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Physical Review A

Semiclassical laser-cooling theory for a trapped multistate ion

Juha Javanainen, Sung Mi Yoo

Published Nov 1, 1993• 8 citations
Publisher

Abstract

We present a semiclassical laser-cooling theory for an ion bound in a quadratic potential well, allowing for an arbitrary internal level scheme of the ion and an arbitrary polarization of the light field. The special case that the ion moves in a region much smaller than the wavelength of the exciting light (Lamb-Dicke limit), as well as an additional low-intensity limit, are worked out in detail. Explicit general expressions are given for the damping and diffusion tensors of the center-of-mass motion of the ion, and the light-induced renormalization of the mechanical oscillation frequencies is discussed. The formalism is implemented analytically using a computer program mathematica for an ion with a j=1/2\ensuremath{\rightarrow}j=3/2 transition moving in a one-dimensional optical molasses consisting of two counterpropagating laser beams with perpendicular linear polarizations (lin \ensuremath{\perp} lin). For a weakly bound ion at negligible saturation, the greatest lower limit of excitation energy turns out to be (33/20${)}^{1/2}$-1/2\ensuremath{\approxeq}0.78 vibrational quanta above the zero-point energy.

DOI
10.1103/physreva.48.3776

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We present a semiclassical laser-cooling theory for an ion bound in a quadratic potential well, allowing for an arbitrary internal level scheme of the ion and an arbitrary polarization of the light field. The special case that the ion moves in a region much smaller than the wavelength of the exciting light (Lamb-Dicke limit), as well as an additional low-intensity limit, are worked out in detail. Explicit general expressions are given for the damping and diffusion tensors of the center-of-mass motion of the ion, and the light-induced renormalization of the mechanical oscillation frequencies is discussed. The formalism is implemented analytically using a computer program mathematica for an ion with a j=1/2\ensuremath{\rightarrow}j=3/2 transition moving in a one-dimensional optical molasses consisting of two counterpropagating laser beams with perpendicular linear polarizations (lin \ensuremath{\perp} lin). For a weakly bound ion at negligible saturation, the greatest lower limit of excitation energy turns out to be (33/20)1/2{)}^{1/2})1/2-1/2\ensuremath{\approxeq}0.78 vibrational quanta above the zero-point energy.
We adapt an earlier semiclassical theory of laser cooling of an arbitrary multistate atom [J. Javanainen, Phys. Rev. A 44, 5857 (1991)] to the limit of low light intensity. The formal theory is implemented analytically on a computer using mathematica. Expressions of light-pressure force and diffusion are provided for j1{\mathit{j}}_{1}j1​=1/2\ensuremath{\rightarrow}j2{\mathit{j}}_{2}j2​=3/2 and j1{\mathit{j}}_{1}j1​=1\ensuremath{\rightarrow}j2{\mathit{j}}_{2}j2​=2 atoms in a one-dimensional optical confinement area in which the pair of counterpropagating waves has orthogonal polarizations. In our models the cooling temperature decreases as the level degeneracy increases.
Laser cooling of a single trapped ion with Zeeman substructure below the Doppler limit is considered theoretically. The laser field consists of two counterpropagating beams linearly polarized in different directions, and the internal atomic transition is Jg{\mathit{J}}_{\mathit{g}}Jg​=1/2\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=3/2. The ion is assumed to be localized to spatial dimensions smaller than the optical wavelength (Lamb-Dicke limit) and placed at a specific position with respect to the laser beams. Under the assumption that the rate for optical pumping between the atomic ground states defines the smallest time constant in the system, analytic expressions for the final energy and the cooling rates are derived, with both a semiclassical and a full quantum treatment. The results show that laser cooling of a trapped ion using polarization gradients leads to very low energies. These energies are insensitive to the precise localization of the ion with respect to the lasers, the angle between the direction of the polarizations of the laser beams, and the detuning of the cooling laser.
We discuss semiclassical polarization gradient cooling of a trapped ion with the transition scheme j1{\mathit{j}}_{1}j1​=1/2\ensuremath{\rightarrow}j2{\mathit{j}}_{2}j2​=3/2 in the limits of small vibration amplitude and low light intensity. The minimum excitation energy of the ion reached by laser cooling is about one vibrational quantum above the zero-point energy.
We describe a numerical implementation of our semiclassical laser-cooling theory [J. Javanainen, Phys. Rev. A 44, 5857 (1991)] for an arbitrary multilevel atom and light field. There is satisfactory agreement between experiments and temperatures obtained from Langevin-equation simulations of the motion of an atom subject to light-pressure force and the accompanying diffusion in three-dimensional (3D) optical molasses in which there are three orthogonal pairs of counterpropagating waves with a common linear polarization for the two beams in a pair, but orthogonal polarizations between the pairs. However, the velocity distribution of the atoms is anisotropic and may deviate strongly from a Gaussian, heterodyne spectra may have a sideband, and the atomic density may be a maximum at the maximum of the potential of the dipole forces of the molasses beams. Subsequent velocity-linearized analysis of atomic damping and diffusion shows that the damping tensor may have a negative eigenvalue corresponding to heating in large regions of space, even though the light is red tuned for cooling. It is also found that the position-averaged damping and diffusion tensors make a poor predictor of the temperature obtained in Langevin-equation simulations and real experiments. Based on additional simulations of a simple 1D model, we formulate a hypothesis that connects our findings. The key idea is that an atom trapped in a minimum of the potential of the dipole forces is subject to a weaker-than-average damping, and therefore tends to boil out of the trap.
Laser cooling of trapped multilevel ions is studied numerically. The ions are assumed to be localized to spatial dimensions smaller than the optical wavelength (Lamb-Dicke regime). A master equation for the center-of-mass motion is used to numerically evaluate cooling rates and final temperatures for arbitrary light field configurations. The results show both well-known cooling mechanisms (Doppler cooling, sideband cooling) and effects introduced by the presence of multiple atomic levels. Quantitative results are given for a trapped Ba+{\mathrm{Ba}}^{+}Ba+ ion. The numerical procedure can easily be adapted for all ions used in today's trapping experiments. \textcopyright{} 1996 The American Physical Society.