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Localization of wave patterns on classical periodic orbits in a square billiard

Y. F. Chen*and K. F. Huang

Department of Electrophysics, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu, 30050, Taiwan Y. P. Lan

Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, Taiwan 共Received 20 June 2002; published 22 October 2002兲

The connection between wave functions and classical periodic trajectories in a square billiard is analytically constructed by using the representation of SU共2兲 coherent states. The analytical function form is modified to show that the wave patterns can be apparently localized on the classical periodic trajectories by superposing a few nearly degenerate eigenfunctions. Based on the analogy between the Schro¨dinger and Helmholtz equa-tions, the features of wave functions are experimentally studied from the transverse pattern formation in a laterally confined microcavity laser. The experimental transverse pattern in a square-shaped microcavity agrees very well with the constructed wave pattern concentrated along classical periodic orbits.

DOI: 10.1103/PhysRevE.66.046215 PACS number共s兲: 05.45.Mt; 03.65.⫺w; 42.60.Jf

I. INTRODUCTION

The two-dimensional 共2D兲 square billiard is one of the simplest billiards that is completely integrable in classical mechanics关1,2兴. One common periodic orbit in a 2D square billiard is usually denoted by共1,1兲. As shown in Fig. 1, the 共1,1兲 periodic orbits can be characterized by a parameter ␾

that is related to the wall positions of specular reflection points关3,4兴. Some examples of periodic orbits are shown in Fig. 1. According to Bohr’s correspondence principle, the classical limit of a quantum system should be achieved when the quantum numbers go to infinity. However, the conven-tional eigenstates of a square billiard in most quantum me-chanics do not manifest the properties of classical periodic orbits even in the correspondence limit of large quantum numbers.

Although semiclassical periodic orbit theory has been used to explain the scarred wave functions in quantum chaos 关5–7兴, the wave functions related to stable periodic orbits seem to have been overlooked. The reason for this disregard is probably that most work has focused on energy levels and energy level statistics 关8–10兴. Furthermore, there are some striking phenomena in open quantum ballistic cavities asso-ciated with the wave functions in terms of classical periodic orbits 关11–13兴. Therefore, to construct the connection be-tween the eigenfunctions and classical periodic trajectories is essentially helpful for understanding quantum-classical cor-respondence as well as quantum transport in mesoscopic sys-tems.

In this paper, we use the representation of SU共2兲 coherent states 关14,15兴 to analytically construct the wave functions related to the classical periodic trajectories in the 2D square billiard. The noticeable finding is that a superposition con-taining only a few nearly degenerate eigenfunctions is al-ready sufficient for localization of the wave function inten-sity on the classical periodic trajectory. This result explains

the reason why the wave functions related to classical peri-odic orbits often appear in weakly perturbed integrable sys-tems关16,17兴. In experiment, the analogy between the Schro¨-dinger and Helmholtz equations 关18兴 enables us to connect the features of wave functions with the transverse modes in a laterally confined microcavity laser. The experimental trans-verse pattern in a square-shaped cavity is generally found to be concentrated along classical periodic orbits. This result confirms that the wave functions related to classical periodic orbits provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.

II. WAVE FUNCTIONS RELATED TO CLASSICAL PERIODIC ORBITS

Recently, Pollet et al. 关19兴 demonstrated that the wave function of the SU共2兲 coherent state for 2D quantum har-monic oscillation is particular simple and well localized on the corresponding classical elliptical trajectory. Mathemati-cally, the SU共2兲 coherent state for 2D quantum harmonic oscillation is a superposition of eigenstates leading to a state with a minimum uncertainty of⌬x⌬y, where x and y are the Cartesian coordinates 关14,15兴. For 2D integrable Cartesian systems, the SU共2兲 minimum-uncertainty states ⌿N(x,y ;␶) can be expressed as the superposition of number eigenstates

