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arXiv:1706.04423 · Journal of Modern Optics

The effect of atomic response time in the theory of Doppler cooling of trapped ions

H. Janacek, Andrew M. Steane, D. M. Lucas, Derek N. Stacey

Published Feb 9, 2018• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1080/09500340.2018.1428771

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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.
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.
This thesis describes work on numerical modelling of the 43Ca+ ion in a Paul trap using the optical Bloch equations. This is a challenging system to study, with many states involved in the internal dynamics. A major outcome is the development of a cooling scheme for the 146.09 gauss atomic clock transitions which makes use of a dark resonance. It is much more effective than methods that avoid coherent effects. The scheme is realised in experiment. Complicated fluorescence data is modelled very well, and predictions for the ion's motional temperature show good agreement with measured values. Data and fits for an ion that has been Doppler cooled below the Doppler limit are presented. I describe GLOBES, a set of routines that simulates an arbitrary ion in the presence of an arbitrary system of laser beams using the optical Bloch equations. Techniques used to efficiently calculate the steady state, analyse fluorescence scans and solve time-dependent problems for small and large times are discussed. A new routine SILVER IMPER that leapfrogs over the initial dynamics to model the approach to the steady state is introduced. Doppler cooling in 40Ca+ is analysed and two extensions made to the basic theory. The ‘excursion method’ of calculation takes account of the non-linear variation with velocity of the scattering rate. The ‘dynamic method’ allows for the fact that the ion may not be in equilibrium with the incident radiation during its oscillations, a necessity as the timescale of the external motion is of order the characteristic timescale of the internal motion for standard secular frequencies. This 'dynamic effect' is a general property of trapped ion systems and is also observed in a two-state system. A two-variable fluorescence scan taken from a four-laser, five-level system in 40Ca+ is analysed. Techniques to fit large data sets and automatically resolve resonant features are demonstrated. A general treatment of resonant behaviour in three, four and five level pump/probe systems is used to describe the data. This is verified by a second, tailor-made set of scans.
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.
We investigate the process of pulsed laser cooling using a self-constructed molecular dynamics simulation (MD-Simulation) program. We simulate the Doppler cooling process and pulsed laser Doppler cooling process of a single 40Ca+ ion, and the comparison with the experimental results shows that this self-constructed MD-Simulation program works well in the weak laser intensity situation. Furthermore, we analyze the pulsed laser Doppler cooling process of a single 27Al+ ion. This program can be used to analyze the molecular dynamic process of various situations of Doppler cooling in an ion trap, which could give predictions and experimental guidance.
We investigate the temporal dynamics of Doppler cooling of an initially hot single trapped atom in the weak-binding regime using a semiclassical approach. We develop an analytical model for the simplest case of a single vibrational mode for a harmonic trap, and show how this model allows us to estimate the initial energy of the trapped particle by observing the time-dependent fluorescence during the cooling process. The experimental implementation of this temperature measurement provides a way to measure atom heating rates by observing the temperature rise in the absence of cooling. This method is technically relatively simple compared to conventional sideband detection methods, and the two methods are in reasonable agreement. We also discuss the effects of rf micromotion, relevant for a trapped atomic ion, and the effect of coupling between the vibrational modes on the cooling dynamics.
We investigate the dynamics of an ion sympathetically cooled by another laser-cooled ion or small ion crystal. To this end, we develop simple models of the cooling dynamics in the limit of weak Coulomb interactions. Experimentally, we create a two-ion crystal of Ca + and Al + by photo-ionization of neutral atoms produced by laser ablation. We characterize the velocity distribution of the laser-ablated atoms crossing the trap by time-resolved fluorescence spectroscopy. We observe neutral atom velocities much higher than the ones of thermally heated samples and find as a consequence long sympathethic cooling times before crystallization occurs. Our key result is a new technique for detecting the loading of an initially hot ion with energy in the eV range by monitoring the motional state of a Doppler-cooled ion already present in the trap. This technique not only detects the ion but also provides information about the dynamics of the sympathetic cooling process.