Physical Review A

Cooling of trapped multilevel ions: A numerical analysis

Dirk Rei\S, Albrecht Lindner, R. Blatt

Published Dec 1, 1996• 21 citations

Abstract

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 ${\mathrm{Ba}}^{+}$ ion. The numerical procedure can easily be adapted for all ions used in today's trapping experiments. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.5133

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Physical Review A

Cooling of trapped multilevel ions: A numerical analysis

Dirk Rei\S, Albrecht Lindner, R. Blatt

Published Dec 1, 1996• 21 citations

Publisher

Abstract

{\mathrm{Ba}}^{+}

ion. The numerical procedure can easily be adapted for all ions used in today's trapping experiments. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.5133

Related papers

Laser cooling of trapped ions in a standing wave

J. I. Cirac, R. Blatt, P. Zoller +1 more

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)].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)].

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J. I. Cirac, R. Blatt, A. S. Parkins +1 more

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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 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.

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John P. Bartolotta, Brian Estey, Michael Foss‐Feig +2 more

Laser cooling methods for trapped ions are most commonly studied at low energies, i.e., in the Lamb-Dicke regime. However, ions in experiments are often excited to higher energies for which the Lamb-Dicke approximation breaks down. Here we construct a non-perturbative, semiclassical method for predicting the energy-dependent cooling dynamics of trapped-ion crystals with potentially many internal levels and motional modes beyond the Lamb-Dicke regime. This method allows accurate and efficient modeling of a variety of interesting phenomena, such as the breakdown of EIT cooling at high energies and the simultaneous cooling of multiple high-temperature modes. We compare its predictions both to fully-quantum simulations and to experimental data for a broadband EIT cooling method on a Raman $S$ - $D$ transition in $^{138}$ Ba ${^+}$ . We find the method can accurately predict cooling rates over a wide range of energies relevant to trapped ion experiments. Our method complements fully quantum models by allowing for fast and accurate predictions of laser-cooling dynamics at much higher energy scales.Laser cooling methods for trapped ions are most commonly studied at low energies, i.e., in the Lamb-Dicke regime. However, ions in experiments are often excited to higher energies for which the Lamb-Dicke approximation breaks down. Here we construct a non-perturbative, semiclassical method for predicting the energy-dependent cooling dynamics of trapped-ion crystals with potentially many internal levels and motional modes beyond the Lamb-Dicke regime. This method allows accurate and efficient modeling of a variety of interesting phenomena, such as the breakdown of EIT cooling at high energies and the simultaneous cooling of multiple high-temperature modes. We compare its predictions both to fully-quantum simulations and to experimental data for a broadband EIT cooling method on a Raman $S$-$D$ transition in $^{138}$Ba${^+}$. We find the method can accurately predict cooling rates over a wide range of energies relevant to trapped ion experiments. Our method complements fully quantum models by allowing for fast and accurate predictions of laser-cooling dynamics at much higher energy scales.

arXiv:2411.18818Nov 27, 20240 citations

{\mathrm{Ba}}^{+}

ion. The numerical procedure can easily be adapted for all ions used in today's trapping experiments. \textcopyright{} 1996 The American Physical Society.

{}^{171}\rm{Yb}^+

ions, we numerically identify optimal parameters for one- and two-ion crystals, and investigate the cooling rate in ion crystals with up to three ions. Our cooling scheme utilizes a single ion addressed with a tweezer to achieve sub-Doppler cooling of the

N

axial motional modes perpendicular to the tweezer propagation direction, does not rely on the Lamb-Dicke regime and uses off-resonant light. We reach

1/e

cooling times for the center of mass mode of

τ_{\rm com}<2

ms, and a steady-state excitation number of

\bar{n}_{\rm{com}}\approx 0.7

. Our scheme is well-suited as an intermediate stage between Doppler cooling and ground-state sideband cooling, and can be readily realized in trapped-ion platforms.

{\mathit{J}}_{\mathit{g}}

=1/2\ensuremath{\rightarrow}

{\mathit{J}}_{\mathit{e}}

=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.

S

D

transition in

^{138}

{^+}

. We find the method can accurately predict cooling rates over a wide range of energies relevant to trapped ion experiments. Our method complements fully quantum models by allowing for fast and accurate predictions of laser-cooling dynamics at much higher energy scales.