How to discretize a quantum bath for real-time evolution

TL;DR

This paper reviews and compares methods for discretizing quantum baths in numerical simulations, highlighting the optimality of orthogonal polynomial strategies for quadratic Hamiltonians and discussing limitations for non-quadratic cases.

Contribution

It clarifies the relation between discretization methods and demonstrates the optimality of orthogonal polynomial strategies for quadratic Hamiltonians.

Findings

Orthogonal polynomial strategy yields exact time evolution for quadratic Hamiltonians up to a maximum time.

No universally best discretization strategy exists for non-quadratic Hamiltonians.

Numerical examples include open quantum systems and dynamical mean-field theory applications.

Abstract

Many numerical techniques for the description of quantum systems that are coupled to a continuous bath require the discretization of the latter. To this end, a wealth of methods has been developed in the literature, which we classify as (i) direct discretization, (ii) orthogonal polynomial, and (iii) numerical optimization strategies. We recapitulate strategies (i) and (ii) to clarify their relation. For quadratic Hamiltonians, we show that (ii) is the best strategy in the sense that it gives the numerically exact time evolution up to a maximum time $t_\text{max}$, for which we give a simple expression. For non-quadratic Hamiltonians, we show that no such best strategy exists. We present numerical examples relevant to open quantum systems and strongly correlated systems, as treated by dynamical mean-field theory (DMFT).