Pseudopath semiclassical approximation to transport through open quantum billiards: Dyson equation for diffractive scattering

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Pseudopath semiclassical approximation to transport through open quantum billiards:

Dyson equation for diffractive scattering

Christoph Stampfer,*Stefan Rotter, and Joachim Burgdörfer

Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria, European Union

Ludger Wirtz

Institute for Electronics, Microelectronics and Nanotechnology, Boîte Postale 60069, 59652 Villeneuve d’Ascq Cedex, France, European Union

共Received 6 April 2005; published 30 September 2005兲

We present a semiclassical theory for transport through open billiards of arbitrary convex shape that includes diffractively scattered paths at the lead openings. Starting from a Dyson equation for the semiclassical Green’s function we develop a diagrammatic expansion that allows a systematic summation over classical and pseudo-paths, the latter consisting of classical paths joined by diffractive scatterings 共“kinks”兲. This renders the inclusion of an exponentially proliferating number of pseudopath combinations numerically tractable for both regular and chaotic billiards. For a circular billiard and the Bunimovich stadium the path sum leads to a good agreement with the quantum path length power spectrum up to long path length. Furthermore, we find excellent numerical agreement with experimental studies of quantum scattering in microwave billiards where pseudo-paths provide a significant contribution.

DOI:10.1103/PhysRevE.72.036223 PACS number共s兲: 05.45.Mt, 73.23.Ad, 73.50.Bk, 03.65.Sq

I. INTRODUCTION

Semiclassical approximations are among the most useful tools in describing and analyzing ballistic transport in meso-scopic systems. On a fundamental level, semiclassical tech-niques allow one to build a bridge between classical and quantum mechanics: the classical paths carry an amplitude which reflects the geometric stability of the orbits and a phase that contains the classical action and accounts for quantum interference关1–3兴.

Ballistic transport through billiards has been studied ex-tensively in the last decade关4–23兴 and a variety of semiclas-sical approximations关7–15,24–26兴 have been introduced in order to provide a qualitative and, in part, also a quantitative description of these systems. In particular, universal conduc-tance fluctuations共UCFs兲 and the “weak localization” 共WL兲 have been studied 关14兴 in order to delineate characteristic differences in the quantum transport of classically chaotic and integrable billiards. Very recently, quantum shot noise 关27兴 in ballistic cavities that are either chaotic 关28,29兴, regu-lar关30兴, or display a mixed phase space 关31兴 has been used as a probe of the quantum-to-classical and chaotic-to-regular crossover关26,28–34兴.

The approach of the共semi兲classical limit of ballistic quan-tum transport is both conceptually as well as numerically nontrivial as it represents, generically, a multiscale problem. The two-dimensional quantum billiard共or quantum dot, see Fig. 1兲 is characterized by an area A or linear dimension

D =

冑

A. The quantum wires共or leads兲 to which the billiard is

attached have the width d. In order to reach sufficiently long dwell times such that differences between transiently regular and chaotic motion become important, the relation d / DⰆ1 should hold. To approach the semiclassical limit for the mo-tion inside the billiard requires␭DⰆD 共␭Dde Broglie wave

length兲 or equivalently kDⰇ2␲. Furthermore, if the 共disor-der兲 potential inside the dot varies over a length scale aP, we

should require␭DⰆaP for a semiclassical approximation to

hold. These conditions pertaining to the dot are necessary but not sufficient. Since the scattering 共S兲 matrix maps asymptotic scattering states onto each other, also the entrance and exit channel states in the quantum wire should reach their classical limit ␭DⰆd or kdⰇ2␲. The latter limit is

virtually impossible to reach, neither experimentally for quantum dots关28,35,36兴 or microwave billiards 关37兴 nor nu-merically关38兴. We will therefore focus in the following on the “intermediate” semiclassical regime pertaining to the in-terior of the billiard␭DⰆD with convex hard-walled

bound-aries such that quantum diffraction in the interior can be neglected, with the understanding, however, that quantum

*Present address: Chair of Micro and Nanosystems, Swiss Federal Institute of Technology Zurich 共ETH Zurich兲, Tannenstr. 3, 8092 Zurich, Switzerland.

FIG. 1. An open arbitrarily convex-shaped billiard with two narrow leads of equal width d: left共L兲 and right 共R兲.

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effects due to the coupling to the asymptotic quantum wires have to be taken into account. Accordingly, the term “semi-classical approximations” refers in the following to approxi-mations to the constant-energy Green’s function for propaga-tion in the interior of the dot G共rជ, rជ

