Submitted by Fernando Augusto Dessotti null (fernandodessoti@hotmail.com) on 2018-01-30T17:37:28Z No. of bitstreams: 1 Tese oficial.pdf: 12552592 bytes, checksum: e44289b205ab39ad49d51a2f976df63c (MD5) / Approved for entry into archive by Cristina Alexandra de Godoy null (cristina@adm.feis.unesp.br) on 2018-01-30T18:39:52Z (GMT) No. of bitstreams: 1 dessotti_fa_dr_ilha.pdf: 12552592 bytes, checksum: e44289b205ab39ad49d51a2f976df63c (MD5) / Made available in DSpace on 2018-01-30T18:39:52Z (GMT). No. of bitstreams: 1 dessotti_fa_dr_ilha.pdf: 12552592 bytes, checksum: e44289b205ab39ad49d51a2f976df63c (MD5) Previous issue date: 2017-12-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O físico italiano Ettore Majorana propôs, no campo da Física de altas energias, a existência de férmions peculiares que têm como característica serem suas próprias antipartículas. No contexto de Física da matéria condensada, tais férmions emergem como quasipartículas de Majorana (MQPs). Da perspectiva da compu- tação quântica, duas MQPs podem compor um férmion regular e atuar como um qubit protegido, que está desacoplado do ambiente e livre do efeito de decoerência. Até onde sabemos, a verificação experimental de uma MQP ainda é questionável, apesar de alguns resultados experimentais, e desta forma, o objetivo desta tese é de propor formas experimentais a fim de ajudar na busca das assinaturas de tais excitações. Como o efeito Fano é um efeito de interferência na qual canais de tunelamento competem entre si pelo transporte eletrônico, ele torna-se uma forma de capturar tais assinaturas das MQPs em sistemas de matéria condensada. Baseado nisto, a ideia é investigar teoricamente três diferentes interferômetros a fim de obter uma assinatura definitiva das MQPs. O primeiro é um interferômetro do tipo Aharanov-Bohm composto por dois quantum dots, sendo um deles acoplado a uma MQP, que se localiza na borda de um fio de Kitaev semi-infinito na fase topológica. Ajustando o nível de Fermi dos terminais e o detuning simétrico dos níveis dos dots, mostrou-se que regimes Fano opostos resultam em uma transmitância caracterizada por distintas regiões condutoras e isolantes, que são marcas de uma MQP isolada. O dispositivo proposto aqui constitui uma alternativa experimental para detectar as MQPs. O segundo interferômetro é composto por pontas de STM e AFM próximas a um dímero de Kitaev de átomos adsorvidos supercondutores, na qual o átomo adsorvido localizado abaixo da ponta de AFM, encerra um par de MQPs. Para uma energia de ligação ∆ do par de Cooper delocalizado nos átomos adsorvidos abaixo das pontas coincidente com a amplitude de tunelamento t entre eles, ou seja, ∆ = t, mostrou-se que somente uma MQP abaixo da ponta de AFM hibridiza com o átomo adsorvido abaixo das pontas de STM, e para esta situação, o padrão de Fano permanece como universal. Mas para o caso das duas MQPs conectadas ao átomo adsorvido abaixo das pontas de STM, foi verificado que tal característica universal foi quebrada. O terceiro e último interferômetro é composto por dois quantum dots assimetricamente acoplados a MQPs isoladas que se localizam em duas cadeias de Kitaev na fase topológica. Este dispositivo habilita a medição da MQP em uma forma distinta do pico zero-bias. Mais importante, o sistema se comporta como um seletor de correntes composto por dois caminhos distintos: (i) para o dot superior conectado a ambas as cadeias, o dispositivo percebe ambas as MQPs como um férmion regular e a corrente atravessa somente o dot inferior, pois a corrente no dot superior é impedida devido a presença de um gap supercondutor; e (ii) pela leve supressão da hibridização do dot superior com a cadeia, a corrente é abruptamente trocada para fluir através deste mesmo dot, uma vez que um elétron é armadilhado como um estado ligado ao contínuo (BIC) surge no dot inferior. Tal seletor de corrente entre os dots inferior e superior caracteriza uma transição de fase quântica, que possibilita não somente a revelação de MQPs, mas também produz um seletor de corrente assistido por elas. / The Italian physicist Ettore Majorana proposed in the field of high-energy Physics the existence of peculiar fermions that constitute their own antiparticles. In the context of condensed matter Physics, these fermions are Majorana quasiparticles (MQPs). From the quantum computing perspective, two MQPs can compose a regular fermion acting as a protected qubit, which is indeed decoupled from the host environment and free of the decoherence effect. To the best of our knowledge, the experimental capture of a MQP up to now is still questionable despite some experimental results, then, the goal of this thesis is to propose helpful experiment manners in revealing signatures from such excitations. As the Fano effect is an interference phenomenon where tunneling paths compete for the electronic transport, it