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Journal of Physics B Atomic and Molecular Physics

The master equation for laser cooling of trapped particles

Markus Lindberg, Stig Stenholm

Published Aug 28, 1984• 31 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/17/16/019

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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.
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.
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 .
We theoretically investigate the quantum dynamics of the center of mass of trapped atoms, whose internal degrees of freedom are driven in a $\ensuremath{\Lambda}$-shaped configuration with the lasers tuned at two-photon resonance. In the Lamb-Dicke regime, when the motional wave packet is well localized over the laser wavelength, transient coherent population trapping occurs, canceling transitions at the laser frequency. In this limit the motion can be efficiently cooled to the ground state of the trapping potential. We derive an equation for the center-of-mass motion by adiabatically eliminating the internal degrees of freedom. This treatment provides the theoretical background of the scheme presented in [G. Morigi et al., Phys. Rev. Lett. 85, 4458 (2000)] and implemented in [C.F. Roos et al., Phys. Rev. Lett. 85, 5547 (2000)]. We discuss the physical mechanisms determining the dynamics and identify parameters regimes, where cooling is efficient. We discuss implementations of the scheme to cases where the trapping potential is not harmonic.
This paper analyzes the cooling of a single particle in a harmonic trap with red-detuned laser light with fewer approximations than previously done in the literature. We avoid the adiabatic elimination of the excited atomic state but are still interested in Lamb-Dicke parameters $\ensuremath{\eta}\ensuremath{\ll}1$. Our results show that the Rabi frequency of the cooling laser can be chosen higher than previously assumed, thereby increasing the effective cooling rate but not affecting the final outcome of the cooling process. Since laser cooling is already a well-established experimental technique, the main aim of this paper is to present a model which can be extended to more complex scenarios, like cavity-mediated laser cooling.
Laser cooling of trapped ions in a standing- and running-wave configuration is discussed theoretically. The ions are assumed to be spatially localized on the scale provided by the wavelength of the laser (Lamb-Dicke limit). A master equation for the center-of-mass distribution of the ion is derived for a multilevel system, and explicit results are presented for two- and three-level systems and harmonic trapping potentials. For the two-level system located at the node of the standing wave, we find final temperatures that are a factor of 2 lower than the limit for a running wave and cooling rates that do not saturate with the laser intensity. At the point of maximum gradient of the standing wave, blue detuned cooling is found that is analogous to the Sisyphus cooling of free atoms. For a three-level system we compare our results with those of Wineland, Dalibard, and Cohen-Tannoudji [J. Opt. Soc. Am. B 9, 32 (1992)].