Hierarchical Schrödinger equations of motion for open quantum dynamics

Abstract

We rigorously investigate the quantum non-Markovian dissipative dynamics of a system coupled to a harmonic-oscillator bath by deriving hierarchical Schrödinger equations of motion (HSEOM) and studying their dynamics. The HSEOM are the equations for wave functions derived on the basis of the Feynman-Vernon influence functional formalism for the density operator, 〈q|ρ(t)|q'〉, where 〈q| and |q'〉are the left- and right-hand elements. The time evolution of〈q| is computed from time 0 to t, and, subsequently, the time evolution of |q'〉 is computed from time t to 0 along a contour in the complex time plane. By appropriately choosing functions for the bath correlation function and the spectral density, we can take advantage of an HSEOM method to carry out simulations without the need for a great amount of computational memory. As a demonstration, quantum annealing simulation for a ferromagnetic p-spin model is studied.