becomes a probe to catch fingerprints of MQPs lying on condensed matter systems. Based on this, the idea is to investigate theoretically three different interferometers in order to obtain a MQP smoking- gun signature. The first one was an Aharonov-Bohm-like interferometer composed by two quantum dots, being one of them coupled to a MQP, which is attached to one of the edges of a semi-infinite Kitaev wire within the topological phase. By changing the Fermi energy of the leads and the symmetric detuning of the levels for the dots, we show that opposing Fano regimes result in a transmittance characterized by distinct conducting and insulating regions, which are fingerprints of an isolated MQP. The setup proposed here constitutes an alternative experimental tool to detect MQPs. The second one is composed by STM and AFM tips close to a Kitaev dimer of superconducting adatoms, in which the adatom placed under the AFM tip, encloses a pair of MQPs. For the binding energy ∆ of the Cooper pair delocalized into the adatoms under the tips coincident with the tunneling amplitude t between them, namely ∆ = t, we find that only one MQP beneath the AFM tip hybridizes with the adatom coupled to the STM tips, and for this situation, the Fano pattern is still universal. But for the case of two MQPs connected to the adatom beneath the STM tips, we verify that such a universality is broken. The third and last one is composed by two quantum dots asymmetrically coupled to isolated MQPs, lying on the edges of two topological Kitaev chains. This setup enables us to probe MQPs in a quite distinct way from the zero-bias peak feature. Most importantly, the system behaves as a current switch made by two distinct paths: (i) for the upper dot connected to both chains, the device perceives both MQPs as an ordinary fermion and the current crosses solely the lower dot, since current in the upper dot is prevented due to the presence of the superconducting gap; and (ii) by suppressing slightly the hybridization of the upper dot with one chain, the current is abruptly switched to flow through this dot, once a trapped electron as a bound state in the continuum (BIC) appears in the lower dot. Such a current switch between upper and lower dots characterizes a quantum phase transition, which enables not only the fundamental revealing of MQPs, but also yields a current switch assisted by them.

Neste trabalho estudamos uma rede hexagonal com uma cadeia de impurezas nas bordas e com supercondutividade induzida, de forma a mostrar a existência de fases com férmions de Majorana. Para tal, começamos introduzindo invariantes topológicos, número de Chern e Z2 e mostramos dois modelos para rede hexagonal. O primeiro, modelo de Haldane, fazemos como motivação histórica. O segundo, modelo de Kane-Mele, é usado como base para todo o trabalho. Seguimos introduzindo supercondutividade e como ela ocorre quando aplicada junto do Modelo de Kane-Mele, o método auto-consistente e quais as condições necessárias para termos supercondutividade apenas nas bordas. Continuamos com efeitos de impurezas magnéticas nas bordas e introduzimos férmions de Majorana que são os alvos principais dos resultados. Mostramos então, que existe fases topológicas em cadeias de impureza magnética, com momentos em espiral, contudo o diagrama de fase depende de várias condições. Por fim, mostramos que a variação da fase topológica se deve a oscilações nos níveis de energia em que o invariante topológico também varia, contrariando resultados obtidos para a rede quadrada. Concluímos esse trabalho com implicações experimentais desse resultado e possíveis caminhos que podem ser seguidos. / In this work, we study a honeycomb lattice with induced superconductivity and edge impurity in order to show the existence of a phase that host Majorana bound state. To do so, we start introducing topological invariants, Chern number and Z2, and we show two models for honeycomb lattice. The first, Haldane\'s Model, due its historical importance. The second, Kane-Mele model, because it will be used during all this work. Then we review superconductivity, showing the self-consistent method, and we apply it to Kane-Mele model, in which we find some necessary conditions to induce superconductivity only at the edges. From this point, we study the effect of magnetic impurities at the edges, and we introduce Majorana bound state, that will be the main objective of our results. In our results, we show the existence of topological non-trivial phases for spiral magnetic chain in the zigzag edge. With this we make a phase diagram. We also find oscillation in the energy spectrum and the topological phase changes with the oscillation, this is different from square lattice in which we should not have a change in the topological phase. We conclude this work with experimental implications of our result and possible developments.

