Solutions for bosonic and fermionic dissipative quadratic open systems

We show how to solve a large class of Lindblad master equations for noninteracting particles on L sites. In the first portion we concentrate on bosonic particles, while in the second we will address fermionic particles. In both cases we show how to reduce the problem to diagonalizing an L x L non-Hermitian matrix. In particular, for boundary dissipative driving of a uniform chain, the matrix is a tridiagonal bordered Toeplitz matrix which can be solved analytically for the normal master modes and their relaxation rates (rapidities). In the regimes in which an analytical solution cannot be found, our approach can still provide a speedup in the numerical evaluation. For bosonic particles, we use this numerical method to study the relaxation gap at nonequilibrium phase transitions in a boundary driven bosonic ladder with synthetic gauge fields. We conclude by showing how to construct the nonequilibrium steady state. The analysis for fermionic particles closely follows that of bosons, but with important differences due to the different commutation rules.