Gap solitons in Rabi lattices

We introduce a two-component one-dimensional system, which is based on two nonlinear Schrödinger or Gross-Pitaevskii equations (GPEs) with spatially periodic modulation of linear coupling (“Rabi lattice”) and self-repulsive nonlinearity. The system may be realized in a binary Bose-Einstein condensate, whose components are resonantly coupled by a standing optical wave, as well as in terms of the bimodal light propagation in periodically twisted waveguides. The system supports various types of gap solitons (GSs), which are constructed, and their stability is investigated, in the first two finite bandgaps of the underlying spectrum. These include on- and off-site-centered solitons (the GSs of the off-site type are additionally categorized as spatially even and odd ones), which may be symmetric or antisymmetric, with respect to the coupled components. The GSs are chiefly stable in the first finite bandgap and unstable in the second one. In addition to that, there are narrow regions near the right edge of the first bandgap, and in the second one, which feature intricate alternation of stability and instability. Unstable solitons evolve into robust breathers or spatially confined turbulent modes. On-site-centered GSs are also considered in a version of the system that is made asymmetric by the Zeeman effect, or by birefringence of the optical waveguide. A region of alternate stability is found in the latter case too. In the limit of strong asymmetry, GSs are obtained in a semianalytical approximation, which reduces two coupled GPEs to a single one with an effective lattice potential.

Figure 1

The bandgap spectrum of the linearized version of Eqs. (1) and (2) with $\epsilon =6$, for $b=0$ (a) and $b=3$ (b), which correspond, respectively, to the symmetric and asymmetric system. Shaded areas designate Bloch bands where gap solitons do not exist. The semi-infinite gap and lowest finite ones are labeled.

Figure 2

(a) A typical example of stable on-site-centered symmetric GSs, found in the first finite bandgap, at $\mu =-2, b=0$, and $\epsilon =6$. Here, and in similar figures below, the background pattern (red sinusoid) represents the scaled underlying Rabi lattice [periodic modulation of the coupling constant, $-\left(\epsilon /6\right)cos\left(2x\right)$, with scaling factor $1/6$ added to keep the sinusoid within boundaries of the panel]. (b) The eigenvalue spectrum for small perturbations around the soliton, which confirms its stability. It is further corroborated by simulations of the evolution of the soliton initially perturbed by random noise at the level of $5\%$ of the soliton's amplitude, which are displayed in panels (c) and (d).

Figure 3

The same as described in the caption of Fig. 2, but for a typical unstable on-site-centered symmetric soliton in the second finite bandgap, at $\mu =3$ and $\epsilon =6$. As seen in panels (c) and (d), development of the oscillatory instability replaces the GS by a “turbulent” pattern, which remains spatially confined.

Figure 4

(a) The numerically found amplitude of fundamental on-site-centered symmetric GSs versus chemical potential $\mu$, for $\epsilon =6$ and $b=0$. The dashed magenta curve is produced by the Thomas-Fermi approximation (TFA) for the single-component GSs: $U(x=0)=V(x=0)=\sqrt{6+\mu }-3{(6+\mu )}^{-3/2}$. Panel (b) shows the total norm $N$ versus $\mu$, along with its dashed-magenta TFA counterpart, $N=2\left[\sqrt{36-{\mu }^2}+\mu {cos}^{-1}(-\mu /6)\right]$. In these panels, as well as in Fig. 7 below, red solid and black dashed segments represent stable and unstable GSs in the first and second bandgaps, respectively, while the green dotted segment designates a region of alternate stability and instability in the second bandgap. The red dotted segment near the right edge of the first bandgap designates a region of alternating stable GSs and ones weakly unstable against oscillatory perturbations. (c) The alternation of the stability and instability in the green dotted segment of (b). Here, red triangles and black circles represents stable and unstable GSs, respectively. (d) The alternation of the stability and weak oscillatory instability in the red dotted segment of (b). Here, and also in Fig. 7, red triangles and blue filled circles represent stable GSs and ones subject to the weak instability, respectively.

