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Physical Review A

Probe transmission in a one-dimensional optical molasses: Theory for circularly-cross-polarized cooling beams

J.-Y. Courtois, G. Grynberg

Published Aug 1, 1993• 11 citations
Publisher

Abstract

We present a detailed theoretical investigation of the absorption spectrum of a probe beam after transmission through a one-dimensional optical molasses obtained with circularly cross-polarized counterpropagating pump beams (corkscrew molasses). The study is performed for a ${\mathit{J}}_{\mathit{g}}$=1\ensuremath{\rightarrow}${\mathit{J}}_{\mathit{e}}$=2 atomic transition and the external degrees of freedom are described by means of a Fokker-Planck equation in the regime where the stationary momentum distribution lies within the range of linearity of the cooling force. We analyze the stimulated Raman transitions occurring between differently populated and light-shifted ground-state Zeeman sublevels. In the case where the probe beam has the same polarization as the copropagating cooling beam, an additional narrow Rayleigh resonance appears for a probe frequency close to the pump frequency. Its width is equal to 2\ensuremath{\alpha}/M, where \ensuremath{\alpha} is the friction coefficient of the cooling force and M is the atomic mass. In the case where the probe beam has the same polarization as the counterpropagating cooling beam, a central recoil-induced resonance is expected. Its width is related to the Doppler width of the optical molasses.

DOI
10.1103/physreva.48.1378

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Journal of the Optical Society of America B•Nov 1, 1989•1,272 citations
We present a detailed theoretical investigation of the absorption spectrum of a probe beam after transmission through a one-dimensional optical molasses obtained with circularly cross-polarized counterpropagating pump beams (corkscrew molasses). The study is performed for a Jg{\mathit{J}}_{\mathit{g}}Jg​=1\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=2 atomic transition and the external degrees of freedom are described by means of a Fokker-Planck equation in the regime where the stationary momentum distribution lies within the range of linearity of the cooling force. We analyze the stimulated Raman transitions occurring between differently populated and light-shifted ground-state Zeeman sublevels. In the case where the probe beam has the same polarization as the copropagating cooling beam, an additional narrow Rayleigh resonance appears for a probe frequency close to the pump frequency. Its width is equal to 2\ensuremath{\alpha}/M, where \ensuremath{\alpha} is the friction coefficient of the cooling force and M is the atomic mass. In the case where the probe beam has the same polarization as the counterpropagating cooling beam, a central recoil-induced resonance is expected. Its width is related to the Doppler width of the optical molasses.
We present a detailed theoretical investigation of the transmission spectrum of a probe beam interacting with atoms in a one-dimensional optical molasses obtained with linearly cross-polarized counterpropagating pump beams. The study is performed for a Jg{\mathit{J}}_{\mathit{g}}Jg​=1/2\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=3/2 atomic transition in the limit where the Hamiltonian part of the atom-field coupling is predominant over the relaxation part. We analyze the stimulated Raman transitions occurring between different vibrational levels of the atoms in the periodic potential created by the light shifts, and we show a dramatic lengthening of the damping time of coherences between such levels due to the Lamb-Dicke effect. Very narrow Rayleigh resonances with a shape sensitive to the probe polarization appear for a probe frequency close to the pump frequency. We interpret these resonances in terms of scattering of the pump waves on density and magnetization gratings, and show that they provide important information about the dynamics and localization of atoms at the bottom of the potential wells. Such information should also be accessed by phase-conjugation experiments. Finally, indications on the treatment of other atomic transitions are given.
The transmission spectrum of a probe interacting with one-dimensional cesium optical molasses made from two circularly cross-polarized cooling beams (corkscrew molasses) displays a narrow (\ensuremath{\approxeq}30--70 kHz) Rayleigh resonance. From the width of this resonance we deduce the friction coefficient of the cooling force to be \ensuremath{\alpha}=(5.8\ifmmode\pm\else\textpm\fi{}0.4)\ensuremath{\Elzxh}k2{\mathit{k}}^{2}k2 for a detuning from resonance equal to 3\ensuremath{\Gamma}. This value is in quantitative agreement with theoretical predictions. We also show that the shape of the Rayleigh resonance depends dramatically on the efficiency of the sub-Doppler cooling mechanism.
We present a calculation of the probe absorption spectrum of atoms in a one-dimensional optical molasses formed by a pair of counterpropagating, cross-polarized fields. When the finite angle \ensuremath{\theta} between the propagation direction of a probe field and that of one of the cooling fields is taken into account, we find that, apart from the contributions to the probe absorption signal that are related to the Raman and Rayleigh resonances between the discretized atomic motional states in the molasses direction, there is also a signal arising from the recoil-induced resonances between atomic states with different center-of-mass momenta in the directions transverse to the molasses beams. The magnitudes of these different contributions are compared.
We present a detailed calculation of the four-wave-mixing signal from one-dimensional optical molasses, in which atoms are cooled by a pair of counterpropagating, cross-polarized fields. The contributions from both Raman and Rayleigh resonances between the discrete motional bound states and the recoil-induced resonances between the motional continuum states are analyzed. It is shown that the recoil-induced resonances can lead to a narrow central resonance peak in the four-wave-mixing signal with a width given by the transverse residual Doppler width. The calculated Raman sidebands also exhibit an asymmetry in the signal strength, a feature observed in recent experiments.
We present a calculation of the probe-transmission spectrum of a two-dimensional optical lattice, in which atoms are cooled by a pair of standing-wave fields with perpendicular polarizations within the plane of incidence. A probe laser is directed along the axis of one of the cooling fields, with a polarization parallel to the incident plane of the cooling fields. The calculated probe-transmission spectrum exhibits both sidebands, which originate from stimulated Raman transitions between different vibrational states of the atoms in the molasses, and a central resonance structure, which originates from Rayleigh resonances among these vibrational states. We also include in our calculation a possible misalignment of the probe field relative to the axis of the cooling field. It is shown that the effrects of the probe misalignment can lead to significant modifications to the central resonance signal.
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.