Effective Hamiltonian theory of the geometric evolution of quantum systems

Omar Gamel, Daniel F. V. James, Timothy C. Ralph +1 more

V. O. Shkolnikov, Guido Burkard

In this work we present an effective Hamiltonian description of the quantum dynamics of a generalized $\mathrm{\ensuremath{\Lambda}}$ system undergoing adiabatic evolution. We assume the system to be initialized in the dark subspace and show that its holonomic evolution can be viewed as a conventional Hamiltonian dynamics in an appropriately chosen extended Hilbert space. In contrast to the existing approaches, our method does not require the calculation of the non-Abelian Berry connection and can be applied without any parametrization of the dark subspace, which becomes a challenging problem with increasing system size.

Gil Elgressy, Lawrence Paul Horwitz

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for “local instability” in the quantum theory.

Andrzej Herdegen

Mikhail Pletyukhov, E. A. Tolkachev

Geometrical effects occurring in the cyclic evolution of twofold degenerate two-level system are considered. A projector representation for the objects involved (Hamiltonian, evolution operator) is constructed. The explicit expressions for connections and curvatures are quoted. Particular cases are considered and found to be in full correspondence with previously obtained results.

Valerij V. Kozlov, Oleg Georgievich Smolyanov