Modelling and structure-preserving discretization of the Schrödinger equation as a port-Hamiltonian system, and simulation of a controlled quantum box

Abstract

The modelling of the Schrödinger Equation as a port-Hamil- tonian system is adressed. We suggest two Hamiltonians for the model, one based on the probability of presence and the other on the energy of the quantum system in a time-independent potential. In order to simulate the evolution of the quantum system, we adapt the model to a bounded domain. The model is discretized thanks to the structure- preserving Partitioned Finite Element Method (PFEM). Simulations of Rabi oscillations to control the state of a system inside a quantum box are performed. Our numerical experiments include the transition between two levels of energy and the generation of Schrödinger cat states.