Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes

Model of commensurate harmonic oscillators with SU(2) coupling interactions:

Analogous observation in laser transverse modes

Y. C. Lin,1T. H. Lu,2K. F. Huang,1and Y. F. Chen1,*

1_{Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan} 2_{Department of Physics, National Taiwan Normal University, Taipei, Taiwan}

(Received 10 February 2012; published 24 April 2012)

We theoretically explore the eigenstates of a coupled p:q commensurate harmonic oscillator with SU(2) coupling interactions under the canonical transformation. The spatial patterns of the high-order eigenstates are found to be markedly localized on Lissajous figures from single to multiple periodic orbits. Controlling the pumping size in large-Fresnel-number degenerate cavities, we have experimentally observed the laser transverse modes that display the wave patterns to be analogous to the derived eigenstates.

DOI:10.1103/PhysRevE.85.046217 PACS number(s): 05.45.−a, 03.65.Fd, 03.65.Ge, 03.65.Vf

I. INTRODUCTION

For the past few decades, models developed from quantum mechanics have been employed progressively to explore the emergent phenomena in numerous different branches of physics because they can be interpreted with the same theoretical forms as quantum formulas [1–5]. One of the most profound similarities is that the electromagnetic wave equation in paraxial approximation is isomorphous to the Schr¨odinger equation [6–9]. Consequently, the electromagnetic radiation modes in the optical resonator or waveguide are analogs of the wave functions of a quantum system [10–12]. The tight connection between the paraxial beam propagation and quantum mechanics has been extensively exploited to study wave chaos phenomena [11,13,14], disorder induced wave localization [15], semiclassical physics [16,17], and transient dynamics of quantum states [18–20].

The coupled harmonic oscillators (HOs) have been em-ployed successfully to explore the hydrogen atom problem [21], charged particles in external field [22,23], states of deformed nucleus in the Nilson model [24], shell effects in nuclei and metallic clusters [25], and orbital magnetism in quantum dots [26]. More recently, the isotropic HOs with SU(2) coupling interactions have been used to investigate the generation and evolution of quantum vortex states [27] and the transformation geometry between Lissajous and trochoidal orbits [28]. It has been shown [29,30] that the commensurate HOs can be mapped into the isotropic HOs via the canonical transformation. Although the isotropic HOs with SU(2) coupling interactions have been verified to be a striking analytical model, the quantum states of canonically mapped commensurate HOs with SU(2) coupling interactions have not been thoroughly explored yet.

In this work we start from a coupled p : q commensurate HO with canonical transformation to develop a quantum mechanical model with SU(2) coupling interactions. We explore the eigenstates and find that the high-order spatial patterns are noticeably concentrated on Lissajous figures from single to multiple periodic orbits. In an earlier work [12], the three-dimensional (3D) coherent lasing modes with transverse patterns corresponding to single Lissajous figures have been

*_{[email protected]}

methodically generated in degenerate cavities with a large off-axis tightly focused pumping scheme. Here we intriguingly verified that the 3D coherent lasing waves can be manipulated to form more intricate transverse patterns corresponding to multiple Lissajous orbits as found in the quantum eigenstates of the developed model. The number of Lissajous orbits in the lasing transverse pattern is experimentally confirmed to be proportional to the pumping spot size. More importantly, the role of the phase factor introduced by the SU(2) coupling interactions can be nicely manifested from the propagating property of the lasing modes. We expect that the findings of controlling lasing transverse modes with spatial patterns to be related to quantum states could open new attractive issues in quantum physics and optical pattern formations.

II. THEORETICAL MODEL

The general form of a p : q two-dimensional (2D) com-mensurate anisotropic HO comprising a weak coupling term can be modeled as

ˆ

H= ˆH0+ ˆHc, (1)

where ˆHcsignifies the coupling characterized by a

vibration-rotational mechanism and the detail will be provided later. With dimensionless spatial operators ˆxand ˆy, the Hamiltonian of the p:q commensurate anisotropic HO ˆH0 can be given

as [29] ˆ H0= 1₂  ˆ p2_x+ ˆp2_y+ ω2₁xˆ2+ ω₂2yˆ2, (2) where ω1= qω and ω2= pω, ω is a common factor of the

oscillation frequencies ω1and ω2, and q and p are integers. In

terms of the ladder operators, the Hamiltonian(2)becomes ˆ H0= ω  1 p  ˆa₁†ˆa1+ 1 2  + 1 q  ˆa†₂ˆa2+ 1 2  , (3) where ω= ωpq, ˆa1= (qω ˆx + i ˆpx)/ √ 2qω, ˆa2 = (pωy + ipˆy)/ √ 2pω, ˆa₁†= (qω ˆx − i ˆpx)/ √

