Creating Extended Landau Levels of Large Degeneracy with Photons
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Kuan-Hao (Waylon) Chen, Physics, M.S.
Graduate Program in Physics
The Ohio State University
2018
Dissertation Committee:
Prof. Tin-Lun Ho, Advisor Prof. Ilya Gruzberg Prof. Stuart Raby Prof. Rolando Valdes-Aguilar c Copyright by
Kuan-Hao (Waylon) Chen
2018 Abstract
Large degeneracy in Landau levels is a key to many quantum Hall phenomena.
Geometric effects on quantum Hall states is another interesting problem that can probe the correlations in the quantum Hall states. A recent experiment has reported a result in creating the energy levels and the wave functions of Landau problem in a cone with photons. Based on their system, we generalize the scheme and discover a way to create extended degenerate levels with considerably larger degeneracy than that of the conventional Landau levels. To fully understand how to achieve this degenerate levels with photons, we also present the relevant topics in optics that are not familiar to condensed matter community to make it self-contained.
The reason of this dramatically large degeneracy is that each degenerate level contains the whole spectrum of a Landau problem in a cone. In another words, we compress the spectrum of a two-dimensional system into one single energy. This considerably large degeneracy is expected to cause dramatic phenomena in quantum
Hall and many-body physics. We suggest experimental measurements that could show this discovery.
ii To the fifteen year-old boy that wants to be a physicist.
iii Acknowledgments
I am the most grateful to my advisor, Dr. Tin-Lun Ho. Every discussion with him ensouls my research. Countless efforts that he has put in to advise me and to shape me into a physicist, and most importantly, a responsible man are invaluable.
I also would like to express my thanks to the committee. I thank Dr. Stuart Raby for the advice he gave in my first year when I studied high energy physics with him.
I thank Dr. Ilya Gruzberg for numerous valuable discussions about my research and the excellent classes he offered to build my knowledge. I also thank Dr. Rolando
Valdes-Aguilar for his input of ideas from an experimentalist’s viewpoints every now and then.
I appreciate the support and resources from the Department of Physics. My research would have been impossible without them. I especially would like to thank
Dr. Jon Pelz for offering the financial support in my third year and this last semester when the funding of my research was short. I also thank the Ministry of Education
Taiwan for the study abroad loan. I also thank the amazing superwoman Mrs. Kris
Dunlap for taking care of every administrative affair for all physics graduate students.
Many teacher’s inspiration, encouragement and support are essential along my path in pursuing a PhD in physics. I thank my physics teacher at my fifteen, Mr.
Ching-Peng Kuo, for the enlightenment and inspiration that made me start my jour- ney in physics. I thank Mrs. Chu-Fong Yen for the everlasting care, encouragement,
iv and setting an example of dedication for me. I thank Dr. Jin-Tzu Chen for sharing his profound knowledge with the young mind and the great support when I applied for the PhD programs.
I enjoyed scientific discussions and friendships with many colleagues. The discus- sions are always meaningful and fun, and sometimes they even turn out to be fruitful.
I thank Dr. Joe McEwen, Bowen Shi and Alex Davis for the fun time we had in the geometry club in my first year. I thank Cheng Li, Jiaxin Wu, and James Roland for many valuable discussions about my research and ideas in physics.
My first year in graduate school was not a total mess only because of the help and company from these people. I especially thank Dr. Xiaolin Zhu for the countless helps with my first year life in a new city. I thank Rachel Hsiao, Yun-Hao Hsiao,
Michelle Lee and Zijie Poh for the friendship since my first year. I thank Dr. Fuyan
Lu for selflessly sharing the information when I applied for the internship.
To the amazing people I became close to after my fourth year in Columbus during my roughest time, your company is indispensable. I thank all fitness classes and the instructors in the RPAC and every amazing friend I made there that made my life wholesome. I thank Dr. Xianyu Yin, Hsiu-Chen Chang, Pei-Zu Hsieh, Dr. Kuyung- min Lee, and Dr. Kyusung Hwang for many good times together. In particular, I thank Jorge Torres, Derek Everett, Noah Charles and Mary-Frances Miller for every laughter and all the delicious food we have shared and the irreplaceable friendship. I also thank Dr. Chris Ehemann for the valuable internship experience and the amazing friendship.
I am the most blessed person to be loved and cared for by all my dear friends, near and far. Love, support and company from them make my life meaningful. To
v the dearest friends I knew since I was in Taiwan, I thank Kuo-Wei Chen and Tang
Lee for a listening ear and sincere advice that are never absent, Yao-Yu Lin for the honest sharing of our cynical attitude to physics and life, Chi Lin for the most candid comments and sarcasm on things, Shao-Yu Chi for the lifelong friendship, and everyone in CK60th 109 class.
I thank Alice Chi for her presence in my life, which has made me a different per- son, and unquestionably a better and happier one. I thank my parents for always trusting me, loving me and giving me the freedom to explore the world in my own way. I thank Kai-Wen Hsiao for starting this journey with me and ever taking the biggest part in my life.
Washington D.C., November 22, 2018 Waylon Chen
vi Vita
November 21, 1989 ...... Born - Taipei, Taiwan
2012 ...... B.S. Physics, Mathematics, The Ohio State University. 2016 ...... M.S. Physics, The Ohio State University. 2013-present ...... Graduate Associate, The Ohio State University.
Fields of Study
Major Field: Physics
vii Table of Contents
Page
Abstract ...... ii
Dedication ...... iii
Acknowledgments ...... iv
Vita...... vii
List of Figures ...... xi
1. Introduction ...... 1
1.1 Motivation ...... 1 1.2 Overview ...... 3
2. Landau problem in cone and anti-cone surfaces ...... 6
2.1 Introduction ...... 6 2.2 Landau problem with electrons in cone surfaces ...... 7 2.2.1 The cone surface ...... 7 2.2.2 The Schr¨odinger’sequation and the solutions ...... 8 2.2.3 Gaussian curvature for cone surfaces ...... 12 2.3 Landau problem in anti-cone surfaces ...... 16 2.3.1 The Schr¨odinger’sequation and the solutions ...... 18 2.3.2 The response of the energy levels to the strength of curvature singularity ...... 20
3. The Landau levels of a rotating harmonic oscillator: the synthetic gauge 22
3.1 Introduction ...... 22 3.2 The quantum Hamiltonian for a uniformly rotating potential . . . . 23
viii 3.3 A rotating quantum harmonic oscillator ...... 26 3.3.1 wave functions in polar coordinate ...... 26 3.3.2 the ladder operator approach ...... 31 3.4 the two labeling schemes of the spectrum ...... 35
4. Realization of rotating quantum harmonic oscillators with paraxial optics 37
4.1 Introduction ...... 37 4.2 Simulating the Schr¨odingerequation of massive particle using laser beam and generation of harmonic oscillator wave functions . . . . . 38 4.2.1 Description of laser beams in paraxial approximation . . . . 38 4.2.2 Laser beams in a two-mirror cavity ...... 43 4.3 Realization of synthetic magnetic fields with the optical resonator . 48 4.3.1 paraxial ray optics ...... 49 4.3.2 paraxial wave optics ...... 59 4.3.3 Summary ...... 75
5. Generalized synthetic Landau levels with photons ...... 80
5.1 Introduction ...... 80 5.2 Simulating Landau levels on a cone with three-fold symmetry . . . 81 5.2.1 The energy levels ...... 81 5.2.2 Wave functions on a cone ...... 85 5.2.3 Detection of the level structure ...... 86 5.3 Create extended landau levels in a cone with dramatically larger degeneracy ...... 88 5.3.1 The energy levels and wave functions of Landau problem for general σ ...... 89 5.3.2 Extend the Landau level degeneracy to simulate the energy level structure of a σ → ∞ anti-cone ...... 90 5.3.3 Detection of the extended level structure ...... 95
5.4 Characterizations of the states in the (χ1 = 2π/3, χ2 = 4π/3) de- generate manifold ...... 97 5.4.1 Finding all degenerate states in an extended level ...... 98
6. The Poincar´ehit patterns ...... 103
6.1 Introduction ...... 103 6.2 The hit patterns from the round-trip matrix of a photonic resonator 105 6.2.1 The formalism ...... 105 6.2.2 Hit patterns of the experimental resonator ...... 107
ix 6.3 The hit patterns from the Hamilton formalism: the stroboscopic evolution ...... 111 6.3.1 The Hamilton equations in matrix form ...... 111 6.3.2 Hamilton equations for a harmonic oscillator ...... 112 6.3.3 Hamilton equation for a rotating harmonic oscillator . . . . 113 6.4 Study the rotational symmetry of the hit patterns with the Gouy phases ...... 118 6.4.1 The three copies of cyclotron motion in stroboscopic dynamics121 6.5 Using hit patterns to find the critical point for extended Landau levels125
Appendices 130
A. Landau problem on cone and anti-cone surfaces embedded in R3 ..... 130
A.1 The Landau problem with magnetic fields on a cone surface in R3 . 130 A.2 the Landau levels on an anti-cone ...... 132 A.2.1 a parametrization for an anti-cone ...... 132 A.2.2 the Landau levels ...... 134
B. More about paraxial optics ...... 137
B.1 the validity of paraxial approximation ...... 137 B.2 Derivation of the Hermite-Gaussian modes ...... 139 B.2.1 spherical wave in paraxial approximation ...... 139 B.2.2 Higher order solutions ...... 140 B.3 Calculating the roundtrip matrices in the twisted resonator . . . . 143 B.3.1 Calculating the rotation matrices ...... 144 B.3.2 Calculating the transfer matrices for non-normal mirror re- flection ...... 145 B.3.3 Laguerre-Gaussian modes: cylindrical symmetric solution to paraxial wave equation ...... 146
C. An example of the generalized Landau level consisting of states from dif- ferent cones ...... 147
Bibliography ...... 149
x List of Figures
Figure Page
2.1 Cut open a cone surface in R3 and flatten it into a sheet of wedge with two sides identified...... 8
2.2 The energy spectrum of a charged particle with uniform magnetic flux on a cone compared to the spectrum on a flat disk...... 11
2.3 Typical examples of surface with different Gaussian curvature and showing cone tip as a limiting case of a K > 0 surface...... 14
2.4 Explaining geodesic curvature...... 15
2.5 2.5a Making an anti-cone surface by inserting a wedge to a disk. 2.5b The resulting anti-cone shown in R3...... 17
2.6 The energy spectrum of a charged particle with uniform magnetic flux on an anti-cone compared to the spectrum on a flat disk...... 19
2.7 The energy levels of Landau problem on an anti-cone when σ → ∞.. 20
2.8 A plot showing shifts in energy levels as a response to the curvature . 21
3.1 The rotation effects on the energy levels for a two-dimensional har- monic oscillator...... 29
3.2 The energy levels of a rotating harmonic oscillator at criticality. . . . 30
4.1 The propagation of a generic laser beam ...... 39
4.2 Two-mirror cavities...... 44
xi 4.3 laser in a two-mirror cavity ...... 45
4.4 Optical axis and transverse planes...... 50
4.5 A paraxial ray and definition of a ray vector...... 51
4.6 Non-normal incidence of rays shown in mirror reflection...... 54
4.7 Astigmatism from non-normal incidence shown in the case of a lens. . 55
4.8 change of the local coordinate system due to reflection of non-normal incidence ...... 58
4.9 The stationary modes of the cavity calculated by the operator method 75
5.1 The Q-tower with bottom states (n1, n2, q) = (0, 0, 0) ...... 83
5.2 3 copies of tower ...... 84
5.3 Level structure with different n2 assuming χ1 < χ2...... 85
5.4 Detection of the stationary modes and their degenerate frequency. . . 88
5.5 Extended degenerate levels from T (0, 3k) Q-towers...... 92
5.6 Extended degenerate levels from all n2 separate into three branches indexed by i = 0, 1, 2...... 93
5.7 Detection of degenerate levels in transmission spectrum...... 96
5.8 Illustration of a systematic way to find all solutions to equation 5.31. 100
5.9 The same procedure as in figure 5.8 with states labeled in (n, m). The pattern shows that we can in principal have all eigenstates of a cone of wedge angle 2π/3 in a single generalized Landau level if we allow q to be infinite...... 101
6.1 Schematic illustrations for hit patterns as a ray optic description of a laser beam in the resonator...... 105
xii 6.2 The twisted resonator of tetrahedral configuration used in experiments in [1]...... 107
6.3 Hit patterns of the experimental resonator with different initial rays. . 108
6.4 The hit pattern of the experimental resonator shows precession when the number of round trip is large...... 109
6.5 Hit patterns of the experimental resonator when the curved mirrors are replaced by planar mirrors...... 110
6.6 Hit patterns representing the stroboscopic dynamics of a rotating har- monic oscillator with parameters matched to simulate the experimental resonator...... 117
6.7 The hit pattern when χ1/2π = 1/2, χ2/2π = 1/3...... 120
6.8 Hit patterns develop into full closed circles gradually as the number of round trips increase ...... 120
6.9 Hit patterns showing the relation between the Gouy phases and the rotational symmetry...... 124
6.10 The hit patterns in a small neighborhood around χ1/2π = 1/3, χ2/2π = 2/3...... 126
6.11 How the hit patterns change when the Gouy phases approach the crit-
ical point χ1/2π = 1/3, χ2/2π = 2/3. –(a) ...... 127
6.12 How the hit patterns change when the Gouy phases approach the crit-
ical point χ1/2π = 1/3, χ2/2π = 2/3. –(b) ...... 128
B.1 The geometry of a plane wave propagating in the direction at angle θ with the optical axis z...... 138
xiii Chapter 1: Introduction
1.1 Motivation
Quantum Hall effect is one of the most remarkable quantum phenomena in con- densed matter physics, and large degeneracy in Landau levels in strong magnetic fields
B is an important feature of the quantum Hall systems in solid-state settings.[2, 3].
The conventional Landau degeneracy for a system of two-dimensional electron gas of
finite area A is
N = gsBA/Φ0, (1.1)
where gs represents the spin degeneracy and Φ0 = hc/e is the magnetic flux quantum.
Studies about the quantized hall effect all require strong magnetic fields in order to create sufficiently high degeneracy in quantized Landau levels so that electrons only occupy a small number of Landau levels characterized by the filling factor ν =
Ne/N[4, 5]. And fractional quantum Hall wave functions are essentially different
filling patterns of Landau levels in the presence of interaction[6, 7, 8, 9, 10, 11, 12, 13].
Hence, being able to create a system that has quantized levels with extraordinarily high degeneracy shall lead to remarkable phenomena in quantum Hall and many-body physics.
1 Geometric effects on quantum mechanical systems is another interesting prob-
lem. There have been increasing attention to the geometric effects in quantum Hall
states[14, 15, 16, 17, 18, 19, 20, 21, 22]. In particular, a recent experiment reports
that they can produce Landau levels with photons[1]. Moreover, the level structure
they produced is that of a Landau problem in a cone surface (specifically a cone
that unfolds to a wedge of angle 2π/3, see figure 2.1) instead of the conventional one
which has electrons in a two-dimensional plane. This result motivated us to study
the Landau level structure in cone-like spaces and also their possible realization in
experiments.
Initially, we found it exciting that the photonic system in reference [1] can ex-
perimentally simulate Landau levels in a cone which is a surface with a curvature
singularity of positive sign at the apex (will be explained in chapter 2). We were
prompted to investigate how to simulate a surface with a curvature singularity of
negative sign that we call “anti-cone” (explained in chapter 2, see figure 2.5b for an
example). We found ways to change the level structure of the photonic system in
reference [1] and tried to mimic the structure of an anti-cone. But our results, to our
best knowledge, do not simulate Landau levels in such anti-cone spaces. However, we
found that our result, surprisingly, can simulate quantized degenerate levels (where
we call each degenerate level an extended Landau level) with degeneracy remarkably greater than the traditional Landau degeneracy. Precisely, if the Landau degeneracy is N, the degeneracy of the extended Landau levels in our system is of the order
O(N 2). Moreover, the “wave functions” we found in one single extended Landau level are those from the entire Landau energy spectrum in a cone. That is to say, we found a way to collapse all traditional Landau levels in a cone with different Landau
2 indices (thus at different energy) into one single degenerate extended level with a dramatically greater degeneracy.
1.2 Overview
This thesis is organized in the order along the line of our motivation; we started with a search for Landau levels in an anti-cone surface, identified the key is to shift the energy levels, but eventually found extended Landau levels with much larger degeneracy. In chapter (2), we show the conventional Landau problem and its energy levels, however, with two generalized versions of electrons in a cone surface or an anti-cone surface that introduce curvature to the spaces. We show that the effects of the curvature singularity in the energy levels is to shift the energy of the levels. And remarkably, in the limit of infinite curvature strength (see section 2.3.2 ), the Landau degeneracy for this anti-cone is already doubled because all the negative angular momentum (m < 0) states are shifted downwards to the same energy as the positive angular momentum (m > 0) branch, and this doubles the degeneracy already.
Chapter 3 and and chapter 4 are the preparation for explaining our main results of this thesis for the large degeneracy in the extended Landau levels. In chapter 3, we introduce another system, a massive particle in a rotating harmonic trap, that can produce the same degenerate level structure as Landau levels. This system is crucial for the understanding of both the scheme in reference [1] and in our results of this thesis. In a rapidly rotating harmonic oscillator, the rotation effect plays the role of magnetic field in Landau problem, and therefore shifts the energy levels and create Landau degeneracy (and thus is coined the term synthetic magnetic fields). In section 3.4, we provide a transformation of the labeling of states that will be used to
3 identify the wave functions of the states in chapter 5, which is crucial to the analysis
in our main result.
The goal of chapter 4 is to explain how to create a rotating harmonic oscillator
with laser beams and mirrors. It is of a larger volume because it takes some theories
in optics that is not familiar to people in the condensed matter community. In section
4.3.3 we summarize the key results that will be used in chapter 5 to understand the
extended degenerate Landau levels.
Chapter 5 is the main results of this thesis. We first show how Landau levels in a
cone is achieved with laser beams shown in reference [1]. This requires all knowledge
from previous chapters. After this, we present our method to create extended Landau
levels by generalizing the same system. The ideas come from our original motivation of
creating Landau levels in an anti-cone. A way to do that is to shift all the energy level
of the m < 0 states. We succeeded in shifting those levels in place, but surprisingly, we realized more levels are brought into the same degenerate levels simultaneously in this process. And this is the origin of the dramatically larger degeneracy. We identify the wave functions of these levels and conclude that we have extended degenerate levels (that we call a degenerate “manifold”) where each degenerate level at a specific energy can contain the states from the entire spectrum of Landau problems in a cone.
We suggest experiments to detect such degeneracy in the end of chapter 5 and chapter
6.
Chapter 6 is a complement to the wave picture description of laser trapped in a resonator. Hit pattern is the ray picture counterpart of the stationary modes (which is
defined in chapter 4) in the resonator. Besides completing the particle-wave duality
of laser, it provides at least three advantages in further clarifying the phenomena
4 seen in wave picture. (1) It shows the correspondence between laser in a four-mirror resonator and a rotating harmonic oscillator in a more straightforward way. (2) It shows that the n-fold rotational symmetry in the state is related to the values of
Gouy phases explicitly. (3) It provides an alternative way to measure and fine tune
Gouy phases at the values that cause degeneracy levels.
5 Chapter 2: Landau problem in cone and anti-cone surfaces
2.1 Introduction
Quantum mechanics on curved surfaces is one of the open questions of funda- mental interests[23, 24, 25, 26, 27, 28]. Recently, there are advances in studying one of the most remarkable quantum phenomena, the quantum Hall effects, in curved spaces[14, 15, 16, 17, 18, 19, 20, 21, 22, 1]. In particular, there is an experiment that synthesizes Landau levels in cone surface with photons [1]. Some experimental advances in strained graphene with ripples [29, 30, 31] also suggest the possibility of probing quantum Hall physics on curved surfaces in nature. Because the degenerate
Landau levels [32] is essential to quantum Hall physics, it is natural to ask the ques- tions: do the degenerate Landau levels persist in curved surfaces? If they do, how is it different from the case in flat space? What are the effects of geometric deformation on the degenerate Landau levels and the corresponding wave functions?
In this chapter, we try to answer these questions through generalizing a system, the so-called Landau problem, that is known to have degenerate levels in flat space to curved surfaces: a charged particle in a two-dimensional plane with constant per- pendicular magnetic fields. The curved surfaces we consider are cone and anti-cone surfaces where the latter we shall define in our discussion in section 2.3. Because they
6 are probably the simplest curved spaces we can generalize from the flat space and
the cone geometry has recent experiment results[1]. Geometrically, cone and anti- cone can be considered as adding an isolated singularity of Gaussian curvature at the center (the apex) to a flat disk (explained in this chapter). The effect of negative curvature singularity on the Landau levels is that a subset of the energy levels are shifted downwards by the negative curvature. In the limit of negative infinite curva- ture strength (denoted by κ → −∞ in this chapter) every Landau level extends to include all negative angular momentum states. And therefore, we can say degeneracy in each level is doubled. This feature is used as the strategy to create Landau levels in an anti-cone with the photonic system described in chapter 5. In chapter 5, we successfully bring all negative angular momentum states into the degenerate level, but realized the degeneracy is surprisingly greater than that of an anti-cone.
2.2 Landau problem with electrons in cone surfaces
2.2.1 The cone surface
The single-particle solutions on cones with constant magnetic flux have been dis- cussed in [18]. Here we summarize the significant results and make remarks relevant to our later discussion. In their treatments, the manifold on which the Hamiltonian is based is a flat sheet of wedge that can be obtained by cutting open a circular cone surface and flattening it as shown in the bottom left in figure 2.1. To formulate the
Landau problems on this flat wedge, it requires constant magnetic fields along the z
direction. Because we have cut the cone open, the two sides of the wedge that were
originally glued together need to be identified and they are marked with the thicker
lines in figure 2.1. Therefore, the domain of the wave functions ψ in terms of complex
7 coordinates z = (x + iy) is restricted to the region ϕ = arg(z) ∈ [0, 2πσ], where
0 < σ < 1 for a regular cone. The identification of the two sides of the wedge is a boundary condition of ψ. The expression in terms of polar coordinate is :
ψ(r, ϕ) = ψ(r, ϕ + 2πσ). (2.1)
Figure 2.1: A circular cone in R3 on the top right corner is cut along the thicker dark line and flattened into a wedge of wedge angle 2πσ where 0 < σ < 1 for regular cones. On the flattened wedge, the two dark edges are identified. To formulate the Landau problem, constant magnetic fields Bzˆ along the z− axis is added to the flattened wedge.
2.2.2 The Schr¨odinger’sequation and the solutions
The Hamiltonian for an electron (q = −e) in a gauge field A~ on the wedge in
figure 2.1, or a cone surface, is the following:
|P~ + e/cA~|2 H = . (2.2) 2M
This expression is the same as the Hamilton we would write down for a Landau problem in a two-dimensional plane. The only difference is that the wave function
8 has a periodicity of 2πσ < 2π in the polar angle ϕ, i.e. ψ(r, ϕ) = ψ(r, ϕ + 2πσ). By
~ 1 choosing the symmetric gauge A = 2 Brϕˆ, the Hamiltonian in polar coordinate (r, ϕ) has the expression
2 −~ 2 1 1 r 2 H = ∂r + ∂r + ( ∂ϕ + i 2 ) (2.3) 2M r r 2lB p where lB = ~c/eB, (e > 0) is the magnetic length. Note that we expand equation (2.2) into equation (2.3) with the following regular relations of polar coordinates.
∂ϕϕˆ ≡ −r,ˆ ∂ϕrˆ ≡ ϕ, ∂rϕˆ = ∂rrˆ = 0 (2.4)
The solutions to the Schr¨odinger’sequation Hψ(r, ϕ) = Eψ(r, ϕ) with the bound-
ary conditions
ψ(r, ϕ) = ψ(r, ϕ + 2πσ), ψ(r, ϕ) → 0 when r → ∞ (2.5)
are
m 1 m −im ϕ −r2/4l2 ( ) 2 2 E = ω (n + ), ψ = N r σ e σ e B L σ (r /2l ), m ≥ 0 nm ~ c 2 nm nm n B |m| |m| 1 |m| −im ϕ −r2/4l2 ( ) 2 2 E = ω (n + + ), ψ = N r σ e σ e B L σ (r /2l ), m < 0. nm ~ c σ 2 nm nm n B (2.6)
(β) The range of the indices are n = 0, 1, 2,... and m ∈ Z. Ln (x) is the associated
Laguerre polynomial of degree n[33]
n X n + β xk L(β)(x) = (−1)k . (2.7) n n − k k! k=0
Therefore we see that n is equal to the number of nodes in the radial wave function.
The index m, on the other hand, is the integer angular momentum quantum number ˆ of the angular momentum operator Lz = −i~∂ϕ whose eigenvalues are Lz = −~m/σ.
9 We use the normalization condition
Z ∞ Z 2πσ 2 dϕ rdr|ψnm| = 1 (2.8) 0 0
to calculate the normalization constant s n! Nnm = m m . (2.9) 2 σ +1 πσ(2lB) Γ(n + σ + 1)
In the above, we also make use of the orthogonal condition of the associated Laguerre
polynomials [33]
Z Γ(n + β + 1) xβe−xL(β)(x)L(β)(x)dx = δ (2.10) n m n! n,m
−imϕ/σ where Γ(x) is the gamma function. Note that the phase term e in ψnm is to ensure that the wave function ψ(r, ϕ) is single-valued under the periodic boundary condition in equation (2.5)
We plot the energy spectrum in figure 2.2 where each state is labeled by a pair of
2 indices (n, m). The energy scale is the cyclotron frequency ωc = ~/(MlB) = eB/(Mc). In figure 2.2, we see that when we change the manifold from a flat disk to a cone, the degenerate Landau levels structure remains the same for the m ≥ 0 states. However, the energy of the states with m < 0 shift upwards. The shifts in energy for the m < 0
−1 states are ~ωc|m|(σ − 1), which is proportional to |m|.
