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Physical Review A

Polarization gradient cooling of a trapped ion

Sung Mi Yoo, Juha Javanainen

Published Jul 1, 1993• 14 citations
Publisher

Abstract

We discuss semiclassical polarization gradient cooling of a trapped ion with the transition scheme ${\mathit{j}}_{1}$=1/2\ensuremath{\rightarrow}${\mathit{j}}_{2}$=3/2 in the limits of small vibration amplitude and low light intensity. The minimum excitation energy of the ion reached by laser cooling is about one vibrational quantum above the zero-point energy.

DOI
10.1103/physreva.48.r30

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We discuss semiclassical polarization gradient cooling of a trapped ion with the transition scheme j1{\mathit{j}}_{1}j1​=1/2\ensuremath{\rightarrow}j2{\mathit{j}}_{2}j2​=3/2 in the limits of small vibration amplitude and low light intensity. The minimum excitation energy of the ion reached by laser cooling is about one vibrational quantum above the zero-point energy.
We report the first experimental demonstration of one-dimensional polarization gradient cooling of bound ions. The trapped particles are only weakly bound along the laser cooling axis. Nevertheless, the particle motion has to be treated quantum mechanically. The minimum kinetic energy is found when the depth of the optical potential approaches a quantum of the vibrational energy of the trapped ions. Cooling to low vibrational states of the trap has been observed. We have also produced the lowest one-dimensional temperature for an ion crystal.
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.
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.
Using a laser polarization gradient, we realize 3D Sisyphus cooling of ^{171}Yb^{+} ions confined in and near the Lamb-Dicke regime in a linear Paul trap. The cooling rate and final mean motional energy of a single ion are characterized as a function of laser intensity and compared to semiclassical and quantum simulations. Sisyphus cooling is also applied to a linear string of four ions to obtain a mean energy of 1-3 quanta for all vibrational modes, an approximately order of magnitude reduction below Doppler cooled energies. This is used to enable subsequent, efficient sideband laser cooling.
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.
Trapped ions provide a highly controlled platform for quantum sensors, clocks, simulators, and computers, all of which depend on cooling ions close to their motional ground state. Existing methods like Doppler, resolved sideband, and dark resonance cooling balance trade-offs between the final temperature and cooling rate. A traveling polarization gradient has been shown to cool multiple modes quickly and in parallel, but utilizing a stable polarization gradient can achieve lower ion energies, while also allowing tailorable light-matter interactions in the sub-wavelength regime. In this Letter, we demonstrate cooling of a trapped ion below the Doppler limit using a phase-stable polarization gradient created using trap-integrated photonic devices. At an axial frequency of 2π×1.45 MHz we achieve ⟨n⟩=1.56±0.07 in 150 μs and cooling rates of ∼0.3 quanta/μs. We examine ion dynamics under different polarization gradient phases, detunings, and intensities, showing reasonable agreement between experimental results and a multilevel model. Cooling is fast and power efficient, with lower average motional Fock state occupation when compared to simulated operation under the corresponding running wave configuration. Our results demonstrate a well-controlled test bed for studying the dynamics of multilevel atomic systems in a phase-stable polarization gradient.