Unitary evolution for a two-level quantum system in fractional-time scenario

The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be non-unitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the non-unitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.