10 Figure 2.2: Both the energy spectrum of a charged particle on a cone with uniform magnetic flux and the spectrum for the same system on a flat disk are shown. Each state ψnm is labeled with a pair of indices (n, m) and their energy are given in equation (2.6). The blue bars are the energy levels with m ≥ 0 that do not change in both cases. The orange bars are the energy levels with m < 0 that shift upwards when we change the geometry from a flat disk to a cone. We show the energy levels on a disk with light orange color and those on a cone with dark orange color. The shift in −1 energy for the m < 0 states ∆E = ~ωc|m|(σ − 1) is also shown in the figure.
Note that although the wave functions are given by the same set of equations for both a flat disk σ = 1 and a cone 0 < σ < 1 as in equation (2.6), different σ’s lead to radial wave functions of different extent. This is reflected in both the power function
|m| |m|/σ ( σ ) term r and the Laguerre polynomials Ln . Roughly speaking, the radial wave function gets pushed outward when σ is smaller.
However, we would like to make a remark here that all the wave functions in the lowest Landau levels on a cone, ψLLL, i.e. n = 0 , when expressed in terms of z and
11 z¯,
z ≡ re−iϕ, z¯ ≡ reiϕ, (2.11)
are still a product of an analytic function f(z) and a Gaussian factor
−|z|2/(4l2 ) ψLLL(z, z¯) = f(z)e B . (2.12)
Because in figure 2.2, we can see that the states in the lowest Landau levels all have the
radial index n = 0. By taking n = 0 in equation (2.6) where the Laguerre polynomial |z|2 |m| − (β) 4l2 L0 ≡ 1 for all β, we have ψ0m(z, z¯) = N0mz σ e B when m ≥ 0. Because every single-particle wave function in the lowest Landau level is only a power function of
z and the Gaussian factor, the many-body wave functions can only be an analytic
function f(z) times the Gaussian. We will see later that this is not true in general
when we extend the discussion to an anti-cone.
2.2.3 Gaussian curvature for cone surfaces
In this section, we show that the Gaussian curvature K is concentrated at the
cone apex, and is related to the deficit angle ∆θ of the wedge cut open from the
original cone as we described in figure 2.1. The deficit angle is the angle difference
between a disk and the wedge, ∆θ ≡ 2π − 2πσ, and it is positive for a cone. The relation between K and ∆θ can be derived from the Gauss-Bonnet theorem, which is an important mathematical theorem about surfaces that connects their geometry to their topology.
For a compact manifold M, the Gauss-Bonnet theorem is [34]
ZZ Z KdA + kgds = 2πχM . (2.13) M ∂M
12 It says, suppose we have a compact two-dimensional manifold M, the integration of Gaussian curvature K over the manifold M and the integration of the geodesic curvature kg (explained below) along the boundary ∂M are related to the Euler characteristic χM of the manifold M. The Euler characteristic χ is a topological invariant of a space that is independent of how the space is bent[35]. It was defined classically for the surfaces of polyhedra by the Euler formula
χ = V − E + F (2.14) where V,E,F are respectively the number of vertices, edges and faces of the polyhe- dron. For general two-dimensional surfaces, the Euler characteristic is generalized as follows. First, we find a polyhedron that is topologically equivalent (homeomorphic) to the original surface. Then we calculate χ of the polyhedron with Euler formula in equation (2.14)[36]. The calculated χ is the Euler characteristic of this surface because χ is independent of the choice of polyhedra as long as they are topologically equivalent to the original surface[36].
In the following, we show what K, κg, χM respectively are in our choice of M on a finite cone, and calculate K with Gauss-Bonnet theorem. First we argue that the
Gaussian curvature K of a cone is concentrated at the apex (~r = 0) and can be represented as K(~r) = κδ(2)(~r), (κ > 0). Figure 2.3a shows some typical examples of surface with different Gaussian curvature. It is known that a convex smooth curved surface like the dome-shaped cap at the bottom left in figure 2.3a has positive
Gaussian curvature K > 0 because K = κI κII and both principal curvature κI , κII are positive [37]. At the left in figure 2.3b, a small region close to the smooth cone tip looks exactly like the K > 0 case in figure 2.3a, but the rest of the surface has
K = 0. Because if we remove the curved cap, the rest of the surface after we cut open
13 and flatten it can be embeded in two-dimensional Euclidean space, see the top right
case in figure 2.3a. Therefore, if we continuously sharpen the smooth cone tip into a
sharp point like what is shown in figure 2.3b, eventually, we obtain a cone which has
a positive singularity of Gaussian curvature K = κδ(2)(~r), singular at the cone apex
(~r = 0) and zero anywhere else as a limiting case.
b)
a)
Figure 2.3: (2.3a) Typical examples of surfaces of different Gaussian curvature K. Figures are copied and modified from [38] to agree with our notation. The Gaussian curvature is a product of two principal curvatures: K = κI κII . The top two figures show the case for K = 0. We can see bending a sheet does not change its Gaussian curvature. The bottom two figures show typical examples for K > 0 and K < 0. (2.3b) A cone can be obtained by sharpening the smooth K > 0 region at the top (left) into a point (right).
Now we use Gauss-Bonnet theorem in equation (2.13) to find the relation between
the Gaussian curvature K and the deficit angle ∆θ in our case of a cone. First of
all, we choose M to be a unit disk centered at the apex O of the cone S. i.e.
M = {x ∈ S| dS(x, O) ≤ 1}, (the region green region in the unfolded cone in figure
2.4 ) and hence, the boundary ∂M is the “unit circle” ∂M = {x ∈ S| dS(x, O) = 1}.
14 Here dS(x, y) denotes the distance between x, y ∈ S with the metric on the cone surface.
Figure 2.4: Calculating the geodesic curvature on a cone . The green region is M, a “unit disk” on the cone surface. The geodesic curvature of ∂M in the space of cone surface is 1 while the curvature in R3 is 1/ sin α
The geodesic curvature κg measures how far a curve on a surface is from being
a geodesic of that surface[39]. Its formal definition can be found in references of
differential geometry such as [34, 40, 41]. However, the curve of the boundary ∂M
in our case is relatively simple. ∂M in R3 is a circle of radius r = sin α on the cone
surface where α is the half opening angle of the cone. This is shown in figure 2.4.
In R3, the curvature of ∂M is the inverse of the radius. If we do not see the cone
but only the circle of ∂M, we would say the curvature of this circle is 1/ sin α. The
geodesic curvature, however, is defined with respect to the cone surface. In the wedge
unfolded from a cone in figure 2.4, the boundary ∂M is an arc of radius r = 1 and of
polar angle 2π − ∆θ where ∆θ is the so-called deficit angle. The geodesic curvature
κgof this arc ∂M is therefore κg = 1.
15 Lastly, the Euler characteristic of M is the same as that of a triangle. The reason
is the following. First, we can smooth out the cone tip. As a result we get a flat
finite disk which is topologically equivalent M. A finite disk can be deformed to a
triangle or any other convex polygon by adding vertices. The Euler characteristic of
any polygon can be easily computed with Euler formular (2.14) and it is χM = 1. To
calculate κ, we use Gauss-Bonnet theorem for M to be the compact region enclosed
by a unit circle centered at the origin on the surface, i.e. ∂M = {x| d(x, O) = 1}.
The Gauss-Bonnet theorem gives
ZZ Z Z 2π−∆θ κ = KdA = 2πχ − kgds = 2π − dθ = ∆θ (2.15) M ∂M 0
where ∆θ = 2π(1 − σ) is the deficit angle of the manifold. We see that the deficit
angle δ is indeed the intensity of the singularity. The range of ∆θ is ∆θ ∈ (−∞, 2π).
Interestingly, the range of ∆θ is asymmetric with respect to zero: ∆θ ∈ (0, 2π) for a cone and ∆θ ∈ (−∞, 0) for an anti-cone because we can insert infinite angle to a disk but can only remove at most 2π of the angle from a disk.
2.3 Landau problem in anti-cone surfaces
Anti-cone surfaces can be constructed with the idea similar to how we construct a circular cone surface. Recall in section 2.2 we have introduced the cone surface defined in [18]. It is a wedge in a two-dimensional plane with the two edges identified and it represents the cone surface in R3. Another way to look at the cone surface is that we first remove a wedge of a deficit angle ∆θ = 2π(1 − σ), (0 < σ < 1) from a disk
such that the remaining wedge if of angle 2πσ, and then glue the two sides together.
Conversely, if we insert an additional wedge of an excess angle ∆θ = 2π(σ−1), (σ > 1)
to a disk, we can obtain a surface that we call anti-cone. This is equivalent to
16 extending the periodicity of the argument of complex number beyond the complex plane to 2πσ > 2π. With this extension of the space, the wave function ψ(r, ϕ) now has the boundary condition ψ(r, ϕ) = ψ(r, ϕ + 2πσ) where σ > 1 is to account for an anti-cone. Hence, the anti-cone surface we discuss corresponds to the region in polar coordinate {(r, ϕ)| r ≥ 0, ϕ ∈ [0, 2πσ]} where σ > 1 is a constant determined by the excess angle ∆θ of the wedge we insert. A figure representing the space of anti-cone surface over which we define the wave functions is depicted in figure 2.5a.
An anti-cone surface can be obtained by cutting papers with the following steps. We
first cut a straight line through a circular shape of paper from the circumference to the center. We then insert a wedge of angle ∆θ by gluing each edge of the wedge to each side of the cut line on the disk. Eventually we get an anti-cone surface that looks like a saddle except that the center is not smooth as shown in 2.5b.
a) b)
Figure 2.5: (2.5a) The anti-cone can be obtained by gluing a wedge to a disk. We first cut along the dashed line from the circumference to the center of the disk, which creates two edges. To insert the wedge of the same radius to the disk, we glue the two edges of the wedge to the edges of the cut open disk. (2.5b). The resulting space is an anti-cone surface embedded in R3. See Appendix A for more details.
17 2.3.1 The Schr¨odinger’sequation and the solutions
The Schr¨odinger’sequation for a charged particle in the space of anti-cone surface
with constant perpendicular magnetic fields is exactly the same as equation (2.2)
~ 1 with the gauge in the polar angle direction: A = 2 Brϕˆ. The Hamiltonian operator in polar coordinate also has the identical expression as in equation (2.3). As a result, the solutions to the Schr¨odinger’sequation Hψ(r, ϕ) = Eψ(r, ϕ) is again identical to equation (2.6. The energy spectrum has the same functional form as we have described before,
1 E = ω (n + ), for m ≥ 0 (2.16) nm ~ c 2 |m| 1 E = ω (n + + ), for m < 0 (2.17) nm ~ c σ 2
However, we are now in the regime of σ > 1. Therefore, the energy for the states of
m < 0 will shift downward instead. Figure 2.6 shows how the single-particle energy
levels flow when we change from a disk to an anti-cone. The energy levels in the
σ → ∞ limit is shown in figure 2.7.
18 Figure 2.6: The energy spectrum of a charged particle on an anti-cone surface with uniform magnetic flux. Each state ψnm is labeled by (n, m). Similar to figure 2.2, only energy levels of states with m < 0 shift but in the opposite direction as compared to on a cone. Blue bars are energy levels of m ≥ 0 that do not change with σ. However, the m < 0 energy levels, shown in orange bars, shift downward when we increase σ, i.e. changing from a flat disk (light orange) to an anti-cone (dark orange).
19 Figure 2.7: The energy levels of Landau problem on an anti-cone when σ → ∞. Note that in this limit, all states with different m but with the same radial index n are degenerate. Each Landau level are extended and can have any values of integer angular momentum index m.
2.3.2 The response of the energy levels to the strength of curvature singularity
Now combining both regimes, we can summarize the response of the energy levels to the isolated singularity of curvature at the apex as the following. Suppose the curvature singularity is K(~r) = κδ(2)(~r) and recall from section 2.2.3 that
κ = ∆θ = 2π(1 − σ), κ ∈ (−∞, 2π) (2.18) with the sign of κ indicating the regime of cone or anti-cone surface. Through the previous study of the energy eigenvalues, we realize only the energy of states with negative angular momentum are affected by the isolated singularity. For curvature
20 K(~r) = κδ(2)(~r), the shift in energy for states with m < 0 is
1 κ ∆E (κ) ≡ E (κ) − E (κ = 0) = ω |m| − 1 = ω |m| (2.19) m m m ~ c σ ~ c 2π − κ
We plot the m independent quantity ∆(κ) = ∆Em(κ)/~ωc|m| in figure 2.8 for clarity.
κ Figure 2.8: The graph of ∆(κ) = ∆Em(κ)/~ωc|m| = 2π−κ with asymptotes κ = 2π and ∆(κ) = −1. The asymptotic behavior ∆(κ) → −1 at κ → −∞ means every (n, m) state on the σ → ∞ anti-cone would have their energy depend on n only.
We see that in the limit of κ → −∞, or equivalently σ → ∞, we have ∆(κ) → −1.
This means the dependence of the |m| dependence for the energy eigenvalues for negative angular momentum states vanishes. We achieve extended Landau levels
1 E = ω (n + ), for m = 0, ±1, ±2,... (2.20) nm ~ c 2
This result is significant because the direct consequence of this set of extended Landau levels is that we can have integer quantum Hall states that are functions of zm and
z¯m with z = r1/σe−iϕ/σ ∈ C
21 Chapter 3: The Landau levels of a rotating harmonic oscillator: the synthetic gauge
3.1 Introduction
In the previous chapter we have seen that Landau levels are formed by the symmet- ric gauge that produces the uniform perpendicular magnetic fields. The energy eigen- functions are circular concentric orbits of different radii called cyclotron orbits[3, 2].
Regardless of different radii, every cyclotron orbit in the same degenerate Landau lev- els goes around the center with the same frequency ωc = eB/(Mc) called cyclotron frequency. The effect of the magnetic fields on a charge particle is essentially intro- ducing a rotation to the charged particle through Lorentz force. Note that implication of the Landau levels is that although the cyclotron orbits of larger radius in the same
Landau levels have greater angular momentum Lz = m~, greater linear velocity v and kinetic energy, they all have the same total energy[3, 2]. This suggests that the symmetric gauge introduce a term that energetically favors the states with positive rotation.The above observation suggests that it is also possible to achieve Landau levels with spatial rotation and they have been studied in the context of cold atom physics[42, 43, 44, 45, 46, 47].
22 In this chapter we first show the quantum mechanical formalism of a system with rigid potential rotating at uniform speed. Then we specialize the case to a harmonic trap potential and demonstrate how the energy levels vary as a function of the rotation frequency and eventually achieve the exact Landau quantization when the rotation approaches the so-called centrifugal limit beyond which the system is unstable. This rotating trap problem is of fundamental importance in this thesis.
Because the photonic system discussed in chapter 4 and 5 is exactly simulating a rotating harmonic oscillator. We show two different approaches to solve the rotating harmonic oscillator and they correspond to two different labeling for the states. In the end we show the conversion between these two labeling sysyems that will be referred to often in chapter 4, and 5 because each has its own advantage for creating the degenerate level and understanding the wave functions.
3.2 The quantum Hamiltonian for a uniformly rotating po- tential
Consider in two-dimensional space, a time-dependent Hamiltonian Hˆ (t) that has a rigid potential Vˆ (~r(t)) that rotates counter-clockwise with a uniform frequency
Ω~ = Ωˆz. i.e. ~r(t) = (x cos Ωt − y sin Ωt, x sin Ωt + y cos Ωt)
Hˆ (t) = Tˆ + Vˆ (~r(t)) (3.1)
Tˆ = P~ 2/2M is the kinetic energy operator. The time-dependent Schro¨odingerequa- tion is
ˆ i∂t |Ψ(t)i = H(t) |Ψ(t)i (3.2) where we take ~ = 1.
23 Consider in the interaction picture, we would like to freeze the rotation by counter-
rotating the state so that the potential and therefore the Hamiltonian are stationary in
that co-rotating frame, and so are the solutions. Thus we choose |ΨI (t)i = U(t) |Ψ(t)i
where U(t) is the unitary operator of the counter-rotation
U(t) = U[R(−Ωt)] = exp(−i(−Ωt)Lz) = exp(iΩtLz) (3.3)
and Lz is the angular momentum operator. The time evolution of |ΨI (t)i is
i∂t |ΨI (t)i = (i∂tU(t)) |Ψ(t)i + U(t)(i∂t |Ψ(t)i) (3.4)
ˆ = (−ΩLz)U(t) |Ψ(t)i + U(t)H(t) |Ψ(t)i (3.5) ˆ † = −ΩLz + U(t)H(t)U(t) |ΨI (t)i (3.6)
The second term in the above equation is can be simplified
U(t)Hˆ (t)U(t)† = U(t)TUˆ (t)† + U(t)Vˆ (~r(t))U(t)† = Tˆ + Vˆ (~r(0)) (3.7)
because the kinetic energy is invariant under rotation and the unitary operator U is
chosen to perfectly cancel the rotation of the potential Vˆ (~r(t)). Therefore we obtain
the Hamiltonian in interaction picture which is stationary
ˆ ˆ ˆ HI = T + V (~r(0)) − ΩLz (3.8)
The above Hamiltonian is the Hamiltonian in interaction picture, and it is also the
Hamiltonian in the rotating frame of rotation frequency Ω. Comparing it to the
stationary case, we realize the Hamiltonian gains an extra term −ΩLz when we change to a uniformly rotating frame. Although this derivation was done in two-dimensional space, the result can actually generalize to three-dimensional space by promoting ~ ~ the−ΩLz term to a inner product −Ω · L. Also note that if we write
Ω~ = Ωˆn, L~ = ~r × ~p (3.9)
24 then the extra term can be rewritten as
−Ωˆn · (~r × ~p) = −~p · (Ωˆn × ~r) (3.10)
ˆ The Hamiltonian in the co-rotating frame HI can be expressed as
(~p − MΩ~ × ~r)2 (MΩ~ × ~r)2 Hˆ = + Vˆ (~r) − (3.11) I 2M 2M
Comparing (3.11) with (2.2), we notice that first of all, the effect of the rotation is equivalent to the combines effect of introducing a symmetric gauge as the gauge potential when we identify qA/c~ = MΩ~ × ~r and a negative quadratic potential
−(MΩ~r)2/2M. When we have the potential V (~r) be the harmonic trap with the trapping frequency equal to Ω, the Hamiltonian HI is identical to the Hamiltonian for a charged particle in the symmetric gauge.
An equivalent and more conventional way to look at it is to start with a rotating
ˆ 2 harmonic oscillator with trapping potential V (~r) = (MωT ~r) /2M and rotating fre- quency Ω. The rotation introduces a synthetic gauge field A~ = MΩ~ × ~r (assuming q = 1) and a centrifugal potential. The Hamiltonian (3.11) has expression
(~p − MΩ~ × ~r)2 M(ω2 − Ω2)r2 Hˆ = + T I 2M 2 1 (qB)syn 2 Mω2 r2 = ~p − zˆ × ~r + eff . (3.12) 2M 2c 2
This Hamiltonian, known as Fock-Darwin Hamiltonian, can be viewed as a particle in a synthetic magnetic field (qB)syn/c = 2MΩ and an effective harmonic potential
p 2 2 with trapping frequency ωeff = ωT − Ω The rotation frequency Ω has an upper limit, the trapping frequency ωT , for the system to stay stable. In the special case when this limit is achieved, Ω → ωT , we say the system is at criticality. The effective harmonic trap is canceled out and the system is identical to a charged particle in the
25 symmetric gauge. We will devote the next section to the solutions for an oscillator in
a rotating harmonic trap.
3.3 A rotating quantum harmonic oscillator
3.3.1 wave functions in polar coordinate
In this section, we focus on the solutions to a rotating harmonic oscillator and
their features. For the convenience of comparison with our discussion for a system of
electron in magnetic field in chapter 2, we choose the rotation to be in the clockwise
direction, i.e. Ω~ = −Ωˆn for consistency.
In the co-rotating frame, the Hamiltonian for a particle of mass M in a two- dimensional rotating isotropic harmonic trap ωT is given by (3.8)
(~p)2 (Mω ~r)2 H = + T + ΩL (3.13) 2M 2M z
where Lz = −i~∂φ is the angular momentum operator. In terms of polar coordinate (r, φ), the Hamiltonian is
2 2 2 " 2 2 # −~ 2 r −~ 2 1 ∂φ r ∂φ H = ∇ − 4 − iΩ~∂φ = ∂r + ∂r + − 2 + i2 2 2M dT 2M r r dT dΩ (3.14) q q where d−1 = MωT and d−1 = MΩ are the inverse length of the trap and the T ~ Ω ~ rotation respectively. Note that each frequency shows up as one term individually
and is proportional to its inverse length squared. Therefore, as we gradually turn off
each frequency, the corresponding term in the Hamiltonian vanishes continuously.
Let’s first consider the limit of Ω → ωT when the system reduce to that in (2.2).
Let d = dΩ = dT , the Hamiltonian (3.14) becomes the same as (2.3) with the identifi-
2 2 cation d = 2lB and noting that φ ∈ [0, 2π] because we are now in a two-dimensional
26 plane " # − 2 1 ∂ 2 r 2 ∂ H = ~ ∂2 + ∂ + φ − + i2 φ (3.15) crit. 2M r r r r d2 d2 " # − 2 1 ∂ r 2 = ~ ∂2 + ∂ + φ + i . (3.16) 2M r r r r d2
We have solved the solutions to the Schr¨odingerequation with this Hamiltonian before
in (2.6). All we have to do is to take σ = 1 in (2.3), and replace ωc with 2ωT
1 2 2 E = (2ω )(n + ), ψ (r, φ) = N rme−imφe−r /2d L(m)(r2/d2), m ≥ 0 nm ~ T 2 nm nm n 1 2 2 E = (2ω )(n + |m| + ), ψ (r, φ) = N r|m|e−imφe−r /2d L(|m|)(r2/d2), m < 0 nm ~ T 2 nm nm n (3.17)
Note that because of the convention we choose here, the angular momentum quantum
number (or helicity) m > 0 meaning the angular momentum is along with the rotation
and vice versa, while n is again the radial quantum number.
We then consider gradually turning down the rotation, and realize that the wave
functions actually don’t change when we turn down the rotation frequency Ω, but
only the corresponding energy eigenvalue for each eigenfunction change. The reason
is because the polar angle dependence of each energy eigenfunction is purely e−imφ
coming from the rotational symmetry. And the only term in the Hamiltonian coupling
to the rotating frequency Ω is the last term in (3.14),
2 −~ i2∂φ · 2 ψnm = (Ω)(−~m)ψnm = −m~Ωψnm. (3.18) 2M dΩ
This agrees with our intuition that the energy eigenvalues is lower when the direction of the angular momentum is along with the rotation (m > 0).
How the shift in energies varies with the rotation is the most transparent when we consider from the other end of the rotation when Ω = 0, and gradually turn it
27 on. Suppose we start with a system with no rotation Ω = 0, we know from the above
discussion that the wave functions are the same as in (3.17). According to (3.13), the
energy eigenvalues have the relation for each eigenfunction ψnm
E(Ω) = E(0) + ΩLz = E(0) − m~Ω (3.19)
Here E(Ω) ≡ Enm(Ω) represents the whole set of energy eigenvalues for each individ- ual eigenfunction {ψnm} at rotation frequency Ω. Consider the special case at Ω = ωT and substitute E(Ω → ωT ) with what we have had in (3.17), we can get the expressions for the spectrum of the two-dimensional harmonic oscillator E(0) = E(ωT ) + m~ωT .
m 1 E (Ω = 0) = (2ω )(n + + ) = ω (2n + m + 1), m ≥ 0 nm ~ T 2 2 ~ T m 1 E (Ω = 0) = (2ω )(n + |m| + + ) = ω (2n − m + 1), m < 0 (3.20) nm ~ T 2 2 ~ T
which is exactly the spectrum we would find if we solve the two-dimensional harmonic oscillator in the polar coordinate. Now using the equation (3.19) again with E(0) given above for a general rotation frequency Ω, we obtain
ωT − Ω 1 Enm(Ω) = ~ωT (2n + m + 1) − m~Ω = ~(2ωT ) n + m + , m ≥ 0 2ωT 2 ωT + Ω 1 Enm(Ω) = ~ωT (2n − m + 1) − m~Ω = ~(2ωT ) n + |m| + , m < 0 2ωT 2 (3.21)
The above equations for energy levels are depicted in figure 3.1 for general Ω, on and
in figure
28 Figure 3.1: The rotation effects on the energy levels for a two-dimensional harmonic oscillator. The dashed arrows indicate the flow of the energy levels when turning up the rotation. The solid rectangular boxes are the energy levels for a harmonic oscillator without rotation where the blue boxes are for m ≥ 0 and the orange are for m < 0.
29 Figure 3.2: The energy levels of a rotating harmonic oscillator at criticality. The energy levels form exactly the identical Landau levels to a charged particle in uniform magnetic fields.
To summarize, the energy spectrum and the eigenfunction for a rotating harmonic oscillator with trap frequency ωT and rotation frequency Ω is
ωT − Ω 1 Enm(Ω) = ~(2ωT ) n + m + , 2ωT 2 2 2 m −imφ −r /2dT (m) 2 2 ψnm = Nnmr e e Ln (r /dT ), m ≥ 0 ωT + Ω 1 Enm(Ω) = ~(2ωT ) n + |m| + , 2ωT 2 2 2 |m| −imφ −r /2dT (|m|) 2 2 ψnm = Nnmr e e Ln (r /dT ), m < 0 (3.22)
We would like to make a remark on the solutions in two different perspectives. For the Hamiltonian (3.13), when we view the system as a rotating harmonic oscillator, we note that length scale of the wave functions is given solely by the harmonic trap
~ natural length dT = We would say the rotation frequency Ω of a rotating har- MωT monic oscillator does not change the wave functions at all including the length scale.