⬘

, k兲 by an approximate semiclassical limit, GSC to be discussed below. The projec-tions of G onto asymptotic scattering states with transverse quantum numbers mL共mR兲 yield the amplitudes for

transmis-sion Tm_L,m_R from the entrance 共left兲 to the exit 共right兲 lead

and reflections Rm_L,m_L⬘. Standard semiclassical

approxima-tions to G face several fundamental difficulties 关10,11,14,18,25,39兴: among many others, unitarity is violated with discrepancies in some cases as large as the conductance fluctuations the theory attempts to describe 关11,12兴. Like-wise, the anticorrelation␦兩T兩2= −␦兩R兩2 between transmission fluctuations ␦兩T兩2 _{and the corresponding fluctuations in}

the reflection ␦兩R兩2 _{as a function of the wave number k is}

broken. Also, the “weak localization” effect is considerably overestimated关16,39兴. These difficulties are due to the fact that hard-walled billiards possess “sharp edges” at the en-trance and exit leads even though the interior of the dot fea-tures a smooth共in the present case, a constant兲 potential. At these sharp edges the contacts to the quantum wires feature spatial variations of the potential where the length scale aP

approaches zero. Consequently the semiclassical limit ␭D/ aPⰆ1 cannot be reached, no matter how small ␭D 共or

large k兲 is. In other words, the quantum properties of the leads influence also the semiclassical dynamics in the inte-rior. This observation is the starting point for diffractive cor-rections such as the Kirchhoff diffraction关7兴 or Fraunhofer diffraction关10兴.

We have recently developed a pseudopath semiclassical approximation 共PSCA兲关15兴 with pseudopaths that result from spawning of classical paths due to diffractive共i.e., non-geometric兲 reflections in the lead mouths 共or point contacts兲. Pseudopaths play an essential role when incorporating inde-terministic features into the semiclassical description of transport. Their existence has also recently been pointed out in Ref. 关40兴, although no explicit numerical investigations were performed. While classical trajectories are either ejected through the exit lead contributing to T or return back to the entrance lead contributing to R, a quantum wave packet will do both. Pseudopaths interconnect otherwise dis-junct subsets of classical paths that exit either through the left or right lead. The lack of this coupling is responsible for violation of the anticorrelation of transmission and reflection fluctuations,␦T2⫽−␦R2, in standard semiclassical approxi-mation共SCA兲. Likewise, the standard SC approximation is expected to fail for quantum shot noise关27兴 that is a signa-ture of this quantum indeterminism. For the special case of the rectangular shaped dot with lead openings placed at the midpoints, summation of exponentially proliferating pseudo-paths could be accomplished by tracing trajectories in an extended zone scheme 关15,41兴. For arbitrarily shaped bil-liards and, in particular, chaotic bilbil-liards where already clas-sical trajectories proliferate exponentially, systematic inclu-sion of paths and pseudopaths up to the same length is considerably more complicated. In the following we address this problem within the framework of a semiclassical Dyson

equation. We present a diagrammatic expansion that allows a systematic summation of classical path and pseudopath con-tributions. Applications to the circular billiard 共as a proto-typical regular system兲 and to the Bunimovich stadium bil-liard 共as the prototype system for chaotic scattering兲 show good agreement with the numerically calculated exact quan-tum path length spectrum. We furthermore apply the PSCA to recent experimental studies of scattering in microwave billiards in which pseudopath contributions could be experi-mentally identified.

The structure of the paper is as follows. In Sec. II we briefly review the standard semiclassical approximation and previous attempts to include diffraction effects. The pseudo-path semiclassical approximation for arbitrarily convex shaped billiards will be presented in Sec. III employing a semiclassical version of the Dyson equation. We develop a diagrammatic expansion of G in terms of paths and pseudo-paths. Its evaluation is numerically facilitated by an algebraic matrix representation that reduces path summations up to infinite order to a sequence of matrix multiplications and inversions, as discussed in Sec. IV. Numerical results and comparison with the full quantum results as well as micro-wave experiments are given in Sec. V, followed by a short summary and outlook onto future applications in Sec. VI. Details on the Fraunhofer diffraction approximation are given in the Appendix.

II. STANDARD SEMICLASSICAL APPROXIMATION The conductance of a ballistic two-terminal system 共as depicted in Fig. 1兲 is determined by the scattering amplitudes

Tmthrough the Landauer formula关42兴

g共k兲 =2e 2 h

冉

m

兺

_L=1 M

兺

m_R=1 M 兩Tm_R,m_L共k兲兩2

冊

, 共2.1兲

where M is the number of open modes in the leads共quantum wires兲 and Tm_R,m_L共k兲 are the transmission amplitudes from

the mLth mode in the entrance lead关referred to in the

follow-ing as left 共L兲 lead兴 to the mRth mode in the exit lead

关re-ferred to as the right共R兲 lead兴. In the following we choose local coordinate systems共xi, yi兲, i=L,R for the leads, where

xidenotes the longitudinal and yithe transverse direction of

lead i. For simplicity we use the coding 共L,R兲 for the en-trance and exit channels throughout this publication, irre-spective of the actual location at which the leads are at-tached. The projection of the retarded Green’s propagator onto the transverse wave functions␾m_L共yL兲 and ␾m_R共yR兲 of

incoming and outgoing modes 关14兴 serves as starting point for most semiclassical theories which approximate transmis-sion amplitudes from mode mLto mode mR:

