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Journal of Physics B Atomic and Molecular Physics

Light-induced motion of trapped ions. III. Expansion around the recoilless solution

Juha Javanainen

Published Apr 28, 1985• 18 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/18/8/013

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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.
We derive a Fokker-Planck equation (FPE) for the center-of-mass Wigner function of an atom with an arbitrary level structure, moving in a given three-dimensional light field. This FPE is obtained by eliminating adiabatically the internal degrees of freedom from the expansion of the equation of motion of the atomic density operator into a power series of photon momentum. The force and the diffusion tensor in the FPE may be computed by solving a set of differential equations closely related to the time-evolution equations of the density matrix for the internal state of the atom, as encountered in the conventional recoilless spectroscopy. By studying spontaneous, induced, and free-evolution terms in the equations of motion of the density operator for various level configurations, we devise algorithms that enable a computer to build all the required equations automatically for an arbitrary atomic level scheme.
We derive an equation of motion for the center-of-mass Wigner function (WF) of a two-level atom interacting with a linearly polarized, monochromatic standing plane wave of light. The derivation is based on a truncation of the spatial Fourier expansion of the WF and is consequently valid only at low intensities of the driving light, but no assumption regarding the size of the recoil velocity need be made. The stationary state of laser cooling is studied analytically and numerically. The asymptotic velocity distribution at large velocities is found to obey a power law, with a power independent of the linewidth of the optical transition.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
We study theoretically Doppler laser cooling of a cluster of two-level atoms confined in a linear ion trap. Using several consecutive steps of averaging we derive, from the full quantum-mechanical master equation, an equation for the total mechanical energy of a one-dimensional crystal, defined on a coarse-grained energy scale whose grid size is smaller than the linewidth of the electronic transition. This equation describes the cooling dynamics for an arbitrary number of ions in the quantum regime. We discuss the validity of the ergodic assumption (i.e., that the phase-space distribution is only a function of energy). From our equation we derive the semiclassical limit (i.e., when the mechanical motion can be treated classically) and the Lamb-Dicke limit (i.e., when the size of the mechanical wave function is much smaller than the laser wavelength). We find a Fokker-Planck equation for the total mechanical energy of the system, whose solution is in agreement with previous analytical calculations that were based on different assumptions and valid only in their specific regimes. Finally, in the classical limit we derive an analytic expression for the average coupling, by light scattering, between motional states at different energies.