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

TL;DR

This paper demonstrates how to achieve unitary evolution in a two-level quantum system governed by fractional-time Schrödinger dynamics by employing a time-dependent metric and Dyson map, reconciling non-Hermitian evolution with standard quantum mechanics.

Contribution

It introduces a method to map non-unitary fractional-time quantum evolution to a unitary form using a time-dependent metric and Dyson map, enabling standard quantum interpretation.

Findings

Successfully constructs a unitary evolution from fractional-time Schrödinger equation

Provides explicit examples with different Hamiltonians and Dyson maps

Reconciles fractional-time dynamics with conventional quantum mechanics

Abstract

The time-evolution operator obtained from the fractional-time Schr\"{o}dinger 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.