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arXiv:quant-ph/0005097 · Journal of Physics B Atomic Molecular and Optical Physics

Cooling of a small sample of Bose atoms with accidental degeneracy

Maciej Lewenstein, J. I. Cirac, Luis Santos

Published Sep 15, 2000• 1 citation
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Abstract

A system of bosons in a harmonic trap is cooled via their interactions with a thermal reservoir. We derive the master equation that governs the evolution of the system and may describe diverse physical situations: laser cooling, sympathetic cooling, etc. We investigate in detail the dynamics of the gas in the Lamb-Dicke limit, whereby the size of the trap is small in comparison with the de Broglie wavelength of the reservoir quanta. In this case, the dynamics is characterized by two time scales. First, an equilibrium is reached on a fast time scale within the degenerated subspaces of the system. Then, an equilibrium between these subspaces is reached on a slow time scale .

DOI
10.1088/0953-4075/33/19/321

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A system of bosons in a harmonic trap is cooled via their interactions with a thermal reservoir. We derive the master equation that governs the evolution of the system and may describe diverse physical situations: laser cooling, sympathetic cooling, etc. We investigate in detail the dynamics of the gas in the Lamb-Dicke limit, whereby the size of the trap is small in comparison with the de Broglie wavelength of the reservoir quanta. In this case, the dynamics is characterized by two time scales. First, an equilibrium is reached on a fast time scale within the degenerated subspaces of the system. Then, an equilibrium between these subspaces is reached on a slow time scale .
The authors consider the laser cooling of a trapped two-level system during its final approach to equilibrium. Then it moves only within one optical wavelength and an expansion in the Lamb-Dicke parameter is possible. This allows an adiabatic elimination of the internal degrees of freedom. There remains a slow time evolution on a new time scale related to the Lamb-Dicke parameter. This scale is determined by an effective time evolution operator, which the authors derive using the method of degenerate perturbation theory. The ensuing master equation can be completely solved both for its time evolution and the ultimate steady state. In addition to providing a complete description of the final stages of the laser cooling, the calculation can be seen to add one more soluble case to the discussion of non-equilibrium statistical mechanics.
A model for cooling a system of bosons in a harmonic trap via their interactions with a thermal bath of other particles is studied. The master equation describing the evolution of the system is derived for an arbitrary number of spatial dimensions. This equation is characterized by transition rates between trap levels. We present an analytic approximation for these rates and compare it with exact formulas, derived for the case of an even number of spatial dimensions. Analytic expressions show very good agreement with the exact ones for a wide range of parameters. We also discuss the cooling dynamics in terms of the approximated rates.
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.
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.
We study the sympathetic cooling of a trapped Fermi gas interacting with an ideal Bose gas below the critical temperature of the Bose-Einstein condensation. We derive the quantum master equation, which describes the dynamics of the fermionic component, and postulating the thermal distribution for both gases we calculate analytically the rate at which fermions are cooled by the bosonic atoms. The particle losses constitute an important source of heating of the degenerate Fermi gas. We evaluate the rate of loss-induced heating and derive analytical results for the final temperature of fermions, which is limited in the presence of particle losses.