2qω, and ˆa†₂= (pω ˆy −

ipˆy)/

√

2pω. The eigenstates of the commensurate anisotropic HO in Eq.(3)are|n1p+ λ1,n2q+ λ2Hˆ0as has been shown

by Louck et al. [29], where (n1, n2) are arbitrary non-negative

integers, and (λ1, λ2) are constants that λ1= 0,1, . . . , p− 1 and λ2= 0,1, . . . ,q − 1. The normalized spatial

representation is the well-known Hermite-Gaussian states [32] x,y|n1p+ λ1,n2q+ λ2Hˆ0 = [2n1p+n2q+λ1+λ2_(n 1p+ λ1)!(n2q+ λ2)!π ]−1/2 × e−(x2_+y2_)/2 Hn1p+λ1(x)Hn2q+λ2(y), (4)

where Hn(·) is the Hermite polynomial of order n. It reveals

the fact that the eigenstates have been divided into pq different subsets of states and the degeneracy holds when n1+ n2is a

constant N for fixed (λ1,λ2) corresponding to the eigenvalues E= ω[(n1+ n2)+ 1/2p+1/2q+λ1/p+ λ2/q] of Eq. (3).

For a particular case (p,q)= (1,1) of the isotropic HO, it is evident that Eq.(3)implies ˆH0= ω[2 ˆJ+ 1], where ˆJ is the

Casmir operator associated with the SU(2) Lie algebra and the corresponding generators derived by Schwinger [32] are

ˆ

J1= (ˆa†1ˆa2+ ˆa†2ˆa1)/2, Jˆ2= (ˆa1†ˆa2− ˆa†2ˆa1)/2i,

(5) ˆ

J₃= (ˆa†₁ˆa1− ˆa2†ˆa2)/2.

The operators ˆJi follow the angular momentum

commuta-tion relacommuta-tion [ ˆJi, ˆJj]= i¯hεij kJˆk [32], where the Levi-Civita

tensor εij k equals + 1 ( − 1) if (i,j,k) is an even (odd)

permutation, and zero otherwise.

With the nonbijective canonical transformation, the com-mensurate anisotropic HO can be mapped onto an isotropic one in a degenerate eigenspace [29]. The mapping suggests Schwinger’s development of SU(2) symmetry represented by the canonically transformed ladder operators and leads to the analytical solutions to the Hamiltonian in Eq.(3). Therefore, under the canonical transformation, the Hamiltonian in Eq.(3) can be transformed into

ˆ H0= ω  ˜a₁†˜a1+1₂  +˜a†₂˜a2+1₂  , (6)

where ˜ai and ˜ai† (i= 1,2) are the canonically transformed

ladder operators which bear the relations [29] ˜a_i†= 1 ξi ( ˆni− λi)[ ˆni( ˆni− 1) · · · (ˆni− ξi+ 1)]−1/2( ˆa†i) ξi_, (7) ˜ai = 1 ξi ( ˆni− λi)[ ˆni( ˆni− 1) · · · (ˆni− ξi+ 1)]−1/2( ˆai)ξi,

with number operator ˆni = ˆai†ˆai and (ξ1,ξ2)= (p,q). The

operation of the ladder operators on particular eigenstates for fixed (λ1,λ2), for instance, are ˜a1†|n1p+ λ1,n2q+ λ2Hˆ0 =

√

n₁+ 1|(n₁+ 1)p + λ₁,n₂q+ λ₂Hˆ0 and ˜a1|n1p+ λ1, n2q+ λ2Hˆ0 =

√

n1|(n1− 1)p + λ1,n2q+ λ2Hˆ0. Obviously,

Eq. (6)is converted into the same form as the isotropic HO when the degeneracy can exist for n1+ n2= N according to

the eigenvalue E= ω(n1+ n2+ 1) to the Hamiltonian ˆH.

The generators of the SU(2) symmetry group can be rewritten in a way that makes them the generators responsible for the anisotropic HO under consideration:

˜

J1= (˜a1†˜a2+ ˜a2†˜a1)/2, J˜2= (˜a1†˜a2− ˜a2†˜a1)/2i,

(8) ˜

J₃= (˜a₁†˜a1− ˜a2†˜a2)/2.

