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Journal of Physics B Atomic and Molecular Physics

Light-induced motion of trapped ions. II. Arbitrary intensity

Juha Javanainen

Published Nov 14, 1981• 17 citations
Publisher

Abstract

For pt.I see ibid., vol.14, no.14, p.2519 (1981). The authors extend their earlier theory for the motion of a trapped ion in a travelling wave of light from the low-intensity in detail. A numerical procedure to give the rate of change of the mechanical energy of the ion due to the light pressure force is presented, and a general analytic expression for the steady-state energy is derived. Exact solutions are compared with existing approximations.

DOI
10.1088/0022-3700/14/21/029

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For pt.I see ibid., vol.14, no.14, p.2519 (1981). The authors extend their earlier theory for the motion of a trapped ion in a travelling wave of light from the low-intensity in detail. A numerical procedure to give the rate of change of the mechanical energy of the ion due to the light pressure force is presented, and a general analytic expression for the steady-state energy is derived. Exact solutions are compared with existing approximations.
For pt.II see ibid., vol.14, p.4191 (1981). The author derives a Fokker-Planck equation for the Wigner function describing the centre-of-mass motion of a harmonically trapped ion in a travelling wave of light, using an expansion in the recoil velocity around the exact time-dependent solution of the recoilless case. The practical calculations are reduced to the scheme presented in the earlier papers of this series. However, the final energy of the cooling differs from his earlier results, and agrees with more recent calculations employing for the centre of mass the representation in the harmonic oscillator states.
Derives a Fokker-Planck equation for the Wigner function describing the centre-of-mass motion of a trapped ion in a monochromatic travelling wave of light. The driving term and the coefficients of diffusion are found by solving certain sets of ordinary differential equations. In the low-intensity limit, the author investigates the rate of change of the mechanical energy of the ion and analyse a linearised Fokker-Planck equation valid in the vicinity of the steady state. The quantum mechanical properties of the harmonic oscillator implicit in the equations for the Wigner functions are discussed.
The mechanical action of light on atoms is nowadays a tool used ubiquitously in cold atom physics. In the semiclassical regime where the atomic motion is treated classically, the computation of the mean force acting on a two-level atom requires in the most general case numerical approaches. Here we show that this problem can be tackled in a pure analytical way. We provide an analytical yet simple expression of the mean force that holds in the most general case where the atom is simultaneously exposed to an arbitrary number of lasers with arbitrary intensities, wave vectors, and phases. This yields a novel tool for engineering the mechanical action of light on single atoms.
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.