Interaction of Quantum Vortex Beams with Matter

by Maria Solyanik-Gorgone

B.S. in Physics, May 2009, Karazin Kharkiv National University, Kharkiv, Ukraine M.S. in Theoretical Physics, May 2010, Karazin Kharkiv National University, Kharkiv, Ukraine

A Dissertation submitted to

The Faculty of The Columbian College of Arts and Sciences of The George Washington University in partial satisfaction of the requirements for the degree of Doctor of Philosophy

August 31, 2019

Dissertation directed by

Andrei Afanasev Associate Professor of Physics The Columbian College of Arts and Sciences of The George Washington University certiﬁes that Maria Solyanik-Gorgone has passed the Final Examination for the degree of Doctor of Philosophy as of May 10, 2019. This is the ﬁnal and approved form of the dissertation.

Interaction of Quantum Vortex Beams with Matter

Maria Solyanik-Gorgone

Dissertation Research Committee: Andrei Afanasev, Associate Professor of Physics, Dissertation Director

Helmut Haberzettl, Professor of Physics, Committee Member

Alexander van der Horst, Assistant Professor of Physics, Committee Member

iii Dedication

To my beloved parents for believing in me unconditionally and loving me selﬂessly.

iv Acknowledgments

I would like to express my deepest appreciation and gratitude to my research advisor, Professor Andrei Afanasev, for sharing his diverse expertise, for his continuous support of my research, patience and encouragement. This work would not be possible without his guidance. I especially thank my collaborators: Professor C. E. Carlson, Professor C. T. Schmiegelow, Professor F. Schmidt-Kaler and Professor V. G. Serbo for a supportive and friendly environment, productive work, stimulating discussions, insightful comments and hard questions throughout the whole time of us working together on projects. I would also like to ac- knowledge the contribution of Professor C. W. Clark on the early stages of my project. His discussions and lectures during my summer internship in 2016 provided the necessary foundation for future theoretical work in Atomic Physics and Spectroscopy. The experience and skills I obtained while working with this group of scientists is unique and irreplaceable in my future endeavors. My sincere thanks go to my readers, Professor H. Haberzettl and Professor A. van der Horst for the valuable comments, insightful questions and kind corrections. I would like to express my deepest gratitude to the department professors and staff for supporting friendly, nurturing but also stimulating and challenging environment for students to develop, mature as specialists and form their scientiﬁc identity. Also, I am grateful for ﬁnancial support that has come from the department funds towards completion of my project. Special thanks go to my graduate student mentor, Professor E. J. Downie. Her great example as a scientist, teacher and leader helped me grow into who I am today and, I am sure, will continue inspiring me in the future. Very special words of thanks go to my dear husband, life mentor and friend, Nicholas M. Gorgone, for his kind support and advice. Thank you for sheltering me when it is tough and sharing happiness in the moments of delight.

v Abstract

Interaction of Quantum Vortex Beams with Matter

The excitations in atomic, nuclear and semiconductor structures with orbital angular momentum carrying photons are analyzed. The violations of the conventional set of angular momentum selection rules are derived and discussed in detail for all three types of structures. In the case of atoms and ions, fine and hyperfine coupling are accounted for by means of the multipole expansion of partial transition amplitudes. The theory is developed for arbitrary beam polarization, alignment of the beam’s optical axis with respect to the target’s quantization axis, and transition multipolarity. The approach is extrapolated to semiconductor structures using the Luttinger formalism. Spin polarization anomalies for electrons photo-excited by twisted light in Γ-point in zinc-blende GaAs are revealed. Unusual sensitivity to the twisted beam polarization content, and radial and axial positioning of the target in the beam profile, are identified and studied in detail. The phenomenon of circular dichroism due to twisted beam topology during propagation in isotropic matter is predicted and analyzed. Three families of modes are considered and analyzed: Bessel, Bessel-Gauss and Laguerre-Gauss. The results are applicable in a wide range of studies: spectroscopy, metrology, quantum computing and communications, cybersecurity and more.

vi Statement of Originality

Herein I certify that the content of this thesis is my own work with the exceptions, listed below. Any resemblance to the results elsewhere, that are not mentioned in this statement and/or throughout this thesis, is accidental. Chapter 1: The overview of the general trends and fundamental breakthroughs in Singular Optics and Photonics, that paved the foundation for modern study of twisted light and matter beams is provided. Chapter 2: The overview of the necessary concepts in Laser Optics and Photonics, and systematic description of the Quantum Mechanical formalism for twisted mode quantization, established in the ﬁeld, is provided. Chapter 3: Professor C. E. Carlson made a substantial integrated contribution in developing the theoretical model for the data in Section 3.5. The data in Chapter 3 is the product of work done by Professor C. T. Schmiegelow, as a part of an experimental collaboration under the supervision of Professor F. Schmidt-Kaler. Chapter 4: The theoretical analysis and main predictions were done by Professor A. Afanasev and Professor C. E. Carlson for the spin-less case. Chapter 5: The formalism and analysis for mesoscopic proton targets for the np-capture discussed here was done by Professor A. Afanasev and Professor V. G. Serbo. Chapter 6: The atomic-like approximation in the eqn. (6.13) was done by Professor A. Afanasev. In the process of the completion of this work, many Mathematica code prototypes with key ideas and plots were shared with me by Professor A. Afanasev before the corresponding theoretical background was developed.

vii Table of Contents

Dedication ...... iv

Acknowledgments ...... v

Abstract ...... vi

Statement of Originality...... vii

List of Figures ...... ix

List of Tables ...... x

List of Abbreviations ...... xi

List of Symbols ...... xii

Chapter 1: Introduction...... 1 1.1 History...... 2 1.2 Thesis Outline...... 6 Chapter 2: OAM Photon States...... 9 2.1 Review of Necessary Fundamentals in Optics...... 9 2.2 Vortex Beams...... 13 2.2.1 Orbital Angular Momentum and Spin of Light...... 16 2.3 Orbital Angular Momentum Modes...... 17 2.3.1 Bessel Beams...... 18 2.3.2 Bessel-Gauss Beams...... 23 2.3.3 Laguerre-Gauss Beams...... 24 2.4 Orbital Angular Momentum Beams and Polarization States.. 26

viii Chapter 3: Quantum Mechanical Transition Matrix for Twisted Light - Atom Inter- actions...... 29 3.1 Plane-wave Transition Amplitude...... 30 3.2 Photo-Excitation by Twisted Beams in Total Angular Momen- tum Basis...... 32 3.3 Twisted Photo-Excitation in Atoms: Hyperfine Structure... 35 3.4 Photo-Excitation with Twisted Light in Cooled Trapped Ions.. 37 3.4.1 40Ca Ion in Parallel Configuration Irradiated by Circularly Po- larized Light...... 37 3.4.2 40Ca Ion in 45◦-Configuration and More...... 46 3.4.3 Ion photo-excitations in Radially and Azimuthally Polarized Orbital Angular Momentum Beams...... 56 3.5 Recoil from Orbital Angular Momentum Photons in Trapped Ions...... 58 Chapter 4: Chirality, Chirooptics and Circular Dichroism in Orbital Angular Mo- mentum Photon-Matter Interactions...... 62 4.1 Chirality in Light...... 62 4.2 Optical Activity and Twisted Photon Beams...... 66 Chapter 5: Electromagnetic Interactions and Neutron-Proton Systems...... 68 5.1 Deuteron Photodisintegration and np-Capture with Orbital An- gular Momentum Beam States...... 69 Chapter 6: Photo-Induced Processes in Semiconductor Materials and Quantum Dots...... 75 6.1 Twisted Photon Interaction with Bulk GaAs...... 76 6.2 Twisted Photon Interaction with Semiconductor Quantum Dots. 86 Chapter 7: Conclusions...... 91 Bibliography...... 94

ix Chapter A: List of Useful Formulas...... 105 Chapter B: Spherical Wave Decomposition of Twisted Modes...... 108

x List of Figures

Figure 2.1. The generic phase proﬁle of the OAM-carrying beam with a screw-like dislocation at the beam center, propagating in the z-direction. ····· 14 Figure 2.2. LG (rows 1 and 2), Bessel (row 3) and BG (row 4) modes’ intensity

proﬁles for different values of OAM `γ , p nodal number, and TAM

mγ . All the axes are in arbitrary units and the columns correspond to

topological charge `γ for LG, and TAM quantum number mγ for BB and BG. ·························· 19 Figure 2.3. Schematic picture of the twisted beam geometry from the prospective of the local observer P’ versus the coordinate system P attached to the beam axis. The red arrow specifies the overall direction of the OAM beam propagation. ······················ 21 Figure 2.4. Schematic visualization of the beam’s polarization state on Poincare sphere and polarization ellipse. ················· 26 Figure 2.5. Polarization profile of the radially and azimuthally-polarized beams in the xy-plane, with the intensity profile on the background. ······ 27

Figure 3.1. Deﬁnition of the relative position of the atom with the nucleus −Ze and photo-excited electron e with respect to the optical axis z of the photon vortex beam in terms of global r and local r0 coordinate systems. ··· 33 Figure 3.2. The scheme of the experiment: (a) The energy levels in 40Ca+ with the E2 transition of interest marked in red. (b)The experimental setup – linear segmented Paul trap (yellow) inside the UHV chamber (gray); the 729 nm OAM mode is formed in the optical setup on the bottom (red) with the help of the holographic phase plate HP (bottom-right). . Further experimental details can be found in the paper by Schmiegelow et al. (2016). The ﬁgure is used with permission of the authors. ···· 38

xi Figure 3.3. Relative position of the trapped ion with respect to the optical axis of the photon vortex beam, represented as an incoming cone of radiation, in the constant quantizing magnetic ﬁeld B. Local to the ion coordinate system {x0,y0,z0} is shown in blue, and global coordinate system {x,y,z}, attached to the beam’s optical axis z, is shown in black. ··· 39 Figure 3.4. Normalized Rabi frequencies as a function of impact parameter b for

initial atomic spin mi = −1/2. Black-dashed curves and purple long- dashed curves are the theory predictions for BG and LG, not accounting for the opposite sign circular polarization admixture; solid curves in black (BG) and long-dashed curves in purple (LG) are the theory predictions with 3% opposite-polarization admixture (by amplitude) for rows 1,3,4 and 6, and 10% admixture for rows 2 and 5. ······· 41 Figure 3.5. Normalized Rabi frequencies as a function of impact parameter b for ini-

tial atomic spin mi = 1/2. Black-dashed curves and purple long-dashed curves are the theory predictions for BG and LG, not accounting for the opposite sign circular polarization admixture; solid curves in black (BG) and long-dashed curves in purple (LG) are the theory predictions with 3% opposite-polarization admixture (by amplitude) for rows 1,3,4 and 6, and 10% admixture for rows 2 and 5. ··········· 42 Figure 3.6. (a) Normalized Rabi frequencies compared with the theory predictions for the Bessel mode (blue dashed and green long-dashed) and BG mode (black solid and black dotted); (b) same transitions versus theory for LG modes with parameters p=0, p=1, and their linear combination (see text for details). ························ 43

xii

Figure 3.7. Similar to Figure 3.4, but here the photons have mγ − Λ = 2. The ﬁts in black dashed (BG) and purple dash-dotted (LG) have no polarization admixture, while the black solid (BG) and purple long-dashed (LG) have 10% by amplitude of opposite helicity photons. ········ 44 Figure 3.8. Contour-plots of the normalized transition strength as a function of

2 2 polarization and impact parameter for 4 S1/2 → 3 D5/2 in a single 40 + Ca . From left to right, ﬁrst and second subplots are for `γ = 0 theory and experimental data correspondingly. The third and fourth are for

`γ = 1 theory and experimental data, respectively. Red lines indicate pure vertical polarization. ·················· 49 Figure 3.9. Contour-plots of the normalized transition strength as a function of

2 2 172 + polarization and impact parameter for S1/2 → F7/2 in a single Yb .

In both cases `γ = 1, θz = 0; but φb = 0 (−0.3 rad) for the left (right) plot. ··························· 50

Figure 3.10. Transition strength Ωr for different atomic multipolarities and beam

types as a function of impact parameter b for parameters: θz = 0 and Λ = 1. See the text for detailed description. ············ 51

Figure 3.11. Transition strength Ωr for different atomic multipolarities and beam types as a function of impact parameter b in case with ∆m = 1 and for vertical and horizontal photon polarization states. The ﬁrst column

shows transition strength for M1 in θz = π/2 magnetic ﬁeld alignment, the second and third columns show E2 and E3 multipoles for alignment

angle θz = π/4, respectively. The columns 1, 2 and 3 increment the

topological charge `γ = 0,1,2, correspondingly. See the text for detailed description. ························ 52

xiii Figure 3.12. Transition strength Ωr for different atomic multipolarities and beam types as a function of impact parameter b. The left column shows the

2 2 20 5+ expected response for a transition P1/2 → D3/2 in HCI Ne . The

parameters used are ∆m = 1, φb = 0 and θz = π/4. Red-solid (blue- dashed) indicates results for light polarized vertically (horizontally) beams. ·························· 54 Figure 3.13. Logarithmic plot of the photo-absorption rates in HCI 10Ne+ with mixed M1+E2 multipolarity. The bold curves correspond to radial (a) and azimuthal (b) polarization states, and light-colored curves are the rates

for LC polarization state and mγ = 2 introduced for the comparison, see Afanasev et al. (2018b). ··················· 57 Figure 3.14. The reference frame setup for the recoil problem, where O is the global coordinate system with respect to the entire beam; O0 is the local coordinate system with respect to the local wavefront. ········· 58

Figure 4.1. Spin asymmetry (ﬁrst column), SAM circular dichroism (second column) and OAM circular dichroism (third column) for the transition

S1/2 → D5/2 (E2) accounting for atomic electron spin degree of freedom as in eqn. (3.19). ···················· 66

(tw) Figure 5.1. Deuteron photodisintegration matrix element MjΛ as a function of the impact parameter b/λ for ∆m = −1 – solid-black, ∆m = 0 – dotted-

purple and ∆m = 1 – dot-dashed-green. Photon states are mγ = 0 and 1 for (a) and (b) correspondingly. ················ 71

tw 0 Figure 5.2. The cross section ratio σ /σ (θk) as a function of the impact parameter b/λ for m = 0 – solid-black, m = 2 – dotted-purple and m = 5 – dot- dashed-green, see Afanasev et al. (2018d). ············ 73

Figure 6.1. GaAs crystal energy band structure, see e.g. Kuwata-Gonokami (2011). 76

xiv Figure 6.2. Allowed transitions for the twisted photons (solid lines) and zero-OAM photons (dadshed lines) with RC polarization in GaAs crystal. Numbers in the circles indicate squares of Clebsch-Gordan coefﬁcients. ···· 78 Figure 6.3. Photo-excitation rate as a function of electron’s distance to the optical

vortex center for the vortex pitch angle θq = 0.1 Rad; long-dashed blue

is mγ = −2, dot-dashed black is mγ = −1, dashed red is mγ = 0, solid

green is mγ = 1, dotted purple is mγ = 2. Incoming photon polarization is (a) – LC, (b) – RC. ···················· 81 Figure 6.4. Photoelectron polarization as a function of the electron’s distance to

the optical vortex center for the vortex pitch angle θq = 0.1 rad; long-

dashed blue is mγ = −2, dot-dashed black is mγ = −1, dashed red is

mγ = 0, solid green is mγ = 1, dotted purple is mγ = 2. GaAs sample and incoming photon polarization, are, respectively (a) – unstrained and left-circular, (b) – unstrained and right-circular, (c) – strained and left-circular polarization and (d) – strained and right-circular. ····· 83 Figure 6.5. Photoelectron polarization averaged over the circular mesoscopic semiconducting target of radius r centered at the beam axis with the pitch

angle θq = 0.1 rad; long-dashed blue is mγ = −2, dot-dashed black

is mγ = −1, dashed red is mγ = 0, solid green is mγ = 1, dotted pur-

ple is mγ = 2. The GaAs sample and incoming photon polarization, are, respectively (a) – unstrained and left-circular, (b) – unstrained and right-circular, (c) – strained and left-circular polarization and (d) – strained and right-circular. Shadowed vertical regions indicate 10x lattice periods for GaAs (r=5.65 nm). ·············· 84 Figure 6.6. Schematic picture of a AlGaAs/GaAs quantum dot (left) and its energy band structure forming the 3D quantum conﬁnement (right). ····· 87

xv Figure 6.7. Transition rates for d → s type E2 transition for single-fermion problem

accounting for spin coupling. The parameters used are Λ = 1, mγ = 1,2,3 for (a), (b) and (c) respectively and λ = 550 nm. ······· 89

huBB( ,t) − uBB( ,t)i n Figure B.1. Average value ¯mγ ρ mγ ρ as a function of the parameter to compare the BB mode eqn. (2.25) and its expression in terms of

spherical harmonics eqn. (B.1) for different values of n and mγ . ··· 109

xvi List of Tables

Table 3.1. Measured Rabi frequencies ·················· 45 Table 3.2. Plane-wave Transition Matrix Elements ············· 47

Table 6.1. Photoelectron polarization for the pitch angle θq → 0 with accuracy

2 O(θq ) for different sets of photon quantum numbers as a function of the parameter x = 2πb/λ. Top two rows are for the unstrained GaAs and bottom two rows are for the strained GaAs sample. ········· 80

xvii List of Abbreviations

1. ASR angular spectrum representation. 2. BB Bessel beam. 3. BG Bessel-Gauss. 4. EM electro-magnetic. 5. HCI highly charged ion. 6. LC left-circular. 7. LG Laguerre-Gauss. 8. OAM orbital angular momentum. 9. QED quantum electrodynamics. 10. QM quantum mechanics. 11. RC right-circular. 12. SAM spin angular momentum. 13. TAM total angular momentum. List of Symbols

1. I nuclear spin angular momentum in hyperﬁne coupling scheme. 2. J total angular momentum.

3. Λ photon helicity. 4. L orbital angular momentum. 5. S spin angular momentum. 6. J the Bessel function of the ﬁrst kind Watson (1995).

7. `γ twisted beam’s OAM quantum number Lz = h¯`γ (winding number, vorticity, topological charge).

8. εkΛ global photon polarization basis with Λ = 0,±1 components.

9. eΛ local photon polarization basis with Λ = 0,±1 components.

10. mγ twisted beam TAM quantum number: Jz = hm¯ γ . 11. σ particle spin.

12. θk the angle of the wave-vector tilt with respect to the beam’s propagation direction (pitch angle).

13. θz the constant quantizing magnetic ﬁeld B alignment angle with respect to the beam’s propagation direction.

14. w0 the Gaussian beam waist Siegman (1986). Chapter 1: Introduction

“The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them.” Sir William Lawrence Bragg (1890 - 1971)

The invention of the laser can certainly be called one of the fundamental breakthroughs of the 20th century in the field of optics. As an excellent probe, it boosted atomic spectroscopy and metrology, enabling new techniques such as optical trapping and cooling. As a medium, it promoted information technologies with quantum entanglement, mode division-multiplexing, D-level protocols with topologically structured beams, and many more. From the point of theoretical research, it inspired such studies as photonics, quantum optics, and the recently emerged field of singular optics. One of the fundamental differences between the classical and quantum description of electro-magnetic (EM) radiation lies in the understanding of the energy exchange mechanism between the radiation field and matter: classical EM theory relates energy to the wave amplitude, while quantum mechanics (QM) fosters frequency-dependent selection rules (e.g. the photoelectric effect). Historically, the two approaches have been developing almost in parallel. This thesis can be related to the emerging field of singular optics that studies topologically stable singular photon beams. From the theoretical point of view, singular optics makes an attempt to unite particle and wave descriptions of light via spatial distribution of radiation fields, and introduces the term topological photon (or topological state of light). The orbital angular momentum (OAM) modes, otherwise called hollow modes due to the peculiar doughnut-like shape, carry well-defined quantized units of rotation. They are

1 known to be structurally stable on propagation in free space and matter. Certain modes, such as Bessel and Bessel-Gauss (BG) modes, are even capable of self-reconstruction. One of the most popular, both in the lab and in theory, are Laguerre-Gauss (LG) OAM beams. They are known to be straightforward to generate. In theory, Gaussian decay suppresses secondary minima in Laguerre polynomials, producing theoretical models that accurately describe the physical OAM laser beams. We start with a review of the theoretical tools of laser optics, such as two fundamental approximations: the plane wave approximation and the paraxial approximation. We discuss the angular spectrum representation (ASR), as it is a crucial step in the plane wave expansion of the Bessel beam (BB). These provide us with fundamental building blocks for our quantum theory of topological states of light. We show how a well-defined OAM can be decomposed into a bundle of Bessel modes, taking advantage of the Bessel functions forming a complete orthonormal basis. This provides us with a sufficient base to develop a QM approach for describing the OAM-photon-matter interactions. Our theory, derived from first principles, reproduces and predicts the global features of the analyzed data. It correctly describes QM absorption processes in photon-matter systems. We were focusing on qualitative reproduction of the QM selection rules. This study is for proof of concept and quantitative assessments of the statistically defined goodness of the fits were considered to be not necessary.

1.1 History

Topological effects have been of interest in theoretical physics and optics for a long time. The non-trivial phase dependence in QM wave functions, arising from field singularities, was brought up by Dirac (1931) in the context of his magnetic monopole theory. He introduced the notion that arbitrariness of the QM wave function may be challenged in a fundamental way, but for a while it remained experimentally unjustified. Years later, Sommerfeld (1950) discusses field singularities when explaining the formation of shadows

2 in light patterns and various exotic diffraction problems. One of the first extensive studies of wave dislocations was done by Nye and Berry (1974). Already then, in the 1970s, they foresaw singular propagation modes being useful in remote sensing. A possibility to generate singular vortex beams in laser cavities was discussed by Coullet et al. (1989), by comparing the light generation in the laser cavity and the spontaneous gauge symmetry breaking in the Ginsburg-Landau theory of superconductivity, e.g. Mineev et al. (1999). From the other perspective, the family of exact solutions of the Helmholtz equation in cylindrical coordinates was also known many years before singular beams were even hypothesized, e.g. Stratton (1941). In particular, the Bessel mode was later extensively analyzed by Durnin (1987) and Durnin et al. (1987). In these papers the authors already predict non-diffractive behavior and a longer propagation range for BB modes. In 1991, a graduate student of Leiden University, the Netherlands, shared his ideas about how to generate a laser beam with a twist. Approximately in a year, that student and Les Allen generated such a laser beam. That student’s name is Han Woerdman, see Allen et al. (1992). The main discovery in this work lies in the realization that states of light with a well-defined propagation direction, such as laser states, are capable of carrying additional rotational quanta. For an overview of the OAM-light history we suggest the recent review article by Mann (2018). Since it has been shown that laser modes can carry a well-defined OAM, a lot of new and exciting physics has been discovered. An extensive study of non-diffractive behavior of cylindrical beams eventually revealed self-reconstructive properties, e.g. Bouchal et al. (1998). These, in turn, resulted in registering the greater average penetration length of such beams in turbid media, see, e.g., Purnapatra et al. (2012) and Mamani et al. (2018). The study of the group velocity of OAM light led to the revelation of subluminal propagation, see Bouchard et al. (2016) and Lyons et al. (2018). Comprehensive historical reviews on the development in the field of singular optics can be found in the papers by Dennis et al.