The operators also satisfy the Lie commutation relation. Particularly, ˜J₁= ˆJ₁, ˜J₂= ˆJ₂, and ˜J₃= ˆJ₃ for the special case of the isotropic HO with (p,q)= (1,1).

Let us now return to our formal considerations of the coupled anisotropic HO of the Hamiltonian given in Eq.(1). The coupling term ˆHc is introduced as an SU(2) coupling

interaction [27,28], which can be modeled as ˆ

H = ˆH0+ ˆHc= ˆH0+ (A ˜J1+ B ˜J2+ C ˜J3), (9)

where A, B, and C are constants indicating the coupling parameters with the assumption A, B, and C ω _{for weak}

coupling. We would like to remark that, in view of the case (p,q)= (1 , 1) for the coupled isotropic HO, the wave functions have been demonstrated previously on a group theory level via the SU(2) transformation [28,31]. Likewise, it enables us to derive the wave functions by employing the transformation of the SU(2) symmetry group.

Now consider our problem of the Hamiltonian in Eq.(9); that is, find the eigenstates to the commensurate coupled HO. Under the SU(2) transformation, the eigenstates to the Hamiltonian in Eq.(9)can be given as

|n1p+ λ1,n2q+ λ2Hˆ

= ˆU|n1p+ λ1,n2q+ λ2Hˆ0

= e−iα ˜J3_e−iβ ˜J2|n

1p+ λ1,n2q+ λ2Hˆ0, (10)

where Uˆ = e−iα ˜J3e−iβ ˜J2 _{is the unitary operator, α}=

− tan−1_{(B/A), and β= − tan}−1₍√_A2_{+ B}2_{/C). The}

Hamil-tonian Hˆ can be diagonalized into Hˆ= ˆU−1H ˆˆU=

ˆ

H0+

√

A2_{+ B}2_{+ C}2_J˜_{3. Therefore, with the operation} on eigenstates |n1p+ λ1,n2q+ λ2Hˆ0, eigenvalues to Hˆ

can be directly achieved as E= ω(n1+ n2+ 1) +

√

A2_{+ B}2_{+ C}2_(n

1− n2)/2, where nearly degenerate occurs

for n1+ n2= N, where N is a constant. Consequently, the

eigenstates to the Hamiltonian ˆHcan be obtained in terms of the Wigner d coefficient [33]:

|n1p+ λ1,n2q+ λ2Hˆ = e iN α/2  _N  m1=0 e−im1αd_mN/_{1−N/2,n1−N/2}2 (β)|m1p+ λ1,m2q+ λ2Hˆ0 , (11) where d_mN/_{1−N/2,n1−N/2}2 (β)=m1!(N− m1)!n1!(N− n1)! min[N−n1,m1] ν=max[0,m1−n1] (−1)ν_[cos(β/2)]n1+m1−2ν_[sin(β/2)]n2−m1+2ν v!(N− n1− v)!(m1− v)!(n1− m1+ v)! , (12)

1 b b2 b3 b4 1 c c2 c3 c4 1 d d2 d3 d4

3,1

3, 2

2,1

, q

p

0

n

n

1

n

2

n

3

FIG. 1. (Color online) (a1)–(a4) Numerical simulations of Wigner d coefficient|d_mN/_1−N/2,n2 _1−N/2(β)|2 _{with respect to m}

1 for various n1.

(b1)–(d4) Numerical wave patterns for the intensities of eigenstates|n1p+ λ1,n2q+ λ2Hˆ. See text for a detailed description of the parameters.

and m₁+ m₂= N. Evidently, eigenstates |n₁p+ λ₁, n₂q+ λ₂Hˆ can be expressed as a linear superposition

of the set of |m1p+ λ1,m2q+ λ2Hˆ0. Figures 1(a1)–

1(a4) show distributions of the Wigner d coefficient |dN/2

m1−N/2,n1−N/2(β)|2 with respect to m1 for n1= 0−3, n₁+ n₂= N, N = 60, and β = π/2, which reveal the

composition of |n1p+ λ1,n2q+ λ2Hˆ with eigenstates

|m1p+ λ1,m2q+ λ2Hˆ0 of different orders. Figures 1(b1)–

1(b4), 1(c1)–1(c4), and 1(d1)–1(d4) illustrate the corre-sponding eigenstates |n1p+ λ1,n2q+ λ2Hˆ with (p,q)=

(2,1), (p,q)= (3,1), and (p,q) = (3,2), respectively, and all with (λ1,λ2)= (0,0), (α,β) = (π/2,π/2), n1+ n2= N,

and N= 60.