K,N⫺K(x,y ) where N is an integer constant and K ⫽0,1,2,...,N:N共x,y;␶兲⫽共1⫹兩␶兩2兲⫺N/2

K⫽0 N

N K

1/2 ␶K K,N⫺K共x,y兲. 共1兲 The parameter ␶is, in general, complex and has a physical meaning in that兩␶兩2 is the ratio of the average values of two quantum numbers. In the limit ␶→0 共or→⬁), the SU共2兲 coherent states becomes the eigenstate ␺0,N(x,y ) 关or *Author to whom correspondence should be addressed. FAX:

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N,0(x, y )]. In terms of the eigenstates of a 2D square bil-liard, ␺K,N⫺K(x, y ) is given byK,N⫺K共x,y兲⫽ 2 asin

共K⫹1兲x a

⫻sin

共N⫺K⫹1兲ay

, K⫽0,1,...,N, 共2兲

where a is the length of the square boundary. The numerical calculation reveals that the classical periodic orbits shown in Fig. 1 can be figured out by setting ␶⫽ei␾. Using Eqs.共1兲 and 共2兲, the condition 兩␶兩2⫽1 can lead to

v

x

/

vy

⫽1 where

vx

and

vy

are the average speeds along the x and y axes. In other words, the general relation⫽ei␾ is consis-tent with the requirement of

vx

/

vy

⫽1 for the classical periodic obits shown in Fig. 1. Substituting Eq. 共2兲 and ␶ ⫽eiinto Eq.共1兲 yields

N共x,y;␾兲⫽

2 a

1 2N/2K

⫽0 N

N K

1/2 eiK␾ ⫻sin

共K⫹1兲ax

sin

共N⫺K⫹1兲y a

. 共3兲 Figure 2 depicts the ␾ dependence of the wave function 兩⌿N(x, y ;␾)兩2for N⫽25. It can be seen that the behavior of 兩⌿N(x, y ;␾)兩2 agrees very well with the classical periodic orbit shown in Fig. 1. Furthermore, the distribution of 兩⌿N(x, y ;␾)兩2 illustrates geometrically Bohr’s correspon-dence principle: the velocity of the classical particle is at a minimum at the specular reflection points of the motion, and therefore the distribution has a peak at these points.

The wave function given in Eq.共3兲 represents a traveling-wave property. The standing-traveling-wave representations can be ob-tained by using ⌿N(x, y ;␾)⫾⌿N*(x,y ;␾). Including the normalization constant, the standing-wave forms can be ex-pressed as ⌿N c 共x,y;␾兲⫽ 共2/a兲

K⫽0 N

N K

cos 2K

1/2

K⫽0 N

N K

1/2 ⫻共cos K␾兲sin

共K⫹1兲ax

⫻sin

共N⫺K⫹1兲ay

共4兲 and ⌿N s共x,y;兲⫽ 共2/a兲

K⫽0 N

N K

sin 2K

1/2

K⫽0 N

N K

1/2 ⫻共sin K␾兲sin

共K⫹1兲ax

⫻sin

共N⫺K⫹1兲ay

. 共5兲 The N dependence of the wave pattern兩⌿Nc(x,y ;␾)兩2is pre-sented in Fig. 3. Here we only show the wave pattern 兩⌿N

c

(x,y ;␾)兩2 because the wave pattern 兩⌿Ns(x, y ;␾)兩2 is generally the same; the value of the parameter ␾is fixed to be ␲/2 for convenient representation. It can be seen that a large quantum number N is not necessary for the localization of the probability density on the classical trajectory. Even so, it should be noted that the wave function in Eqs. 共3兲–共5兲 is not a stationary state because the eigenstate components are not degenerate for the Hamiltonian H. Nevertheless, the cal-culation result shown in Fig. 4 reveals that⌬H/

H

is pro-portional to 1/N. In other words, ⌬H/

H

→0 as N→⬁.