30 All the rotation does is shifting the energy levels. However, if we see the system as a harmonic trapped particle in a synthetic field as the Fock-Darwin Hamiltonian in equation (3.12), would say the wave function has a length scale dependent on both
syn the effective harmonic trap ωeff and the synthetic field (qB) . The length scale of the wave function is given by q 2 (qB)syn 2 2 1 M ωeff + ( ) /M = 2c (3.23) d2 ~ " #1/2 1 2 1 2 = 2 + 2 . (3.24) deff 2lB Or equivalently,
1 1 1 = √ + (3.25) 4 4 4 d ( 2lB) deff
2 ~ p syn where d = and the synthetic magnetic length lB = c/(qB) . The intuition eff Mωeff ~ is that when we have a system in a synthetic fields and we gradually turn on an external harmonic trap with trap frequency ωeff, the length scale of the states decreases from the magnetic length lB with increasing harmonic trap frequency. Interestingly, let us compare both pictures to understand the invariance of the length scale d when we tune only the parameter Ω. When we increase only Ω, in the synthetic field perspective we decrease the magnetic length and increase the effective trap length.
And they coincidentally cancel each other and preserve the length scale of the states.
3.3.2 the ladder operator approach
We can also solve the system of a rotating harmonic oscillator in an algebraic approach using ladder operators as the famous example of a harmonic oscillator.
This approach helps us to see clearly the positive and negative polarization modes when we later discuss the Landau quantization of synthetic photons.
31 We take the Hamiltonian of a rotating harmonic oscillator in (3.13) where the
rotation is Ω~ = −Ωˆz. To diagonalize the Hamiltonian, we follow the steps in [43].
First, we express the Hamiltonian in terms of the conventional raising and lowering
operators in x and y axes 1 x dT px † 1 x dT px ax = √ + i , ax = √ − i (3.26) 2 dT ~ 2 dT ~ p where dT = ~/(mωT is the natural length of the harmonic trap as we defined before.
† The definition for ay, ay is similar, and recall that the ladder operators satisfy the
† † canonical commutation relations: [ax, ax] = [ay, ay] = 1. The position and momentum operators in terms of the ladder operators are
† † (ax − ax) (ax + ax) ~ x = √ dT , px = √ (3.27) i 2 2 dT
With these equations, the last term ΩLz in the Hamiltonian (3.13) can be expressed
as
† † ΩLz = Ω(xpy − ypx) = i~Ω(axay − ayax) (3.28)
The Hamiltonian (3.13) in terms of the ladder operators can now be written as
† † † † H = ~ωT (axax + ayay + 1) + i~Ω(axay − ayax) ω Ω = T (a† a + a a† + a† a + a a† ) + i (a† a + a a† − a† a − a a† ) (3.29) ~ 2 x x x x y y y y ~ 2 x y y x y x x y
The second line in (3.29) is in a symmetric form that is convenient for us to diagonalize
with matrix notation. Consider in terms of a row vector of ladder operator A† =
† † (ax ax ay ay), the Hamiltonian in the matrix notation is now
† ωT 0 −iΩ 0 ax ~ † † 0 ωT 0 iΩ ax ~ † H = ax ax ay ay † = A MA (3.30) 2 iΩ 0 ωT 0 ay 2 0 −iΩ 0 ωT ay
32 where M is the Hermitian 4 × 4 matrix in the above equation. The Hermitian matrix
M can be diagonalized with a unitary matrix U. The eigen-decomposition is
M = UDU † (3.31)
where √ √ 0√ i/ 2√ 0 −i/ 2 ωT − Ω 0 0 0 −i/ 2 0 i/ 2 0 0 ω − Ω 0 0 U = √ √ ,D = T 0 0 ω + Ω 0 0√ 1/ 2 0√ 1/ 2 T 1/ 2 0 1/ 2 0 0 0 0 ωT + Ω (3.32)
From the diagonal matrix D, we denote the two normal mode frequencies ω± =
ωT ∓ Ω to simplify the notation. The plus subscript in ω+ means a positive helicity
frequency that rotate along the rotating of the harmonic trap and vice versa. With
this decomposition, we can diagonalize the Hamiltonian
h i H = ~(A†U)D(U †A) = ~ (a a† + a† a )ω + (a a† + a† a )ω (3.33) 2 2 + + + + + − − − − −
The ladder operators for the normal modes, sorting out from A†U, are
iax + ay −iax + ay a+ = √ , a− = √ (3.34) 2 2
The ladder operators of the normal modes also satisfy the canonical communication
relations
† [aα, aβ] = δα,β, α, β = +, −. (3.35)
We can now recast the Hamiltonian in terms of the number operators of normal modes
1 1 H = (a† a + ) ω + (a† a + ) ω . (3.36) + + 2 ~ + − − 2 ~ −
33 The ground state wave function and all other eigenfunctions can be obtained by the
conventional prescription for a harmonic oscillator.
† † (a )n+ (a )n− a ψ = 0, ψ = + − ψ (3.37) ± 00 n+n− p p 00 n+! n−!
We choose the complex variables to be z = x−iy, z¯ = x+iy to solve the eigenfunctions with ladder operators. We choose z = x−iy to be our convention because the rotation is in clockwise direction to match the case for electron in chapter 2. The annihilation operators of the normal modes in terms of z, z¯ are
i z −i z¯ a+ = + 2dT ∂z¯ , a− = + 2dT ∂z , (3.38) 2 dT 2 dT
and the ground state wavefunction is
1 −zz/¯ 2d2 ψ00(z, z¯) = √ e T . (3.39) πdT
All other eigenstates can be derived by acting the raising operators on the ground state. The eigenfunctions we obtain here have one-to-one correspondence to the eigenfunctions in (2.6). We will demonstrate the correspondence in the next section by comparing these two schemes of labeling the state.
In this form, we see the energy levels of the system are quantized and there are two normal modes of different frequencies. The positive helicity mode carries energy
~ω+ = ~(ωT − Ω), and the negative helicity mode carries ~ω− = ~(ωT + Ω)
1 1 E = (n + ) ω + (n + ) ω (3.40) n+n− + 2 ~ + − 2 ~ − 1 1 = (n + ) (ω − Ω) + (n + ) (ω + Ω) (3.41) + 2 ~ T − 2 ~ T
where n± are the eigenvalues of the number operators of the positive/negative helicity ˆ † modes N± = a±a± with range n± = 0, 1, 2, ... We would like to make the remark that
34 the ladder operators do not depend on the rotation frequency Ω. This again justifies
our previous result that the eigenfunctions will not change with the rotation frequency
Ω. The effects of the rotation is shifting the energy levels.
3.4 the two labeling schemes of the spectrum
We have introduced two different schemes to label the energy levels and eigen-
functions of the rotating harmonic oscillator: the first one (n, m) is labeled with the
radial quantum number n and the angular momentum quantum number m; the other
is with the number of two normal modes (n+, n−). The first labeling scheme with
(n, m) is obtained through solving the differential equations of the Schr¨odingerequa- tion in polar coordinates. It has the advantages that it can be easily generalized to the solutions on cone and anti-cone surfaces so long as the system has two-dimensional rotational symmetry. Besides, the eigenfunctions can be easily read out and plotted
(see equation (3.17). The second scheme, on the other hand, comes from solving the system algebraically. It shows the two independent normal modes and can be more easily related to the convention in terms of complex variables (z, z¯) that is commonly adopted in quantum Hall physics. Although its generalization to cone and anti-cone surfaces is not as transparent as the first scheme, it has generalization to systems with a broken rotational symmetry, for example a rotating anisotropic harmonic trap is discussed in [43].
In this section, we would like to establish the correspondence between both schemes.
Because the methodology we adopt for the synthetic Landau levels for photons in chapter 4 is equivalent to the normal modes scheme (n+, n−). However, to see the
35 structure and comprehend the wave functions clearly, the (n.m) scheme provides more direct information because the forms of the wave functions are explicitly expressed.
Let’s consider the system a rotating harmonic oscillator on a disk with rotation frequency Ω < ωT to establish the correspondence. Recall the energy levels in (n, m) scheme in (3.21) can be combined and rearranged into
Enm = (2n + |m| + 1)~ωT − m~Ω (3.42)
With the normal modes scheme (n+, n−), remember that ω± = ωT ∓ Ω the energy levels in (3.40) can be rearrange as
En+n− = (n+ + n− + 1)~ωT − (n+ − n−)~Ω (3.43)
Because ωT and Ω are two independent parameters that one can control, by comparing the two equations we immediately have the relations
m = n+ − n− , 1 n = (n + n − |n − n |) = min(n , n ). (3.44) 2 + − + − + − That is to say, the radial quantum number n is determined by the smaller integer among the two normal mode quantum number, and the angular momentum quantum number m is the difference between the positive and negative normal mode number.
We can also verify the equation for m from the angular momentum operator in terms of the normal modes ladder operators with (3.28) and (3.34)
† † † † ˆ ˆ Lz = i~(axay − ayax) = −i~(a+a+ − a−a−) = −i~(N+ − N−) (3.45)
Combine the above with the definition of Lz = −i~∂φ in polar coordinate, we have the following relation for each eigenfunction ψnm as in (3.17),
† † Lzψnm = −i~(a+a+ − a−a−)ψnm = −i~∂φψnm = −m~ψnm (3.46)
36 Chapter 4: Realization of rotating quantum harmonic oscillators with paraxial optics
4.1 Introduction
In the conventional solid-state settings, Landau levels are generated by strong perpendicular magnetic fields in a system of two-dimensional electron gas [32]. Re- cently, it is shown by Johnathan Simon’s group that it is possible to generate Landau levels with a photonic system in a cavity (a.k.a. an optical resonator)[1]. They have found ways to simulate massive particles in a rotating harmonic trap with photons generated by laser beams in a cavity. This success in simulating a rotating harmonic oscillator in turn leads to the synthetic Landau levels in the energy spectrum, which we have discussed in chapter 3. While Simon’s experiment has received a lot of atten- tion, his scheme is very difficult to understand. Perhaps because of this, there have been no following experiments. The goal of this chapter is to make Simon’s scheme much more accessible.
Two ingredients are needed to simulate Landau levels of a massive particle with photons. The first is to produce a set of harmonic oscillator wave functions with the usual harmonic oscillator spectrum. The second is to introduce effective rotation in the system to change the harmonic oscillator spectrum into a set of Landau levels.
37 The first goal can be achieved by trapping photons in a two-mirror cavity[48]. As we shall see, the transverse modes of the laser beams in the cavity are precisely those of harmonic oscillators with the frequencies of the modes given by the usual harmonic oscillator spectrum. To achieve an effective rotation, one generalizes the two-mirror cavity to a four-mirror cavity (with mirrors arranged in a tetrahedral configuration.) The rest of this chapter is to show how these arrangements lead to the desired spectrum of a rotating harmonic oscillator.
4.2 Simulating the Schr¨odingerequation of massive particle using laser beam and generation of harmonic oscillator wave functions
4.2.1 Description of laser beams in paraxial approximation
We first discuss the description of a propagating laser beam as shown in figure
4.1.
38 Figure 4.1: The propagation of a generic laser beam. A general laser beam has a Gaussian transverse profile as shown by ψ(x, y, z) that spreads out as it propagates. In the figure, L1 is the length scale for the envelope of the transverse profile in the ∂ψ transverse direction. L2 is the length scale for the variation of the slope ∂z in z direction. λ is the wavelength. R(z) is the radius of curvature of the wave front and w(z) is the beam width at the transverse plane at z. zR is the distance from the beam waist at which the beam width diverges significantly.
A monochromatic EM wave in free space is governed by the following Maxwell’s wave equation
1 ∇2 − ∂2 E(x, y, z, t) = 0. (4.1) c2 t
For a linearly polarized monochromatic wave with wavelength λ that primarily propagates along z−axis, the electric field is
E(x, y, z, t) = ˆψ(x, y, z)ei(kz−ωt) (4.2)
2π where k = λ is the wave number along z, ˆis the polarization direction, and ψ(x, y, z), called the transverse field profile, is a complex scalar wave amplitude that specifies
39 the transverse profile of the field on each transverse plane. Substituting (4.2) into
(4.1), we obtain an equation of ψ:
∂2ψ ∂2ψ ∂2ψ ∂ψ + + + 2ik = 0 (4.3) ∂x2 ∂y2 ∂z2 ∂z
Next, we make the so-called paraxial approximation, which is designed to describe
propagating laser beams in figure 4.1. The paraxial approximation is
2 2 2 ∂ ψ ∂ ψ ∂ ψ , , (4.4) ∂z2 ∂x2 ∂y2 2 ∂ ψ ∂ψ k . (4.5) ∂z2 ∂z
Condition (4.4) means that the length scale of the envelope in the transverse direction
(L1 in figure 4.1 ) is much less than the scale of variation of the wave in z− direction
∂ψ (L2 in 4.1). Condition (4.5) means the slope ∂z varies very little over a wavelength λ in the z-direction. The derivation and validity of the paraxial approximation can
be found in appendix B.1.
After applying the paraxial approximation, we obtain the paraxial wave equation:
∂2ψ ∂2ψ ∂ψ + + 2ik = 0. (4.6) ∂x2 ∂y2 ∂z
or equivalently, −∇2 i∂ ψ = ⊥ ψ. (4.7) z 2k
Paraxial wave equation is a differential equation of the transverse field profile,
ψ(x, y, z). Its absolute value square |ψ(x, y, z)|2 gives the field intensity on the trans-
2 ∂2 ∂2 verse plane (z = constant). ∇⊥ = ∂x2 + ∂y2 is the Laplacian in transverse dimensions. Note that equation (4.7) is identical to the two-dimensional Schr¨odingerequation for a
40 massive particle in free space with z identified as time. The transverse field ψ(x, y, z)
plays the role of the wave function in quantum mechanics, and the mass of the particle
is given by the wave number k.
To find a solution that describes the propagating beam in figure 4.1, we look for solutions that vanish rapidly in the transverse plane. In appendix B.2, we show that the general solutions of equation (4.6) for a beam of wavelength λ = 2π/k with circular cross section is
k k k ψnx,ny (x, y, z) = ψnx (x, z)ψny (y, z), (4.8)
1 1 √ ! 4 −i(nx+ )[χ(z)−χ0] 2 2 k 2 e 2 2x ikx x ψ (x, z) = Hn exp − (4.9) nx p n x 2 π 2 x nx!w(z) w(z) 2R(z) w (z) 1 1 √ ! 4 −i(ny+ )[χ(z)−χ0] 2 2 k 2 e 2 2y iky y ψ (y, z) = Hn exp − . (4.10) ny p n y 2 π 2 y ny!w(z) w(z) 2R(z) w (z)
nx, ny = 0, 1, 2,..., and Hn(x) is the Hermite polynomial of order n.
ψnx,ny is called Hermite-Gaussian modes, also denoted as HGnxny in the litera-
ture of optics . R(z), w(z), χ(z) are all real-valued functions. They are related to a
complex-valued function q(z) as
1 1 2i = − . (4.11) q(z) R(z) kw(z)2 i χ(z) = − arg( ) (4.12) q(z)
where q(z) is linear in z
q(z) = z − ζ (4.13)
and ζ ∈ C is a complex number and its meaning is explained in appendix B.2.1. It
is clear from equations (4.8-4.10) that w(z) is the width of the Gaussian beam (also
41 known as the spot size of the beam in some literature of optics), and R(z) is the radius of curvature of the wave front.
Let us first consider a laser beam collimated at z = 0 (see figure 4.1.) This means that at z = 0, the wave front of the beam is a plane, i.e. R(z = 0) → ∞, perpendicular to the direction of propagation z. In this case, equation (4.11) implies q(0) is imaginary because
1 1 2i 2i = − = − . (4.14) q(0) R(0) kw2(0) kw2(0)
Hence, we write
q(z) = z + izR (4.15) kw2(0) z = . (4.16) R 2
Equations (4.15,4.16) with equations (4.11, 4.12) give
z2 R(z) = z + R (4.17) z r z 2 w(z) = w0 1 + ( ) (4.18) zR πw2(z) z χ(z) = tan−1 = tan−1( ). (4.19) λR(z) zR
It can be seen that the spot size w(z) takes minimum at w0 = w(0) when z = 0, so w0 is also called the beam waist of the Gaussian beam. zR characterizes the distance from the beam waist at which the beam starts to diverge significantly and is called
Rayleigh range[48]. As can be seen in equation (4.18), at z = zR, the beam width is √ 2 times of the beam waist w0, and the cross section area of the beam is doubled.
Also, χ(z) = 0 when z = 0, so we would have χ0 = 0 in equations (4.9,4.10).
42 k We finally remark that for a given Hermite-Gaussian beam (ψnxny ) with mode
orders (nx, ny), it is totally characterized by the wavelength (λ = 2π/k) and the beam waist w0 only.
4.2.2 Laser beams in a two-mirror cavity (a) Characterization of a laser beam in a cavity
Although a single Gaussian beam in free space can produce the harmonic-oscillator- like wave functions, as shown in equation (4.8- 4.10) in the previous section, We shall now consider trapping a laser beam in a two-mirror cavity and later generalize it to the four-mirror cavity that generates rotation effects on these wave functions. First we introduce the concept of a “good” two-mirror resonator. A two-mirror resonator is shown in figure 4.2, which shows three cases. In the first case in figure 4.2a , the laser beam in the cavity leaks out every time it hits the mirror (also called diffraction loss) because the beam widths at the mirrors, w1, w2 are comparable to the transverse dimensions of the mirrors L1,L2. A good resonator with no diffraction loss would
satisfy w1, w2 L1,L2 . The second case is shown in figure 4.2b where the beam after bouncing back and forth many times in the cavity, eventually escapes from it; such cavities are called unstable resonators. The third case is that the laser beam is completely trapped inside the cavity with no loss of the first and second kind. Such
“good” two-mirror cavity is the case that we shall consider from now on.
43 a)
b) c)
Figure 4.2: (4.2a) A resonator with significant diffraction loss. (4.2b) An unstable resonator. (4.2c) A stable resonator.
We now show the inobvious fact that given a laser with wavelength λ and beam waist w0, a stationary mode can only be achieved by placing mirrors with specific radii of curvature at the right locations. Consider two mirrors with radius of curvature
R1,R2 located at z1, z2 individually as shown in figure 4.3. The location where the laser beam is collimated is at z = 0,
44 Figure 4.3: A collimated laser beam with wave number k = 2π/λ and beam waist w0 enters the two-mirror cavity at z = 0 and form stationary modes in the resonator. The two mirrors with radius of curvature R1,R2 are placed at z1, z2 respectively, and the laser beam has beam widths w1, w2 on the mirrors.
From equation (4.17) we have
2 zR R(z1) = z1 + = −R1 (4.20) z1 2 zR R(z2) = z2 + = R2. (4.21) z2
2 Since zR = kw0/2 is given, for mirrors with radius of curvature R1,R2, they must be placed at z1, z2 that satisfy equations (4.20, 4.21).
In optics, it is found convenient to introduce a set of “resonator g parameters” for the resonator in place of R1,R2, z1, z2. They are defined as
L gi ≡ 1 − ,L = z2 − z1. (4.22) Ri
The g parameters can be used to determine the stability of a two-mirror resonator
[48]that we mentioned previously in this section. A two-mirror resonator is stable if
0 ≤ g1g2 ≤ 1. (4.23)
45 Using these g parameters, and equations we can express positions of the mirrors z1, z2, the Rayleigh range zR, and the beam widths w0, w1 = w(z1), w2 = w(z2) in terms of
them. Explicitly, we have
g2(1 − g1) g1(1 − g2) z1 = −L , z2 = L , (4.24) g1 + g2 − 2g1g2 g1 + g2 − 2g1g2
2 2 g1g2(1 − g1g2) zR = L 2 , (4.25) (g1 + g2 − 2g1g2) s 2 Lλ g1g2(1 − g1g2) w0 = 2 , (4.26) π (g1 + g2 − 2g1g2) r r 2 Lλ g2 2 Lλ g1 w1 = , w2 = (4.27) π g1(1 − g1g2) π g2(1 − g1g2)
(b)Characterization of stationary modes
With the discussion in section 4.2.1, the electric field of the laser with wavelength
λ = 2π/k can also be decomposed in terms of the HG modes
k k i(kz−ωt) Enxny (x, y, z, t) = ˆψnxny (x, y, z)e (4.28)
k where ψnxny is given by equation (4.8).
k In the cavity, the wave vector k of an HG mode ψnxny is quantized by the boundary conditions imposed by the mirrors, and these quantized wave vectors k in turn specify
the frequencies of the modes ω = ck as we argue in the following. A stationary mode
is defined by electric fields such that
E(x, y, z, t) = E(x, y, z + 2L, t). (4.29)
which describes the electric field remains unchanged after a round trip. Equation
(4.29), together with equations(4.28 ,4.8) implies
1 1 ∆φ = k · (2L) − (n + )χ(x) − (n + )χ(y) = 2πq, (4.30) x 2 y 2 (α) (α) (α) χ ≡ 2 χ (z2) − χ (z1) , α = x, y, (4.31)
46 where q is an arbitrary integer that labels the longitudinal mode. χ(α) is called the
round-trip Gouy phase of the resonator, or just Gouy phases whenever we refer
to a resonator.
(x) (y) In equation (4.30), we write the Gouy phases associated with nx, ny as χ , χ
even though they are the same in equations (4.9,4.10). We do this in order to il-
lustrate the separate propagation of the nx and ny modes. With this separation, our
results derived below can also be applied to the general case where the mirrors are
elliptical instead of spherical.
From equation 4.30, we have 1 1 1 k = k = (n + )χ(x) + (n + )χ(y) + 2πq , (4.32) nx,ny,q 2L x 2 y 2 c 1 1 ω = ck = (n + )χ(x) + (n + )χ(y) + 2πq (4.33) nx,ny,q nx,ny,q 2L x 2 y 2
k where c is the speed of light. The stationary modes of the electric fields Enxny (x, y, z, t) in the cavity is specified by the quantum numbers (nx, ny, q) because k is now quan- tized. Using equation (4.19), the round-trip Gouy phases can be expressed in terms of the resonator g parameters: q (α) (α) (α) cos χ /2 = ± g1 g2 , α = x, y (4.34)
where the ± sign follows the sign of g1 (and remember that g1, g2 have the same sign
because the stability condition requires 0 ≤ g1g2 ≤ 1).
Our discussion above also implies that the transverse fields ψnx,ny at the beginning
and the end of a round trip is connected through the round-trip Gouy phases
1 (x) 1 (y) −i[(nx+ 2 )χ +(ny+ 2 )χ ] ψnxny (x, y, z + 2L) = e ψnxny (x, y, z). (4.35)
47 In summary, the stationary mode electric fields are labeled by (nx, ny, q). If we choose the point where the laser is collimated as z = 0, we have
i(knxnyqz−ωnxnyqt) Enxnyq(x, y, z, t) =ψ ˆ nxny (x, y, z) × e (4.36) 1 −i (n + 1 )χ(x)(z)+(n + 1 )χ(y)(z) √ ! √ ! 2 2 e [ x 2 y 2 ] 2x 2y = ˆ Hn Hn p n n x y π 2 x nx!2 y ny!wx(z)wy(z) wx(z) wy(z) x2 y2 x2 y2 i(knxnyqz−ωnxnyqt) exp iknxnyq + − 2 2 × e . 2Rx(z) 2Rx(z) wx(z) wy(z) (4.37) The zeroth transverse mode especially, which we list here for later reference to com- pare, is
s (x) (y) −i[χ (z)+χ (z)] x2 y2 x2 y2 2 e ik 2R (z) + 2R (z) − 2 − 2 k x y wx(z) wy(z) ψ00(x, y, z) = e , (4.38) π wx(z)wy(z)
4.3 Realization of synthetic magnetic fields with the optical resonator
To achieve the synthetic magnetic fields for photons, we introduce image rotation that leads to a Coriolis force which is equivalent to a Lorenz force. This process is more conveniently described by geometric optics, which we shall use in this section.
In geometric optics,
The image rotation can be realized through the non-planar geometry arrangement of mirrors in the resonator, and it manifests itself as abrupt changes of coordinate systems upon reflection on the mirror. The rotation can be directly formalized in terms of of paraxial ray optics [48, 1]. Additionally, the paraxial ray optics is crucial to derive the stationary modes in a general resonator like those in Simon’s experiment
[49].
In this section, we first give an introduction to paraxial ray optics, that is, how a paraxial ray transforms when it propagates in space and optical elements. They are
48 the building blocks ot explain the stationary modes that propagate in a resonator.
Later we examine closely how the non-planar geometry leads to the image rotation in this context. The discussion of the formalisms follows closely with those in the references [48, 49].
4.3.1 paraxial ray optics The formalism
In ray optics, the laser beam is considered as a ray that has no width. Given a stable resonator (described in section 4.2.2), the optical axis of the resonator is a unique path along which the light ray makes a closed path in a single round trip as shown in figure 4.4. Recall previously in a stable two-mirror resonator where the mirrors are facing each other and well-aligned, the optical axis is simply the path that connects the centers of the two mirrors (see figure 4.4a). For a stable four-mirror resonator of a tetrahedral configuration that generates rotation, the optical axis is shown in figure 4.4b. We parametrize the optical axis with z, and for each z we have a transverse plane that is perpendicular to the optical axis. From here it is clear why we call such path the “optical axis” of the resonator. Because in reality, a laser beam is of finite width which can be thought of as a bunch of light rays around the optical axis. We can study the bunch of rays in each transverse plane where every ray corresponds to a point in the transverse plane.
49 a)
b)
Figure 4.4: (4.4a) Optical axis and transverse planes of two-mirror resonator. (4.4b) Optical axis and transverse planes of a twisted four-mirror resonator in tetrahedral configuration.
