Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics

Abstract

A wide range of quantum systems are time-invariant and the corresponding dynamics is dictated by linear differential equations with constant coefficients. Although simple in mathematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limiting factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge-Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we apply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.