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arXiv:1902.00302

Determining temperature and Rabi frequency regarding trapped ions in Doppler cooling: An analytic investigation

L.-L. Yan, Shi‐Lei Su, Feng, Mang

Published Feb 1, 2019• 0 citations
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Abstract

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 $^{40}$Ca$^{+}$, 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.

DOI
10.48550/arxiv.1902.00302

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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.
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 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.
The optimization of the parameters of laser cooling and the investigation of the heating rate in ion traps require the measurement of the temperature of ion chains for which the Lamb–Dicke regime is satisfied. A novel method based on the investigation of the dynamics of Rabi oscillations at a narrow electron transition in an individual ion of the chain has been suggested for such measurement. An analytical expression for the population of the upper state as a function of the excitation time is derived taking into account the thermal distribution of phonons over the vibrational modes of a chain with an arbitrary number of ions. The method is tested experimentally for the chain of five 171Yb+^{{171}}{\text{Y}}{{{\text{b}}}^{ + }}171Yb+ ions using the quadrupole transition at 435 nm, as well as for a single ion. The heating rate measured for the axial vibrational mode in the implemented trap is 8 × 10 3 phonons/s.
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.
We demonstrate sympathetic Doppler cooling and our progress towards sideband cooling of a CaH+^{+}+ ion co-trapped with a Ca+^{+}+ atomic ion in a linear Paul trap. Molecular ions are generally difficult to laser cool due to a lack of closed electronic transitions as a result of vibrational and rotational states. Despite this challenge, they can be cooled indirectly through their Coulombic interaction with a fluorescent atomic ion that is being directly laser cooled. Ions are firstly Doppler cooled to get to the Lamb-Dicke regime, where the ion motion is small relative to the excitation wavelength and then sideband cooled reaching temperatures below 1 μ\muμK. All the ions' axial modes (center of mass and breathing mode) and radial modes ( two center of mass and two tilt modes) are addressed, and the temperature is determined by examining the ratio of sidebands.
We present a technique for measuring ion temperature using cavity-based electromagnetically induced transparency (EIT) applicable for cavity QED systems. This method enables efficient extraction of the ion's phonon occupation number following sub-Doppler cooling close to the motional ground state. The proposed method requires operation in the resolved-sideband regime, where individual motional states can be selectively addressed for all relevant transitions either by selecting appropriate energy levels for the three-level system or by employing strong confinement with high secular frequencies (∼10MHz\sim 10 MHz∼10MHz). It relies on monitoring the cavity probe transmission while scanning the probe laser frequency to establish cavity-induced EIT using a control beam, thereby significantly simplifying the measurement procedure. We establish a theoretical model that demonstrates the influence of the thermal state of the trapped ion vis-à-vis the EIT linewidth measured. We show through numerical simulations how the cavity-induced EIT transmission may be used as a thermometry tool to deduce the ion temperature as well as its motional state in the sub-Doppler cooling regime, even for systems that are in the weak coupling regime.
We have determined the temperature and heating rate of laser-cooled ions in a Penning trap using Doppler laser spectroscopy. Between and ions are trapped in a Penning trap and Doppler laser cooled to temperatures of a few millikelvin, where they form ion crystals. This system is an example of a strongly coupled one-component plasma. The ion temperature was measured as a function of time after turning off the laser-cooling. In the solid phase, we measured a heating rate of . Information about possible heating mechanisms was obtained directly from temperature measurements, and also from measurements of the rate of radial expansion of the ion plasma. We determined that the observed heating is due to collisions with the residual gas of our vacuum system.