Wannier functions using a discrete variable representation for optical lattices.

Area of Science:

• Atomic, Molecular, and Optical Physics
• Computational Physics
• Condensed Matter Physics

Background:

• Wannier functions are crucial for describing electron localization in periodic potentials.
• Accurate construction of real-valued Wannier functions is essential for condensed matter and atomic physics calculations.
• Optical lattices provide controllable systems for studying quantum phenomena with ultracold atoms.

Purpose of the Study:

• To develop and apply a numerical method for constructing real-valued Wannier functions.
• To investigate the applicability of the method to both symmetric and asymmetric periodic potentials, particularly in optical lattices.
• To assess the accuracy of the constructed Wannier functions and their impact on calculating interaction energies.

Main Methods:

• Utilizing the discrete variable representation (DVR) for constructing localized Wannier functions.
• Employing a uniform Fourier grid DVR to obtain real-valued eigenstates and band structure.
• Calculating tunneling energies and two-body interaction energies within the Bose-Hubbard model.

Main Results:

• The proposed DVR method successfully constructs real-valued Wannier functions for symmetric and asymmetric potentials.
• Wannier functions exhibit high accuracy (better than 10 significant digits) in double-precision arithmetic.
• Calculated two-body interaction energies in the Bose-Hubbard model show dependence on lattice asymmetry.

Conclusions:

• The DVR method provides an accurate and efficient way to generate localized Wannier functions for ultracold atoms in optical lattices.
• The method's ability to handle asymmetric potentials is critical for realistic experimental simulations.
• This work advances the computational tools for studying quantum many-body systems in optical lattice potentials.