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 crossover 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.
Highlights
There has been a strong interest in driven-dissipative manybody systems, nonequilibrium photonic systems
These systems are typically computationally more challenging since the total number of photons is typically not conserved resulting in a much larger effective Hilbert space. They are in general in a mixed state described by a density matrix while a closed system at low temperature is described by the groundstate wavefunction
For a given cutoff order NC the number of equations grows according to a power law as a function of the size of the system. This is a consequence of the global cutoff: for a local cutoff, instead, the number of equations grows exponentially with the system size. This shows that truncation schemes based on a global cutoff can be very efficient for problems that involve a low order of correlation
Summary
There has been a strong interest in driven-dissipative manybody systems, nonequilibrium photonic systems. Due to the inherent out-of-equilibrium nature of these systems many of the numerical approaches developed for systems at equilibrium are not necessarily applicable These systems are typically computationally more challenging since the total number of photons is typically not conserved resulting in a much larger effective Hilbert space. The second truncation scheme is based on a self-consistent ’soft’ cutoff, which corresponds to an expansion around coherent states and applies when the number of excitations per site is large. This paper is structured as follows: in Section II we introduce the theoretical approach in a general way; in Section III, we specialize the theory to the driven-dissipative Bose-Hubbard model; in Section IV the numerical results are presented and critically discussed.
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