Magnetically generated spin-orbit coupling for ultracold atoms with slowly varying periodic driving

The spin-orbit coupling (SOC) affecting the center of mass of ultracold atoms can be simulated using a properly chosen periodic sequence of magnetic pulses. Yet such a method is generally accompanied by micro-motion which hinders a precise control of atomic dynamics and thus complicating practical applications. Here we show how to by-pass the micro-motion emerging in the magnetically induced SOC by switching on and off properly the oscillating magnetic fields at the initial and final times. We consider the exact dynamics of the system and demonstrate that the overall dynamics can be immune to the micro-motion. The exact dynamics is shown to agree well with the evolution of the system described by slowly changing effective Floquet Hamiltonian including the SOC term. The agreement is shown to be the best when the phase of the periodic driving takes a specific value for which the effect of the spin-orbit coupling is maximum.