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We present a method to describe driven-dissipative multi-mode systems byconsidering a truncated hierarchy of equations for the correlation functions.We consider two hierarchy truncation schemes with a global cutoff on thecorrelation order, which is the sum of the exponents of the operators involvedin the correlation functions: a 'hard' cutoff corresponding to an expansionaround the vacuum, which applies to a regime where the number of excitationsper site is small; a 'soft' cutoff which corresponds to an expansion aroundcoherent states, which can be applied for large excitation numbers per site.This approach is applied to describe the bunching-antibunching transition inthe driven-dissipative Bose-Hubbard model for photonic systems. The resultshave been successfully benchmarked by comparison with calculations based on thecorner-space renormalization method in 1D and 2D systems. The regime ofvalidity and strengths of the present truncation methods are criticallydiscussed.

Truncated correlation hierarchy schemes for driven-dissipative multimode quantum systems