Truncated correlation hierarchy schemes for driven-dissipative multimode quantum systems

TL;DR

This paper introduces two correlation hierarchy truncation schemes for modeling driven-dissipative multimode quantum systems, effectively capturing phenomena like the bunching-antibunching transition in photonic Bose-Hubbard models.

Contribution

It develops and compares 'hard' and 'soft' cutoff schemes for correlation hierarchies, extending the analytical tools for multi-mode driven-dissipative quantum systems.

Findings

Successfully describes the bunching-antibunching transition in the Bose-Hubbard model.

Benchmarked results agree with corner-space renormalization in 1D and 2D.

Discusses the regimes where each truncation scheme is valid.

Abstract

We present a method to describe driven-dissipative multi-mode systems by considering a truncated hierarchy of equations for the correlation functions. We consider two hierarchy truncation schemes with a global cutoff on the correlation order, which is the sum of the exponents of the operators involved in the correlation functions: a 'hard' cutoff corresponding to an expansion around the vacuum, which applies to a regime where the number of excitations per site is small; a 'soft' cutoff which corresponds to an expansion around coherent states, which can be applied for large excitation numbers per site. This approach is applied to describe the bunching-antibunching transition in the driven-dissipative Bose-Hubbard model for photonic systems. The results have been successfully benchmarked by comparison with calculations based on the corner-space renormalization method in 1D and 2D systems. The regime of validity and strengths of the present truncation methods are critically discussed.