1.4 Mesoscopic Wave Functions
1.4.2 Disordered Wave Functions in Random Media
Wave behavior in Random medium is a popular subject that has gone through a remarkable transformation in the past decades. The transformation was initiated by Anderson who suggested the possibility of electron localization inside a semiconductor [35]. The issue is now an important area of research which includes a variety of problems such as wave localization (weak [36-45] or strong [36,46-48]),
Classical
Mechanics Ray Optics
Quantum
Mechanics Wave Optics
Schrödinger’s completion Hamilton’s Analogy
0 0
Fig. 1.3.1 Optical-Mechanical analogy.
wave diffusion [49-52], intensity fluctuations [53-57], and correlation [58]. Since disorder phenomena are not restricted to a specific kind of wave, various approaches [36-38,59] have been developed individually in condensed matter physics, optics, acoustics, and atomic physics. It could be found that the localization phenomenon is still an important issue and deserves further investigations.
In this thesis, we experimentally acquire the disordered wave functions from the conical second harmonic generation to explore the variation of weak localization from extended to pre-localized states. We numerically verify that the experimental density distributions with different extents of weak localization can be excellently analyzed with a reduced version of the nonlinear sigma model (RV-NLS model). Moreover, we demonstrate that the chi-square distributions with fractional degrees of freedom are practically equivalent to the density distributions of the RV-NLS model. Our finding indicates that the concept of fractional degrees of freedom can be applied to the statistical properties of disordered wave functions. It is believed that the present work can bring more insight into the localization phenomena of diverse disordered systems.
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Chapter 2
Coherent Wave Transformation in Quantum Harmonic Oscillators and Spherical Laser Resonators
2.0 Introduction
Transformation in coupled isotropic harmonic oscillators
Numerous recent researches on optical spatial modes have come out in modern physics [1-3] ranging from classical simulators of quantum entanglement [4-6] to parallel information [7,8]. The transverse Hermite-Gaussian (HG) modes are emitted by most laser cavities and are formally identical to the eigenstates of two-dimensional (2D) quantum harmonic oscillator (HO) [9]. Consequently, HG modes are often used to represent the spatial quantum photon states within the paraxial regime [10].
Recently, a variety of quantum Lissajous states formed by the coherent superposition of HG eigenstates has been analogously generated from the degenerate laser resonators, which exhibit wave patterns resembling Lissajous figures [11].
Constructing wave states with spatial morphologies well localized on the particle orbits has become one of the most fundamental features in different branches of physics such as solid-state physics, nuclear and atom physics, and laser physics [12,13].
Likewise, the Laguerre-Gaussian (LG) modes correspond to circular eigenstates of the 2D HO and play a prominent role in singular optics [14]. In the early 1990s, it was shown that a high-order HG mode can be converted into a LG mode by using
astigmatic lenses [15,16]. Since this discovery, researchers have made tremendous progress in manipulation [17], detection [18], and application [19,20] of the orbital-angular-momentum states of light. The generation of optical coherent states with intensities localized on intriguing periodic orbits might be an enabling tool to explore further possibilities for creating a new class of quantum light-matter entangled states.
In section 2.1.1 and 2.2.2, we theoretically and experimentally present the continuous transformation between the HG and LG modes. Furthermore, in section 2.1.2 and 2.2.3, we theoretically verify that converting the HG modes into the LG can lead to the spatial morphologies of the two-dimensional (2D) coherent states to be transformed from Lissajous figures to trochoidal curves. With this transformational relationship, we experimentally generate various structured lights by exploiting a cylindrical-lens mode converter to transform the optical Lissajous modes. The present investigation manifests a notable method to generate optical coherent waves with various orbital spatial morphologies.