Tm_L,m_R共k兲 = − i

冑

kx_R,m_Rkm_L,x_L

冕

dyR

冕

dyL␾m*_R共yR兲

⫻ G共xR,yR,xL,yL,k兲␾m_L共yL兲. 共2.2兲

Here and in the following we use atomic units 共ប=兩e兩=me

= 1兲. The transverse wave-functions␾m共y兲 are given for zero

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␾m共y兲 =

冑

2 d

冦

cos

冉

m␲ d y

冊

, m odd, sin

冉

m␲ d y

冊

, m even,

冧

共2.3兲

where d denotes the width of the leads. We assume, for sim-plicity, that the two leads have identical width. The scattering state in the lead is the product of the transverse lead wave function␾m共y兲 and a plane wave in the longitudinal direction

with wave vector kx,m=

冑

k2−兩m␲/ d兩2. Both leads have a total

of 共m=1, ... ,M兲 open 共i.e., transmitting兲 modes. Analo-gously, the amplitude for reflection from the incoming mode

mL into a different mode mL

⬘

in the same quantum wire is

given by Rm_L,m_L⬘=␦m_L,m_L⬘− i

冑

kx_L,m_L⬘km_L,x_L

冕

dyL

⬘

The factor兩kx_L兩共兩kx

⬘

_L兩兲 denotes the longitudinal component of

the momentum of the incoming共outgoing兲 wave function at the left lead.

The semiclassical approximation to the scattering ampli-tudes is obtained by approximating the Green’s function

G共rជ2, rជ1, k兲 in Eq. 共2.2兲 by the semiclassical Green’s

propa-gator GSC共rជ2, rជ1, k兲. The standard semiclassical Green’s

propagator GSC_共rជ

2, rជ1, k兲, the Fourier-Laplace transform of

the van Vleck propagator evaluated in stationary phase ap-proximation 共SPA兲, describes the probability amplitude for propagation from rជ1to rជ2at a fixed energy, E = k2/ 2. It can be

expressed in terms of a sum over all classical paths of energy

E 共or, equivalently, wave vector k兲 connecting these two

points关1兴, GSC_共r_ជ 2,rជ1,k兲 =

兺

␣:rជ1→rជ2 G_␣SC=

兺

␣:rជ1→rជ2 兩D␣共rជ2,rជ1,k兲兩1/2 共2␲i兲1/2 ⫻ exp

冋

i

冉

S_␣共rជ2,rជ1,k兲 − ␲ 2␮␣

冊

册

. 共2.8兲 Here, S_␣共rជ2, rជ1, k兲=kL␣ is the action of the path␣ of length

L_␣. D_␣共rជ2, rជ1, k兲 is the classical deflection factor 关1兴

which describes the stability of the paths and ␮_␣ denotes the Maslov index of the path␣. In line with the semiclassical approximation, the double integrals in Eqs. 共2.2兲 and 共2.4兲 are frequently evaluated in stationary phase approximation. Physically this means that the paths are entering and exiting the cavity only with the discrete angles

␪m= arcsin关m␲/共dk兲兴 due to the quantization of the

trans-verse momentum in the leads. For completeness we mention at this point that the path sum in Eq. 共2.8兲 also contains, as a subset, those orbit pairs described in Ref. 关25兴 which yield a weak-localization correction beyond the diagonal approximation.

Several strategies have been proposed to introduce dif-fractive effects in order to quantitatively improve the semi-classical theory for transport through open quantum billiards. A straightforward way is to eliminate the SPA for the double integral in Eq.共2.4兲. Expanding the action in the semiclassi-cal Green’s function Eq.共2.8兲 to first order in the transverse coordinate, the integral takes the form of a Fraunhofer dif-fraction integral and can be evaluated analytically关10兴. On this level of approximation, diffraction effects are thereby automatically included, however, only upon entering and ex-iting the cavity, not during propagation inside the cavity. Schwieters et al. 关7兴 employed Kirchhoff diffraction theory to calculate the diffractional weight of paths entering and exiting the billiard. In addition, they introduced the concept of “ghost paths”: paths that are specularly reflected at the lead opening due to diffractive effects. The proper use of a diffractive weight for classical paths allowed the quantitative determination of the peak heights in the power spectra of the transmission and reflection amplitudes 关7,10兴 - at least for short path lengths. Ghost orbits could account for some of the peaks that were missing in the semiclassical spectra关7兴. Several deficiencies remained, however, unresolved: unitar-ity of the semiclassical S matrix is, typically, violated; the weak localization peak is significantly underestimated, and the semiclassical pathlength共l兲 spectrum 关17兴

P_mSC_⬘_,m共l兲 =

冏

冕

dkeikl_T m⬘,m SC _共k兲

冏

2

, 共2.9兲

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III. DYSON EQUATION FOR THE PSCA

Starting point is a semiclassical version of the Dyson equation for the Green’s function where the standard semi-classical Green’s function GSC plays the role of the unper-turbed Green’s function and the diffractive scatterings at the lead openings 共or point contacts兲 are the perturbation. Ac-cordingly, we have

GPSC= GSC+ GSCVGPSC, 共3.1兲 where the perturbation “potential” V is given in terms of the projectors Eqs.共2.5兲 and 共2.6兲 as

V = PL+ PR. 共3.2兲

Iterative solution by summation

GPSC= GSC

兺

i=0 ⬁ 共VGSC_兲i = GSC

兺

i=0 ⬁ 关共PL+ PR兲GSC兴i 共3.3兲

includes diffractive scatterings into the GPSCto all orders. The key to the multiple diffractive scattering expansion is that within the semiclassical expansion each projection op-erator onto the L and R point contacts selects classical tra-jectories emanating from or ending up at the leads and at the same time spawns new generations of classical trajectories.

