Quantum Phase Transition in an Ion Trap

English This thesis considers the interaction between N ions confined in an ion trap and a set of laser fields, described by the Dicke model. The Dicke Hamiltonian for the two different models, used by the research group at the University of Innsbruck (the Quantum Optics and Spectroscopy group) and the Quantum Optics group at the National Institute of Science and Technology in Boulder, Colorado, respectively. The Innsbruck model uses an effective two-level system, while the Boulder model uses a ∆-system, which is reduced to a two-level system by eliminating the excited level adiabatically. This thesis shows that the system undergoes a phase transition at a critical value of the Lamb-Dicke parameter η, where the system goes from the ground state into a state characterized by super radiance, where the atomic ensemble spontaneously emits radiation with intensity proportional to N. The existence of the phase transition is shown from a mean-field description of the Dicke model and from the Holstein-Primakoff representation of the model in the thermodynamic limit N →∞. This representation enables a physical interpretation of, and a detailed description of, the phase transition. A direct numerical diagonalization of the ground state of the Dicke Hamiltonian have been carried out to test the behaviour of a finite number of ions. This diagonalization also indicates that the system needs to prepared in such a way that the interaction with the laser fields excite more oscillations in the center-of-mass mode than in the atomic spin. The possibility of measuring the phase transition has also been discussed by numerically solving the time-dependent Schrödinger equation for various linear models of the Lamb-Dicke parameter. It is shown that the Lamb-Dicke parameter needs to be adiabatically driven to properly measure the phase transition and that the deviation from the phase transition decreases as the Lamb-Dicke parameter is driven more slowly. Lastly the universality of the phase transition is discussed. Dansk Denne bacheloropgave omhandler interaktionen mellemN ioner i en ionfælde og en række lasere, beskrevet ved hjælp af Dicke-modellen. Dicke Hamiltonen er blevet opstillet for to forskellige modeller, der bliver brugt af henholdsvis Quantum Optics and Spectroscopy group fra University of Innsbruck og kvanteoptikgruppen fra National Institute of Science and Technology i Boulder, Colorado. Innsbruck-modellen bruger et to-niveau atom, mens at Boulder-modellen bruger et ∆−system, der reduceres til et to-niveau system ved at eliminere the tredje niveau adiabatisk. Denne opgave viser, at der findes en faseovergang i system ved en kritisk værdi af Lamb-Dicke parameteren η for systemets grundtilstand. Denne faseovergang går fra grundtilstanden til en tilstand, hvor det atomare ensemble er karakteriseret ved begrebet "super-radiance", hvor ensemblet udsender spontant emitteret stråling med en intensitet proportional til N. Faseovergangens eksistens bevises gennem en middelfelts-beskrivelse til nulte og første orden, blandt andet ved hjælp af Holstein-Primakoff repræsentationen af modellen i den thermodynamiske grænse N → ∞. Denne repræsentation muliggør desuden en fysisk fortolkning af faseovergangen. Grundtilstanden for Dicke Hamiltonen er blevet diagonaliseret numerisk for at teste om hvordan antallet af ioner i fælden påvirker faseovergangen. Diagonaliseringen viser også, at systemet skal forberedes sådan, at interaktionen danner flere ekscitationer i oscillationen i ionernes massemidtpunkt end i det atomare spin. Den tidsafhængige Schrödingerligning er blevet løst numerisk for at undersøge muligheden for at måle faseovergangen i et eksperiment. Dette er gjort for forskellige lineære modeller for Lamb-Dicke parameteren. I den forbindelse vises det, at Lamb-Dicke parameteren skal køres adiabatisk langsomt i tid , od det vises desuden, at afvigelsen fra grundtilstanden samtidig bliver mindre. Universaliteten af faseovergangen diskuteres til slut.