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PHYSICAL REVIEW A VOLUME 48, NUMBER 3 SEPTEMBER 1993 Interaction of atoms with a magneto-optical potential C. S. Adams, T. Pfau, Ch. Kurtsiefer, and J. Mlynek Fakultiit fiir Physik, Universitiit Konstanz, D-7750 Konstanz, Germany (Received 22 February 1993) A theoretical study of the coherent interaction of multilevel atoms with a magneto-optical potential is presented. The potential is formed by counterpropagating linearly polarized laser beams whose polariza- tion vectors intersect at an angle q:> and a static magnetic field applied parallel to the laser propagation direction. For a particular ratio of the light and magnetic field amplitudes, the light shift at positions of purely circularly polarized light is equal to the Zeeman splitting. In this case, for a three-level atom, one of the eigenvalues has a triangular spatial form. The diffraction of atoms from this triangular phase grat- ing is an efficient beam splitter. The splitting is symmetric for q:>= 90' and asymmetric for q:> < 90'. In ad- dition we show that at well-defined positions in the light field, the atom undergoes nonadiabatic transi- tions and thus by using state-selective detection, one could observe an interference pattern produced by an array of double slits. PACS number(s): 32.90.+a, 32.80.-t, 42.50.-p I. INTRODUCTION The diffraction of two-level atoms from a standing- wave light field is interesting both in the context of our general understanding of light-matter interactions and because of possible applications as coherent beam splitters, which are a key component in the realization of atom interferometers. The first convincing demonstra- tion of the transfer of individual photon momenta be- tween light and atoms was reported by Moskowitz et af. in 1983 [1]. Improved results were reported in 1986 [2]. The coherent diffraction process (i.e., no spontaneous emission) can be described either in terms of discrete momentum transfer caused by the scattering of photons, or by refraction of a matter wave from an optical phase grating (or optical potential) arising from the spatial modulation of the light shift. If the transverse motion of the atom is small compared to the optical wavelength, then the wave function accumulates a position-dependent phase shift proportional to the energy of the populated eigenstate. The final momentum distribution is given by the Fourier transform of the phase-shifted wave function. For a standing wave, the eigenvalues are sinusoidal func- tions of position, i.e., the atom is diffracted by a sinusoidal phase grating, and the final momentum distri- bution is given by a Bessel-function distribution [3]. For a small phase modulation, e.g., one absorption-stimulated emISSIon cycle, standing-wave diffraction produces efficient scattering into states with ±2fzk. However, for a large phase modulation (many absorption-stimulated emission cycles) a large number of diffraction orders are populated and there is broad spreading rather than a clear splitting of the beam. For this reason, standing- wave diffraction is far from being the ideal beam splitter with just two outputs and a large splitting in momentum space. In this paper we show that by introducing a further de- gree of freedom in the light-atom interaction (namely, polarization-selective absorption and emission), it is pos- 1050-2947/93/48(3)/2108(9)/$06.00 48 sible to gain more control of the momentum-transfer pro- cess, and for example create an effective scheme for an atomic beam splitter [4]. Two extensions of the normal standing-wave interaction are proposed. First, we allow the atom to distinguish between the counterpropagating laser beams which form the standing wave. This can be achieved using beams with different polarizations and a multilevel atom where the transitions are polarization selective. Second, we introduce a magnetic field in order to switch the atomic coupling from one beam to the oth- er. To illustrate the momentum-transfer process in this magneto-optical interaction, consider a J=O to J' = 1 transition. The quantization axis is chosen parallel to the magnetic field. The level scheme is shown in Fig. l(a). The excitation of the atom by linearly polarized light in- duces an equal superposition of the m J' = ± 1 levels known as an alignment [depicted schematically by an el- lipsoid in Fig. l(b)]. The direction of the alignment is parallel to the polarization direction. The alignment states does not couple to light polarized perpendicular to the alignment direction. A magnetic field induces a mix- ing of the excited-state sublevel coherences leading to a precession of the alignment. Consider an interaction formed by counterpropagating, linearly polarized beams, whose polarization vectors in- tersect at an angle cp [as shown in Fig. l(b)]. The preces- sion of the alignment changes probability of absorbing or emitting photons from one beam or the other. If the fre- quency of the absorption and emission processes is matched to the precession frequency, the atom repeats cycles of absorption from one beam and emission into the other, or vice versa. In this case, the direction of momen- tum transfer (determined by the first absorption process) is preserved. For orthogonally polarized laser beams, a symmetric splitting of the beam is expected from the symmetry of the laser fields. The physical mechanism of this effect is analogous to the magneto-optical force proposed and demonstrated by 2108 ©1993 The American Physical Society 48 INTERACTION OF ATOMS WITH A MAGNETO-OPTICAL POTENTIAL 2109 (a) Ig> (b) FIG. 1. (a) Level scheme for a J=Q to J'= I transition with the quantization axis chosen parallel to the magnetic field. (b) The configuration of the laser fields Ex and E"" and the magnet- ic field Bz> relative to the atomic beam direction (y). The momentum-transfer process is controlled by the Larmor preces- sion of the excited-state alignment (shown schematically as an ellipsoid) and the polarization-dependent selection rules for transitions to the excited state. Grimm et al. [5]. However, the unidirectional magneto- optical force occurs in the regime where the interaction time is much longer than the spontaneous-decay time. In this case the transverse motion of the atom through the potential becomes significant, and the atom experiences an averaged dipole force. The combined effect of spon- taneous emission and linearly polarized beams with po- larization vectors at an angle