Role of transverse excitations in the instability of Bose-Einstein condensates moving in optical lattices

The occurrence of energetic and dynamical instabilities in a Bose-Einstein condensate moving in a one-dimensional (1D) optical lattice is analyzed by means of the Gross-Pitaevskii theory. Results of full 3D calculations are compared with those of an effective 1D model, the nonpolynomial Schrödinger equation, pointing out the role played by transverse degrees of freedom. The instability thresholds are shown to be scarcely affected by transverse excitations, so that they can be accurately predicted by effective 1D models. Conversely, transverse excitations turn out to be important in characterizing the stability diagram and the occurrence of a complex radial dynamics above the threshold for dynamical instability. This analysis provides a realistic framework to discuss the dissipative dynamics observed in recent experiments.

Figure 1

Excitation spectrum of a cylindrical condensate at rest in the optical lattice (quasimomentum of the condensate $p=0$) as a function of the quasimomentum $q$ of the excitations, for lattice intensity $s=5$. The first two Bloch bands are shown and for each band we plot the first 20 radial branches. The lowest branch in each band (thickest line) corresponds to axial excitations with no radial nodes. In the inset we show a magnification of the low-energy spectrum.

Figure 2

Bogoliubov sound velocity $c$ as a function of the lattice intensity $s$. Points: slope of the lowest phononic branch in the Bogoliubov spectrum of the cylindrical condensate. Dashed line: analytic prediction $c={\left(\kappa m^{*}\right)}^{-1∕2}$ (see text).

Figure 3

Real part of the excitation spectrum of a cylindrical condensate in a lattice with $s=5$ and for different values of the condensate quasimomentum $p\left(p∕q_B=0,0.25,0.5,0.55,0.75,1\right)$, calculated from Eqs. (12, 13).

Figure 4

Real (top) and imaginary (bottom) parts of the frequencies obtained from Eqs. (12, 13) for $s=5$ and $p∕q_B=0.55$. The vertical (dotted) lines correspond to the points where a conjugate pair of complex frequencies appears.

Figure 5

Excitation spectrum obtained from the NPSE for a cylindrical condensate in a lattice with $s=5$ and for different values of the quasimomentum $p$, as in Fig. 3 $\left(p∕q_B=0,0.25,0.5,0.55,0.75,1\right)$.

Figure 6

Stability diagrams obtained from the GP spectra (left) and NPSE spectra (right) as a function of the quasimomenta $p$ and $q$ of the condensate and of the quasiparticles, respectively, and for $s=1,5,10$. Shaded areas represent the regions where the system is dynamically (dark) and/or energetically (both light and dark) unstable. Dashed lines correspond to the modes having the largest imaginary part (most unstable modes).

Figure 7

Thresholds for energetic (circles) and dynamical (squares) instability as a function of the lattice intensity $s$. Up to $s\simeq 8$, the results of GP (filled symbols) and NPSE (empty symbols) nicely agree. Shaded areas represent the regions where the system is dynamically (dark) and/or energetically (both light and dark) unstable, according to the GP.

Figure 8

Growth rates of the most dynamically unstable modes for different values of the lattice intensity $s$ ($s=0.1$ triangles, $s=1$ squares, $s=5$ circles) obtained with both GP and NPSE (filled and empty symbols, respectively).

Figure 9

Top: GP stability diagram for a cylindrical condensate which approximates the experimental elongated condensate of Ref. [1] in its central region. Light and dark shaded areas are unstable regions, as in Fig. 6. Bottom: center-of-mass velocity as a function of the condensate quasimomentum $\left(v_B=\hslash q_B∕m\right)$. The critical velocities for the energetic $\left(v_E\right)$ and dynamical $\left(v_D\right)$ instabilities of the cylindrical condensate are shown as horizontal dotted lines. The horizontal dashed line is the velocity at which the instability occurs in the GP simulation of the experimental 3D condensate of Ref. 1.

Figure 10

Center-of-mass velocity of the condensate during its evolution in the harmonic trap after an initial displacement of $60\mu \mathrm{m}$. The velocity initially increases, as expected for the harmonic dipole oscillation of the whole condensate, up to a critical value at which the condensate breaks up. The corresponding quasimomentum $p$ lies in the dynamical unstable region of Fig. 9. In the insets we plot the linear density of the condensate along $z$ at different times. The horizontal ruler has a width of $280\mu \mathrm{m}$ and the center of the trap is at midpoint.

Figure 11

Density plots in the $(z,x)$ plane for different evolution times, corresponding to the inset pictures of Fig. 10 $\left(t=0,30,35,40,50\mathrm{ms}\right)$, after an initial displacement $\Delta z=60\mu \mathrm{m}$. The parameters are those of the experiment in Ref. 1.