Ultracold Atomic System Of Adiabatic Geometric Phase

In quantum mechanics, the Berry’s phase is one of the most important concepts, which is associated with a phase shift in adiabatic processes and reveals the gauge struc-ture of the system. Recently, the study on this quantum phase has received renewed inter-est because of its important applications in quantum computing gates and in condensed-matter physics. It is well known that, in classical mechanics, the Berry’s geometric phase has a classical analogue——the Hannay’s angle. This classical geometric angle is deter-mined by the variation of the canonical angle variable and measures the holonomy effects of the system. In1985, at a semiclassical level, Berry established the connection between the above quantum phase and classical angle. From then on, a number of research works on the Berry’s phase and Hannay’s angle have been addressed. Among these research works, two representative results should be mentioned:one is the vacuum field-induced Berry’s phase, the other is the connection between the Berry’s phase and the quantum phase transition. However, so far most of the previous investigations in this field concern mainly the single-particle systems and the single quantum systems. Actually, the many-body systems and the composite quantum systems are of more interest and dominate the real physical world. A Bose-Einstein condensate (BEC) is a typical many-body system and a BEC coupled to an optical cavity is a composite system, both will be considered in this thesis. In these systems, the interaction between coherent particles is very important, which can be changed precisely via a Feshbach resonance technique. At present, for the above complicated systems, two theoretical approaches have been employed in studying the geometric phase issue——the mean-field method and the full quantum description. Based on these two approaches, in the present thesis, the adiabatic geometric phases in three ultracold boson systems are investigated theoretically. The thesis includes one in-troduction (see chapter1) and four chapters, and the main contents are shown in chapters2-4.In chapter1, a brief introduction on the research background of the work is given. Moreover, some important concepts closely related to the study including classical and quantum adiabatic theorem, Berry’s phase, Hannay’s angle, and quantum phase transition are reviewed in detail.In chapter2, the Berry’s phase and Hannay’s angle in an interacting two-mode boson system are studied. The analytic expressions for the phase and angle are obtained in explicit forms. Particularly, the connection between the quantum geometric phase and the classical Hannay’s angle is established. It is found that, the classical angle is equal to the derivative of the quantum phase with respect to the particle number except for a sign in the large-particle-number limit. For the many-body boson systems that the total particle number is conserved and the coherent-state description is available, the above relationship is also valid.In chapter3, the quantum phase transition from a mixed atom-molecule phase to a pure molecule phase in an ultracold atom-homonuclear-molecule conversion system is investigated. A coherent state is constructed, which is found to be a good approximation of the quantum ground state. By viewing the coherent state as a variational trial state, the critical point is derived analytically. The scaling laws for the phase transition are built and the corresponding critical exponents are obtained. The signature of Berry’s curvature for the transition is discussed as well. The results show that, the total particle number derivatives of the Berry’s curvature intersect at the critical point.In chapter4, a model of a Bose-Einstein condensate coupled to an optical cavity is considered. In this system, both the condensate and the cavity are described by the coherent states. By slowly varying the argument of the atom-cavity coupling parameter from zero to2n, the geometric phase of the ground state accumulated during the adia-batic process is calculated analytically. It is shown that, at a critical coupling strength, the ground-state geometric phase jumps from zero to π. In fact, this critical point just corresponds to the point of the normal-superradiant quantum phase transition. The vir-tual magnetic flux interpretation for the novel behavior of the geometric phase is also addressed.