Noting that GPSC_{will only be evaluated in the domain of}

PL or PR, Eq. 共3.3兲 can be reorganized in terms of a 2⫻2

matrix Dyson equation. We decompose GSC_{as follows:}

PLGSCPL⇒ GLLSC=

兺

␣LL G_␣ LL SC _共3.4a兲 PLGSCPR⇒ GLR SC =

兺

␣LR G_␣ LR SC , 共3.4b兲 PRGSCPL⇒ GRL SC =

兺

␣RL G_␣ RL SC , 共3.4c兲 PRGSCPR⇒ GRRSC=

兺

␣RR G_␣ RR SC _. _共3.4d兲

In Eq. 共3.4兲 the index ␣ij refers to paths that emanate

from the point contact j共j=L,R兲 and end up at point contact

i共i=L,R兲. Each of the four disjunct subsets of classical paths

is, in general, infinite. In the following we denote the trun-cated number of trajectories of the corresponding class by

Nij. Each GijSC containing a large finite number Nij or an

infinite number of trajectories is diagrammatically repre-sented by a double line, each contribution of an individual trajectory by a single line共Fig. 2兲. The trajectories emanate or end on vertices representing the L or R point contacts.

. 共3.5兲

Its solution GPSC _{denoted by thick solid lines can now be}

diagrammatically represented 共Fig. 3兲 as a sum over all pseudopaths that result from the couplings of classical-path Green’s functions by successive diffractive scatterings at point contacts. Only pseudopath combinations共or diagrams兲 contribute that are connected at vertices L or R.

, 共3.6兲

resulting in a sum over共up to兲 infinitely long pseudo paths with 共up to兲 an infinite number of diffractive scatterings. Note that the number of classical paths between two diffrac-tive “kinks” and their lengths may reach infinity as well. The handling of this double limit plays an important role in the numerical implementation as destructive interference of pseudopaths and classical paths of comparable length must be properly taken into account. The evaluation of Eq.共3.5兲 and共3.6兲 is not straightforward as each operator product in Eq. 共3.3兲 contains multiple two-dimensional integrals over R2_{. For illustrative purposes we explicitly give the}

first-FIG. 2. Diagramatic expansion of paths. Double line: G_ijSC, single line: contribution of individual classical trajectories

G_␣ ij

SC_{, dots: replica of point contacts along the propagation direction}

kˆ共“time”兲. 共a兲 Typical term appearing in the expansion of GPSC_._共b兲

Expansion of G_ijSCin terms of classical path contributions.

FIG. 3. Thick solid line: GPSC_{; first terms of the expansion of}

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correction term 共GSC_P

LGSC+ GSCPRGSC兲 of Eq. 共3.5兲. We

have to calculate the double integral, e.g.,

GLRSCPRGRLSC=

冕

R2 d2_r

_⬘

lead共L→R兲 while␣LR represents paths共R→L兲. In order to

solve the integral in Eq. 共3.9兲 analytically, we expand the path length L_␣ of the classical paths␣leading from point rជi

to the lead mouth rជR 共here we set yR= 0 for simplicity and

only consider regular billiards兲:

L_␣ RL共rជR,rជi兲 = L␣RL+ sin共␪␣RL兲y + cos共␪␣RL兲 L_␣ RL y2¯ , 共3.10兲 where L_␣

RL= L␣RL共rជR, rជi兲 is the length of the path that reaches

the center of the lead mouth.

Taking into account only the first-order correction in the lateral displacement, the integral in Eq.共3.9兲 takes the form of a Fraunhofer diffraction integral and can be solved ana-lytically关10,15兴共see the Appendix also兲,

I =

冕

−d/2

d/2

dy

冑

兩kx_R共y兲兩兩kx

⬘

_R共y兲兩 ⫻ exp兵ik关L␣_LR共y兲 + L␣_RL共y兲兴其

⬇ k

冑

cos共␪␣LR兲cos共␪␣

⬘

RL兲

冕

−d/2

d/2

dy exp兵ik关L_␣

LR+ L␣RL兴

+ ik关sin共␪␣LR兲 + sin共␪␣

⬘

RL兲兴y其

= exp关ik共L␣LR+ L␣RL兲兴r共␪␣LR,␪␣

⬘

RL,k兲, 共3.11兲

where␪_␣

RL

⬘

is the ending angle of the incoming path and␪_␣

LR

is the starting angle of the exiting path. Furthermore,

r共␪_␣

␣LR

兺

␣RL G_␣ LR SC_r_共_␪ ␣LR,␪␣

⬘

RL,k兲G␣RL SC_. _共3.13兲

Note that classical paths contributing to GSCdo not incorpo-rate the finite width of the point contact since we take the limit d→0 in the simulation of classical paths. Instead they are specularly reflected at the open lead mouth R as if there were a hard wall. Such a path will interfere with the pseudo-path that has almost the same length and topology but expe-riences a nonspecular, i.e., diffractive reflection at the same point contact.

