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This is a work in two parts in which we show how to solve a large class ofLindblad master equations for non-interacting particles on $L$ sites. In part Iwe concentrate on bosonic particles. We show how to reduce the problem todiagonalizing an $L \times L$ non-Hermitian matrix. In particular, for boundarydissipative driving of a uniform chain, the matrix is a tridiagonal borderedToeplitz matrix which can be solved analytically for the normal master modesand their relaxation rates (rapidities). In the regimes in which an analyticalsolution cannot be found, our approach can still provide a speed-up in thenumerical evaluation. We use this numerical method to study the relaxation gapat non-equilibrium phase transitions in a boundary driven bosonic ladder withsynthetic gauge fields. We conclude by showing how to construct thenon-equilibrium steady state.

Solutions for dissipative quadratic open systems: part I - bosons