3 (2009) and Yao and Padgett (2011); and books by Torres and Torner (2011) and Andrews (2011). A great variety of ways to generate twisted photon beams in the lab has been invented over the past 27 years. The most utilized and available ones are the diffraction gratings with fork dislocations, see, e.g., Bazhenov et al. (1992), Bekshaev and Karamoch (2008) and Roux (1993). Another important technique uses helical undulators, described, for instance, by Sasaki et al. (2007) and Afanasev and Mikhailichenko (2011). There have been fundamental proposals in the ﬁeld, that suggest the possibility to generate twisted X-rays, see Jentschura and Serbo (2011b) and Taira et al. (2017). OAM-beams in the hard X-ray range have been generated by Taira et al. (2019) using spiral phase plates, and the soft X-rays were recently generated by Lee et al. (2019) with diffraction gratings. Production

of twisted γ-rays has been proposed by Taira et al. (2017). Among the hypothesized natural sources of OAM photons are rotating black holes, see, e.g., Harwit (2003) and Tamburini et al. (2011), and synchrotron radiation, see Molina-Terriza et al. (2007) and Katoh et al. (2017). Ion trapping and cooling has triggered a revolution in atomic spectroscopy and metrology, and later caused a wave of engineering applications in quantum computing and communications, see e.g. Cirac and Zoller (1995), Häffner et al. (2008) and Wineland et al. (1998). The possibilities of integrating topological photon beams into these technologies is also actively discussed Nagali et al. (2009), Woerdman et al. (2010). The so-called D-level communication protocols offer the advantages of high communication bandwidth and cryptographic security, e.g. Cerf et al. (2002) and Calvo et al. (2006). The entangled pairs are created in the lab by spontaneous parametric down-conversion, e.g. Pearsall (2017). It has been long known that photons can be entangled in frequency and polarization. It was conﬁrmed in Mair et al. (2001) that one can also entangle the photon OAM, which enabled entanglement in high-dimensional Hilbert space of photon degrees of freedom, given by OAM and spin angular momentum (SAM). Nowadays, a lot of effort is being put towards

4 both theoretical descriptions of OAM-related phenomena and experimental advances in this realm, e.g. Leach et al. (2012), Erhard et al. (2017), Babazadeh et al. (2017) and Krenn et al. (2017). It is worth mentioning that these developments and technological advancements are intimately connected with the progress in studying OAM phenomena in solid state physics. The focus of this work is the interaction of the OAM light and matter beams with atomic and nuclear matter. On the photonics side, absorption of OAM photon beams by atoms, solids and nuclear matter are discussed. The interactions with nuclear matter are considered in this thesis for the first time. Understanding these processes has been of interest across many areas of science: in nuclear, atomic and molecular spectroscopy, nanotechnology, quantum computing, communication technologies, astrophysics, to name a few. OAM photon-atom interactions have been a focus in singular optics for a while, see e.g. Franke-Arnold (2017). The QM formalism of the OAM photo-excitation in hydrogen-like ions was studied extensively in Afanasev et al. (2013), followed by a series of papers: Afanasev et al. (2014a), Scholz-Marggraf et al. (2014), Afanasev et al. (2016) and more. These works laid the foundation of my research project. Substantial effort has been put towards understanding the QM mechanism of photoionization and radiative recombination with photon vortex beams by Matula (2014) in his dissertation, and the corresponding series of publications. The LG photon mode, scattering off the hydrogen atom in the Born approximation and to the first order in perturbation theory, has been considered in Davis et al. (2013). The authors’ results for magnetic quantum number selection rules are of the same nature as the ones to be discussed in this thesis. As for many-electron atoms, I would like to highlight the following works: Surzhykov et al. (2015) and Rodrigues et al. (2016). In the first paper, the detailed description of scattering in the relativistic regime is given. The authors note the peculiar sensitivity of the electrons’ photo-excitation process with respect to the relative phase, coming from the topology in the beam profile. This result is

5 reciprocal to the discussion of the phase effects in Chapter 3 of this thesis. Rodrigues et al. (2016) accounted for hyperfine interaction in Rydberg atoms, and noted that higher-order multipolar contributions are expected to be enhanced in twisted photon-atom interactions. Chirality of light has been an irreplaceable tool to extract morphological information from chiral organic samples (molecules, proteins) for at least the past 50 years. As the solid state technology progressed, the same techniques were adopted for analyzing chiral nanostructures, metamaterials and plasmonic devices. The torques, induced by chiral matter on light, are well described, but the question of optical chirality remains open on both fundamental and experimental levels, see Bliokh and Nori (2011), Bliokh and Nori (2015) and Tang and Cohen (2010). Recent discoveries in singular optics have drawn a new wave of attention to this topic, see, e.g., Andrews et al. (2004), Brullot et al. (2016), Forbes and Andrews (2018) and Forbes (2019). Also, dichroic effects of solely photon-OAM origin have become a new trend, see, e.g., Zambrana-Puyalto et al. (2014) and Zambrana-Puyalto et al. (2016). Evident chirality of OAM-carrying modes intuitively inspired physicists to search for the photon OAM coupling to chiral structures. Surprisingly, negative results have been reported, e.g. Löffler et al. (2011). However, later work identified that the diameter and the helical pitch of the chiral structures and the OAM light have to be in a certain proportion, and then giant effects of 200 times dichroism enhancement can be observed, Ni et al. (2018). As for the interactions with solid state systems, big effort has been put towards understanding the electron photo-excitation and subsequent spin dynamics in the conduction zone of a bulk semiconductor and various nano-systems by Cygorek et al. (2015) and Quinteiro and Tamborenea (2009a) in their series of works. They use the semiclassical formalism, namely, describe a classical OAM-beam interacting with a quantum semiconductor structure. The experimental research on this has been conducted by Clayburn et al. (2013). It did not take scientists too long to extrapolate the results in singular optics onto matter

6 beams. The results obtained by McMorran et al. (2011) were revolutionary in electron microscopy. This research laid the foundation for new techniques in superresolution electron microscopy. The proof of concept showed that it is possible to experimentally generate twisted electron beams, which triggered interest in adjacent ﬁelds, see, e.g., Matula et al. (2014). In this thesis, neutron beams will play an important role. Twisted cold neutrons were generated for the ﬁrst time at the National Institute of Standards and Technology (NIST), see Clark et al. (2016). Over the course of the past three years, the group at NIST has developed a theoretical framework and experimental techniques for twisted neutron interferometry.

1.2 Thesis Outline

This thesis presents cross-disciplinary research on twisted beams interacting with matter. In the process, we were looking for prominent effects that have been registered, or are feasible to register, in experiments. In terms of theoretical tools, the framework is laid out in Chapters 2 and 3. The rest of the thesis shows the generality of QM formalism to be applied to various systems and scales. Chapter 2: This chapter is devoted to establishing the QM framework for the description of OAM states of light. The discussion goes over the main concepts, jargon and deﬁnitions of singular optics and photonics. QM observables for different families of OAM modes, relying on symmetry arguments and axiomatics, are discussed. The modes of interest for my research are introduced and described in ASR , as well as deﬁnitions for photon polarization and exotic polarization states, which are further discussed in subsequent chapters. Chapter 3: Atomic photo-absorption and photo-luminescence are crucial processes to explore on the way to understanding the photon-matter OAM coupling. In Chapter 3, the QM OAM photo-absorption matrix is derived using the multipole expansion in vector spherical harmonics, accounting for the electron spin degree of freedom. The

7 theory of dipolar interactions is extended to higher-order multipolar absorption processes in Hydrogen and Hydrogen-like atoms. Fine and hyperfine interactions have been taken into account. The theory for treating the recoil in the framework of QM of cold trapped ions is set up. The formalism describes the enhancement of the multipolar contributions, higher than electric dipole, and violation of QM photo-absorption selection rules. The experimental results, confirming these phenomena, were produced by our collaborators in Mainz in a setup with cold 40Ca+ ions trapped in a Paul trap and exposed to a non-zero- OAM LG laser beam. A comparative analysis of the data versus theory is presented in the last section of Chapter 3. Chapter 4: The necessary phenomenology and terminology for a description of the chirality of matter and EM radiation are introduced. The phenomenon of magnetic circular dichroism in non-chiral matter (Hydrogen gas) has been predicted and the corresponding theory has been developed by Afanasev et al. (2017) with no account for electron spin. The results presented in Chapter 4 account for electron spin using the coupling scheme developed in Chapter 3. The theory predicts that isotropic matter absorbs OAM-photons with opposite chiralities with different rates. This effect has been validated on the level of individual multipolar contributions to the overall amplitude. Chapter 5: The nuclear processes of proton-neutron capture and deuteron photodisintegration are considered using time-dependent first-order perturbation theory. The Fermi’s Golden rule is used to calculate the matrix element and extract the OAM selection rules for the magnetic quantum numbers. The twisted fermion (neutron) state is introduced to describe the OAM neutron-proton interaction. Analogous to the case with OAM photo- absorption by ions and atoms, nuclear selection rules are predicted to change as well due to the strong gradients of the EM-fields in singular photons. Chapter 6: The general approach for OAM transfer to a bulk semiconductor structure is developed using the Luttinger formalism for total angular momentum (TAM) coupling in

the near-states of a photo-excited electron. The class of vertical transitions in the Γ-point of

8 I-III direct bandgap semiconductors is considered, and the corresponding set of modiﬁed selection rules is discussed. The polarization of photo-excited electrons immediately after the photo-absorption event is studied in detail. The possibility to observe the photon’s OAM-related anomalies in mesoscopic semiconductor targets is predicted. In the last section, the magnetic selection rules for photo-excited GaAs quantum dots are derived. Chapter 7: This chapter summarizes the main contributions that this thesis has made in the cross-disciplinary ﬁeld of quantum photonics and topological states of light and matter. The outlook to potential future developments is also given.

9 Chapter 2: OAM Photon States

2.1 Review of Necessary Fundamentals in Optics

To fully define the system of EM-field interacting with medium, one needs five equations in electrodynamics, which are the set of Maxwell’s equations:

1 ∇ ·B = 0; ∇ ·E = ρ(r,t); (2.1) ε ∂B 1 ∂E 1 ∇ ×E = − ; ∇ ×B = + j(r,t) (2.2) ∂t µε ∂t ε

and the Newton-Lorentz equation

d2 h i m r(t) = q E(r(t),t) +v(t) ×B(r(t),t) (2.3) dt2 valid for each charged particle in the system (in SI units). Here µ is the magnetic permeability and ε is the electric permittivity of a medium, ρ(r,t) and j(r,t) are the charge and current density, respectively; m is the particle mass, q is the charge, and v(t) is the velocity. The EM-field is parametrised in terms of electric field strength E and magnetic flux density B, generally given as functions of a position vector r and time t. The wave equation for EM fields in vacuum is the direct consequence of the eqs (2.1)-(2.3): 1 ∂ 2 ∇2u(r,t) − u(r,t) = 0. (2.4) c2 ∂t2

For a monochromatic source in reciprocal space, the full Helmholtz equation is

(∇2 + k2)u(r) = 0. (2.5)

10 The simplest solution is a plane wave propagating in the z-direction in free space:

i(kz−ωt) √ i(kz−ωt) E = e1E0e , B = e2 µ0ε0E0e , (2.6)

where ei is a polarization basis and µ0 and ε0 are the magnetic permeability and electric permittivity constants in vacuum, respectively. In general, one can ﬁnd a solution by separation of variables in 11 coordinate systems, see, e.g., Korotkova (2013). However, to obtain a beam-like solution one needs to impose the preferred direction, which leaves us with only 4 solutions: rectangular-, circular- , elliptical- and parabolic-cylindrical. When solving eqn. (2.5) in optics or quantum electrodynamics (QED), one usually assumes harmonic time-dependence e−iωt. It means we effectively neglect non-locality of the matter, which allows a simpler form of linear

response functions: ε(k,ω) → ε(ω) and µ(k,ω) → µ(ω). This assumption, however, might have to be reconsidered for certain cases of attenuation in materials with non-simple response (e.g. hysteresis) or high enough ﬁeld strengths for optical applications. The issue of non-locality in QED is an area of ongoing debate, and we will not discuss this subject here. The plane wave, most commonly used in physics, is the solution of the EM-wave equation in Cartesian coordinates. Despite its simplicity, it is proven to be an extremely useful theoretical tool that allows expansion of generally complicated multimodal ﬁelds into an ensemble of oscillations with local planar wave fronts. Citing Jackson (1999), "the simplest and most fundamental electromagnetic waves are transverse, plane waves". In laser optics the approximation, associated with the plane-wave solution implies,

∂ 2u(r) ∂ 2u(r) ∂ 2u(r) , << (2.7) ∂x2 ∂y2 ∂z2

meaning that the transverse deviations in the beam proﬁle on propagation are much smaller than longitudinal, e.g. Siegman (1986).

11 Making one more step, the theoretical description of laser optics is often based on the notion of light bundles of plane-wave-like EM-radiation, propagating in a well-deﬁned direction. For such tightly focused rays of light, the frequently used ansatz is

u(x,y,z) = W(x,y,z)e−ikz (2.8) with a plane wave part traveling in the z-direction, and the spatial proﬁle described by the function u(x,y,z). Technically, all it implies is an explicit factorization of a rapidly varying plane wave in the expression for the amplitude. This approximation is widely used in Gaussian laser optics. If eqn. (2.8) is plugged into the Helmholtz equation eqn. (2.5), one gets ∂ ∇2 + 2ik W(x,y,z) = 0, (2.9) ∂z where ∇ is a 3D-derivative. The solution to this equation spreads and disperses with the propagation, but the envelope changes slowly with propagation, so that the beam stays well collimated over large ∆z, Siegman (1986). This requirement is necessary to obtain the beam-like behavior of the mode. Hence, we need to make the following approximation

∂ 2W(x,y,z) ∂ 2W(x,y,z) ∂ 2W(x,y,z) << , (2.10) ∂z2 ∂x2 ∂y2 which is called the paraxial approximation. In diffraction theory, the paraxial approximation in the reciprocal space is

s k2 k2 k = k 1 − ⊥ ≈ k − ⊥ . (2.11) z k2 2k

It is interesting to note is that one of the ﬁrst theoretical papers, see Lax et al. (1975), where the paraxial approximation was explicitly applied to the Helmholtz equation, arrives at the LG-like mode as the result. It is worth noting that after applying the paraxial approximation to the EM ﬁelds,

12 they do not satisfy Maxwell equations, and hence the Helmholtz equation anymore, e.g. Korotkova (2013). In this limit, Fourier optics and ASR shown in eqn. (2.12), are essentially equivalent to each other. Continuing thinking in terms of bundles of rays building up complicated beam profiles, we may now apply the main mathematical tool of Fourier optics, the Fourier transform, and obtain the ASR of a beam profile, see, e.g., Sherman (1982) and Goodman (2005) for detailed fundamental description, or Korotkova (2013) for applications in singular optics in turbid media. Given a mode propagating in a homogeneous and isotropic medium, ASR allows to decompose any beam-like solution of the Helmholtz equation in an ensemble of plane and evanescent waves, propagating in a well-defined direction. The ASR integral is usually expressed as the 2D-Fourier transform:

ZZ +∞ ikk⊥·ρ ikzz u(ρ;z) = dkk⊥ a(k⊥;z)e e , (2.12) −∞

where k⊥ = ksinθk is the normal spatial frequency, such that θk is the pitch angle of the p directional tilt of the local wave-vector, see Durnin et al. (1987); |r| = ρ2 + z2 is the

position-vector in cylindrical coordinates. Here a(k⊥;z) is the Fourier kernel, such that

ZZ dρ a(k ;z) = u(x,y,z)e−ikk⊥·ρ (2.13) ⊥ (2π)2 which gives the amplitudes of the plane waves. The exponential factor eikzz is called a response function (Fourier optics) or ﬁlter function (signal processing). It should be emphasized that the longitudinal wavenumber is deﬁned as a function of the transverse q 2 2 spatial frequency kz = k − k⊥, not a parameter. The decomposition allows for the wave-vectors such that

 q 2 2 2 2 2 2  k − kx + ky , kx + ky ≤ k ; homogeneous waves kz = (2.14) q 2 2 2 2 2 2  kx + ky − k , kx + ky ≥ k ; evanescent waves

13 The evanescent nature of singular beams in the realm of ASR is an open question to be addressed, Berry (1994). However, it is clear that the transformation kernel accounts for the decay of evanescent fields in the Fresnel diffraction limit. The beauty of ASR is that in far field it transforms into the limit of geometrical optics, and in the paraxial approximation it leads to the Fresnel diffraction integral, e.g. Goodman (2005). One of the most important advantages of defining laser beams via ASR is that if

q 2 2 the resulting ﬁelds obey the condition kx + ky << k, and their Fourier kernel is much smaller than 1, it results in highly directional beams, Korotkova (2013). The proof can be found in the book by Mandel and Wolf (1995), Sec. 3.3. To compare, working with the families of solutions of the paraxial equation does not guarantee directionality.

2.2 Vortex Beams

Typically, when one characterizes a beam of light, one deﬁnes it in terms of polarization, direction of propagation, frequency, coherence, focusing, etc. The phase-related effects in waves of different nature, though repeatedly discussed for many years, e.g. Findlay (1951), Nye and Berry (1974) and Wesfreid and Zaleski (1984), have triggered a renaissance era in the study of wave phenomena only relatively recently. Dislocations in light and matter beams gave birth to a new branch of physics: singular optics. Mathematically, singularities are represented by discontinuities in functional mapping on a smooth parameter space. One can deﬁne a singular function as function f (X) :

{x1,...,xi,...,xn} ∈ X, such that f has at least one xi on X where its value f (xi) is ill deﬁned. One of the simplest mathematical examples is a function, deﬁned on the domain of complex numbers: p x + iy = x2 + y2 eiφ , (2.15)

−1 y where φ = tan x . While the left side is well defined everywhere in space, the right side carries a phase factor, ill defined at x = y = 0 – no definitive phase can be assigned to

14 Figure 2.1: The generic phase proﬁle of the OAM-carrying beam with a screw-like dislocation at the beam center, propagating in the z-direction.

u(0;0) = 0 + i0. One of the types of dislocations of special interest is called a screw-dislocation (solid state) or vortex (hydrodynamics, photonics, superfluidity). Our intuition is well trained to recognize vortices: rotation of water in the sink, tornadoes, etc. Optical vortices can be described as the rotational motion of optical rays in the laser mode bundle around the zero intensity region. An optical vortex can be defined as a type of EM laser mode, characterized by an annular intensity profile and carrying well-defined units of OAM. Hence, the OAM

projection on the beam propagation direction z is Lz = h¯`γ , where `γ = 0,±1,±2,.... The

quantity `γ is called topological charge (adopted from abstract geometry) or vorticity, as deﬁned by Dennis (2001). Topological charge, according to Coullet et al. (1989), is a gradient circulation around the closed loop enclosing the dislocation. Dennis (2001) deﬁnes a topological charge as:

I drr ` = · ∇φr, (2.16) C 2π where C is the closed contour around the singularity. The laser modes carrying a well-deﬁned OAM contain a phase-dependent multiplier

ei`γ φr , which introduces the screw-like topology into the phase proﬁle, Figure 2.1, and

15 alters parity. This can be seen from the structure of eqs (2.5) and (2.9) – second-order differential equations, when assuming the problem to be cylindrically symmetric. The physical signiﬁcance of phase singularities and the observable effects, related to their presence in the wave front, were brought up for the ﬁrst time by Nye and Berry (1974).

The authors introduced two essential criteria that deﬁne dislocations in waves: the phase φr

of the wave should change by a multiple `γ of 2π on a closed path around it; the dislocation is localized in the area of zero amplitude of the wave. The OAM-carrying beams appear to have a hollow intensity profile, such that zero- intensity coincides with the region of singularity, e.g. Andrews (2011). This fact used to make scientists skeptical of any effects being observable in the vicinity of the vortex. Zero intensity used to be associated with zero photons being observed in the region of singularity. This was a serious drawback, especially because most exciting phenomena indeed happen near the beam center. The first solid arguments and theoretical evidence of “hollow beams not being hollow” were provided by Klimov et al. (2012). They explicitly associate the possible OAM-triggered effects to the presence of non-zero magnetic fields and strong field gradients in the vortex region. They state that even though interactions, driven by an electric dipole E1, appear to be insensitive to the OAM-carrying field structure due to zero-electric field component, higher-order multipoles do respond to the EM-stimulation and can be observed experimentally. The aforementioned can be extrapolated to matter beams due to the particle-wave duality principle. The semi-classical formalism for free propagation of electron vortex beams has been developed by Bliokh et al. (2007). The well-defined trajectories of the electrons in such matter beams have been proven to spiral around the beam singularity in the same way as optical rays circle around the annular area of an OAM laser mode. In this work, Bliokh and collaborators had foreseen applications in electron microscopy if such beams would have become possible to generate in the lab. OAM electron beams were successfully generated by McMorran et al. (2011), and the related revolutionary

16 techniques in electron microscopy, such as contrast enhancement, were achieved for the ﬁrst time. Analogous effects have been foreseen by the experimental group at NIST, where twisted neutron beams have been experimentally observed for the ﬁrst time Clark et al. (2015). This direction is currently a subject of ongoing research.

