Dirac theory of ring-shaped electron distributions in atoms

There are two main optical methods by which an atomic electron can acquire relativistic speeds. The first one in- volves an extremely powerful laser pulse, where the large force associated with the electric-field component of the laser is primarily responsible for accelerating the electron during a single cycle of the field. A second method to excite atoms into relativistic orbits is based on exploiting cyclotron-type resonances of the electron interacting with a combined laser and magnetic field. Atomic resonances, however, cannot be exploited directly to bring an electron's speed up to the relativistic regime be- cause an unlimited increase in an electron's speed at reso- nance is typically avoided by the nonlinearity of the atomic potential encountered by the large-amplitude motion. This limitation typically sets in at velocity scales much smaller than the speed of light. However, for an electron in a suffi- ciently strong magnetic field nonlinear atomic effects are not so important, and the velocity can grow to extremely large values until relativistic effects that limit the speed from growing beyond bound become important. If the cyclotron frequency V associated with a static mag- netic field is approximately commensurate with the laser fre- quency v L ~nV'mv L , n,m51,2,3 ,... !, the interaction of the electron with the combined laser and magnetic field be- comes resonant. In this regime a wide variety of relativistic phenomena have been investigated recently @1#. The elec- tronic charge distribution, for instance, can develop into ring @2#, figure-8@3#, and propellerlike structures @4# whose center rotates around the nucleus. These relativistic charge-cloud distributions emerge from initial atomic states after a few laser cycles. The absence of these distributions in the corre- sponding nonrelativistic solution for the same parameters demonstrates that this ring structure is a genuinely relativis- tic phenomenon. To understand the formation of these struc- tures, a simplified model based on the spiral orbits of indi- vidual classical trajectories was proposed which associates the relativistic dephasing with a strong velocity dispersion enhanced by the resonance between the magnetic and laser fields @1#.