Extended Non-Markovian Stochastic Schr\"odinger Equation with Complex Frequency Modes for General Basis Functions

TL;DR

This paper extends the non-Markovian stochastic Schr"odinger equation to incorporate arbitrary basis functions, enabling accurate simulation of open quantum systems with complex spectral densities beyond exponential forms.

Contribution

The authors develop an extended cNMSSE that uses non-exponential basis sets, allowing for flexible modeling of environments with complex spectral features, including higher-order poles.

Findings

Accurate simulations for diverse spectral densities.

Excellent agreement with HFB-SSE and HEOM benchmarks.

Efficient implementation with matrix product states.

Abstract

We introduce an extended formulation of the non-Markovian stochastic Schr\"odinger equation with complex frequency modes (extended cNMSSE), designed for simulating open quantum system dynamics under arbitrary spectral densities. This extension employs non-exponential basis sets to expand the bath correlation functions, overcoming the reliance of the original cNMSSE on exponential decompositions of the spectral density. Consequently, the extended cNMSSE is applicable to environments beyond those characterized by Debye-type spectral densities. The flexibility to employ general basis functions is particularly advantageous for handling spectral densities with higher-order poles, for which exponential decompositions are often inaccurate or unavailable. The extended cNMSSE is implemented in a pseudo-Fock space using conventional ladder operators and solved efficiently via matrix product state (MPS) techniques, preserving the favorable linear-scaling and wavefunction-based nature of the original method. Benchmark simulations across four representative cases, including discrete spectral density, Ohmic spectral density with exponential and algebraic cutoffs, and critically damped Brownian spectral density, demonstrate excellent agreement with results of hierarchy of forward-backward stochastic Schr\"odinger equations (HFB-SSE) and extended hierarchical equation of motion (HEOM).