where R and P are the nuclear position and momenta, respectively, of the bath coordinates, M is the mass of the bath particles, and {|a⟩, |b⟩} are the quantum DOFs of the system (e.g. the electronic excitation and photonic exciton of the molecule-cavity hybrid system). As the total number of states increases, quantum dynamics simulations can become expensive due to the exponential scaling of the calculation with the number of degrees of freedom. Over the years, there has been an enormous development of many semi-classical mixed quantum-classical , and quantum master equation approaches , each with their own strengths and shortcomings. Quantum master equation approaches are often based on perturbative approaches which are restricted to weak system-bath couplings and often fail to account for non-Markovian effects from bath feedback to the quantum subsystem DOFs. Semi-classical approaches, on the other hand, have been shown to accurately treat non-Markovian feedback from the nuclear bath to the quantum subsystem and can accurately treat a wide range of system-bath couplings. Here, we use the PLDM approach to compute optical spectra. The time evolution of the matrix element of the operator Ô is given by