2.2.1 Orbital Angular Momentum and Spin of Light

OAM of light has always been at the very foundation of electricity and magnetism, introduced by Maxwell (1861). It is common knowledge that spherical waves can be expanded in TAM basis and carry OAM, together with the spin degree of freedom. It has also been known for a long time in QED, see Akhiezer and Berestetskii (1959), and spectroscopy, see Corney (1978), that photons, absorbed or emitted as the result of higher- order multipolar transitions in atoms, can transfer or carry multiple units of TAM, where only ±1 may come from polarization. For spherical waves of this sort, TAM J is a QM observable, while OAM L and SAM S are not. For twisted beams, the problem of separation of TAM of light beams into OAM and SAM is a question of ongoing debates in the field of singular optics and quantum photonics. Phenomenologically, it is obvious that this separation should be unphysical at the very least because SAM is defined as the TAM of a particle in its rest-frame, which does not exist for a photon. There is another, more rigid, argument, that the spin and orbital parts of TAM of classical free EM-field, as defined, for instance, by Cohen-Tannoudji et al. (1997),

Z j Jtrans = ε0 drr E⊥(r · ∇)A⊥ j +E ⊥ ×A⊥ (2.17)

do not satisfy the rules of standard angular momentum algebra. One preserves either gauge invariance or rotational group symmetry. In this sense, strictly speaking, only TAM J is a valid observable. These arguments are summarized by Van Enk and Nienhuis (1994).

17 A beautiful introduction to the study of OAM-carrying light is laid out in the review by Bliokh and Nori (2015). As mentioned before, it was ﬁrst pointed out by Allen et al. (1992) that twisted beams in the paraxial limit are eigenstates of OAM and SAM independently. Starting from the EM-ﬁeld in its general form:

A(ρ,z) = (αρˆ + βφˆ)u(ρ,z)eimφ eikzz, (2.18) where {ρˆ,φˆ,zˆ} are the unit-vectors of cylindrical coordinate system. Writing out the ﬁelds in the Lorentz gauge: c2 E = −iω A + ∇(∇ ·A), iω (2.19) B = ∇ ×A,

the authors arrive at the following expression,

2 2 rΛ ∂|u(ρ,z)| Jz = ε ω `|u(ρ,z)| − , (2.20) 0 2 ∂r where Λ = ±1. Hence, under the conditions of the paraxial limit eqn. (2.10), the convenient separation J = L +S is valid for photon beams with OAM. In other cases one would need to work in TAM basis. One of the important, and easy to infer, consequences of this discovery is that light beams with non-zero OAM must carry non-negligible momenta in the plane, transverse to the propagation direction, e.g. Bekshaev et al. (2011). However, these momenta are known to be very small in comparison to their longitudinal counterparts. As was pointed out by Berry (1998), in paraxial beams one can even neglect the net transverse momentum contribution and treat OAM of light as an intrinsic property of twisted beams, in the same sense as the photon spin.

18 2.3 Orbital Angular Momentum Modes

In this section we consider three modes of twisted-light beams: BB, BG and LG modes, see Afanasev et al. (2018a,b). Even though all of them represent optical beam-like ﬁelds, they belong to fundamentally different families. The Bessel mode, Figure 2.2, row 3, is an example of a non-paraxial mode, structurally stable under propagation. The BG mode is the Helmholtz-type beam which satisﬁes the paraxial wave equation and is characterized by the Bessel-like behavior close to the beam axis and the Gaussian-like decay on the periphery, see Figure 2.2, row 4. The LG mode, Figure 2.2, rows 1 and 2, plays a fundamental role in photonics, laser optics and resonators, see Siegman (1986) and Teich and Saleh (1991). It belongs to the family of Gaussian solutions to the scalar paraxial equation. The fundamental difference between the non-paraxial BBs and paraxial BG beams is that the former satisfy Maxwell’s equations, while the latter one, strictly speaking, does not. However, one can still apply conventional QED methods to BG modes for the case of not-tightly focused paraxial beams. Photo-excitations by BBs is the convenient place to start, due to the elegance of the mathematical representation and the property of Bessel functions to form a complete orthonormal basis. These allow us to simply expand the other beam-like solutions in terms of Bessel modes with further application of the formalism developed for BBs.

2.3.1 Bessel Beams

The Bessel mode is the solution of the Helmholtz equation, belonging to the family of diffraction-free beams. One of the ﬁrst detailed descriptions can be found in the paper by Durnin (1987). The kernel of these modes, the Bessel function, is frequently encountered in theoretical physics. A short and comprehensive overview of the properties and applications of Bessel functions is given in, e.g., Arfken and Weber (2005). A more complete and

19 Figure 2.2: LG (rows 1 and 2), Bessel (row 3) and BG (row 4) modes’ intensity proﬁles for different values of OAM `γ , p nodal number, and TAM mγ . All the axes are in arbitrary units and the columns correspond to topological charge `γ for LG, and TAM quantum number mγ for BB and BG.

20 mathematically advanced treatment of Bessel functions is offered by Watson (1995). Some of the relations, expansions and limits of Bessel functions are listed in Appendix A. We start with a brief and standard overview of the Helmholtz equation in cylindrical coordinates and the resulting Bessel modes. Starting with (2.5) and using the ansatz

u(r,t) = R(ρ)Φ(φ)e−ikzzeiωt (2.21) one arrives at the Bessel equation

d2R 1 dR m2 + + κ2 − γ R = 0, (2.22) dρ2 ρ dρ ρ2

2 2 2 where κ = k − kz . The general solution is

R(ρ) = AJmγ (κρ) + BNmγ (κρ), (2.23)

where Jm(x) is the Bessel function as in eqn. (A.9), and Nm(x) is the Neumann function, e.g. Abramowitz and Stegun (1964); A and B are constant coefﬁcients. Neumann functions are typically considered to be unphysical. Using the normalization condition

Z 2πδ(k −k0) u†(ρ,t)u(ρ,t)dV = (2.24) V 2ω

one can calculate A = pκ/2π. Together with the azimuthal part, the solution is

r BB κ i(m φ−kzz) iωt u (ρ,t) = Jm (κρ)e γ e + h.c. (2.25) 2π γ

where h.c. stands for hermitian conjugate of the ﬁrst term. Here mγ represents the total angular momentum J projection on the direction of propagation of the physical beam. In the paraxial regime one may express it as a function of the vorticity as mγ =`γ +Λ, where

21 Λ stands for the photon helicity ±1,0. Bessel-like beams, generated in the lab, are exclusively stable with respect to diffraction effects and exhibit beam reconstruction properties, see Durnin (1987) and Bouchal et al. (1998). The central core of the beam can be estimated as

ρ0 r0 = , (2.26) ksinθk

such that Jmγ (ρ0) = 0. For example, for mγ = 0 we get ρ0 = 2.4048 and so on, which can be used to assess of the OAM beam radius. As the Bessel mode belongs to the family of the exact solutions of the Helmholtz equation, it can be expressed in ASR eqn. (2.12) by taking a spatial component of eqn. (2.25) at z = 0 being an initial ﬁeld proﬁle traveling through the halfspace z > 0

Z dkk BB −i(kzz−ωt) ⊥ ikk ·ρ u (ρ,z;t) = e a m (k )e ⊥ + h.c., (2.27) (2π)2 κ γ ⊥

where aκmγ (k⊥) is a 2D Fourier transform of u(ρ;z = 0,t = 0):

r m 2π im φ a m (k ) = (−i) γ e γ k δ(k − κ). (2.28) κ γ ⊥ κ ⊥

Here, eqn. (A.2) has been used. We have used the known integral representation of the Bessel function (A.9). Here we have not used the paraxial approximation, taking the integral over all homogenous and evanescent components of the radiation field. Rapid extinction of the evanescent contributions in the far field of a laser beam allows to use the decomposition of this form for the most general analysis. At this point we are ready to move on to the quantization of the photon field, carrying the Bessel-beam topology. The corresponding QM formalism has been developed by Jentschura and Serbo (2011a) and Afanasev et al. (2013, 2014b, 2016). The main idea is to decompose the total (global) field of the mode into its local

22 Figure 2.3: Schematic picture of the twisted beam geometry from the prospective of the local observer P’ versus the coordinate system P attached to the beam axis. The red arrow speciﬁes the overall direction of the OAM beam propagation.

counterparts. In Figure 2.3 the local point-like observer P0, located at the position r in the beam wavefront does not experience the effects of the beam topology. Instead, they see the plane-wave-like behavior of EM-ﬁeld, with the local wave-vector k, tilted by the

angle θk, deﬁned after eqn. (2.12). We adopt the convention of the “local” polarization

state eΛ being deﬁned with respect to the observer’s plane-wave-like photon’s propagation

direction as   √1 {−Λ;−i;0}; Λ = ±1  2 eΛ = (2.29)  {0;0;1}; Λ = 0

In the global beam basis, however, it would be

j −imφk εkΛ = ∑ dΛ,m(θk)e em, (2.30) m=±1,0

j where dΛ,m(θk) is the Wigner rotation matrix, as deﬁned by, for instance, Edmonds (1957). Some of the useful properties of this matrices can be found in Appendix A. This

23 polarization vector can be expressed as, Afanasev et al. (2018b)

−iΛφ 2 θk iΛφ 2 θk Λ εkΛ = e cos eΛ + e sin e−Λ + √ sinθke0. (2.31) 2 2 2

Combining this result with the scalar solution, eqn. (2.27), one gets the vector potential

Z d2k BB j ⊥ imφke ikk⊥·ρ † −i(kzz−ωt) AkΛ;ω (ρ,z;t) = ∑ dΛm(θk) 2 {aκmγ (k⊥)e eΛe aˆ e + h.c}, m=±1,0 (2π) (2.32) where aˆ† is the photon creation operator. This expression describes the quantum mechani-

cal state of a twisted Bessel photon of energy E = h¯ω, carrying the TAM quantum number

mγ and helicity Λ .

Due to the diverging norm of the Bessel function Jm(x), the Bessel mode effectively carries an inﬁnite power and has the non-zero intensity everywhere in space. The paraxial

limit of non-paraxial BBs can be taken as κ = ksinθk ≈ kθk = 2/w0 in eqn. (2.32) and earlier in this section.

2.3.2 Bessel-Gauss Beams

BBs are known to be an accurate model of twisted laser modes in the central region of the beam. However, due to their diverging nature, they overﬁt physical modes on the periphery. BG modes offer a more realistic model of an experimentally generated optical beam, where the Bessel-like behavior is strongly suppressed by the Gaussian-like decay, see Figure 2.2. The BG mode was ﬁrst obtained by Sheppard and Wilson (1978), and later directly introduced by Gori et al. (1987). The generalization to the higher-order OAM can be found in the papers by Borghi et al. (2001) and Kiselev (2004):

2 2 BG imγ φρ i(kzz−ωt) −ρ /w0 u (ρ,z;t) = Ae Jmγ (κρ)e e + h.c., (2.33)

24 p where w0= π/λzR is the Gaussian waist of the beam, Λ the wavelength and zR the Rayleigh range. A is the overall normalization constant, that can be calculated from the Fresnel expansion, see Bagini et al. (1996). After taking the 2D-Fourier transform, one can obtain the following form for the plane wave expansion of the scalar BG mode:

2 Z d k (BG) uBG(ρ,z;t) = ei(kzz−ωt) ⊥ a (k )e−ikk⊥·ρ , (2.34) (2π)2 κmγ ⊥ where the integral is taken over the entire reciprocal space, similar to the angular spectrum representation technique, described by Novotny and Hecht (2012), and the contribution

coming from evanescent waves (k⊥ ∈ [k,∞)) is negligible. The corresponding Fourier kernel is

2 2 2 (BG) h κ + k i κw a (k ) = Aπimγ eimγ φk w2 exp − ⊥ w2 I 0 k . (2.35) κmγ ⊥ 0 4 0 mγ 2 ⊥

mγ The function Imγ (z) = i Jmγ (iz) is the modiﬁed Bessel function. Applying the formalism in subsection 2.3.1, we get exactly the form in eqn. (2.32) for the EM vector potential, but with the kernel, derived above. As is shown by Gori et al. (1987), the BG mode, being a mode carrying the Gaussian proﬁle, retains the diffraction-free properties of the Bessel mode.

2.3.3 Laguerre-Gauss Beams

The LG mode is the solution of the paraxial equation eqn. (2.9) in cylindrical coordinates, which is known to be one of the most used modes to fit experimental vortex beams’ profiles. To visually compare the typical intensity profiles of the LG to the BB and the BG for the

25 same parameter set see Figure 2.2. The explicit scalar form is

√ 2 ρ2 ρ 2|`γ | |` | 2ρ − uLG(ρ,z;t = 0) = L γ e w2(z) w(z) p w2(z) kρ2z (2.36) −i c 2(z2+z2 ) i|`γ |φρ +ikzz iφG × e R e q e + h.c. 2 2 1 + z /zR

Here `γ is the beam vorticity factor that coincides with its OAM projection in paraxial

p 2 approximation; wz =w0 1 + (z/zR) is its spotsize; φG = arctan(z/zR) is the Gouy phase of the LG mode. The associate Laguerre polynomial is given by (see, e.g. Abramowitz and Stegun (1964) and Gradshteyn and Ryzhik (2014))

2 p 2 j |`γ |2ρ j (|`γ | + p)! 2ρ Lp 2 = ∑ (−1) 2 , (2.37) w j=0 (p − j)!(|`γ | + j)! j! w where p is the number of radial nodes, which results in p + 1 concentric circles in the beam intensity proﬁle, see Figure 2.2. Bessel function, being a complete set of orthogonal functions, can be used as an expansion basis. Here we will consider the mode in focus z = 0 and perform the Hankel transform given as

Z f (x) = ξF(ξ)Jγ (ξx)dξ (2.38) Dξ Z F(ξ) = x f (x)Jγ (ξx)dx (2.39) Dξ

to obtain the expression for the LG mode, expanded in Bessel modes (2.25), where ξ ∈ Dξ

and x ∈ Dx. We assume that the transformation kernels are symmetric, such that

K(x;ξ) = ξJγ (ξx); K(ξ;x) = xJγ (ξx). (2.40)

26 After applying (??) to (2.36) and using the transform, see Magnus et al. (1954),

ν+1 ν − 1 px2 2 ξ − 2/p H [xν e 4 ;ξ] = e ξ (2.41) ν pν+1 we get the following expression for the scalar LG mode

p √ LG |`γ | u (ρ,z = 0;t = 0) = ∑ Bp j 2π j=0 (2.42) Z ∞ 1 2 j+|`γ |+ 2 2 2 −k⊥w0/4 BB × k⊥ e u (ρ,z = 0;t = 0)dk⊥ 0

|`γ | where the Bessel state is given as in (2.25) and Bp j is the expansion coefﬁcient deﬁned as j 2 j+|` |+2 |`γ | (−1) (|`γ | + p)! w0 γ Bp j = √ (2.43) (p − j)!(|`γ | + j)! j! 2

The vector solution would be

LG j −i(kzz−ωt) AkΛ;ω (ρ,z;t) = ∑ dΛm(θk)e × m=±1,0 (2.44) p √ Z ∞ 1 n |`γ | 2 j+|`γ |+ 2 2 o 2 −k⊥w0/4 BB imφk † × ∑ Bp j 2π dk⊥ k⊥ e u (ρ,z;t)e eΛaˆ + h.c. j=0 0

The integral over k⊥ can be calculated analytically as in eqn.(6.643 4) of Gradshteyn and 2 Ryzhik (2014), with the variable substitution x = k⊥. A similar result was obtained by Peshkov et al. (2017) by performing a 2D Fourier transform.

2.4 Orbital Angular Momentum Beams and Polarization States

Polarization is one of the fundamental properties of light. It plays a crucial role in the description of EM-radiation in Electrodynamics and in QM, it deﬁnes the photon state in QED, etc. In the lab, it allows polarization-based mode multiplexing – one of the widely-used techniques in modern optical communications; non-invasive tools in medical diagnostics; contrast enhancement techniques in spectral analysis, and many more. For

27 Figure 2.4: Schematic visualization of the beam’s polarization state on Poincare sphere and polarization ellipse. our purposes, polarization is a degree of freedom, complementary to the OAM of light, that all-together describes units of rotation that photon beams can carry when propagating through space.

Polarization state can be completely deﬁned by the set of two angles {θ,ψ} or {φ,∆}, see Figure 2.4, such that ellipticity can be deﬁned as

B π tanψ = ± = tan( − γ) (2.45) A 4

for ψ ∈ [−π/4,π/4] and γ ∈ [0,π/2]. Another deﬁning parameter set consists of ∆ = δx −

δy, the relative phase shift of oscillations along x and y axes and tanφ = X/Y vibrational amplitude along X relative to Y in Figure 2.4. However, the ﬁrst set of angles is usually the preferred one due to its independence of the choice of coordinate system. The description of the polarization state in terms of the orientation of the polarization ellipse, outlined above, can be mapped onto the Poincare sphere, see Figure 2.4 and Jones et al. (2016). The generic polarization state in terms of the parameters {θ,γ} and photon helicity states Λ = ±1, as in eqn. (2.29), can be expressed as

−i2θ eˆ = eˆ− cosγ − eˆ+ sinγ e (2.46)

28 Figure 2.5: Polarization proﬁle of the radially and azimuthally-polarized beams in the xy-plane, with the intensity proﬁle on the background.

where the phase retardation θ changes in the range [0,π]; and e− stands for the Λ= −1

and right-circular (RC) state, and e+ for the Λ= 1 and left-circular (LC) state. We will mostly use the latter description and the definition above in the following chapters. We want to point out the radial and azimuthal polarization states, see e.g. Ornigotti and Aiello (2013), and Figure 2.5. Radially polarized beams were first generated by Mushiake et al. (1972) by fine-tuning the optical cavity. Tidwell et al. (1990) provide a detailed description of the theoretical and experimental details for generating these modes using interferometry with linearly and circularly polarized sources. Azimuthal polarization was first routinely produced in axicon resonators, see Chodzko et al. (1980), with subsequent developments in beam interferometry. We are interested in obtaining radial and azimuthal states based on two Bessel-like modes’ interference as in eqn. (2.25). The desired resulting state can be described by

ˆ BB BB Ur = u1 (ρ,t) η−1 − u−1(ρ,t) η1; (2.47) ˆ BB BB Ua = −i u1 (ρ,t) η−1 + u−1(ρ,t) η1 .

Our interest is governed by the spectroscopic effects, related to the peculiar ﬁeld

29 distributions near the axis of such beams. Some of the phenomena will be discussed in the subsequent chapters.

30 Chapter 3: Quantum Mechanical Transition Matrix for Twisted Light - Atom Interactions

It has been long known that photon helicity can be transferred into the internal degrees of freedom of the atomic structure in the photo-excitation process. After the possibility of twisted beams to carry a well-defined OAM had been confirmed by Allen et al. (1992), another question became of a high importance: how exactly will the optical rotational quanta, coming from the beam topology, be distributed among the target’s degrees of freedom? What would be the coupling mechanism between the electron momenta and photon TAM? We will answer these and other related questions by means of QM and QED. We assume that the non-relativistic first-order time-dependent perturbation theory suffices to describe the photon-matter system to a good enough accuracy. Under these conditions, one can use the Golden Rule of time-dependent perturbation theory, or Fermi’s Golden Rule, to calculate atomic photo-absorption matrix elements. In our calculations, we take into account fine and hyperfine interactions. After establishing the theoretical formalism, we apply it to physical systems, where the corresponding measurements can be taken: cold trapped ions. We consider low probability forbidden transitions in calcium, ytterbium, argon and highly charged ion (HCI) neon for RC, LC and linear photon polarization states. The violations of the magnetic number selection rules due to the topological structure of the incoming photon beam are theoretically described, extensively tested and discussed. The unusual sensitivity of the experimentally registered Rabi frequencies, e.g. Cohen-Tannoudji (1990), with respect to the position in the beam profile is revealed and investigated. The closing subsection is dedicated to the predictions of the ions’ behavior, when being radiated by the radially and azimuthally polarized beams of the resonant frequency. Due to the peculiar field distribution in these beams, we predict effects that can be useful

31 in atomic spectroscopy, material sciences, and beam diagnostics.