A ray of the kind we mentioned above is called a paraxial ray in optics. A paraxial T ray in a resonator is represented by a four-dimensional vector x y θx θy .(x, y) specify the transverse positions in the transverse plane, and (θx, θy) are two angles that specify the propagation of the ray in the transverse direction x, y relative to the optical axis. The term paraxial means that the ray is “almost parallel to the optical axis” of the resonator (see 4.5). Hence the paraxial approximation in ray optics is expressed as
θx, θy 1 (4.39) and consequently,
∂x ∂y (θ , θ ) ≈ (tan θ , tan θ ) = ( , ). (4.40) x y x y ∂z ∂z
50 A ray that goes along the optical axis is represented in terms of a paraxial ray vector as 0 0 0 0T .
T Figure 4.5: A paraxial ray is denoted by a four-component ray vector (x, y, θx, θy) . (x, y) denotes the transverse position of the ray displaced from the optical axis, and (θx, θy) denotes the angle or slopes of propagating direction in the transverse plane. This figure shows the ray vector in one dimension only to avoid clutter. i.e. x = 0 r, θx = r .
Paraxial ray optics talks about how the ray transforms when it propagates in space or passes through optical devices. The transformations are linear under paraxial approximation, and thus can be represented by a 4 × 4 real matrix M, called the
T transfer matrix (or ABCD matrix), acting on a 4 × 1 ray column vector (x, y, θx, θy) x0 x y0 y 0 = M . (4.41) θx θx 0 θy θy We use the following notations for the ray vector as well x ! R~ y µ = = , (4.42) θ~ θx θy
51 and R,~ θ~ represent the transverse coordinates and transverse angles of the ray respec- ~ tively. The angles (θx, θy) are related to the transverse momentum as P = (Px,Py) = ~ (kθx, kθy) = kθ. For simplicity of discussion, we adopt the convention that the lon-
gitudinal coordinate z parametrize the path of a ray that defines the optical axis in
the resonator, that is to say, the longitudinal coordinate keeps increasing along the
way regardless of the fact that it bends when it hits a mirror and that it comes back
to the same point after a round trip. It can also be considered that z ≈ ct is a
parametrization of time, and different transverse planes are referred to by z = const.
We also keep in mind the identification that z ∼ z + Lrt because we refer to the same
transverse plane in the resonator with the only difference that we’ve gone through a
round trip that hits every mirror once in the resonator for the latter. In the following,
we introduce the minimal set of transfer matrices we need in order to understand the
resonator in Simon’s experiment.
The propagation of the ray along the longitudinal coordinate by distance z in free
space can be described by
1 z1 M (z) = (4.43) PROP. 0 1
hereafter 1 = 12 unless specified otherwise. The transfer matrix for a spherical mirror
(or a Gaussian thin lens) with normal incidence (i.e. the incident ray parallel to the
normal line of the mirror/lens) is
1 0 MMIRROR(f) = 1 (4.44) − f 1 1
where f = r/2, f is the focal length and r the radius of the spherical mirror. This transformation can be understood by examining the condition that the ray parallel
52 to the optical axis (θ~ = ~0) will pass through the focus, i.e.
1 0 R~ R~ 1 ~ = ~ . (4.45) − f 1 1 0 −R/f
However, to form a non-planar geometry, we unavoidably have non-normal incidence which causes astigmatism. And the transfer matrix is modified as
1 0 0 0 0 1 0 0 MMIRROR(f, Φ) = (4.46) −1/fs(f, Φ) 0 1 0 0 −1/ft(f, Φ) 0 1
where Φ is the incident angle of ray along the optical axis, i.e. the angle between the
optical axis and the normal of the mirror (see figure 4.6,4.7 ). ft(f, Φ) = f cos Φ is
the tangential focal length that lies in the plane of incidence, and fs(f, Φ) = f/ cos Φ
is the sagittal focal length that lies in the plane perpendicular to both the plane
of incidence and the transverse plane. And here we’ve assumed the y transverse
dimension is chosen to be in the plane of incidence. The difference between the focal
length along the tangential and sagittal plane is referred to as astigmatism from non-
normal incidence. Astigmatism means parallel rays don’t focus at the same point and
in general could have many different causes, non-normal incidence is one of them.
53 Figure 4.6: Non-normal incidence of rays shown in mirror reflection. The tangential plane is defined with respect to each incident light ray. For the ray marked by a thinner line, because the tangential plane is define to be the incidence plane, the tangential is exactly the plane where these lines are in. The sagittal plane, on the other hand is defined to be the plane that is perpendicular to both the tangential and the transverse plane. The effective focal length in either plane follow the equations in the figure.
54 Figure 4.7: Astigmatism from non-normal incidence shown in lens refraction. The lens is shown in the figure as the dashed line that ends with hollow triangles. The lens’ axes are along y0 and x, where y0 is slanted from y while x remains parallel to another x. AB and DC are rays that intersect with the y0 axis of the lens and they converge at ft. On the other hand, EF and HG intersect with x axis and converge at fs. The phenomenon that foci ft and fs are not the same point is the astigmatism. The astigmatism in this figure is a consequence of non-normal incidence of the rays in lens refraction. The dashed line N is the normal of the lens and is perpendicular to the axes of the lens y0 and x by definition. The tangential plane is defined to be the plane that the normal line and incident lines lie in, which is ABCD in the figure. The sagittal plane is the plane perpendicular to both the normal line and the tangential line, being EF GH in the figure.
A general property of the transfer matrices that we introduced and will be useful
in later discussion is
M T GM = G (4.47)
where M is any of the transfer matrices we’ve introduced and
0 1 G = . (4.48) −1 0
This property can easily be checked by direct calculation.
55 Also, the resonators we are interested in are stable resonators, and the stabil-
ity is defined in terms of eigenvalues of the roundtrip transfer matrix of the res-
onator, M rt. A resonator is stable if each eigenvalue has complex modulus
one. Given a round trip transfer matrix M rt ∈ M(4, R), suppose it has eigenvalues
∗ ∗ ∗ {m1, m1, m2, m2} and eigenvectors{µ1, µ1, µ2, µ2∗}. For a paraxial ray column vector r0 = (a1µ1 + a2µ2) + c.c., we say the resonator is stable if paraxial rays don’t diverge
in positions or angles after many round trips in the resonator
rt n n n (M ) r0 = (m1 a1µ1 + m2 a2µ2) + c.c. (4.49)
Therefore the eigenvalues can only have unit modulus, or the ray would be magnified
or demagnified.
iχ1 iχ2 m1 = e , m2 = e (4.50)
image rotation from the non-planar geometry
The synthetic magnetic fields in the cavity system come from the image rotation
which is created by the tetrahedral arrangement of mirrors in the system. The analysis
in this section follows those in the supplementary information of the reference [1].
The image rotation is a consequence of
• reflection with non-normal incidence,
• the non-planar arrangement of mirrors inducing a rotation when going from one
plane of incidence to the other,
• the necessary matching of transverse coordinates after a roundtrip is traversed
by the ray along the optical axis.
56 The first results in an inversion in the transverse coordinate that lies in the plane
of incidence only (ˆy, tangential) while it leaves the other coordinate (ˆx, sagittal) invariant (see figure 4.8 ) i.e. a transformation (x, y) 7→ (x, −y). The choice of
coordinate system is defined as the following. As shown in figure 4.8, the unit vectors
that define the propagation of beams, or the optical axis, are
X~ ~zOUT = ~zIN = j+1,j , (4.51) j j+1 ~ |Xj+1,j| ~ ~ ~ where IN/OUT and j marks whether it’s into or out of the mirror j, Xj,k = Xj −Xk, ~ and Xj are the positions of the center of the j-th mirrors. We also define the unit
IN OUT vectors ~xj = ~xj perpendicular to the plane of incidence at mirror j by
X~ × X~ ~zIN × ~zOUT ~xIN = ~xOUT = j,j−1 j+1,j = j j (4.52) j j ~ ~ IN OUT |Xj,j−1 × Xj+1,j| |~zj × ~zj |
The remaining transverse coordinate that lies in the plane of incidence is naturally defined by the right hand rule as
IN/OUT IN/OUT IN/OUT ~yj = ~zj × ~xj (4.53)
With this coordinate system, it’s clear that for non-normal incident light rays parallel to the optical axis, which is the case in collimated laser beams, the reflection results in an inversion in y while leaves the x invariant since we preserve the right hand rule.
This is shown in figure 4.8
57 Figure 4.8: change of the local coordinate system due to reflection of non-normal incidence
Secondly, we notice that the non-planar arrangement of the mirrors rotates the plane of incidence abruptly every time the ray reflects on a mirror. This is illustrated mathematically as the following: though the ray coming out of mirror j is the same as
OUT that going into the mirror j + 1, the former adopts the coordinate system (~x,~y, ~z)j
IN while the later adopts (~x,~y, ~z)j+1. A rotation transformation in x, y is necessary because the next transfer matrix to account for the reflection at mirror j + 1 (see equation 4.46), refers to the local tangential and sagittal planes.
Finally, due to the fact that the optical axis makes a closed loop after hitting each mirror once, it forces any ray parallel to the optical axis to be compared with the original ray before the round trip. Upon matching the coordinate systems at the same transverse plane before and after a round trip, we realized that the rotation of image after each round trip cannot be just the choice of coordinate systems, and is
58 necessary so long as there’s no accidental degeneracy in the rotation angle (i.e. overall
not by multiples of 2π.)
The effect of the rotation in between mirror j and j + 1 can be accounted for by
the following transfer matrix:
Rj 0 MROTj = (4.54) 0 Rj where the 2 × 2 matrix Rj is given by
OUT IN OUT IN ~xj · ~xj+1 ~yj · ~xj+1 Rj = OUT IN OUT IN (4.55) ~xj · ~yj+1 ~yj · ~yj+1
Take the tetrahedral non-planar resonator in Simon’s experiment for example, and
we choose the midway between mirror 4 and 1 to be the origin. The round trip matrix
can be written as
4 rt Y ~ ~ M = MPROP.(|Xj+1,j|/2) · MROTj · MMIRRORj · MPROP.(|Xj,j−1|/2) (4.56) j=1
4.3.2 paraxial wave optics
Now we turn back to the wave picture of the laser beams in the resonator. The
main purpose to introduce paraxial wave optics is to obtain the stationary modes
rt ψn1n2 in a general resonator from only the roundtrip transfer matrix M , and the
stationary modes ψn1n2 in a twisted resonator with the tetrahedral configuration are
the counterpart of single particle wave functions of the Landau levels. Alternatively,
we could have solved the paraxial wave equations with proper boundary conditions
but it is mathematically highly complicated, if even tractable. The paraxial wave
optics provides a practical and systematic methods to solve for the stationary modes
of the resonator.
59 The formalism
Recall that the object of interest in paraxial wave is the transverse field ψ which is characterized by equations 4.2 and 4.6. Let us introduce the Dirac notation of the transverse field in this section ψ(x, y, z) ≡ hx, y|ψ(z)i where (x,y) is the transverse
coordinates. According to the paraxial equation (4.6), the free space propagation of
ψ from one transverse plane to another in Dirac notation is given by
ˆ − i Pˆ2z |ψ(z)i = UPROP (z) |ψ(0)i = e 2k |ψ(0)i (4.57)
where we define the momentum operators in the transverse directions to be
ˆ P = (ˆpx, pˆy) = (−i∂x, −i∂y) (4.58)
and they are proportional to the transverse angles in paraxial optics
Pˆ −i∂ −i∂ Θˆ = (θˆ , θˆ ) = = ( x , y ) (4.59) x y k k k
We automatically have the canonical commutation relation in the transverse dimen-
sions
i [ˆx , pˆ ] = iδ , [ˆx , θˆ ] = δ (4.60) i j ij i j k ij
Besides the free space propagation, the unitary operator for the effect of mirrors or lenses is
ˆ −ik RFˆ −1Rˆ |ψOUT (z)i = UMIRROR(F ) |ψIN (z)i = e 2 |ψIN (z)i (4.61)
where F is in general a 2 × 2 matrix that encodes the orientation and focal lengths
of the mirror or lens (see the bottom left block of equation 4.46). F is diagonal when
the mirror is perfectly spherical, while off diagonal elements could emerge to account
60 for astigmatism from asymmetry. We can convince ourselves by applying the unitary operator on a momentum eigenstate eiP~ ·R~ , and we see that this operator transform the transverse momentum in the following manner:
−ik RFˆ −1Rˆ iP~ ·R~ −1 −ik RFˆ −1Rˆ iP~ ·R~ Pˆ e 2 e = (P~ − kF R~) e 2 e (4.62) which agrees with the effect of mirror in ray optics in (4.44). (Recall that ~p = kθ~)
Given this unitary operator representation of these optical elements, we can represent the evolution of the transverse field in Dirac notation:
|ψ(z)i = Uˆ(0, z) |ψ(0)i (4.63) where Uˆ(0, z) is the product of a sequence of unitary operators Uˆ for each optical element in between (0, z).
Finally, we would like to introduce the analogy of “Heisenberg picture” for the paraxial waves because it is essential for solving the stationary modes with operator Rˆ(z) method. We define the z-dependent operators in transverse plane z as Θ(ˆ z)
Rˆ(z) Rˆ = Uˆ †(0, z) Uˆ(0, z) (4.64) Θ(ˆ z) Θˆ such that
Rˆ(z) Rˆ hψ(0)| |ψ(0)i = hψ(0)| Uˆ †(0, z) Uˆ(0, z) |ψ(0)i Θ(ˆ z) Θˆ Rˆ = hψ(z)| |ψ(z)i (4.65) Θˆ
Last but not least, we require the expectation values of the operators (Rˆ Θ)ˆ = ˆ ˆ (ˆx yˆ θx θy) transform as in ray optics for consistency
hRˆ(z)i hRˆi = M(0, z) (4.66) hΘ(ˆ z)i hΘˆ i
61 Therefore we have
Rˆ(z) Rˆ Rˆ = Uˆ †(0, z) Uˆ(0, z) = M(0, z) (4.67) Θ(ˆ z) Θˆ Θˆ
This equation defines the counterpart of time-dependent operators in Heisenberg pic- ture in paraxial wave optics and integrate in the consistency requirement in paraxial optics.
Stationary modes calculated with operator method: the harmonic oscilla- tor
As have been explained in section 4.2.2, the HG modes are the stationary modes for a stable two-spherical-mirror resonators.
The fact that HG modes can be the stationary modes for the two-mirror resonator suggests that the underlying Hamiltonian describes a harmonic oscillator. It should be noted that the interpretation is every transverse plane has a copy of stationary harmonic oscillator of different trapping frequencies that are dependent on the beam spot size w(z) in (4.18), and the stationary modes in each transverse plane should be understood in terms of the stroboscopic dynamics because it takes a finite round-trip distance for the EM wave to reproduce itself. Aside from the approach of solving the
Maxwell’s equation with the boundary conditions to find the stationary modes, we can also find the ladder operators to create HG modes of every order algebraically[50, 49]. r r k pˆ k aˆ (z) = K (z)ˆx + iB (z) x = K (z)ˆx + iB (z)θˆ (4.68) x 2 x x k 2 x x x r r k pˆ k aˆ (z) = K (z)ˆy + iB (z) y = K (z)ˆy + iB (z)θˆ . (4.69) y 2 y y k 2 y y y
ˆ pˆi −i ∂ where θi = = . Note that our ladder operators are defined for each transverse k k ∂i plane at z, which means we have a copy of harmonic oscillator in every transverse
62 plane. This fact has been seen in equations (4.8, 4.9, 4.10) where the HG modes have z -dependent spot sizes w(z) and radii of curvatures R(z). The ladder operators creates photons that obey bosonic commutation relations
† [ˆai(z), aˆj(z)] = δij, i, j = x, y (4.70) which turns out imposing constraints on the coefficients B’s and K’s (we omit the z parameter henceforth to avoid clutter in notation)
∗ ∗ KxBx + BxKx = 2 (4.71)
∗ ∗ KyBy + ByKy = 2 (4.72)
∗ ∗ ∗ ∗ KxBy + BxKy = KyBx + ByKx = 0 (4.73)
Note that the coefficients K,B’s are complex in general as apposed to the quantum harmonic oscillator that we’re familiar with. This has been seen in the nontrivial phases in the HG modes in equations (4.8, 4.9, 4.10).
We obtain the higher order HG modes by acting the raising operators multiple times on the zeroth modes
[ˆa† (z)]nx [ˆa† (z)]ny |ψ (z)i = x y |ψ (z)i (4.74) nxny p 00 nx! ny! where the zeroth mode |ψ00(z)i is defined as
aˆx(z) |ψ00(z)i =a ˆy(z) |ψ00(z)i = 0. (4.75)
The zeroth mode in coordinate basis is
k Kx 2 Ky 2 − 2 B x + B y hx, y|ψ00(z)i = ψ00(x, y, z) = N00 e x y . (4.76)
63 where the normalization constant N00 is determined as follows Z hψ00|ψ00i = hψ00(z)|x, yi hx, y|ψ00(z)i dxdy (4.77) Z ∗ = ψ00(x, y, z)ψ00(x, y, z)dxdy = 1 (4.78)
for all value of z in the sense that the mode is normalized in every transverse plane.
The normalization constant can be calculated and further simplified with the help of
the constraints on B and K’s from (4.71), (4.72):
∗ K∗ k Kx Kx 2 Ky y 2 Z − + ∗ x + + ∗ y 2 2 Bx Bx By By hψ00|ψ00i = |N00| e dxdy (4.79)
Z − k 2 x2+ 2 y2 2 2 |Bx|2 |By|2 = |N00| e dxdy (4.80) r π2|B |2|B |2 = |N |2 x y = 1. (4.81) 00 k2
Hence, we found the normalized zeroth mode in the transverse coordinate basis
s k k Kx 2 Ky 2 − 2 B x + B y ψ00(x, y, z) = e x y (4.82) πBxBy
Up to this point, we have the expressions for the stationary modes once we are given
the coefficients B’s and K’s. However, we have not talked about how to find the
coefficients B’s and K’s. The reason is that we have not used any assumptions about
the stationary modes, and all we did was simply solving the modes based on the
general ladder operators for HG modes (4.68), (4.69). To determine the coefficients
B’s and K’s, we need to take into account the periodic conditions that define the
stationary modes.
A stationary mode of a resonator is defined such that the paraxial EM field is
identical after a round trip
k k E (x, y, z, t) = E (x, y, z + Lrt, t). (4.83)
64 where Lrt is the round-trip path length of the ray in the resonator. Recall the defini- tion of transverse field ψ in (4.2)
Ek(x, y, z, t) = ˆψk(x, y, z)ei(kz−ωt), (4.84) this allows for a phase change in the transverse field ψ plus a constraint on the total phase change of the electric field E is multiples of 2π
iα |ψ(z + Lrt)i = e |ψ(z)i (4.85)
kLrt + α = 2πl (4.86) for any stationary mode |ψi and some phase eiα to be determined.
First, let us compare the zeroth mode ψ00 obtained in two different approaches
(4.38),(4.82) and discuss the necessary conditions for consistency. As
s k k Kx(z) 2 Ky(z) 2 − 2 B (z) x + B (z) y ψ00(x, y, z) = e x y (4.87) πBx(z)By(z)
s 2 2 2 2 −i[χx(z)+χy(z)] x y x y 2 e ik 2R (z) + 2R (z) − 2 − 2 = e x y wx(z) wy(z) , (4.88) π wx(z)wy(z) we compare the two expressions. The exponents that are quadratic in x, y should be identical; hence we can express the ratios Kx and Ky in terms of the beam spot sizes Bx By wi(z) and the radii of curvature of the wavefront Ri(z) " # Ki 2 Re = 2 (4.89) Bi (z) kwi (z) " # K 1 Im i = − i = x, y (4.90) Bi (z) Ri(z) where we use the definition of Rayleigh range (4.16) and spot size (4.18) in the first equation. We also restore the z dependence in K’s and B’s to remind the reader.
Ki The above suggests that the ratio’s only depend on the spot sizes wi(z) and the Bi 65 radii of curvature Ri(z) of the transverse field. Since the modes are stationary after each round trip, the spot sizes and radii of curvature stay invariant:
wi(z + Lrt) = wi(z) (4.91)
Ri(z + Lrt) = Ri(z). (4.92)
This leads to the first necessary condition about the coefficients K’s and B’s that the
ratios
K (z + L ) K (z) i rt = i (4.93) Bi(z + Lrt) Bi(z)
are invariant under round-trip propagation.
By comparing the parts in the square root, we also realize that the Bi’s change up to a phase after each round trip. i.e.
s k k Kx(z) 2 Ky(z) 2 − 2 B (z) x + B (z) y ψ00(x, y, z + Lrt) = e x y (4.94) πBx(z + Lrt)By(z + Lrt)
s 2 2 2 2 −i[χx(z+Lrt)+χy(z+Lrt)] x y x y 2 e ik 2R (z) + 2R (z) − 2 − 2 = e x y wx(z) wy(z) , (4.95) π wx(z)wy(z)
and
iα Bi(z + Lrt) = e Bi(z) (4.96)
for some phase eiα to be determined.
To figure out the extra phase the stationary modes gain after each round-trip
as shown in (4.85), let us first consider how the ladder operator transform with a
round-trip propagation. Consider the |ψ01i after one round trip,
66 † |ψ10(Lrt)i = U(Lrt) |ψ10(0)i = U(Lrt)ax(0) |ψ00(0)i (4.97)
On the other hand, by definition,
† † |ψ10(Lrt)i = ax(Lrt) |ψ00(Lrt)i = ax(Lrt)U(Lrt) |ψ00(0)i , (4.98)
where the unitary propagation U(z) has been defined in 4.3.2.
From the above, we see that
† † † ai (Lrt) = U(Lrt)ai (0)U (Lrt), (4.99)
† ai(Lrt) = U(Lrt)ai(0)U (Lrt), i = x, y, (4.100)
where the transformation of the lowering operator can be obtained by applying the
same argument. We can now insert the definition of the ladder operators (4.68,4.69)
to see how the coefficients B’s and K’s transform. It is convenient to introduce the
following expressions to relate back to the consistency of paraxial ray optics. r k aˆ (z) = K (z)ˆx + 0y ˆ + iB (z)θˆ + 0 θˆ (4.101) x 2 x x x y r k Rˆ = i B (z) 0 iK (z) 0 G (4.102) 2 x x Θˆ r k Rˆ = i [µ (z)]T G , (4.103) 2 x Θˆ and similarly, r k aˆ (z) = 0x ˆ + K (z)ˆy + 0 θˆ + iB (z)θˆ . (4.104) y 2 y x y y r k Rˆ = i 0 B (z) 0 iK (z) G (4.105) 2 y y Θˆ r k Rˆ = i [µ (z)]T G , (4.106) 2 y Θˆ
67 where
T µx = Bx(z) 0 iKx(z) 0 (4.107)
T µy = 0 By(z) 0 iKy(z) . (4.108)
Expanding the RHS of (4.100) as an example, we get r k Rˆ aˆ (L ) = i [µ (0)]T GUˆ(L ) Uˆ †(L ) (4.109) j rt 2 j rt Θˆ rt r k Rˆ = i [µ (0)]T G(M rt)−1 (4.110) 2 j Θˆ r k Rˆ = i [µ (0)]T (M rt)T G (4.111) 2 j Θˆ r k Rˆ = i [M rtµ (0)]T G (4.112) 2 j Θˆ stationary modes calculation with operator methods: a rotating harmonic oscillator
The coefficients B’s and K’s can be determined by the round-trip transfer matrix of the resonator. We show the general formalism to obtain the ladder operators and it specializes to both the resonators for the harmonic oscillator and that for the Landau levels. For a general resonator with the underlying Hamiltonian being quadratic in ~ ~ ˆ ˆ positions R = (ˆx, yˆ) and momenta P = (ˆpx, pˆy) = k(θx, θy). The ladder operators are linear combinations of positions and momenta and are of the following form: r k aˆ (z) = K(1)(z)ˆx + K(1)(z)ˆy + iB(1)(z)θˆ + iB(1)(z)θˆ (4.113) 1 2 x y x x y y r k aˆ (z) = K(2)(z)ˆx + K(2)(z)ˆy + iB(2)(z)θˆ + iB(2)(z)θˆ (4.114) 2 2 x y x x y y
68 The ladder operators can be reorganized into the following expressions for the conve- nience of calculation r k Rˆ aˆ (z) = K(j)(z) K(j)(z) iB(j)(z) iB(j)(z) j 2 x y x y Θˆ r k Rˆ = i [µ (z)]T G 2 j Θˆ r k Rˆ = i B(j)(z) B(j)(z) iK(j)(z) iK(j)(z) G (4.115) 2 x y x y Θˆ
0 1 where G = was defined in equation (4.48), and −1 0
(j) (j) (j) (j) µj(z) = Bx (z) By (z) iKx (z) iKy (z) (4.116) is introduced to simplify the notation. Also, the ladder operators must obey the bosonic commutation relation
† [ˆai(z), aˆj(z)] = δij (4.117)
This is equivalent to a normalization condition on µj(z)
∗ µi(z) (−iG)µj(z) = 2δij (4.118) or explicitly,
(i)∗ (j) (i)∗ (j) (i)∗ (j) (i)∗ (j) Kx Bx + Ky By + Bx Kx + By Ky = 2δij (4.119) where the z dependence in each K,B’s are omitted to avoid clutter.
The consistency condition across different transverse planes requires that
ˆ aˆj(z) |ψ(z)i = U(0, z) [ˆaj(0) |ψ(0)i] (4.120) or equivalently,
ˆ ˆ † aˆj(z) = U(0, z)ˆaj(0)U (0, z) (4.121)
69 Substituting this into (4.115), we have r k Rˆ aˆ (z) = i [µ (0)]T GUˆ(0, z) Uˆ †(0, z) (4.122) j 2 j Θˆ r k Rˆ = i [µ (0)]T GM(0, z)−1 (4.123) 2 j Θˆ r r k Rˆ k Rˆ = i [µ (0)]T M(0, z)T G = i [M(0, z)µ (0)]T G (4.124) 2 j Θˆ 2 j Θˆ where we used (4.67) for the second equality, and (4.47) for the third. This equation implies that the coefficients for the ladder operators at different transverse plan, µ(z) and its complex conjugate µ(z)∗ transform as the paraxial ray vector.
