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Journal of Physics B Atomic Molecular and Optical Physics

Laser cooling by sigma+- sigma-circularly polarized beams of unequal intensities

J Werner, Hartmut Wallis

Published Sep 28, 1993• 16 citations
Publisher

Abstract

The force on a multilevel atom in a 1D magneto-optical molasses with imbalanced laser intensities is theoretically investigated. The authors employ a full quantum treatment to obtain the stationary momentum distribution of the atoms, for arbitrary parameters. The force exerted the counterpropagating laser beams balances for a nonzero locking velocity. The dependence of this velocity on the intensity imbalance is found to be reduced by polarization gradient cooling. For certain values of the imbalance there are two stable velocities, and a splitting of the velocity distribution occurs. A small imbalance, or a magnetic field along the laser beams, can cause the subDoppler cooling to be lost. However, by the correct choice of parameters these effects can be made to cancel, and the sub-Doppler cooling is regained.

DOI
10.1088/0953-4075/26/18/016

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The force on a multilevel atom in a 1D magneto-optical molasses with imbalanced laser intensities is theoretically investigated. The authors employ a full quantum treatment to obtain the stationary momentum distribution of the atoms, for arbitrary parameters. The force exerted the counterpropagating laser beams balances for a nonzero locking velocity. The dependence of this velocity on the intensity imbalance is found to be reduced by polarization gradient cooling. For certain values of the imbalance there are two stable velocities, and a splitting of the velocity distribution occurs. A small imbalance, or a magnetic field along the laser beams, can cause the subDoppler cooling to be lost. However, by the correct choice of parameters these effects can be made to cancel, and the sub-Doppler cooling is regained.
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.
Xenon atoms are released from a magneto-optical trap and equilibrate in a one-dimensional, ${\mathrm{\ensuremath{\sigma}}}_{+}$-${\mathrm{\ensuremath{\sigma}}}_{\mathrm{\ensuremath{-}}}$ optical molasses, along which a uniform magnetic field is applied. In the rest frame of the ``corkscrew'' pattern of laser polarization, the magnetic field induces a nonzero average atomic velocity. Introducing a moving molasses through a relative difference in the two laser-beam frequencies allows straightforward, accurate measurements of the molasses-frame velocity distribution over a large range of magnetic fields. The measurements reveal the loss of polarization-gradient cooling in fields higher than \ensuremath{\sim}0.2 mT, in quantitative agreement with theoretical predictions. \textcopyright{} 1996 The American Physical Society.
We have studied theoretically and experimentally the effect of a relatively strong magnetic field on sub-Doppler laser cooling in a one-dimensional optical molasses. We used the operator description of laser cooling with the Larmor precession frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{Z}}$ being much higher than the optical pumping rate. We found velocity-selective resonances (VSR) in the force at velocities vr{\mathit{v}}_{\mathit{r}}vr​=n${\mathrm{\ensuremath{\omega}}}_{\mathit{Z}}$, with n=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2 for both the scattering and redistribution force operators. These depend on the relative direction of the magnetic field and the polarization vectors of the light beams. Analytical results for the force on the atom are obtained in two cases that illustrate the effect of the VSR on the force. These formulas are compared with numerical calculations of the force. We also discovered a redistribution mechanism that relies on the gradient of the eigenstates of the light-shift operator, with eigenvalues that are independent of position so that a ``Sisyphus cooling'' picture does not apply. The theory is compared with many experimental results and excellent agreement is found. We believe that all essential features of laser cooling at low intensity are well described by this operator theory.
We investigate the effect of a longitudinal magnetic field on the orientational cooling mechanism that operates both in σ+–σ− optical molasses and in a magneto-optical trap. For an atom with different Landé g factors in its ground and excited states, single-photon scattering and stimulated Raman scattering are resonant at different field-dependent velocities. The sharp polarization-gradient feature in the force-versus-velocity curve may therefore be translated to a velocity at which the Doppler force is dominant. In addition to this, we find that the momentum diffusion coefficient can grow as the field is increased. By numerically solving a semiclassical Fokker–Planck equation, we show that these effects lead to a field-induced inhibition of efficient sub-Doppler cooling for atoms with unequal g factors. Our treatment also enables us to discuss other phenomena that are accessible only if both Doppler and polarization-gradient effects are included in the calculation.
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 laser cooling of a two-level atom with magnetic sublevels in the presence of polarization gradients. The optical Bloch equations for the multilevel system are solved numerically for four combinations of polarizations in one-dimensional optical molasses. The light-pressure force on the atom as given by a simple two-level theory is recovered in the absence of polarization gradients, whereas a spatial variation of the polarization is found to lead to a strong cooling force for slow atoms. The increased cooling force is responsible for the recent observations of atoms cooled in optical molasses to temperatures an order of magnitude below the Doppler limit.