Dynamical stabilization of a superfluid motion in the presence of an ac force

We study the superfluid behavior motion of interacting particles in a periodic potential subject to an external ac force. We observe a dynamical stabilization of the superfluid motion in the presence of ac forces around the boundary of the Brillouin zone. We analyze relevant nonlinear Floquet-Bloch states within the mean field approximation that display loops in the energy spectrum. In particular, we investigate with some detail the transport properties of these states at the edge of the Brillouin zone, showing the existence of a nonvanishing motion of particles. These findings are corroborated by a nonlinear generalization of the Hellmann-Feynman expression for the group velocity. Our results shed new light on the observation of stable persistent motion of particles in periodic potentials.

Figure 1

(a) Quasienergy spectrum as a function of the quasimomentum $\kappa$ for $g=0$. (b) Nonlinear Floquet-Bloch bands for various values of the interaction strength $g$. These bands are the continuation in the nonlinear regime of the Floquet band for $g=0$, which is depicted in panel (a): red band. The parameters are $\mathfrak{h}=0.2$, $v_0=1$, $L=2\pi$, $e_0=3.2$, and $\omega =3$. The quasimoment $\kappa$ is in units of the recoil momentum $\mathfrak{h}k$.

Figure 2

Upper panel: Absolute values of the eigenvalues of the matrix (8) vs quasimomentum $\kappa$. (a) $\omega =3$ and $g=0.16$; (b) $\omega =6$ and $g=0.22$. In the inset in panel (b), the quasienergy vs momentum is depicted for $g=0.22$. In this inset, the red circle indicates the point for $\kappa =0.5$ where the PO is stable. Lower panel: Evolution of the wave packet of the nonlinear Floquet-Bloch state for $\kappa =0.5$. (c) Using the parameters of panel (a). (d) Using the parameters of panel (b). The other parameters are the same as in Fig. 1.

Figure 3

(a) Loop in the energy spectrum for $\omega =6$, $e_0=3.2$, and $g=0.34$. The loop is formed by three quasienergy bands depicted in red, black, and brown. (b) Absolute values of the eigenvalues of the matrix (8) vs quasimomentum $\kappa$ for the quasienergy band in red in panel (a). (c) Husimi representation for states with $\kappa =0.48$ and $\kappa =0.515$. Bright color in panel (c) corresponds to high intensity while the dark region implies low intensity. $\mathcal{S}\left(p\right)={\int }_0^{2\pi }I(x,p)dx$ is the overall momentum integrated over the space of the quantum state where $I$ is the Husimi function (11). (d) ${\mathcal{S}}_2\left(p\right)-{\mathcal{S}}_1\left(p\right)$ vs $p$. Here ${\mathcal{S}}_1\left(p\right)$ and ${\mathcal{S}}_2\left(p\right)$ correspond to the overall momentum for $\kappa =0.48$ and $\kappa =0.515$, respectively.

Figure 4

First column: Mean momentum vs quasimomentum $\kappa$. Second column: $\frac{\partial \epsilon }{\partial \kappa }$ vs quasimomentum $\kappa$. Third column: Function $v_{\mathcal{P}}$ vs quasimomentum $\kappa$. Fourth column: Group velocity $\frac{\partial \epsilon }{\partial \kappa }-v_{\mathcal{P}}$ vs $\kappa$. First row: panels (a), (d), (g), and (j) show $g=0$. Second row: panels (b), (e), (h), and (k) show $g=0.22$. Third row: panels (c), (f), (i), and (l) show $g=0.34$. The other parameters are the same as in Fig. 1.