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Physical Review Letters

Energy bands and Bloch states in 1D laser cooling and their effects on the velocity distribution

M. R. Doery, M. T. Widmer, J. Bellanca, Edgar J. D. Vredenbregt, T. H. Bergeman, H. Metcalf

Published Apr 18, 1994• 20 citations
Publisher

Abstract

We describe calculations that predict and analyze distinctive features in the velocity distribution in 1D laser cooling for light shift potential well depths only a few times the recoil energy. These features can be interpreted in terms of populations of energy bands or even Bloch states in the periodic potential. They occur with a ${\mathrm{\ensuremath{\sigma}}}^{+}$ standing wave, with and without a small B field, and for lin\ensuremath{\perp}lin laser cooling. We have observed these features experimentally in a beam of metastable helium atoms cooled on the ${2}^{3}$${\mathit{S}}_{1}$\ensuremath{\leftrightarrows}${2}^{3}$${\mathit{P}}_{2}$ transition, with velocity resolution \ensuremath{\sim}0.3 recoil.

DOI
10.1103/physrevlett.72.2546

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We describe calculations that predict and analyze distinctive features in the velocity distribution in 1D laser cooling for light shift potential well depths only a few times the recoil energy. These features can be interpreted in terms of populations of energy bands or even Bloch states in the periodic potential. They occur with a \ensuremathσ+{\mathrm{\ensuremath{\sigma}}}^{+}\ensuremathσ+ standing wave, with and without a small B field, and for lin\ensuremath{\perp}lin laser cooling. We have observed these features experimentally in a beam of metastable helium atoms cooled on the {2}^{3}$${\mathit{S}}_{1}\ensuremath{\leftrightarrows}{2}^{3}$${\mathit{P}}_{2} transition, with velocity resolution \ensuremath{\sim}0.3 recoil.
We report computational results for the time evolution of the velocity distribution P(V,t) for two-level and multilevel ``Doppler'' laser cooling. We compare results obtained from the semiclassical (SC) Fokker-Planck equation and from generalized optical Bloch equations applied to density matrices over a basis of products of internal and quantized translational states (QDM). Computer memory requirements are optimized to make large-scale QDM calculations feasible. QDM and SC agree well except for two cases: (a) atoms in the wells of the light-shift potential (with kinetic energy less than the well depth, U0{\mathit{U}}_{0}U0​), and (b) atoms with recoil energy ER{\mathit{E}}_{\mathit{R}}ER​ comparable to or greater than the natural linewidth \ensuremath{\Elzxh}\ensuremath{\Gamma}. Transient dips occur in P(V,t) at V=0 in QDM results due to slow cooling of atoms in the light-shift potential wells. Dips in P(V,t) occur at velocity-tuned-resonance (Doppleron) velocities but disappear over long interaction times as atoms accumulate near points where the force is zero. When ER{\mathit{E}}_{\mathit{R}}ER​\ensuremath{\approxeq}\ensuremath{\Elzxh}\ensuremath{\Gamma}, sharp peaks occur in P(V,T) at V=\ifmmode\pm\else\textpm\fi{}VR{\mathit{V}}_{\mathit{R}}VR​ from velocity-selective population quasitrapping not previously found in a two-level transition. Sharp features in P(V,t) occur also for J\ensuremath{\rightarrow}J+1 transitions with J>0, small U0{\mathit{U}}_{0}U0​/ER{\mathit{E}}_{\mathit{R}}ER​, and sufficiently large detuning, from transitions between individual quantum states in the periodic potential.
Quantum calculations for magnetic field induced laser cooling have shown that cooling is less efficient for atoms near the bottom of the light shift potential wells. Inversion of population among the lowest quantum states is found when the pumping rate is less than the well depth. To see whether this occurs in lin {perpendicular} lin polarization gradient cooling, we are performing quantum density matrix calculations. Here the light shift potential matrix obtained in the weak excitation approximations is not diagonal for J{sub g} {>=} 1. We compute eigenfunctions by Fourier transform methods. The band structure is novel and puzzling. The dispersion relation (E vs. Bloch index {nu}) has 2 branches for either 2 or 3 coupled channels (coupled m{sub F} values). Degeneracies occur at {nu}=0 for 2 channels, and {nu}=1 for 3 channels. We are computing optical excitation and decay rates and the temporal evolution of the density matrix so as to obtain the velocity distribution of the cooled atoms.