Neste trabalho investigamos a fusão entre estados ligados de Majorana em nanoestruturas compostas por um ponto quântico conectado a contatos metálicos e acoplado lateralmente a dois fios quânticos supercondutores que sustentam modos de Majorana em suas pontas. Modelando cada fio quântico por uma cadeia de Kitaev, nós adotamos duas abordagens: inicialmente usando as funções de Green do ponto obtidas através do método recursivo calculamos a condutância e a densidade local de estados (LDOS), posteriormente diagonalizamos o sistema no formalismo de Bogoliubov-de Gennes (BdG) e obtemos o espectro completo dos autoestados. Como descrito em (1), o LDOS do ponto quântico acoplado a uma única cadeia de Kitaev mostra claramente o vazamento do modo de Majorana inicialmente presente na ponta da cadeia para o ponto quântico, onde este modo surge fixo na energia de Fermi dos contatos metálicos (εƒ). A condutância de dois terminais medida através do ponto mostra uma assinatura dos estados de Majorana neste sistema, uma ressonância fixa mesmo quando o nível do ponto está vazio ou não. Interessante ressaltar que mesmo na presença de interações no ponto essa assinatura de Majorana é válida como mostrado em (2). Motivados por estes resultados anteriores estamos particularmente interessados em investigar a hibridização (aqui denominada de fusão) entre dois modos de Majorana resultando em um modo fermiônicos ordinário dentro do ponto quântico. Nossos resultados demonstram que controlando a diferença de fase supercondutora entre os fios e a voltagem de gate do ponto quântico somos capazes de controlar a emergência e fusão dos modos de Majorana. Além disso nós reforçamos a proposta de se utilizar o efeito Josephson a.c. de período 4π para identificar os modos de Majorana pela reprodução dos resultados obtidos por (3). / In this work we investigate the fusion between Majorana bound states in nanostructures composed of a quantum dot connected to source and drain leads and side coupled to two topological superconducting nanowires sustaining Majorana end modes. Modeling the nanowire via a Kitaev chain, we have used two approaches: first using a recursive Greensfunction approach we calculate the conductance and local density of states (LDOS) and then by the diagonalization using the Bogoliubov-de Gennes (BdG) formalism we obtain the full spectrum of eigenstates. As described in (1) the LDOS of quantum dot coupled to a single wire clearly shows a leakage of the Majorana end mode from the wire into the dot, where it emerges as a unique dot level pinned to the Fermi energy of the leads (εƒ). The calculated two-terminal conductance through the dot displays an unambiguous signature of the Majorana bound states, i. e., a pinned resonance occurring even when the dot level is far above εƒ . Interestingly this Majorana signature remains even in the presence of interactions within the dot as showed in (2). Motivated by these earlier results we are particularly interested to investigate the fusion of Majonana end modes into ordinary fermionic modes within the dot. Our results demonstrate that by tuning the superconducting phase difference between the wires and the quantum-dot gate voltage we are able to control the emergence and splitting of Majorana modes. Furthermore we reinforce the proposal of using the 4π periodic a.c Josephson effect to identify Majorana modes by reproducing the results obtained by (3).

