Polaritonic Solitons in a Bose-Einstein Condensate Trapped in a Soft Optical Lattice

We investigate the ground state (GS) of a collisionless Bose-Einstein condensate (BEC) trapped in a soft one-dimensional optical lattice (OL), which is formed by two counterpropagating optical beams perturbed by the BEC density profile through the local-field effect (LFE). We show that LFE gives rise to an envelope-deformation potential, a nonlocal potential resulting from the phase deformation, and an effective self-interaction of the condensate. As a result, stable photon-atomic (polaritonic) lattice solitons, including an optical component, in the form of the deformation of the soft OL, in a combination with a localized matter-wave component, are generated in the blue-detuned setting, without any direct interaction between atoms. These self-trapped modes, which realize the system’s GS, are essentially different from the gap solitons supported by the interplay of the OL potential and collisional interactions between atoms. A transition to tightly bound modes from loosely bound ones occurs with the increase of the number of atoms in the BEC.

Figure 1

The chemical potential of the numerically found ground state vs the number of atoms, $N$. The inset shows details for relatively small values of $N$.

Figure 2

Wave functions of loosely bound ground states for different numbers of atoms.

Figure 3

The ground-state wave functions for larger $N$ than in Fig. 2, in the form of tightly bound solitons.

Figure 4

The deformation of the envelope of the right-traveling electromagnetic wave for different values of $N$.

Figure 5

The numerically found LFE-induced potential, $\delta V$ for $N={10}^5$ and $2\times {10}^5$ (a), and $N=3\times {10}^5$ and $6\times {10}^5$ (b). The approximate potential $\delta V^{\mathrm{appr}}$, given by Eq. (7), is shown, for $N=2\times {10}^5$ and $6\times {10}^5$, by the dotted lines.

Figure 6

The potential terms accounting for the envelope-deformation effect, $\delta V^{\mathrm{def}}$, the effective nonlocality, $\delta V^{\mathrm{nonloc}}$, and the light-induced self-interaction, $\delta V^{\mathrm{SI}}$, are shown for different $N$.