µi(z) = M(0, z)µi(0) (4.125)
Now we can see how the condition for stationary modes of the resonator translate in this formalism. Recall that a stationary mode is defined such that the paraxial
EM field is identical after a round trip
k k E (x, y, z, t) = E (x, y, z + Lrt, t). (4.126)
where Lrt is the round trip path length of the ray in the resonator. This amounts to a phase change in the paraxial transverse field ψ and a constraint on k that turns out to be a quantization condition later
iα |ψ(z + Lrt)i = e |ψ(z)i (4.127)
kLrt − α = 2πq (4.128)
For the ease of the notation, we denote the state |ψn1,n2 (z)i ≡ |n1, n2, zi Most gener-
iα ally we assume the round-trip phases e n1n2 are dependent on the mode numbers
iαn n |n1, n2, z + Lrti = e 1 2 |n1, n2, zi (4.129)
70 First consider raising the mode |n1, n2,Lrti
† √ a1(Lrt) |n1, n2,Lrti = n1 + 1 |n1, n2,Lrti (4.130) √ iαn +1,n = n1 + 1 e 1 2 |n1, n2, 0i (4.131) √ a†(0) iαn +1,n 1 = n1 + 1 e 1 2 √ |n1, n2, 0i (4.132) n1 + 1 † iαn1+1,n2 = e a1(0) |n1, n2, 0i . (4.133)
On the other hand, from (4.129), we have
† † iαn1n2 a1(Lrt) |n1, n2,Lrti = a1(Lrt)e |n1, n2, 0i . (4.134)
Combining the above two equations and remembering that the z dependence of the ladder operators comes totally from µi(z) as explicitly shown in equation (4.124), we arrive at
† † i(αn1+1,n2 −αn1n2 ) a1(Lrt) = e a1(0) (4.135)
Making use of equation 4.124 and the above, we have
rt ∗ i(αn1+1,n2 −αn1n2 ) ∗ M µ1(0) = e µ1(0) (4.136)
The above equation is true for all n1. It suggests that
∗ (j)∗ (j)∗ (j)∗ (j)∗ µ1(0) = Bx (0) By (0) −iKx (0) −iKy (0) (4.137)
rt is an eigenvector ( and so is µ1(0)) of the real matrix M = M(0,Lrt) with the corresponding eigenvalue
−iχ i(α −α ) i(α −α ) e 1 ≡ e n1+1,n2 n1,n2 = ··· = e 10 00 (4.138)
71 † The same argument applies to a2 as well, and combined we have
αnm = α00 − nχ1 − mχ2 (4.139)
In fact, the coefficients for the ladder operators µi of the stationary modes trans- form as the paraxial ray vectors (x y θx θy)[49, 50]. To summarize, the relation between the coefficients and the transfer matrix is the following: the coefficients
µi for the lowering operator aˆi as in equation (4.115) are the eigenvectors of the transfer matrix of the resonator with eigenvalues e−iχi
rt −iχi rt ∗ iχi ∗ M µi = e µi,M µi = e µi (4.140)
where χi are the round-trip Gouy phases that satisfy
−i[(n1+1/2)χ1+(n2+1/2)χ2 |n, m, Lrti = e ] |n1, n2, 0i (4.141)
and µi are normalized with the condition
T ∗ µi(z) (iG)µj(z) = 2δij (4.142) that is equivalent to the canonical commutation relation of the ladder operators. the zeroth mode and higher order modes
To find α00, we solve the zeroth mode. Consider the following operator: r (1) (1) (1) (1) ! ˆ ˆ aˆ1(z) k Bx (z) By (z) iKx (z) iKy (z) R A(z) = = i (2) (2) (2) (2) G ˆ (4.143) aˆ2(z) 2 Bx (z) By (z) iKx (z) iKy (z) Θ r k ∇ = (KRˆ + B ) (4.144) 2 k where
(1) (1) ! (1) (1) ! ∂x Bx (z) By (z) Kx (z) Ky (z) ∇ = , B(z) = (2) (2) , and K(z) = (2) (2) ∂y Bx (z) By (z) Kx (z) Ky (z) (4.145)
72 ˆ The zeroth mode has to satisfy A(z) |ψ00(z)i = 0. We can solve the zeroth mode in
the coordinate basis and we get s k − k rT [B(z)−1K(z)]r hx, y|ψ (z)i = ψ (x, y, z) = e 2 (4.146) 00 00 π det[B(z)]
where rT = (x, y). Now recall equation (4.129) and that
(1) (1) ! (1) (1) ! eiχ1 B (0) eiχ1 B (0) eiχ1 K (0) eiχ1 K (0) B(L ) = x y , and K(L ) = x y rt iχ (2) iχ (2) rt iχ (2) iχ (2) e 2 Bx (0) e 2 By (0) e 2 Kx (0) e 2 Ky (0) (4.147)
from the eigenvalue equations. We obtain
χ + χ α = − 1 2 (4.148) 00 2 1 1 α = −[(n + )χ + (n + )χ ] (4.149) n1n2 1 2 1 2 2 2
Here we summarize the condition of the stationary modes for a general resonator in
terms of the transverse fields
1 1 −i[(n1+ 2 )χ1+(n2+ 2 )χ2] |ψn1n2 (Lrt)i = e |ψn1n2 (0)i (4.150) 1 1 kL + (n + )χ + (n + )χ = 2πq (4.151) rt 1 2 1 2 2 2
And we recognize that in equation (4.150), χ1, χ2 is actually the round trip Gouy
phase in the same spirit we defined in equation (4.129). And again we see that the
longitudinal wave number is quantized and given by
1 1 1 kn1,n2,q = (n1 + )χ1 + (n2 + )χ2 + 2πq (4.152) Lrt 2 2
where n1, n2 are the quantum number for the transverse modes and q is that for the longitudinal modes. And the frequency of the modes can be immediately obtained
73 by ωn1n2q = ckn1n2q where c is the speed of light,
c 1 1 ωn1,n2,q = (n1 + )χ1 + (n2 + )χ2 + 2πq (4.153) Lrt 2 2 1 1 = ν (n + )χ + (n + )χ + 2πq . (4.154) FSR 1 2 1 2 2 2
The term c/Lrt is the frequency spacing of the longitudinal modes. It is referred to as free spectral range
c νFSR = (4.155) Lrt in the context of quantum optics as well as in reference [1]. This definition will be used frequently in the discussion in chapter 5.
We calculate a few lowest order stationary modes of the cavity with parameters reported in Simon’s experiment [1]. They are cyclotron orbits in traditional Landau levels and they show agreement with those in Simon’s paper.
74 a) ψ00 b) ψ10
c) ψ20 d) ψ11
Figure 4.9: The stationary modes of the cavity calculated by the operator method, with cavity parameters reported by Simon’s experiment [1]. The the on-axis length ◦ La = 1.816 cm, opening angle θ = 16 , and the radii of curvature of the four mirrors R = (2.5, 5, 5, 2.5) cm
4.3.3 Summary
We now summarize the algorithm to use paraxial photons to simulate massive particles in a rotating harmonic trap where the Landau levels are labeled by
ωn1,n2 = n1(ω + Ω) + n2(ω − Ω), (4.156)
75 where ω and Ω are the frequency of the harmonic trap and the rotation respectively, and n1, n2 are non-negative integers. The algorithm consists of a sequence of steps.
1. To create a four-mirror cavity in tetrahedral configuration with the light ray of
the stationary modes shown as in figure 4.4b
2. The electric field of the stationary modes in this resonator is
i(kn1n2qz−ωn1n2qt) En1n2q(x, y, z, t) = ˆψn1n2 (x, y, z)e (4.157)
where n1, n2 are non-negative integers labeling the order of the transverse mode,
and q is an integer that labels the longitudinal mode. ψn1n2 (x, y, z) can be
† † generated from ψ00 by operators a1(z), a2(z) as defined in equations (4.143).
† † The summary of the algorithm to determine a1(z), a2(z), a1(z), a2(z) will be given below.
3. The wave vector kn1,n2,q and the frequency ωn,n2,q = ckn1,n2,q are given by
1 1 1 kn1,n2,q = (n1 + )χ1 + (n2 + )χ2 + 2πq , (4.158) Lrt 2 2 c 1 1 ωn1,n2,q = (n1 + )χ1 + (n2 + )χ2 + 2πq (4.159) Lrt 2 2
~ ~ ~ ~ ~ ~ ~ ~ where Lrt = |X2 − X1| + |X3 − X2| + |X4 − X3| + |X1 − X4| as in figure 4.4b is
the length of the round-trip path, and χ1, χ2 are the round-trip Gouy phases of
the resonator. The algorithm for determining these phases are also summarized
below.
† † 4. The algorithm for determining the operators a1(z), a2(z), a1(z), a2(z) and the
Gouy phases χ1, χ2 is as follows.
76 4.1. First we construct the round-trip matrix M rt(~r) at a point ~r on the “optical
axis” (defined in section 4.3.1). Here ~r specifies the starting point of the
roundtrip matrix M rt. The matrix M rt(~r) is a 4 × 4 matrix acting on T the ray vector µ = x y θx θy that describes the transverse positions
and directions relative to “the optical axis” of the ray in the paraxial ray
optics. The matrix M rt(~r) is composed of a product of transfer matrices
MPROP.(z2 − z1),MMIRRORj and MROTj , and it represents the propagation
of paraxial rays in a round trip. The definition and meaning of each transfer
matrix listed above were shown in section 4.3.1. For example, in figure
4.4b, the propagation of paraxial ray from z = 0 to z = z3 is
M(z3, 0) = MPROP.(z3 − zM2 ) MMIRROR2 MROT2 MPROP.(zM2 − 0). (4.160)
The round-trip matrix M rt(~r) is constructed in the same manner with a
single round trip path that starts and ends at ~r as we have described in
equation (4.56) where we have chosen ~r at the midpoint of mirror 1 and
mirror 2.
4 rt Y ~ ~ M (~r) = MPROP.(|Xj+1,j|/2) · MROTj · MMIRRORj · MPROP.(|Xj,j−1|/2) j=1 (4.161)
4.2. Once M rt(~r) is constructed, we find its eigenvalues and eigenvectors with
equation
M(~r)rtµ = λµ (4.162)
Because the resonator is stable and M rt is a real matrix, the eigenval-
ues of M rt must have unit modulus and must come in conjugates. i.e.
77 the eigenvalues are eiχ1 , e−iχ1 , eiχ2 , e−iχ2 with corresponding eigenvectors
∗ ∗ µ1, µ1, µ2, µ2. When normalization conditions T ∗ 0 12 µi (iG)µj = 2δij, where G = (4.163) −12 0
0 are imposed on these eigenvectors µis, µi can be used to determine both
† † the Gouy phases χ1, χ2 and the operators a1(0), a1(0) , a2(0), a2(0) at ~r.
The importance of the normalization in equation (4.163) is two-fold. It eliminates the ambiguity between χ and 2π − χ in the Gouy phases and
† ensures the canonical commutation relations [ai, aj] = δij. (a)The Gouy phases: suppose one of the solutions to equation (4.162) is λ = eiχ and µ. Because its conjugate λ∗ = e−iχ and µ∗ is also a solu- tion, it seems unclear whether the Gouy phase for this pair of solution is
χ or 2π − χ. (Without loss of generality, we choose the range of the Gouy phases to be [0, 2π]. ) However, this ambiguity is eliminated by the nor- malization condition in equation (4.163) because only the correct λ, µ that corresponds to the Gouy phase can lead to a positive “norm” defined by equation (4.163). i.e. if χ is the Gouy phase, its corresponding eigenvector
µ would satisfy µT iGµ∗ = 2, and the conjugate solution 2π − χ, µ∗ would satisfy (µ∗)T iGµ = −2.
(b)The ladder operators: Once the Gouy phases χ1, χ2 are determined from above, the ladder operators can be determined merely by the corre- sponding normalized eigenvectors. Suppose one of the Gouy phase is χ1
(1) (1) (1) (1) and its corresponding eigenvector is µ1 = Bx By iKx iKy . The
78 annihilation operator is given by
x r k (1) (1) (1) (1) y a1(0) = i Bx By iKx iKy G (4.164) 2 θx θy x r k T y = i µ1 G . (4.165) 2 θx θy
a2(0) is defined in the same manner. The creation operators are the Hermitian
conjugate respectively.
5. The stationary transverse modes at ~r can now be generated by acting the cre-
ation operators repeatedly on the zero transverse mode |ψ00(0)i that is defined
by a1(0) |ψ00(0)i = a2(0) |ψ00(0)i = 0. A stationary transverse mode of order
(n1, n2) is given by
† nx † ny [a1(0)] [a2(0)] |ψn1n2 (0)i = √ |ψ00(0)i . (4.166) n1! n2!
79 Chapter 5: Generalized synthetic Landau levels with photons
5.1 Introduction
In the previous chapter, we have shown that the frequency of stationary modes in the four-mirror resonator is
1 1 E = ν (n + )χ + (n + )χ + 2πq (5.1) n1,n2,q FSR 1 2 1 2 2 2 where χ1 and χ2 are the Gouy phases that can be adjusted by varying the parameters of the resonator, and νFSR = c/Lrt is the free spectral range defined before. (For the convenience of comparison, we take ~ = 1 and denote frequencies of the stationary modes as E in this chapter.) This is already of the form of a massive particle in a rotating trap
En1,n2 = n1(ωT − Ω) + n2(ωT + Ω) + const. (5.2)
To achieve a completely flat Landau level as in the case of electrons in two-dimensional space, Simon et. al use the following trick: making χ1/2π rational and χ2/2π irra- tional. The latter was not emphasized in reference [1], but it is crucial for the con- struction. For otherwise, we may have some other intricate level structure, which we
2π shall discuss later in this chapter. Particularly, they consider the case χ1 = 3 . In
80 this case, equation (5.1) becomes
h n i E = ν ( 1 + q)2π + n χ + const. (5.3) n1,n2,q FSR 3 2 2
The index-independent constant term will be ignored from now on when similar fre- quency equations are shown. For each n2, there is a family of degeneracy states with
fixed n1/3 + q . Despite the simplicity of equation (5.3), its level structure is in fact quite subtle and is not discussed in reference [1]. In this chapter, we shall first dis- cuss this structure. After this discussion, we shall extend this scheme to attempt to simulate the case of negative curvature space by pulling negative angular momentum states in to the degenerate level. However, we will see that our scheme in fact lead to an interesting case that a lot more states are pulled in inevitably during this process.
We examine the wave functions of the states we pulled in and realize that the Landau levels on a two-dimensional cone with positive curvature are all flattened down into a degenerate manifold. This degeneracy is far greater than the usual Landau levels and is a remarkable regime for future study of many-body physics in this setting.
5.2 Simulating Landau levels on a cone with three-fold sym- metry
5.2.1 The energy levels
By choosing χ1/2π = 1/3, and χ2/2π irrational, the frequency levels are given in equation (5.3). The choice of 2π/3 is for experimental convenience [1]. In principle, it can be any rational ratio. For a given n2, there are three families of degenerate
81 levels
∗ ∗ (n1, q) = (3k, q − k), at E = νFSR [2πq + χ2n2] , (5.4) 1 (n , q) = (3k + 1, q∗ − k), at E = ν 2π(q∗ + ) + χ n , (5.5) 1 FSR 3 2 2 2 (n , q) = (3k + 2, q∗ − k), at E = ν 2π(q∗ + ) + χ n . (5.6) 1 FSR 3 2 2 where k = 0, 1, 2, . . . q∗ and q∗ is some integer. Let us first focus on one family. In equation (5.4) q∗ is related to the energy as
∗ ∗ En2,n1 (q ) = νFSR(2πq + χ2n2) (5.7)
∗ ∗ with degeneracy q + 1 given by different integer values of n1 satisfying n1/3 + q = q .
Here we intentionally denote the energy indices in such order for a reason that will come clear soon.
This is the analog of the usual Landau levels of two-dimensional electrons.
En,m = ωc · n, (5.8)
where ωc is the cyclotron frequency, n is the Landau level index, and l is the degen- erate quantum number which is the angular momentum index associated with the coordinate z = x + iy. The quantum number n1 in equation (5.7) is the analog of l in equation (5.8). These are the degeneracy of the level as they do not appear in the energy equations (5.7), (5.8).
In figure 5.1, we show the level structure of equation (5.4) with n2 = 0. (This inverted triangular tower is a set of levels that we later will refer to as a Q-tower.)
As the energy (or q∗) increases, the degeneracy increases. For very large q∗, ( say q∗ = 104), the level structure approaches to a bulk system. In reference [1], the authors operate at the frequency associated with a fixed large q∗, and is therefore
82 looking at a small subset of levels of this tower. There are also other level structures
∗ ∗ correspond to (n1, q) = (3k + 1, q − k) and (n1, q) = (3k + 2, q − k) as shown in equations (5.5), (5.6).
Figure 5.1: The Q-tower with bottom state (n1, n2, q) = (0, 0, 0). In the figure the states are labeled with (n1, q). This family of degeneracy levels occur in the resonator when one Gouy phase χ is tuned to exactly 2π/3.
The entire level structure for n2 = 0 therefore includes three sets that are shown in figure 5.2. Note that when these three sets of level are put together, all the levels are equally spaced.
83 Figure 5.2: The energy level structure with different i = n1 mod 3. For each given n2, all states with n2 are listed in these three copies of the Q-tower. Also note that these three Q-towers stagger in a way that all degenerate levels in the figure are equally spaced because χ1 = 2π/3.
Still, we have only considered the level structure for a fixed n2, (which is the analog of the higher Landau level index.) A different n2 will have another similar tower. It will be too cumbersome to display the level structure including n2. However, it is
∗ useful to display it in the neighborhood of the degenerate level E = νFSR2πq (see
figure 5.3).
84 Figure 5.3: Level structure with different n2 assuming χ1 < χ2. Note that the whole picture is still three copies of Q-towers indicated with similar color codes as in 5.2. We show here only n2 = 0, 1, 2. But there is no constraint on the value of n2. Also our figure shows only the case when χ1/2π χ2/2π for simplicity. Depending on the values of the Gouy phases, the levels in the figure may stagger.
5.2.2 Wave functions on a cone
We now discuss the wave functions of these levels. Before proceeding, let us recall the Landau level wave functions of the usual two-dimensional electron gas (planar).
∗ The wave function Ψn,l(z, z ) with energy En,l = ωcn.
l Ψ0,l ∝ z , l = 0, 1,... (5.9)
l l+1 Ψ1,l ∝ z L1(¯zz) ∼ zz¯ , l = −1, 0, 1,... (5.10)
l n l+n Ψn,l ∝ z Ln(¯zz) ∼ z¯ z l = −n, −n + 1,..., 0, 1,...... (5.11)
85 wherez ¯ is the complex conjugate of z, and Ln(z) is a polynomial of degree n. We shall only pay attention to the leading power as it is the term that controls the maximum angular momentum of the state.
Now we look at the wave functions of photonic states (i.e. the transverse stationary
∗ modes profile ψn1,n2 introduced in section 4.3.2) with energy En2,n1 (q ). As we have discussed, there are three families with energies
∗ En2,3k+i(q ), i = 0, 1, 2. (5.12)
The wave function ψn1,n2 (z, z¯) to the leading power is
n1 n2 ψn1,n2 (z, z¯) ∝ z z¯ . (5.13)
This can be obtained by combining the transformation (3.44) and the σ = 1 solutions
∗ in equation (2.6). We denote the wave function of the state (n1 = 3k + i, n2, q ) as
ψn2,n1=3k+i(z, z¯). To the leading power term, they are
3k+i ∗ ψn2=0,n1=3k+i ∝ z , k = 0, 1, 2, . . . , q (5.14)
3k+i ∗ ψn2=1,n1=3k+i ∝ z¯ · z , k = 0, 1, 2, . . . , q (5.15)
2 3k+i ∗ ψn2=2,n1=3k+i ∝ z¯ · z , k = 0, 1, 2, . . . , q (5.16)
5.2.3 Detection of the level structure
The level structure of the stationary modes in equation (5.3) can be detected experimentally by the transmission spectrum of these modes. The result of reference
[1] is shown in figure 5.4b. The horizontal axis denotes the tuning of the round-trip length Lrt. Their variation will lead to changes in Gouy phases, and hence can vary
χ1/2π in the neighborhood of the rational ratio 1/3 which corresponds to ∆L = 0 at
86 the center in the figure. The peaks of the blue curves in the figure correspond to the stationary modes in the lowest Landau level n2 = 0 for the case that χ1/2π > 1/3.
As χ1/2π approaches 1/3 from above, all the peaks converge into one single peak.
This indicates that all the levels becomes a degenerate Landau level. As χ1/2π falls below 1/3, the degenerate levels begin to separate from each other with increasing separation as χ/2π moves farther away from 1/3. The labeling of the peaks of the red curve is precisely the reverse of the blue curve.
Another way to detect these modes is to look directly at the profile of these transverse stationary modes by placing a screen perpendicular to one of the beams.
The mode profiles from [1] are shown in figure 5.4a.
87 a) b)
Figure 5.4: Detection of the stationary modes and their degenerate frequency. (5.4a) The intensity profile of the stationary modes in the resonator from [1]. (5.4b) The transmission spectrum of stationary modes in the lowest Landau level from [1] The transmission spectrum of stationary modes in the lowest Landau level from figure 2.b in [1]. The main panel shows the spectrum at different on-axis length where ∆L is the deviation from the critical value of L that gives degeneracy. ∆L = 0 can be regarded as tuning χ1 = 2π/3 at the best resolution of the experimental resonator. The top inset is a close-up of the main panel that shows the lower order modes are degenerate within the range the resonator linewidth of ∼ 200kHz.
5.3 Create extended landau levels in a cone with dramati- cally larger degeneracy
In chapter 1, we have studied the energy spectrum and wave functions of a two-
dimensional electron in a cone and anti-cone. We note that the curvature changes
88 the angular momentum of the wave functions as well as its spectrum. If one wants to simulate this Hamiltonian exactly, one would have to produce the same wave functions and the same spectrum. In Simon et. al’s experiment, they focus on simulating the positive angular momentum branch of the cone states. The negative branch which would have higher energy than the planar Landau levels, however, was not considered.
We note that the anti-cone spectrum has the important features that the negative angular momentum states move down towards the lowest Landau level as curvature becomes more negative. In the limit of negatively infinite curvature, we have the unusual situation that all Landau levels are extended to include the negative angular momentum states (see figure2.7.)
In this thesis, our goal is to produce these extended Landau levels. We will demon- strate a way to bring the negative angular momentum states that are originally non- degenerate with the Landau levels to become degenerate with them. It is in this sense that we say we have achieved the anti-cone Landau levels at negatively infinite curva- ture. However, as we shall see, the actual wave functions remain to be those of a cone with positive curvature. Hence, the simulation is for the energy spectrum only. Still, we find a very remarkable situation that the extended degenerate Landau level in fact include all the wave functions in a cone. In other words, our scheme is to put all the cone states (with positive and negative angular momentum) into a single degenerate level, similar to the situation of an anti-cone with negative infinite curvature.
5.3.1 The energy levels and wave functions of Landau prob- lem for general σ
First, we summarize the Landau level structure and the wave functions in terms of complex coordinate z, z¯ for the cases of a plane, a cone and an anti-cone altogether.
89 For wave functions, we are only concerned with the leading power of z andz ¯, for
they represent the maximum angular momentum of the sate, and angular momentum
correlations are essential to quantum Hall states. [51, 7, 11, 10, 52, 13, 12, 8, 53, 6, 9,
54, 55]. For general curvature specified by σ, the Landau level wave functions behave as
n m/σ Ψn,m(z, z¯) ∝ (¯zz) z , m ≥ 0 (5.17)
n−|m| |m|/σ Ψn,m(z, z¯) ∝ (¯zz) z¯ , m < 0 (5.18) where z = re−iϕ is on the cone wedge ϕ ∈ [0, 2πσ], and m is the integer angular momentum index such that Lz = ~m/σ. These equations can be easily obtained by identifying z = re−iϕ in equation (2.6) and remembering that the Laguerre polynomi- als Ln(x) has degree n. The planar case corresponds to σ = 1. The comparison of the behavior between the planar case and the general curvature case is shown in figure
2.6. We see clearly that the negative angular momentum states are bent down by negative curvature. In the limit of σ → ∞, each Landau level is extended to include all negative momentum states. This was shown in figure 2.7 before.
5.3.2 Extend the Landau level degeneracy to simulate the energy level structure of a σ → ∞ anti-cone
Remember in figure 5.1, we already have a triangular tower of degenerate levels for each fixed n2 when χ1/2π = 1/3 is exacy. Recall that for a given n2, such tower contains all the states that satisfy 3n1 + q = const. Therefore, we can uniquely label a tower with a pair (i, n2) where i = 0, 1, 2. The pair index (i, n2) can be thought of
as the single level at the bottom of each tower that uniquely specify it. We will call it
a Q-tower with index (i, n2) and denote it as T (i, n2). For example, the tower shown
90 in figure 5.1 is denoted as T (0, 0), and in figure 5.2 are T (0, 0),T (1, 0),T (2, 0). The entire set of the spectrum in equation (5.1), say Σ, can be represented as
[ Σ = T (i, n2) (5.19) i=0,1,2 n2=0,1,2,...
However, the optical trick to be performed is not to pull the negative angular momentum states in a Q-tower towards the Landau level. Rather, it pulls the negative angular momentum states from other towers. We now explain this process. The frequencies of the stationary modes are given by equation (5.1). Let us consider
χ1 = 2π(1/3), χ2 = 2π(2/3 + ). That is to say, in addition to χ1/2π being a rational number, we also make χ2/2π close to a rational number. These rational number can be chosen arbitrarily. Here, we choose
χ 1 χ 2 1 = , 2 ≈ (5.20) 2π 3 2π 3
for simplicity. As we shall see, even this simple choice leads to very rich structure.