This result guarantees the coherent state in Eqs.共3兲–共5兲 to be a stationary state in the classical limit.

FIG. 1. Some classical periodic orbits. The periodic orbits are in terms of the parameter ␾ which is related to the wall positions of specular reflection points.

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FIG. 2. The ␾ dependence of the wave pattern 兩⌿N(x,y ;␾)兩2 from Eq. 共3兲 for N⫽25. The wave patterns correspond to those shown in Fig. 1.

FIG. 3. The N dependence of the wave pattern 兩⌿N c

(x,y ;␾)兩2 from Eq.共4兲 obtained by fixing␾ to be ␲/2 to show the standing-wave property.

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Equations 共3兲–共5兲 indicate that the wave function repre-sentation consists of N⫹1 Hamiltonian eigenstates. How-ever, the numerical analysis reveals that a superposition of only a few eigenstates is already sufficient to result in the localization on the classical trajectory. To reflect this prop-erty, we define the partially coherent states corresponding to Eqs. 共3兲–共5兲 as ⌿N, M共x,y;␾兲⫽ 共2/a兲

K⫽q N⫺q

N K

冊册

1/2

K⫽q N⫺q

N K

1/2 exp共iK␾兲 ⫻sin

共K⫹1兲ax

sin

共N⫺K⫹1兲y a

, 共6兲 ⌿N, M c 共x,y;兲⫽ 共2/a兲

K⫽q N⫺q

N K

cos 2K

1/2

K⫽q N⫺q

N K

1/2 ⫻共cos K␾兲sin

共K⫹1兲ax

⫻sin

共N⫺K⫹1兲ay

, 共7兲 and ⌿N, M s 共x,y;兲⫽ 共2/a兲

K⫽q N⫺q

N K

sin 2K

1/2

K⫽q N⫺q

N K

1/2 ⫻共sin K␾兲sin

共K⫹1兲ax

⫻sin

共N⫺K⫹1兲ay

, 共8兲

FIG. 4. The calculation result shown for ⌬H/H典 versus the quantum number N.

FIG. 5. The M dependence of wave patternN, M c

(x,y ;␾) from Eq.共7兲 obtained by fixing␾ to be 0.55␲ to show the dependence of wave localization on the number of eigenstates.

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where the index M⫽N⫺2q⫹1 represents the number of eigenstates used in the wave function. All partially coherent states in Eqs. 共6兲–共8兲 have similar density localization; we conveniently choose⌿N, Mc (x,y ;) to demonstrate the M de-pendence of the wave pattern, as shown in Fig. 5. Here we fix ␾ to be 0.55␲only for the presentation. It can be seen that only 5–9 eigenstates are adequate to localize the wave pattern on the classical trajectory. The eigenvalue of the en-ergy corresponding to the eigenstate ␺m⫺1,n⫺1(x,y ) ⫽(2/a)sin关m(x/a)兴sin关n(y/a)兴 is given by

E⫽ ប 2 2m

a

2 共m2⫹n2兲. 共9兲

The degenerate eigenstate depends on the sum of two integer squares (m2⫹n2). Figure 6 shows that the eigenstates of the partially coherent states in Fig. 5 are not exactly degenerate but nearly degenerate. The partially coherent states in Eqs. 共6兲–共8兲 may often become the eigenstates in weakly per-turbed 2D square billiards关16,17兴 because they can be com-posed of only a few nearly degenerate eigenstates. In fact, the wave patterns, like partially coherent states to be local-ized on classical periodic orbits, have been discussed exten-sively in ballistic quantum dots 关20,21兴. In the following section, we demonstrate that the wave patterns of the par-tially coherent states can be experimentally observed using the transverse pattern formation of a confined microcavity laser.

