Wannier functions for one-dimensional $s-p$ optical superlattices

The physics of one-dimensional optical superlattices with resonant $s\text{−}p$ orbitals is reexamined in the language of appropriate Wannier functions. It is shown that details of the tight-binding model realized in different optical potentials crucially depend on the proper determination of Wannier functions. We discuss the properties of a superlattice model which quasiresonantly couples $s$ and $p$ orbitals and show its relation with different tight-binding models used in other works.

Figure 1

Energy bands for the double-well potential (3) with parameters $V_0=32E_r$, $\epsilon =2$, and ${\varphi }_1={\varphi }_2=0$. The lowest energy band is not shown, the red (dashed) second band crosses with the green (solid) third band, leading to a linear “relativistic” dispersion around $k=0$.

Figure 2

(a) Two localized squared Wannier functions of the double-well potential (3) with parameters $V_0=32E_r$, $\epsilon =2$, and ${\varphi }_1={\varphi }_2=0$. Functions $W_s$ (red dashed line) and $W_p$ (black solid line) come from Bloch functions transformed by (6). The dotted blue line depicts the lattice potential (not in scale). (b) The modulus squared of the same Wannier functions in the logarithmic scale, revealing their exponential localization within double precision arithmetics [44, 45].

Figure 3

(a) Plot of the $s$- (red dashed line) and $p$-like (black solid line) MLWFs for $\epsilon =1$, corresponding to a large detuning from the resonance. The potential is represented by the dotted (blue) line, whereas the horizontal orange stripes represents the Bloch bands (on the same scale as the potential). (b) The density plot of the same composite-band MLWFs is shown here in logarithmic scale. Note the exponential decay of the tails [44, 45].

Figure 4

Same as Fig. 3, but for $\epsilon =3$.

Figure 5

On-site energies and tunneling amplitudes obtained from the composite-band MLWFs approach as a function of $\varepsilon$. (a) On-site energies $E_s$ (black solid line) and $E_p$ (red dashed line). (b) Tunneling amplitudes. The blue (dotted) line corresponds to the amplitude $|J_l|=|J_r|$ for nearest-neighbor tunneling, i.e., between $s$ and $p$ orbitals; the black (solid) line corresponds to the $s\text{−}s$ tunneling amplitude $J_s$, while the red (dashed) line represents the $p\text{−}p$ tunneling $J_p$. Thin lines represents the values of $J_s$ and $J_p$ obtained from a single-band calculation (see text). The (blue and black) dots correspond to the prediction of the analytic expressions in Eq. (9) for the resonant case ($\varepsilon =2$).

Figure 6

(a) Density plot of the $p$-like single-band Wannier functions (solid line) for $\epsilon =1$, compared with that obtained from the composite-band approach (dashed line). (b) The case of the $s$-like MLWF, as above.

Figure 7

Same as Fig. 6, but for $\epsilon =3$.

Figure 8

Plot of $\delta E_{\alpha }$ ($\alpha =s,p$; see text) as a function of $\varepsilon$, as obtained from the single- and composite-band approaches (indicated in the label as “sb” and “cb,” respectively).

Figure 9

A sketch of the various tunneling and interaction terms of the extended Bose-Hubbard model. The color code is the same as that in Figs. 5 and 10. $J_{sp}$ schematically denotes two types of density-dependent tunnelings, mediated by $s$ and $p$ orbitals densities. Remember that the nearest-neighbor tunneling coefficient $J$ is strictly vanishing within the single-band approach.

Figure 10

Plot of the (modulus of) the integrals $I_{iskp}$ characterizing the amplitude of various interaction terms of the extended Bose-Hubbard Hamiltonian. Lines with symbols correspond to the two-band approach, whereas plain lines refer to the single-band case. Observe that the two-band approach yields generally much smaller corrections to standard Bose-Hubbard terms, showing superiority over single-band Wannier functions.