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Journal of Physics B Atomic and Molecular Physics

Light-induced motion of trapped ions. I. Low-intensity limit

Juha Javanainen

Published Jul 28, 1981• 16 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/14/14/022

Related papers

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

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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.

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In this paper we derive the Fokker-Planck equation that describes the motion of a charged particle, trapped in a static harmonic potential and cooled by radiation pressure forces exerted by a laser beam. This equation is then discussed in both the classical and quantum regimes. The model adopted here is a simple one, in which one-dimensional motion and a two-level atom are assumed, and the time evolution of the system is described in the quasiclassical limit, in which the quantum nature of the photon exchanged between the particle and the field plays no role. We report relevant graphs for the loss of kinetic energy suffered by the particle in the cooling process and estimates of the lowest kinetic temperature that can be reached in these experiments.

Physical Review A•Sep 1, 1989•1 citation
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.
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.
We derive a Fokker-Planck equation for the center-of-mass Wigner function of an atom moving in a Zeeman neutral-atom trap. The Zeeman structure of atomic sublevels is taken into account. Our analysis is limited to optical transitions between two sets of sublevels corresponding to angular momenta jb{\mathit{j}}_{\mathit{b}}jb​=0 and ja{\mathit{j}}_{\mathit{a}}ja​=1. Expressions for the light pressure, friction, and diffusion forces in the case of low intensity of the laser light are presented. An analytical solution for the stationary center-of-mass Wigner function as well as the size and geometry of the trap are found and discussed.
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.
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.
In this paper we derive the Fokker-Planck equation that describes the motion of a charged particle, trapped in a static harmonic potential and cooled by radiation pressure forces exerted by a laser beam. This equation is then discussed in both the classical and quantum regimes. The model adopted here is a simple one, in which one-dimensional motion and a two-level atom are assumed, and the time evolution of the system is described in the quasiclassical limit, in which the quantum nature of the photon exchanged between the particle and the field plays no role. We report relevant graphs for the loss of kinetic energy suffered by the particle in the cooling process and estimates of the lowest kinetic temperature that can be reached in these experiments.