Nonequilibrium Green's function method for the harmonic oscillator in a heat bath

Using the nonequilibrium Green's function method, the problem of thermal relaxation of systems interacting with a heat bath is investigated. We apply this method to the harmonic oscillator interacting with a heat bath which consists of many harmonic oscillators. Using with the Keldysh nonequilibrium Green function technique we derive expressions for the the absorption and emission spectra. We also establish a connection to the the solution of the Heisenberg equation based on a fully quantum-mechanical Hamiltonian.