With this choice of χ1, χ2, the frequency spectrum becomes
1 2 E = 2πν n + n + q + n . (5.21) n1,n2,q FSR 3 1 3 2 2
∗ ∗ Let us consider now a degenerate level at frequency E = 2πνFSRq , where q is
some large integer. This degenerate level belongs to the (i = 0, n2 = 0) Q-tower,
T (0, 0). The two ends of this degenerate level are (3q∗, 0, 0) and (0, 0, q∗) as shown as in figure 5.5, and we call it the q∗-level of T (0, 0) tower. Let > 0 a small positive number, we can see in figure 5.5 that the(q∗ −2)-level of T (0, 3) tower and the
(q∗ − 4)-level of T (0, 6) tower are almost degenerate to it with the energy difference
3 and 6 respectively.
91 That is to say, the energy difference between (q∗ − 2k)-level of T (0, 3k) tower and
(q∗)-level of T (0, 0) tower is 3k for k = 1, 2,..., [q∗/2] where [x] is the floor function.
q∗ When = 0, this sequence of levels from {T (0, 3k)}k=0 towers becomes degenerate.
Figure 5.5: Extended degenerate levels from n2 = 3k,(k=1,2,. . . ) Q-towers. Each inverted triangle is a Q-tower uniquely labeled by the pair index (n1, n2) of the state at the bottom,( or equivalently (i, n2), i = n1 mod 3). Each dot is a state in the tower ∗ labeled by (n1, n2, q). When operating at frequency E = 2πνFSRq = E with Gouy phases χ1 = 2π/3, χ2 = 2π(2/3 + ), the thicker lines are the degenerate levels that have frequencies of the form E = E + 3k. The extended degeneracy happens when the Gouy phase χ2 is fine-tuned such that → 0. The negative angular momentum states (n1 − n2 < 0) on the thicker lines are marked with red dots. The degeneracy of each thicker level is marked under the curly lower bracket.
This is not the end of the story yet. Parallel to the sequence (where i = 0)
stated above in figure 5.5, we still have two other branches with i = 1, 2 that can be
degenerate when = 0. They correspond to the sequence of levels from {T (1, 3k +
q∗−1 q∗−2 1)}k=0 and from {T (2, 3k + 2)}k=0 . This is shown in figure 5.6. Combining these two dimension, the overall structure is now clear. Each sequence of degenerate level
92 in figure 5.5 forms a family labeled by i. There are three families (i = 1, 2, 3) in total. Within each i family, the allowed n2 aren2 = 3k + i because of the degeneracy condition
1 2 n + n + q = const. (5.22) 3 1 3 2
Figure 5.6: Extended degenerate levels from all n2 separate into three branches in- dexed by i = 0, 1, 2. The first stack of Q-towers are exactly those in figure 5.5. The other two stacks correspond to the case when the Q-towers satisfy n2 = 3k + 1, 3k + 2 respectively. Each stacks terminates when n2 is too large such that even the bottom state of the Q-tower is above the degenerate frequency.
The degeneracy when = 0 can now be evaluated. For each q∗-level has degeneracy
q∗ + 1, the degeneracy in figure 5.5 is
∗ ∗ Di=0(q ) = (q ∗ +1) + (q − 1) + .... (5.23)
93 The entire degenerate manifold is a union of the three families i = 0, 1, 2. Therefore, the total degeneracy is
∗ ∗ ∗ ∗ D(q ) =D0(q ) + D1(q − 1) + D2(q − 2) (5.24)
=(q∗ + 1) + (q∗ − 1) + ... (5.25)
+ q∗ + (q∗ − 2) + ...
+ (q∗ + 1) + (q∗ − 1) + ...
( 3(q∗+1)2 ∗ 4 , when q odd = 3q∗(q∗+2) ∗ (5.26) 4 , when q even
In summary, in the limit of → 0 such that χ1 = 2π/3, χ2 = 4π/3 exactly, we
have the degenerate manifold described by the integer solutions (n1, n2, q) to n1/3 +
2n2/3 + q = E where E is a constant specifying the frequency. This manifold consists
of q∗-levels from different Q-towers, and all negative angular momentum states come
∗ from those n2 > 0 towers (specifically k states from the right in each (q −2k)-level of
the T (0, 3k) tower.) The negative angular momentum states are marked as red dots
in figure 5.5.
We also see that besides the negative angular momentum states that come into
the degeneracy in this process, we have a lot more states that have positive angular
momentum. Exactly what these states are will be explained later in another scheme.
What is striking about this result is that the degeneracy of the levels is consid-
erably larger than those of the ordinary Landau levels, as the degeneracy occupies a
two-dimensional plane rather than a one-dimensional line. This considerable increase
in degeneracy has never been encountered in condensed matter and will surely lead
to dramatic many-body phenomena when interaction effects are included.
94 5.3.3 Detection of the extended level structure
We give a prediction of how experimentally this extended degeneracy can be
detected. Similar to figure 5.4b, we can look at the transmission frequency spec-
trum of the stationary modes when we tune the round-trip length Lrt of the cavity.
However, tuning Lrt change both Gouy phases simultaneously. Suppose we oper-
ate in a neighborhood of the exact degeneracy point (where 1 = 2 = 0), say
χ1/2π = 1/3 + 1, χ2/2π = 2/3 + 2, where 1, 2 change with Lrt, the frequency
spectrum En1,n2,q is
1 2 E = 2πν n + n + q + n + n . (5.27) n1,n2,q FSR 3 1 3 2 1 1 2 2
= 2πνFSR(E + 1n2 + 2n2) (5.28)
Depending on how 1, 2 approach zero when we increase Lrt, the frequency spectrum
may show three different patterns. When 1, 2 > 0, we approach both Gouy phases
from above. When we increase Lrt to approach the degeneracy point, the frequency
spectrum should have peaks (There will be O(N 2) peaks if there are O(N) peaks in
figure 5.4b ) all above the degeneracy frequency, and the peaks converge to a single
peak at degeneracy point (∆L = 0).
Similar arguments can be applied to describe the behaviors for other cases when
1, 2 approach zero from different quadrants. If 1, 2 are of different signs, the peaks
will converge from both sides towards degeneracy. If they are both negative, the
peaks converge from above. These are summarized in figure 5.7.
95 a) The quadrants. b) The frequency spectrum for case 1.
c) The frequency spectrum for case 2. d) The frequency spectrum for case 3.
Figure 5.7: Detection of degenerate levels in transmission spectrum. Depending on how we approach the degenerate point when we tune Lrt, there are in general three cases of the transmission spectrum. In each spectrum, the dots indicate the peaks in figure 5.4b
96 5.4 Characterizations of the states in the (χ1 = 2π/3, χ2 = 4π/3) degenerate manifold
Recall one feature of the degenerate manifold we have obtained with (χ1 =
2π/3, χ2 = 4π/3) is that we have negative angular momentum states (m = −3k, k =
1, 2, . . . , q∗) in it. We have noticed that there are a lot more states also come in the
degenerate manifold that we did not characterize. In this section we take a different
approach that discuss the case right at the extended degeneracy point, i.e. 1 = 2 = 0
in equation (5.28), and provide a systematic way to find every states in the degenerate
manifold. We also characterize the states by examining the wave functions. For this
specific choice of Gouy phases, we conclude, surprisingly, that all the wave functions
are those of a σ = 1/3 cone, including states in higher Landau levels. Finally, we also
study the case for another combination of Gouy phases that are rational multiples of
2π as an attempt to give a general description.
At the extended degeneracy point of (χ1 = 2π/3, χ2 = 4π/3), the entire frequency
spectrum is given by all non-negative integer solutions to the equation
1 2 E = 2πν n + n + q (5.29) n1,n2,q FSR 3 1 3 2
For simplicity, we first consider the extended degeneracy manifold at a specific fre- quency that contains the state (0, 0, E) for some large integer E. This frequency is
E = 2πνFSRE. Finding all the states degenerate at this level is equivalent to finding
all non-negative integer solutions (n1, n2, q) to
1 2 n + n + q = E (5.30) 3 1 3 2
97 Our goal is to find all the solutions. We can achieve this by starting with a known
solution (which is (0, 0, E) in this case) and find all other solutions with
∆n1 + 2∆n2 + 3∆q = 0 (5.31)
In the following we give a systematic way to find all the solutions to equation (5.31).
the notation
We use ∼ for the equivalence relation that two states have the same frequency. If
0 0 0 0 0 0 (n1, n2, q) is a solution to equation (5.31) and (n1, n2, q) ∼ (n1, n2, q ), then (n1, n2, q ) is also a solution. For example, starting with the solution (0, 0, E) and use equation
(5.31), we can have
(0, 0, E) ∼ (3, 0, E − 1) ∼ (6, 0, E − 2) ∼ ...,
(0, 0, E) ∼ (0, 3, E − 2) ∼ (0, 6, E − 4) ∼ ... (5.32)
5.4.1 Finding all degenerate states in an extended level
A systematic algorithm to exhaust all solutions is as follows. First we run through
the states with condition ∆n2 = 0 until q hit the bottom. We have
∆n2 = 0 :
(0, 0, E) ∼ (3, 0, E − 1) ∼ (6, 0, E − 2) ∼ · · · ∼ (3E, 0, 0) (5.33)
(Note that this procedure is equivalent to finding all the states in E-level of the
T (0, 0) Q-tower that we had previously.) For each state we have gotten here, we run through another procedure with q fixed, which impose the condition ∆n1 +2∆n2 = 0.
98 Therefore, we have
∆q = 0 :
(3, 0, E − 1) ∼ (1, 1, E − 1) (5.34)
(6, 0, E − 2) ∼ (4, 1, E − 2) ∼ (2, 2, E − 2) ∼ (0, 3, q − 2) (5.35) . .
Each sequence terminates when n1 or n2 cannot be further decrease (n1, n2 ≥ 0).
(Note that in this procedure, we in fact explore all other possible n2 for each state in
the E-level in T (0, 0) tower that we have obtained. It can be understood as the steps in figure 5.5 and figure 5.6 combined. ) Repeating such process we can eventually exhaust all the solution in this extended degenerate manifold. The procedure of this recipe is illustrated in figure 5.8.
99 Figure 5.8: A schematic illustration of the procedure for finding all the stationary modes in the extended degenerate level at frequency E = 2πνFSRE with the Gouy phases χ1 = 2π/3, χ2 = 4π/3. Stationary modes are labeled with (n1, n2). The q quantum number is indicated with the orange arrows along which q is fixed. We start with the (0, 0) state and goes along the black vertical arrow with n2 fixed. For each state in the first column (n2 = 0) we fix q and find all other states. The states in the shaded boxes correspond to those n = 0 states plotted in figure 5.8.
The (n1, n2) labeling is convenient for finding the solutions to the degeneracy condition. However, it is not as transparent to see the angular momentum contents and to identify the wave function. Therefore, we convert the labeling to (n, m) with equation (3.44) that we derived in section 3.4.
n = min(n1, n2), m = n1 − n2, where χ1 < χ2 (5.36)
With this conversion, in figure 5.9 we see clearly that every state in our degenerate levels is an eigenstate on a σ = 1/3 cone and no states repeat. As long as we have large enough E accessible, in principle we can obtain an extended degenerate level that contain every single-particle eigenstate in a σ = 1/3 cone.
100 Figure 5.9: The same procedure as in figure 5.8 with states labeled in (n, m). The pattern shows that we can in principal have all eigenstates of a cone of wedge angle 2π/3 in a single generalized Landau level if we allow q to be infinite. .
Conclusion
Originally we wanted to realize Landau levels on an infinite σ anti-cone which has all negative angular momentum states degenerate in the extended Landau level.
With our scheme, although we can achieve degenerate levels that contain states with negative angular momentum, the wave functions are not the same as those in an anti-cone in our understanding. However, we found each degenerate level in this system extends to a degenerate manifold that actually contains “every eigenstate” for a particle in a σ = 1/3 cone in the Landau problem. This large degeneracy has not been seen before and could lead to interesting fractional quantum Hall states if interaction effects can be created. Recently, some effort in creating interaction effect for this system is reported with Rydberg polaritons[56, 57]. This could be a potential
101 platform to study fractional quantum Hall physics in this degenerate manifold with
an extraordinarily larger degeneracy compared to the conventional Landau levels.
Another literature has reported that there is a way to select the Laguerre-Gaussian
modes by their radial quantum number n [58]. This discovery suggests more control
over the states we can inject into the resonator and could enrich the study of quantum
Hall physics (and many-body physics in general) with this system.
Last but not least, we notice that by making a different choice of Gouy phases,
the extended Landau levels can have even richer structure in the states that are
degenerate. We shown another example with Gouy phases χ1 = 2π/5, χ2 = 2π/3 in
appendix C, where the resulting degenerate states are from both σ = 1/3 and σ =
1/5 cones. Further analysis and a systematic classification is required to thoroughly understand the different kinds of extended Landau levels that can be realized with the photonic resonator system.
102 Chapter 6: The Poincar´ehit patterns
6.1 Introduction
In this chapter we introduce the ray picture description of laser in the resonator,
the Poincar´ehit patterns, as a complement to the wave description we discussed
previously. For the four-mirror resonator, hit patterns also provide a direct corre-
spondence between the laser in the four-mirror resonator and a rotating harmonic
oscillator. This correspondence is less transparent in the wave picture description.
Moreover, hit patterns are directly relevant to our study of the extended Landau
levels because of the following reasons. First, they explicitly explain the n-fold rota-
tional symmetry for specific choices of Gouy phases. Secondly, they show the choice
of Gouy phases for the extended Landau levels can be realized by experiment. Lastly,
they suggest experimental signatures for the degenerate point we suggested. These
will be discussed in detail in this chapter.
In the context of paraxial ray optics, the Poincare hit patterns are the set of hit points that the ray trajectory intersects with a particular transverse plane in each round-trip. In figure 6.1, we show examples of the transverse wave profiles and the hit patterns observed at the transverse plane depicted in the resonator. Figure 6.1a shows the case for a beam going along the optical axis, and figure 6.1b shows the beam
103 that is slightly off from the optical axis where the hit pattern is three points. In the ray picture, the laser is treated as a ray. Therefore, the details of the transverse profile are smeared out and we represent it with a point. On the other hand, in the context of classical Hamilton equation, a hit pattern is the set of points that represent the stroboscopic dynamics of the system. They are generated by discrete time evolution in the phase space which is equivalent to observing the system at constant discrete time steps only. The two formalisms are equivalent and the special case for the four-mirror resonator and the rotating harmonic trap is shown explicitly in this chapter. Hit patterns generated by the classical Hamilton equation are used to study the degenerate point of Gouy phases for the extended Landau levels.
104 a) The transverse profile and corre- b) The transverse profile and corresponding hit sponding hit pattern at the transverse pattern at the transverse plane shown in the res- plane shown in the resonator when the onator when the laser beam is slightly off the op- laser beam is along the optical axis. tical axis.
Figure 6.1: Schematic illustrations for hit patterns as a ray optic description of a laser beam in the resonator. 6.1a shows a laser beam along the optical axis, which repeats itself after a single round-trip. 6.1b shows a laser beam that propagates slightly off the optical axis and repeats itself after three round-trips. In general cases, the off- axis laser may never repeat itself in finite number of round trips. The transverse field profile represents a wave optic description of the laser and the hit pattern is a ray optic description.
6.2 The hit patterns from the round-trip matrix of a pho- tonic resonator
6.2.1 The formalism
Suppose we start with a ray vector µ(0) at a specific transverse plane
x(0) (0) (0) y µ = (0) . (6.1) θx (0) θy
105 Its propagation in a resonator after each round trip can be calculated by acting the
round-trip matrix M rt on it.
µ(1) = M rtµ(0) (6.2)
is the subsequent ray vector after µ(0) goes through a round trip in the resonator and
comes back to the same transverse plane.
The Poincare hit pattern is the set of hit points R~ (n) in a transverse plane given
by: x(n) R~ (n) = , (6.3) y(n) where x(n) x(0) (n) (0) (n) y rtn y µ = (n) = M (0) (6.4) θx θx (n) (0) θy θy That is to say, the hit pattern is generated by repeatedly applying the round-trip matrix on the initial ray vector and plotting only the position components. The general procedure to calculate the round-trip matrix is outlined in equation (4.56)
4 rt Y ~ ~ M = MPROP (|Xj+1,j|/2) · MROTj · MMIRRORj · MPROP (|Xj,j−1|/2) (6.5) j=1
The details of each matrix in the product can be found in chapter 4. And the exact numerical calculation of the round-trip matrices for the experimental resonator with tetrahedral configuration used in [1] can be found in detail in appendix.B.3. In summary, the round-trip matrix M rt is fully determined by the following parameters of the resonator as illustrated in figure 6.2: the on-axis length, L, the opening half angle, θ and the radii of curvature of the four mirrors (R1,R2,R3,R4).
106 Figure 6.2: The twisted resonator of tetrahedral configuration used in experiments in [1]. Mirror 1, 3 are in the y − z plane, and mirror 2, 4 are in the x − y plane. The coordinate of each mirror is marked in the figure. The on-axis length L is measured ~ ~ ~ ~ from the midpoint of X1 and X3, (0, 0, 0), to the midpoint of X2 and X4, (0, L, 0). The half opening angle θ is measured from the y-axis to each mirror and are all equal. Each mirror has its own radius of curvature Ri.
6.2.2 Hit patterns of the experimental resonator
Let us look at some examples of the hit patterns of the four-mirror resonator of tetrahedral configuration used in the experiments by Simon et al. Generic hit patterns of this resonator are shown in figure 6.3 with two different initial ray vectors. A key feature here is the three-fold symmetry in the hit patterns shown by three fuzzy circles.
These circles can be understood as the cyclotron orbits of the rotating harmonic oscillator because we have shown that the tetrahedral configuration generates the effects of image rotation in section 4.3.1. We show how the rotation and harmonic trap effects manifest themselves in the hit patterns in the following.
107 a) µ(0) = (.01, 0, 0, 0), with curved mirrors b) µ(0) = (.01, 0, 0,.01), with curved mirrors
Figure 6.3: The Poincare hit patterns of the resonator with parameters of the round- trip matrix M rt set to be same as the experimental resonator of Simon et al. The ◦ parameters are L = 1.816 cm, θ = 16 and (R1,R2,R3,R4) = (2.5, 5, 5, 2.5) cm as provided in [1]. The two figures start with different initial ray vectors µ(0). Both hit pattern are generated with 300 round trips.
108 Figure 6.4: The hit pattern of the experimental resonator shows precession when the number of round trip is large. The hit pattern is generated with ν(0) = (.01, 0, 0, 0), 3,000 round-trips. The Gouy phases with the parameters of the cavity are χ1/2π = 0.333372, χ2/2π = 0.236401. The precession effect is explained in (6.46).
First, we show that the harmonic trap effects come from the mirror curvature and that the rotation effects come from the tetrahedral configuration. We remove the mirror curvature in the simulation and keep all other parameters the same as before, i.e. we replace the mirror reflection matrices in (4.56) with identity matrices to account for planar mirror reflection. Looking at the hit patterns, we find that the resonator is not stable anymore. Because in figure 6.5a, when the initial ray has no transverse momentum, the hit points orbit around the origin in clockwise direction.
However, when we add a small transverse momentum in, the hit pattern in figure
6.5b spirals out and diverges. This shows that the image rotation is introduced by the tetrahedral configuration of the mirrors because the hit pattern is in a rotating non-inertial frame and any small perturbation in the transverse momentum of the
109 initial ray leads to both Coriolis and centrifugal force. In conclusion, we can interpret the mirror curvature as the harmonic trap confinement in the corresponding rotating trap system to make it stable, and the tetrahedral configuration introduces the spatial rotation.
a) µ(0) = (.01, 0, 0, 0), the hit points orbit around b) µ(0) = (0, 0, 0,.01), the hit points are divergent the center when planar mirrors are used when planar mirrors are used
Figure 6.5: The Poincare hit patterns of the resonator with the same parameters as in figure 6.3 except we replace all mirrors with planar mirrors. i.e. R1 = R2 = R3 = R4 = ∞. In figure 6.5a the initial ray vector has no transverse momentum and the hit points orbit the origin in clockwise direction. In figure 6.5b a small perturbation in the transverse momentum leads to a divergent hit pattern. Note that the range of the axes in figure 6.5b is 103 times larger than the range in figure 6.5a. Both hit patterns are generated with 300 round trips.
110 6.3 The hit patterns from the Hamilton formalism: the stro- boscopic evolution
In this section, we show the analysis of stroboscopic dynamics of a classical ro-
tating harmonic oscillator in the Hamilton equation formalism. This formalism has
one-to-one correspondence with the round-trip matrices formalism of the resonators
we introduced in the last section. Our purpose for introducing this formalism is to use
it as a tool, first of all, to understand the rotational symmetry in the hit patterns of
the experimental resonator, and secondly, to justify the validity of our idea of tuning
the Gouy phases. Therefore, an exact transformation between the two systems is
not necessary for our discussion because it is sufficient to investigate the connection
between the rotational symmetry of the hit patterns and the Gouy phases with the
know correspondence.
6.3.1 The Hamilton equations in matrix form
A generic quadratic Hamiltonian takes the form
1 H = νT (GT Q)ν (6.6) 2 T 0 12 where ν = x y px py is the phase space column vector, G = , −12 0 and GT Q is a 4 × 4 real matrix that defines the Hamiltonian. It is useful to know that the inverse of G is its transpose
GT G = GGT = 1 (6.7)
We cast the Hamiltonian in this form as it is convenient to write down the Hamilton
equations in the matrix form. The Hamilton equations are
∂H ∂H q˙i = , p˙j = − (6.8) ∂pj ∂qj 111 In the matrix form we can write it as
∂ ∂ ∂px ∂x ∂ν ∂ ∂ = ∂py H = G ∂y H = G∂ H (6.9) ∂t −∂ ∂ ν ∂x ∂px −∂ ∂ ∂y ∂py where ∂ = ( ∂ ∂ ∂ ∂ )T is a column vector of the differential operators in phase ν ∂x ∂y ∂px ∂py space. The convenience of writing the Hamiltonian in this form is now obvious because we can rewrite the above equation
T ∂νH = G Qν (6.10) when we plug in (6.6) for H. This immediately leads to a neat expression for the
Hamilton equations in terms of the phase space vector ν using (6.9)
∂ν = Qν (6.11) ∂t where we used the identity GGT = 1.
The solution to it is simply
ν(t) = etQν(0) (6.12)
6.3.2 Hamilton equations for a harmonic oscillator
We first consider the classic example of a harmonic oscillator from which we can gain intuition about this phase space description. We later add another term to account for the rotation by considering the system in a rotating non-inertial frame which also known as synthetic magnetic fields and discuss its consequent Poincare hit patterns. The Hamiltonian for a harmonic oscillator
p2 + p2 1 H = x y + mω2(x2 + y2) (6.13) h.o. 2m 2
112 is a diagonal quadratic form, where
2 1 T mω 12 0 0 m 12 G Qh.o. = 1 ⇒ Qh.0. = 2 (6.14) 0 m 12 −mω 12 0
For the convenience of solving the phase space vector ν, we use a trick and choose the following rescaling
T ν˜ = mωx mωy px py (6.15)
the corresponding Q˜ matrix is
˜ 0 ω12 Qh.o. = (6.16) −ω12 0
˜ such thatν ˜(t) = etQh.o. ν˜(0) is equivalent to (6.12).
The convenience of this rescaling is in evaluating the matrix exponential ˜ cos ωt 1 sin ωt 1 etQh.o. = 2 2 (6.17) − sin ωt 12 cos ωt 12
In this form, it is essentially a rotation matrix except that this rotation mixes across
the position and momentum as opposed to the real-space rotation that mixes coordi-
nates within position and momentum respectively. Therefore we can say the harmonic
oscillator Hamiltonian induces a phase space rotation in the dynamics. The solution
for the phase space vector is
1 cos ωt 12 mω sin ωt 12 νh.o.(t) = ν(0) (6.18) −mω sin ωt 12 cos ωt 12
6.3.3 Hamilton equation for a rotating harmonic oscillator
The next step is to introduce the synthetic gauge field by going into a rotating
non-inertial frame where we only need to add an extra term −ΩLz to the Hamiltonian
p2 + p2 1 H = x y + mω2(x2 + y2) − Ω(xp − yp ) = H + H (6.19) syn. 2m 2 y x h.o rot. 113 The corresponding Q matrix is
1 0 Ω m 0 1 1 −Ω 0 0 m Ωiσy m 12 Qsyn. = 2 = 2 (6.20) −mω 0 0 Ω −mω 12 Ωiσy 0 −mω2 −Ω 0
0 −i where σ = is one of the 2 × 2 Pauli matrices. The exponential of Q is y i 0 syn. easier to calculate than it may seem as it also separates into two parts
Ωiσy 0 Qsyn. = Qh.o. + = Qh.o. + Qrot. (6.21) 0 Ωiσy
and the two separable parts actually commute with each other
[Qh.o,Qrot.] = 0. (6.22)
The exponential can be calculated by using the Baker-Campbell-Hausdorff formula
cos ωt 1 1 sin ωt 1 R(−Ωt) 0 etQsyn. = etQh.o etQrot. = 2 mω 2 −mω sin ωt 12 cos ωt 12 0 R(−Ωt) cos ωt R(−Ωt) 1 sin ωt R(−Ωt) = mω (6.23) −mω sin ωt R(−Ωt) cos ωt R(−Ωt)
cos Ωt − sin Ωt where R(Ωt) = is the 2×2 rotation matrix. The classical solution sin Ωt cos Ωt to this Hamiltonian is
cos ωt R(−Ωt) 1 sin ωt R(−Ωt) ν (t) = mω ν(0) (6.24) syn. −mω sin ωt R(−Ωt) cos ωt R(−Ωt)
From (6.23), we see that the time evolution matrix etQsyn. is essentially a tensor
product of two 2 × 2 rotation matrices. Hence, we introduce the notation
etQsyn. = R˜ (−ωt) ⊗ R(−Ωt), (6.25)
where 1 0 mω 0 R˜ (−ωt) = mω R(−ωt) (6.26) 0 1 0 1
114 Since we know that the eigenvalues of a 2 × 2 rotation matrix R(θ) are {eiθ, e−iθ}, we realize immediately that the eigenvalues for this tensor product matrix etQsync.
are the products of the two sub-sectors due to the property of tensor product, say
{ei(ω+Ω)t, ei(ω−Ω)t, e−i(ω+Ω)t, e−i(ω−Ω)t}. This consequence reminds us of the property of the round-trip matrices for a stable resonator (4.50) where the eigenvalues of M rt must be of unity modules and in complex conjugate pairs, i.e. {eiχ1 , eiχ2 , e−iχ1 , e−iχ2 }.