All higher-order corrections are evaluated analogously. Each additional vertex connecting two GSC’s gives rise to an

additional Fraunhofer integral with an interior reflection am-plitude r共␪

⬘

,␪兲.

IV. MATRIX REPRESENTATION OF PSEUDOPATH SUM A numerical representation of the Dyson equation 关Eqs. 共3.5兲 and 共3.6兲兴 requires the truncation of the number of contributing paths in each of the four basic Green’s functions 关Eq. 共3.4兲兴 to large but finite numbers. Let

NLL denote the number of paths leading from rជL0 back to

r ជL

0_{共L→L兲. Accordingly, N}

RR is the number of paths of the

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num-ber for共R→L兲. Note that NRL= NLR for systems with time

reversal symmetry. The total number of contributing paths is

N = NLL+ NRR+ NLR+ NLR. In order to numerically solve the

semiclassical Dyson Equation共3.5兲, we write the semiclassi-cal Green’s function GSCas a diagonal N⫻N matrix where each diagonal matrix element represents the contribution of one particular classical path. Distinguishing the four dif-ferent subclasses of paths, the matrix can be written in the following form: G=SC_{共k兲 =}

冢

G=LL SC_共k兲 ₀ G=RLSC共k兲 G=RR SC_共k兲 0 G=LRSC共k兲

冣

, 共4.1兲

where the matrix elements of the submatrices are given by, e.g., G=␣LL SC_{共k兲 =}兩D␣LL兩 1/2 共2␲i兲1/2 exp

冋

i

冉

kL␣LL− ␲ 2␮␣LL

冊

册

, 共4.2兲

and␣LL= 1 , . . . , NLL. The point to be emphasized is that the

reduction of the共2⫻2兲 block structure of the matrix Dyson integral equation关Eq. 共3.5兲兴 into a purely algebraic equation requires an extension to a 4⫻4 block structure 关Eq. 共4.1兲兴. The latter is the minimum size required to accomodate the eight independent block matrices representing the vertices 关see Eq. 共4.4兲 below兴. Due to diffractive coupling at the lead mouths, the matrix G=PSC_{that represents the Green’s function}

in the pseudopath semiclassical approximation also contains nondiagonal matrix elements. A specific matrix element 共␣

⬘

,␣兲 consists of a sum of all path combinations that con-tain the classical path ␣ as first segment and the classical paths␣

⬘

兺

i=0

⬁

关r=共k兲G=SC_共k兲兴i _共4.5a兲

=G=SC_{共k兲关1 − r=共k兲G=}SC_共k兲兴−1_. _共4.5b兲

Equation 共4.5b兲 represents the “exact” summation over pseudopaths with up to an infinite number of diffractive scat-terings. The accuracy of the result is, however, limited by the fact that we can only take into account a finite number N of classical paths.

Finally, calculation of the S matrix elements requires the projection of G=PSC _{onto the asymptotic scattering states in}

the left and right quantum wire关Eqs. 共2.7a兲 and 共2.7b兲兴. Fol-lowing the same line of reasoning as in Eqs.共3.9兲–共3.12兲, the projections PLand PR give rise to a transmission amplitude

in Fraunhofer diffraction approximation. The transmission amplitude from incoming mode m to a classical path␣inside the cavity with launching angle␪_␣ is given by关15兴

tm共␪␣,k兲 =

冑

2 cos␪_␣ kd

冤

sin

冋

冉

k sin␪_␣+m␲ d

冊

d 2

册

sin␪_␣+m␲ kd + sin

冋

冉

k sin␪_␣−m␲ d

冊

d 2

册

sin␪_␣−m␲ kd

冥

. 共4.6兲

Likewise, the transmission amplitude for a trajectory ␣ approaching the point contact with angle␪_␣to exit in mode

m is also given by Eq.共4.6兲.

With Eq. 共4.6兲 we form now amplitude matrices to map the asymptotic scattering states onto the N⫻N representation of GPSC_{. The N⫻2M matrix for the incoming state, where M}

is the number of open transverse modes共mL,R= 1 , . . . , M兲, is

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A=in_{共k兲 =}

冢

tm_L共␪␣_LL,k兲 0 tm_L共␪␣_RL,k兲 0 0 tm_R共␪␣_RR,k兲 0 tm_R共␪␣_LR,k兲

冣

. 共4.7兲

The corresponding projection amplitude for the outgoing scattering state is a 2M⫻2N matrix and reads

A=out_共k兲 =

冉

tm_L共␪␣_LL,k兲 0 0 tm_L共␪␣_LR,k兲 0 tm_R共␪␣_RL,k兲 tm_R共␪␣_RR,k兲 0

冊

. 共4.8兲 The 2M⫻2M dimensional S matrix follows now from

S=PSC_{共k兲 = A=}out_共k兲G=PSC_共k兲A=in_{共k兲 =}

冉

RmL⬘,mL TmL⬘,mR Tm R ⬘,mL RmR⬘,mR

冊

. 共4.9兲 With Eq.共4.9兲 the semiclassical calculation of the S matrix of a hard-walled quantum billiard with two point contacts is reduced to a sequence of matrix multiplications. System-specific input are the data 共length, angle of incidence and emission from point contact and Maslov index兲 of each of the four classes共L→L,L→R,R→L,R→L兲 of classical tra-jectories.

