Cooling the motion of a trapped atom with a cavity field

We theoretically analyze the cooling dynamics of an atom which is tightly trapped inside a high-finesse optical resonator. Cooling is achieved by suitably tailored scattering processes, in which the atomic dipole transition either scatters a cavity photon into the electromagnetic field external to the resonator, or performs a stimulated emission into the cavity mode, which then dissipates via the cavity mirrors. We identify the parameter regimes in which the atom center-of-mass motion can be cooled into the ground state of the external trap. We predict that, in particular, for high cooperativities, interference effects mediated by the atomic transition may lead to higher efficiencies. The dynamics is compared with the cooling dynamics of a trapped atom inside a resonator studied by Zippilli and Morigi [Phys. Rev. Lett. 95, 143001 (2005)] where the atom, instead of the cavity, is driven by a laser field.

Figure 1

The cooling dynamics of the center-of-mass motion of a trapped atom is studied, under the assumption that the atom is confined within a high-finesse resonator. The motion is cooled by scattering of photons of the cavity mode, which is pumped by a laser.

Figure 2

Basic processes of photon scattering leading to the contributions (a) $\mathcal{D}$ and (b) ${\mathcal{A}}_{\pm }^{\left(\gamma \right)}$ and ${\mathcal{A}}_{\pm }^{\left(\kappa \right)}$ in Eq. (21). The processes start from state $|g,0_c,m\rangle$ and connect it to states $|g,0_c,m\pm 1\rangle$ via a scattering path, the intermediate states of which are the dressed states $|\pm ,m\rangle$ and $|\pm ,m\pm 1\rangle$. The straight arrows indicate coherent processes, the wiggled arrows the incoherent processes. One notes that there is no direct excitation $|g,0_c,m\rangle \to |\pm ,m\pm 1\rangle$, differing from the case in which the atom is driven by the laser [25, 26]. This property leads to a major difference between the cooling dynamics of the two situations.

Figure 3

Cooling rate $\Gamma$ [Eq. (2)] and mean phonon number $\langle m\rangle$ [Eq. (3)] as a function of the detunings $\delta$ and $\Delta$ (in units of the trap frequency $\nu$) and in the good-cavity limit, assuming $\gamma \gg \kappa$. The parameters are $\gamma =10\nu$, $\kappa =0.025\nu$, cooperativity $C=3$ (left panel) and $C=15$ (right panel). White areas denote parameter regions where the motion is heated, the dashed curve in the contour plots indicates Eq. (26), the dotted curves Eq. (27). The bottom plot displays $\langle m\rangle$ along the dashed and dotted curves for $C=15$. The other parameters are ${\Omega }_{\mathrm{P}}=0.01\nu$, $\mathcal{D}=1$, $\phi =0$, $\varphi =0.45\pi$.

Figure 4

Same as Fig. 3, except that $\kappa =0.1\nu$.

Figure 5

Same as Fig. 3, but in the bad-cavity limit $\kappa \gg \gamma$. The parameters are $\gamma =0.15\nu$, $\kappa =4.5\nu$, $C=5$ (left panel) and $C=25$ (right panel). The other parameters are ${\Omega }_{\mathrm{P}}=0.1\nu$, $\mathcal{D}=1$, $\varphi =\pi /3$, and $\phi =0$.

Figure 6

Excitation paths contributing to the heating transition $|g,0_c,m\rangle \to |g,0_c,m+1\rangle$, here depicted in the lowest order of the Born expansion in the cavity-atom coupling $g$ and for $\kappa \gg \gamma$. The amplitude of each path scales with $1/(\Delta +{\delta }_c)$ (left) and $1/(\Delta +{\delta }_c-\nu )$ (right). For $\delta =\Delta +{\delta }_{\mathrm{c}}=\nu /2$, the energy defect has opposite sign and the two processes interfere destructively.