Nonadiabatic Geometric Phase Research Articles

Non-Abelian geometric phase and generalized invariant method.

We quantitatively formulate the nonadiabatic non-Abelian geometric phase by generalizing the Lewis invariant method to a degenerate case, which is reduced to Wilczek and Zee's [Phys. Rev. Lett. 52, 2111 (1984)] non-Abelian geometric phase in the adiabatic limit. We also derive the effective propagator accommodating the nonadiabatic non-Abelian geometric phase and the semiclassical quantization rule in the adiabatic limit.

Relation between “phases” and “distance” in quantum evolution

We discuss the dynamical phase and the geometric phase in relation to the geometric distance function for cyclic evolution of quantum states. For all cyclic evolution of quantum states, we have shown that the non-adiabatic geometric phase is the integral of the contracted length of the curve which the system traverses.

Quaternionic gauge fields and the geometric phase

The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang’s SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov’s spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess–Zumino term unlike the one in the complex case.

Non-adiabatic non-abelian geometric phase

The non-integrable geometric phase factor is generalized to a non-abelian phase factor which in the adiabatic limit is the Wilczek-Zee generalization of the Berry phase. It is then extended to the evolution of mixed states. A theorem of Narasimhan and Ramanan is used to relate the gauge field connection to the generalized geometric connection.