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Physical Review A

Dressed-atom description of the bichromatic force

Леонід Петрович Яценко, Harold J. Metcalf

Published Dec 2, 2004• 41 citations
Publisher

Abstract

We develop a dressed-atom picture of the bichromatic force in two standing waves using a Floquet approach. It is based on previous work, but the approach allows for an interpretation of the velocity range of the force. It is limited to two-level atoms and one dimension, and the Floquet frequency is the beat between the two bichromatic optical fields. The force is mediated by Landau-Zener transitions between the dressed states of the Floquet Hamiltonian. Related topics have been addressed before in the literature, but not applied to this particular case.

DOI
10.1103/physreva.70.063402

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We develop a dressed-atom picture of the bichromatic force in two standing waves using a Floquet approach. It is based on previous work, but the approach allows for an interpretation of the velocity range of the force. It is limited to two-level atoms and one dimension, and the Floquet frequency is the beat between the two bichromatic optical fields. The force is mediated by Landau-Zener transitions between the dressed states of the Floquet Hamiltonian. Related topics have been addressed before in the literature, but not applied to this particular case.
The simulated magneto-optical force, which in the presence of a longitudinal static magnetic field is exerted on atoms resonant with the high-intensity light field of two counterpropagating, linearly polarized laser beams with an angle of \ensuremath{\sim}45\ifmmode^\circ\else\textdegree\fi{} between their directions of polarization, is described in the dressed-atom picture. This allows an easy calculation of the force on atoms at rest, illustratively explains its nonzero wavelength average, and provides an intuitive understanding of its dependence on the atomic velocity. For a simple atomic model transition (J=0\ensuremath{\leftrightarrows}J'=1) the velocity-dependent force is calculated by considering the motion along the dressed states and using the Landau-Zener model for the nonadiabatic transitions at the avoided level crossings. It is shown that for a certain relation between Larmor and optical Rabi frequency a perfectly diabatic motion takes place, leading to a disappearance of the velocity-tuned (doppleron) resonance structure.
The dressed atom approach is used to investigate the evolution of atoms moving in an intense standing wave laser field. When coherences between the dressed states can be neglected, a simple expression is obtained for the radiative force acting on the atoms. The Landau-Zener model, well-known in the theory of slow atomic collisions, is here used to include coherences in an analytic treatment. At high velocities, where the simplest approach breaks down, the results are in good agreement with exact numerical calculations. Applications outside the field of laser cooling are discussed.
We show that the dressed-atom approach provides a quantitative understanding of the main features of radiative dipole forces (mean value, fluctuations, velocity dependence) in the high-intensity limit where perturbative treatments are no longer valid. In an inhomogeneous laser beam, the energies of the dressed states vary in space, and this gives rise to dressed-state-dependent forces. Spontaneous transitions between dressed states lead to a multivalued instantaneous force fluctuating around a mean value. The velocity dependence of the mean force is related to the modification, induced by the atomic motion, of the population balance between the different dressed states. The corresponding modification of the atomic energy is associated with a change of the fluorescence spectrum emitted by the atom. The particular case of atomic motion in a standing wave is investigated, and two regimes are identified in which the mean dipole force averaged over a wavelength exhibits a simple velocity dependence. The large values of this force achievable with reasonable laser powers are pointed out with view to slowing down atoms with dipole
The limit of $\ensuremath{\Elzxh}k\ensuremath{\gamma}/2$ imposed on the magnitude of radiative forces by the spontaneous decay rate $\ensuremath{\gamma}$ of the excited state can be overcome by coherent control of the momentum exchange between atoms and the light field. This can be implemented with two light beams, each containing two frequencies whose phases, amplitudes, and frequency difference are carefully chosen. We have made precise measurements of the extremely large magnitude and velocity range of this bichromatic force, and have shown that its velocity dependence near the edge of its range is suitable for atomic beam slowing and laser cooling. Our measurements have corroborated various models and calculations of this bichromatic force.
In this paper we present a new calculation of the force on a stationary three-level atom in a bichromatic standing-wave light field (see Fig. 1). Contrary to other theoretical treatments,1 which ignore optical pumping into the trapped state, we find that a stationary atom experiences no force when there is no difference frequency detuning (i.e., when Δ = 0). However, when there is a nonzero differential detuning (Δ ≠ 0) there can be force components that vary in both the wavelength of the optical fields and their beat wavelength. This stimulated force can be much larger than the maximum spontaneous force associated with this system. To obtain a simple physical interpretation of this force, the problem is treated in a basis that is formed from symmetric |+〉 and antisymmetric |−〉 linear combinations of the atomic ground states2 (see Fig. 1).