Algebraic approach to dynamics of open multimode bosonic systems

Quantum dynamics of open continuous variable systems is a multifaceted, rapidly evolving field of both fundamental and technological significance. An important example is a multimode photonic system coupled to a Markovian bath (the environmental correlation times are shorter than the system's relaxation or decoherence time) so that the density matrix dynamics is governed by the master equation of the Lindblad form. Using currently available methods, theoretical analysis of the Lindblad dynamics complicated by intermode couplings can be rather involved even for exactly soluble models. The algebraic approach suggested in this paper simplifies both quantitative and qualitative analysis of the intermode-coupling-induced effects in multimode systems by reducing the Lindblad equation to the form determined by the effective Hamiltonian. Specifically, we develop the algebraic method based on the Lie algebra of quadratic combinations of left and right superoperators associated with matrices to study the Lindblad dynamics of multimode bosonic systems coupled with a thermal bath and described by the Liouvillian superoperator that takes into account both dynamical (coherent) and environment mediated (incoherent) interactions between the modes. Our algebraic technique is applied to transform the Liouvillian into the diagonalized form by eliminating jump superoperators and solve the spectral problem. The temperature independent effective non-Hermitian Hamiltonian, $\hat{H}_{\text{eff}}$, is found to govern both the diagonalized Liouvillian and the spectral properties. It is shown that the Liouvillian exceptional points are represented by the points in the parameter space where the matrix, $H$, associated with $\hat{H}_{\text{eff}}$ is non-diagonalizable. We use our method to derive the low-temperature approximation for the superpropagator and to study the special case of a two mode system representing the photonic polarization modes,e.g. optical polarization qubit.For this system, we describe the geometry of exceptional points in the space of frequency and relaxation vectors parameterizing the intermode couplings and, for a single-photon state, evaluate the time dependence of the density matrix.