3.1 Plane-wave Transition Amplitude

The QM Hamiltonian for an atomic system interacting with EM-ﬁeld can be expressed as

h¯ 2 ihe¯ H = − ∆ +U(r) − (A · ∇) − µs · (B +BLS + BI) − µI · (B +BLS) (3.1) 2me me

The term quadratic in the vector ﬁeld A has been dropped since it is negligible. Here B is

the external static magnetic field; BLS is the electron spin-orbit coupling field; BLI is the nuclear spin-orbit coupling field. This Hamiltonian accounts for the fine and magnetic hyperfine structures in atomic spectra. The photo-absorption matrix element in non-relativistic QM can be written as:

Z Mfi = dth f |A · ∇|i;kΛi (3.2)

First we solve the photo-excitation problem for the case, when nuclear magnetic moment I is zero and the proper eigensystem for operator eqn. (3.1) is {n; j,m}. This corresponds to the transition amplitude

Z M(pw) (r) = drrhn j m |(pˆ ·e )eikk·r|n j m ;kΛi (3.3) m f miΛ f f f kΛ i i i

In the case when k z we can use the expansion

√ ∞ j+1 √ ikkkz ` jΛ ekΛe = 4π ∑ ∑ i 2` + 1 j`(kr)C`01ΛY j,`,Λ(Ω) (3.4) j=1 `= j−1 where, with the deﬁnition in eqn. (A.24), the convention used for Clebsch-Gordan coefﬁ-

32 cients is   jm p j f ji j C = (−1) j f − ji+m 2 j + 1  (3.5) jimi j f m f   m f mi −m

One can expand the sum over ` as

r √ ∞ n j + 1 e eikkkz = 4π i j−1 j (kr)Y (Ω) kΛ ∑ 2 j−1 j j−1Λ j=1 (3.6) r r j 2 j + 1 o + i j+1 j (kr)Y (Ω) − i j Λ j (kr)Y (Ω) 2 j+1 j j+1Λ 2 j j jΛ where the following identities for Clebsch-Gordan coefﬁcients were applied:

s s j + 1 j Λ C j,Λ = ; C j,Λ = ; C j,Λ = −√ . j−1,0,1,Λ 2(2 j − 1) j+1,0,1,Λ 2(2 j + 3) j,0,1,Λ 2

Now one can use the deﬁnitions for the vector-potential in terms of vector spherical harmonics as in Arfken and Weber (2005)

M A jm(k,r) = j j(kr)Y j jm(Ω) (3.7)

s s ! j + 1 j AE (k,r) = j (kr)Y (Ω) − j (kr)Y (Ω) (3.8) jm 2 j + 1 j−1 j, j−1,m 2 j + 1 j+1 j, j+1,m to rewrite eqn. (3.6) as the superposition of electric and magnetic contributions:

√ ∞ r ikkkz (2 j + 1) jn E M o ekΛe = − 4π ∑ i iAA jm(k,r) + ΛA jm(k,r) ; (3.9) j=1 2

We get back to the deﬁnition of the transition amplitude in eqn. (3.3) and rewrite it plugging

33 in the result above:

r √ ∞ (2 j + 1) n Z M(pw) (0) = − 4π i j i drrhn j m |(pˆ ·AE (k,r))|n j m ;kΛi m f miΛ ∑ f f f jm i i i j=1 2 Z M o + Λ drrhn f j f m f |(pˆ ·A jm(k,r)|ni jimi;kΛi (3.10) We reduce this expression utilizing the Wigner-Eckart theorem as:

∞ s 2π(2 j + 1) j m n Z M(pw) (0) = − C f f i j+1 drrhn j |(pˆ ·AE(k,r))|n j ;kΛi m f miΛ ∑ jimi jm f f j i i j=1 (2 j f + 1) Z j M o + i Λ drrhn f j f |(pˆ ·A j (k,r)|ni ji;kΛi (3.11) At this point we introduce the notation for multipolar contributions to the photo-absorption matrix elements: Z µ Mjµ = drrhn f j f |(pˆ ·A j (k,r))|ni ji;kΛi (3.12) where µ = 1 stands for electric and µ = 0 for magnetic multipolarity. This leads us to the ﬁnal form, as in eqn. (13) of the paper by Afanasev et al. (2018b):

√ ∞ r (pw) j+µ (2 j+1) µ+1 j f ,m f M (0) = − 4π i Λ C Mj (3.13) m f miΛ ∑ (2 j f +1) ji,mi, j,Λ µ j=1

This is the photo-absorption amplitude for a plane wave travelling in the z direction with

helicity Λ= ±1. Similar expansions in terms of spherical harmonics have been performed in nuclear theory, e.g. Akhiezer and Pomeranchuk (1950).

3.2 Photo-Excitation by Twisted Beams in Total Angular Momentum Basis

In this section describe of the OAM coupling scheme in the photo-absorption process. A detailed derivation of the photo-absorption amplitudes in spinless case is given by Scholz- Marggraf et al. (2014). Hence, only the ﬁrst three terms in eqn. (3.1) were considered, and

34 Figure 3.1: Deﬁnition of the relative position of the atom with the nucleus −Ze and photo-excited electron e with respect to the optical axis z of the photon vortex beam in terms of global r and local r0 coordinate systems. the corresponding matrix element was derived:

r (BB) m f −mi−2m i(m +mi−m f )φ κ M = i γ e γ b Jm +m −m (κb) m f miΛ 2π γ i f (3.14) ` f `i (pw) × d 0 (θk)d 0 (θk)M 0 0 (0,0) ∑ m f m mim m m 0 0 f i f i m f mi

To obtain this expression, Scholz-Marggraf et al. (2014) used the ASR, deﬁned in eqn. (2.12), to decompose the non-trivial topological proﬁle of the Bessel mode in eqn. (2.25) into the ensemble of the plane waves, as derived in eqn. (2.27). The Wigner ro-

tation matrices, as in Edmonds (1957), are responsible for compensating for the angle Ωk = 0 {φk,θk,0}, by rotating the incoming and outgoing photon states by Ωk = {0,−θk,−φk}. Here we will go over a similar derivation, but including the electron spin contribution and for the photon state, deﬁned in eqn. (2.32). We start with an absorption matrix element:

Z M(tw) (r) = drrhn j m |pˆ ·A(tw) (ρ,z;t)|n j m ;kΛi (3.15) m f miΛ f f f kΛ;ω i i i

where { ji,mi} and { j f ,m f } are the initial and ﬁnal states of the bound electron with the

corresponding projections of the electron TAM ji and j f . We plug the expression for the twisted Bessel-like state, obtained in eqn. (2.32):

35 M(BB) (r) = d j (θ )e−i(kzz−ωt) m f miΛ ∑ Λm k m=±1,0 Z Z 2 d k⊥ imφ ikk ·(ρ−b) × drr a m (k )e k hn j m |(pˆ ·e )e ⊥ |ni jimi;kΛi (2π)2 κ γ ⊥ f f f Λ (3.16) where b = {bcosφb,bsinφb,0} is the atom’s impact parameter. In Figure 3.1 one can see the two coordinate systems, global {x,y,z} and local {x0,y0,z0}. Local is attached to the ion’s center of mass, while global is associated with the optical center of the incoming photon vortex, schematically represented as a red cone. At this point, one of the key assumptions made is that the atom’s wave packet is well localized. By that we mean that

the exponent eikk⊥·r behaves effectively like a constant in the integral over reciprocal space. With this assumption one can separate the two integrals as

M(BB) (r) = d j (θ )e−i(kzz−ωt) m f miΛ ∑ Λm k m=±1,0 Z Z 2 ikk ·ρ d k⊥ imφ ikk ·b × drrhn ` m |(pˆ ·e )e ⊥ |ni`imi;kΛi a m (k )e k e ⊥ f f f Λ (2π)2 κ γ ⊥ (3.17) The integral over the position space is the QM absorption amplitude for the plane-wave photon interacting with a hydrogen-like ion, as in eqn. (3.13). The integral over the momentum space is the topological correction due to the presence of the non-zero photon OAM. With a bit of algebra and applying eqn. (A.9), we get

r j (BB) 2π M (r) = im−2mγ ei(mγ −m)φb J (κb)d j (θ )M(pw) (0) (3.18) m f miΛ ∑ mγ −m mΛ k m f miΛ κ m=− j

or, in terms of active rotation and analogous to the derivation in the paper by Scholz-

36 Marggraf et al. (2014):

r 2 (BB) π m f −mi−2m i(m +mi−m f )φ M (r) = i γ e γ b Jm −m +m (κb) m f miΛ κ γ f i (3.19) j f ji (pw) × d 0 (θk)d 0 (θk)M 0 0 (0) ∑ m f m mim m m Λ 0 0 f i f i m f mi

The Wigner rotation matrices, resulting from the local tilt in the photon propagation direction, and the Bessel-factor, are the two novel factors. They are the consequence of the non-trivial topology in the incoming state. For instance, in direct proximity of

the optical vortex, the Bessel function has a limit of lim Jm −m +mi (κb), which imposes b→0 γ f the TAM conservation as mγ = m f − mi. Hence, these terms modify the conventional set of spectroscopic selection rules, resulting into relative enhancement of the higher-order multipolar contributions.

3.3 Twisted Photo-Excitation in Atoms: Hyperﬁne Structure

Atomic nuclei are known to have non-trivial internal structure, described by electric charge and current distributions, that can result in non-zero nuclear spin I. Its presence gives rise to the interaction that perturbs the electron states in an atom, inducing hyperﬁne shifts in atomic spectra. Nuclear spin effects are typically at least three orders of magnitude weaker than electronic. However, the effects resulting from the hyperﬁne interaction are very important in metrology, spectroscopy and for fundamental understanding of atomic and nuclear matter.

1 Nuclear spin is a cumulative effect coming from the intrinsic spin 2 h¯ of each of the constituents, and the internal orbital motion, quantized in units of h¯. For this reason, in nuclear physics the quantity I is often referred as nuclear OAM, rather than spin. However, in atomic spectroscopy, one typically neglects the internal nuclear structure. A simpliﬁed model of a nucleus as an elementary particle with non-zero magnetic moment has proven itself to work remarkably well.

37 Most of the effects can be described by nuclear magnetic dipole and electric quadrupole coupling. The electric quadrupole contribution comes in for deformed nuclei, often in their excited states. In QM, nuclear electric quadrupole does not contribute unless I≥ 1. The atomic structures that we are going to consider possesses only magnetic dipole contributions, making magnetic hyperﬁne structure a priority in this section. A nucleus, modeled as an elementary particle with non-zero magnetic moment, gener- ates the associated ﬁelds

µ0 µ I ×r AI = ; BI = ∇ ×AI. (3.20) 4π r3

Hence, the following terms that lead to hyperﬁne interaction can be inferred from the Hamiltonian eqn. (3.1): ihe¯ Hˆhf = − AI · ∇ −µ s ·BI (3.21) mr

The first term describes nuclear spin-orbit interactions, the second term contains magnetic dipole-dipole interaction and the Fermi contact term. One of the main limitations to remember is that fine splitting should strongly dominate over the hyperfine structure for the theory to work in the leading order of the perturbation theory. This will allow coupling of the TAM F directly to J, and not to L and S independently. In this case F =J+I forms an (2F + 1)-dimensional space, and all possible values

of F form a coupled basis |n jIFm f i. In analogy to eqn. (3.22), the absorption matrix accounting for hyperﬁne interactions can be written as

Z M(pw) (r) = drrhn j IF m( f )|Hˆ |n j IF m( f );kΛi (3.22) m f miΛ f f f f hf i i i i

38 One can reduce the matrix element applying the Wigner-Eckart theorem as

√ ∞ r ( f ) Z (pw) j+µ (2 j+1) µ+1 Ff ,m f M (0) = − 4π i Λ C drrhn f j f IFf |Hˆhf|ni jiIFi;kΛi m f miΛ ∑ (2 j f +1) F ,m( f ), j, j=1 i i Λ (3.23) and use the cascade rule to allow even further reduction:

  L0 J0 I 0 0 J+L0+I−k 0   hL IJ |Tk|LIJi = (−1) hL |Tk|Li (3.24)  J L k where the Racah coefficients have been used, as defined in, e.g. Edmonds (1957). With the definition in eqn. (3.12), we get the final form as was shown by Solyanik-Gorgone et al. (2019):

∞ (pw) √ p M (0) = − 4π i j f +I+Fi+µ (2 j + 1)(2F + 1)Λµ+1 m f miΛ ∑ i j=1   (3.25) ( f )  j F I Ff ,m f  f f  ×C ( f ) Mjµ Fi,m , j,Λ i Fi ji j

This matrix element describes photon-atom interactions for zero-OAM optical beams and reduces to eqn. (3.13) when I is zero. To generalize this description for the case with the twisted beams, we can insert this equation into eqn. (3.18) or eqn. (3.19).

3.4 Photo-Excitation with Twisted Light in Cooled Trapped Ions

From the point of view of fundamental physics, cooled trapped ions are a perfect tool for photon beam diagnostics, due to the high sensitivity of optical response to the location of the ion in the optical beam. Along with that, the need to study optical transitions with high multipolarity makes trapped ions an excellent target for high-precision atomic spectroscopy with OAM beams. The semi-classical theory of twisted photon – ion interactions has been developed by Schmiegelow and Schmidt-Kaler (2012) and Quinteiro et al. (2017c). Our

39 Figure 3.2: The scheme of the experiment: (a) The energy levels in 40Ca+ with the E2 transition of interest marked in red. (b)The experimental setup – linear segmented Paul trap (yellow) inside the UHV chamber (gray); the 729 nm OAM mode is formed in the optical setup on the bottom (red) with the help of the holographic phase plate HP (bottom-right). . Further experimental details can be found in the paper by Schmiegelow et al. (2016). The ﬁgure is used with permission of the authors.

QM approach is based on using the photo-absorption matrices to evaluate the transition strengths, or Rabi frequencies, that are proportional to the norm of the corresponding matrix element. The theory developed above was successfully applied to explain the violation in magnetic selection rules observed in experiments conducted by Schmiegelow et al. (2016).

3.4.1 40Ca Ion in Parallel Conﬁguration Irradiated by Circularly Polarized Light

In this experiment, a single 40Ca+ ion was trapped and Doppler cooled in a segmented Paul trap with sub-nanometer positioning precision. The experimental setup is schematically shown in Figure 3.2. A Ti:Sa laser, shown in red, was tuned to the resonance E2 transition at 729 nm. The resulting beam was twisted with a fork-dislocation diagram, like the ones shown on the bottom-right, ﬁltered and focused on the trapped ion. The beam waist was

about 5 µm on the output. The Paul trap, shown in yellow, was positioned inside an Ultra

40 Figure 3.3: Relative position of the trapped ion with respect to the optical axis of the photon vortex beam, represented as an incoming cone of radiation, in the constant quantizing magnetic ﬁeld B. Local to the ion coordinate system {x0,y0,z0} is shown in blue, and global coordinate system {x,y,z}, attached to the beam’s optical axis z, is shown in black.

High Vacuum (UHV) chamber, shown in gray. A constant magnetic ﬁeld B = 13 mT was applied to Zeeman-split the levels while the external magnetic ﬁelds were shielded. The

2 2 complete set of measurements of the 4 S1/2 → 3 D5/2 transition amplitudes was obtained

in the parallel conﬁguration, i.e. θz = 0 in Figure 3.3. For more details on the experimental setup, see papers by Schmiegelow et al. (2016) and Afanasev et al. (2018a). The photo-absorption matrix element for the BB mode was obtained earlier, see (3.19). The theoretical model for this case contains two ﬁtting parameters: (a) the overall constant;

(b) the pitch angle θk . Even though the parameter space is small, compared to the models, discussed further, there is a payoff: BBs do not behave as the beams produced in the lab far from the vortex, as we have discussed in the previous section. Therefore, the model is expected to overshoot the experimental results on the beam periphery. Using the same approach as for BBs one gets the amplitude for BG beams. By substituting eqn. (2.35) into eqn. (2.32) to obtain the BG vector potential, and then into

41 the eqn. (3.15) for the twisted photo-excitation amplitude, one obtains the expression

2 (BG) Aw (pw) 2 2 0 −κ w0/4 M (b) = M 0 0 (0)e m f miΛ ∑ m m Λ 4π 0 0 f i m f ,mi Z j f j 1 2 2 i 2 −w0k⊥/4 × k⊥dk⊥ dm m0 (θk)dm m0 (θk)Jmγ −m f +mi (k⊥b)Imγ w0κk⊥ e f f i i 2 (3.26)

(pw) where the plane-wave amplitude M 0 0 (0) is given by eqn. (3.13). This integral can be m f miΛ calculated, e.g. eqn. (6.633 1) in Gradshteyn and Ryzhik (2014), involving an inﬁnite sum

over hypergeometric functions. When the Gaussian waist w0 is large compared to other dimensional quantities, such as the wavelength, we can evaluate the Wigner function at the

pitch angle θκ and take it out of the integral. The approximated expression for eqn. (3.26) is then

(BG) −b2/ 2 M (b) = e w0 m f miΛ (3.27) A j f ji (pw) × Jm −m +m (κb) d 0 (θk)d 0 (θk)M 0 0 (0) . γ f i ∑ m f m mim m m Λ 2π 0 0 f i f i m f mi

For this BG model one has one extra parameter of the beam waist w0, coming from the Gaussian spread overlaid on top of the oscillating Bessel proﬁle. For LG modes one needs to substitute eqn. (2.44) for the corresponding vector potential in the expression for the amplitude, eqn. (3.15). The resulting LG amplitude is

r p κ |` | j (LG) ˜ γ f ji |M (b,z)| = Bp j d 0 (θk)d 0 (θk) m f miΛ ∑ ∑ m f m f mimi 2π j=0 m0 m0 f i (3.28) 2 (pw) b|η| 2 2 |η| b −b /w0 × M 0 0 (0) e L . m miΛ j+ξ 2 f 2 w0

42 m=-2 m=-1 m=0 m=1 m=2

20 100 8 3.0 3.0 1 2 15 80 6 2.5 2.5

=- 2.0 2.0

=- 60 Λ 10 4 1.5 1.5 , 40 1.0

R 1.0 m 5 20 2

Ω 0.5 0.5 0 0 0 0.0 0.0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 12 200 12 20 0.5 1 10 10 1 150 15 0.4 =- 8 8 0.3 Λ

=- 6 100 6 10 , 4 4 0.2 R 50 5 m 2 0.1 Ω 2 0 0 0 0 0.0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 10 140 20 6 1 120 5 4 8 100 15

=- 4 3 0 6 80 Λ = 10 3 2

, 60 4 2 R 40 5 m 2 1

Ω 20 1 0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 2.0 4 20 100 4 1 1.5 3 15 80 3 =- 0 60 Λ = 43 1.0 2 10 2 , 40

R 0.5 5 m 1 20 1 Ω 0.0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 20 12 150 1 2.0 10 6

1 15 =- 1.5 8 100 =

Λ 4 1.0 10 6 , R m 0.5 5 4 50 2

Ω 2 0.0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 8 10 10 1 3.0 100 8 2 2.5 6 80 8 =- = 2.0 6 60 6 Λ 1.5 4 , 4 40 4 m R 1.0 2 20 2 Ω 0.5 2 0.0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm)

Figure 3.4: Normalized Rabi frequencies as a function of impact parameter b for initial atomic spin mi = −1/2. Black-dashed curves and purple long-dashed curves are the theory predictions for BG and LG, not accounting for the opposite sign circular polarization admixture; solid curves in black (BG) and long-dashed curves in purple (LG) are the theory predictions with 3% opposite-polarization admixture (by amplitude) for rows 1,3,4 and 6, and 10% admixture for rows 2 and 5. Δm=-2 Δm=-1 Δm=0 Δm=1 Δm=2 8 1 8 80 3.0 0.25 2 6 2.5 0.20 =- 6 60 2.0 =- Λ 4 0.15

γ 1.5 , 4 40 0.10

R 1.0 m 2 20 2 0.05

Ω 0.5 0 0 0 0.0 0.00 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 5 150 12 1.0 1 4 10 20 1 0.8 =- 3 100 8 15 0.6 Λ =- 6 10 , γ 2 50 4 0.4 R 5 m 1 2 0.2 Ω 0 0 0 0 0.0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 20 6 5 1 80 5 6 15 4

=- 60 4 0 3

Λ 4 = 40 10 3 ,

γ 2 R 2 20 5 2 m Ω 1 1 0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 7 20 1 0.8 6 120 8 5 15 100 =- 6 0 0.6 80

Λ 4 = 44 10 60

, 0.4 4 γ 3 R 2 40 m 0.2 5 2 Ω 1 20 0.0 0 0 0 0 10 5 0 5 10 -10 -5 0 5 10 - - -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b m b (μm) (μ ) b (μm) b (μm) b (μm)

1 2.0 12 10 10 10 150 1 8 8 =- 1.5 8 =

Λ 6 100 6 γ 1.0 6 , 4 4 R m 4 50 0.5 2 2 Ω 2 0.0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 4 5 10 20 1 120 100 2 3 4 8 15 =- = 3 6 80 Λ γ 2 60 10 , 2 4 m R 40 1 2 5

Ω 1 20 0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm)

Figure 3.5: Normalized Rabi frequencies as a function of impact parameter b for initial atomic spin mi = 1/2. Black-dashed curves and purple long-dashed curves are the theory predictions for BG and LG, not accounting for the opposite sign circular polarization admixture; solid curves in black (BG) and long-dashed curves in purple (LG) are the theory predictions with 3% opposite-polarization admixture (by amplitude) for rows 1,3,4 and 6, and 10% admixture for rows 2 and 5. 120 BG,Δm=-1 p=0&1,Δm=-1 100 ● ● ● (a) ● ●● (b) BB,Δm=-1 ● ● p=0,Δm=-1 ● ● ● ● ● ● ● 100 ● ● BG,Δm=-2 ● p=1,Δm=-1 ● ●● ● ● ● 80 ● ● ● ● BB,Δm=-2 ● ● p=0&1,Δm=-2 ● ● ● ● ● ● ● ● ● 80 p=0,Δm=-2 ● ● 60 ● ● ● p=1,Δm=-2 ● ● ●

, Λ =- 1 ● 60 ● ● , Λ =- 1 R ● ● R ● Ω 40 ● ● ● ● Ω ● 40 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ●● ■■ ■■■■ ● ● ● ● ● ● ■ ■ ■ ● ● ● ● ■ ■ ● ● ■ ■●■■ ● ● ● ● ●● ■■ ■■ ● 20 ● ●● ■ ■ ■ ● ■ ●● ● ■■ ■ ■ ■ ■ ■ ● ■ ■■■ ■■■ ■ ■ ■ ■■■■■ ■■ ● ●● ■ ■■ ● ■■■■ ■■■ ■ ■■ ■ ■ ■■■■■ ■■ ● ■■ ●● ■■ ● ● ■ ● ■■ ■ ■ ■ ■ ■ ■■■ ■■■ ■ ■ ■ ■■■■■ ■■ 0 ■■■■ ■■■ ■ ■■ ■ ■ ■■■■■ ■■ 0 ■■ ● ● ■ ● ■■ ■ -10 -5 0 5 10 -10 -5 0 5 10 b(μm) b(μm)

Figure 3.6: (a) Normalized Rabi frequencies compared with the theory predictions for the Bessel mode (blue dashed and green long-dashed) and BG mode (black solid and black dotted); (b) same transitions versus theory for LG modes with parameters p=0, p=1, and their linear combination (see text for details).

The parameters are ξ = 0.5(|`γ | − |η|) and η = mγ + mi − m f . The new coefﬁcient is

η j |` | sgn(η) (−1) (|` | + p)!b( j + ξ)e! 2 |η| ˜ γ = γ (3.29) Bp j − j−(|` |+|η|)/2 2 γ (p − j)!(|`γ | + j)! j! w0 where sgn(•) denotes the signum of the number, b•e! is the Roman factorial, see Roman |ν| (1992), and Ln (•) is the extended Laguerre polynomial

|ν| b(n + |ν|)e! Ln (x) = F (−n,α + 1,x). (3.30) bne!b|ν|e! 1 1

The parameter space includes the overall constant, the pitch angle θk , the beam waist w0 and the number of zeros of Laguerre polynomial p. For additional information and remarks on the derivation of the transition amplitudes the reader is referred to the papers by Afanasev et al. (2018a) and Solyanik and Afanasev (2018).