Figure 5

An example of a weakly unstable GS with $\mu =0.32$, belonging to the red dotted segment near the right edge of the first finite bandgap in Fig. 4. (a) Unstable eigenvalues of small perturbations are complex with small imaginary parts. (b, c) Direct simulations corroborate the weak oscillatory instability of the soliton, which keeps its localized shape. (d) The weak instability is additionally illustrated by the time dependence of squared amplitudes of both components of the same solutions.

Figure 6

Panels (a) and (b) display a typical stable off-site-centered antisymmetric spatially even GS for $\mu =-2$, which falls into the first finite bandgap. (c) Direct simulations (with the initial random-amplitude perturbation at the $5\%$ level) of the evolution of the same soliton, which corroborate its stability. (d) Simulations of the perturbed evolution of the GS of the same type, but belonging to the second finite bandgap, with $\mu =2$, demonstrate that this unstable soliton is finally transformed into a spatially confined turbulent state.

Figure 7

(a) Total norm $N$ versus chemical potential $\mu$ for off-site-centered antisymmetric spatially even GSs, cf. Fig. 4 for the on-site centered symmetric solitons. (b) The $N\left(\mu \right)$ curve for the family of off-site-centered antisymmetric spatially-odd GSs. Here, the red and black dashed curves refer to unstable solitons which evolve, respectively, into robust breathers [see Fig. 8], or into a confined turbulent state shown in Fig. 8. (c) The detailed structure of the red-dotted segment with alternate stability and instability in panel (a). The red triangles and blue filled circles label stable and weakly unstable solutions, respectively.

Figure 8

(a, b) An example of an unstable off-site-centered spatially-odd antisymmetric GS for $\mu =-3$ (which belongs to the first finite bandgap) with $b=0$ and $\epsilon =6$. (c) The perturbed evolution (with initial random perturbations at the $5\%$ level) of the same soliton shows its transformation into a persistent breather. (d) Simulations of the perturbed evolution of the GS of the same type, but with $\mu =2$ (which falls into the second finite bandgap) show its transformation into a confined turbulent mode.

Figure 9

Examples of stable on-site-centered asymmetric GSs with $\epsilon =6$, and fixed chemical potential $\mu =-3.5$ : (a) $b=0.2$; (b) $b=1$; (c) $b=3$. Fields $U\left(x\right)$ and $V\left(x\right)$ with larger and smaller amplitudes are shown by blue and green lines, respectively.

Figure 10

(a, b) Direct simulations of the perturbed evolution of a stable asymmetric on-site-centered GSs with $(b,\mu )=(3,-3.5)$ . (c, d) The same for an unstable asymmetric soliton, with $(b,\mu )=(3,-1.5)$.

Figure 11

(a) Total norm $N$ of asymmetric on-site-centered GSs versus chemical potential $\mu$, at $\epsilon =6$ and fixed values of the asymmetry coefficient: $b=1,b=3$, and $b=5$. (b) Asymmetry ratio, $R$, defined as per Eq. (9), versus $N$, for the same soliton families. The family with $b=0$, which has $R\equiv 0$, is included too, for the completeness' sake. In these panels, solid and dashed lines represent stable and unstable GSs, respectively, while the dotted segments designate regions of alternate stability and instability.

Figure 12

Panels (a), (b), and (c) show the detailed structure of regions with alternating stable and unstable on-site-centered asymmetric GSs in the regions shown by the dotted line in Fig. 11, for $b=1,3,$ and 5, respectively. Red triangles and black circles represent, severally, stable and unstable solitons.

Figure 13

Stability diagram for on-site-centered asymmetric GSs in the planes of $(b,\mu )$ (a) and $(N,b)$ (b). The red and gray colors designate, respectively, the stability area, and the one of alternate stability and instability. GSs do not exist in the top blank areas in panels (a). In the bottom blank area in (a) and blank area in (b), there exist completely unstable GSs.

Figure 14

Panels (a1, a2) and (b1, b2) present typical examples of stable GSs produced, respectively, by the full system of Eqs. (1) and (2), and by the semianalytical approximation which amounts to the single Eq. (14), for $(b,\mu )=(10,-9.1)$. Results of direct simulations, initiated by the full two-component solution, and by the single-component approximation, are displayed, for the $u$ component, in panels (a3) and (b3), respectively.