Transformation in coupled commensurate harmonic oscillators
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 [2,21-24]. One of the most profound similarities is that the electromagnetic wave equation in paraxial approximation is isomorphous to the Schrödinger equation [25-28]. Consequently, the electromagnetic radiation modes in the optical resonator or waveguide are analogs of the wave functions of a quantum system [9,11,29]. The tight connection between the paraxial beam propagation and quantum mechanics has been extensively exploited to study wave chaos phenomena
[29-31], disorder induced wave localization [32], semiclassical physics [33,34], and transient dynamics of quantum states [35-37].
The coupled HOs have been employed successfully to explore the hydrogen atom problem [38], charged particles in external field [39,40], states of deformed nucleus in the Nilson model [41], shell effects in nuclei and metallic clusters [42], and orbital magnetism in quantum dots [43]. More recently, the isotropic HOs with SU(2) coupling interactions have been used to investigate the generation and evolution of quantum vortex states [44] and the transformation geometry between Lissajous and trochoidal orbits [45]. It has been shown [46,47] 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 section 2.1.3, we theoretically explore the eigenstates of a commensurate HO 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 to trochoidal curves from single to multiple periodic orbits. In section 2.1.4, 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. Moreover, by exploiting the cylindrical-lens mode converter, we have experimentally presented the beam transformation from multiple Lissajous orbits to the multiple trochoidal curves.
2.1 Coupled Quantum Harmonic Oscillators
2.1.1 Eigenstates : SU(2) Transformation
It is well-known that the Hamiltonian for the 2D isotropic HO with the dimensionless spatial variables x~ and y~ is given by
where
0 is the angular frequency of the HO. Furthermore, Eq. (2.1.1) can be rewritten in terms of the ladder operators, and hence it becomes
† †
be derived to be the two-mode Fock state 1 20 0
0,0 H denotes the ground state. The corresponding eigenvalues are
ˆ0( , ) (1 2 1 2 1) 0
E
Hn n
n
n
, where n1 and n2 are positive integers. Moreover, the normalized spatial representation is expressed as
The general form of a 2D HO with the SU(2) coupling can be modeled as
0 1 1 2 2 3 3
ˆ ˆ ˆ ˆ ˆ
H
H
J
J
J
, (2.1.4)where the coupling parameters (i
i
1, 2,3) are assumed to be real constants andˆ
iJ are the Casmir operators associated with the SU(2) Lie algebra and the corresponding generators derived by Schwinger [48] are
† †
The operators Jˆi follow the angular-momentum commutation relation ˆ ˆi, j ijk ˆk
J J i
J
[48], where the Levi-Civita tensor
ijk equals +1 (-1) if i j k
, ,
is an even (odd) permutation, and zero otherwise. With the dimensionless spatial variables,
Before solving the quantum eigenstates of Hˆ , let first investigate the classical representation of the Hamiltonian Hˆ . The classical equation of motion for the Hamiltonian Hˆ can be found to be motion for the Hamiltonain Hˆ in the classical system possesses the same form as the Schrödinger equation when considering the case of a 2-level system. By employing the SU(2) algebra, the general solution for Eq. (2.1.7) can be derived to be
/2 /2
where
tan1
2 1
,
3
To explore the quantum eigenstates of the Hamiltonian Hˆ, we employ the same SU(2) algebra as in the classical dynamics to define a new pair of operators
/2 /2 thus obtain the Hamiltonian Hˆ to be transformed into a separable 2D HO
1 1 1 2 2 2
Therefore, the eigenstates and eigenvalues of the Hamiltonian Hˆ yield to be
1 2 relation in Eq. (2.1.9) and
ˆ ˆ0
0, 0
H 0, 0
H , the eigenstates to the Hamiltonian Hˆ can be derived in terms of the Wigner d-coefficient [49]:
The detailed derivation can be found in Appendix A. Evidently, the eigenstates
ˆ 1
,
2 Hn n can be expressed as a linear superposition of the set of
ˆ0 transformation from the HG to the LG modes can be continuously obtained by changing the parameter or
, which suggests different extent of coupling effect.The intermediate states, the Ince-Gaussian modes, are accessibly acquired through the SU(2) algebra associated with the coupled isotropic HO. Notably, the LG modes presented in Fig. 2.1.2(e) and Fig. 2.1.2(e’) possess fairly large orbital angular momentum per photon [19,20] of l
5
and l 52
, respectively. Since light beams with well-defined orbital angular momentum have a number of developing applications [19,20], generation of such optical beams should be an important issue for further studies. Moreover, in Fig. 2.1.3, we present the comparison between the traveling-wave and the standing-wave forms of the eigenstates n n1,
2 Hˆ for
2
. The standing-wave forms are obtained by taking the real part of the eigenstates n n1,
2 Hˆ .2.1.2 Coherent States : Single Periodic Orbits
According to Eq. (2.1.8), we can obtain the classical orbits for Hˆ with the parametric equations
x t
( ) Re v t
1
andy t
( ) Re v t
2
, where 0 6 3 2 5 2
0 , 20
2 , 18
4 , 16
6 , 14
10 , 10
... ...
