In the Λ-system, non-adiabatic holonomic quantum phases are used to construct holonomic quantum gates. An interesting approach would be to implement open path holonomies in the Λ-system. By dividing the loop into two curve segments with a unitary transformation between them, universality can be reached. In doing so the exibility of the scheme has been increased by the fact that one single full pulse is now enough for universality, and we have achieved a clearer proof of the geometric property of the Λ system. / I ett Λ-system så används icke-adiabatiska holonoma kvantfaser för att bygga holonoma kvantgrindar. I detta arbete undersöker vi om holonomier för öppna kurvor kan implementeras i Λsystemet. Genom att dela upp en loop i Λ-systemet i två sekvenser med en unitär transformation emellan så kan vi konstruera en universell holonom kvantgrind. Med detta så har vi ökat exibiliteten för systemet genom att vi nu bara behöver ta en loop för att nå universalitet, och vi har även erhållit en klarare bild över den geometriska egenskapen hos Λ-systemet.

Note that equations and expressions has been omitted here and is instead presented in the work.  This thesis focuses on the geometric phase in pure and mixed quantum states. For the case of a pure quantum state, Berry's adiabatic approach (4.1.10) and Aharonov & Anandan's non-adiabatic generalization of Berry's approach (4.2.8) are included in this work. Mixed quantum state involves Uhlmanns approach (5.1.42), which is used extensively in Section 7 and Sjöqvist's et al. approach (5.2.22), used extensively in Section 6. Sjöqvist's approach states that the Uhlmann phase is an observable and provides the experimental groundworkusing an interferometer.􀀀This was later proven, by Du et al.[45] to reproduce experimental data (Figure 19) on page 56. The Uhlmann phase can be used to observe the behaviour of topological kinks. This was tested on 3 models, the Creutz-ladder, the Majorana chain andthe SSU-model. It is found that the Uhlmann phase is split into two regimes with the dividing parameter being the temperature. This temperature is called the critical temperature, Tc. If the temperatureis is below the critical temperature, the Uhlmann phase yields Π and if thetemperature is above the critical temperature, the Uhlmann phase yields zero. / Observera att ekvationer och andra uttryck har exkluderats här och är presenterade i själva arbetet. Detta examensarbete behandlar geometriska faser i rena och blandade kvanttillsånd. I rena kvanttillstånd finner man Berrys adiabatiska behandling av den geometriska fasen (4.1.10) och Aharonov & Anandan icke-adiabatiska generaliseringav Berry fasen (4.2.8). I det blandade kvanttillstånden har Uhlmann introducerat en förlängning av den geometriska fasen som sträcker sig till det blandadekvanttillstånden (5.1.42), detta finner man i sektion 7. Senare har Sjöqvistet al. introducerat ett alternativ till att angripa geometriska faser (5.2.22) som beskrivs i sektion 6. Sjöqvist konstaterade att Uhlmannfasen är observerbar,i kvantmekanisk mening, och presenterade ett  experimentelt upplägg där han visade just detta med hjälp av en interferometer. vilket senare bevisades av Du et al.[45] där de experimentella mätvärdena stämde överens med dem teoertiska (se figur 19 på sidan 56). Uhlmannfasen kan även användas för att observera topologiska "kink"-lösningar. Detta testades för 3 olika modeller; Creutz stege formationen, Majorana kedjan och SSU modellen. Det visade sig att Uhlmannfasen delades up i två regioner och var starkt beroende av temperaturen. Denna temperaturen kallades för den kritiska temperaturen Tc. Om temperaturen liggerunder eller över den kritiska temperaturen får man att Uhlmannfasen ger Π eller 0 respective.

We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.