III. EXPERIMENTAL RESULTS AND DISCUSSION Recently, vertical-cavity surface emitting semiconductor lasers共VCSEL’s兲 of large transverse section and short cavity

length have been used to study the transverse pattern forma-tion关22–25兴. VCSEL’s inherently emit a single-longitudinal-mode wave because of their extremely short cavity length. The single-longitudinal-mode laser is a useful laboratory to study transverse phenomena without the influence of other degrees of freedom. Hegarty et al. 关24兴 reported interesting transverse mode patterns from oxide-confined square-shaped VCSEL’s with large aperture. Their experimental results re-vealed that a wave incident upon the current-guiding oxide boundary would undergo total internal reflection because of large index discontinuities between the oxide layer and the surrounding semiconductor material. In other words, the VCSEL cavity can be considered as a planar waveguide with a dominant wave vector along the vertical direction.

According to the waveguide theory关26兴, the electromag-netic fields with a predominantly z direction of propagation can be approximated as

E共x,y,z,t兲⫽E共x,y兲ei␤z⫺␻t兲, 共10兲 where ␻ is the angular frequency and␤ is the propagation constant along the z direction. Using expression 共10兲 in the Maxwell equations for a uniform medium gives the well-known Helmholtz equation 关26兴

关ⵜt

2

⫹共k22兲兴E⫽0, 共11兲

whereⵜt2is the transverse part of the Laplacian operator, k is the total propagation constant related to the angular fre-quency by k⫽␻/c, and c is the wave speed. In fact, lasing modes in a conventional laser are usually characterized by near-paraxial propagation normal to the resonator mirrors, with polarization in the plane of the mirrors 关27,28兴. Within the framework of the scalar paraxial approximation, the mag-nitude of the longitudinal field 兩Ez兩 is very much smaller than that of the transverse field 兩Et兩 关27兴. Therefore, trans-verse modes in a vertical-cavity laser can be determined by the Helmholtz equation for the transverse field. The solutions to the Helmholtz equation with total internal reflection boundaries (Et⫽0 at the boundary兲 are equivalent to the solutions of the 2D Schro¨dinger equation with hard wall boundaries (⌿N⫽0 at the boundary兲 of the same geometry. Recently, Doya et al. 关29,30兴 have introduced the paraxial approximation to establish an analogy between light propa-gation along a multimode fiber and quantum confined sys-tems. Actually, the guiding character in the oxide-confined VCSEL’s is similar to that in optical fibers. Even so, the transverse patterns of VCSEL’s under cw operation corre-spond truly to stationary states of the system.

Due to the analogy between the Schro¨dinger and Helm-holtz equations 关18兴, it is essentially feasible to use the oxide-confined VCSEL cavities like microwave cavities 关31,32兴 to represent quantum mechanical potential wells. In this case, the transverse patterns can reveal the probability density of the corresponding wave functions in the 2D quan-tum billiards. Here we experimentally study the transverse pattern formation in a square-shaped VCSEL with large ap-erture to compare with the wave functions in the 2D square billiards.

FIG. 6. A diagram illustrating the eigenspectrum of the square billiard. Each gray point represents an eigenvalue; the solid line indicates the curve of the equation n2⫹m2⫽272⫹272; the circles are the eigenstates that are selected for the wave patterns shown in Fig. 5.

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Square-shaped VCSEL’s with large apertures are fabri-cated by metal organic chemical vapor deposition to emit at a wavelength around␭z⫽795 nm. The wafers were wet oxi-dized at 425 °C and the oxidation time is controlled to fab-ricate a 40 ␮m oxide aperture in a 110␮m mesa structure. The device structure of these oxide-confined VCSEL’s is similar to that described by Ref. 关24兴. Experimental results show that the transverse patterns of VCSEL’s can be evi-dently divided into two regimes of low-divergence and high-divergence emissions. Hereafter we will concentrate on the high-divergence emission, which appears only at reduced temperature and near-threshold operation. It is expected that the thermal-lensing effect will switch the device into the low-divergence regime because joule heating induces a tem-perature rise across the device cross section. Typically, high-divergence patterns are very symmetric and those of low divergence are more irregular. Therefore it is easy to differ-entiate the regimes in which the lasers are being operated.