2 2 In Figures 3.4 and 3.5 one can see the complete set of transitions 4 S1/2 → 3 D5/2 40 in the Zeeman-split Ca, ﬁtted with BG and LG models with mγ − Λ = 0,±1. Fig-

ure 3.4 shows the results for the atomic spin projection on the quantization axis mi =

−1/2, and Figure 3.5 shows the same for mi = 1/2. Columns 1 through 5 correspond to the change of the magnetic quantum number by ∆m = −2,−1,0,1,2, respec-

45 tively. Rows 1 through 6 correspond to the photon’s quantum numbers {mγ ,Λ} = {−2,−1},{−1,−1},{0,−1},{0,1},{1,1},{2,1}. The cases where the Rabi frequen-

cies have maxima in the central region m f = mi +mγ are highlighted in red. Initially, when analyzing the data, we assumed that the beam was 100% LC- or RC-polarized since the polarization admixture was known to be very small in the experiment. However, in the process we have discovered that the twisted light-matter interactions are highly sensitive to the polarization content. Even admixtures on the level of 3-10% cause noticeable effects in the amplitudes of Rabi oscillations. One can see the two cases (with and without polarization admixture) presented in the plots in Figures 3.4 and 3.5. The polarization admixtures in rows 1, 3, 4 and 6 were inferred from the theoretical analysis and held at 3%, and in rows 2 and 5 at 10% due to the variations in the experimental setup for OAM 6= 0 versus zero-OAM cases. In the case with BG and LG beams we looked at the first two plots in the first row with ∆m = −2,−1 to fix the waist w0, the node number p and the overall constant factor. The quality of the fits was assessed relying on the locations of the minima and the relative amplitudes of the central peaks, that appear to be especially sensitive to the choice of parameters. For these models we got an agreement with the experiment for w0 = 10 µm≈ 10λ and θk = 0.095 rad, see Figures 3.4 and 3.5. The LG-fit is sensitive to the waist and the node number. As one can see, the fit systematically underestimates the

data in the second and ﬁfth rows in the ﬁgures. This may be caused by the OAM `γ being

a well-deﬁned quantum number in paraxial laser modes (while TAM mγ is not); and the subsequent coupling of the photon OAM to TAM J of the bound electron in the ion target. This OAM-modes’ quantization problem has been discussed in Section 2.3.

The results for the BB and BG models with the parameter values θk = 0.095 rad and w0 = 10 µm are shown in Figure 3.6a. In these plots we compare experimental data for Rabi frequencies as a function of the impact parameter b, generated by the OAM light of vorticity mγ = −2 and Λ = −1. The probed transitions are ∆m = −2 and ∆m = −1 from

46 Δm=-2 Δm=-1 Δm=0 Δm=1 Δm=2

8 8 10 10

1 50 6 6 8 40 8 =- 1 6 6 Λ 4 30 =- 4 , γ 4 20 4 R m 2 2 Ω 2 10 2 0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm) 20 80 6 2.0 1 5 2.0 15 60 1.5 =-

3 4 1.5 Λ =- 10 40 3 1.0 1.0 γ , R m 5 20 2 0.5 0.5

Ω 1 0 0 0 0.0 0.0 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 b (μm) b (μm) b (μm) b (μm) b (μm)

Figure 3.7: Similar to Figure 3.4, but here the photons have mγ − Λ = 2. The ﬁts in black dashed (BG) and purple dash-dotted (LG) have no polarization admixture, while the black solid (BG) and purple long-dashed (LG) have 10% by amplitude of opposite helicity photons.

the ion’s ground state with mi = −1/2. BG and BB behave essentially identical in the region b ≤ 3 µm, but BG ﬁts best on the periphery. The Bessel function dominates at the center, deﬁning the magnetic selection rules in the limit b → 0: one can see the “open”

∆m = mγ (red) and “closed” ∆m 6= mγ (black) amplitudes next to each other. Figure 3.6b was generated with two model ﬁts: (1) a pure LG beam with either

p = 0 or 1, and w0 = 4 µm; (2) two LG modes with parameters {w01 = 4 µm, p = 0},

and {w02 = 6.5 µm, p = 1} were intermixed with a ratio LGp=1/LGp=0 = 0.43 in the amplitude. As one can infer from the plots, the ﬁrst model ﬁts as well as the second one in

the case with ∆m = −2. However, the second model behaves much better for ∆m = −1. The polarization admixture was held at 3%, as well as in the previous case.

In Figure 3.7 the data for the case mγ − Λ = ±2 was analyzed. The quantum numbers

are mγ = −1,Λ = 1 (OAM and spin anti-aligned in upper plots) and mγ = −3,Λ = −1

(OAM and spin aligned in lower plots), mi = −1/2. In this case, the polarization sensitivity

is especially noticeable: 10% admixture in row 1, column 2 (the case with mγ = ∆,m = −1)

turned maximum into minimum, obscuring the true magnetic selection rules mγ = m f −mi,

see Afanasev et al. (2018a). It is important to remember that for paraxial beams, mγ −Λ ≡

`γ is the orbital angular momentum.

47 Table 3.1: Measured Rabi frequencies

Fit BB BG LG Data BB BG LG Data ∆m mi = −1/2 mi = −1/2 mi = −1/2 mi = −1/2 mi = 1/2 mi = 1/2 mi = 1/2 mi = 1/2 ∆m = −2 2.92 2.92 2.92 2.92 (8) 1.24 1.24 1.24 1.33 (4) ∆m = −1 27.1 29.7 21.7 31.21 (87) 18.2 19.9 14.5 23.89 (66) ∆m = 0 2.76 3.11 2.76 2.78 (8) 2.62 2.95 2.62 2.87 (8) ∆m = 0 2.76 3.11 2.76 2.78 (7) 2.62 2.95 2.62 2.61 (8) ∆m = 1 19.2 21. 15.3 19.22 (62) 25.7 28.2 20.6 34.08 (92) ∆m = 2 1.3 1.31 1.31 1.26 (4) 2.77 2.77 2.77 2.77 (8)

Theoretical values compared to the data in the supplementary materials in the paper by Schmiegelow et al. (2016), in units of kHz/µW, taken at zero impact parameter b → 0. The overall normalization was ﬁxed at the case of ∆mγ = −2.

The intensity proﬁles for all three ﬁts as functions of the impact parameter are similar for the choice of parameters, discussed above, see Figure 7 in the paper by Afanasev et al. (2018a). In this paper, the authors discuss the peculiar azimuthal dependence of the atomic transition amplitudes, generated by linearly polarized twisted beams. The observed effect

happens due to the states with non-zero OAM and opposite helicities λ = ±1 not being mirror-symmetric to each other anymore, as this is the case for OAM-zero photon states. This leads to the accumulation of relative phase in the linear superposition of LC and RC states of light, when forming a linear state. This effect will be re-iterated in subsection 3.4.2. The relative amplitudes of the Rabi oscillations are controlled by two factors: (1) the Wigner rotation matrices, see eqs (3.19), (3.27) and (3.28); (2) the Clebsch-Gordan coefﬁcients, coming from the plane-wave matrix element reduction (3.13). The peak values of non-vanishing photo-absorption amplitudes, eqs (3.19), (3.27) and (3.28), in the limit b → 0 are listed in the Table 3.1 together with the data of the measured Rabi frequencies, taken from the supplementary materials in the paper by Schmiegelow et al. (2016). This assessment was previously done by using the formalism outlined in the earlier papers, see Quinteiro et al. (2015) and Quinteiro et al. (2017a). The results shown here are consistent with those obtained by Quinteiro et al. (2017c).

48 Table 3.2: Plane-wave Transition Matrix Elements

(pw) Model Transition Plane-wave matrix element M (0) Coupling m f ,miΛ Ion scheme

5/2m 40Ca+ S → D C f E2 eqn. (3.13) 1/2 5/2 1/2 mi 2 Λ

40 + √ Ar P1/2 → P3/2 ±Λ π M1 eqn. (3.13)

√ 7/2 m 172Yb+ S → F −2 14π C f E3 eqn. (3.13) 1/2 7/2 1/2 mi 3 Λ √ m( f ) 171 + 3 f Yb S1/2 → F7/2 2 7π C ( f ) E3 eqn. (3.25) 0 mi 3 Λ √ 3/2 m √ 3/2 m 10Ne+ P → D 5 C f E3 − Λ 3 C f M1 eqn. (3.13) 1/2 3/2 1/2 mi 2 Λ 1/2 mi 1 Λ

3.4.2 40Ca Ion in 45◦-Conﬁguration and More

In this subsection we discuss the case with constant quantizing magnetic ﬁeld B being

arbitrarily aligned with respect to the beam quantization axis θz 6= 0, see Figure 3.3. We test our theoretical model against the data for 40Ca presented in the paper by Schmiegelow et al. (2016) for a linearly polarized photon probe, and then proceed with the analysis of the new data for varying photon polarization. We also make predictions for other ions, such as 171Yb+ and 172Yb+, 40Ar13+ and 20Ne4+, which are popular in quantum computing. We develop the theory to include the effects coming from magnetic hyperﬁne structure.

We start from the theoretical description of the case with B ∦ z. For the BB interacting with a trapped calcium ion we apply the general photo-absorption amplitude of eqn. (3.19) to get (BB) M (b) ' A im f −mi−2mγ ei(mγ +mi−m f )φb J (κb) m f miΛ mγ −m f +mi 0 (3.31) j f ji j f mi+Λ × d 0 (θk)d 0 (θk)C 0 E2 ; ∑ m f ,m +Λ mi,m ji m 2 Λ 0 i i i mi see Table 3.2 and the papers by Afanasev et al. (2018b) and Solyanik-Gorgone et al. (2019). When the static magnetic ﬁeld that deﬁnes the quantization axis of the atomic states points

49 in a general direction {0,−θz,−φz}, the resulting photo-absorption amplitude reads

(BG) −b2/ 2 −i(m −m ) M (b;θ ) = e w0 e f i φz m f miΛ z (3.32) j f ji (BB) × d 0 (θz)d 0 (θz)M 0 0 (r;θz = 0) ∑ m f m mim m ,m Λ 0 0 f i f i m f mi

(BB) where M 0 0 (r) is the BB absorption matrix element calculated earlier, see eqn. (3.19). m f ,miΛ The geometry of the problem is schematically presented in Figure 3.3, where an ion in the wavefront of the OAM laser beam is shown to be an impact parameter b away from the optical axis. The local to the ion coordinate system {x0,y0,z0} is shown in blue color and the topological state of light is represented as an incoming cone of radiation. Here we perform a set of two rotations on the initial plane-wave absorption matrix, see eqn. (3.13):

ﬁrst we rotate the electron states by {0,−θk,−φk} to align them with the direction of the

local wave vector k; and then we rotate the entire system one more time by {0,−θz,−φz} to account for the alignment B ∦ z. The next step is to form a state of arbitrary polarization. We use the deﬁnitions from section 2.4, in particular eqn. (2.46), where the photo-absorption amplitude in eqn. (3.32) is substituted into eqn. (3.32), so that

(BG) (BG) −i2θ (BG) M (b;θz) = cos(γ) M (b;θz) − sin(γ) e M (b;θz). (3.33) m f miH/V m f ,mi −1 m f ,mi 1

The convention for LC and RC polarized light is the same as in eqn. (2.29), such that

◦ ◦ e1 ≡LC and e−1 ≡RC. The horizontal state is obtained with γ = 45 and θ = 0 , and the vertical state with γ = 45◦ and θ = 90◦, see Figure 2.4. The experimental data we used to test the theory, developed above, were obtained by the research group at the University of Mainz, Germany (PI: Professor Schmidt-Kaler). The same setup as described by Schmiegelow et al. (2016) was used. The constant quantizing magnetic ﬁeld B was oriented at 45◦ to the direction of the optical axis of

50 0 0.2 0.4 0.6 0.8 1.0 Figure 3.8: Contour-plots of the normalized transition strength as a function of polarization 2 2 40 + and impact parameter for 4 S1/2 → 3 D5/2 in a single Ca . From left to right, ﬁrst and second subplots are for `γ = 0 theory and experimental data correspondingly. The third and fourth are for `γ = 1 theory and experimental data, respectively. Red lines indicate pure vertical polarization.

51 Figure 3.9: Contour-plots of the normalized transition strength as a function of polarization 2 2 172 + and impact parameter for S1/2 → F7/2 in a single Yb . In both cases `γ = 1, θz = 0; but φb = 0 (−0.3 rad) for the left (right) plot. the laser beam. To control the photon polarization state, a set of wave-plates was used. By rotating a half-wave-plate, the polarization state was varied along a meridian in the Poincaré sphere, see Figure 2.4. In Figure 3.8 the contour-plots of the normalized transition strength as a function of

2 2 40 + polarization and impact parameter for 4 S1/2 → 3 D5/2 in a single Ca are shown. The polarization angles γ were extracted from the measurements by comparing the optical response in experiments and those generated in the simulation. The sets of parameters for the plots are {θk = 0.075 rad, b = −0.62 rad} and {θk = 0.095 rad,φb = −0.3 rad} for `γ = 0 and 1 correspondingly. The overall normalization constant was picked for each profile independently, while the waist w0 = 9 µm/λ was fixed throughout. We obtain ◦ ◦ ◦ ◦ θz = 45 ± 5 and θ = 0 ± 0.02 . We see that the theoretical model reproduces all the main features, including the breaking of radial symmetry in the polarization patters (figure eight-like patterns in the coutour-plots). Since the generic polarization state, as shown in eqn. (3.33), is introduced by the linear

52 Figure 3.10: Transition strength Ωr for different atomic multipolarities and beam types as a function of impact parameter b for parameters: θz = 0 and Λ = 1. See the text for detailed description. combination of two circularly polarized twisted photon states, the relative phase factor gets generated exp[i(mγ + mi − m f )φb]. As was mentioned in subsection 3.4.1, it leads to observable effects, see Figure 7 in the paper by Afanasev et al. (2018a). The phase φb is responsible for the azimuthal orientation of the ion with respect to the plane, formed by the beam symmetry axis and the constant magnetic ﬁeld. For instance, if one changes the sign of the phase to the opposite in Figure 3.8, so that φb = [0.62,0.3], the resulting patterns will be mirror symmetric. Hence, the ﬁt parameters of this model are the overall normalization constant, the beam waist w0, the pitch angle θk and the azimuthal angle φb. 2 2 172 + Adapting the developed model to the case of S1/2 → F7/2 in a single Yb , we produced a set of predictions, shown in Figure 3.9, for the normalized transition strength as a function of polarization and impact parameter. We consider `γ = 1 for two very close values of the azimuthal phase factor φb = 0 and −0.3 rad. This is remarkable how

53 Figure 3.11: Transition strength Ωr for different atomic multipolarities and beam types as a function of impact parameter b in case with ∆m = 1 and for vertical and horizontal photon polarization states. The ﬁrst column shows transition strength for M1 in θz = π/2 magnetic ﬁeld alignment, the second and third columns show E2 and E3 multipoles for alignment angle θz = π/4, respectively. The columns 1, 2 and 3 increment the topological charge `γ = 0,1,2, correspondingly. See the text for detailed description. such small variations in phase cause a very noticeable effect on the level of the oscillator strength.

In the Figure 3.10 we show the excitation proﬁles for alignment angle θz = 0 and left- circular polarization Λ = 1, with the amplitude as in eqn. (3.32). We show the predictions for three multipoles: M1, E2 and E3, see Table 3.2. The line-types indicate the magnetic transition ∆m = 1,2,3 for solid-green, dash-dot-brown and dot-red. For the M1 transition in argon, Figure 3.10 (a, d, g), there is one open transition

{∆m = 1, `γ = 0}, graph (a) (i.e. open transitions correspond to central peaks in plots of transition strengths). In case with calcium (E2), Figure 3.10 (b, e, h), we get open transitions for {∆m = 1, `γ = 0} in graph (a), and {∆m = 2, `γ = 1} in graph (b). And

54 ﬁnally for E3 in ytterbium there are three: {∆m = 1, `γ = 0}, {∆m = 2, `γ = 1} and

{∆m = 3, `γ = 2}, in full accordance with the magnetic selection rules for twisted beams

∆m = mγ , extracted from eqn. (3.32). In ﬁgure (h) we contrast our predictions to the results presented by Schmiegelow et al. (2016).

In Figure 3.11 we plot the transition strengths Ωr as a function of the impact parameter for ∆m = 1 for horizontal and vertical photon polarization states, see eqn. (3.33). Red- solid (blue-dashed) indicates results for light polarized vertically (horizontally). For E3 transitions in ytterbium, Table 3.2, we also show the corresponding results for nuclear spin I = 1/2 in dotted-purple (dash-dot-black). Transitions E2 and E3 are for alignment angle

θz = π/4, while M1 is shown in θz = π/2 conﬁguration. The M1 transition in 40Ar+ (Table 3.2) shows qualitatively similar behavior to the electric quadrupole in 40Ca+ and octupole Yb+, but with the polarizations inverted, Figure 3.12. In the case of Yb+, see Figure 3.11 (c, f, i), there is also a dependence of the relative interaction strengths on the spin content of the nucleus. However, the correction due to the presence of the nuclear spin is only on the order of 10%. We studied HCI neon separately in the paper by Afanasev et al. (2018b), as a unique case of an ion with the transition of mixed multipolarity; see Table 3.2 and consult the paper by Rynkun et al. (2012). This transition is also peculiar as the photo-absorption amplitude is simply a linear combination of M1 and E2 with no interference terms present. The resulting amplitude for a transition with a mixture of M1 and E2 is shown in the ﬁrst column of Figure 3.12. One can clearly see that the quadrupole contribution (second column) looks identical to the E2 transition discussed above for calcium ions (Figure 3.8). The magnetic dipole contribution (third column) is identical to the M1 transition discussed above for argon HCIs. It is instructive to analyze the behavior in the proximity of the optical axis of the beam where the magnetic selection rules violation is the most apparent. For that we expand the

55 Figure 3.12: Transition strength Ωr for different atomic multipolarities and beam types as a function of impact parameter b. The left column shows the expected response for 2 2 20 5+ a transition P1/2 → D3/2 in HCI Ne . The parameters used are ∆m = 1, φb = 0 and θz = π/4. Red-solid (blue-dashed) indicates results for light polarized vertically (horizontally) beams.

absorption matrices around b → 0. For 40Ca we get

(BG)(`γ =0) 2 (BG)(`γ =0) 2 M3/2 1/2 H ∝ i(5θk − 4)cos(2θz), M3/2 1/2 V ∝ i(5θk − 4)cos(θz),

(BG)(`γ =1) (BG)(`γ =1) M3/2 1/2 H ∝ 2θk(1 + 4cosθz)sinθz, M3/2 1/2 V ∝ 2θk(2cosθz − 1)sinθz,

(BG)(`γ =2) 3 2 (BG)(`γ =2) (tw) M ∝ i √ θ (cosθz + cos2θz), M = −M (`γ = 2). 3/2 1/2 H 2 k 3/2 1/2 V m f mi H (3.34)

Hence, the Bessel function collapses into the δ-function, and the projection of the photon’s

total angular momentum mγ is transferred into the internal degrees of freedom of the target

atom, ∆m = mγ , as was discussed by Afanasev et al. (2013, 2016, 2018a). In such a case,

the interaction strengths only depend on the pitch angle θk and the alignment angle θz. From the equations above one can see that the horizontal polarization is completely

suppressed in the vortex center when θz = π/4 and `γ = 0. In a 45-degree alignment this is a signature of transitions with dominant E2, which is due to their sensitivity to the ﬁeld

56 gradients, see Schmiegelow et al. (2016). Another factor is the generation of geometry- dependent terms due to the rotation of the quantization axis (3.32). The dependence on alignment angle was previously discussed in papers by James (1998), Roos (2000) and Schmiegelow and Schmidt-Kaler (2012). For 40Ar+ in this limit one gets

2 2 (BG)(`γ =0) θk (BG)(`γ =0) θk M3/2 1/2 H ∝ −i(1 − ), M3/2 1/2 V ∝ i(1 − )cos(θz), 4 4 (3.35) (BG)(` =1) (BG)(` =2) θ M γ ∝ θ sinθ , M γ ∝ θ 2 cos2 z . 3/2 1/2 H/V k z 3/2 1/2 H/V k 2

For the case of `γ = 0 and θz = 0, both H and V interactions strengths are equal. This is identical to the behavior of E1 and E2 transitions. However, when varying θz to π/2, we see that the value of the horizontally polarized beam does not change, while the vertical goes to zero. The behavior of M1 at `γ = 0 and θz = π/2 is indeed similar to E2 at

θz = π/4, as can be seen from the asymptotic formulas above, as was emphasized earlier in the analysis of Figure 3.11. The matrix elements at the vortex center for Yb+ are:

(BG)(`γ = 0) 2 M1 0 H ∝ i(4 − 11θk )(cosθz + 15cos3θz),

(BG)(`γ = 0) 2 M1 0 V ∝ −i(4 − 11θk )(3 + 5cos2θz),

(BG)(`γ = 1) M1 0 H ∝ −4θk (23 + 20cosθz + 45cos2θz)sinθz, (3.36) (BG)(`γ = 1) M1 0 V ∝ −4θk (13 − 20cosθz + 15cos2θz)sinθz, (BG)(` = 2) 3 M γ ∝ i θ 2 (22 + 7cosθ + 10cos2θ + 25cos3θ ), 1 0 H 2 k z z z (BG)(` = 2) θ M γ ∝ i6θ 2 (21 − 40cosθ + 35cos2θ )cos2 z . 1 0 V k z z 2 Hence, the horizontal polarization is completely suppressed for the impact parameter b → 0, for θz = π/2.

57 For the highly charged neon in proximity to the beam vortex for `γ = 0 one gets:

(BG) √ M3/2 1/2 H ∝ i( 3M1 − E2(1 − 2cosθk)cos2θz), (3.37) (BG) √ M3/2 1/2 V ∝ i( 3M1 − E2(1 − 2cosθk)cosθz).

