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.

In this supplementary material, we provide details on the exact solution of BEC density distribution after a short lattice pulse, and compare exact results to those from classical treatment (Eqs. 3, 6 in main text) for various lattice configurations.

EXACT SOLUTION OF MOMENTUM DISTRIBUTION
The atoms in a two-dimensional (2D) optical lattice can be described by the Hamiltonian: Here H i and H o are, respectively, the Hamiltonian for in-plane and out-plane lattices. k r = 2π/λ is the recoil momentum, which defines the recoil energy as E rec = 2 k 2 r /(2m). According to the Bloch theorem, the eigen-state wave function can be expressed as: here k lies within the first Brillouin zone (BZ) and n is the band index; G x,y are the reciprocal vectors: G x,y = 2N k r for in-plane lattice and G x,y = N x,y k r for out-plane lattice (N, N x , N y are all integers). Given the initial state as a Bose condensate at |p 0 = (0, 0) >, at time t the atomic density distribution in momentum space can be derived as: here E nk is the eigen-energy of state |nk >. In our practical numerical calculations, we have set a cutoff momentum to both G x and G y as 20k r . Accordingly, the single-particle Hamiltonian is expanded as a (2N c + 1) 2 × (2N c + 1) 2 matrix with N c = 20. We have checked that such cutoff can give convergent results to the momentum distribution.

COMPARISON BETWEEN DIFFERENT LATTICE CONFIGURATIONS: EXACT SOLUTION VS. CLASSICAL TREATMENT
The density distribution in momentum space by the classical treatment (as shown in Eqs. 3, 6 in main text): here F means taking the Fourier transform, and U (x, y) is the 2D lattice potential. In Figs. 1-3, we show the momentum distributions from exact solutions and from classical treatments for various lattice pulse configurations, using the same parameters as in the experiments (see Fig. 4 and Fig. 5 in the main text).
1. In-plane lattice: In Fig. 1  It is found that the density distributions calculated by these two methods are qualitatively consistent with each other, and also consistent with the experimental results [ Fig. 4(a1)-(a4) in main text]. Moreover, in this case the distributions are same for red-detuning(V < 0) and blue-detuning(V > 0) lattice potentials.
2. Out-plane lattice with V < 0: In Fig. 2 we show the density distributions for out-plane lattice with red-detuning lattice potential V < 0. We can see that in this case the exact results and those form classical treat are again qualitatively consistent with each other, which both show the ring structures. Meanwhile, both the size and the number of rings increase with the lattice depth. These properties are consistent with the experimental findings [ Fig. 4(c1)-(c4) in main text].
3. Out-plane lattice with V > 0: In Fig. 3 we show the density distributions for out-plane lattice with blue-detuning lattice potential V > 0.
We can see that in this case the exact results and those form classical treat show large deviations as increasing V : the classical results still give rise to bigger ring structure as V increases, identical to the V < 0 case in Fig. 2; while the ring structure in exact solutions gradually vanish as increasing V . Such deviation can be attributed to the breakdown of classical treatment in describing the matter-wave dynamics under this type of potential, where the quantum fluctuations play important role in such process, as analyzed in the main text.