Nearly-one-dimensional self-attractive Bose-Einstein condensates in optical lattices

Within the framework of a mean-field description, we investigate atomic Bose-Einstein condensates, with attraction between atoms, under the action of a strong transverse confinement and periodic [optical-lattice (OL)] axial potential. Using a combination of the variational approximation, one-dimensional (1D) nonpolynomial Schrödinger equation, and direct numerical solutions of the underlying 3D Gross-Pitaevskii equation, we show that the ground state of the condensate is a soliton belonging to the semi-infinite band gap of the periodic potential. The soliton may be confined to a single cell of the lattice or extended to several cells, depending on the effective self-attraction strength $g$ (which is proportional to the number of atoms bound in the soliton) and depth of the potential, $V_0$, the increase of $V_0$ leading to strong compression of the soliton. We demonstrate that the OL is an effective tool to control the soliton’s shape. It is found that, due to the 3D character of the underlying setting, the ground-state soliton collapses at a critical value of the strength, $g=g_c$, which gradually decreases with the increase of $V_0$; under typical experimental conditions, the corresponding maximum number of ${}^7\mathrm{Li}$ atoms in the soliton, $N_{\max }$, ranges between 8000 and 4000. Examples of stable multipeaked solitons are also found in the first finite band gap of the lattice spectrum. The respective critical value $g_c$ again slowly decreases with the increase of $V_0$, corresponding to $N_{\max }\simeq 5000$.

Figure 1

(Color online) Axial length $\eta$, transverse width $\sigma$, and energy per particle, $E$, of the quasi-1D soliton versus the self-attraction strength; $g\equiv 2\mid a_s\mid N∕a_{\perp }$, as predicted by the variational approximation based on ansatz (4) and Eqs. (7, 8). The dependences are displayed at fixed values of depth $V_0$ of the periodic potential (3), with $k_L=1$. On the right side, all curves terminate at a critical point $g=g_c\left(V_0\right)$, beyond which the collapse occurs (the upper curve in the top panel, which is cut by the panels’s frame, actually extends up to $g=0$, like its counterpart in the bottom panel).

Figure 2

Continuation of Fig. 1: chemical potential $\mu$ of the quasi-1D soliton versus $g$, as predicted by the variational approximation for the same three values of $V_0$, and $k_L=1$.

Figure 3

The critical value of the effective self-attraction strength $g_c$, above which the soliton collapses, versus wave number $k_L$ of the periodic potential (3). The results are obtained from the numerical solution of variational equations (7, 8).

Figure 4

The axial density profile ${\mid f\left(z\right)\mid }^2$ of the soliton in the periodic potential (3), with $k_L=1$ and four different values of $V_0$. The self-attraction strength is fixed at $g=0.5$.

Figure 5

Axial length of the ground-state bright soliton, ${⟨z^2⟩}^{1∕2}$, as a function of self-attraction strength $g$, for $V_0=0.4$ and $k_L=1$ in Eq. (3). Displayed are results provided by the variational approximation based on the Gaussian ansatz—i.e., Eqs. (7, 8)—and by the nonpolynomial Schrödinger equation (NPSE), Eq. (13).

Figure 6

The axial density profile $\rho \left(z\right)$ of the soliton in the potential (3), with $k_L=1$ and $V_0=0.2$. Comparison between results provided by the different equations—NPSE, 3D GPE, and 1D GPE (the latter two with the cubic nonlinearity)—is presented. In the case of the 3D equation (1), the axial density is defined as $\rho \left(z\right)=\int \int {\mid \psi \left(\mathbf{r}\right)\mid }^{2}dxdy$, while in the other cases it is simply ${\mid f\left(z\right)\mid }^2$.

Figure 7

The axial-density profile ${\mid f\left(z\right)\mid }^2$ of the lowest state in the first finite band gap of the periodic potential (3) with $k_L=1$. The self-attraction strength is $g=0.5$. The top, middle, and bottom panels correspond, respectively, to $V_0=0.2$, $V_0=0.6$, and $V_0=1$.

Figure 8

Eigenvalues ${ϵ}_j$ found from Eq. (15) with $V_0=1$ and $k_L=1$, as functions of self-attraction strength $g$.