Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability

In an example of Bose-Einstein condensates embedded in two-dimensional optical lattices, we show that in nonlinear periodic systems modulational instability and interband tunneling are intrinsically related phenomena. By direct numerical simulations we find that tunneling results in attenuation or enhancement of instability. On the other hand, instability results in asymmetric nonlinear tunneling. The effect strongly depends on the band gap structure and it is especially significant in the case of the resonant tunneling. The symmetry of the coherent structures emerging from the instability reflects the symmetry of both the stable and the unstable states between which the tunneling occurs. Our results provide evidence of the profound effect of the band structure on the superfluid-insulator transition.

Figure 1

The two lowest bands of the potential (2). In case I, where $V_0=-0.2$, the bands overlap: $E_X-E_M\approx -0.045$. When $V_0=-0.2575$, the band structure transforms into that shown in the case II: the gap edges acquire practically the same value: $E_X-E_M\approx {10}^{-5}$. Further increase of $|V_0|$ leads to opening the gap, as is shown in case III, where $V_0=-0.27$ and $E_X-E_M\approx 0.1025$. The inset in case I shows notations of the high symmetry points in the Brillouin zone.

Figure 2

(a) Dynamics of the density maximum. (b) Time evolution of the occupation rates $r_M(t)$ (red, bottom lines) and $r_X(t)$ (blue, top lines). (c) Snapshots of the atomic density distributions after instability is developed. All simulations are done with the initial distributions $r_M(0)=0.2$ and $r_X(0)=0.8$. The cases I, II, and III correspond to those in Fig. 1.

Figure 3

The same as in Fig. 2, but with initial particles distribution between points $M$ and $X$ taken as follows: $r_M(0)=0.8$ and $r_X(0)=0.2$. To accelerate modulational instability we impose an initial perturbation $0.1|{\Psi }_{\max }|\sin (0.314x)\sin (0.314y)$ on the numerically obtained linear Bloch states.