We see that by switching one or the other multipole, one obtains the results in eqs (3.34) and (3.35). This is exactly as expected theoretically and also can be seen from Figure 3.12. It is also evident that for the case with mixed multipolarity the horizontal component cannot be

completely suppressed at any α, unless the magnetic contribution is negligible or the pitch

angle θk is very small (plane-wave limit). Hence, in the case with mixed multipolarity, the presence of a strong M1 multipole drastically changes the optical response in the proximity to the beam center. Comparing the result for argon and neon, one may notice that if non-zero E2 in M1-dominated argon can be enhanced by the beam topology enough to be registered experimentally, then one would expect the strongest effect at the beam center, where the minima would be affected in the same way as in neon. However, at this stage we do not expect it to be easily seen in experiments with argon, since the effect would be too localized around the beam center and too weak to be reliably extracted from noise.

3.4.3 Ion photo-excitations in Radially and Azimuthally Polarized Orbital Angular Momentum Beams

In section 2.4 we introduced non-uniform exotic polarization states – azimuthal and radial. Radially polarized modes are known to produce strong longitudinal fields near the center of the beam, while azimuthal modes produce no field in z-direction. To check this statement we consider the vector potential of the twisted EM-field, as was derived by eqn. (11) in

58 5 5 (a) (b)

0.500 0.500

, ( a.u. ) total 0.050 total

j μ 0.050

tw M M1

Γ M1 E2 E2 0.005 0.005

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 b/λ b/λ Figure 3.13: Logarithmic plot of the photo-absorption rates in HCI 10Ne+ with mixed M1+E2 multipolarity. The bold curves correspond to radial (a) and azimuthal (b) polarization states, and light-colored curves are the rates for LC polarization state and mγ = 2 introduced for the comparison, see Afanasev et al. (2018b).

the paper by Afanasev et al. (2013):

  Λ iΛφr 2 θk −Λ −iΛφr 2 θk  Λ(i e sin Jm +Λ(κρ) − i e cos Jm −Λ(κρ))   2 γ 2 γ  µ (tw)   imγ φr Λ+1 iΛφr 2 θk −Λ+1 −iΛφr 2 θk AkΛω ∝ e −(i e sin Jm +Λ(κρ) + i e cos Jm −Λ(κρ)) .  2 γ 2 γ     √Λ eimγ φr sinθ J (κρ)   2 k mγ  (3.38) Then we construct the azimuthal and radial polarization as it was deﬁned in eqn. (2.47);

    −2icosφrJ1(κρ)cosθk −2sinφrJ1(κρ)     µ (tw; r)   µ (tw; a)   AkΛω ∝ −2isinφrJ1(κρ)cosθk ; AkΛω ∝ −2cosφrJ1(κρ) . (3.39)      √     2sinθkJ0(κρ)   0 

µ (tw; r) One can see that when b → 0 the only state that survives is the radial state AkΛω ∝ √ {0,0, 2sinθk}. On the other hand, in the case with azimuthal polarization, the longitudinal component of the vector ﬁeld is identical zero. This means that no M1 transitions are possible for the photo-absorption process when the incoming azimuthal beam is in the parallel conﬁguration. We tested this conclusion in the case with the OAM photo-absorption in the ion of mixed multipolarity 10Ne+. We used the absorption matrix element as in eqn. (3.31), and

59 the corresponding expression for the plane-wave amplitude from the Table 3.2, to calculate (m ) (m ) the photo-excitation rates Γ γ = f ·σ γ . Here the cross section for the hydrogen-like n f j f Λ n f j f Λ atom is m f = j f (m ) σ γ = 2πδ(E − E − ω ) |M(tw) (r)|2/ f (3.40) n f j f Λ f i γ ∑ m f miΛ m f =− j f The photo-excitation rates as a function of the impact parameter, normalized to the photon wavelength, are shown in Figure 3.13. The light-colored curves are the impact parameter dependence of the rates for the LC photon polarization state in both (a) and (b), while the bold-lines represent the same observable for radial polarization in (a) and

azimuthal polarization in (b). The pitch angle used is θk = 0.2 rad. One can see that radial polarization state causes strong enhancement of the magnetic multipolarity in the beam center, as compared to the LC polarization state, see Afanasev et al. (2018b). Azimuthal polarization, however, suppresses M1 completely at the center, leaving E2 the dominating transition. The conclusion follows that twisted radially polarized beams can be used for selective enhancement of magnetic transitions in atomic spectroscopy. We obviously can expand the argument on other systems, like artiﬁcial atoms, mesoscopic targets, etc. This effect can also be used in reverse – as a beam diagnostic tool.

3.5 Recoil from Orbital Angular Momentum Photons in Trapped Ions

As was stated at the end of Section 2.4, twisted modes carry momenta in the plane transverse to the direction of the beam propagation. The question arises: how would these transverse momenta influence the optical cooling and pumping processes? To shed some light on this subject, we apply a QM technique developed for optical cooling. We will calculate the transition amplitudes, accounting for recoil at trapped ions when the external EM-field carries a well-defined OAM. Semi-classical and QM approaches to describe the transfer into external degrees of freedom were discussed by Jauregui (2004)

60 Figure 3.14: The reference frame setup for the recoil problem, where O is the global coordinate system with respect to the entire beam; O0 is the local coordinate system with respect to the local wavefront.

and Afanasev et al. (2014b). Mondal et al. (2014) considered the transfer of OAM from a LG mode into external atomic degrees of freedom in the Bose-Einstein condensate using the Power-Zineau-Wooley scheme. In this setup, we simplify a generally complicated ﬂuorescence cycle and consider a

fundamental two-level system: the ion is initially at its ground state Eg. Then it absorbs

a quantum of EM energy h¯ωa and transfers to the excited state Ee. The frequency ωa is often called the resonance atomic frequency. The Hamiltonian of the system, accounting for recoil reads:

H = Hc.m. + Ha + Hint (3.41) which consists of the center-of-mass contribution; the Hamiltonian of the electron motion in the atomic system; and the interaction part written to the ﬁrst order, H = e A · p, int me respectively. Assuming that the electronic degrees of freedom are decoupled, and that

the perturbation is small, we can solve the Schrödinger equation by separation: |Ψi = |inti |ti |radi. We assume that the ion is free for simplicity:

|ti = |Ki = CeiKK·R (3.42)

such that hK|P|Ki = hK¯K and hK 0|Ki = δ(K 0,K); C is the normalization constant.

61 For the schematic representation of the quantum system with respect to the photon

beam, see Figure 3.14. Here, mc is the mass of the atomic core, me is the mass of the electron interacting with the external ﬁeld, and R and R0 are the position-vectors of the center of mass of the atomic system in global O and local O0 coordinate systems,

respectively. Assuming that mc me, electron position and momentum vectors in the global coordinate system can be expressed as

re = R +r −b; (3.43) m p = 1P + p e M where P is the center of mass momentum and p is the electron momentum relative to the

core, with M = mc + me. With this setup, the photo-excitation amplitude with recoil in the zero-OAM case can be expressed as Mrec (R,r;k, ) = hK,e|A · p |K,gi (3.44) m f miΛ Λ e where |gi and |ei are the ground and excited states of the atomic two-level system. For more details on the conventional QM photo-excitation formalism with recoil, see papers by Lam and Berman (1976) and Wineland and Itano (1979). For the case of photo-excitation with a twisted BB, one can rewrite eqn. (2.32) as:

nm Mrec (R,r;k,Λ) = Ae−iωt 1 m f miΛ M Z dkk ikz(Rz+z) ⊥ ikk ·(R +r −b) × hKe,e|e P · a m (k )εˆ e ⊥ ⊥ ⊥ |Kg,gi (2π)2 κ γ ⊥ kΛ Z dkk⊥ ikk ·(R +r −b) o + hKe,e|p · a m εˆ e ⊥ ⊥ ⊥ |Kg,gi (2π)2 κ γ kΛ (3.45) We assume that not only the transition amplitude that describes the change in internal state of the ion can be separated from the topological part, but the same trick can be done on the external term of the amplitude, responsible for the energy-momentum transfer into the

62 vibrational modes of the ion. Hence, we assume that the atom is sufﬁciently cold. In that case, the characteristic dimensions of the wave-packet are much smaller than the waist of the beam. The factorization of the amplitude in the external term is valid only when ions are very cold and not localized in the beam penumbra. This is because close to the vortex the gradients of the EM-ﬁelds are so strong that even a sub-nanometer target “feels” the topology. By comparison, in the works by Jauregui (2004) and Afanasev et al. (2014b), the topological terms were held in the external terms which lead to a more detailed and complicated analysis. Under the assumptions given here we get:

If for a certain mγ the transition to the internal degrees of freedom is allowed (second term), then transfer into the external degrees of freedom is usually neglected (ﬁrst term) in the non-relativistic case due to

h¯(k ·r)(K · eˆ ) v g m ∼ , (3.47) Mωa(eˆm ·r) c where v is the electron velocity relative to the atomic core, see Wineland and Itano (1979), and c is the speed of light. However, if the transition into the internal state is prohibited by the TAM selection rules, then the second term is zero, and we can not neglect the ﬁrst term anymore. Hence, TAM goes into external vibrational modes, and this transfer will be

63 described by:

r rec −iωt j m−2m κ M (R,r;k,Λ) = Ae d (θ )(−i) γ Jm −m(κρ) m f miΛ ∑ Λ,m k 2π γ m=±1,0 (3.48) nihm¯ o × e (k ·r)(K · eˆ )δ(K +k,K ) . M g m g e

This expression depends on the kinematics of the problem: (1) the orientation of the initial center of mass wave-vector with respect to the local polarization vector of the EM-ﬁeld; (2) the direction of the local photon wave-vector, as one would expect. We cross-check these conclusions with the experimental results in Figures 3.4 and 3.5, subsection 3.4.1. In that case, the main attention has been paid to the transfer into the internal degrees of freedom. For instance, one sees at Figure 3.4, line 1, row 2: the electron

magnetic number changes by -1. Hence, the maximum mγ that can be transferred to the

atom is -1. If TAM mγ = −2 could be split in half, and −1 could go into the internal TAM of the electron, and another −1 into the vibrational degrees of freedom, we would see the same kind of maximum as in line 2, row 2 of this panel. Instead, the experimental group was able to observe the selection rules with no transfer into internal degrees of freedom, as the expressions above suggest. This picture is also phenomenologically similar to the picture described by Mondal et al. (2014), where external and internal contributions are only competing in second order terms, while in ﬁrst order the internal contribution is dominating by several orders of magnitude, basically putting TAM transfer into vibrational modes to zero. The rest of this calculation can be carried out similar to the conventional QM formalism.

64 Chapter 4: Chirality, Chirooptics and Circular Dichroism in Orbital Angular Momentum Photon-Matter Interactions

The effects of optical activity, see Arago (1811), and circular dichroism, see Cotton (1895), have been known for more than a century. These phenomena provided chemists and biologists with indispensable tools for spectral analysis of chemical compounds and biological materials. The introduction of the concept of chirality in physics is often credited to Lord Kelvin (1904), see, e.g., Cahn et al. (1966). However, the study of the chiro-optical effects in physics started actively gaining momentum only around the 1960’s, with the advent of photomultiplier tubes. It is hard to overestimate the importance of chirooptics these days. Its contribution ranges from fundamental chemistry to state-of-the- art medical applications. Furthermore, chiral structures are used in signal processing and nanophotonic devices. In chemistry, two chiral forms which are mirror-reﬂections of one another are called enantiomers. It is understood that enantiomers behave in an absolutely identical manner when interacting with non-chiral matter or light. However, their physical and chemical properties are distinctive when encountering other structures with non-zero chirality. In this chapter we go over the key concepts of matter and optical chirality with the emphasis on the effects coming from photon topology. We make predictions for circular dichroism in isotropic matter coming from optical OAM, followed by the discussion of a possible experimental conﬁrmation and applications of this effect.

4.1 Chirality in Light

Fresnel’s theory of optical activity describes the rotation of the polarization plane as a consequence of different propagation velocities of the left- versus right-circularly polarized

65 components of a linearly polarized light source

π α = (n+1 − n−1), (4.1) α where the difference between the refractive indices is called circular birefringence. One may reasonably argue that refraction and photo-absorption are intimately related. This implies that there should be a substantial difference in the absorption of LC versus RC light in birefringent media as well (ﬁrst observed by Haidinger in 1847). Indeed, linearly- polarized light becomes elliptical under propagation in an optically active sample

(E−1 − E1) tanψ = L L 6= 0, (4.2) −1 1 (EL + EL)

R L where EL and EL are the amplitudes of the beam components with RC and LC polarization after propagating a distance L in the medium. The incoming and outgoing ﬁeld amplitudes are related through the attenuation coefﬁcient as

−2πσΛL/λ EL = E0e , (4.3)

where σΛ is the absorption coefﬁcient. With a little math, one arrives at

hπ` i tanψ = tanh (σ − σ ) , (4.4) λ 1 −1 where the difference between the two photo-absorption coefﬁcients is called circular dichroism. Dichroism and birefringence are two main concepts of theories of optical activity. These phenomena are traditionally associated with a chiral structure of the medium. Chirality of the medium may be coming from the bulk properties (crystals of quartz) and low symmetry properties of individual molecules (organic molecules, proteins). Only

66 recently it was realized that effects similar to optical activity can also arise due to the topological structure of the light sources, see Zambrana-Puyalto et al. (2014). Since the discovery of birefringence and dichroism, it has been generally understood that circularly polarized light must be chiral, while linear polarization states are not. However, until now, chirality of light itself is not well defined and often inferred via light-matter interactions. For instance, Bliokh and Nori (2015) in their paper on the foundations of OAM-carrying states of light directly relate optical chirality to the photon helicity. Hence, in this case, chirality is a purely SAM-related quantity. However, they also discuss the effect of magnetic circular dichroism coming from the presence of an external magnetic field. It is related to the circular dichroism due to photon OAM, which will be discussed below. In this work the following definition of optical chirality will be used, see Tang and Cohen (2010) and Bliokh and Nori (2011):

ε 1 C = 0E · ∇ ×E + B · ∇ ×B. (4.5) 2 2µ0

Rewriting the eqs in (2.6) for a corresponding polarization basis, one can easily see that linearly polarized plane waves are not carrying any chirality, as they always can be represented by one-component vector ﬁelds. However, circular polarization is characterized by

2 chirality C = ε0|E0| k according to this deﬁnition, as has been stressed in Tang and Cohen (2010). Applying this same logic to the twisted light, we arrive at a somewhat unexpected conclusion. We start with the vector potential of the twisted Bessel-type EM-ﬁeld in the paraxial regime, written as

   −√Λ i−Λe−iΛφρ J (kθ ρ)  r  2 mγ −Λ k  kθ   −i(ωt−kzz) k imγ φρ i−Λ−1 −Λ −iΛφ AΛ(ρ,z) = e e √ i e ρ Jm −Λ(kθkρ) . (4.6) 2π  2 γ     √Λ J (kθ ρ)   2 mγ k 

67 Consider states with linear polarization:

  i(e−iφρ J (kθ ρ) ± eiφρ J (kθ ρ)) r  mγ −Λ k mγ +Λ k  kθ   −i(ωt−kzz) k imγ φρ −iφ iφ AH/V (ρ,z) = e e (e ρ J (kθ ρ) ± e ρ J (kθ ρ)) . 4π mγ −Λ k mγ +Λ k      Jmγ (kθkρ)(1 ∓ 1)  (4.7) In the general case, neither of the linear polarization states can be expressed by only one non-zero component of the vector field, as twisted light always carries at least one longitudinal and one transverse component, see Yao and Padgett (2011). Consequently, even in the case when a twisted beam is linearly polarized, which corresponds to a non- chiral plane-wave, OAM provides non-zero chirality of the optical fields. In other words, the beams with different SAM but the same non-zero OAM are not enantiomers, adopting the term from chemistry. Hence their superposition does not correspond to zero chirality, as it is the case for beams with zero OAM. In this light, one of the main results in the paper by Tang and Cohen (2010) of interest to us is that an overall asymmetry in the excitation rates in their samples were dependent on both the chirality of the medium and the chirality of the photon beam. These considerations make it necessary to distinguish between chirality due to the photon SAM and OAM. The corresponding definitions have been recently proposed by Kerber et al. (2018): SAM ∆σ` = σ`γ ,1 − σ`γ ,−1; γ (4.8) ∆ OAM = − . σ`γ σ+`γ ,Λ σ−`γ ,Λ The authors also introduce an important notion of two beam classes: parallel OAM ↑↑ SAM and anti-parallel OAM ↑↓ SAM. The significance of these classes was noted earlier experimentally by Zambrana-Puyalto et al. (2016) and pointed out by Quinteiro et al. (2017c) in their theoretical description of OAM-carrying modes.

68 Figure 4.1: Spin asymmetry (ﬁrst column), SAM circular dichroism (second column) and OAM circular dichroism (third column) for the transition S1/2 → D5/2 (E2) accounting for atomic electron spin degree of freedom as in eqn. (3.19).

4.2 Optical Activity and Twisted Photon Beams

Afanasev et al. (2017) deﬁne two quantities to characterize optical activity in isotropic sample. These are the coefﬁcient of circular dichroism itself

(`γ +1) (`γ −1) σn j 1 − σn j −1 CDSAM = f f f f , (4.9) `γ , j f (` +1) (` −1) σ γ + σ γ n f j f 1 n f j f −1 and the photon-spin asymmetry

(`γ +1) (`γ −1) ` , j Γn j 1 − Γn j −1 A γ f = f f f f , (4.10) Λ (` +1) (` −1) Γ γ + Γ γ n f j f 1 n f j f −1

(m ) (m ) where an excitation rate is deﬁned as Γ γ = f · σ γ , cross section was deﬁned earlier n f j f Λ n f j f Λ (tw) in eqn. (3.40), and Mm f mi (r) is the OAM beam photo-absorption matrix element. In analogy to Kerber et al. (2018), we also introduce OAM circular dichroism as

(|`γ |+1) (−|`γ |+1) σn j 1 − σn j 1 CDOAM = f f f f . (4.11) `γ , j f (|` |+1) (−|` |+1) σ γ + σ γ n f j f 1 n f j f 1

69 We assume summation over ﬁnal spins and averaging over initial spins. When plugging in the photo-absorption amplitude, eqn. (3.19), and considering E2-transitions such as

S1/2 → D5/2, we get the dependence as shown in Figure 4.1. We plot the spin asymmetry, SAM and OAM circular dichroism as functions of the impact parameter b for different values of vorticity `γ = 0,±1,±2,±3 in S1/2 → D5/2 (E2) transition. One can immediately see that the absorption rates for twisted photons whose SAM and OAM are collinear are higher, as was experimentally observed by Zambrana-Puyalto et al. (2016). The effect is most prominent in the beam’s penumbra. The underlying mathematical justiﬁcation for circular dichroism in non-chiral matter interacting with OAM light can be seen when going back to the amplitudes in the limit b → 0, as shown in eqs (3.34-3.36). For instance, for the E2 transition in calcium, eqn. (3.34),

one ﬁnds that for `γ = 0 and 1, the LC and RC amplitudes contribute symmetrically.

However, for `γ = 2, only the RC component survives. For octupolar transitions in ytterbium, see eqn. (3.36), the dichroic amplitude extinction happens at `γ ≥ 1. For M1- transitions, eqn. (3.35), absorption contributions from the optical components of different

helicity become uneven at `γ = 1. These effects are impact parameter dependent and get stronger for higher beam vorticities, until eventually all the amplitudes become extinct at the beam center. For instance, for E3 in Yb+ all the amplitudes become extinct in the

proximity of the beam’s optical axis when `γ ≥ 5, see Solyanik-Gorgone et al. (2019). This theory does not predict dichroic effects in electric dipole interactions with OAM-light. Afanasev et al. (2017) highlight that circular dichroism in an atom photo-absorption

is only present if: (1) high-order OAM-beams `γ ≥ 0 are considered; (2) higher-order

multipolar contributions with OAM of the excited electron state ` f ≥ 1 are allowed. It is

also predicted in this paper that at moderately small pitch angles θk < 0.25 rad the rate

asymmetries become independent of θk , which can be used as a beam diagnostic tool to determine the topological charge of the vortex.

70 Chapter 5: Electromagnetic Interactions and Neutron-Proton Systems

The discovery of the neutron wave nature in late 1930’s has become a foundation of

modern neutron optics and interferometry. Thermal and cold neutrons (E . 0.025 eV; 100 ≤ v ≤ 103 m/s; λ ∼ 0.2 nm) are used in nuclear and biological sciences, material sciences, condensed matter, the Bose-Einstein condensate and many more. In a similar way as the EM-wave described by the Helmholtz equation, eqn. (2.5), the neutron matter wave is described by the Schroedinger equation:

pˆ2(r) +V(r;t) Ψ(r;t) = EˆΨ(r;t), (5.1) 2mn where mn ∼ 939.57 MeV is the neutron rest mass and Ψ(r;t) is the neutron wave function. The same fundamental effects – scattering, diffraction, interference, refraction – govern neutron-matter interactions. Neutron beam coherence is of a similar if not greater importance in neutron optics due to engineering challenges. Hence, it is only logical that a lot of experimental techniques in optics are being adopted in neutron optics. Despite the highlighted similarities, the neutron, being a massive composite fermion, interacts with the environment through all four forces of the Standard Model. Hence, we expect a lot of new physics being revealed in OAM neutron-matter interactions and interferometry, complementary to the achievements in singular optics and photonics. Being completely electrically neutral and massive, neutrons penetrate much deeper into the target enabling deep scanning of the isotope content and structure of the target material. An extrapolation of photon-OAM-related phenomena onto matter beams and cold neutron interferometry was brilliant in its phenomenological simplicity. But obtaining a twisted neutron beam was neither a simple challenge, nor obvious to overcome. Due to the low beam intensity in cold neutron sources, the typical signal collection time is measured in days, which turns noise and coherence control into a big challenge.