HG Ince-Gaussian LG
m m1, 2
0 , 20
2 ,16
4 ,12
6 , 8
10 , 0
p l, n n
1, 2
Fig. 2.1.1 The intensity distribution of the eigenstates n n1
,
2 Hˆ with different indices n n , various values of
1, 2
values and constant value of 2
.(a)
(c)
(e) (b)
(d) β=0°
β=47.4°
β=90°
β=25.8°
β=78°
β=0°
β=36°
β=90°
β=60°
β=25.8°
(a’)
(c’)
(e’) (b’)
(d’)
Fig. 2.1.2 The intensity distribution of the eigenstates n n1
,
2 Hˆ with different indices n n
1, 2
, various values of
and constant value of 2
: (a)-(e) n n
1, 2
15,10
, (a’)-(e’) n n
1, 2
55,3
.m m1, 2 0, 20 2,18
4,16
6, 14
10,10
Fig. 2.1.3 The intensity distribution of the eigenstates n n1
,
2 Hˆ with different indices n n
1, 2
, various values of and constant value of 2
; (Upper) traveling-wave form; (Lower) standing-wave form.1 1 1 2 2 2
according to Eq. (2.1.13). The periodic orbits are shown to be associated with a continuous transformation between Lissajous curves and hypotrochoids for differing values of and
. In addition, Fig. 2.1.5 displays the periodic orbits the case of1 2 9 /1
for different and
values with the parameters of
A A1,
2 60,150
and
1, 2
,0
according to Eq. (2.1.13). The classical trajectories are found to be a continuous transformation between Lissajous figures and epitrochoids with varying values of and
. It is obvious that a hypotrochoid or an epitrochoid depends on the ratio of
1 2 to be positive or negative with2
/
and
/2.The periodic orbits for the Hamiltonian ˆ
H in the classical mechanics have been
clearly demonstrated as mentioned above. Here we wonder whether the quantum states corresponding to the Schrödinger equation reveal the same morphologies localized on the periodic orbits. To construct such quantum states for mimicking the localized curves in classical dynamics, we apply the concepts of wave-packet coherent states first proposed by Schrödinger in 1926.The wave-packet coherent states of 1D HO developed by Schrödinger are given by
α＝0 π/6 π/4 π/3 π/2
π/2 π/3 π/4 π/6 β＝0
Fig. 2.1.4 Classical periodic orbits for the case
1 2 8 /1, A A
1, 2
35,100
, and
1, 2
,0
corresponding to Eq. (2.1.13).α＝0
π/2
π/8 π/4 2π/5 π/2
2π/5 π/4
π/8 β＝0
Fig. 2.1.5 Classical periodic orbits for the case
1 2 9 /1, A A
1, 2
60,150
, and
1, 2
,0
corresponding to Eq. (2.1.13).