Note that it is valid for us to choose a specific eigenspace of (λ1,λ2)= (0,0) since, in the classical limit (N large enough),

[34] has confirmed that the choice of the eigenspace does not affect the final results. Therefore, parameters (λ1,λ2) are

set to be (0,0) in the following discussions. Moreover, (α, β) are chosen for specific parameters. α signifies an additional phase shift between the two HOs in x and y directions, and

β corresponds to the coupling strength arising from ˆHc. The

distribution|d_mN/_{1−N/2, n1−N/2}2 (β)|2 _{shown in Figs.2(a1)–2(a8)}

are varied with β, which indicates different composition of the corresponding eigenstates|n1p,n2qHˆ as that depicted in

Figs.2(b1)–2(b8) with (p,q)= (2,1), α = π/2, n1= 1, and N = 60. For β = 0 and β = π, the eigenstates can be seen to

project precisely onto particular eigenstates |p,q(N − 1)Hˆ0

and|p(N − 1),qHˆ0, respectively. While β is determined, the

conversion of α can be illustrated as shown in Figs.3(a1)–3(a5) with n1= 1, β = 0.4π, and N = 60, and in Figs.3(b1)–3(b5)

with n1= 3, β = 0.74π, and N = 60. The morphologies

transform since different relative phases are introduced into the superposition of states |n1p,n2qHˆ with the set of states

|m1p,m2qHˆ0.

Theoretical results disclose intriguing geometric patterns localized on an ensemble of periodic Lissajous orbits, which suggests a kind of quantum-classical analog. It is evidenced that the number of peaks of|dmN/_{1−N/2, n1−N/2}2 (β)|2is consistent

with the number of Lissajous orbits of|n1p,n2qHˆ for various n₁. This fact implies that each orbit of the multi-Lissajous patterns is formed by the superposition of a particular group of the set |m1p,m2qHˆ0 with distribution centered on the

m1 0. 00 0. 02 0. 04 0. 06 0. 08 m₁ 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 1 b b2 b3 b4 5 b b6 b7 b8 m₁ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 m₁ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 m₁ 0.00 0.02 0.04 0.06 0.08 0.10 m₁ 0.00 0.02 0.04 0.06 0.08 0.10 m1 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 m₁ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2 2 3 5 6 5 a a6 a7 a8 0 6 4 3 1 a a2 a3 a4 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60

FIG. 2. (Color online) (a1)–(a8) Numerical simulations of Wigner d coefficient|d_mN/2

1−N/2,n1−N/2(β)|

2 _{with respect to m}

1 for various β.

(b1)–(b8) Corresponding numerical wave patterns for the intensities of eigenstates|n1p,n2qHˆ. corresponding peak of |d_mN/_{1−N/2,n1−N/2}2 (β)|2_{. A relation l =}

min(n1,n2)+ 1 can be given, where l denotes the number of

orbits. While the magnitude of min(n1,n2) becomes larger,

the related excited states display more complex causticlike

0 0.08 0.18 0.28 0.5 1 a a2 a3 a4 a5 1 b b2 b3 b4 b5 0 0.18 0.25 0.35 0.5

FIG. 3. Numerical wave patterns for the intensities of eigenstates|n1p,n2qHˆ with respect to varyingα. (a1)–(a5) β= 0.4 π. (b1)–(b5)

1

6

n

n

14

22

n

n

30

10

n

n

18

26

n

n

34

FIG. 4. Numerical wave patterns for higher indices n1followed by the case in Figs.1(b1)–1(b4).

geometric patterns as shown in Fig.4, followed by the case in Figs.1(b1) and1(b2). Additionally, the symmetry is held for (n1,n2)⇔ (n2,n1), e.g., eigenstates of (n1,n2)= (26,34) and

(n1,n2)= (34,26) shown in Figs.4(f)and4(h)possess

identi-cal morphology for equal distribution |d_mN/_{1−N/2,n1−N/2}2 (β)|2_.

