Memory effect in time fractional Schrödinger equation

Abstract

A significant obstacle impeding the advancement of the time fractional Schrödinger equation lies in the challenge of determining its precise mathematical formulation. In order to address this, we undertake an exploration of the time fractional Schrödinger equation within the context of a non-Markovian environment. By leveraging a two-level atom as an illustrative case, we find that the choice to raise i to the order of the time derivative is inappropriate. In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment, the time fractional Schrödinger equation, when devoid of fractional-order operations on the imaginary unit i, emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects. Meanwhile, we also prove that it is meaningless to study the memory of time fractional Schrödinger equation with time derivative 1 < α ≤ 2. It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrödinger equation. This will be the focus of future research. Our study might provide a new perspective on the role of time fractional Schrödinger equation.