Stroboscopic painting of optical potentials for atoms with subwavelength resolution

We propose and discuss a method to engineer stroboscopically arbitrary one-dimensional optical potentials with subwavelength resolution. Our approach is based on subwavelength optical potential barriers for atoms in the dark state in an optical A system, which we use as a stroboscopic drawing tool by controlling their amplitude and position by changing the amplitude and the phase of the control Rabi frequency in the A system. We demonstrate the ability of the method to engineer both smooth and comblike periodic potentials for atoms in the dark state, and establish the range of stroboscopic frequencies when the quasienergies of the stroboscopic Floquet system reproduce the band structure of the time-averaged potentials. In contrast to usual stroboscopic engineering which becomes increasingly accurate with increasing the stroboscopic frequency, the presence of the bright states of the Lambda system results in the upper bound on the frequency, above which the dynamics strongly mixes the dark and the bright channels, and the description in terms of a time-averaged potential fails. For frequencies below this bound, the lowest Bloch band of quasienergies contains several avoided-crossing coming from the coupling to high-energy states, with widths decreasing with increasing stroboscopic frequency. We analyze the influence of these avoided crossings on the dynamics in the lowest band using Bloch oscillations as an example, and establish the parameter regimes when the population transfer from the lowest band into high bands is negligible. We also present protocols for loading atoms into the lowest band of the painted potentials starting from atoms in the lowest band of a standard optical lattice.