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Physical Review A

Laser cooling of trapped ions in a squeezed vacuum

J. I. Cirac, P. Zoller

Published Mar 1, 1993• 17 citations
Publisher

Abstract

Laser cooling a trapped ion damped by an electromagnetic reservoir in a squeezed-vacuum state is investigated. The cooling rate and the final temperature are given for the case in which the ion is located at the node of a laser standing wave, on resonance with its internal transition. In particular, we find that the ion can have final temperatures below the Dopper limit.

DOI
10.1103/physreva.47.2191

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Laser cooling a trapped ion damped by an electromagnetic reservoir in a squeezed-vacuum state is investigated. The cooling rate and the final temperature are given for the case in which the ion is located at the node of a laser standing wave, on resonance with its internal transition. In particular, we find that the ion can have final temperatures below the Dopper limit.
Laser cooling of trapped ions in a standing- and running-wave configuration is discussed theoretically. The ions are assumed to be spatially localized on the scale provided by the wavelength of the laser (Lamb-Dicke limit). A master equation for the center-of-mass distribution of the ion is derived for a multilevel system, and explicit results are presented for two- and three-level systems and harmonic trapping potentials. For the two-level system located at the node of the standing wave, we find final temperatures that are a factor of 2 lower than the limit for a running wave and cooling rates that do not saturate with the laser intensity. At the point of maximum gradient of the standing wave, blue detuned cooling is found that is analogous to the Sisyphus cooling of free atoms. For a three-level system we compare our results with those of Wineland, Dalibard, and Cohen-Tannoudji [J. Opt. Soc. Am. B 9, 32 (1992)].
The process of laser cooling in a quasi resonant standing laser wave is one of the principal technics of laser cooling of atoms. Recent advances in this field include breaking both the "Doppler and the recoil limits". On another front modification of the quantum statistical properties of the "vacuum" fluctuations by squeezing has been achieved in the laboratory. It is, therefore, interesting to find what is the nature of the modification of the laser cooling process when the atoms are also embedded in squeezed vacuum. Such a state can be produced in principle by a degenerate parametric amplifier whose input are vacuum fluctuations. It is shown that due to the modified decay rates of an atom which is embedded in such a state, profound modification of the cooling processes in a standing wave is achieved . Among the important results obtained is the change of sign of the stimulated force from a heating force in normal vacuum to a cooling force in squeezed vacuum. This phenomenon can result in cooling forces more than two orders of magnitude larger than the maximum force in normal vacuum. In this talk the method for calculating the force in squeezed vacuum will be given. The modified force will be compared to the force in normal vacuum and finally the practical implications of these results for laser cooling will be discussed.
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.
Limits for the laser cooling of a trapped ion and some experimental observations are summarized. The final temperature is low when a dipole‐forbidden line is two‐photon excited, e.g. by way of an electronic Raman process. Now, the two light modes also support dispersive interaction by stimulated Raman‐like scattering. A simplified model in terms of the vibrational Raman effect identifies processes analogous to those known from atoms in a standing light wave, and it reveals novel parametric interaction of ion vibration and light. In the diabatic limit, when dephasing by spontaneous decay is small, the ion approaches the dynamics of an electron in a free‐electron laser.