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Physical Review A

Simple theory of sideband cooling

Ann Laurie Wells, Richard J. Cook

Published Apr 1, 1990• 3 citations
Publisher

Abstract

The problem of laser cooling of a single ion or atom in a harmonic trap is considered. A simple theory of sideband cooling is developed. In the limit that the particle's secular motion can be treated semiclassically the theory allows the calculation of the logarithmic cooling rate either numerically or, for two special cases, analytically.

DOI
10.1103/physreva.41.3916

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The problem of laser cooling of a single ion or atom in a harmonic trap is considered. A simple theory of sideband cooling is developed. In the limit that the particle's secular motion can be treated semiclassically the theory allows the calculation of the logarithmic cooling rate either numerically or, for two special cases, analytically.
Abstract : The problem of laser cooling of a single ion or atom in a harmonic trap was considered. A simple theory of sideband cooling has been developed. In the limit that the particle's secular motion can be treated semiclassically, the theory allows the calculation of a logarithmic cooling rate either numerically or, for two special cases, analytically. The theory could be used to optimize the parameters of a cooling experiment. The spectroscopy of a single atom in the ladder configuration has been treated theoretically. A dressed atom approach was used to provide qualitative information about the system.
We study laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasicontinuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of dark states which may slow down the cooling is considerably reduced. We show that cooling to the ground state is also possible outside the Lamb-Dicke regime. We develop the model and use quantum Monte Carlo calculations for specific examples. We show that a rate equation treatment is a good approximation in all cases.
Laser (light-pressure) cooling of two-level ions stored in a Penning trap is treated theoretically, in the limit that the frequencies of motion of the ions are much smaller than the natural linewidth of the optical transition. Rate equations for the mean-squared amplitudes of motion are derived from a semiclassical analysis, which is confirmed by quantum-mechanical perturbation theory. Simultaneous cooling of all three modes of motion is shown to require a spatially nonuniform laser-beam profile, unlike the case of the harmonic trap. Comparison is made with recent experiments. Also, the three-dimensional harmonic trap is treated by two simple methods. One is based on the energy rate equations and the other on Langevin-type equations. Identical results are obtained by the two methods for the steady-state energies. These results are compared with the works of others.
An experimental and theoretical study of laser sideband cooling of a trapped ion in a radiofrequency trap is presented. The influence of the micromotion in the time-dependent potential of a Paul trap can lead to a counterintuitive situation, where sideband cooling is possible for positive values of the laser detuning, i.e., for a laser frequency that is higher than the resonance frequency of the ion at rest. The cooling rate and the capture range are calculated in a semiclassical model. Experimental results of laser sideband cooling at positive detuning are demonstrated with a single trapped In+{\mathrm{In}}^{+}In+ ion stored in a miniature Paul trap.
We propose a cooling scheme for realizing single-photon sideband cooling on particles trapped in a statedependent optical potential.We develop a master rate equation from an ab initio model and find that in experimentally feasible conditions it is possible to reduce the average occupation number of the vibrational levels in one dimension by more than 90% by applying a frequency sweep on the cooling laser that sequentially cools all the motional states.Notably, this cooling scheme works also when a particle experiences a deeper trap in its internal ground state than in its excited state, a condition for which conventional single-photon sideband cooling does not work.In our analysis, we consider two cases: a two-level particle confined in an optical tweezer, and Li atoms confined in an optical lattice, and we find conditions for efficient cooling in both cases.The results from the model are confirmed by a full quantum Monte Carlo simulation of the system Hamiltonian.Our findings provide an alternative cooling scheme that can be applied in principle to any particle, e.g., atoms, molecules, or ions, confined in a state-dependent optical potential.