Therefore, to make the stroboscopic dynamics of the rotating harmonic oscillator correspond to the round-trip hit pattern of the resonator, we choose our time step to be the time it takes to make a round trip. That is ∆t = (2π)/ωFSR, or equivalently a sequence of discrete time {tn} such that,
2π tn = n (6.27) ωFSR
where ωFSR = 2πνFSR is the free spectral range angular frequency we have introduced
in section 4.3.2. By setting the unit of ω and Ω such that ωFSR = 2π, we can say the round-trip matrix M rt is a similar matrix to the matrix eQsyn. . Therefore, the hit patterns of the two systems can be transformed to one another through a invertible linear transformation.
To produce the hit patterns of the stroboscopic dynamics of this rotating harmonic oscillator, we take discrete integral values of t = 1, 2, 3, ... to account for the “round- trips”, and match the frequencies (ω±Ω) with the Gouy phases χ1, χ2 of the resonator.
Without loss of generality we can assign
χ1 = ω + Ω, χ2 = ω − Ω, (6.28)
115 such that the eigenvalues of eQsyn. , which we refer to as the stroboscopic evolution
matrix hereafter are equal to the eigenvalues of the round-trip matrix M rt of our in-
terest. We calculate the eigenvalues of M rt from the parameters of the experimental
resonator of Simon et al. that give the hit pattern with three-fold symmetry. Follow-
ing the summary in section 4.3.3, we find that the Gouy phases of the experimental
resonator are χ1 = 0.333372 × 2π, χ2 = 0.236401 × 2π. Identifying them with the
frequencies in rotating traps χ1 = (ω+Ω), χ2 = (ω−Ω) and using (6.24), we can gen-
erate the hit patterns. The results of the simulation are shown in figure (6.6). They
share the same feature of the three-fold symmetry with those in figure (6.3a,6.3b)
though the hit patterns are not identical because the two formalisms are related by
some non-trivial invertible linear transformation.
We notice an advantage for the formalism of stroboscopic evolution for a rotating
∗ ∗ harmonic oscillator is that the eigenvalues {λ1, λ1, λ2, λ2} of the stroboscopic evolution
matrix eQsyn. have simple relation with the trapping and rotation frequencies.
i(ω+Ω) i(ω−Ω) λ1 = e , λ2 = e (6.29)
In the simulation of Simon’s cavity, we see that the three-fold rotational symmetry is directly determined by the eigenvalues of the round-trip matrices, or more specif- ically, the Gouy phases. Therefore, the hit patterns generated by the stroboscopic evolution of a rotating harmonic oscillator provide us with an alternative approach to study the relation between the symmetry of the hit patterns and the the Gouy phases in the cavity system. And the eigenvalues are very easy to tune in this formalism as compared to the round trip matrices formalism of the resonator. A quick but crucial observation is that when ω ± Ω are both commensurate with 2π, the stroboscopic hit
116 a) ν(0) = (.01, 0, 0, 0), generated with eQsyn. b) ν(0) = (.01, 0, 0,.01), generated with eQsyn.
Figure 6.6: Hit patterns of the stroboscopic dynamics of a rotating harmonic oscillator with parameters such that ω + Ω = 0.333372 × 2π and ω − Ω = 0.236401 × 2π where setting ωFSR = 2π defines the unit for ω and Ω. Each hit point ν(n)can be related back to a hit point µ(n) in figure 6.3 through an invertible linear transformation. The two subfigures are generated with different initial phase vector ν(0), and both figures went through 300 round-trips.
117 patterns must be simply a finite number of dots regardless of the number of round-
trips, as (eQsyn )n = 1 for some integer n, and the hit points come back to itself after n round-trips. This suggests that the three copies of the seemingly continuous orbits in
Simon’s cavity comes from the small deviation in Gouy phases from a perfect fraction of 2π.
6.4 Study the rotational symmetry of the hit patterns with the Gouy phases
We have seen the hit patterns generated by Simon’s cavity are three copies of circular orbits. The three-fold rotational symmetry is related to one of the round-trip
Gouy phases χ1 that is very close to 2π/3. Because we have shown the equivalence between the round-trip hit patterns of the experimental resonator and the strobo- scopic dynamic of a rotating harmonic oscillator, we can make use of the approach in the latter to investigate the relation between the Gouy phases and the symmetry of the hit patterns. The reason we choose the stroboscopic dynamics of a rotating harmonic oscillator is clearly due to the direct control of the Gouy phases we have as we have shown before.
To start with a simple case, let both round-trip Gouy phases be some rational
1 1 fractions of 2π, say χ1 = 2 × 2π, χ2 = 3 × 2π. Note that we could have chosen
χ1 = 2π/3, χ2 = 2π/5 as we did in the examples in appendix C. To avoid cluttering
1 1 in the hit patterns, we choose χ1 = 2 ×2π, χ2 = 3 ×2π, while the idea of the rotational symmetry is the same. Following the identification in equation (6.28),
χ1 = ω + Ω, χ2 = ω − Ω (6.30)
118 χ1+χ2 5 χ1−χ2 1 we just set our parameters ω = 2 = 12 × 2π, Ω = 2 = 12 × 2π. We can read directly from equation (6.24) that the hit patterns will be only a set of finite
number (it is exactly 6 distinct points as shown in figure 6.7) of points for any
non-zero initial ray vector ν(0). For the convenience of studying the rotational symmetry, we would like to generate continuous trajectory as those hit patterns in
figure 6.6. Therefore, we come up with the idea to make one of the Gouy phases slightly deviated from the perfect fraction of 2π, say
1 1 χ = × 2π + , χ = × 2π. (6.31) 1 2 2 3
This way the hit points cannot come back to the exact position it was before the
round trip due to the deviation in the Gout phase. We show in figure 6.8a, 6.8b
how the hit patterns develop from small arches to a full circle as they go through
more round trips. The hit patterns eventually develop into three copies of full circle.
It is not obvious from merely equation (6.24) that the hit patterns would develop into
three full circles, so we investigate the hit patterns and explain the reason for the
circular orbit in the following.
119 Figure 6.7: The hit pattern when χ1/2π = 1/2, χ2/2π = 1/3 shows six fixed points only. The plot is generated with ν(0) = (.01, 0, 0, 0), 150 round-trips
a) ν(0) = (.01, 0, 0, 0), 500 round-trips, and Gouy b) ν(0) = (.01, 0, 0, 0), 5,000 round-trips, and 1 1 1 1 phases ( 2 × 2π + .001, 3 × 2π) Gouy phases ( 2 × 2π + .001, 3 × 2π)
Figure 6.8: Figures 6.8a-6.8b show a sequential process of how the closed orbits are form. One of the Gouy phase has to be slightly deviated from a simple fraction.
120 6.4.1 The three copies of cyclotron motion in stroboscopic dynamics
The two-dimensional rotation matrix R(θ) has eigenvalues e±iθ and corresponding eigenvectors (1, ±i). The eigenvectors for the matrix eQsyn. in (6.25) are the tensor products of the eigenvectors for each matrix {v˜i ⊗ vj} and has the following represen- tation in the phase space basis
1 1 1 1 1 1 i i i i ν = = , ν = = , ν∗, ν∗ (6.32) 1 1 imω 2 1 −imω 1 2 imω −imω i −mω i mω
∗ ∗ where ν1 , ν2 are the corresponding complex conjugate. Given arbitrary real phase space vector ν(0) it can be decomposed into
ν(0) = c1ν1 + c2ν2 + c.c. , (6.33)
where ci are complex numbers in general. The hit point after n ∈ N round-trips is
iχ1n iχ2n ν(n) = c1e ν1 + c2e ν2 + c.c. (6.34)
Plugging in (6.31) and taking n = 3k for some integer k such that ν2 components are stationary, we find
i3kχ1 ν(3k) = c1e ν1 + c2ν2 + c.c (6.35)
We further substitute the eigenvectors in (6.32) and focus only on the spatial (first two) components
x 2Re(c ei3kχ1 ) 2Re(c ) = 1 + 2 (6.36) y 2Re(ic ei3kχ1 ) 2Re(ic ) (3k) 1 2
121 iφ1 Let c1 = |c1|e
x 2|c | cos(3kχ + φ ) 2Re(c ) = 1 1 1 + 2 (6.37) y −2|c | sin(3kχ + φ ) −2Im(c ) (3k) 1 1 1 2 2|c | cos(φ + 3kπ + 3k) 2Re(c ) = 1 1 + 2 (6.38) −2|c1| sin(φ1 + 3kπ + 3k) −2Im(c2)
It becomes clear that the first term in equation (6.38) represents a circular motion and the second term is the center of the circular orbit. Note that we have chosen
χ1 = 2π/2 + , hence 3kχ1 = 3kπ + 3k. This means that in every 3 round-trip,
(3k 7→ 3k +3), the hit points jump in between two opposite sides of the circumference of the circle centered at (2Re[c2], −2Im[c2]) and rotate with a small angle 3 to develop the circular orbit. This process can be seen clearly in our simulation in figure 6.7-
6.8b where we gradually increase the number of the round-trips from a small number where the hit points have not yet form a closed circle, to a large enough number that complete the full circles. In addition, with (6.37), we can very easily generalize it to the cases of n = 3k + p, where p = 0, 1, 2:
x 2|c | cos[3kχ + φ + pχ ] 2Re(eip2π/3c ) = 1 1 1 1 + 2 (6.39) y −2|c | sin[3kχ + φ + pχ ] ip2π/3 (3k+p) 1 1 1 1 −2Im(e c2)
2π where the centered of the orbits are rotated by p 3 while the points in the orbits are rotated by an angle pχ1, (p = 0, 1, 2) with respect to the optical axis.
It is also instructive to represent c1, c2 in terms of ν(0) such that we can calculate the center of the orbit and the radius easily. From (6.33), we have x0 1 1 1 1 c1 ∗ y0 i −i i −i c = 1 (6.40) px0 imω −imω −imω imω c2 ∗ py0 −mω −mω mω mω c2
122 Taking the inverse of the matrix we get
1 i i 1 c1 4 − 4 − 4mω − 4mω x0 ∗ 1 i i 1 c1 4 4 4mω − 4mω y0 = 1 i i 1 (6.41) c2 4 − 4 4mω 4mω px0 ∗ 1 i i 1 c2 4 4 − 4mω 4mω py0
We obtain c1, c2 in terms of ν(0):
1 p 1 p c = x − y0 − i y + x0 , (6.42) 1 4 0 mω 4 0 mω 1 p 1 p c = x + y0 − i y − x0 . (6.43) 2 4 0 mω 4 0 mω
We now can read the center and the radius of the cyclotron motion easily.
1 py0 xc 2Re(c2) x0 + = = 2 pmω (6.44) y −2Im(c ) 1 y − x0 c (3k) 2 2 0 mω 1 p 2 p 21/2 r = 2|c | = x − y0 + y + x0 (6.45) c 1 2 0 mω 0 mω
We would like to make a conclusive statement about the relation between the Gouy phases and the rotational symmetry. When one of the two Gouy phases is exactly
2π/n and the other is close to some rational fraction of 2π, the hit pattern would
show n copies of circular orbits and n-fold rotational symmetry. However, in real
experiments it’s impossible to fine tune the parameters to get the absolute accuracy
of the Gouy phases. In the case of Simon’s experimental resonator, the Gouy phase
χ1 = 0.333372(2π) closer to 2π/3 than the other Gouy phase χ2 = 0.236401(2π) to
2π/5. Therefore, we can make the following conclusion: the rotational symmetry
of the hit patterns would follow the Gouy phase that is closer to 2π/n and
show n−fold rotational symmetry.. It is χ1 = 2π/3 in this case and consequently
showing three-fold rotational symmetry. We show the hit patterns with Gouy phases
χ1 = 2π/3 + , χ2 = 2π/3 and χ1 = 2π/3, χ2 = 2π/3 + to justify this statement. In
123 figure 6.9a the hit pattern shows three-fold symmetry because the the closer simple
fraction is 2π/3, while in figure 6.9b we see two-fold symmetry instead.
a) ν(0) = (.01, 0,.01, 0), 150 round-trips, and b) ν(0) = (.01, 0,.01, 0), 150 round-trips, and 1 1 1 1 Gouy phases ( 2 × 2π + , 3 × 2π), = .01 Gouy phases ( 2 × 2π, 3 × 2π + ), = .01
Figure 6.9: The hit patterns of rotating traps in stroboscopic dynamics shows n-fold 2π rotational symmetry when one of the Gouy phase is a simple fraction of 2π, say n , 2π and the other is a simple fraction of 2π plus a small deviation, m + . We see the hit patterns are forming circular orbits that has n-fold rotational symmetry and form the circular orbits in m segments. We intentionally show the figures with less roundtrips to point out the m segments in an orbit.
But now the centers of the each circular orbits also precess around the optical axis because now equation (6.44) has to be modified:
x 2Re(ei3k2 c ) c = 2 , (6.46) y −2Im(ei3k2 c ) c (3k) 2 where 2 = χ2 − 2π/3 is the difference between the other Gouy phase and another simple fraction 1/q and q is some positive integer. We see the precession of the
124 cyclotron orbits in the hit patterns of the experimental resonator when the number of round trips are increased to 3,000.
6.5 Using hit patterns to find the critical point for extended Landau levels
In this section we show how hit patterns can be used to help find the critical points of Gouy phases that results in the extended Landau levels we suggested in chapter 5. For clarity, we take the case χ1/2π = 1/3, χ2/2π = 2/3 as an example in our discussion. However, hit patterns for other more critical values of Gouy phases can be easily generalized with this idea.
The key to generate the extended Landau levels is, as we have seen, is to accurately tune the Gouy phases at simple fractions of 2π. The hit pattern as shown previously in this chapter is also sensitive to the actual values of the Gouy phases. The hit pattern is a description of the laser in geometric optics. Therefore, it shall be realized experimentally in the condition that the dimension of the resonator is much larger than the wavelength of the laser. Also we need a high shutter speed so that the round-trip number in the hit pattern is controllable. Based on these two assumption, we show the hit patterns when the resonator is tuned close to the critical point in
Gouy phases at χ1/2π = 1/3, χ2/2π = 2/3 where the extended Landau levels occur.
First we show in figure 6.10 the hit patterns for χ1/2π = 1/3+1, χ2/2π = 2/3+2
p 2 2 −3 on a circle = 1 + 2 = 10 around the critical point (1, 2) = (0, 0). The patterns are repeated in some regions and they can be easily identified with continuity consideration. We show only some representatives to avoid the cluttering in the
figures. The key features are the following. At the critical value where the Gouy phases are the exact simple fractions, the hit patterns is three fixed point regardless
125 of the number of round-trip, or the exposure time of the camera. On the circle,
the hit patterns show different patterns depending on the values of 1, 2. But all
of them have the three-fold rotational symmetry. Note that when only either of the
Gouy phase is exact, the hit patterns are always three circular orbits for the reasons
we have discussed.
Figure 6.10: The hit patterns in a small neighborhood around χ1/2π = 1/3, χ2/2π = 2/3. Note that at the center where the extended degenerate Landau levels occur, the hit patterns are three fixed points only. And on the 1, 2 axes, the hit patterns are always three copies of cyclotron orbits. All hit patterns show three-fold rotational symmetry.
126 Next we show how the hit patterns change when we approach the critical point
χ1/2π = 1/3, χ2/2π = 2/3. In figure 6.11, we show only the patterns along the
45◦, 180◦, 315◦ lines, and in figure 6.12 we show only the cases along 22.5◦, −22.5◦ lines. Because of the repetition, these patterns should qualitatively represent all cases around the critical point. In all cases it shows the hit patterns degrade into three fixed points as we approach the critical point. These patterns can be used to tune the resonator to find the parameters that produce the extended Landau levels we suggested.
Figure 6.11: How the hit patterns change when the Gouy phases approach the crit- ical point χ1/2π = 1/3, χ2/2π = 2/3. We show in the figure how the hit patterns would change when we fine tune the resonator parameters such that the Gouy phase approach the critical point along 45◦, 180◦, 315◦ lines.
127 Figure 6.12: How the hit patterns change when the Gouy phases approach the critical point χ1/2π = 1/3, χ2/2π = 2/3 . We show in the figure how the hit patterns would change when we fine tune the resonator parameters such that the Gouy phase approach the critical point along 22.5◦, −22.5◦ lines.
In summary, hit patterns is another experimental signature with two importance in turns of the extended Landau levels. First of all, it explains the relation between the n-fold rotational symmetry and the Gouy phases values, and this in turns shows the states we obtain are those on a cone. Secondly, the hit pattern is experimentally convenient to find the critical point for the extended Landau levels. Combining with the transmission spectrum and mode profiles, we can test experimentally to see the extended Landau levels.
128 Appendices
129 Appendix A: Landau problem on cone and anti-cone surfaces embedded in R3
A.1 The Landau problem with magnetic fields on a cone sur- face in R3
The Hamiltonian of a particle on a 2-d surface of a cone where the curvature is
flat everywhere but positively singular at the tip with uniform magnetic flux is [59]
|nˆ × (~p + e/cA~)|2 H = (A.1) 2m
We choose spherical coordinate to solve the system. The momentum operator is
1 1 ~p = ~∇ = ~ r∂ˆ + θˆ ∂ + φˆ ∂ (A.2) i i r r θ rsinθ φ
The vector potential for uniform magnetic flux is chosen to be
1 1 A~ = Brφˆ ⇒ B~ =r ˆ(B cot θ) + θˆ(−B) (A.3) 2 2
The surface of a cone (half lower branch) can be parametrized as the following.
x2 + y2 = (z tan α)2, (A.4)
z < 0 (A.5)
130 Or in spherical coordinate:
r > 0, (A.6)
θ = π − α, (A.7)
φ ∈ [0, 2π) (A.8)
and the unit normal vector isn ˆ = −θˆ.
The Hamiltonian in spherical coordinate is
2 2 2 −~ ˆ ie ~ −~ 2 1 b r 2 H = θ × (∇ + A) = ∂r + ∂r + ( ∂φ + i 2 ) (A.9) 2m ~c 2m r r 2lB
1 where b = sin α ∈ [1, ∞). In order to generalize to the negative curvature singularity case, we have to re-
parametrize the space. However, the Hamiltonian turns out to be the same except
for the change in the range of b to b ∈ (0, 1). Altogether, the Hamiltonian (A.9) is valid for both positive(b ∈ [1, ∞)) and negative(b ∈ (0, 1)) curvature.
A short argument for the correspondence between 0 < b < 1 and the concentrated negative curvature surfaces is as following. For a cone of half opening angle α, the
2 πr2 surface area is πr sin α = b as can be easily seen by unfolding the cone to a flat
2π wedge of angle b . For a surface with negative curvature singularity at the tip instead, the surface has an wedge angle greater than 2π after unfolding it. We may abuse the
2π notation and define this angle to be b again and extend the range of it to b ∈ (0, 1]. And the aforementioned formulas follow.
131 A.2 the Landau levels on an anti-cone
A.2.1 a parametrization for an anti-cone
Consider the following parametrization for an anti-cone: X~ (u, v) = (x(u, v), y(u, v), z(u, v))
x(u, v) = u sin(α cos βv) cos v
y(u, v) = u sin(α cos βv) sin v
z(u, v) = u cos(α cos βv)
u ≥ 0, v ∈ [0, 2π) (A.10) where (u, v) are the parameters and α, β can be arbitrary constants in general. How- ever, to avoid the complication of self-intersecting surface and to be specific for the discussion, we restrict 0 < α < π/2 and β being integers. The parametrized surface with the choice α = π/4, β = 2 is plotted in figure (2.5b). We will show that this sur- face has zero Gaussian curvature everywhere except at the center explicitly. Gaussian curvature K on a two-dimensional surface can be calculated from the coefficients of the first and second fundamental forms
eg − f 2 K = (A.11) EG − F 2 where E,F,G are the first fundamental form coefficients defined by the metric
ds2 = dx2 + dy2 + dz2 = Edu2 + 2F dudv + Gdv2. (A.12)
~ ~ ~ ~ ~ ~ 2 0 2 2 E = Xu · Xu = 1,F = Xu · Xv = 0,G = Xv · Xv = u [θ (v) + sin θ(v)] (A.13)
132 ~ ∂X~ where we adopt a short-handed notation Xu = ∂u . e, f, g on the other hand are defined as follows:
~ ~ ~ ~ ~ ~ e = N · Xuu, f = N · Xuv, g = N · Xvv (A.14)
~ ~ ~ where N = Xu ×Xv is the normal vector of the surface. Readers may have recognized that the parametrization we chose is based on the spherical coordinate, specifically with the polar angle θ:
dθ(v) θ = θ(v) = α cos βv, θ0 = = −αβ sin βv (A.15) dv
The other two spherical coordinates as functions of (u, v) are ρ(u, v) = u, φ(u, v) = v The first order derivatives are
~ Xu = (sin θ cos v, sin θ sin v, cos θ) =ρ, ˆ (A.16)
~ 0 0 0 Xv = u(θ cos θ cos v − sin θ sin v, θ cos θ sin v + sin θ cos v, −θ sin θ) (A.17)
= uθ0θˆ + u sin θφ,ˆ (A.18)
0 ∂θ where θ = θ(u, v) and θ = ∂v are the shorthanded notation we adopt and will use through out the calculation, andρ, ˆ θ,ˆ φˆ are the natural basis for the spherical coordinates. The normal vector N~ is
~ ~ ~ 0 ˆ ˆ N = Xu × Xv = ρθ φ − ρ sin θθ. (A.19)
The unit vectors of the curvilinear coordinates in terms of spherical coordinates are:
X~ uˆ = u =ρ ˆ (A.20) ~ |Xu| X~ θ0θˆ + sin θφˆ vˆ = v = √ (A.21) ~ 02 2 |Xv| θ + sin θ θ0φˆ − sin θθˆ nˆ =u ˆ × vˆ = √ (A.22) θ02 + sin2 θ
133 The second derivatives are
~ Xuu = 0 (A.23)
~ ~ 0 ˆ ˆ Xuv = Xvu = θ θ + sin θφ. (A.24)
(A.25)
The coefficients of second fundamental form is
~ ~ ~ ~ 0 0 e = N · Xuu = 0, f = N · Xuv = uθ sin θ − uθ sin θ = 0 (A.26) and we find out the Gaussian curvature
eg − f 2 K = = 0 (A.27) EG − F 2 vanishes everywhere on the surface except for the center. This result shows that the surface is identical to a flat disk locally, and therefore, can be unfolded to a flat disk, however, with a circular angle greater than 2π. The deficit angle of this surface can also be calculated by consider the path length of a unit circle, say we impose a restriction γ = {X~ (u, v)|u = 1}.
Z Z 2π q δ = 2π − ds = 2π − (θ0)2 + sin2 θ dv < 0 (A.28) γ 0
A.2.2 the Landau levels
To generate Landau levels on the anti-cone, we mimic the idea similar to that for
a disk and a cone by introducing a gauge potential that result in constant magnetic
magnetic flux. A natural choice for the gauge potential is ! 1 1 uθ0 u sin θ A~ = Buvˆ = B θˆ + φˆ (A.29) 2 2 p(θ0)2 + sin2 θ p(θ0)2 + sin2 θ
134 The magnetic field is
" 2 !# 2 0 ! ~ ~ ˆ B ∂ u sin θ ˆ B ∂ u θ B = ∇ × A =ˆρAρ + θ − + φ 2u ∂u p(θ0)2 + sin2 θ 2u ∂u p(θ0)2 + sin2 θ (A.30)
=ρA ˆ ρ +nB ˆ (A.31)
wheren ˆ = N~ is the unit normal vector following the definition in equation (A.19), |N~ |
and Aρ is some function of (u, v) that we ignore because it doesn’t contribute to the
flux.
