Quantum dynamics of open continuous variable systems is a multifaceted rapidly evolving field of fundamental and technological significance. An important example is a multimode photonic system coupled to a Markovian bath so that temporal evolution of the density matrix is governed by the master equation of the Lindblad form. Theoretical analysis of the Lindblad dynamics complicated by intermode couplings such as dynamical (coherent) and environment-mediated (incoherent) interactions between the modes can be rather involved, even for exactly soluble models. Our algebraic technique uses the Lie algebra of quadratic combinations of left and right superoperators to simplify both quantitative and qualitative analysis of intermode coupling-induced effects in multimode systems by eliminating jump superoperators and reducing the Lindblad equation to the form governed by the effective non-Hermitian Hamiltonian ${\widehat{H}}_{\mathrm{eff}}$. We show that the spectral properties of the thermal bath Liouvillian are determined by the effective temperature-independent Hamiltonian and employ jump-eliminating transformations for computing its eigenvalues and eigenoperators. It is found that the Liouvillian exceptional points (EPs) are the points in the parameter space where the matrix $H$ associated with ${\widehat{H}}_{\mathrm{eff}}$ is nondiagonalizable. For the case of photonic polarization modes modeled as a bimodal bosonic system, we describe the geometry of EPs and apply our method to study how the intermode-couplings-induced effects such as EPs and temperature influence the dynamics of the speed of evolution that governs the quantum speed limit times of polarization qubit states.

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• Open quantum systems
• Open quantum systems & decoherence
• Nonequilibrium systems
• Lindblad equation
• Master equation