V. NUMERICAL RESULTS

In the following we will apply the pseudopath semiclas-sical approximation to the calculation of transmission and reflection amplitudes in different regular and chaotic struc-tures 共i.e., circle, rectangle, and stadium兲. We compare the PSCA calculations with the results of the standard semiclas-sical approximation 关10兴, exact quantum calculations 关43兴, and microwave experiments关44兴.

In order to evaluate the scattering matrix elements Tm⬘,m

and Rm⬘,m numerically we have used the truncated form of

Eq. 共4.5a兲 with i艋K, where K is the maximum number of diffractive scatterings共or “kinks”兲. We therefore relate K to the maximum pathlength lmaxof classical paths included. In

order to include all pseudopaths with at least the same length, we require that the length of the shortest trajectory in the system l_minmultiplied by K + 1 has to be larger than l_max to guarantee that all pseudopaths up to lmax are included.

Hence we require

K艌

冋

lmax lmin

册

− 1, 共5.1兲

where the bracket stands for the largest integer less than the argument. This requirement only assure that each classical path is shadowed by a pseudopath of comparable length. The converse is evidently not the case. Pseudopaths are permitted with pathlength up to l_max共K+1兲 for which no classical coun-terpart of comparable length is included. We account for this deficiency by Fourier filtering the power spectra P共l兲 of the

S-matrix elements for l⬎lmax. The classical input data are

determined analytically for the square and circle billiards while for the chaotic stadium billiards the data are generated by calculating the classical trajectories numerically.

A. Regular structures

For regular structures classical and pseudopaths can be easily enumerated and calculations up to very high path length 共l=40兲 have been performed as a sensitive test of semiclassical approximations. We present in the following results for the path-length power spectrum关Eq. 共2.9兲兴. Fig-ures 4 and 5 show the power spectra of the transmission and reflection amplitudes for the second mode, T22and R22, in the

circular billiard with perpendicular leads. Note that all the scattering geometries which we investigate in the following have the same cavity area A = 4 +␲, or, accordingly, a linear dimension D =

冑

4 +␲⬇2.7.

Here we mainly concentrate on the improvement of the PSCA in comparison to the SCA. For T₂₂, the overestimation FIG. 4. Power spectra兩T˜₂₂共l兲兩2_{of the transmission amplitude in}

the circle billiard with perpendicular leads with R =

冑

1 + 4 /␲ and

d = 0.25共see inset Fig. 5兲 for a finite window of k, 1艋k艋6 in units

of␲/d.

FIG. 5. Power spectra兩R˜22共l兲兩2of the transmission amplitude in

the circle billiard with perpendicular leads with R =

冑

1 + 4 /␲ and

d = 0.25 for a finite window of k, 1艋k艋6 in units of␲/d. For more

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of long paths共l⬎18兲 in the SCA calculation is clearly vis-ible: compare the peaks labeled by black arrows in Fig. 4共a兲 with those in Fig. 4共b兲. The contributions of the additional pseudopaths included in the PSCA are responsible for the cancellation effects which suppress the peak heights. The peak heights of the PSCA agree well with the quantum cal-culations for long paths. Also the lack of individual peaks 共i.e., pseudopaths兲 present in the quantum spectrum but miss-ing in the SCA spectra共e.g., at l⬇27.5; see white arrows in Fig. 4兲 can be addressed in the transmission amplitude.

This defect of the SCA can be seen more clearly in the plot for R₂₂ 共see arrows in the upper part of Fig. 5兲. The pseudopath semiclassics provides for the additional peaks in agreement with the quantum calculation. The insets in the lower part of Fig. 5 schematically show the geometry of dominant pseudopaths, which experience at least one diffrac-tive reflection at an open lead. The number next to the insets gives the information on how often the segment of a classical path has to be traversed to build up the corresponding pseudopath.

