Reexamination of the variational Bose-Hubbard model

For strongly interacting bosons in optical lattices, the standard description using the Bose-Hubbard model becomes questionable. The role of excited bands becomes important. In such a situation, we compare results of simulations using the multiband Bose-Hubbard model with a recent proposition based on a time-dependent variational approach. It is shown that the latter, in its original formulation, uses a too small variational space, often leading to spurious effects. Possible expansion of the variational approach is discussed.

Figure 1

Ground-state energy for six particles in a four-site system with periodic boundary conditions calculated within the MBH model for ${\mathcal{N}}_M=1–5$ Bloch bands included (black curves with number of bands indicated). Results for the variational ansatz of Ref. [28] are shown as a dashed red curve—they converge for ${\mathcal{N}}_V>2$. The energy is shown with respect to the ground-state energy of the standard Bose-Hubbard model as a function of the coupling constant $g.$ Clearly, for $g>1$, the TDV approach based on (11) fails to approximate the ground-state energy.

Figure 2

Population of lattice sites in time for the initial Fock state $|2,2,1,1\rangle$ (all particles in the lowest band) for different interaction strengths: panel (a) $g=0.2$, panel (b) $g=2$, and panel (c) $g=4$. Results from the TDV approach are represented by black solid curves; MBH predictions are shown as colored curves with stars and circles. The first two and second two sites are equivalent due to assumed periodic boundary conditions.

Figure 3

Populations of different lattice sites in time for [panel (a)] the initial Fock state $|2^{+},2,1,1\rangle$ and [panel (b)] $|2^{++},2,1,1\rangle$, where “+” (“++”) denote occupation by both particles of the first (second) excited Bloch band. Black curves show the occupation of site 1, and red curves show the occupation of site 2. Solid (dashed) lines represent results obtained with the help of TDV (MBH) models. The site in which particles originally resided in the excited band is not depleted at all within the TDV approach if the excited band has opposite parity (the case shown in the top plot with $g=0.2$) or if the atoms are noninteracting (bottom plot, $g=0$).

Figure 4

Final energy after a linear quench of interaction strength $g$ during time $\tau$, in panel (a), starting from $g_{\mathrm{ini}}=0.2$ up to $g_{\mathrm{fin}}=1.0$ and in panel (b), $g_{\mathrm{ini}}=1.0,g_{\mathrm{fin}}=5.0$. Red (upper) solid curves show results obtained using the TDV method (for five bands), whereas, black thicker lines correspond to the simulation using the MBH model (with three bands). Horizontal dashed lines show ground-state energies for corresponding methods (see text).

Figure 5

Transfer efficiency from the ground state during a modulation of total duration $T=400\hslash /E_r$. The top panel shows the MBH results, and the lower panel (in the mirror image) corresponds to the TDV model calculations.

Figure 6

Ground-state energy for four particles distributed on three sites calculated within the MBH model for ${\mathcal{N}}_M=1,...,4$ Bloch bands included (black curves). Results for the variational ansatz with $D=1,2$ and ${\mathcal{N}}_V=5$ are shown in red. The energy is presented with respect to the ground-state energy of the standard BH model.