employing the generating function of the Hermite polynomials, the probability distributions of the Schrödinger coherent states can be derived to be
It can be obviously seen that the center of the coherent state moves along the path of the classical trajectory x t
( ) 2 Re[ ( )]
u t 2 cos(
A t
)
. In Eq. (2.1.10) we have performed that the coupled HO with HamiltonianHˆ can be transformed into a
separable 2D HO through the SU(2) algebra. The Schrödinger coherent states for a 2D coupled HO can be expressed as the product of two 1D wave-packet coherent states:
cumbersome algebra, Eq. (2.1.16) is given by
1 2
(2.1.8). According to Eq. (2.1.15) and Eq. (2.1.17), we find that the probability distribution of the coherent states
t v v
; ,1 2
can be expressed as on the same trajectories represented by the parametric equations in Eq. (2.1.13). To provide a comprehensive study in the corresponding quantum coherent states localized on the periodic orbits, we are here to find out the time independent stationary coherent states extracted out from the wave-packet coherent states t v v
; ,1 2
.
Consider a general expression for the stationary coherent state, the wave-packet coherent states
t v v
; ,1 2
can be expressed as the double finite sum with fairly large A1 and A2 values. For the corresponding 2D Poisson distribution, the probability of the coherent state t v v
; ,1 2
in the eigenstate n1,
n2 Hˆ can therefore be written as
1 2
ˆ 1 2
1 2
11 1 2 2 2be given by
n
1 a a
ˆ ˆ1 † 1 A
12 andn
2 a a
ˆ ˆ2 † 2 A
22. Since the values of the means n1 and n2 are sufficiently large, according to the central limit theorem, the distribution P n n
1,
2
becomes normally distributed and the standard deviations are given by n1 and n2 . As a result, Eq. (2.1.17) can be rewritten as signifies the nearest integer to a at the lower side. Considering the case for a commensurate HO with a frequency ratio of
1 2 q p
, we let
1q
and2
p
, where p and q are relatively prime and positive integers. The eigenstateswith indices
s s1,
2
of the commensurate HO in Eq. (2.1.20) hence can be dividedwith indices k1
(
s
k) / 2
and k2 (
s
k) / 2
, where the sign ± of the index k is spatial morphology of the stationary states1 2 and
n large enough). Moreover, the amplitude coefficient of
21 2
dominates when the index s equals to zero. Thus, the stationary coherent state
0
(2.1.24) suggest the periodic orbits that the wave-packet coherent states move along.
Figure 2.1.6 depicts the transformation between the Lissajous curves and hypotrochoids for varying values of
with the parameters of
1 2 5 / 2, n n
1, 2
9,80
,
4
, and 2
. The continuous transformation between the Lissajous curves and the epitrochoids is clearly shown in Fig. 2.1.7 for different
values with the parameters of
1 2 5 / 2 n n
1, 2
29,60
, 4
, and
2
. Moreover, Fig. 2.1.8 displays the transformation from Lissajous curves to hypotrochoids for various values of with the parameters of
1 2 5 / 2 n n
1, 2
29,60
,
2
, and 2
. Exploiting the concepts of the Schrödinger coherent states, we have successfully constructed the quantum states that mimic the classical dynamics in the coupled HOs. Via the SU(2) algebra, we also demonstrated the continuous transformation of the spatial morphologies between the two sort of distinct classical orbits, the Lissajous curves and the trochoids. The present results provide a comprehensive survey of the quantum-classical correspondence and might stimulate further ideas concerning the intriguing non-classical behavior in the mesoscopic regime.β=48°
Fig. 2.1.6 Theoretical results for the intensity distribution of the stationary states
1 2
(a)
Fig. 2.1.7 Theoretical results for the intensity distribution of the stationary states
1 2
(a)
Fig. 2.1.8 Theoretical results for the intensity distribution of the stationary states
1 2
2.1.3 Coherent States : Multiple Periodic Orbits
In this section we start from the coupled commensurate HO with the Hamiltonian ˆ
H
