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arXiv:quant-ph/0105150 · Physical Review A

Doppler cooling of a Coulomb crystal

Giovanna Morigi, Jürgen Eschner

Published Nov 16, 2001• 26 citations
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Abstract

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.

DOI
10.1103/physreva.64.063407

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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.
We discuss laser cooling of a string of two-level atoms confined in a linear ion trap. The equations for the dynamics are first analysed in the Lamb–Dicke regime (i.e. when the size of the mechanical wavefunction is much smaller than the laser wavelength), and then in general, for Doppler cooling. In the latter case, using several consecutive steps of averaging we derive from the full quantum mechanical master equation an equation for the total mechanical energy of the 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 analogy and the scalability of the string's laser-cooling dynamics with that of a single trapped ion.
We present a master equation that describes the dynamics of laser cooling of a trapped ion. It is valid in the Lamb–Dicke limit and rests on an adiabatic elimination combined with a degenerate perturbation treatment. It describes relaxation of probabilities and coherences in the harmonic-trap degrees of freedom. The eigenvalue spectrum the of the time-evolution operator is derived, and it follows that only one zero eigenvalue exists, giving the unique steady-state probability distribution. The coherences all decay to zero with time. The ultimate steady state is distribution, which can be characterized by a temperature. We also report a numerical calculation that a Planck supports our analytical work. The final energy of the cooling is given and discussed. Finally there is a comparison between the present results and our earlier, approximate, treatments.
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.
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.
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.
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.