This thesis investigates the long-standing Abraham-Minkowski controversy concerning the momentum of light inside a dielectric medium. A revealing connection to the optical He-McKellar-Wilkens (HMW) phase is found upon studying the Langrangian describing the classical laser-atom interaction. This connection is further highlighted by moving into a semi-classical model in which the phase arises as a result of the transformation between the Abraham and Minkowski Hamiltonians. The HMW along with the Aharonov-Casher phases are found to be both dynamic and geometric depending on the representation. It is shown that an optical version of the HMW phase is acquired by a dipole moving in a laser beam, and I propose several interferometric schemes in order to observe the optical HMW effect. Finally, by moving into a cavity system, it is possible to account for the back action of the atoms on the light which changes the electromagnetic mode structure. This increase in model sophistication grants an alternative vantage from which to interpret the Abraham-Minkowski problem. / Thesis / Doctor of Philosophy (PhD)

<p>Quantum holonomies are investigated in different contexts.</p><p>A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions.</p><p>A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies are</p><p>provided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting.</p><p>An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmann's parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators.</p><p>Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed.</p><p>The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation.</p><p>A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.</p>

Quantum holonomies are investigated in different contexts. A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions. A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies are provided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting. An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmann's parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators. Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed. The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation. A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.

Made available in DSpace on 2015-05-14T12:14:03Z (GMT). No. of bitstreams: 1 arquivocompleto.pdf: 1077441 bytes, checksum: 7e40786e3a70b55afbfcd9bdc28c5ffa (MD5) Previous issue date: 2010-12-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / One of the foundations of the actual technology is the Quantum Mechanics. Through Quantum Mechanics, with the development of semiconductors physics, we are able to build devices which, manipulate particles for certain purposes. As an example of these devices, we have the diode and transistor. In this thesis we seek physical systems, to obtain some quantum effects, theorically. The quantum effects which we want to obtain, are a band structure for photons and charged particles, the Aharonov-Bohm effect for photons and geometric phases for relativistic particles described by non-Hermitian Hamiltonian. For this purpose, the issues addressed here are: the study of photons in photonic crystals formed by topological insulators, the description of systems of charged particles of varying mass, the study of dynamics of the charged particles with periodic mass, the study of the Aharonov- Bohm for the photon through a viscous uid, and the appearance of geometric phases for relativistic particles described by non hermitian Hamiltonian. The main contributions of our work was the proposal for a Hamiltonian which describes particles with variable mass, obtaining a structure of energy bands for a charged particle with periodic mass, obtaining a structure of frequency bands for photons in a photonic crystal formed by topological insulators , showing the Aharonov-Bohm effect for the photon, where the vorticity of a viscous uid is the role of magnetic fields confined, and show that relativistic particles subject to non-Hermitian Hamiltonian which varies slowly, has complex geometric phases. / A tecnologia que usamos nos dias de hoje tem como uma de suas bases principais, a Mecânica Quântica. Através da Mecânica Quântica, com o desenvolvimento da física dos semicontutores, somos capazes de construir dispositivos os quais, manipulam partículas para determinados fins. Como exemplo destes dispositivos, temos o diodo e o transistor, que são o coração da eletrônica. Nesta tese, buscamos teoricamente sistemas físicos para a obtenção de certos efeitos quânticos. Tais efeitos que desejamos obter são, uma estrutura de bandas para fótons e partículas carregadas, o efeito Aharonov-Bohm para fótons, e fases geométricas para partículas relativísticas descritas por hamiltonianos não hermitianos. Para tal finalidade, os problemas tratados aquisição: o estudo de fótons em cristais fotônicos formados por isolantes topológicos, a descrição de sistemas de partículas carregadas de massa variável, o estudo da dinâmica de partículas carregadas de massa periódica, o estudo do efeito Aharonov-Bohm para o fóton através de um fluido viscoso, e o surgimento de fases geométricas para partículas relativísiticas sujeitas a hamiltonianos não hermitianos e que variam lentamente. As principais contribuições do nosso trabalho, foram a proposta para um hamiltoniano que descreve partículas com massa variável, a obtenção de uma estrutura de bandas de energia para uma partícula carregada com massa periódicas, a obtenção de uma estrutura de bandas de frequência para fótons num cristal fotônico formado por isolantes topológicos, á obtenção de um análogo do efeito Aharonov-Bohm para o fóton, onde a vorticidade de um uido viscoso faz o papel do campo magnético confinado, e mostrar que partículas relativísticas sujeitas a hamiltonianos não hermitianos e que variam lentamentem, apresentam fases geométricas complexas.

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