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Ab initio calculation of Hubbard parameters for Rydberg-dressed atoms in an one-dimensional optical lattice

We obtain ab initio the Hubbard parameters for Rydberg-dressed atoms in an one-dimensional sinusoidal optical lattice in the basis of maximally localized wannier states. Finite range, soft-core inter-atomic interactions become the trait of Rydberg admixed atoms, which can be extended over many neighbouring lattice sites. On contrary to dipolar gases, where the interactions follow an inverse cubic law, the key feature of Rydberg-dressed interactions being the possibility of making neighbouring couplings to the same magnitude as that of the onsite ones. The maximally localized Wannier functions are typically calculated via spread minimization procedure [Phys. Rev. B 56, 12847 (1997)] and always found to be real functions apart from a trivial global phase when considering an isolated set of Bloch bands. For an isolated single Bloch band, the above procedure reduces to a simple quasi-momentum dependent unitary phase transformation. Here, in stead of minimizing the spread, we employ a diagonal phase transformation which eliminates the imaginary part of the Wannier functions. The resulting Wannier states are found to be maximally localized and in exact agreement with those obtained via spread minimization procedure. Using that we calculate the Hubbard couplings from the Rydberg-admixed interactions, including dominant density assisted tunnelling coefficients. In the end we provide realistic lattice parameters for the state of the art experimental Rydberg dressed Rubidium setup.