Dissipative motion of an atom with transverse coherent driving in a cavity with many degenerate modes

We study the cavity-induced cooling of the center-of-mass motion of a transversally coherently pumped atom moving along the axis of a high-$Q$ multimode cavity. Including spontaneous emission and quantum fluctuations of the cavity field, we analytically calculate the linear friction and the diffusion coefficients. Efficient cavity-induced motional damping leads to steady-state temperatures well below the Doppler limit and lower than for an equivalent setup with direct cavity field pumping. The cooling process is strongly enhanced in the case of a multimode field, yielding a significantly lower temperature and a small number of spontaneous emissions per cooling time, which can even drop below 1.