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Journal of Physics B Atomic Molecular and Optical Physics

Polarization gradient cooling in a one-dimensional sigma+- sigma-configuration for any atomic transition

Andrew M. Steane, G. Hillenbrand, Christopher J. Foot

Published Nov 28, 1992• 16 citations
Publisher

Abstract

The authors discuss the radiation force in a one-dimensional sigma + - sigma - laser standing wave, for transitions J to J+ Delta J, for any J. and Delta J=+or-1.0. in the limit of low saturation of the atomic transition. The interaction between the radiation field and the atom produces a motion-induced orientation of the atomic ground state which grows rapidly with J. The radiative force on the atom is sensitive to this orientation. For J to J+1 the dissipative part of the total force is much larger than the reactive part, at low velocities, while for the other cases the two contributions are roughly equal and opposite. By using a simple method of calculation, the authors gain much physical insight into the mechanism of induced orientation, and give quantitative results for the size of the radiation force, including both the low-velocity friction coefficient and the velocity capture range.

DOI
10.1088/0953-4075/25/22/010

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The authors discuss the radiation force in a one-dimensional sigma + - sigma - laser standing wave, for transitions J to J+ Delta J, for any J. and Delta J=+or-1.0. in the limit of low saturation of the atomic transition. The interaction between the radiation field and the atom produces a motion-induced orientation of the atomic ground state which grows rapidly with J. The radiative force on the atom is sensitive to this orientation. For J to J+1 the dissipative part of the total force is much larger than the reactive part, at low velocities, while for the other cases the two contributions are roughly equal and opposite. By using a simple method of calculation, the authors gain much physical insight into the mechanism of induced orientation, and give quantitative results for the size of the radiation force, including both the low-velocity friction coefficient and the velocity capture range.
A theoretical analysis is given of the interaction of a laser field consisting of two orthogonal, linearly polarized, low-power standing waves, with an ensemble of J=1/2 ground-state atoms. The atoms are assumed to move with velocities below the Doppler limit of laser cooling. The two-dimensional light-induced atomic force is calculated for the specific case when the field polarization directions differ by \ensuremath{\pi}/2. The effective optical-pumping time is shown to increase strongly near the nodes of the laser field, leading to a force that can be much larger than that in the one-dimensional case. The force is not isotropic, and can lead to the bunching of the atoms in velocity space.
In the process of cooling and trapping neutral atoms, the authors investigate a new laser configuration, which consists of two counterpropagating laser beams with orthogonal sigma + and sigma - polarisations. They show that such a configuration looks more promising than an ordinary standing wave (where the two counterpropagating waves have the same polarisation), and they explain this result as being due to angular momentum conservation which prevents any coherent redistribution of photons between the two waves. The authors' conclusions are based on a quantitative calculation of the various parameters (potential depth, friction coefficient, diffusion coefficient) describing the mean value and the fluctuations of the radiative forces experienced, in such a laser configuration, by an atom with a J=0 ground state and a J=1 excited state.
Simple atomic models (1/2→1/2 and 1/2→3/2 transitions) were taken as an example to consider, in the sub-Doppler cooling approximation, influence of the spatial rotation of a polarization plane on the radiative friction force at arbitrary field configurations of dimensionality D <1. Spatial gradients of the angles determining this rotation additionally contribute to the friction force. This contribution is comparable in magnitude with other forces if the detuning δ is on the order of the radiative relaxation constant γ. For the j → j +1 transitions, the contribution promotes sub-Doppler cooling at δ<0, whereas for the j → j transitions (half-integer j ) it induces anisotropic heating and cooling processes.
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 give a general theoretical description of radiative forces on atoms in a monochromatic radiation field. The driven transition has a lower level consisting of several substates, and we consider the limit of low velocities and weak intensities. This situation comprises the schemes where cooling to temperatures below the Doppler limit is possible. After expressing the atomic dipole in terms of the polarizability tensor, we obtain an expression of the force as the sum of the radiation pressure, the dipole force, and a contribution from the gradient of the polarization. This latter contribution contains a part resulting from redistribution of photons between the plane waves that compose the field, and a part resulting from fluorescence. We express the average force and the momentum-diffusion tensor in terms of a closed evolution equation for the lower-state density matrix. We show by a scaling argument that in any case of cooling, the final temperature goes down linearly with the intensity, until the recoil limit is approached. The description is valid for a weak and monochromatic, but otherwise arbitrary field, and it allows for the presence of an external magnetic field, and of hyperfine splitting. It is applicable to cooling in one, two, or three dimensions. We give numerical results for the average force in a number of specific cases of two counterrunning plane waves.
In relation to the newly discovered laser cooling mechanisms, the authors have studied the momentum diffusion of an atom with a j (ground)-j+1 (excited) state transition (j not=0), interacting with two counterpropagating sigma +- sigma - plane waves. A correlation between successive photon absorptions via the multi-sublevel internal atomic structure leads to a large increase of the momentum diffusion coefficient. In a simple probabilistic picture this enhanced diffusion is interpreted as a random walk with effective steps much larger than the one-photon recoil momentum k. For j=1 the authors report the results of a complete calculation (i.e. with a quantum description of the internal atomic state) showing a marked dependence upon the velocity of the atom as well as on the parameters of the laser field (detuning and power).