B. Chaotic structures (the Bunimovich stadium) Up to now we have compared the SCA and PSCA for open billiards with regular classical dynamics. We turn now to chaotic 共i.e., nonregular兲 billiards. The Bunimovich sta-dium 关38,45,46兴 serves as prototype system for structures with chaotic classical dynamics. Before we discuss our nu-merical results, we point out some of the characteristic dif-ferences between the regular and chaotic structures. In the stadium billiard the number of path bundles 关10兴 up to a fixed length increases exponentially. The exponential prolif-eration represents a major challenge for the summation of paths. This leads to technical problems for the calculation of transport quantities in chaotic structures关38兴 due to the limi-tation of the number of paths that realistically can be taken into account. Furthermore, the path length distribution in chaotic structures differs qualitatively from that of regular structures. For the latter we find an algebraically decaying for the classical path-length distribution in contrast to the exponential decay for classically chaotic structures. This is to be distinguished from the exponential decay behavior for pseudopaths. Wirtz et al.关15兴 have shown that for the rect-angular billiard the SCA leads to a linear共algebraic兲 decay of the path length power spectra, while the PSCA gives rise to an exponential decay when pseudopaths are included. How-ever, in chaotic systems, already classical paths proliferate exponentially as a function of the path length and can ac-count for exponential suppression of large path lengths. Therefore, the lack of pseudopaths which also proliferate ex-ponentially is less dramatically felt than in regular systems where the exponential proliferation of pseudopaths competes with only linear proliferation of classical paths. This obser-vation is key to the surprising findings in previous semiclas-sical calculations that the agreement between the semiclassi-cal and the quantum pathlength spectrum is better for chaotic rather than for regular systems 关10,14,17,22,38兴. However, also in chaotic systems, the effects of diffractive pseudopaths can be clearly seen. Figure 6 shows the power spectra of

transmission amplitudes for the stadium with perpendicular leads. For T11, remarkably, the path-length spectra within

PSCA displays fewer pronounced peaks than the SCA in agreement with the quantum calculation. The reason is that in a chaotic system the high density of pseudopaths effec-tively causes path shadowing of true classical paths. As a result, some of the classical peaks are drastically reduced by destructive interference even for comparatively short path length.

C. Comparison with microwave experiments

As a third application we discuss the comparison between experimental studies of geometry-specific quantum scatter-ing in microwave billiards关44兴 and semiclassical approxima-tions共SCA and PSCA兲. The physics and modeling of micro-wave cavities are conceptually similar to that of semiconductor quantum dots due to the equivalence of the time-independent Schrödinger and Helmholtz equations关37兴. Moreover, for microwave frequencies ␯⬍␯max= c / 2h 关37兴,

where h is the height of the microwave billiard, only a single transverse mode is supported by the cavity. This reduces the electromagnetic boundary conditions to Dirichlet-boundary conditions allowing for an exact correspondence between electrodynamics and quantum mechanics. Accordingly, the component of the electrical field perpendicular to the plane of the microwave billiard corresponds to the quantum me-chanical wave function.

A direct measurement of transmission and reflection am-plitudes 共i.e., electrical field amplitudes兲 becomes possible. Moreover the semiclassical analysis presented above is di-rectly applicable to describe scattering in microwave bil-liards.

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13 GHz艋␯艋18 GHz, where only one mode is propagating in the waveguides. Figures 7 and 8 show the experimental and calculated data for the power spectra of the transmission and reflection amplitude, 兩T˜11共l兲兩2 and 兩R˜11共l兲兩2. The

impor-tant role of pseudopaths in reproducing the correct peak pat-tern in good agreement with experimental results can be seen in both the transmission and the reflection spectra. Especially in兩R˜11共l兲兩2 distinct pseudopaths关see insets in Fig. 8共b兲兴

ap-pear which were noted in Ref.关44兴. For example, the peak at

l⬇9 关right white arrow in Fig. 8共a兲兴 is caused by the

trajec-tory with the length l⬇4.5 which, after one revolution in the billiard, is reflected back at the exit by the lead mouth, so that it continues for one more revolution. Thus its total length l⬇2⫻4.5=9. Of course, such nonclassical trajecto-ries are not included in the standard semiclassical approxi-mation关Fig. 8共a兲兴. In this special case, the reflection at the open lead mouth is specular or geometric. This peak would therefore also be present in the semiclassical approximation suggested by Schwieters et al. 关7兴. By contrast, non-geometric reflections 共“kinks”兲 by diffractive scattering are also present, e.g., the peak at l = 7.5关see inset in Fig. 8共b兲兴. The latter class is only contained in the PSCA.

VI. CONCLUSIONS

We have presented an extension of the pseudopath semi-classical approximation 共PSCA兲关15兴 to billiards with arbi-trary convex shape. A diagrammatic expansion of the semi-classical Dyson equation for the Green’s function in terms of diffractive scattering diagrams is developed. The unperturbed Green’s function represents the sum over classical paths. Each encounter with the lead 共point contact兲 spawns new trajectories. By joining disjunct classical paths due to non-geometric共diffractive兲 scattering a large number of trajecto-ries representing pseudopaths results. The Dyson integral equation can be converted to an algebraic matrix equation which can be solved by power series expansion or inversion. Using the examples of a circular and a stadium billiard we have shown numerically that the path-length power spectra calculated by the PSCA overcome shortcomings of standard FIG. 9.共a兲 Test of unitarity U共k兲 on different levels of semiclas-sical approximations关SCA and PSCA 共K_max= 10兲兴 to the rectangu-lar billiard 共D=L=

冑

4 +␲ and d=0.25兲. The staircase function shows the QM result where U共k兲=N共k兲 and N共k兲 is the number of open modes.共b兲 shows the discrepancy between the numerical in-tegration of I and the FDA corresponding to Eq.共3.12兲共solid line兲 and共c兲 the discrepancy in the phase angle.

