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Physical Review A

Quantum calculations for one-dimensional laser cooling: Temporal evolution

T. H. Bergeman

Published Nov 1, 1993• 17 citations
Publisher

Abstract

Temporal integration of density-matrix equations with quantized translational motion gives alternative results for laser cooling of atoms. For polarization gradient cooling, the average kinetic energy in the cold-atom peak after 1000 radiative lifetimes decreases with increasing ${\mathit{F}}_{\mathit{g}}$, and is 1 recoil energy at low laser intensity. With magnetically induced laser cooling (MILC) and low optical pumping rates, atoms cool slowly within the potential wells, giving minima in the velocity distribution at \ensuremath{\nu}=0. MILC is seen as a cyclic process; optical pumping cools and B-field mixing slightly heats.

DOI
10.1103/physreva.48.r3425

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Temporal integration of density-matrix equations with quantized translational motion gives alternative results for laser cooling of atoms. For polarization gradient cooling, the average kinetic energy in the cold-atom peak after 1000 radiative lifetimes decreases with increasing Fg{\mathit{F}}_{\mathit{g}}Fg​, and is 1 recoil energy at low laser intensity. With magnetically induced laser cooling (MILC) and low optical pumping rates, atoms cool slowly within the potential wells, giving minima in the velocity distribution at \ensuremath{\nu}=0. MILC is seen as a cyclic process; optical pumping cools and B-field mixing slightly heats.
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 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 present two cooling mechanisms that lead to temperatures well below the Doppler limit. These mechanisms are based on laser polarization gradients and work at low laser power when the optical-pumping time between different ground-state sublevels becomes long. There is then a large time lag between the internal atomic response and the atomic motion, which leads to a large cooling force. In the simple case of one-dimensional molasses, we identify two types of polarization gradient that occur when the two counterpropagating waves have either orthogonal linear polarizations or orthogonal circular polarizations. In the first case, the light shifts of the ground-state Zeeman sublevels are spatially modulated, and optical pumping among them leads to dipole forces and to a Sisyphus effect analogous to the one that occurs in stimulated molasses. In the second case (σ+−σ− configuration), the cooling mechanism is radically different. Even at very low velocity, atomic motion produces a population difference among ground-state sublevels, which gives rise to unbalanced radiation pressures. From semiclassical optical Bloch equations, we derive for the two cases quantitative expressions for friction coefficients and velocity capture ranges. The friction coefficients are shown in both cases to be independent of the laser power, which produces an equilibrium temperature proportional to the laser power. The lowest achievable temperatures then approach the one-photon recoil energy. We briefly outline a full quantum treatment of such a limit.
We present a calculation of one-dimensional sub-Doppler cooling of neutral atoms, in which the atomic center-of-mass motion is quantized. The cooling field consists of a pair of counterpropagating, linearly polarized laser beams, with their polarizaton vectors at an angle \ensuremath{\theta} with respect to each other. The steady-state atomic density matrix is analyzed as a function of \ensuremath{\theta}. It is shown that, depending on the internal atomic level scheme, the equilibrium population of the quantized ground state of atomic center-of-mass motion can either increase or decrease significantly when \ensuremath{\theta} is reduced from \ensuremath{\pi}/2. These effects are interpreted based on the spatial localization of atoms in the light-induced potential wells.
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.
In sub-Doppler laser cooling of atoms, the lowest temperatures are known to occur at low (but not too low) excitation, and the temperature increases roughly linearly with laser intensity. However, under conditions for which the lowest temperatures are obtained, the small velocity capture range and low optical pumping rate limit the number of atoms that can be collected into the cold sample. In this study, we present measurements for laser cooling of Rb and metastable He atoms for two counterpropagating laser beams with orthogonal linear polarization (lin\ensuremath{\perp}lin) over a wide range of saturation parameters and laser detunings, together with results of semiclassical (Fokker-Planck) and quantum density matrix calculations. We find that at higher laser intensity, a larger number of atoms can be collected into a final velocity distribution that is significantly narrower than that given by a linear extrapolation vs the square root of laser intensity. Two cooling mechanisms are at work: the sub-Doppler Sisyphus mechanism and also Doppler cooling. Under certain conditions that we discuss, Doppler cooling aids Sisyphus cooling by collecting high-velocity atoms beyond the Sisyphus capture range. He∗{\mathrm{He}}^{*}He∗ presents a rather unusual situation in that the minimum average kinetic energy (Eav){(E}_{\mathrm{av}})(Eav​) for purely Sisyphus cooling is comparable to the minimum value of Eav{E}_{\mathrm{av}}Eav​ with Doppler cooling alone.