The Hamiltonian of the system is identical to (A.1) with the normal vectorn ˆ = N~ |N~ | Here we demonstrate the calculation in more detail. The momentum operator in terms
of the canonical basis is
~ P =uP ˆ u +vP ˆ v +nP ˆ n (A.32)
where
P =ˆu · P~ = ~∂ , (A.33) u i u 0 ~ θ 1 ~ 2 Pv = √ ∂θ + √ ∂v = √ ∂v (A.34) i u θ02 + sin2 θ u θ02 + sin2 θ i ρ θ02 + sin2 θ
Here we take a digression to introduce the the circular arc length l(u, v) in the aniti-
cone for later convenience in calculation. Recall from (A.12) (A.13), the metric on
the anti-cone is
ds2 = du2 + [θ0(v)2 + sin2 θ(v)]dv2 (A.35)
Let
q dl = u θ0(v)2 + sin2 θ(v)dv. (A.36)
135 It follows
1 ∂l = ∂v (A.37) upθ0(v)2 + sin2 θ(v)
we define the circular arc length of radius u as
Z v q l(u, v) = u θ0(¯v)2 + sin2 θ(¯v)dv¯ (A.38) 0 and the polar angle ϕ
l(u, v) Z v q ϕ(v) = = θ0(¯v)2 + sin2 θ(¯v)dv¯ (A.39) u 0
The Hamiltonian is
2 2 |nˆ × (P~ + e/cA~)| 1 ~ u = vPˆ u + (−uˆ) Pv + i 2 (A.40) 2M 2M i 2lB 2 ( 2) −~ 2 u u = ∂u + (−uˆ)(2∂l + i 2 ) · v∂ˆ u + (−uˆ)(2∂l + i 2 ) 2M 2lB 2lB (A.41) 2 " 2# −~ 2 2 u = ∂u + ∂u + 2∂l + i 2 (A.42) 2M u 2lB
136 Appendix B: More about paraxial optics
B.1 the validity of paraxial approximation
Here we show an argument for the validity of the paraxial approximation that
leads to the paraxial equation (4.6). i.e. the second derivative of ψ with z is much
smaller than other terms in (4.3) as long as the wave is paraxial, to say θ is small. For simplicity, we consider the case in only one transverse dimension. The scalar field of a plane wave propagating in the direction at a small angle θ to the optical axis z can
be written as (see figure B.1):
E(x, z, t) = ei(kx sin θ+kz cos θ−ωt) (B.1)
Again we would like to extract the transverse profile as we did in (4.2):
E(x, z, t) = ψ(x, z)ei(kz−ωt), (B.2)
where ψ the transverse profile is:
ψ(x, z) = ei[kx sin θ−kz(1−cos θ)] (B.3)
When the beam is paraxial, the angle θ is small. The derivatives of ψ in (4.3) can be
approximated by Taylor expansion to the first significant order in θ :
2 ∂ψ 2 2 θ k = k (1 − cos θ) ≈ k , (B.4) ∂z 2
137 2 ∂ ψ 2 2 2 2 = k sin θ ≈ k θ , (B.5) ∂x2
and
2 2 2 4 ∂ ψ 2 2 2 θ 2 θ = k (1 − cos θ) ≈ k = k . (B.6) ∂z2 2 4
2 Because ∂ ψ is to the fourth order in θ, it is negligible compared to two other terms ∂z2 2 that are to the second order of θ. More specifically, when θ < 1 ≈ 14◦, ∂ ψ will 4 ∂z2 ∂ψ be two order of magnitude smaller than k ∂z . The is true for most optical cavity system; thus the paraxial wave equation can approximate the physics well enough
before significant corrections are required.
Figure B.1: The geometry of a plane wave propagating in the direction at angle θ with the optical axis z.
138 B.2 Derivation of the Hermite-Gaussian modes
A standard choice of the set of general solutions to the paraxial equation in Lasers
is the Hermite-Gaussian modes.
1/2 s 2 1 1 1 1 k k k i[(n+ 2 )χx(z)+(m+ 2 )χy(z)] ψn,m(x, y, z) = ψn(x, z)ψm(y, z) = n m e π 2 n!2 m! wx(z)wy(z) √ ! √ ! 2x 2y x2 y2 x2 y2 Hn Hm exp −ik + − 2 − 2 wx(z) wy(z) 2Rx(z) 2Ry(z) wx(z) wy(z) (B.7)
This set of functions is the solution to the paraxial equation in free space.
∂2ψ ∂2ψ ∂ψ + − 2ik = 0 (B.8) ∂x2 ∂y2 ∂z
This section is dedicated to show the derivation of this solution.
B.2.1 spherical wave in paraxial approximation
We propose a trial solution to paraxial equation by applying the paraxial approx-
imation to the spherical wave solution that solves the scalar Maxwell equation:
eikρ ∇2 + k2 E˜(x, y, z) = 0 ⇒ E˜(x, y, z) = (B.9) ρ
p 2 2 2 where ρ(x, y, z) = (x − x0) + (y − y0) + (z − z0) . The paraxial approximation
assume the wave propagate mostly along the longitudinal direction, say z − z0 x − x0, y − y0, and
2 2 (x − x0) + (y − y0) ρ(x, y, z) ≈ (z − z0) + (B.10) 2(z − z0)
(x−x )2+(y−y )2 1 ikρ 1 ik(z−z ) ik 0 0 E˜(x, y, z) = e ≈ e 0 e 2(z−z0) (B.11) ρ z − z0
139 The transverse profile is thus
2 2 1 ik (x−x0) +(y−y0) ψ(x, y, z) = e 2(z−z0) (B.12) z − z0
One can verify that (B.12) solves the paraxial equation (B.8). The first step of the generalization for the solution is to generalize z0, the z−position of the source, to a
complex variable. Consider
2 2 1 ik (x−x0) +(y−y0) ψ(x, y, z) = e 2q(z) (B.13) q(z)
where
q(z) = z − z0 + q0 (B.14)
is the complex radius function, and q0 ∈ C . This proposed transverse field continues to be a solution to equation (B.8). It’s convenient to identify the real and imaginary part of 1/q(z) as the following
1 1 2 = + i (B.15) q(z) R(z) kw(z)2
With this identification, the transverse profile is
2 2 (x−x )2+(y−y )2 1 ik (x−x0) +(y−y0) − 0 0 ψ(x, y, z) = e 2R(z) e w(z)2 (B.16) q(z)
1 We see that the function R(z) is the radius of curvature of the spherical wave, and w(z) is the spot size of the laser beam at z.
B.2.2 Higher order solutions
To find the general solutions, we use the tricks in solving differential equations.
140 the zeroth order solution
We start with the trial solution
2 2 ik x +y ψ(x, y, z) = A(z)e 2q(z) (B.17)
After plugging it into the paraxial equation 4.6, we obtain " # k 2 dq 2ik q dA − 1 (x2 + y2) − + 1 A = 0 (B.18) 2 dz q A dz
For this to be valid for arbitrary x, y, the term in the bracket vanishes identically.
dq = 1, or q(z) = z − z + q (B.19) dz 0 0
dA A A(z) q = − , or = 0 (B.20) dz q A0 q(z)
We reproduce the exact same solution as we applied the paraxial approximation to
the spherical wave solution. This is the zeroth order Hermite-Gaussian solution.
higher order Hermite-Gaussian solutions
Solving the equation in rectangular coordinate, we assume the transverse coordi-
nates separate.
ψn,m(x, y, z) = ψn(x, z) × ψm(y, z) (B.21)
Each of them satisfies the one-dimensional paraxial equation.
2 ∂xψn(x, z) + 2ik∂zψn(x, z) (B.22)
We consider a trial solution of the form
2 x ik x ψ (x, z) = a(z) × h ( ) × e 2q (B.23) n n p(z)
141 dq where p(z) is a scaling factor dependent on z. We assume dz = 1 because we want h0(u) = 1, and we’re solving a(z), hn(u), and p(z). Substituting the trial solution into the paraxial equation gives
p dp ikp2 2q da h00 − 2ik − xh0 − 1 + h = 0 (B.24) n q dz n q a dq n
Recall that Hermite polynomials Hn(x/p) satisfies
x H00 − 2 H0 + 2nH = 0 (B.25) n p n n
Therefore we impose the conditions
dp p i = + (B.26) dz q kp 2q da 2inq = − 1 (B.27) a dq kp2
A choice of p(z) that solves equation (B.26) is
w(z) p(z) = √ (B.28) 2
This choice of p(z) makes transverse profile have the same normalized shape at every
transverse plane and they scale like w(z), as can be seen in √ ! 2 2 2x ik x − x ψ (x, z) ∝ h e 2R(z) w2(z) (B.29) n n w(z)
With the choice of p(z), we can solve for a(z)
da 1 dq n dq∗ dq = − + − (B.30) a 2 q 2 q∗ q
1/2 ∗ 1/2 q0 q0 q (z) a(z) = a0 ∗ (B.31) q(z) q(z) q0
142 The normalized transverse profiles are √ 1/4 r 1/2 ∗ n/2 ! 2 2 1 q0 q0 q (z) 2x x ψn(x, z) = n ∗ Hn exp ik π 2 n!w0 q(z) q(z) q0 w(z) 2q(z) (B.32)
q(z) where q0 = q(z0), and the relation w(z) = w0 was used. The normalization q0 condition is
Z ∞ ∗ ψn(x, z)ψm(x, z)dx = δmn (B.33) −∞ Equation B.32 can be recast into a more widely used form involving the spot size
w(z) and a phase angle χ(z). We associate a phase to q(z) in the following way
i 1 πw2(z) = e−iχ(z) ⇒ tan χ(z) = (B.34) q(z) |q(z)| λR(z)
χ(z) is known as Gouy phase in optics, and it’s chosen to have χ(z) = 0 at the “waist”
of the Gaussian beam. One can show that
q0 w0 = e−i(χ(z)−χ0) (B.35) q(z) w(z) q q∗(z) 0 −i2(χ(z)−χ0) ∗ = e (B.36) q(z) q0 Therefore we can rewrite equation (B.32) into √ 1/4 s ! 2 2 2 1 −i(n+ 1 )[χ(z)−χ ] 2x x x ψ (x, z) = e 2 0 H exp ik − n π 2nn!w(z) n w(z) 2R(z) w2(z) (B.37) HG modes provide an orthonormal set of complete basis at any given transverse plane
z0.
B.3 Calculating the roundtrip matrices in the twisted res- onator
Here we show the nontrivial calculation for the round-trip matrix in the twisted
resonator with mirrors arranged in a tetrahedral configuration that is used in Simon’s
143 experiment. Recall the round-trip matrix in (4.56):
4 rt Y ~ ~ M = MPROP (|Xj+1,j|/2) · MROTj · MMIRRORj · MPROP (|Xj,j−1|/2) (B.38) j=1
The calculation of the matrix for rotation MROTj and mirror reflection with non-
normal incidence MMIRRORj is nontrivial due to the geometry of the elongated tetra- hedron. We reproduce the details of the calculation for the convenience of simulation.
B.3.1 Calculating the rotation matrices
We reproduce in figure 6.2 the tetrahedron configuration of mirrors in Simon’s
figure (1c) in [1] with the following notations: L is the on-axis length, and t = tan θ is the tangent of the is the opening half angle θ. We choose a coordinate system ~ ~ such that the location of the four mirrors can be expressed as X1 = (0, 0, −tL), X2 = ~ ~ (tL, L, 0), X3 = (0, 0, tL), X4 = (−tL, L, 0).
The transfer matrix for the rotation was mentioned before in (4.54) and (4.55),
Rj 0 MROTj = (B.39) 0 Rj where
OUT IN OUT IN ~xj · ~xj+1 ~yj · ~xj+1 Rj = OUT IN OUT IN (B.40) ~xj · ~yj+1 ~yj · ~yj+1
OUT IN We start with calculating the first matrix elements: ~xj · ~xj+1. Without loss of
OUT IN generality, consider j = 2. For ~x2 and ~x3 , recall the definitions from (4.52)
X~ × X~ (tL, L, tL) × (−tL, −L, tL) (t, −t2, 0) ~xOUT = 2,1 3,2 = = √ (B.41) 2 ~ ~ ~ ~ 2 4 |X2,1 × X3,2| |X2,1 × X3,2| t + t
X~ × X~ (−tL, −L, tL) × (−tL, L, −tL) (0, −t2, −t) ~xIN = 3,2 4,3 = = √ (B.42) 3 ~ ~ ~ ~ 2 4 |X3,2 × X4,3| |X3,2 × X4,3| t + t
144 An observation worth mentioning is that these two unit vectors are exactly the normal
vectors of the triangles ∆123 and ∆234, and it shows that the image rotation is
induced by the non-coplanarity of the four faces of the tetrahedron.
The first matrix element in R2 in (4.55) is
t4 t2 ~xOUT · ~xIN = = (B.43) 2 3 t2 + t4 1 + t2
OUT IN The second matrix element ~y2 · ~x3 can be computed similarly. We omit the procedure and show only the result: √ − 2t2 + 1 ~yOUT · ~xIN = . (B.44) 2 3 t2 + 1
R2 is essentially a rotation matrix, therefore we can write it as cos φ2 − sin φ2 R2 = (B.45) sin φ2 cos φ2
√ tan2 θ − 2 tan2 θ+1 where cos φ2 = 1+tan2 θ , and sin φ = tan2 θ+1
B.3.2 Calculating the transfer matrices for non-normal mir- ror reflection
Another non-trivial part of the calculation of the round-trip matrices comes from the non-normal incident angles at the mirrors in the tetrahedral configuration where the calculation is not complicated but may be ignored if not careful. Recall in (4.46) the focal lengths have to be modified to the sagittal and tangential components.
~ IN ~ IN In our choice of coordinate, Xj and Yj lie in the sagittal and tangential planes individually. The reflection matrix reads
1 0 0 0 0 1 0 0 MMIRROR(Rj, Φ) = −2 cos Φ 0 1 0 (B.46) Rj 0 −2 0 1 Rj cos Φ 145 The only complication here is the angle of the non-normal incidence Φ which can be
calculated from the geometry of the tetrahedral configuration. Figure 6.2 shows the
geometry and we can see tan θ tan Φ = √ (B.47) 1 + tan2 θ where θ is again the opening half angle of the tetrahedron mentioned before.
Up to this point, we are ready to calculate the round-trip matrices for this twisted resonator of tetrahedral configuration. Recall (4.56), we now have an explicit expres- sion as follows.
4 rt Y ~ ~ M = MPROP (|Xj+1,j|/2) · MROTj · MMIRRORj · MPROP (|Xj,j−1|/2) (B.48) j=1
~ √ |Xj+1,j | L 1+2 tan2 θ where 2 = 2 . MROTj and MMIRRORj are given in detail previously. The round-trip matrix has now been put in terms of L and θ that determines the
tetrahedron.
B.3.3 Laguerre-Gaussian modes: cylindrical symmetric so- lution to paraxial wave equation
We list the cylindrical symmetric solution for comparison and reference. If we
solve the paraxial equation in cylindrical coordinate[48],
1 1 i2k∂ ψ = ∂2 + ∂ + ∂2 ψ (B.49) z r r r r2 θ
the general solutions take the form of Laguerre-Gaussian modes: s √ !m 2p! 2r i(2p+m+1)(χ(z)−χ0) imθ ψp,m(r, θ, z) = e e (B.50) (1 + δ0m)π(m + p)! w(z) 2r2 r2 r2 Lm exp −ik − p w(z)2 2R(z) w(z)2
146 Appendix C: An example of the generalized Landau level consisting of states from different cones
We show another example with the choice of Gouy phases χ1 = 2π/5, χ2 = 2π/3.
The states in this degenerate levels are from both σ = 1/3 and σ = 1/5 cones.
Staring with (n1, n2, q) = (0, 0, q), every states in this degenerate level satisfies
3∆n1 + 5∆n2 + 15∆q = 0 (C.1)
For the convenience of later discussion, we call G1 = 3,G2, 5,D = 15. Following the procedure in equations (5.33), with ∆n2 = 0 and ∆n1 = 0 individually, we have the states in the same degenerate levels with the following relations:
(0, 0, q) ∼ (5, 0, q − 1) ∼ (10, 0, q − 2) ∼ ... (C.2)
∼ (0, 3, q − 1) ∼ (0, 6, q − 2) ∼ ... (C.3)
The above immediately implies we have states {1, z5, z10,..., z¯3, z¯6 ... } in the degen- erate level of frequency. For the higher Landau level states, we continue with the
147 steps in equation (5.34),(5.35) but now use the condition 3∆n1 + 5∆n2 = 0. We get
(5, 0, q − 1) ∼ (0, 3, q − 1) (C.4)
(10, 0, q − 2) ∼ (5, 3, q − 2) ∼ (0, 6, q − 2) (C.5)
(15, 0, q − 3) ∼ (10, 3, q − 3) ∼ (5, 6, q − 3) ∼ (0, 9, q − 3) (C.6) . .
Surprisingly, we see that those higher Landau level sates n > 0 states in this case is no more restricted to m = 3k or m = 5k (k ∈ N). In this example it turns out the n > 0 states can have any possible m. This can be seen when consider the change in m in the procedure we provided to find all the stationary modes. The step in equation (C.2) satisfies ∆m = 5k, and the steps in equations (C.4), (C.5), (C.6) satisfies ∆m = −8l. It turns out the total change in m is ∆m = 5k − 8l,(k, l ∈ N)
which can be any integer. Through this example, we can conclude that if the two
conditions
(G1,D) = (G2,D) = 1 (C.7)
G1 + G2 = D (C.8)
are both satisfied, where (a, b) is the greatest common divisor of a, b, the stationary
modes in the generalized Landau level are from the same cone with wedge angle 2π/D.
148 Bibliography
[1] N. Schine, A. Ryou, A. Gromov, A. Sommer, and J. Simon, “Synthetic landau levels for photons,” Nature, vol. 534, pp. 671 EP –, 06 2016.
[2] D. Yoshioka, The Quantum Hall Effect. Springer Series in Solid-State Sciences, Springer Berlin Heidelberg, 2013.
[3] R. Prange, M. Cage, K. Klitzing, S. Girvin, A. Chang, F. Duncan, M. Haldane, R. Laughlin, A. Pruisken, and D. Thouless, The Quantum Hall Effect. Graduate Texts in Contemporary Physics, Springer New York, 2012.
[4] K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy de- termination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett., vol. 45, pp. 494–497, Aug 1980.
[5] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotrans- port in the extreme quantum limit,” Phys. Rev. Lett., vol. 48, pp. 1559–1562, May 1982.
[6] R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett., vol. 50, pp. 1395– 1398, May 1983.
[7] G. Dev and J. K. Jain, “Band structure of the fractional quantum hall effect,” Phys. Rev. Lett., vol. 69, pp. 2843–2846, Nov 1992.
[8] J. K. Jain, “Composite-fermion approach for the fractional quantum hall effect,” Phys. Rev. Lett., vol. 63, pp. 199–202, Jul 1989.
[9] R. Laughlin, “Excitons in the fractional quantum hall effect,” Physica B+C, vol. 126, no. 1, pp. 254 – 259, 1984.
[10] F. D. M. Haldane, “Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states,” Phys. Rev. Lett., vol. 51, pp. 605–608, Aug 1983.
149 [11] M. Greiter, “Microscopic formulation of the hierarchy of quantized hall states,” Physics Letters B, vol. 336, no. 1, pp. 48 – 53, 1994.
[12] R. Haussmann, “Microscopic many-particle wave functions for the fractional quantum hall effect in periodic geometry,” Phys. Rev. B, vol. 51, pp. 10755– 10773, Apr 1995.
[13] B. I. Halperin, “Statistics of quasiparticles and the hierarchy of fractional quan- tized hall states,” Phys. Rev. Lett., vol. 52, pp. 1583–1586, Apr 1984.
[14] A. Gromov and A. G. Abanov, “Density-curvature response and gravitational anomaly,” Phys. Rev. Lett., vol. 113, p. 266802, Dec 2014.
[15] A. G. Abanov and A. Gromov, “Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field,” Phys. Rev. B, vol. 90, p. 014435, Jul 2014.
[16] R. R. Biswas and D. T. Son, “Fractional charge and inter-landau–level states at points of singular curvature,” Proceedings of the National Academy of Sciences, vol. 113, no. 31, pp. 8636–8641, 2016.
[17] T. Can, M. Laskin, and P. Wiegmann, “Fractional quantum hall effect in a curved space: Gravitational anomaly and electromagnetic response,” Phys. Rev. Lett., vol. 113, p. 046803, Jul 2014.
[18] Y.-H. Wu, H.-H. Tu, and G. J. Sreejith, “Fractional quantum hall states of bosons on cones,” Phys. Rev. A, vol. 96, p. 033622, Sep 2017.
[19] A. Gromov, G. Y. Cho, Y. You, A. G. Abanov, and E. Fradkin, “Framing anomaly in the effective theory of the fractional quantum hall effect,” Phys. Rev. Lett., vol. 114, p. 016805, Jan 2015.
[20] N. Read and E. H. Rezayi, “Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum hall systems,” Phys. Rev. B, vol. 84, p. 085316, Aug 2011.
[21] X. G. Wen and A. Zee, “Shift and spin vector: New topological quantum numbers for the hall fluids,” Phys. Rev. Lett., vol. 69, pp. 953–956, Aug 1992.
[22] J. E. Avron, R. Seiler, and P. G. Zograf, “Viscosity of quantum hall fluids,” Phys. Rev. Lett., vol. 75, pp. 697–700, Jul 1995.
[23] Q. H. Liu, “Geometric momentum for a particle constrained on a curved hyper- surface,” J. Math. Phys., vol. 54, p. 122113, 2013.
150 [24] P. Dirac, Lectures on Quantum Mechanics. Dover Books on Physics, Dover Publications, 2013.
[25] B. J. Bernard and L. C. L. Y. Voon, “Notes on the quantum mechanics of particles constrained to curved surfaces,” European Journal of Physics, vol. 34, no. 5, p. 1235, 2013.
[26] R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A, vol. 23, pp. 1982–1987, Apr 1981.
[27] M. Ikegami, Y. Nagaoka, S. Takagi, and T. Tanzawa, “Quantum mechanics of a particle on a curved surfacecomparison of three different approaches,” Progress of Theoretical Physics, vol. 88, no. 2, pp. 229–249, 1992.
[28] B. Podolsky, “Quantum-mechanically correct form of hamiltonian function for conservative systems,” Phys. Rev., vol. 32, pp. 812–816, Nov 1928.
[29] J.-D. Debus, M. Mendoza, and H. J¨urgenHerrmann, “Shifted Landau levels in curved graphene sheets,” ArXiv e-prints, Mar. 2018.
[30] U. Ludacka, M. R. A. Monazam, C. Rentenberger, M. Friedrich, U. Stefanelli, J. C. Meyer, and J. Kotakoski, “In situ control of graphene ripples and strain in the electron microscope,” npj 2D Materials and Applications, vol. 2, no. 1, p. 25, 2018.
[31] J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, “The structure of suspended graphene sheets,” Nature, vol. 446, pp. 60 EP –, 03 2007.
[32] L. D. Landau and E. M. Lifshits, Course of theoretical physics. Vol. 3, Vol. 3,. Oxford: Pergamon P., 1965.
[33] G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists: A Comprehensive Guide. Elsevier Science, 2013.
[34] M. do Carmo, Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
[35] Wikipedia contributors, “Euler characteristic — Wikipedia, the free encyclope- dia,” 2018. [Online; accessed 22-October-2018].
[36] M. Nakahara, Geometry, Topology and Physics, Second Edition. Graduate stu- dent series in physics, Taylor & Francis, 2003.
[37] Wikipedia contributors, “Gaussian curvature — Wikipedia, the free encyclope- dia,” 2018. [Online; accessed 23-October-2018].
151 [38] E. A. P. Hernandez, B. Kiefer, D. J. Hartl, A. Menzel, and D. C. Lagoudas, “Analytical investigation of structurally stable configurations in shape memory alloy-actuated plates,” International Journal of Solids and Structures, vol. 69-70, pp. 442 – 458, 2015.
[39] Wikipedia contributors, “Geodesic curvature — Wikipedia, the free encyclope- dia,” 2017. [Online; accessed 23-October-2018].
[40] J. Lee, Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, Springer New York, 1997.
[41] B. O’Neill, Elementary Differential Geometry. Academic Press, 1997.
[42] N. Cooper, “Rapidly rotating atomic gases,” Advances in Physics, vol. 57, no. 6, pp. 539–616, 2008.
[43] A. L. Fetter, “Lowest-landau-level description of a bose-einstein condensate in a rapidly rotating anisotropic trap,” Phys. Rev. A, vol. 75, p. 013620, Jan 2007.
[44] G. Baym and C. J. Pethick, “Vortex core structure and global properties of rapidly rotating bose-einstein condensates,” Phys. Rev. A, vol. 69, p. 043619, Apr 2004.
[45] N. R. Cooper, S. Komineas, and N. Read, “Vortex lattices in the lowest landau level for confined bose-einstein condensates,” Phys. Rev. A, vol. 70, p. 033604, Sep 2004.
[46] T.-L. Ho, “Bose-einstein condensates with large number of vortices,” Phys. Rev. Lett., vol. 87, p. 060403, Jul 2001.
[47] G. Watanabe, G. Baym, and C. J. Pethick, “Landau levels and the thomas-fermi structure of rapidly rotating bose-einstein condensates,” Phys. Rev. Lett., vol. 93, p. 190401, Nov 2004.
[48] A. Siegman, Lasers. University Science Books, 1986.
[49] S. J. M. Habraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A, vol. 75, p. 033819, Mar 2007.
[50] G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A, vol. 48, pp. 656–665, Jul 1993.
[51] S. Das Sarma and A. Pinczuk, Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures. Wiley, 2008.
[52] B. I. Halperin, P. A. Lee, and N. Read, “Theory of the half-filled landau level,” Phys. Rev. B, vol. 47, pp. 7312–7343, Mar 1993.
152 [53] J. Jain, Composite Fermions. Cambridge University Press, 2007.
[54] A. Lopez and E. Fradkin, “Fractional quantum hall effect and chern-simons gauge theories,” Phys. Rev. B, vol. 44, pp. 5246–5262, Sep 1991.
[55] G. Murthy and R. Shankar, “Hamiltonian theories of the fractional quantum hall effect,” Rev. Mod. Phys., vol. 75, pp. 1101–1158, Oct 2003.
[56] R. O. Umucal ılar, M. Wouters, and I. Carusotto, “Probing few-particle laughlin states of photons via correlation measurements,” Phys. Rev. A, vol. 89, p. 023803, Feb 2014.
[57] A. Sommer, H. P. B¨uchler, and J. Simon, “Quantum Crystals and Laughlin Droplets of Cavity Rydberg Polaritons,” ArXiv e-prints, May 2015.
[58] X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: Concepts and experiments,” Phys. Rev. Lett., vol. 120, p. 103601, Mar 2018.
[59] F. D. M. Haldane, “Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states,” Phys. Rev. Lett., vol. 51, pp. 605–608, Aug 1983.
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