Hamiltonianos não-Hermitianos dependentes do tempo.

Abstract

The main goal of this work is to investigate time-dependent Hamiltonians through the pseudo-Hermitian representation of quantum mechanics. In this formulation a dynamic Hilbert space defined in terms of a time-dependent metric defined in terms of the Hamiltonian is established, in relation to this metric the system evolves unitarily, and we can assign an adequate quantum interpretation according to the usual quantum theory. In this context, we study three interesting problems: Pseudo-Hermitian Dynamic Casimir Effect; two-level systems described by the time-fractional Schrödinger equation; and entanglement between two apparently uncoupled bosonic modes modulated by complex frequency, in the context of the dynamic symmetries SU(1, 1) and SU(2). The pseudo-Hermitian Dynamic Casimir Effect problem was constructed in an ad hoc way from Law’s Hamiltonian, which describes the Hermitian Dynamic Casimir Effect, and we show that through this description it is possible to obtain a rate of creation of photons much higher when compared to the Hermitian case. We also use the timedependent pseudo-Hermitian formalism to find a suitable metric in terms of which the solution of the time-fractional Schrödinger equation evolves unitarily. Furthermore, we consider the dynamic symmetries SU(1, 1) and SU(2) and their respective two-way bosonic representations, for which we consider a generic Dyson map, which takes non-Hermitian Hamiltonians that apparently do not have interaction terms presenting quantum correlations through the non-trivial metric of Hilbert space.