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CRC Press eBooks

Dynamics of a Trapped Ion Driven by Stochastic Optical Processes

John A. Hoffnagle, Luis G. Reyna, JORGE R. SOBEHART

Published Oct 8, 2018• 1 citation
Publisher

Abstract

The probability distributions for the ground and excited states of a two-level ion in a harmonic trap are studied. The incoming photons excite the ion which relaxes back to its ground state by either spontaneous or stimulated emission. The chapter argues that the arrival times of the photons are described by a Poisson process. It considers the simplified case of the semiclassical description of a two-level ion trapped in a static, one-dimensional harmonic potential. The solutions of the model describes the dynamics of dilute ion ensembles in terms of a coupled system of stochastic differential equations, correct to all orders in the momentum transfer between the ion and the radiation field. The numerical solution of the steady state of the both levels and the results of a Monte Carlo simulation of the equation of motion confirm the analytical estimation derived for the ion energy.

DOI
10.1201/9780429503573-26

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The probability distributions for the ground and excited states of a two-level ion in a harmonic trap are studied. The incoming photons excite the ion which relaxes back to its ground state by either spontaneous or stimulated emission. The chapter argues that the arrival times of the photons are described by a Poisson process. It considers the simplified case of the semiclassical description of a two-level ion trapped in a static, one-dimensional harmonic potential. The solutions of the model describes the dynamics of dilute ion ensembles in terms of a coupled system of stochastic differential equations, correct to all orders in the momentum transfer between the ion and the radiation field. The numerical solution of the steady state of the both levels and the results of a Monte Carlo simulation of the equation of motion confirm the analytical estimation derived for the ion energy.
The probability distributions of dilute ion systems and two-ion systems trapped in time-dependent potential wells and driven by a radiation field are studied. Ions are excited by electromagnetic radiation and relax back due to either spontaneous or stimulated emissions. The photon statistics are considered Poissonian and the momentum transfer between the electromagnetic field, and the ions are assumed discrete. In the present classical treatment, the photon–ion interaction is simulated, with an effective noise acting on the ions, which produces a stochastic transfer of momentum. The present results are closely related to the quantum treatment in the heavy particle limit as well as to those derived from previous semiclassical models.
The probability distribution of a two-level ion confined in a time-dependent nonlinear trapping field and driven by a radiation field is calculated using an alternative treatment based on a singular asymptotic expansion. It is assumed that the ion is excited by electromagnetic radiation and relaxes back irreversibly by spontaneous emission. The present calculations reproduce the quantum results of the laser cooling of ions in static traps in the heavy-particle limit, and previous classical treatments for the confinement and cooling of ions in radio-frequency traps.
We finalize the dynamics of confined ions driven by a quantized radiation field. The ions can absorb photons from an incident laser beam and relax back to the ground state by either induced emissions or spontaneous emissions. Here we assume that the absorption of photons is immediately followed by spontaneous emissions, resulting in single-level ions perturbed by the exchange of momentum with the radiation field. The probability distribution of the ions is calculated using singular expansions in the low noise asymptotic limit. The present calculations reproduce the quantum results in the limit of heavy particles in static traps, and the classical results of ions in radio-frequency confining wells.
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.
Based on the Langevin equation, a stochastic formulation is implemented to describe the dynamics of a trapped ion in a bath of ultracold atoms, including an excess of micromotion. The ion dynamics is described following a hybrid analytical-numerical approach in which the ion is treated as a classical impurity in a thermal bath. As a result, the ion energy's time evolution and distribution are derived from studying the sympathetic cooling process. Furthermore, the ion dynamics under different stochastic noise terms is also considered to gain information on the bath properties' role in the system's energy transfer processes. Finally, the results obtained from this formulation are contrasted with those obtained with a more traditional Monte Carlo approach.
Laser (light-pressure) cooling of two-level ions stored in a Penning trap is treated theoretically, in the limit that the frequencies of motion of the ions are much smaller than the natural linewidth of the optical transition. Rate equations for the mean-squared amplitudes of motion are derived from a semiclassical analysis, which is confirmed by quantum-mechanical perturbation theory. Simultaneous cooling of all three modes of motion is shown to require a spatially nonuniform laser-beam profile, unlike the case of the harmonic trap. Comparison is made with recent experiments. Also, the three-dimensional harmonic trap is treated by two simple methods. One is based on the energy rate equations and the other on Langevin-type equations. Identical results are obtained by the two methods for the steady-state energies. These results are compared with the works of others.