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Journal of Physics B Atomic and Molecular Physics

Steady state of a laser-cooled trapped ion in the Lamb-Dicke limit

Markus Lindberg

Published May 28, 1984• 20 citations
Publisher

Abstract

The author formulates the problem of the steady state for a laser-cooled trapped ion using a generalisation of the method of moments. The problem is solved using a perturbation expansion and he obtains a steady-state solution which is exact in the Lamb-Dicke limit. The stead-state energy is compared with earlier results. The populations of the occupation number eigenstates of the harmonic trap are also obtained.

DOI
10.1088/0022-3700/17/10/016

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Physical Review A•Jan 1, 1994•77 citations
The author formulates the problem of the steady state for a laser-cooled trapped ion using a generalisation of the method of moments. The problem is solved using a perturbation expansion and he obtains a steady-state solution which is exact in the Lamb-Dicke limit. The stead-state energy is compared with earlier results. The populations of the occupation number eigenstates of the harmonic trap are also obtained.
We present a master equation that describes the dynamics of laser cooling of a trapped ion. It is valid in the Lamb–Dicke limit and rests on an adiabatic elimination combined with a degenerate perturbation treatment. It describes relaxation of probabilities and coherences in the harmonic-trap degrees of freedom. The eigenvalue spectrum the of the time-evolution operator is derived, and it follows that only one zero eigenvalue exists, giving the unique steady-state probability distribution. The coherences all decay to zero with time. The ultimate steady state is distribution, which can be characterized by a temperature. We also report a numerical calculation that a Planck supports our analytical work. The final energy of the cooling is given and discussed. Finally there is a comparison between the present results and our earlier, approximate, treatments.
The resonance fluorescence spectrum of a laser-cooled trapped ion is theoretically studied in the Lamb-Dicke limit. Angular averaged spectrum around the elastic peak and the motional sidebands is calculated. The widths of the sidebands are shown to be given by the cooling rate of the ion in the trap. In the limit when the trapping force is strong a possibility of dispersive line shapes is predicted. The spectrum derived is also shown to reproduce in the low-intensity limit the earlier perturbative results.
This paper reviews our current understanding of the dynamics of a laser-cooled trapped particle in the Lamb–Dicke regime, where the quantum structure of the energy levels cannot be ignored. The derivation and validity of a master equation are surveyed, and its physical interpretation is discussed in some detail. The structure and physical nature of the ultimate steady state are discussed. Using a generating function method, we can solve for both the complete eigenvalue spectrum and the general time-dependent solution of the master equation. These results are derived and interpreted physically. They have earlier been scattered in our various publications and are presented here in a coherent way for the first time. Also included are some new results and a physical discussion of the situation. The paper concludes with a discussion of the validity and limitations of the model as treated so far.
Trapping ions outside the Lamb–Dicke limit have been proven to be useful for the laser-cooling and quantum computing. Under the supposition of the Rabi frequency much smaller than the Lamb–Dicke parameter, we can use a simple method to analytically solve the system with a single cold ion trapped and radiated beyond the Lamb– Dicke limit, in the absence of the rotating-wave approximation (RWA). Discussion has been made for the limitation of our approach and the comparison of our results with the solutions under the RWA.
Trapping ions outside the Lamb-Dicke limit have been proven to be useful for the laser-cooling and quantum computing.Under the supposition of the Rabi frequency much smaller than the Lamb Dicke parameter,we can use a simple method to analytically solve the system with a single cold ion trapped and radiated beyond the Lamb Dickc limit,in the absence of the rotating-wave approximation (RWA).Discussion has been made for the limitation of our approach and the comparison of our results with the solutions under the RWA.
Laser cooling of a single trapped ion in a Paul trap is discussed theoretically in the Lamb-Dicke limit, with full consideration of the time dependence of the trapping potential. Resulting mean kinetic energies are defined as time averages over one period of the micromotion and are compared with final temperatures expected from the laser-cooling treatment with harmonic traps. For laser-atom detunings close to the micromotion frequency the results differ significantly from those expected for a harmonic trap potential. A physical interpretation is given and simple formulas are derived for the strong confinement case.