Dark solitons and vortices in the intrinsic bistability regime in exciton polariton condensates

We report on a class of dark solitons and vortices existing in the exciton-polariton condensates and having discontinuity in their excitonic component. These solutions exist due to an intrinsic bistability of the exciton density in the given optical field and for nonzero detuning between the cavity and excitonic resonances. We specify a well defined energy boundary where they transform into previously known polaritonic dark solitons and vortices.

Figure 1

Areas shaded in green and black are the existence domain of the spatially homogeneous and modulationally stable background. The green part corresponds to the parameters where a stable background coexists with the intrinsic bistability and the black one is where it does not.

Figure 2

Potential functions $F_m$, Eq. (10), for the intrinsic bistability ($q=0, q=-1.8$) and no intrinsic bistability ($q=-2.2$) cases. The detuning is fixed to $\delta =-2$.

Figure 3

The phase planes for the potentials (a) $F_{2,3}$ and for (b) $F_{1,2,3}$: $q=0, \delta =-2$. Bold full and dashed lines show possible heteroclinic connections and jumps between the different potentials corresponding to the dark solitons with (a) one or (b) two discontinuities.

Figure 4

The soliton solutions corresponding to the phase trajectories shown in Fig. 3

Figure 5

Transition from the intrinsic bistability dark solitons to the conventional ones. (a) shows the excitonic density profiles of the dark solitons with discontinuities at $x=0$ ($q=-1.7, q=-1.9$) and of the conventional ones ($q=-2.1$): $\sigma =0$. (b) shows how the background excitonic density ($max\left|\psi \right|$, red line) and the density at the center of the dark soliton ($min\left|\psi \right|$, black line) vary with the energy parameter $q$ for $\delta =-2$. The transition between the two types of solitons happens at $q=\delta$. The dashed vertical lines mark the $q$ values at the lower and upper polariton branches. The full vertical line marks the intrinsic bistability boundary.

Figure 6

(a) Exciton density profiles for the dark soliton solutions. Parameters are the same as in Fig. 5, but $\sigma =0.0025$. (b) Line 1 is the width of the photonic core for the dark soliton solutions across the full range of the soliton existence. Line 2 is the width of the broad part of the excitonic density in the soliton core. The dashed red line shows the width of the narrow part of the excitonic density in the soliton core. The dashed vertical lines mark the $q$ values at the lower and upper polariton branches. The full vertical line marks the intrinsic bistability boundary.

Figure 7

(a) Photonic and excitonic components of the intrinsic bistability dark solitons with $\sigma \ne 0$. The other parameters are $\delta =1.55, q=-0.45$. (b) The dark soliton instability growth rate (full lines) as a function of $q$ for $\sigma =0.25,0.025,0.0025$, and the same for the soliton with two discontinuities separated by a distance 1, $\sigma =0$ (dashed curve).

Figure 8

Radial profiles of intrinsic bistability vortices with $m=1$ (solid curves) and $m=2$ (dashed curves). In panel (a) vortex solitons with the discontinuity in $\mathrm{\Psi }$ field in the center ($r=0$) are shown. The vortex solitons in panel (b) have the discontinuity along the ring ($r=1$). The parameters are $\sigma =0, q=0, \delta =-2$.

Figure 9

The variation of the pump amplitude $E_p\left(t\right)$ in the numerical experiment. The inset shows the spatial distribution of the intensity of the pump.

Figure 10

Field evolution in the numerical experiment: fields (a) $E$ and (b) $\mathrm{\Psi }$, and renormalized fields (c) $\varepsilon =\left|E\right|/max\left|E\right|$ and (d) $\varphi =\left|\mathrm{\Psi }\right|/max\left|\mathrm{\Psi }\right|$. Parameter values are $\sigma =0, q_p=0, \delta =-2, {\gamma }_1={\gamma }_2=0.1$. The inset in panel (b) shows profiles of the $\left|\mathrm{\Psi }\right|$ field at $t-t_0=6$ and $t-t_0=15$; the curves are marked correspondingly.

Figure 11

Field profiles in the numerical experiment for the times (a) $t-t_0=6$ and (b) $t-t_0=15$. Dashed curves show the stationary vortex solution for the conservative problem.