Observation of sub-wavelength phase structure of matter wave with two-dimensional optical lattice by Kapitza-Dirac diffraction

We report an experimental demonstration of generation and measurement of sub-wavelength phase structure of a Bose-Einstein condensate (BEC) with two-dimensional optical lattice. This is implemented by applying a short lattice pulse on BEC in the Kapitza-Dirac (or Raman-Nath) regime, which, in the classical picture, corresponds to phase modulation imprinted on matter wave. When the phase modulation is larger than 2π in a lattice cell, the periodicity of phase naturally forms the sub-wavelength phase structure. By converting the phase information into amplitude, we are able to measure the sub-wavelength structure through the momentum distribution of BEC via the time-of-flight absorption image. Beyond the classical treatment, we further demonstrate the importance of quantum fluctuations in the formation of sub-wavelength phase structure by considering different lattice configurations. Our scheme provides a powerful tool for exploring the fine structure of a lattice cell as well as topological defects in matter wave.