Momentum diffusion for coupled atom-cavity oscillators

It is shown that the momentum diffusion of free-space laser cooling has a natural correspondence in optical cavities when the internal state of the atom is treated as a harmonic oscillator. We derive a general expression for the momentum diffusion, which is valid for most configurations of interest: The atom or the cavity or both can be probed by lasers, with or without the presence of traps inducing local atomic frequency shifts. It is shown that, albeit the (possibly strong) coupling between atom and cavity, it is sufficient for deriving the momentum diffusion to consider that the atom couples to a mean cavity field, which gives a first contribution, and that the cavity mode couples to a mean atomic dipole, giving a second contribution. Both contributions have an intuitive form and present a clear symmetry. The total diffusion is the sum of these two contributions plus the diffusion originating from the fluctuations of the forces due to the coupling to the vacuum modes other than the cavity mode (the so-called spontaneous emission term). Examples are given that help to evaluate the heating rates induced by an optical cavity for experiments operating at low atomic saturation. We also point out intriguing situations where the atom is heated although it cannot scatter light.

Figure 1

(Color online) An atom placed between two mirrors is excited from the side (laser-atom Rabi frequency $2\eta$). The cavity mode is also coherently pumped by a laser source (coupling strength $E$). The mirrors dissipate the field at a rate $\kappa$, which is the analogue of the relaxation rate $\gamma$ of the atom’s dipole moment. The atom-cavity vacuum Rabi frequency is denoted by $2g$. Forces exist mainly along the cavity axis (due to $E$ and/or $\eta$), and along the side-laser beam.

Figure 2

Diffusion $D_{\mathit{\text{atom}}}+D_{\mathit{\text{mode}}}$ (solid) along the laser axis when the atom is excited from the side, for a running wave laser, as a function of the laser frequency ${\omega }_L$ $(E=0,\mid \eta \mid =0.1\gamma ,\kappa =\gamma ,g_0=6\gamma ,{\omega }_{eg}-{\omega }_{\mathit{cav}}=9\gamma )$. Diffusion due to spontaneous emission is equal to $D_{\mathit{\text{atom}}}$. The peaks display the normal modes, $D_{\mathit{\text{atom}}}$ (dashed) dominating for ${\omega }_L\approx {\omega }_{eg}$ and far from the cavity resonance ${\omega }_{\mathit{cav}}$ (fluctuating atom), and vice versa for $D_{\mathit{\text{mode}}}$ (dotted) dominating for ${\omega }_L\approx {\omega }_{\mathit{cav}}$ (fluctuating mode).

Figure 3

Diffusion $D_{\mathit{\text{atom}}}+D_{\mathit{\text{mode}}}$ (solid) along the side laser, resonant to the atom ${\omega }_L={\omega }_{eg}$, vs the cavity frequency ${\omega }_{\mathit{cav}}$ $(E=0,\mid \eta \mid =0.1\gamma ,\kappa =\gamma ,g_0=3\gamma )$. The diffusion is well suppressed compared to free space for the same intensity (horizontal line). $D_{\mathit{\text{mode}}}$ (dotted) reaches a maximum for ${\omega }_{\mathit{cav}}={\omega }_L$, where one has $D_{\mathit{\text{mode}}}=2C{D}_{\mathit{\text{atom}}}⪢D_{\mathit{\text{atom}}}$ (dashed), and decreases when the light hardly builds up inside the cavity ${\omega }_{\mathit{cav}}\ne {\omega }_L$.