Dans cette thèse, nous étudions d’un point de vue théorique les propriétés physiques de nanostructures couplées à des cavités micro-ondes. L’électrodynamique quantique (QED) en cavité en présence d’une boîte quantique s’est révélée être une technique expérimentale puissante, permettant d'étudier cette dernière par des mesures photoniques en plus des mesures de transport électronique conventionnelles. Dans cette thèse, nous proposons d'utiliser le champ micro-ondes de la cavité afin d’extraire des informations supplémentaires sur les propriétés des conducteurs quantiques : le coefficient de transmission optique est directement lié à la susceptibilité électronique de ces conducteurs quantiques. Nous appliquons ce cadre général à différents systèmes mésoscopiques couplés à une cavité supraconductrice micro-ondes comme  une jonction tunnel, une boîte quantique couplée à des réservoirs, un fil topologique et un anneau supraconducteur. La QED en cavité peut être utilisée pour sonder, par l'intermédiaire de mesures photoniques, la dépendance en fréquence de l’admittance du puits quantique couplé à la cavité micro-ondes. En ce qui concerne le fil topologique, nous avons montré que la cavité permet de caractériser la transition de phase topologique, l'émergence de fermions de Majorana, ainsi que la parité de l'état fondamental. Pour l'anneau supraconducteur, nous étudions par l'intermédiaire de la réponse optique de la cavité l’effet Josephson et le passage à l'effet Josephson fractionnaire, qui est associé à l'apparition de fermions de Majorana dans le système. Le cadre théorique proposé dans cette permet de sonder de manière non-invasive un large éventail de nanostructures, des boîtes quantiques aux supraconducteurs topologiques. En outre, il donne de nouvelles informations sur les propriétés de ces conducteurs quantiques, informations non accessibles via des expériences de transport. / In this thesis, we study theoretically various physical properties of nanostructures that are coupledto microwave cavities. Cavity quantum electrodynamics (QED) with a quantum dot has been proven to be a powerful experimental technique that allows to study the latter by photonic measurements in addition to electronic transport measurements. In this thesis, we propose to use the cavity microwave field to extract additional information on the properties of quantum conductors: optical transmission coefficient gives direct access to electronic susceptibilities of these quantum conductors. We apply this general framework to different mesoscopic systems coupled to a superconducting microwave cavity, such as a tunnel junction, a quantum dot coupled to the leads, a topological wire and a superconducting ring. Cavity QED can be used to probe the finite frequency admittance of the quantum dot coupled to the microwave cavity via photonic measurements. Concerning the topological wire, we found that the cavity allows for determining the topological phase transition, the emergence of Majorana fermions, and also the parity of the ground state. For the superconducting ring, we propose to study the Josephson effect and the transition from the latter to the fractional Josephson effect, which is associated with the emergence of the Majorana fermions in the system, via the optical response of the cavity. The proposed framework allows to probe a broad range of nanostructures, including quantum dots and topological superconductors, in a non-invasive manner. Furthermore, it gives new information on the properties of these quantum conductors, which was not available in transport experiments.

Le sujet de cette thèse est sur les propriétés non-locales des états symétriques invariant sous les permutations des systèmes et les usages potentiels de ces états dans le domaine de traitement d'information quantique. La non-localité de presque tous les états sysmétriques, hors les états de Dicke, est établie par une version étendue du paradoxe de Hardy. Grâce à la représentation de Majorana pour les états symétriques, des paramètres de mesure avec lesquels toutes les conditions du paradoxe sont satisfaites peuvent être trouvés. Une version étendue de l'inégalité de CH peut être dérivée à partir des conditions probabilistes de ce paradoxe. Cette inégalité est violée par tous les états symétriques. Les propriétés de la non-localité et les propriétés de l'intrication sont aussi discutées et comparées, notamment par rapport à la persistance et la monogamie. Des résultats indiquent que la dégénérescences de certains états symétriques est liée à la persistance, qui donne une façon d'inventer des tests qui sont indépendants des dispositifs visés pour séparer les différentes classes de non-localité. Il est aussi montré que l'inégalité utilisée pour démontrer la non-localité de tous les états symétriques n'est pas monogame dans le sens strict. Néanmoins, une nouvelle inégalité pour les états de Dicke est proposée, qui est monogame quand le nombre de participants tends vers l'infinité. Malheureusement, toutes ces inégalités sont incapables de détecter la non-localité authentique. Des applications de la non-localité à la complexité de communication et aux jeux bayésiens sont discutées. / This thesis is about the nonlocal properties of permuation symmetric states and the potential usefulness of such properties in quantum information processing. The nonlocality of almost all symmetric states, except Dicke states is shown by constructing an $n$-party Hardy paradox. With the help of the Majorana representation, suitable measurement settings can be chosen for these symmetric states which satisfy the paradox. An extended CH inequality can be derived from the probabilistic conditions of the paradox. The inequality is shown to be violated by all symmetric states. The nonlocality properties and entanglement properties of symmetric states are also discussed and compared, natbly with respect to persistency and monogamy. It is shown that te degeneracy of some symmetric states is linked to the persistency, which provides a way to use device independent tests to separte nonlocality classes. It is also shown that the inequalities used to show the nonlocality of all symmetric states are not strictly monogamous.A new inequality for Dicke states is shown to be monogamous when the number of parties goes to infinity. But all these inequalites can not detect genuine nonlocality. Applications of nonlocality to communication complexity and Bayesian game theory are also discussed.