We present a theoretical analysis of a new one-dimensional laser-cooling scheme that was recently demonstrated on a beam of metastable 4He atoms. Both internal and translational degrees of freedom are treated quantum mechanically. Unlike semiclassical approaches, such a treatment can be applied to situations in which the atomic coherence length is of the same order of or larger than the laser wavelength, which is the case for atoms cooled below the one-photon recoil energy. We introduce families of states that are closed with respect to absorption and stimulated emission, and we establish the generalized optical Bloch equations that are satisfied by the corresponding matrix elements. The existence of velocity-selective trapping states that are linear combinations of states with different internal and translational quantum numbers is demonstrated, and the mechanism of accumulation of atoms in these trapping states by fluorescence cycles is analyzed. From a numerical solution of the generalized optical Bloch equations, we study in detail how the final atomic-momentum distribution depends on the various physical parameters: interaction time, width of the initial distribution, laser detuning, laser power, and imbalance between the two counterpropagating waves. We show that the final temperature decreases when the interaction time increases, so that there is no fundamental limit to the lowest temperature that can be achieved by such a method. Finally, possible extensions of this method to two-dimensional cooling are presented.
We present the first detailed investigation of the line shape of the momentum distribution of atoms cooled far below the recoil limit. This distribution is deduced from a direct measurement of the atomic spatial correlation function of metastable helium atoms cooled by velocity-selective coherent population trapping. The measured line shape is then compared to the prediction of an analytical model of this cooling based on L\'evy statistics. A very good agreement is found between experiment and theory and fundamental features such as self-similarity and nonergodicity are identified.
A theoretical analysis is given for 1-D laser cooling below the Doppler limit of J = 1/2 ground- state atoms. The laser field consists of a pair of counterpropagating, linearly polarized low- power beams, whose polarization directions differ by an angle θ (0 ≤ θ ≤ π/2). For θ ≪ 1, the effective optical pumping time is shown to increase strongly near the nodes of the standing wave and the cooling force can be much larger than that for θ ~1. Moreover, for θ ≪ 1, it can be shown that the stimulated part of the atomic diffusion is reduced considerably as compared with that for θ ~ 1. As a consequence it is possible to achieve an equilibrium atomic distribution that, for θ ≪ 1, is characterized by a mean kinetic energy lower than that predicted to occur for θ = π/2. The equilibrium velocity distribution is not necessarily Maxwellian and thus the temperature of the atomic ensemble may not be well defined. The achievable kinetic energy is so small that the cooled atoms are trapped in the vicinity of the laser field nodes. An explanation for these effects is given in terms of Sisyphus cooling.
A theoretical analysis is given for one-dimensional laser cooling below the Doppler limit of J=1/2 ground-state atoms. The laser field consists of a pair of counterpropagating, linearly polarized, low-power beams, whose polarization directions differ by an angle \ensuremath{\theta} (0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}\ensuremath{\pi}/2). For \ensuremath{\theta}\ensuremath{\ll}1, the effective optical-pumping time is shown to increase strongly near the nodes of the standing wave, and the cooling force can be much larger than that for \ensuremath{\theta}=\ensuremath{\pi}/2. Moreover, for \ensuremath{\theta}\ensuremath{\ll}1, it can be shown that the stimulated part of the atomic diffusion is reduced considerably as compared with that for \ensuremath{\theta}=\ensuremath{\pi}/2. As a consequence it is possible to achieve an equilibrium atomic distribution that, for \ensuremath{\theta}\ensuremath{\ll}1, is characterized by a mean kinetic energy that is lower than that predicted to occur for \ensuremath{\theta}=\ensuremath{\pi}/2. The equilibrium velocity distribution is not necessarily Maxwellian, and thus the temperature of the atomic ensemble may not be well defined. The achievable kinetic energy is so small that the cooled atoms may be trapped in the vicinity of the laser-field nodes.