Fases geométricas e invariantes dinâmicos em mecânica quântica
Abstract
The increasing interest to understand the geometric phases (mainly due to the possi-
bility to achieve geometric quantum computation robust against some kind of errors) and
the search for new techniques to solve time-dependent problems in Quantum Mechanics,
are the topics approached in this thesis.
i) Within nowadays technology, it is presented a scheme to control and measure the
nonadiabatic geometric phases in cavity quantum electrodynamics. In this context, it is
possible to generate superposition states of the cavity mode which acquire relative phases
of purely geometric character, termed the geometric Schrödinger cat-like states.
ii) For two interacting Bose-Einstein condensates in the two-mode approximation,
modeled by a Hamiltonian whose parameters are time dependent, the nonadiabatic and
noncyclic geometric phase acquired by the state of the system is analyzed. For this
purpose, analytical solutions of the Schrödinger equation are obtained in di¤erent regimes
of parameters. Connections between the constants of motion associated to each solution
and the geometric phases are established. The e¤ects of the time-dependent parameters
on the geometric phase as well as the population imbalance and relative phase between
the two condensed components are analyzed.
iii) Starting only from the parallel transport condition, it is presented a general de…-
nition of the geometric phase acquired by the basis states of a system. The de…ned phase
generates gauge invariant observables which apply to a general scenario of adiabatic or
nonadiabatic, cyclic or noncyclic, and transitional or nontransitional evolutions of pure
or mixed states of the system. Several results presented in the literature are recovered.
iv) Finally, together with some considerations about the dynamical invariant tech-
nique, it is presented an alternative method to obtain the density operator of two level
systems. Preliminary results show that this method can be extended to dissipative as well
as interacting two level systems, such as spins-1=2 chains