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Limit of Doppler cooling

We present a quantum theory of one-dimensional laser cooling of free atoms using a transition with a J = 0 ground state and a J = 1 excited state. This treatment is valid both for broad lines (recoil energy small compared with the energy width ℏ Γ of the excited level) and for narrow lines. For broad lines we recover the well-known cooling limit for a two-level transition (∼ℏΓ/2), whereas for a narrow line the cooling limit is found to be of the order of the recoil energy. The stationary momentum distribution is obtained for both cases and is found to be close to the one obtained by Monte Carlo simulations.