The objective of this thesis is to use the Majorana description of a spin-1 system to give a geometrical construction of a maximal set of Mutually Unbiased Bases (MUBs) and Symmetric Informationally Complete Positive Operator Valued Measures (SIC-POVMs) for this system. In the Majorana Approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the Reimann sphere, or a pair of unit vectors (known as Majorana vectors or M-vectors). Spin-1 states can be of three types: those whose vectors are parallel, those whose vectors are antiparallel and those whose vectors make an arbitrary angle. The types of bases possible for a spin-1 system are thus geometrically much more varied than for a spin-half system or qubit, which is the standard unit of information storage in most quantum protocols. Our derivation of the MUBs and SIC-POVMs proceeds from a recently derived expression for the squared overlap of two spin-1 states in terms of their M-vectors and the minimal additional set of assumptions that are needed. These assumptions include time-reversal invariance in the case of the MUBs and the requirement of three-fold symmetry in the case of the SIC-POVMs. The applications of these results to problems in quantum information are mentioned.

Les résultats récents des expériences d'oscillations ont montré que le neutrino est une particule massive; jusqu'à présent la valeur absolue de sa masse reste inconnue, même si les expériences de cosmologie ou de mesure directe donnent des contraintes. D'autre part, comme le neutrino est le seul fermion qui n'est pas chargé électriquement, il peut être identique à son antiparticule, c'est-à-dire être une particule de Majorana. La recherche de la double désintégration bêta $\mathrm{\beta \beta 0 \nu}$ est actuellement la seule technique expérimentale susceptible de mettre en évidence le neutrino de Majorana. Le détecteur NEMO-3, qui recherche la double désintégration bêta du Molybdène 100 et du Sélénium 82, prend des données au Laboratoire Souterrain de Modane depuis février 2003. La première phase de prise de données a permis de mettre en évidence une contamination de la chambre à fils trop élevée en Bismuth 214. Afin d'éviter qu'une faible fraction du radon de l'air du laboratoire ne diffuse dans le détecteur, une tente a été installée autour de celui-ci et une installation de déradonisation de l'air permet d'appauvrir en radon l'air qui entre dans la tente. Le détecteur NEMO-3 peut mesurer pour chaque prise de données sa contamination interne en Bismuth 214; des modèles permettent de décrire comment le radon peut diffuser dans le détecteur, ou comment l'activité en radon de l' air du laboratoire peut varier lors d'une coupure de ventilation. Après installation de la tente et de l'installation de déradonisation de l'air, plusieurs hypothèses sont proposées pour expliquer la contamination résiduelle, sachant que compte-tenu de la longue demi-vie du Radon 222, l'analyse des désintégrations du Bismuth 214 ne permet pas de remonter avec certitude à l'origine du matériau où le radon émane. Enfin, les désintégrations du Bismuth 214 rendent possible un test global, sur lénsemble du détecteur et pour une prise de donnée assez longue, de l'étalonnage en énergie. La recherche de la double désintégration bêta est basée sur une méthode de maximum de vraisemblance, qui utilise l'ensemble des informations mesurées sur les événements à deux électrons par NEMO-3 : non seulement la somme des énergies des électrons, mais également leurs énergies individuelles et l'angle d'émission entre eux. Les distributions sont ajustées sur des simulations pour les signaux et les bruits de fonds; ensuite les contributions des bruits de fond autres que la $\mathrm{\beta \beta 2 \nu}$ sont fixées, grâce à des canaux dédiés de plus haute statistique. Après 2,6 ans de prises de données pour la période à teneur en radon réduite, aucun signal n'a été observé et les contraintes sur les demi-vies de $\mathrm{\beta \beta 0 \nu}$ sont : $$\mathrm{T_{1/2}^{\beta \beta 0 \nu,~} > 8,3~10^{23}~ann\acute{e}es (90\% CL)~^{100}Mo}$$ $$\mathrm{T_{1/2}^{\beta \beta 0 \nu,~} > 4,9~10^{23}~ann\acute{e}es (90\%CL)~^{82}Se}$$ L'utilisation de l'information sur les énergies individuelles et sur l'angle d'émission entre les électrons permet d'améliorer la sensibilité au processus de $\mathrm{\beta \beta 0 \nu}$ généré par les courants droits : $$\mathrm{T_{1/2}^{\beta \beta 0 \nu,~V+A} > 3,5 ~10^{23} ~years~ (90\% CL)~^{100}Mo}$$ $$\mathrm{T_{1/2}^{\beta \beta 0 \nu,~V+A} > 2,7~ 10^{23} ~years~ (90\% CL)~^{82}Se}$$

