University of Michigan - Ann Arbor Physics Department
My Advisor, Professor P.R. Berman
Atom-Optics Bragg Scattering
The Kapitza-Dirac Effect, first suggested by P.L. Kapitza and P.A.M. Dirac in 1933 at the Cambridge Physical Society, offered an analog between matter and light in the form of matter-wave diffraction through a light crystal. The reciprocity of matter and light in quantum electrodynamics suggests that one could perform a Bragg-like diffraction of matter from a periodic light source. Kapitza and Dirac suggested an electron beam diffracting from a strong light source in a cavity of mirrors where the spatial periodicity of the light would act like the atoms of a crystal and the electon beam like an x-ray source. In the energy conserving regime or Bragg regime, the effect of this diffraction is spatially quantized diffraction of the electron beam satisfying Bragg conditions specific to the beam and light crystal.
With the advent of lasers, such an experiment has been reported in 2003 by H. Batelaan et. al., however, many more experiments have been performed with atomic beams begining with D.E. Pritchard in 1988. In these experiments, couterpropagating lasers are used to create a light crystal or grating. With atoms, the physics is complicated by the internal structure of the atom. In the simplest of cases, the atom comes in at the Bragg angle and exiting at the opposite angle. In the Laue geometry, this scattering process can lead to Pendulosung in the outgoing beams. In the atomic frame, the interaction is that of two pulses mediated by the laser profile and atomic velocity. Since the atom carries some transverse velocity, the pulses are Doppler shifted blue and red. On resonance, this energy difference must equal the change in the mechanical energy of the atomic beam. From the frame of the incoming atom, the outgoing atom has gained energy equal to the difference in the observed fields. The process is stimulated and with the geometry a gain accompanies each diffracted atom. From the lab frame, the momentum of the diffraction is conserved in the change in the atoms momentum. The energy in the field and in the atom is left intact. However, the system can be generalized by introducing a detuning in one of the couterpropagating lasers. Here the light is approximately a standing wave with a velocity. The atomic beam can be perpendicular to the lasers and the scattering is now a stimulated Raman transition as the final state of the atom is in a momentum state with an larger energy. The lowest order scattering is a two-photon process and the internal state should not become populated, so these processes are far off resonance with respect to the electronic states of the atom. Given the two features of the system which can be considered resonant or off resonant and the fact that the internal electronic transitions are always off resonant, we choose to use the term resonant to refer to the outgoing momentum states of the atomic beam. If undeflected or Bragg scattered, resonant, if higher order in momentum, off resonant.
Given the non-adiabatic contributions of the equations of motion, it is easy to find higher order momentum states. For large enough detuning, the off resonant populations can be made exponentially small, however, the effect of the higher order momentum states are in fact necessary for correctly predicting the Pendulosung of the resonant states. The equations of motion are coupled, term by term, to higher off resonant momentum states and are infinite. Finding the dependence of the number of states with respect to field strength and detuning is necessary as the phase and frequency of the Pendulosung is radically altered with the inclusion of higher order momentum states.