Surface gap solitons in exciton polariton condensates

A gap soliton is a solitonic state existing inside the band gap of an infinite-periodic exciton-polariton condensate (EPC). The combination of surface states and gap solitons forms the so-called surface gap solitons (SGSs). We analyze the existence of SGSs near the interface between uniform and semi-infinite periodic EPCs. We find that SGSs exist only when the system is excited by a pump with a smaller width. As the width of the pump increases, SGSs become unstable.

Figure 1

Energy dispersion relations without pump and loss effects for $V_1=2$ and $V_2=3$. Energies of $E_1(k=0)$, $E_1(k=\pi /2)$, and $E_2(k=\pi /2)$ are 0.900, 1.710, and 2.707, respectively.

Figure 2

Density profiles of surface gap solitons with $V_2=3$, ${\alpha }_0=0.5$, and $w=0.5$. (red solid curves): (a) a hyperbolic-secant-type SGS with $V_1=2$ and $E=2.110$, (b) a hyperbolic-secant-type SGS with $V_1=4$ and $E=2.542$, (c) a hyperbolic-cosecant-type SGS with $V_1=2$ and $E=0.700$, (d) a hyperbolic-cosecant-type SGS with $V_1=4$ and $E=1.420$. The black dashed curve is the relative strength of the external potential.

Figure 3

Density profiles of surface gap solitons with $V_2=3$, ${\alpha }_0=1.5$, and $w=0.5$. (red solid curves): (a) a hyperbolic-secant-type SGS with $V_1=2$ and $E=2.110$, (b) a hyperbolic-secant-type SGS with $V_1=4$ and $E=2.542$, (c) a hyperbolic-cosecant-type SGS with $V_1=2$ and $E=0.700$, (d) a hyperbolic-cosecant-type SGS with $V_1=4$ and $E=1.420$. The black dashed curve is the relative strength of the external potential.

Figure 4

Peak amplitude ${\left[\right(\left|\psi \left(x\right)\right|)}_{\mathrm{peak}}$, solid line] and position ($x_{\mathrm{peak}}$, dashed line) of SGSs with $V_1=2$ and $V_2=3$. (a) A hyperbolic-secant-type SGS with $\delta =0.4$ and $w=0.3$; (b) a hyperbolic-cosecant-type SGS with $\delta =-0.2$ and $w=0.5$.

Figure 5

Phase boundaries between stable and unstable SGSs with $V_1=2$ and $V_2=3$. (a) A hyperbolic-secant-type SGS with $\delta =0.4$ and $E=2.110$; (b) a hyperbolic-cosecant-type SGS with $\delta =-0.2$ and $E=0.700$.

Figure 6

The numerically identified stable regions for SGSs are located on the left-hand side of borders, which are shown for ${\alpha }_0=1$ and $V_2=3$. (a) Hyperbolic-secant-type SGSs: The black solid curve is the borders of $\delta$ versus $w$ with $V_1=2$. The blue dashed curve is the border of $V_1$ versus $w$ with $\delta =0.4$. (b) Hyperbolic-cosecant-type SGSs : The black solid curve is the borders of $\delta$ versus $w$ with $V_1=2$. The blue dashed curve is the border of $V_1$ versus $w$ with $\delta =-0.2$.

Figure 7

The numerically identified stable regions for SGSs are located on the right-hand side of borders, which are shown for $V_1=2$, $V_2=3$ and $w=0.5$. (a) Hyperbolic-secant-type SGSs with $\delta =0.4$. (b) Hyperbolic-cosecant-type SGSs with $\delta =-0.2$.