Though the same morphology is notified, the eigenstates (n1,n2) and (n2,n1) are characterized by distinct features of the

quantum probability currentJ , where J (x,y) = Im(∗ ∇)

[33] and n1p,n2q(x,y)= x,y|n1p,n2qHˆ. Taking the cases of

(n1,n2)= (1,59) and (n1,n2)= (59,1) as an example, it can

be seen that the probability currentJ (x,y) flows in counter

directions for the two states as depicted in Figs. 5(b) and 5(c). Note that the vector fieldJ (x,y) has been normalized to

J (x,y)/|J (x,y) | for observing the detailed structures, and the

constants ¯h and particle mass are set to be unity.

Figure6further displays the phase structures for the case in Figs.4(a)and4(e). The enlarged figures of the box region in Figs.6(a2) and6(b2) are presented, respectively, in Figs.6(a3)

and6(b3), where the complicated phase distribution indicates promising development in quantum physics such as quantum entanglement and quantum information as long as the quantum states |n1p,n2qHˆ can be accessibly prepared [35]. As we

will see in the following, the correlated optical modes can be successfully generated in an astigmatic large-Fresnel-number laser cavity [31]. The certification is based on the reconciliation between the wave equation for laser transverse modes in the paraxial approximation and the Schr¨odinger equation for the 2D quantum confined systems [10–12]. Most importantly, Nienhuis et al. [10] has clarified high correlation between the quantum operator algebra and manipulation in the laser cavity. Consequently, based on the acts of the quantum operators

e−iα ˜J3e−iβ ˜J2_{in Eq.(10), we are able to generate the analogous}

wave patterns by the correlated operation in a laser resonator. The presented research will not be restricted to the theoretical viewpoint of quantum physics and, intriguingly, practical correspondence will be explicitly provided with optical waves.

a b c

1, 2 1, 59

n n 59 , 1

FIG. 5. (Color online) (a) Numerical wave patterns for the intensities of eigenstates|n1p,n2qHˆfor (n1,n2)= (1,59) and (n1,n2)= (59,1);

1

6

n

22

n

FIG. 6. (Color online) (a1),(b1) Theoretical results in Figs.4(a)and4(e). (a2),(b2) Phase distribution of (a1) and (b1), respectively. (a3), (b3) Enlarged figures of the box region in (a2) and (b3), respectively.

III. ANALOGOUS OBSERVATION IN LASER TRANSVERSE MODES

The experiment mainly consists of a laser resonator, a pumping source, and an imaging system as shown in Fig.7(a). The laser resonator is composed of a spherical mirror and a large-aperture gain medium. The gain medium is an a-cut 2.0-at. % Nd:YVO4crystal with a length of 2 mm and a 10 ×

10 mm2 cross section. Two sides of the crystals are coated for high reflection and antireflection, respectively, at 1064 nm. The radius of curvature of the spherical mirror is R= 10 mm

and its reflectivity is 99.7% at 1064 nm. The pump source is a 3 W 808 nm fiber-coupled laser diode with a pump core of 100 μm in radius. A focusing lens with a focal length of 20 mm and 90% coupling efficiency is employed to reimage the pump beam into the crystal.

The length of the present resonator can be set to form various degenerate cavities in which a resonance frequency with a high-order transverse mode is equal to another reso-nance frequency with fundamental transverse modes [36]. It has been found [12,31] that the lasing modes in degenerate

c b d e Cavity mirror Gain medium Screen CCD camera a Focusing lens

FIG. 7. (Color online) (a) Experimental setup for the generation of laser modes with large off-axis defocusing pumping. (b)–(e) Experimental far-field patterns corresponded to the numerical results in Figs.1(b1)–1(b4).

1 a a2 a3 a4 a5 1 b b2 b3 b4 b5 0 z L z 0.05L z 0.11L z 0.17L z 0.33L 0 z L z 0.12L z 0.15L z 0.22L z 0.35L

FIG. 8. (Color online) Experimental tomographic transverse patterns observed along the propagation direction from the beam waist. (a1)–(a5) (x,y)= (0.21 mm,0.10 mm). (b1)–(b5) (x,y) = (0.57 mm,0.10 mm).

cavities with a large off-axis tightly focused pumping spot of ∼25 μm are usually dominated by the 3D coherent waves with transverse patterns corresponding to single Lissajous figures. In this work we systematically find that the 3D coherent lasing waves can be manipulated by enlarging the pumping spot size to form more intricate transverse patterns corresponding to multiple Lissajous orbits as shown in the quantum eigenstates of the developed model. Figures7(b)–7(c) depict the experimental observations for the cavity length L of 7.5 mm with the pumping size∼50–100 μm. It can be seen that the experimental observations agree very well with the numerical results in Figs.1(b1)–1(b4), which is associated with our theoretical analysis that groups of eigenstates can be excited simultaneously to compose the corresponding Lissajous patterns. Note that the indices (p,q) are determined from the cavity length L and the degenerate conditions [12]. We verify that the number of Lissajous orbits in the lasing transverse pattern is governed by the spot size of the pumping beam. The larger the pumping size, the greater the number of Lissajous orbits that can be effectively excited.