71 A new polarized neutron interferometry facility has been recently installed at NIST, see Shahi et al. (2016). It hosted a ﬁrst of a kind experiment proving that neutrons, just like photons and electrons, can carry a well-deﬁned OAM, see Clark et al. (2015). With the help of a single-crystal Mach-Zehnder interferometer, and a ∼15 mm – diameter spiral

phase plate, Clark et al. (2015), a cold neutron beam λ = 0.271 nm was split into reference and test beams. The test beam was twisted and then interfered with the reference beam conﬁrming the presence of the non-zero OAM. The design has been patented, see Clark et al. (2016), and new experiments have already been proposed, e.g. Nsoﬁni et al. (2016). In this chapter we focus on nuclear electromagnetic interactions. We consider the QED framework with time-dependent perturbation theory and Fermi’s Golden Rule, where

the ﬁne structure constant α = 1/137 is a small parameter. We derive the deuteron photodisintegration matrix element and consider the reverse reaction of neutron-proton

(np) capture np → dγ for the case of a cold neutron beam, such as the one described by Shahi et al. (2016).

5.1 Deuteron Photodisintegration and np-Capture with Orbital Angular Momen- tum Beam States

The nuclear electromagnetic interactions can be described by the QED interaction Hamil- tonian in radiation gauge as

e Z H = − p drr J (r) ·A(r). (5.2) int c N

Here JN(r) is the nuclear current, which is a sum of the convection current Jc arising from the proton orbital motion and the magnetization current due to magnetic moments of

the constituents: µp = 2.7934 eh¯/2mpc and µn = 1.9135 eh¯/2mpc. For the problem of a

72 pn-system in an EM-ﬁeld one can rewrite this Hamiltonian as

e Hint = − p p ·A(r p) + µp · (∇p ×A(r p)) + µn · (∇n ×A(rn)) , (5.3) mpc

and the photon vector potential is usually expressed as:

† i(k·r−ωt) Ak,Λ;ω (r) = ∑Ak{ekΛaˆkΛe + h.c.}. (5.4) k

If the system in its initial state |ii absorbs (emits) a quantum of the EM-ﬁeld and gets excited (de-excited) into a ﬁnal state | f i, according to Fermi’s Golden Rule, the corresponding transition probability is

2 2 dω = 2πk |h f |Hint|ii| dΩ f , (5.5)

where h f |Hint|ii is the transition matrix element and Ω f is the solid angle of the ﬁnal nucleon state. For more details on the formal QED nuclear theory the reader is referred to the books by Blatt and Weisskopf (1991), Walecka (2004) and Byrne (2013) As the framework has been formulated, we proceed to the description of the deuteron photodisintegration process. The ground state of a deuteron is an S-like conﬁguration with 4% admixture of a D-state. Depending on the relative orientation of proton and neutron

3 1 1 spins, there are two possibilities: a triplet S1 and singlet S0. However, the S0-state is isospin-prohibited. Hence, when the deuteron absorbs a photon of energy h¯ω ≥ 2.226

3 3 MeV, it may split into a neutron and a proton in an M1-dominated transition S1 → P0,1,2. To solve the problem of the incoming twisted γ-ray interacting with the deuteron ground state, we do the perturbative expansion to ﬁrst order in the ﬁne structure constant

in its general form. We use the ASR to describe the twisted BB γ-photon, eqn. (2.32),

ikk·r in TAM basis. The OAM-zero radiation mode ekΛe can be expanded in the vector

73 spherical harmonics, eqn. (3.9), and substituted in eqn. (2.32) to obtain

M(BB) = − (−i)2mγ −m f +mi J (κb)ei(mγ −m f +mi)φb m f miΛ mγ −m f +mi (5.6) ji j f × d 0 (θk)d 0 (θk)M 0 0 (0). ∑ mim m f m m f miΛ 0 0 i f mim f

Here Mm f miΛ(0) is the amplitude for a photoexcitation with zero OAM

√ ∞ r ep j+µ (2 j + 1) µ+1 Mm f miΛ(0) = 4π ∑ i Λ h j f m f |Tjm| jimii, (5.7) c j=1 2 where Tjm is a spherical tensor (see, e.g., Walecka (2004) for the proof):

Z µ Tjm = drr (A jm(k,r) ·JN(r)) (5.8)

Λ and Y j`1(Ωk) are vector spherical harmonics, see eqn. (A.15). We can apply the Wigner- Eckart theorem to extract the coupling selection rules explicitly:

∞ s ep √ 2 j + 1 j m M (0) = 4π i j+µ Λµ+1C f f h j ||T || j i (5.9) m f miΛ ∑ jimi jm f j i c j=1 2 j f + 1 which results in

| ji − j f | ≤ | j| ≤ ji + j f ; (5.10) m f − mi = λ

In eqn. (5.6), these rules are modiﬁed by the topological factor lim Jm −∆m(κb) = b→0 γ

δmγ ,∆m, resulting in the magnetic selection rule mγ = m f − mi in the vicinity of the singu-

ji larity. The Wigner rotation matrices dmm0 (θk) in eqn. (5.6) take care of the local photon wave-vector being tilted with respect to the overall beam propagation direction, as was discussed in section 2.3. One can infer the selection rules from Figure 5.1, where we plot the deuteron photodisintegration matrix element as a function of the target’s position in

the OAM γ-ray beam. Additional information on this matter can be found in the papers by

74 0.7 0.4 (a) (b) 0.6 0.3 0.5 0.4 ( a.u. ) ( tw ) j Λ 0.2 0.3 M 0.2 0.1 0.1 0.0 0.0 -10 -5 0 5 10 -10 -5 0 5 10 b/λ b/λ

(tw) Figure 5.1: Deuteron photodisintegration matrix element MjΛ as a function of the impact parameter b/λ for ∆m = −1 – solid-black, ∆m = 0 – dotted-purple and ∆m = 1 – dot- dashed-green. Photon states are mγ = 0 and 1 for (a) and (b) correspondingly.

Afanasev et al. (2018d,c). As we have extracted the selection rules for the photodisintegration, we consider the reverse process of np-capture. We assume a cold, ∼ 10 meV twisted neutron being

captured by a proton with the simultaneous emission of a γ-ray and a deuteron: np → dγ. In this case, the reaction is S-wave dominated with M1 multipolarity. The OAM state of a neutron can be generically written as:

|mκ pzσi = Amκ pz (p;r)|σi, (5.11)

where |σi is the spinor state and Amκ pz (p;r) is a spatial part of the mean eigenstate, see Nsoﬁni et al. (2016) and Afanasev et al. (2018d). According to Afanasev and Serbo,

for BBs this state is the eigenstate of the energy Hˆ , the longitudinal momentum pˆz, and

the projection of OAM Lˆ z. Hence, by solving the Schrödinger equation in cylindrical coordinates one gets

imφr ipzz |mκ pzσi = Jm(κρ)e e |σi, (5.12)

p 2 where the transverse momentum can be expressed as κ = 2mnE − pz . The spatial part of this equation is identical in structure to eqn. (2.25) for twisted photon beams. Hence, it

75 is convenient to apply ASR in the similar manner as before. The ﬁnal expression is

Z dpp ipzz ⊥ |mκ pzσi = e a m(p ·ρ)|σi, (5.13) (2π)2 κ ⊥ where the Fourier-transform kernel can be cast into:

2π −m imφp aκm = i δ(p⊥ − κ)e . (5.14) p⊥

The angles φp and φr deﬁne the position and direction of the neutron momentum in the transverse xy-plane of the beam in cylindrical coordinates. One can deﬁne LG and BG neutron states in a similar manner, mimicking the procedure for photon states in Chapter 2. The density of the OAM neutron states carries the same doughnut-like structure as the corresponding photon states, see Figure 2.2. According to Afanasev et al. (2018d), the corresponding amplitude can be expressed as

(m) −ikk b (0) imφ M (κ, pz;k,b) = e ⊥ M (k, p)Jm(κb)e b , (5.15) where M(0)(k, p) is the amplitude for the case when the incoming cold neutron beam does not carry OAM, see, e.g., Byrne (2013). As for the probability of the nucleon capture process per unit time, it is proportional to the density of the OAM-neutron state. One of the important conclusions is that the scattering cross section of both reactions considered in this chapter is very sensitive to the density distribution of the incoming twisted neutrons:

Z ∞ m 1 2 2 −(b2+b2)/(2w2) bdb σ ∝ R = J (κb)I (bbt/w )e t . (5.16) 2 2 −κ2w2 m 0 2 I0(κ w )e 0 w

tw 0 The parameter R ∝ σ /σ (θk) is shown in Figure 5.2 as a function of the impact parameter

b in units of the probe wavelength λ. The plots are generated for the pitch angle θk = 0.1 rad. The parameter w = 1/κ is the Gaussian dispersion coefﬁcient. One may notice that

76 1.0

0.8

0.6 R 0.4

0.2

0.0 0 2 4 6 8 10 b/λ

tw 0 Figure 5.2: The cross section ratio σ /σ (θk) as a function of the impact parameter b/λ for m = 0 – solid-black, m = 2 – dotted-purple and m = 5 – dot-dashed-green, see Afanasev et al. (2018d).

the behavior of the parameter R is strongly OAM-dependent. This property can be used in neutron beam diagnostics. If supplied a sensitive enough experimental setup, one can take advantage of the neutron-OAM enhancement of the higher multipoles and probe the D-wave contribution into the deuteron structure, in a similar manner as it was done for photon-atom interactions.

77 Chapter 6: Photo-Induced Processes in Semiconductor Materials and Quantum Dots

Solid state materials play an extraordinarily important role for modern electronics. The continuously accelerating race in efficiency, compactness and innovation in modern technology forces material science and engineering to constantly push the boundaries of solid state physics and semiconductor technology. Starting from the early 80’s and until recently, Moore’s law had been imposing the engineering challenge to reduce the size of computer microchips and circuits. It eventually led to the formation of a new direction in modern science and technology – semiconductor nanostructures and nanodevices. In the framework of solid-state-photon interactions, the breakthroughs were not less impressive: LEDs, diodes, hardware, fiberoptics, and many more. We are interested in a new trend in this field, which integrates advances in modern engineering semiconductor technologies and availability of twisted light sources. The ability to work in high-dimensional Hilbert space, provided by OAM-modes, requires new hardware designs in quantum communications and information processing, such as OAM-generating microlasers and single-photon sources, preserving the beam topology microcavities and single OAM-photon detectors. The possibility to use metamaterials to generate OAM photon sources has been actively explored in the field, see, e.g., Zeng et al. (2013), Miao et al. (2016) and Seghilani et al. (2016). From the fundamental point of view, we anticipate a plethora of novel effects on the interface of topological phenomena in solid state and laser light. In this chapter we show that the photon OAM can be transferred to electron degrees

of freedom in a semiconductor in vertical v − c-transitions in the Γ-point. We derive the set of photo-absorption selection rules in a bulk direct band-gap I-III type semiconductor, such as GaAs, and predict anomalies in conduction electron polarization. We also describe

78 Figure 6.1: GaAs crystal energy band structure, see e.g. Kuwata-Gonokami (2011).

a possible experimental design with a mesoscopic semiconductor target, to minimize smearing of effects due to electron diffusion in the conduction band.

6.1 Twisted Photon Interaction with Bulk GaAs

The process of light interacting with solid state materials such as semiconductors is one of the objectives of optoelectronics. This branch of solid state physics emerged and matured in the late 50’s to early 60’s, after transistors and photovoltaic cells were invented. Functioning of modern ﬁber optic communication schemes is based on the developments in semiconductor optoelectronics, where GaAs and AlGaAs get most of the attention. GaAs is a direct-band (1.42eV) III-V compound semiconductor with the energy band structure shown in Figure 6.1. There the lowest conduction and valence bands are marked

in red, and the red arrow represents the vertical transition in the Γ-point. They provide good light emission efficiency, high electron mobility and irradiation resistance. Another field of applications is polarized electron sources and field emitters, where GaAs has become a material of choice due to its high polarization yield. When illuminated with a 860 nm circularly polarized photon laser source, GaAs emits polarized electrons

79 with the theoretical limit of 50% polarization Hernandez-Garcia et al. (2008). In this subsection we focus on the electron polarization in GaAs excited by twisted photons. The semiclassical formalism was previously developed by Quinteiro and Tamborenea (2009b); Quinteiro and Berakdar (2009); Quinteiro and Tamborenea (2010) and further developed by Cygorek et al. (2015). The predictions are in agreement with experiment Clayburn et al. (2013). However, the authors in all these studies averaged the electron response over the spatial location of the photo-excited electron in the conduction band. In contrast, we keep the impact parameter dependence and show that the OAM transfer can be seen in a high resolution experiment (of several photon wavelengths). The lowest conduction band in GaAs is S-like with even parity, while the closest valence band is P-like odd parity. The corresponding allowed transitions are shown in

the energy level diagram, see Figure 6.2 in bold. This results in the lattice symmetry Γ25, which does not allow for time reversal. For more reviews on III-V semiconductors and devices, see, e.g., Look (1989), Voon and Willatzen (2009) and Aliofkhazraei (2014). Re- cently, there have been updates on possible corrections to zincblende-type semiconductor symmetries, see Elder et al. (2011). In QM the one-electron state in a crystal can be described by the Hamiltonian:

h¯ 2 h¯ 2 p H = − ∇ + 2 2 (σ × ∇U) · p +U(r). (6.1) 2me 4mec

When an electron interacts with a photon, the energy gets modiﬁed and the corresponding Hamiltonian describes the coupling to the EM-ﬁeld in a simple form

ih¯ Hint = − (A · ∇), (6.2) me where me is the free electron mass and A is the incoming photon vector potential. We consider the simplest case of one-electron approximation and neglect higher-order effects. When such an electron in a valence band gets photo-excited by a high-energy photon

80 Figure 6.2: Allowed transitions for the twisted photons (solid lines) and zero-OAM photons (dadshed lines) with RC polarization in GaAs crystal. Numbers in the circles indicate squares of Clebsch-Gordan coefﬁcients.

source into a conduction band, then, if supplied enough energy, it may overcome the work function and escape into vacuum. The absorption rate is proportional to the squared norm of the matrix element

µ 2 Wvc ∝ ∑ |hc|A pµ |v;kΛi| δ(EcΛ − EvΛ), (6.3) f ,Λ where Λ goes over the photon modes. In the case of a non-twisted photon source, the vector-potential can be expressed as:

(pw) † i(q·r−ωt) Aq,Λ;ω = A∑{eqΛaˆqΛe + h.c.}, (6.4) q where q is the photon wavenumber. We consider the vertical transitions in the Γ-point, when q kv, kc. This allows us to neglect excitonic contributions, see Naka et al. (2016). In the description of the twisted light, see Chapter 2, we expressed the photon state as an eigenfunction of J . Hence, coupling of the twisted photons to matter needs to be

81 explicitly expressed through the TAM exchange with the electron spin taken into account. This can be done in the framework of the theory of invariants, first introduced by Luttinger and Kohn (1955). In this case, instead of implicitly imposing the symmetries when defining the number of independent parameters, as in perturbation theories, one initially assumes that the QM Hamiltonian must be invariant with respect to the semiconductor crystal symmetry group. It is natural to start with a TAM basis, which is the proper basis for atomic structures. However, the remote periodic band structure in semiconductors alters the topology of the entire crystal, reducing the symmetry in the lattice nodes. To solve this problem, Löwdin (1951) proposed starting out with the “right” symmetry group and then performing the perturbative expansion to account for the correction terms. This way he separated the eigenstates of the Hamiltonian |αi;ii into near states – eigenstates of J ; and remote states – perturbative corrections. In this manner, in the leading order, the initial and final electron states can be expressed in terms of the near states:

mn where Sn are the expansion coefﬁcients which depend on the symmetry group of the

crystal, and | jv,mvi are the TAM eigenstates; αv and αc stand for all the other “good” quantum numbers.

The next step is to cast Cartesian tensor pµ in eqn. (6.3) into the spherical tensor of rank 1, see A.3.2. of Voon and Willatzen (2009) for the detailed procedure:

ν (1) pµ = ∑αµ Tν (6.6) ν

82 2 Table 6.1: Photoelectron polarization for the pitch angle θq → 0 with accuracy O(θq ) for different sets of photon quantum numbers as a function of the parameter x = 2πb/λ. Top two rows are for the unstrained GaAs and bottom two rows are for the strained GaAs sample.

mγ −2 −1 0 1 2 σ

4 4 2 36−x 4−x − x − / − / 1 72+36x2+2x4 2(4+8x2+x4) 4+2x2 1 2 1 2

2 4 4 − x − 4−x − 36−x 1 1/2 1/2 4+2x2 2(4+8x2+x4) 72+36x2+2x4

4 4 36−x 4−x − − − 1 36+x4 4+x4 1 1 1

4 4 − − 4−x − 36−x 1 1 1 1 4+x4 36+x4 with αν given as:  √ √  −i/ 2 1/ 2 0     αν =  0 0 −i. (6.7)    √ √  i/ 2 1/ 2 0

Hence, the matrix element for a particular mode q can be expressed as:

(PW) ν µ i(q·r−ωt) (1) mc∗ mv MvcΛ = A αµ ∑ eqΛe h jc,mc|Tν | jv,mviSc Sv , (6.8) mc, mv

see Voon and Willatzen (2009) and Elder et al. (2011) for more details. We use the expansion in vector spherical harmonics, as it has been done in eqs (13) and (15) of the paper by Solyanik-Gorgone and Afanasev (2019), and plug those into eqn. (6.8):

r r 2 2 j + 1 M(PW) = A D j∗ (0,−φ ,−θ )Λη+1i j+η αν vcΛ 3 ∑ 2 Λm q q µ j,m (6.9) µ (1) mc∗ mv × ∑ j0(qr)∑ χi h jc,mc|Tν | jv,mviSc Sv . mc,mv i

83 Figure 6.3: Photo-excitation rate as a function of electron’s distance to the optical vortex center for the vortex pitch angle θq = 0.1 Rad; long-dashed blue is mγ = −2, dot-dashed black is mγ = −1, dashed red is mγ = 0, solid green is mγ = 1, dotted purple is mγ = 2. Incoming photon polarization is (a) – LC, (b) – RC.

For the case of photo-excitations with the twisted BB laser mode, eqn. (2.32), we get:

r r 2 (BB) 2 2 j + 1 Z d q M = A d j (θ )Λη+1i j+η αν ⊥ a (q )eiqq⊥·r vcΛ ∑ Λm q µ ∑ ( )2 κmγ ⊥ 3 j,m 2 mc,mv 2π

imφq µ (1) mc∗ mv × j0(qr)e ∑ χi h jc,mc|Tν | jv,mviSc Sv . i (6.10) And, ﬁnally, applying the Wigner-Eckart theorem we get

r s 2 2 j + 1 Z d2q (BB) j η+1 j+η ν ⊥ q MvcΛ = A ∑ dΛm(θq)Λ i αµ ∑ 2 aκmγ (q⊥) 3 2 jc + 1 (2π) j,m mc,mv (6.11) (1) × eiqq⊥·r j (qr)eimφq C jcmc χ µ h j ||T || j iSmc∗Smv . 0 jvmv; jm ∑ i c ν v c v i

The integral over the photon reciprocal space was discussed in the paper by Solyanik- Gorgone and Afanasev (2019), see eqn. (17) in particular. With this result being substituted in the expression above one gets

s r 2 j + 1 (BB) κ 2mγ −m η+1 j+η ν 0 MvcΛ = A ∑(−i) Λ i αµ j0(q r) 3π 2 jc + 1 jm (6.12) × d j (θ )J (κr ) C jcmc χ µ h j ||T (1)|| j iSmc∗Smv . Λm q mγ −m ⊥ ∑ jvmv; jm i c ν v c v mcmvi

84 This is the expression for the photo-excitation rate in terms of the single-electron near (1) states. The reduced matrix element h jc||Tν || jvi and the accompanying symmetry- related coefﬁcients are the contributions independent of the TAM exchange. The terms j dΛm(θq)Jmγ −m(κr⊥) are responsible for the magnetic selection rules’ violation in the same way as in Chapter 3 for atoms and Chapter 5 for nuclei. The convenient factorization of the plane-wave contributions at the level of the matrix element has been achieved. One can further develop this approach, calculating the higher-order contributions (remote electron states) by applying perturbation theory, see, e.g., Voon and Willatzen (2009) and Elder et al. (2011).

For the calculations in the direct vicinity to the Γ-point, when the photon wavelength

is λ ∼ 871nm, we extract instructive results even in the leading order. By neglecting the higher-order terms in the electron wave functions, we effectively neglect the OAM-transfer into the lattice. This approximation is valid in the case with OAM-zero photo-excitations, e.g. Pierce and Meier (1976), and for twisted light. Hence, we approximate the transition amplitude by the corresponding atomic-like one, see Afanasev et al. (2018a,b):

r h (tw) κ ¯ 2mγ −m−η− j η+1 j MvcΛ = − ∑(−i) Λ dΛm(θq)Jmγ −m(κr) 3π m0 jm s 2 j + 1 jcmc × C h jc||T|| jvi, (6.13) ∑ 2 j + 1 jvmv jm mcmvi f where h jc||T|| jvi is the reduced matrix element for the OAM-zero atomic-like p → s transition. Here r is the position radius-vector of a particular electron with respect to the beam’s axis. The corresponding allowed transitions for the non-zero OAM are shown in

Figure 6.2 in dashed lines (∆m = 0,−1), see Solyanik-Gorgone and Afanasev (2019).