as derived in Eq. (2.1.10) and develop a quantum model by adding a new coupling term to the the coupled commensurate HO. We explore the eigenstates and find that the high-order spatial patterns are noticeably concentrated on Lissajous figures to trochoidal curves from single to multiple periodic orbits.The general form of a two-dimensional (2D) commensurate HO comprising weak coupling term can be modeled as
ˆ ˆ
H
H H
c, (2.1.25)where H
ˆ
c signifies the SU(2) coupling term characterized by a vibration-rotational mechanism and the detail will be provided later. The Hamiltonian ˆH of the
commensurate HO is given byH
ˆ H
ˆ0 1 1J
ˆ 2 2J
ˆ 3 3J
ˆ as has been shown in Eq. (2.1.4). By the use of the SU(2) algebra and in terms of the quantum ladder operators, the Hamiltonian has been transformed into Eq (2.1.10) with
1 1
1
2 2
2ˆ ˆ ˆ† 1 2 ˆ ˆ† 1 2
H
a a
a a
, where
1q
,
2p
,
is a common factor of the oscillation frequencies
1 and
2, q and p are integers, and theladder operators ˆ
a and ˆ
i†a
i
i 1, 2
follow the SU(2) transformation in Eq. (2.1.9).The eigenstates of the commensurate HO can be divided into subsets given by
ˆ
1 1, 2 2 H
n p
n q
as has been presented by Louck [46] and also discussed in section 2.1.2, where
n n1,
2
are arbitrary nonnegative integers, and
are 1,
2
constants that
10,1, ,p
1 and
20,1, ,q
1. According to Eq. (2.1.11), the eigenstatesn p
1
1,n q
2
2 Hˆ can be written as
s s
are the eigenstates for the 2D isotropic HO given in Eq. (2.1.2),1 1 1 1 1 2
It reveals the fact that the eigenstates have been divided into pq different subsets of states and the degeneracy holds when n1
n2 is a constant N for fixed
1,
2
corresponding to the eigenvalues E
n1
n2 1 2
p 1 2
q
1 p
2 q
. For aparticular case
p q, 1,1
of the isotropic HO, it is evident that ˆH
2J
ˆ1 , where ˆJ is the Casmir operator associated with the SU(2) Lie algebra and the corresponding generators derived by Schwinger [48] are shown in Eq. (2.1.5).With the non-bijective canonical transformation, the commensurate HO can be mapped on to an isotropic one in a degenerate eigenspace [46]. 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
H . Therefore, under the canonical transformation, the Hamiltonian in Eq. (2.1.10)
ˆ can be transformed into† †
1 1 2 2
1 1
ˆ 2 2
H
a a
a a
, (2.1.27)where
a and
i a†i are the canonically transformed ladder operators which bear the relations [46]ladder operators on particular eigenstates for fixed
1,
2
, for instance, are has been converted into the same form as the isotropic HO when the degeneracy can exist for n1
n2
N according to the eigenvalue E
n1
n21
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 commensurate HO under consideration:
† †
1 1 2 2 1
2
J
a a
a a
, J
2
a a
1† 2
a a
†2 1 2
i, J
3
a a a a
1† 1
†2 2 2
. (2.1.29) The operators also satisfy the Lie commutation relation. Particularly, J
1
Jˆ
1 ,2
ˆ
2J
J , and J
3
Jˆ
3 for the special case of the isotropic HO with
p q, 1,1
.Let us now return to our formal considerations of the coupled HO of the Hamiltonian
H given in Eq. (2.1.25). The Hamiltonian ˆ H is expressed in the
c form of the SU(2) coupling interactions [44,45] and hence the Hamiltonian H can be modeled as demonstrated previously on a group theory level via the SU(2) transformation in section 2.1.1. Likewise, it enables us to derive the wave functions by employing the transformation of SU(2) symmetry group. It can be seen that Eq. (2.1.30) possessesthe same mathematical interpretation as Eq. (2.1.4). Therefore, the eigenstates with distributions of the Wigner d-coefficient 12 2, 1 2
2N 2.1.9(b1)-2.1.9(b4), 2.1.9(c1)-2.1.9(c4), and 2.1.9(d1)-2.1.9(d4) illustrate the corresponding eigenstates 1
,
2; ,
1 2m m H for different indices
m m
1, 2
with
p q, 2,1
,
p q, 3,1
, and
p q, 3, 2
, respectively, and all with theparameters
1,
2 0, 0
,
,
2,
2
,
, 0,0
, and N 60
.Note that it is valid for us to choose a specific eigenspace of
1,
2 0, 0
since, in the classical limit (N large enough), [50] has confirmed that the choice of theeigenspace does not affect the final results. Therefore, parameters
are set 1,
2
to be
0, 0
in the following discussions and we simplify the denotation of the eigenstates 1,
2;0, 0
m m H to 1
,
2m m H. Moreover,
,
are chosen for specific parameters.