In the preceding section, we demonstrated the effect of the parameter β which signifies the degree of coupling mechanism and governs the distribution of the states |m1p,m2qHˆ0

employed in the superposition of eigenstates|n1p,n2qHˆ. In

a laser resonator, the influence of β corresponds fairly to the amount of astigmatism arising from the off-axis pumping. While β is chosen, the parameter α can be realized as the Gouy phase shift [12,37], which differs along the propagation direction for Gaussian beams. As shown in Fig.8, transverse patterns of different positions along the propagation direction are visibly consistent with the theoretical results in Fig. 3 for the evolution of HOs. Obviously, a three-dimensional evolutional parametric surface can be exploited to interpret the transformation of the spatial patterns inside the cavity. The same clarification for single periodic Lissajous figures had been primarily provided [12] to show noticeable localization on the 3D parametric surface by observing the tomographic transverse patterns inside the cavities. Note that Figs. 8(a1)–8(a5), and Figs.8(b1)–8(b5) are generated with off-axis pumping (x,y)= (0.21 mm, 0.10 mm), and

1

a b1 c1 d1

, 3 , 2

p q _{3 ,1 4 , 3 5 , 2}

a b c _d

FIG. 9. (Color online) (a)–(d) Numerical wave patterns for the intensities of|n1p,n2qHˆ with different (p,q). (a1)–(d1) Experimental

0 0.08 0.38 0.5 1.17 1 a a2 a3 a4 a5 1 b b2 b3 b4 b5 0 z L z 0.02L z 0.08L z 0.11L z 0.30L

FIG. 10. (Color online) (a1)–(a5) Numerical wave patterns for the intensities of|n1p,n2qHˆ with (p,q)= (3,2) and varying α. (b1)–(b5)

Experimental tomographic transverse patterns observed along the propagation direction from the beam waist for (p,q)= (3,2).

(x,y)= (0.57 mm, 0.10 mm), where (x,y) are measured relative to the optical axis of the laser cavity and an objective lens is employed to reimage the near-field patterns on the screen.

In Figs.9(a1)–9(d1), experimental observation of different (p,q) are displayed corresponding to numerical calculations of|n1p,n2qHˆ, where (p,q)= (3,2), (α,β) = (π/2,π/2) for

Fig. 9(a), (p,q)= (3,1), (α,β) = (π/2,π/2) for Fig. 9(b), (α,β)= (0,π/2), (p,q) = (4,3) for Fig. 9(c), and (α,β)= (π/2,π/2), (p,q)= (5,2) for Fig.9(d), and all with n1= 1,

and N= 60. Experimental results of the patterns (p,q) = (3,2), (p,q)= (3,1), (p,q) = (4,3), and (p,q) = (5,2), are observed at L = 9.0 mm, L = 4.9 mm, L = 6.1 mm, and L = 7.4 mm, respectively. In Figs. 10(a1)–10(a5), the transverse patterns along the propagation direction of (p,q)= (3,2) in Fig. 9(a1) are reconstructed agreeably, as depicted in Figs.10(a1)–10(a5) for varying α. This agreement suggests that our quantum operator model of the coupled commensurate HO is applicable to the ubiquitous laser modes.

IV. CONCLUSION

In summary, we have systematically investigated the quan-tum signatures of the eigenstates corresponding to the coupled commensurate HO with SU(2) coupling interactions. Further-more, we have explored the analogous observation of the laser transverse modes from large-Fresnel-number degenerate cavities via varying pumping size. It has been experimentally verified that the 3D coherent lasing waves corresponding to the quantum states with multiple Lissajous orbits can be systematically generated by enlarging the pumping spot size. Finally, we employ the propagating property of the lasing modes to manifest the role of the phase factor introduced by the SU(2) coupling interactions.

ACKNOWLEDGMENT

The authors acknowledge the National Science Council of Taiwan (NSCT) for their financial support of this research under Contract No. NSC-100-2628-M-009-001-MY3.

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