Closed form solution of Lindblad master equations without gain

We present a closed form solution to the eigenvalue problem of a class of master equations that describe open quantum system with loss and dephasing but without gain. The method relies on the existence of a conserved number of excitation in the Hamiltonian part and that none of the Lindblad operators describe an excitation of the system. In the absence of dephasing Lindblad operators, the eigensystem of the Liouville operator can be constructed from the eigenvalues and eigenvectors of the effective non-Hermitian Hamiltonian used in the quantum jump approach. Open versions of spin chains, the Tavis-Cummings model and coupled Harmonic oscillators without gain can be solved using this technique.