Population dynamics in sideband cooling of trapped ions outside the Lamb-Dicke regime

Abstract

We present the results of simulations of optical sideband cooling of atomic ions in a trap with a shallow potential well. In such traps, an ion cannot be Doppler cooled near to the Lamb-Dicke regime $\mathrm{[}{\ensuremath{\eta}}^{2}(2\ensuremath{\langle}n\ensuremath{\rangle}+1)\ensuremath{\ll}1]$. Outside the Lamb-Dicke regime, the sideband cooling dynamics are altered by the existence of various Fock states with weak coupling where the cooling becomes very slow. A $^{40}\mathrm{Ca}^{+}$ ion trapped in our Penning trap realizes such a situation; hence single stage cooling is inefficient to prepare the ion in the motional ground state. For these systems, it is necessary to study the cooling dynamics in detail and we show that it is possible to implement an optimized cooling sequence to achieve efficient ground-state cooling. We also present the simulated cooling dynamics of two ions trapped in a Penning trap, where the presence of an additional motional mode requires a complicated cooling sequence in order to cool both axial modes to the ground state simultaneously. Additionally, we demonstrate the dissipative preparation of Fock states outside the Lamb-Dicke regime by sideband heating a single ion in a Penning trap.