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Physical Review A

Semiclassical theory of sub-Doppler forces in an asymmetric magneto-optical trap with unequal laser detunings

Heung‐Ryoul Noh, Wonho Jhe

Published May 17, 2007• 9 citations
Publisher

Abstract

We present a semiclassical theory of the sub-Doppler forces in an asymmetric magneto-optical trap where the trap-laser frequencies are unequal to one another. To solve the optical Bloch equations, which contain explicit time dependence, unlike in the symmetric case of equal laser detunings, we have developed a convenient and efficient method to calculate the atomic forces at various oscillating frequencies for each atomic density matrix element. In particular, the theory provides a qualitative understanding of the array of sub-Doppler traps (SDTs) recently observed in such an asymmetric trap. We find that the distances between SDTs are proportional to the relative detuning differences, in good agreement with experimental results. The theory presented here can be applied to a dynamic system with multiple laser frequencies involved; the number of coupled equations to solve is much reduced and the resulting numerical calculation can be performed rather simply and efficiently.

DOI
10.1103/physreva.75.053411

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We present a semiclassical theory of the sub-Doppler forces in an asymmetric magneto-optical trap where the trap-laser frequencies are unequal to one another. To solve the optical Bloch equations, which contain explicit time dependence, unlike in the symmetric case of equal laser detunings, we have developed a convenient and efficient method to calculate the atomic forces at various oscillating frequencies for each atomic density matrix element. In particular, the theory provides a qualitative understanding of the array of sub-Doppler traps (SDTs) recently observed in such an asymmetric trap. We find that the distances between SDTs are proportional to the relative detuning differences, in good agreement with experimental results. The theory presented here can be applied to a dynamic system with multiple laser frequencies involved; the number of coupled equations to solve is much reduced and the resulting numerical calculation can be performed rather simply and efficiently.
The theoretical and experimental studies on the observation of adjustable multiple sub-Doppler traps in the six-beam magneto-optical trap (MOT) with unequal laser detunings are presented. When the detuning of the longitudinal lasers along the magnetic field axis is different from that of the transverse lasers, we obtain an array of three sub-Doppler traps; one usual but suppressed trap at the origin and two additional traps symmetrically but oppositely located at equal and controllable distances from the origin. We could explain the observed sub-Doppler traps and their behavior by the calculation of the sub-Doppler forces where multiple oscillating frequencies are involved.
We have realized adjustable multiple sub-Doppler traps (SDT's) in a six-beam magneto-optical trap (MOT) under asymmetric trap conditions. In the simplest case, one obtains an array of three SDT's, one usual trap at the origin and two additional traps symmetrically but oppositely located at equal and controllable distances from the origin, in good agreement with theoretical calculations. The easily adjustable array of SDT's readily available in the asymmetric MOT may open the possibility of novel atom optics or quantum optics experiments such as atom interferometer or quantum information.
We theoretically investigate the damping and trapping forces in a three-dimensional magneto-optical trap (MOT), by numerically solving the optical Bloch equations.We focus on the case where there are dark states because the atom is driven on a 'type-II' system where the angular momentum of the excited state, F¢, is less than or equal to that of the ground state, F. For these systems we find that the force in a three-dimensional light field has very different behaviour to its one dimensional counterpart.This differs from the more commonly used 'type-I' systems (F F 1 ¢ = + ) where the 1D and 3D behaviours are similar.Unlike type-I systems where, for red-detuned light, both Doppler and sub-Doppler forces damp the atomic motion towards zero velocity, in type-II systems in 3D, the Doppler force and polarization gradient force have opposite signs.As a result, the atom is driven towards a non-zero equilibrium velocity, v 0 , where the two forces cancel.We find that v 0 2 scales linearly with the intensity of the light and is fairly insensitive to the detuning from resonance.We also discover a new magneto-optical force that alters the normal MOT force at low magnetic fields and whose influence is greatest in the type-II systems.We discuss the implications of these findings for the laser cooling and magneto-optical trapping of molecules where type-II transitions are unavoidable in realising closed optical cycling transitions.
We have measured the trap frequency as well as the damping coefficient of a magneto-optical trap by using a transient oscillation method. The dependence of such trap properties on the various experimental parameters such as the cooling laser intensity, detuning, and magnetic field gradient is investigated. We find that the measured trap frequency is in excellent agreement with the simple rate-equation analysis based on the Doppler cooling theory. In contrast, the damping coefficient is about twice as large as the calculated one, which is attributed to the existence of the sub-Doppler trap near the trap center.
Current dual-frequency magneto-optical traps for ultracold molecules are plagued by sub-Doppler heating effects, making them vastly inferior to standard atomic MOTs. Here we demonstrate theoretically that the sub-Doppler effects in such a MOT can be engineered to provide cooling instead of heating. We give an intuitive picture how to achieve such cooling and show the cooling and trapping force results of the 16 level optical Bloch equations for the case of CaF molecules. From three-dimensional Monte Carlo simulations we estimate the temperature and density of our MOT to be 40μK40 μK40μK and 4×108cm−34 \times 10^8 cm^{-3}4×108cm−3 respectively for a molecule number of 10510^5105.
We investigate radiation-imprisonment (RI) phenomena in a system of cold atoms trapped in magneto-optical traps (MOTs) when cooling lasers are switched off for a short period. The problem is formulated within the framework of the Streater-Cooper-Sandle equation that belongs to the Markov class and accounts for photon frequency redistribution in multiple scattering processes. On the basis of a semiclassical approach, we derive explicit analytical solutions for the spectral problem of the RI equation neglecting Doppler effects. The analysis reveals peculiarities in emission profiles under initial partial saturation of the resonance transition by the probe laser pulse. Comparison between theoretical and experimental data on effective radiative lifetimes obtained in Cs and Rb MOTs confirms the validity of the discussed nonstationary model of RI in cold samples. For small detunings, the Streater-Cooper-Sandle equation agrees with the experimental data, while in the case of large detunings, the theory does not agree, probably due to non-Markovian effects.