 K ᐉ⌬␣K⬘⌬K⬘ (␥) , ␥⫽1,•••,n 共6.2兲 with (␥)⫽兺
K⬘⫽1 K ⌬K⬘ (␥) , ␣(␥)⫽兺
K⬘⫽1 K ⌬␣K⬘ (␥) , 共6.3兲and whose Maslov indices differ from the zero-kink path. The exponential growth of pseudopaths with pseudoran-domly varying phases assures the approximately exponential suppression of the contributions of long paths.
This observation also clears up another puzzle found in previous semiclassical calculations. The agreement between the semiclassical and the quantum path-length spectrum is much better for chaotic rather than for regular systems
关4,7,5,12,26兴 contrary to many other observables for which
standard semiclassical approximation performs much better for regular rather than for chaotic systems. In chaotic sys-tems, already classical paths exponentially proliferate as a function of the path length and can account for exponential suppression of large path lengths. Therefore, the lack of pseudopaths which also proliferate exponentially is less dra-matically felt than in regular systems where the exponential proliferation of pseudo paths competes with only linear pro-liferation of classical paths.
VII. CONCLUSIONS
We have presented a semiclassical theory for ballistic transport that goes beyond the standard semiclassical ap-proximation by including an ascending order of diffractive scatterings in the interior of the ballistic cavity correspond-ing to certain classes of contributions in increascorrespond-ing order of
ប. As the present formulation requires an additional and
in-dependent approximation for the elementary diffractive am-plitudes for injection, emission, and internal reflection, inclu-sion of all contribution to a given order in បn cannot be expected. Using the example of a rectangular billiard, we have shown numerically that an exponentially proliferating number of pseudopaths with diffractive kinks converges to the quantum multiple scattering series. In the rectangular bil-liard, the leads are the only sources of diffraction. For other geometries with concave walls or diffractive edges additional sources of diffraction are present. While the numerical result presented pertain to the rectangular billiard, we find numeri-cal evidence that convergence towards quantum transport can be found for other geometries, specifically for the circle and the Bunimovich stadium关35兴. We expect that our PSCA will also resolve other unresolved issues of semiclassical bal-listic transport such as the problem of weak localization
关14,15兴 and the breakdown of symmetry of the
autocorrela-tion funcautocorrela-tion in reflecautocorrela-tion and transmission.
ACKNOWLEDGMENTS
Support by the FWF and by the FWF-SFB 016 is grate-fully acknowledged.
关1兴 M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics 共Springer Verlag, New York, 1991兲, and reference therein. 关2兴 W. Miller, Adv. Chem. Phys. 25, 69 共1974兲.
关3兴 W.A. Lin, J.B. Delos, and R.V. Jensen, Chaos 3, 655 共1993兲. 关4兴 R.A. Jalabert, H.U. Baranger, and A.D. Stone, Phys. Rev. Lett.
65, 2442 共1990兲; H.U. Baranger, R.A. Jalabert, and A.D.
Stone, Chaos 3, 665共1993兲.
关5兴 H. Ishio and J. Burgdo¨rfer, Phys. Rev. B 51, 2013 共1995兲. 关6兴 C.D. Schwieters, J.A. Alford, and J.B. Delos, Phys. Rev. B 54,
10 652共1996兲.
关7兴 L. Wirtz, J.-Z. Tang, and J. Burgdo¨rfer, Phys. Rev. B 56, 7589 共1997兲.
关8兴 L. Wirtz, J.-Z. Tang, and J. Burgdo¨rfer, Phys. Rev. B 59, 2956 共1999兲.
关9兴 T. Blomquist and I.V. Zozoulenko, Phys. Rev. B 64, 195301 共2001兲; T. Blomquist and I.V. Zozoulenko, Phys. Scr., T T90, 37共2001兲.
关10兴 R. Akis, D.K. Ferry, and J.P. Bird, Phys. Rev. Lett. 79, 123 共1997兲.
关11兴 E. Persson, I. Rotter, H.-J. Sto¨ckmann, and M. Barth, Phys. Rev. Lett. 85, 2478共2000兲.
关12兴 R.G. Nazmitdinov, K.N. Pichugin, I. Rotter, and P. Seba, Phys. Rev. E 64, 056214共2001兲.
关13兴 R.G. Nazmitdinov, K.N. Pichugin, I. Rotter, and P. Seba, Phys. Rev. B 66, 085322共2002兲.
关14兴 M. Hastings, A.D. Stone, and H. Baranger, Phys. Rev. B 50, 8230共1994兲.
关15兴 A.D. Stone, Les Houches Session LXI (1994), edited by E. Akkermans et al.共North-Holland, Amsterdam, 1995兲, p. 329. 关16兴 L.A. Bunimovich, Funct. Anal. Appl. 8, 254 共1974兲. 关17兴 G. Benettin and J.M. Strelcyn, Phys. Rev. A 17, 773 共1978兲. 关18兴 E. Heller, Phys. Rev. Lett. 53, 1515 共1984兲.
关19兴 H. Friedrich, Theoretical Atomic Physics 共Springer-Verlag, Berlin, 1998兲.
关20兴 M.V. Berry and K.E. Mount, Rep. Prog. Phys. 35, 315 共1972兲. 关21兴 J. Burgdo¨rfer, C.O. Reinhold, J. Sternberg, and J. Wang, Phys.
Rev. A 51, 1248共1995兲.
关22兴 R. Landauer, IBM J. Res. Dev. 1, 223 共1957兲.
关23兴 See, e.g., A. Erde´ly, Asymptotic Expansions 共Dover, New York, 1956兲, p. 51.
关24兴 R.P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals共MacGraw-Hill, New York, 1965兲.
关25兴 M.C. Gutzwiller in Chaos and Quantum Physics, edited by M.-J. Giannoni, A. Voros, and J. Zinn-Justin, 共North-Holland, Amsterdam, 1991兲.
关26兴 S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgdo¨rfer, Phys. Rev. B 62, 1950共2000兲.
关27兴 See e.g., E. Merzbacher, Quantum Mechanics, 3rd ed. 共Wien, New York, 1996兲.
关28兴 P. Pichaureau and R.A. Jalabert, Eur. Phys. J. B 9, 299 共1999兲. 关29兴 The Poisson sum formula 共PSF兲 兺n⬁⫽1f (n)
⫽兺l⬁⫽⫺⬁兰⫺⬁⬁ f ()e i2l
d follows from the identity 兺l⬁⫽⫺⬁ei2l⫽兺n⬁⫽1/2␦(⫺n). If f () is assumed to be
neg-ligible for ⫽0, the PSF can be written as 兺n⬁⫽1f (n) ⫽1
2兺l⬁⫽⫺⬁兰⫺⬁⬁ f ()ei2ld. For a summation over odd
inte-gers only, one obtains 兺n⬁⫽1f (2n⫹1) ⫽1
4兺l⬁⫽⫺⬁兰⫺⬁⬁ f ()e ild.
关30兴 A. Sommerfeld, Vorlesungen u¨ber Theoretische Physik 共Verlag H. Deutsch, Frankfurt, 1985兲, Vol. 5.
关31兴 J.B. Keller, J. Opt. Soc. Am. 52, 116 共1962兲.
关32兴 G. Vattay, J. Cserti, G. Palla, and G. Sza´lka, Chaos Solitons Fractals 8, 1031共1997兲.
关33兴 M. Sieber, N. Pavloff, and C. Schmit, Phys. Rev. E 55, 2279 共1997兲.