signifies an additional phase shift between the two HOs in x and y directions, and
corresponds to the coupling strength arising from ˆH . The
cdistribution shown in Fig. 2.1.10(a1)-2.1.10(a8) are varied with
, which indicates different composition for the corresponding eigenstates 1,
2m m H as depicted in Fig.
2.1.10(b1)-2.1.10(b8) with
p q, 2, 1
,
2,
, 0,0
,m
1 , and 160
N
. For
and 0
, the eigenstates can be seen to project precisely onto particular eigenstates p q N, 1
H andp N
( 1) ,q
H , respectively. While
is determined, the conversion of
can be illustrated as shown in Fig.2.1.11(a1)-2.1.11(a5) with m1
1
, 0.4
, 0,0
andN
60, and inFig. 2.1.11(b1)-2.1.11(b5) with m1
3
,
0.74
,
, 0,0
and N60 .The morphologies transform since different relative phases are introduced into the superposition of states 1
,
2m m H with the set of states 1
,
2 ˆ n p n q H.Theoretical results disclose intriguing geometric patterns localized on an ensemble of periodic Lissajous orbits, which suggests a kind of quantum-classical analog. It is evident that the number of peaks associated with 12 2, 1 2
2N
n N m N
d
is consistent with the number of Lissajous orbits in 1,
2m m H according to various m1. The fact implies each orbit of multi-Lissajous patterns is formed by the superposition of a particular group of the set 1
,
2 ˆn p n q H with distribution centered on the
b1
b2
b3
b4Fig. 2.1.9 (a1)-(a4) Numerical simulations of the Wigner d-coefficient with respect to
n for various
1m ; (b1)-(d4) numerical wave patterns for the intensities of
1 eigenstates 1,
2;0,0
m m H . Detailed description of the parameters; see text.
m1
Fig. 2.1.10 (a1)-(a8) Numerical simulations of Wigner d-coefficient with respect to n1
for various
; (b1)-(b8) corresponding numerical wave patterns for the intensity distribution of eigenstates 1,
2m m H.
0
0.08
0.18
0.28
0.5
a1
a2
a3
a4
a5
b1
b2
b3
b4
b5
0
0.18
0.25
0.35
0.5
Fig. 2.1.11 Numerical wave patterns for the intensities of eigenstates 1
,
2m m H with respect to varying
; (a1)-(a5)
0.4
; (b1)-(b5)
0.74
.corresponding peak of 12 2, 1 2
2N
n N m N
d
. A relation l min
m m1,
2 1
can be given, where l denotes the number of orbits. While the magnitude ofmin
m m1,
2
becomes larger, the related excited states display more complex caustic-like geometric patterns as shown in Fig. 2.1.12 followed by the case in Fig. 2.1.9(b1)-2.1.9(b2).
Additionally, the symmetry is held for
m m1,
2
m m2,
1
, e.g., eigenstates of
m m1,
2 26,34
and
m m1,
2 34, 26
shown in Fig. 2.1.12(f) and 2.1.12(h) possess identical morphology for equal distribution 12 2, 1 2
2N
n N m N
d
. Though the same morphology is notified, the eigenstates m m
1, 2
and m m
2, 1
arecharacterized by distinct features of the quantum probability current J x y