Spatial period doubling in Bose-Einstein condensates in an optical lattice

We demonstrate that there exist stationary states of Bose-Einstein condensates in an optical lattice that do not satisfy the usual Bloch periodicity condition. Using the discrete model appropriate to the tight-binding limit we determine energy bands for period-doubled states in a one-dimensional lattice. In a complementary approach we calculate the band structure from the Gross-Pitaevskii equation, considering both states of the usual Bloch form and states which have the Bloch form for a period equal to twice that of the optical lattice. We show that the onset of dynamical instability of states of the usual Bloch form coincides with the occurrence of period-doubled states with the same energy. The period-doubled states are shown to be related to periodic trains of solitons.

Figure 1

Energy per particle in units of $K$ as a function of wave number $k$ for $p=0,1,2$, and 3 (indicated by labels). (a) $U\nu ∕K=2$. (b) $U\nu ∕K=6$. The $p=1$ phase states are indicated by the solid circle at $k=\pi ∕2d$.

Figure 2

Energy per particle in units of $E_R$ as a function of wave number $k$ for the lowest bands as obtained from the wave function (13). The bold curves correspond to spatial period-doubled states, and the thin curves to the usual Bloch states (see text).

Figure 3

Particle density ${\mid \psi \left(x\right)\mid }^2∕n$ as a function of position for the lower- (full line) and higher-energy period-doubled states (dashed line) of Fig. 2a at $\mid k\mid =\pi ∕2d$. Lower panel: the periodic potential, Eq. (10).