Non-adiabatic Berry phases for periodic Hamiltonians
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
A method for the calculation of Berry phases for periodic, but not necessarily adiabatic, Hamiltonians is reported. This method is based on a novel factorisation of the evolution operator and is in the spirit of the theory of systems of linear differential equations with periodic coefficients. The use of this approach in practical situations is greatly facilitated by exploiting the Fourier decomposition of the Hamiltonian. This converts the problem into an equivalent time-independent form. The solution to the problem is then expressible in terms of the eigenvectors and eigenvalues of a certain self-adjoint operator called the Floquet Hamiltonian. This operator can be calculated from the Fourier decomposition of the original Hamiltonian. Our formalism has several calculational advantages over the other methods used in the literature. These advantages are best seen by considering standard quantum optical systems such as the semi-classical model of a two-level atom strongly irradiated by a near resonant laser beam. The utility of our formalism is not confined to systems of this type however. For example it can be used to great advantage in the study of systems with time-odd electron-phonon coupling. Apart from its calculational utility, our formalism also has important theoretical applications. Here it is used to clarify the relationship between Berry phases and the time dependence of the Hamiltonian.