Input-Output Hierarchical Equations Of Motion

TL;DR

This paper extends the hierarchical equations of motion (HEOM) framework to model non-Gaussian input states and output observables in bosonic environments, enabling non-perturbative analysis of non-Markovian open quantum systems.

Contribution

The authors develop an extended HEOM formalism that incorporates non-Gaussian input states and output observables, broadening the applicability of HEOM to more general quantum environments.

Findings

Derived an input-output Lindblad equation in the Markovian limit.

Bounded the index range for input states and observables, ensuring computational feasibility.

Demonstrated the formalism's ability to handle non-Gaussian states and non-Markovian dynamics.

Abstract

We derive an extended version of the hierarchical equations of motion (HEOM) to compute output physical properties of a bosonic environment, which is allowed to be initially prepared at an earlier time in a non-Gaussian input state and then non-perturbatively interact with a quantum system with a linear environmental operator. While spectral assumptions analogous to the ones used in the regular HEOM are imposed to compute dynamical output bath observables, they are not required to model input states or output observables at a fixed time, in this case leading to time-dependent contributions to the equations. In the Markovian limit, we use this formalism to derive an input-output Lindblad equation which can be used to extend the applicability of the regular version. For a given desired input state and output observable, the range of the indexes extending the regular HEOM is, by construction, bounded. Overall, the aim of this formalism is to take advantage of the efficiency and generality of the HEOM framework to model non-Gaussian input states and the dynamics of environmental observables in bosonic, non-Markovian open quantum systems.