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Physical Review A

Analytical investigation of one-dimensional Doppler cooling of trapped ions with Λ -type configuration

L.-L. Yan, Shi‐Lei Su, Mang Feng

Published Sep 23, 2019• 9 citations
Publisher

Abstract

Doppler cooling with lasers is essential to ion trapping and is also a preliminary step toward achieving ultracold ions. It requires substantial technological effort to experimentally monitor the ion temperature in the cooling process. Based on the simplified three-level $\mathrm{\ensuremath{\Lambda}}$ configuration, we analytically explore an one-dimensional Doppler cooling process of trapped ions, exemplified by ${}^{40}{\mathrm{Ca}}^{+}$, by solving the friction coefficient in Doppler cooling with respect to a thermal bath; in particular, for a bath with large heating rate. We propose a practical method for measuring the Rabi frequency by the Doppler cooling window, which helps for monitoring the final temperature of the Doppler-cooled ions. With our approach, the obtained results are analytical in the case of a weak thermal bath, whereas for the strong thermal bath numerical treatment is partially needed due to the involvement of the Doppler shift and a large Lamb-Dicke parameter. Our analytic treatments are applied to understand many experimental observations, such as configuration phase transition, phonon lasers, and thermodynamics regarding hot trapped ions, and could help for further our understanding three-dimensional Doppler cooling in realistic ion-trap experiments.

DOI
10.1103/physreva.100.033418

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Doppler cooling with lasers is essential to ion trapping and is also a preliminary step toward achieving ultracold ions. It requires substantial technological effort to experimentally monitor the ion temperature in the cooling process. Based on the simplified three-level \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ configuration, we analytically explore an one-dimensional Doppler cooling process of trapped ions, exemplified by 40Ca+{}^{40}{\mathrm{Ca}}^{+}40Ca+, by solving the friction coefficient in Doppler cooling with respect to a thermal bath; in particular, for a bath with large heating rate. We propose a practical method for measuring the Rabi frequency by the Doppler cooling window, which helps for monitoring the final temperature of the Doppler-cooled ions. With our approach, the obtained results are analytical in the case of a weak thermal bath, whereas for the strong thermal bath numerical treatment is partially needed due to the involvement of the Doppler shift and a large Lamb-Dicke parameter. Our analytic treatments are applied to understand many experimental observations, such as configuration phase transition, phonon lasers, and thermodynamics regarding hot trapped ions, and could help for further our understanding three-dimensional Doppler cooling in realistic ion-trap experiments.
Doppler cooling with lasers is essential to ions' trapping and also a preliminary step towards achievement of ultracold ions. Due to lack of effective tools, experimentally monitoring the ions' temperature and the laser-ion coupling is difficult in Doppler cooling. Here we analytically explore the Doppler cooling process of trapped ions, exemplified by 40^{40}40Ca+^{+}+, by solving the friction coefficient in the Doppler cooling with respect to a thermal bath, particularly, to a bath with large heating rate. We show four regions for cooling and heating induced by the three-level electromagnetically induced transparency and propose a practical method for measuring the Rabi frequency by the Doppler cooling window. In addition, the final temperature of the laser-cooled ions can be obtained analytically in the case of a weak thermal bath, whereas for the strong thermal bath this requires numerical treatment due to involvement of the Doppler shift and large Lamb-Dicke parameter. Our analytic results would help for understanding many experimental observations, such as configuration phase transition, phonon laser and thermodynamics regarding hot trapped ions.
Efficient Three-Dimensional Sub-Doppler Cooling of 40^{40}40Ca+^++ in a Penning Trap
We demonstrate efficient sub-Doppler laser cooling of the three eigenmodes of a 40^{40}40Ca+^++ ion confined in a compact Penning trap operating with a magnetic field of 0.91 T. Using the same set of laser beams as required for the initial Doppler laser cooling operation, we detune the laser frequencies to produce a narrow two-photon dark resonance. The process achieves a 1/e cooling time constant of 108(8) μμμs, ultimately reducing the mean thermal axial mode occupation from 72(23) to 1.5(3) in 800 μμμs as measured by resonantly probing an electric quadrupole transition near 729 nm. A parametric drive is applied to the trap electrodes which coherently exchanges the axial mode occupation with that of each radial mode, allowing for three-dimensional sub-Doppler cooling using only the axially-propagating laser beams. This sub-Doppler cooling is achieved for an axial oscillation frequency of $ω_z = 2π~\times~$221 kHz, which places the motion well outside of the Lamb Dicke confinement regime at the Doppler laser cooling limit. Our measured cooling rate and final mode occupation are in good agreement with a semiclassical model which combines a Lindblad master equation solution for ion-photon interactions with classical harmonic oscillator motion of the trapped ion.
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.
Laser cooling of three-level ions of the cascade, vee, and lambda configuration is considered with simultaneous excitation on a weak and strong transition. Choosing appropriate parameters for the Rabi frequencies and the respective detunings allows one to eliminate adiabatically one of the levels. The thus designed two-level ion can acquire decay rates \ensuremath{\Gamma}' smaller than the trap frequency \ensuremath{\nu} such that sideband cooling becomes possible and accordingly allows cooling of the ion to its lowest oscillator state 〈n〉=0 in the trap. Explicit expressions for the detunings and the Rabi frequencies for sideband cooling of the effective two-level system are derived.
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.
We describe a simple approach to the problem of incorporating the response time of an atom or ion being Doppler-cooled into the theory of the cooling process. The system being cooled does not in general respond instantly to the changing laser frequencies it experiences in its rest frame, and this ‘dynamic effect’ can affect significantly the temperatures attainable. It is particularly important for trapped ions when there is a slow decay out of the cooling cycle requiring the use of a repumping beam. We treat the cases of trapped ions with two and three internal states, then apply the theory to 40Ca+. For this ion experimental data exist showing the ion to be cold under conditions for which heating is predicted if the dynamic effect is neglected. The present theory accounts for the observed behaviour.