One-to-one correspondence between Hierarchical Equations of Motion and Pseudomodes for Open Quantum System Dynamics

TL;DR

This paper establishes a formal equivalence between the Hierarchical Equations of Motion and pseudomode approaches for modeling non-Markovian open quantum systems, providing explicit transformations and insights.

Contribution

It proves a constructive, explicit correspondence between HEOM and pseudomodes for any exponential bath correlation function, unifying two major methods.

Findings

Every exponential BCF corresponds to a pseudomode model with Lindblad damping.

A linear transformation maps HEOM evolution to pseudomode dynamics and vice versa.

The approach offers insights and derivations for the HOPS and nuHOPS methods.

Abstract

We unite two of the most widely used approaches for strongly damped, non-Markovian open quantum dynamics, the Hierarchical Equations of Motion (HEOM) and the pseudomode method by proving two statements: First, every physical bath correlation function (BCF) that can be written as a sum of $N$ exponential terms can be obtained from a physical model with $N$ interacting pseudomodes which are damped in Lindblad form. Second, for every such BCF there exists a non-unitary, linear transformation which mirrors the evolution of the system-pseudomode state onto the HEOM hierarchy, and vice versa. Our proofs are constructive and we give explicit expressions for the mirror transformation as well as for the pseudomode Lindbladian corresponding to a given exponential BCF. This approach also gives insight and provides elegant derivations of the corresponding Hierarchy of stochastic Pure States (HOPS) method and its nearly-unitary version, nuHOPS. Our result opens several avenues for further optimization of non-Markovian open quantum system dynamics methods.