TW Since the P → S transition is of the E1 type, the transition rates ∑mcmv |Mvc | are proportional to the photon beam intensity proﬁles. This can be seen from the plots of the transition amplitude as a function of the impact parameter b normalized to the photon wavelength λ in Figure 6.3. We show the amplitudes for several values of the TAM

85 100 100 (a) (c) 50 50

0 0 P (%) -50 -50

-100 -100 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2

100 100 (b) (d) 50 50

0 0 P (%) -50 -50

-100 -100 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 b/λ b/λ

Figure 6.4: Photoelectron polarization as a function of the electron’s distance to the optical vortex center for the vortex pitch angle θq = 0.1 rad; long-dashed blue is mγ = −2, dot-dashed black is mγ = −1, dashed red is mγ = 0, solid green is mγ = 1, dotted purple is mγ = 2. GaAs sample and incoming photon polarization, are, respectively (a) – unstrained and left-circular, (b) – unstrained and right-circular, (c) – strained and left-circular polarization and (d) – strained and right-circular.

quantum number mγ = 0,±1,±2 for LC (a) and RC (b) polarized radiation. One can also

see, that only solid green mγ = 1 for LC and dot-dashed black mγ = −1 for RC survive in

full agreement with the modiﬁed selection rules ∆m = mγ . An interesting dependence on the impact parameters is predicted for photo-excitation spin polarization content in the conduction band, which can be expressed as

(|M(tw) |2 − |M(tw) |2) ∑mv v;1/2 v;−1/2 P = Λ Λ . (6.14) (|M(tw) |2 + |M(tw) |2) ∑mv v;1/2Λ v;−1/2Λ

The corresponding plots can be found in Figure 6.4, where photoelectron polarization is plotted as a function of the electron’s distance to the optical vortex center for the vortex

pitch angle θq = 0.1 rad. Here one sees the case of an unstrained GaAs in the left column

86 100 100 (a) (c) 50 50

0 0 P (%) -50 -50

-100 -100 0 200 400 600 800 1000 0 200 400 600 800 1000

100 100 (b) (d) 50 50

0 0 P (%) -50 -50

-100 -100 0 200 400 600 800 1000 0 200 400 600 800 1000 r(nm) r(nm)

Figure 6.5: Photoelectron polarization averaged over the circular mesoscopic semiconducting target of radius r centered at the beam axis with the pitch angle θq = 0.1 rad; long-dashed blue is mγ = −2, dot-dashed black is mγ = −1, dashed red is mγ = 0, solid green is mγ = 1, dotted purple is mγ = 2. The GaAs sample and incoming photon polarization, are, respectively (a) – unstrained and left-circular, (b) – unstrained and right-circular, (c) – strained and left-circular polarization and (d) – strained and right-circular. Shadowed vertical regions indicate 10x lattice periods for GaAs (r=5.65 nm).

and strained in the right (c. f . Ref. Pierce and Meier (1976) and Ref. Maruyama et al. (1991), respectively). In Table 6.1 we show the calculations for photoelectron polarization eqn. (6.14) for

small pitch angles θq as a function of the photo-excited electron’s position in the photon beam proﬁle. The matrix element in eqn. (6.13) has been used in calculations. We would like to point out several important observations that can be made both from Figure 6.4 and from Table 6.1: (1) the polarization maxima are localized in low intensity region of the

twisted beam; (2) when OAMΛ the polarization remains unchanged, while for OAM ↑↓Λ it is ﬂipped near the beam axis; (3) the result converges to the one for a zero-OAM

beam incident at a pitch angle θq to the quantization axis when averaging the electron

87 polarization over the entire transverse beam proﬁle (hPi = 0.5cosθq for unstrained and twice that value for strained). The second conclusion was repeatedly pointed out in the ﬁeld in various situations: in interactions with low-dimensional objects by Quinteiro and Kuhn (2014), Quinteiro et al. (2017c) and Zambrana-Puyalto et al. (2016). The last conclusion in the previous paragraph directly points towards the necessity for subwavelength position resolution for experimental observations of the predicted effects. One may use a mesoscopic target – subwavelength sample of GaAs. We make predictions for the average degree of polarization for such a sample, centered at the beam’s axis. In Figure 6.5 we show the photoelectron polarization averaged over the circular mesoscopic semiconducting target of radius r

centered at the beam axis with the pitch angle θq = 0.1 rad. One can see that the sample sizes of ∼200 nm potentially allow to clearly observe the polarization effects due to non- zero photon OAM. Shaded areas correspond to the limit when dimensional conﬁnement effects become strong, and the bulk GaAs approach is no longer valid. Calculations for spin-relaxation effects are necessary to describe the electron kinetics after the photo-absorption event. We have shown here the results for polarization of

photo-excited in the Γ - point and spatially localized electrons in the conduction band right after the transition had happened. It is important to note that the band structure and symmetries in crystals are a subject of an ongoing research. For instance, Elder et al. (2011)provide arguments towards a more exact model of the GaAs band structure, but the resulting observable effects might be too weak to be registered with conventional Gaussian-beam spectroscopy. However, utilizing the ability of OAM-photon states to enhance higher-order multipoles, one may be able to extract the necessary information using twisted beam interactions with GaAs.

88 6.2 Twisted Photon Interaction with Semiconductor Quantum Dots

The invention of the semiconductor superlattices and quantum wells in the 1970’s was the beginning of a new era of semiconductor heterostructures. The fabrication of low- dimensional semiconductor systems became possible due to the advent of epitaxial growth techniques. It has been noted that introducing confinement into a semiconductor structure causes dramatic changes in their optical, thermodynamical, mechanical and other properties, e.g. Harrison (2016). Nowadays, quantum wires and quantum dots are already integrated in photonic devices, optical and electron chip circuits. A quantum dot, also called “artificial atom” or “mesoscopic atom”, can be defined as an sequence of junctions in a semiconductor structure which can confine electrons/holes in a three-dimensional well. Consequently, the energy bands in quantum dots become energy levels. A schematic picture of a typical quantum dot and its band structure are presented in Figure 6.6, where a AlGaAs/GaAs nanostructure was specified only as an example. There are no hard limitations on the size of a quantum dot. Though quantum dots have a lot in common with atoms, they do exhibit unique properties. In modern labs, a typical size of a quantum dot can go down to 1-2 nm in diameter. Due to such tight spatial confinement, its thermal activation energy becomes large enough to enable fabrication of single photon sources, which are in high demand in quantum computing and communications, e.g. Shields (2010). For more information on history, fabrication and applications of quantum dots, we refer the reader to Lakhtakia (2004) and Ihn (2010). OAM photo-excitations in semiconductor quantum dots have previously been extensively studied in the series of works by Quinteiro and Tamborenea (2009a), Quinteiro et al. (2010) and Quinteiro and Kuhn (2014) and others. These authors explore various systems, mostly focusing on two-band semiclassical models of photo-excitation in strained samples. The dependence on the impact parameter is considered by Quinteiro et al. (2010). The

89 Figure 6.6: Schematic picture of a AlGaAs/GaAs quantum dot (left) and its energy band structure forming the 3D quantum conﬁnement (right).

main distinctive assumption in many of the considered models was that the twisted photon’s vector potential is a slowly-varying function. In this work we avoid this assumption

since we are focusing on sub-wavelength size quantum dots d w0 , located close to the beam’s singularity, where field gradients are very strong. We use the single-particle QM formalism for OAM photo-excitations in semiconductor quantum dots, neglecting the effects coming from confinement and Coulomb interaction, quasi-local processes and a band structure morphology. In the process considered here, a single electron absorbs a photon and gets excited into the conduction band leaving behind a hole, similar to the semiclassical approach developed by Quinteiro and Tamborenea (2009a). The formation of such electron-hole pairs define the spectroscopic selection rules in quantum dots, which result in discrete lines in absorption and emission spectra, resembling atomic structures. The Hamiltonian of an electron in the quantum dot can be written as p2 H = +U(r), (6.15) 2m where e and m are the free electron charge and mass (the same is true for confined holes); p is the electron momentum and A is the photon vector potential; U(r) is the lattice potential of the heterostructure containing the quantum dot. The leading-order term responsible for the interaction with external EM-field can be written as

e H = A · p. (6.16) int m

90 The electron wave function written in terms of lattice-periodic Bloch-functions uc/v(r),

and the corresponding envelope contributions ψe/h(r) is

(r,t) = 1 {u (r) (r)bˆ (t) + u (r) ∗(r)cˆ† (t)}, (6.17) Ψn jm 3/2 c ψe pe,αe v ψh −ph,αh R0 where R0 is the radius of the quantum dot. Indices c and v stand for the conduction and valence bands (levels), and e and h for electron and hole charge carriers. The operators bˆ (t) and cˆ† (t) are the electron annihilation and hole creation operators, pe,αe −ph,αh respectively. The position vector r deﬁnes the spatial location of the charge carrier. The set {n, j,m} are the principal, azimuthal and magnetic quantum numbers, similar to the QM model of an atom. The envelope functions are local to the quantum dot, while Bloch functions describe the charge carrier behavior in the entire nanostructure. Using the interaction, eqn. (6.16), and the electronic states, eqn. (6.17), we proceed to calculating the matrix element in a conventional way

1 Z M = u∗(r) ∗ (r)(A · p)u (r) (r)drr (6.18) cvΛ 3 c ψne jeme v ψnh jhmh R0

substituting the vector potential of choice. The corresponding transition rates can be calculated as je 2 ΓcvΛ = 2πδ(Ee − Eh − ωγ ) ∑ |McvΛ(r)| . (6.19) me=− je For twisted beams, we work with the BB laser mode, eqs (2.32) and (2.30), to obtain

r (tw) κ je jh pw |M (b)| = Jm −m +m (κb) d 0 (θk)d 0 (θk)M (0) . cvΛ γ e h ∑ meme mhm cvΛ (6.20) 2π 0 0 h memh

Due to the factorization property, this matrix element is similar in its form to the one obtained by Afanasev et al. (2016) and Scholz-Marggraf et al. (2014). The plane-wave

91 2.0 (a) Δm=-2 Δm=-1 1.5 Δm=0 Δm=1

( a.u. ) 1.0 Δm=2 cv

M Δm=3 0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 b/λ

1.2 (b) Δm=-2 1.0 Δm=-1 Δm=0 0.8 Δm=1

( a.u. ) 0.6 Δm=2 cv

M 0.4 Δm=3

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 b/λ

1.0 (c) Δm=-2 0.8 Δm=-1 Δm=0 0.6 Δm=1

( a.u. ) Δm=2

cv 0.4

M Δm=3 0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 b/λ

Figure 6.7: Transition rates for d → s type E2 transition for single-fermion problem accounting for spin coupling. The parameters used are Λ = 1, mγ = 1,2,3 for (a), (b) and (c) respectively and λ = 550 nm.

92 matrix element can be approximated as:

Z Z (PW) 0 ∗ 00 iqqqrr0 00 ∗ McvΛ (0) ∝ drr uc(r )e (eΛ · p)uv(r ) dRR ψe (R)ψh(R), (6.21) where the first integral contains the cell-periodic carrier wavefunctions that rapidly vary on the quantum dot length scale. Since we consider an isolated quantum dot, this integral is analogous to the one in eqn. (3.13). The second integral will not contribute to the magnetic selection rules in the case of strong confinement since it does not depend on the internal degrees of freedom of the quantum dot. Hence, we effectively treat it as unity. The set of selection rules defined by this matrix element:

∆ j = 0,±1,±2, ∆m = 0,±1,±2. (6.22)

In the case of a plane-wave photon coming in, one gets ∆m = Λ. However, we see that

the topology-dependent term Jmγ −me+mh (κb) modiﬁes the conventional set in the vicinity

of the optical beam center, as before, leading to ∆m =mγ . This enhances high-order multipolar transitions in the same way as it has been shown for the ions in Chapter 3. A similar selection rule was obtained by Quinteiro et al. (2010) for the disk-shaped quantum dots and rings, and later for magnetic transitions of light holes, see Quinteiro et al. (2017b). The absorption rates for the quadrupole transitions in quantum dots photo-exited by LG, BG and BB modes without spin interaction taken into account were calculated in Solyanik and Afanasev (2017) earlier. In Figure 6.7 the absorption rates for d → s type transition eqn. (6.20) are shown for the LC-polarized OAM-photon. The plane-wave contribution

(PW) 1/2,m f to the photo-absorption matrix element in this case is Mcv ∝ C E2. We have 5/2,mi jm

considered several cases: mγ = 1,2, and 3. One can clearly see how ∆m =Λ selection rule

dominates on periphery in all three cases. However, the topology-imposed rule ∆m =mγ wins over in the penumbra. This effect is possible to observe in high precision experiments, analogous to the one described by Schmiegelow et al. (2016) for trapped and cooled ions.

93 Chapter 7: Conclusions

The results of the theoretical study of angular momentum selection rules in atomic, nuclear and semiconductor structures upon the absorption of quantum vortex beams of photons and neutrons were demonstrated. Electron and nuclear spin coupling to the TAM of the BB mode have been taken into account within the framework of QM. ASR for the BB-photon quantization in combination with the multipolar OAM-mode expansion in vector spherical harmonics were used together for the first time. The explicit photon-to-electron TAM coupling coefficients were obtained. We would like to stress that, independently, the application of ASR as a tool to quantize a twisted photon and multipolar expansion of photon modes are well-known techniques in photonics and QED. Relying on the analysis of the experimental data for a 40Ca ion, the predictions for photo-excitation in cooled and trapped 40Ar, 171Yb, 172Yb and 20Ne have been made using the previously derived formalism. The developed theory accounts for arbitrary constant quantizing magnetic field alignment, probe multipolar content, photon TAM and SAM, fine and hyperfine coupling. The ASR expansion of LG and BG beam modes with subsequent application for photon-matter interactions was performed. The results suggest that TAM gets transferred to atoms, nuclei, and solid state from paraxial and non-paraxial twisted photon beams, despite doubts coming from the OAM and SAM nature for the former modes. However, BG and LG describe the coupling better on the photon beam’s periphery due to the suppression by the Gaussian envelope function, while BB performs best at the penumbra area. Relying on theoretical analysis and experimental results, this study confirms that the dark region at the penumbra of the vortex beam carries EM-fields with non-trivial topology, characterized by strong gradients and complicated multipolar content. Particularly, these properties of the OAM-carrying fields lead to dramatic enhancement of higher-order mul-

94 tipolar contributions in atomic, nuclear and semiconductor optical response. In transitions with mixed multipolarity, the balance of partial rates in relative contributions depends on the probe’s location in the beam profile, both in radial and azimuthal dimensions. This may allow for the separation of the transitions of electric and magnetic type for the purpose of fundamental studies of the multipolar content in a high-accuracy regime. These phenomena are predicted to be the most apparent in transitions different in multipolarity but compatible in rates. Photon-OAM-driven (magnetic) circular dichroism is explained on the level of factor- ized multipolar contributions. This allows one to directly relate the symmetry degradation to the strength and character of the dichroic effects. Predictions are made for the case of OAM-carrying modes propagating in an isotropic hydrogen gas. It has been independently verified that there exist two families of beams: OAM and SAM aligned and anti-aligned. A theoretical approach for describing the photo-nuclear reactions with OAM-carrying probes is presented, based on QM time-dependent perturbation theory. The corresponding modified set of magnetic selection rules is derived. Rates and cross sections appear to be very sensitive to the location of the target in the beam profile, which can be used as a precise and powerful diagnostic tool. Our model for OAM photo-exited electron spin is the first and so far the only available theoretical description. The calculations have been performed for vertical transitions into

the valence band of zincblende GaAs in the Γ-point. For the ﬁrst time, the Luttinger approach has been used to express coupling of the twisted photon TAM to the valence electron state, providing the foundation for further study of the electron kinetics in the

conduction band. The selection rules for photo-excitation in the Γ-point have been derived. The predictions for twisted photo-electron polarization experiments with a beam-centered mesoscopic target have been made, and the discussion of a possible experimental setup is provided. Due to the non-trivial and spatially- and temporally-stable topology, twisted beams

95 have shown themselves to be a powerful, versatile and precise tool on the sub-nanometer level. They can be used across a wide range of studies, including atomic and nuclear spectroscopy, metrology, polarimetry, analysis of morphology and multipolar content of biological samples and metamaterials, etc. On the industry and engineering side, there are foreseeable applications in quantum computing, storage and communication in D-level systems, quantum encryption and high bandwidth data transfer protocols. The theory can be expanded to use for systems of multiple ions or artiﬁcial atoms, which would pave the road for the study of quantum entanglement with OAM-carrying modes for the purpose of quantum computing and communications.

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107 Appendix A: List of Useful Formulas

Properties of a delta-function

∞ ` ∗ δ(cosθ1 − cosθ2)δ(φ1 − φ2) = ∑ ∑ Y`,m(θ1,φ1)Y`,m(θ2,φ2), (A.1) `=0 m=−` 1 Z 0 δ (d)(r −r0) = dkk eikk·(r−r ), (A.2) (2π)d 2 1 − x δ(x) = lim √ e 2σ2 , (A.3) σ→0 σ 2π Z ∞ π 0 2 0 2 δ(k − k ) = dr r j`(kr) j`(k r). (A.4) 2k 0

Integral representations of Khroneker symbol

Z 1 n−m−1 δnm = z dz, (A.5) 2πi |z|==1 Z 2π 1 i(n−m)t δnm = e dt. (A.6) 2π 0

Spherical Bessel-function

sin(ρ − `π/2) lim j`(ρ) = , (A.7) ρ→∞ ρ 1 d n sinx j (x) = (−x)n . (A.8) n x dx x

Bessel-function

Z 2π m i(mφ±xcosφ) dφ Jm(x) = (∓i) e , (A.9) 0 2π n J−n(x) = (−1) Jn(x). (A.10)

108 Assymptotic forms Abramowitz and Stegun (1964): for ν → ∞ through real positive values, while other parameters are ﬁxed

1 ez ν Jν (z) ∼ √ , (A.11) 2πν 2ν ρ −` 1 lim J`(ρ) = . (A.12) ρ→0 2 Γ(` + 1)

A plane wave is expanded in the basis of Bessel functions, Erdélyi et al. (1953):

∞ ∞ izcosφ n inφ izsinφ inφ e = ∑ i e Jn(z); e = ∑ Jn(z)e , n=−∞ n=−∞ ∞ ∞ (A.13) −izcosφ n inφ −izsinφ n inφ e = ∑ (−i) e Jn(z); e = ∑ (−1) Jn(z)e . n=−∞ n=−∞

A “remarkable" identity, discovered by Dominici et al. (2012):

∞ ∞ Z Jν (at)Jν (bt) Jν (an)Jν (bn) dt = ∑ εn , (A.14) 0 t n=0 n where εn = 1 for n ≥ 1 and zero otherwise. Spherical harmonics, see Arfken and Weber (2005):

s 2n + 1 (n − m)! m imφ Ynm(θ,φ) = P (cosθ)e , (A.15) 4π (n + m)! n

(0) Y nm(θ,φ) = −(k × ∇)Ynm(θ,φ), (A.16)

(1) Y nm(θ,φ) = ∇Ynm(θ,φ), (A.17)

such that Z (λ) (λ 0) 0 Y nm (Ω)Y nm (Ω )dΩ = δnn0 δmm0 δλλ 0 , (A.18)

109 (λ) n+1−λ (λ) and changes under parity as Pˆ{Y nm (Ω)} = (−1) Y nm (Ω). Normalization of Laguerre polynomials:

Z ∞ −x k k k (n + k)! e x Ln(x)Lm(x)dx = δmn. (A.19) 0 n!

Wigner-D matrices and their symmetries:

j −imφ j −im0ψ Dmm0 (φ,θ,ψ) = e dmm0 (θ)e , (A.20) j j±m dmm0 (±π) = (−1) δm0,−m, (A.21) j j dm0m(−β) = dmm0 (β), (A.22) j m0−m j ∗ Dm0m(α,β,γ) = (−1) D−m0,−m(α,β,γ) . (A.23)

The Racah formula for the Clebsch-Gordan coupling coefﬁcients:

  j j j  1 2    m1 m2 −m

j1− j2+mp p = (−1) ∆( j1, j2, j) ( j1 + m1)!( j1 − m1)!( j2 + m2)!( j2 − m2)!( j + m)!( j − m)! (−1)` × ∑ , ` `!( j − j2 + ` + m1)!( j − j1 + ` − m2)!( j1 + j2 − j −t)!( j1 − ` − m1)!( j2 − ` + m2)! (A.24) where (α + β − γ)!(β + γ − α)!(γ + α − β)! ∆(αβγ) = . (α + β + γ + 1)!

110 Appendix B: Spherical Wave Decomposition of Twisted Modes

A great amount of problems in physics can be approximated as spherically symmetric. That is why expansion in terms of spherical harmonics, eqn. (A.15), established itself as one of the fundamental tools of ultimate importance. This argument makes it attractive to work in terms of vector spherical harmonics, when tackling the problems with OAM beams interacting with TAM-conserving systems. Such attempts were made previously by e.g. Mitri (2014), Chen et al. (2010). Here we offer a simple and straightforward method to implement this expansion for the case of the Bessel mode. We start with the expression for the scalar Bessel mode in eqn. (2.25). One can prove, see e.g. Abramowitz and Stegun (1964), that:

r s κ 4π (n − mγ )! m κρ uBB( ,t) = e−ikzzeiωt (−n)mγ Y γ , . ¯mγ ρ lim n φ (B.1) 2π n→∞ 2n + 1 (n + mγ )! n + 1/2

One can see that for the purpose of numerical calculations, this expression converges well starting from n ∼ 103, see Figure B.1, where the convergence of the eqn. (B.1) to the Bessel mode eqn. (2.25) is investigated as a function of n. We apply the deﬁnitions for vector spherical harmonics:

r BB (0) BB κ −ikzz iωt uˆm (ρ,t) = − (k × ∇)u (ρ,t) = − e e γ mγ 2π s 4 (n − m )! mγ π γ (0) κρ × lim (−n) Y nmγ ,φ , n→∞ 2n + 1 (n + mγ )! n + 1/2 r (B.2) BB (1) BB κ −ikzz iωt uˆm (ρ,t) = ∇u (ρ,t) = e e γ mγ 2π s 4 (n − m )! mγ π γ (1) κρ × lim (−n) Y nmγ ,φ . n→∞ 2n + 1 (n + mγ )! n + 1/2

111 0.0 -0.2 > γ -0.4 BB m mγ=0 - u -0.6 γ BB m -0.8 mγ=1 < u -1.0 mγ=2 -1.2 mγ=3

0 200 400 600 800 1000 n huBB( ,t) − uBB( ,t)i n Figure B.1: Average value ¯mγ ρ mγ ρ as a function of the parameter to compare the BB mode eqn. (2.25) and its expression in terms of spherical harmonics eqn. (B.1) for different values of n and mγ .

This allows to construct the magnetic (µ = 0) and electric (µ = 1) ﬁeld components:

Z BB (µ) ikk·r E ∝ E0 uˆmγ (ρ,t)e dΩk. (B.3)

Here, n is not a TAM, but a mathematical parameter. For the details on further steps towards the multipole expansion in QED, the reader is suggested to check Akhiezer and Berestetskii (1959).

112