The near-field patterns are measured with a charge-coupled device camera 共Coherent, Beam-Code兲 and an opti-cal setup similar to that described in Ref. 关24兴. The trans-verse mode spectral information of the laser is monitored by an optical spectrum analyzer共Advantest Q8347兲. The present spectrum analyzer employs a Michelson interferometer with a Fourier spectrum system to reach a resolution of 0.002 nm. The transverse mode spacing can be derived as ⌬␭t ⬇␭z

3

/(4a2)⫽0.0785 nm. Since the resolution of the spec-trum analyzer is 0.002 nm, the transverse mode spectral in-formation can be clearly resolved. We cooled the device to a temperature around 0–10 °C. Near lasing threshold the trans-verse pattern emitted from the present VCSEL device is found to be linearly polarized and highly concentrated along the classical orbit, as shown in Fig. 7. The measurement of the optical spectrum for the laser beam is depicted in Fig. 8. The result reveals that the linearly polarized transverse pattern is a single-frequency oscillation; namely, it is a sta-tionary state. The excellent similarity between the

experi-mental transverse pattern and the wave pattern of the par-tially coherent state shown in Fig. 6 indicates that the experimental transverse pattern can be described in terms of a few nearly degenerate eigenstates of a perfect square bil-liard. The optical spectrum information shown in Fig. 8 im-plies that the nearly degenerate modes are phase synchro-nized to a common frequency by the mechanism of cooperative frequency locking 关33兴. Previous laser experi-ments have proved cooperative frequency locking to be an important process in transverse pattern formation 关34–36兴. As seen in Fig. 6, a lot of nearly degenerate modes can be selected in the process of cooperative frequency locking. However, the mode selection rule is based on the criterion that the resultant field structure should have the minimum mode volume for the lowest lasing threshold. The criterion of a minimum-volume mode that corresponds to the minimum free energy is equivalent to wave localization along the clas-sical trajectory. This is why the experimental transverse pat-terns have a connection with the partially coherent states related to classical trajectories. This result confirms that the wave functions related to classical periodic orbits provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.

IV. CONCLUSIONS

We have analytically connected the wave function with the classical periodic trajectory in a square billiard using the representation of SU共2兲 coherent states. We have further modified the analytical form to demonstrate that only a few nearly degenerate eigenfunctions are already adequate to re-sult in the localization of the wave pattern on the classical periodic trajectory. In the experiment, we use the analogy between the Schro¨dinger and Helmholtz equations to study the features of wave functions from transverse pattern forma-tion in a laterally confined microcavity laser. In a square-shaped microcavity, the experimental transverse pattern agrees very well with the theoretical wave pattern concen-FIG. 7. The experimental result for the near-field pattern of the

square-shaped VCSEL device near lasing threshold.

FIG. 8. A plot of the optical spectrum of the transverse pattern shown in Fig. 7.

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trated along classical periodic orbits. The experimental result evidences that the wave function obtained as a linear super-position of a few nearly degenerate eigenstates can provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.

ACKNOWLEDGMENT

The authors thank the National Science Council for their financial support of this research under Contract No. NSC-91-2112-M-009-030.

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數據

FIG. 1. Some classical periodic orbits. The periodic orbits are in terms of the parameter ␾ which is related to the wall positions of specular reflection points.
FIG. 3. The N dependence of the wave pattern 兩⌿ Nc
FIG. 4. The calculation result shown for ⌬H/ 具 H 典 versus the quantum number N.
FIG. 6. A diagram illustrating the eigenspectrum of the square billiard. Each gray point represents an eigenvalue; the solid line indicates the curve of the equation n 2 ⫹m 2 ⫽27 2 ⫹27 2 ; the circles are the eigenstates that are selected for the wave patt
+2

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