Effective Dirac dynamics of ultracold atoms in bichromatic optical lattices

We study the dynamics of ultracold atoms in tailored bichromatic optical lattices. By tuning the lattice parameters, one can readily engineer the band structure and realize a Dirac point, i.e., a true crossing of two Bloch bands. The dynamics in the vicinity of such a crossing is described by the one-dimensional Dirac equation, which is rigorously shown beyond the tight-binding approximation. Within this framework we analyze the effects of an external potential and demonstrate numerically that it is possible to demonstrate Klein tunneling with current experimental setups.

Figure 1

The lowest Bloch bands for the case $V_1=5$ and $V_2=1.56$. For $\phi =\pi$ one observes a true crossing between the first and the second excited bands, a Dirac point, which can be used to simulate the Dirac equation.

Figure 2

(a, b) A fit of the relativistic dispersion relation (3) to the Bloch bands $\alpha =1,2$ in the center of the Brillouin zone for $\phi =0$ and $\phi =\pi$, respectively. (c, d) The resulting fit values for the effective parameters $m$ and $c$ as a function of the phase $\phi$. The remaining parameters are the same as in Fig. 1.

Figure 3

Squared modulus of the Wannier states $|w_{\alpha ,n}{\left(x\right)|}^2$ in (top) the ground ($\alpha =0$) and (bottom) the first excited band ($\alpha =1$) at the lattice site $n=0$ in a semilogarithmic scale. We assume a bichromatic optical lattice with $V_1=5$, $V_2=1.56$, and (left) $\phi =0$ and (right) $\phi =\pi$.

Figure 4

Matrix elements $V_{\alpha ,n;\beta ,m}$ of a linear potential $V\left(x\right)=x$ in a Wannier basis as a function of the lattice depth $V_1$. We have plotted the matrix elements for $\alpha =1$, $\beta =2$, and $|n-m|\le 2$, which are the leading corrections to the diagonal approximation (19). As in Fig. 3, we assume $V_2/V_1=1.56/5$ and (left) $\phi =0$ and (right) $\phi =\pi$.

Figure 5

Quantum simulation of Klein-Tunneling out of a dipole trap. Full time evolution of the Schrödinger equation in a bichromatic optical lattice for the case $V_1=5$ and $V_2=1.56$ and a relative phase of (a1) $\phi =0$, (a2) $\phi =0.8\pi$, and (a3) $\phi =\pi$. Dynamics of the effective Dirac equation (20) with effective mass and speed of light given by (b1) $m{c}^2=0.78$, (b2) $m{c}^2=0.24$, and (b3) $m{c}^2=0$.