FIG. 7. Power spectra 兩T˜11共l兲兩2 of the experimental and

calculated共QM, SCA, and PSCA兲 transmission amplitude for the rectangular billiard 共L=225 mm, D=237 mm, and d=15.8 mm兲 with opposite leads 共not centered兲 for a finite window of ␯, 13 GHz艋␯艋18 GHz. The characteristic peaks are identified in terms of classical transmitted trajectories 共insets兲. Experimental data by Schanze关44兴.

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semiclassical approximations by including an exponentially proliferating number of pseudopaths and converges towards quantum transport. Moreover, we have presented a compari-son between microwave billiard experiments关44兴, the SCA and the PSCA calculation and find evidence for contributions by pseudopaths of PSCA to be present in the experimental data. We expect that our pseudopath semiclassical approxi-mation will be able to address unresolved issues of semiclas-sical ballistic transport such as the problem of weak localiza-tion 关16,39兴, the breakdown of symmetry of the autocorrelation function in reflection and transmission, and the semiclassical description of quantum shot noise关27兴.

ACKNOWLEDGMENTS

We thank H. Schanze, U. Kuhl, and H.-J. Stöckmann for providing the experimental data to allow a comparison be-tween the microcavity experiment and our pseudopath semi-classical approximation. Support by the Grants Nos. FWF-SFB016 and FWF-P17359 is gratefully acknowledged. L. W. acknowledges support by the European Community Network of Excellence NANOQUANTA共Grant No. NMP4-CT-2004-500198兲.

APPENDIX: FRAUNHOFER APPROXIMATION AND ITS LIMITATIONS

The integrals along the transverse coordinate 共y兲 across the opening of the point contact are evaluated in Fraunhofer diffraction approximation. The dependence of the action in zero magnetic field kL_␣共y兲 on the transverse coordinate y is taken into account to first-order Taylor series expansion,

L_␣

i共y兲 = L␣i+ sin共␪␣i兲y +

cos共␪␣i兲 L_␣

i

y2¯ , 共A1兲

where L_␣= L_␣共y=0兲 is the length of the path that starts from the center of the lead mouth. Keeping all three terms in Eq. 共A1兲 leads to Fresnel diffraction integrals. Dropping the third term results in a Fraunhofer diffraction approximation with the fundamental integral

IFDA共␪,n兲 =

_冑

1

2d

冕

−d/2

d/2

ei共kn+k sin␪兲y_{dy ,} 共A2兲

which, unlike Fresnel integrals, can be readily solved ana-lytically in terms of elementary functions

IFDA共␪,n兲 =

冑

2 d

冉

sin关共kn i + k sin␪兲d/2兴 kn i + k sin␪

冊

. 共A3兲 In terms of Eq.共A3兲 the reflection amplitude 关Eq. 共3.12兲兴 is given by

by

tm共␪,k兲 =

冑

k cos␪关IFDA共␪,m兲 + IFDA共␪,− m兲兴. 共A5兲

The validity of the FDA hinges on the condition that the third term in Eq. 共A1兲 is negligible. This term is of the order共kd兲共d/L_␣兲ⱗkd共d/D兲 for short paths. Since in quan-tum billiards the asymptotic far-field regime is never reached, the FDA, somewhat counterintuitively, will fail for large kd. Indeed, in Fig. 9共a兲 we show a test of unitarity

U共k兲=R共k兲+T共k兲 on different levels of semiclassical

ap-proximations to the rectangular billiard 关inset of Fig. 9共a兲兴 where the breakdown of the FDA and consequently of the PSCA for high modes m⬎10 共or kdⲏ30兲 can clearly be seen. To highlight the failure of the approximation used in Eq. 共A1兲 or 共3.12兲 we plotted in Figs. 9共b兲 and 9共c兲共solid lines兲 the discrepancy between the exact solution of the in-tegral I 关LHS of Eq. 共3.12兲兴 and the corresponding Fraun-hofer approximation. For L_␣

1= L␣2= L and␪␣1=␪␣2= 0 as a

function of k. Figure 9共b兲 shows the difference in the abso-lute values兩I兩−兩IFDA_{兩 and Fig. 9共c兲 the absolute discrepancy}

in the arguments兩arg共I兲−arg共IFDA_{兲兩. Since the phase}

discrep-ancy reaches a fraction of unity for mⲏ16 关see Fig. 9共c兲兴 random phases in the scattering amplitudes destroy path shadowing by pseudopaths and therefore cause the violation of unitarity 关Fig. 9共a兲兴. The failure of the FDA has more dramatic consequences for the PSCA than for the SCA. This is due to the fact that in PSCA the pseudopath coupling leads to an exponentially growing number of FDAs with an in-creasing number of diffractive scatterings, in contrast to the linear scaling of the number of FDAs involved within SCA. The point to be stressed is that this failure is not due to the PSCA but due to the additional FDA. Applying more accu-rate diffraction integrals in this regime is expected to remedy the problem.

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