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

## TL;DR

This paper provides an analytical study of Doppler cooling in trapped ions, offering methods to determine temperature and Rabi frequency, and analyzing different cooling regimes to aid experimental understanding.

## Contribution

It introduces an analytical framework for Doppler cooling, including a practical method to measure Rabi frequency and insights into temperature behavior under various thermal bath conditions.

## Key findings

- Identified four regions of cooling and heating due to electromagnetically induced transparency.
- Proposed a practical method for measuring Rabi frequency via the Doppler cooling window.
- Derived analytical expressions for the final temperature in weak thermal baths.

## 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.