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Physical Review A

Laser cooling of atoms with broadband real Gaussian laser fields

A. S. Parkins, P. Zoller

Published May 1, 1992• 16 citations
Publisher

Abstract

We describe a theoretical approach to laser cooling with broadband real Gaussian laser fields that is based on adiabatic elimination methods and the Fokker-Planck equation for the Wigner function. This approach is used to model the recent broadband cooling experiment of Zhu, Oates, and Hall [Phys. Rev. Lett. 67, 46 (1991)] and gives correct order-of-magnitude results. The theory is then applied to J=0\ensuremath{\rightarrow}J=1 and J=1\ensuremath{\rightarrow}J=2 transitions in a ${\mathrm{\ensuremath{\sigma}}}_{+\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\sigma}}}_{\mathrm{\ensuremath{-}}}$ optical molasses configuration with correlated broadband laser fields. An instability in the cooling scheme at zero velocity is found for red laser detunings, and sub-Doppler-width velocity distributions are predicted for J=1\ensuremath{\rightarrow}J=2 transitions with both red and blue laser detunings.

DOI
10.1103/physreva.45.6522

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Sub-Doppler laser cooling of potassium atoms

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arXiv:1107.2337•Physical Review A•Oct 28, 2011•77 citations
We describe a theoretical approach to laser cooling with broadband real Gaussian laser fields that is based on adiabatic elimination methods and the Fokker-Planck equation for the Wigner function. This approach is used to model the recent broadband cooling experiment of Zhu, Oates, and Hall [Phys. Rev. Lett. 67, 46 (1991)] and gives correct order-of-magnitude results. The theory is then applied to J=0\ensuremath{\rightarrow}J=1 and J=1\ensuremath{\rightarrow}J=2 transitions in a {\mathrm{\ensuremath{\sigma}}}_{+\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\sigma}}}_{\mathrm{\ensuremath{-}}} optical molasses configuration with correlated broadband laser fields. An instability in the cooling scheme at zero velocity is found for red laser detunings, and sub-Doppler-width velocity distributions are predicted for J=1\ensuremath{\rightarrow}J=2 transitions with both red and blue laser detunings.
We investigate theoretically laser cooling with broadband ${\mathrm{\ensuremath{\sigma}}}_{+\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\sigma}}}_{\mathrm{\ensuremath{-}}}$ lasers acting on J=0\ensuremath{\rightarrow}J=1 and J=1\ensuremath{\rightarrow}J=2 transitions. For cross-correlated laser fluctuations an instability in the friction force at zero velocity is predicted for red laser detunings. The dependence of this instability on the cross-correlation and spectral line shape is investigated.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
We describe a numerical implementation of our semiclassical laser-cooling theory [J. Javanainen, Phys. Rev. A 44, 5857 (1991)] for an arbitrary multilevel atom and light field. There is satisfactory agreement between experiments and temperatures obtained from Langevin-equation simulations of the motion of an atom subject to light-pressure force and the accompanying diffusion in three-dimensional (3D) optical molasses in which there are three orthogonal pairs of counterpropagating waves with a common linear polarization for the two beams in a pair, but orthogonal polarizations between the pairs. However, the velocity distribution of the atoms is anisotropic and may deviate strongly from a Gaussian, heterodyne spectra may have a sideband, and the atomic density may be a maximum at the maximum of the potential of the dipole forces of the molasses beams. Subsequent velocity-linearized analysis of atomic damping and diffusion shows that the damping tensor may have a negative eigenvalue corresponding to heating in large regions of space, even though the light is red tuned for cooling. It is also found that the position-averaged damping and diffusion tensors make a poor predictor of the temperature obtained in Langevin-equation simulations and real experiments. Based on additional simulations of a simple 1D model, we formulate a hypothesis that connects our findings. The key idea is that an atom trapped in a minimum of the potential of the dipole forces is subject to a weaker-than-average damping, and therefore tends to boil out of the trap.
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.
We investigate the sub-Doppler laser cooling of bosonic potassium isotopes, whose small hyperfine splitting has so far prevented cooling below the Doppler temperature. We find instead that the combination of a dark optical molasses scheme that naturally arises in this kind of system and an adiabatic ramping of the laser parameters allows us to reach sub-Doppler temperatures for small laser detunings. We demonstrate temperatures as low as $25\ifmmode\pm\else\textpm\fi{}3$ $\ensuremath{\mu}$K and $47\ifmmode\pm\else\textpm\fi{}5$ $\ensuremath{\mu}$K in high-density samples of the two isotopes 39{}^{39}39K and 41{}^{41}41K, respectively. Our findings should find application to other atomic systems.