Special parameters to treat atoms in optical lattices

MCTDH-X offers two ways of dealing with atoms in optical lattices. Lattice Hamiltonians can be treated using an exact diagonalization treatment for one-, two-, and three-dimensional lattices. Especially, in the two- and three-dimensional cases, the dimensionality of the Hilbert space and the matrix that has to be diagonalized in the exact diagonalization approach is exploding rapidly and the problem size can no longer be handled. For the larger systems, one has to use the so-called MCTDBH approach, where the Bose-Hubbard Hamiltonian is expanded in a multi-configurational basis with a number of effective single-particle states than there is lattice sites. With this approach, a systematic improvement beyond mean-field theories such as the discrete non-linear Schrödinger equation is possible. Contrary to the approaches like time-evolved block decimation, matrix product states or time-dependent density-matrix renormalization group, MCTDBH can provide accurate predictions in time and for higher dimension than $D=1$, because the partitioning of the Hilbert space is done by the variational principle and not artificially introduced. The parameters to select the former exact diagonalization or the latter MCTDBH approach are collected in the following table 4.2.

System Parameters Namelist
Parameter	Meaning	Options
Bose_Hubbard	Do an exact diagonalization of a Bose Hubbard Hamiltonian? Attention: Has to be used together with JOB_TYPE = 'FCI' !	Logical, default .F.
Periodic_BH	Has the treated lattice system periodic boundary conditions?	Logical, default .F.
DVR Parameters Namelist
DVR_<I>	If DVR_<I> = 6, MCTDHB is applied to a lattice of that many sites in I = X/Y/Z direction. Attention: has to be used together with JOB_TYPE = 'BOS' !	Integer, default 4.

It is important to note that in the case that an exact diagonalization of a Bose-Hubbard Hamiltonian is done, the one-body potential offset is obtained from the routine Get_BH_Offset. If MCTDHB applied to the Bose-Hubbard Hamiltonian the default routine Get_1bodyPotential is used to compute the on-site potential energy offset. Both routines can be found in the file ./source/ini_guess_pot/Get_1bodyPotential.F .