This thesis describes aspects of dynamics of solid state systems which are relevant to quantum metrology and computation. It may be divided into three research directions (parts). For the first part, a new method to enhance precision measurements that makes use of a sensor’s environment to amplify its response to weak external perturbations is described. In this method a “central” spin is used to sense the dynamics of surrounding spins, which are affected by the external perturbations that are being measured. The enhancement in precision is determined by the number of spins that are coupled strongly to the central spin and is resilient to various forms of decoherence. For polarized environments, nearly Heisenberg-limited precision measurements can be achieved. The second part of the thesis focuses on the decoherence of Majorana fermions. Specializing to the experimentally relevant case where each mode interacts with its own bath we present a method to study the effect of external perturbations on these modes. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with individual Majorana modes - components of a qubit. We present examples with both static and dynamic perturbations (noise), and derive a rate of information loss for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath. For the third part of the thesis we discuss vortices in topological superconductors which we model as closed finite systems, each with an odd number of real fermionic modes. We show that even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator [1] as well as additional linearly independent zero modes, one of which is continuously connected to the Majorana mode in the non-interacting limit. We also show that in the situation where there are two or more well separated vortices their zero modes have non-Abelian Ising statistics under braiding. / Physics

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20889号 / 理博第4341号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 戸塚 圭介, 教授 川上 則雄, 教授 佐藤 昌利 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Majorana fermions promise potential applications in quantum computing, superconductivity, and related fields. In this thesis, an analysis of A. Y. Kitaev's “Kitaev Chain”, a quasi-one-dimensional quantum wire in contact with a p-wave superconductor, designed as a model exhibiting unpaired Majoranas, is performed. Described by tunneling of spinless fermions between quantum dots, and formation of Cooper pairs on neighboring dots, Kitaev's chain Hamiltonian serves as a basis for emergent Majorana Zero Modes (zero energy Majorana fermions localized at either end of the chain) and artificial gauges (phases) to appear. By exact diagonalization, energy spectra and wavefunctions of a chain of spinless fermions on discrete quantum dots described by Kitaev's Hamiltonian are generated. By transforming the system into a basis of Majorana fermions and "bond fermions", where Majoranas on neighboring dots are paired, emergent Majorana Zero Modes (MZMs) are found at the ends of the chain. These emergent MZMs are paired in a non-local, zero energy bond fermion, which is found to allow degenerate energy states of the system to occur. Joining the ends of the chain by allowing tunneling and pairing of fermions on end sites, a ring topology is considered, where an "artificial gauge" emerges. This artificial gauge, or phase, causes a phase change on tunneling and Cooper pairing Hamiltonian matrix elements as a result of operator ordering within the Hamiltonian's ring terms. These required operator orderings are derived by comparison of energy spectra of the Kitaev ring in the fermion and bond fermion bases. Matching of calculated energy spectra in the Majorana and fermionic bases is used to confirm the presence of the artificial gauge, where this phase is found to be necessary in order to maintain a consistent energy spectra across the transformation between bases. This analysis is performed in order to understand the concept of Majorana Zero Modes and the emergence of Majorana fermions in 1D chains. By doing so, it is determined what Majorana fermions are, where they come from, and why Majorana Zero Modes are considered to be zero energy. These results contribute to the understanding of Kitaev chains and rings, as well as serve as a starting point for discussions regarding physical implications of the artificial gauge's effect, fermion statistics, and the emergence of Majorana Zero Modes in quasi-one-dimensional systems.