Coupled spatial multimode solitons in microcavity wires

A modal expansion approach is developed and employed to investigate and elucidate the nonlinear mechanism behind the multistability and formation of coupled multimode polariton solitons in microcavity wires. With pump switched on and realistic dissipation parameters, truncating the expansion up to the second-order wire mode, our model predicts two distinct coupled soliton branches: stable and unstable. Modulational stability of the stationary homogeneous solution and soliton branches stability are studied. Our simplified 1D model is in remarkably good agreement with the full 2D mean-field Gross-Pitaevskii model, reproducing correctly the soliton existence domain upon variation of pump amplitude and the onset of multistability.

Figure 1

(a) Scheme of the microcavity polaritonic wire structure with a tilted in-plane cw pump; (b) linear polariton dispersion with parabolic fit (dashed curves): All parameters are as in Ref. [36], and coefficients for parabolic fits are listed in Appendix pp2, Eqs. (B1, B2, B3). (c) Modal profiles for $q=1.4237$ (which corresponds to pump inclination at 20 degrees). All modes are normalized such that $N_j=1$ in Eq. (6). Dashed curve indicates scaled profile of the potential ${\mathrm{\Omega }}_R\left(y\right)$.

Figure 2

Bi- and multistability of the homogeneous solution for $\gamma =0.01$, $\mathrm{\Delta }=-0.1$ (left column) and $\mathrm{\Delta }=0$ (right column), cf. Figs. 2(b) and 2(c) in Ref. [36]. Solution in terms of $Q_0$ and $Q_2$ is shown in the top row, and the corresponding conversion in terms of the full $\mathrm{\Psi }$ field is illustrated in the bottom row ($\mathrm{\Psi }$-norm, $N=\int {\left|\mathrm{\Psi }\right|}^{2}dy$).

Figure 3

(a) Coupled stable (type 1) and unstable (type 2) soliton branches, superimposed on the multistability curves of the stationary nonlinear coupled ($Q_0,Q_2$) modes vs $E_p$ (homogeneous solution stability indicated) at $\mathrm{\Delta }=0$ and ${\gamma }_0=0.04$; note that the stable soliton branches sit on a stable background (black curve), while the unstable soliton branches sit on a modulationally unstable background (green curve). Soliton branches corresponding to $Q_0$, $Q_2$, and $Q_4$ soliton components, inferred from the inverse transform [Eq. (4)] of the full-model 2D solitons, computed by a dynamical (split-step) model from [Eqs. (1) and (2)], are shown with open circles connected by magenta line; (b) Stable type 1 $|Q_0|$ and $|Q_2|$-soliton profiles in the middle of the stable soliton branch at $E_p=0.0715$; this soliton profile remains unchanged from the left stable soliton branch edge at $E_p=0.06886$ to the right edge at $E_p=0.07366$. Unstable type2 profiles at the (c) left edge of unstable soliton branch at $E_p=0.07781$; (d) right edge before kink at $E_p=0.08157$; (e) left edge after kink at $E_p=0.08190$; (f) at $E_p=0.09$; (g) right edge of unstable soliton branch at $E_p=0.1038$.

Figure 4

(a) Coupled stable/unstable soliton branches vs $E_p$, superimposed on the homogeneous stationary solution multistability curves (stability indicated) at $\mathrm{\Delta }=-0.1$ and ${\gamma }_0=0.04$. (b) Stable/unstable type1 soliton profiles at the left edge of the soliton branch at $E_p=0.03988$; (c) at $E_p=0.045$ at the edge of the homogeneous solution stable region (black curve); (d) at $E_p=0.0535$. (e) Stable/unstable type1 profiles at the right edge of the soliton branch at $E_p=0.06699$.

Figure 5

Full eigenvalue spectrum of the soliton branch [Fig. 3] $\mathrm{\Delta }=0,{\gamma }_0=0.04$: (a) at $E_p=0.068865$; (b) at $E_p=0.0712$: the solitons are stable for these pump amplitudes $[\lambda =Im(eig)<0]$. Unstable solitons (at least one $\lambda >0$): (c) at the right edge of type1 stable soliton branch at $E_p=0.073665$; (d) at $E_p=0.07781$ (left edge of unstable soliton branch); (e) at $E_p=0.08157$: right edge before kink; (f) at $E_p=0.08190$ after kink; (g) $E_p=0.09$ middle of unstable soliton branch; (h) at $E_p=0.1038$: right edge of unstable soliton branch.

Figure 6

Full eigenvalue spectrum for the soliton branch [Fig. 4] at $\mathrm{\Delta }=-0.1,{\gamma }_0=0.04$ at the: (a) left edge $E_p=0.039882$; (b) right edge of the homogeneous stable solution background $E_p=0.043384$; (c) at $E_p=0.05$; (d) right edge of modulationally unstable background region $E_p=0.058951$; (e) right edge of soliton branch $E_p=0.066993$; the solitons are slightly unstable for all pump amplitudes ($\lambda \approx 0>0$).

Figure 7

Snapshot at $t=80\mathrm{ps}$ of a (a),(b) single-hump soliton for $E_p=0.0672,Es=0.34114$; (c),(d) double-hump soliton for $E_p=0.0762$, $Es=0.34114$ at $\mathrm{\Delta }=0$.

Figure 8

Soliton branch inferred from dynamical computation of Eqs. (1) and (2) superimposed on the coupled multistability curves of the reduced model and the full model (black dash-dotted curve) at $\mathrm{\Delta }=0,{\gamma }_0=0.04$: single-hump solitons persist up to the first maximum, above which double-hump solitons are formed.

Figure 9

3D plot of the reconstructed 2D soliton at $\mathrm{\Delta }=0,{\gamma }_0=0.04$ (see Fig. 3): (a) $\left|E\right|$; (b) $\left|\mathrm{\Psi }\right|$ from type 1 stable soliton ($Q_0,Q_2$) profiles at $E_p=0.0715$; (c),(d) full-model soliton solutions [Eqs. (1) and (2)] at $E_p=0.0715$; (e),(f) reconstructed $\left|E\right|$ and $\left|\mathrm{\Psi }\right|$ 3D plots from the unstable type 2 soliton at $E_p=0.08572$.

Figure 10

3D plot of the reconstructed 2D soliton at $\mathrm{\Delta }=-0.1$, ${\gamma }_0=0.04$ (a) $\left|E\right|$; (b) $\left|\mathrm{\Psi }\right|$ from type $1/2$ stable/unstable soliton ($Q_0,Q_2$) profiles (see Fig. 4) at the (a) left edge of the soliton branch $E_p=0.03988$. (c),(d) Full-model 2D soliton solutions [Eqs. (1) and (2)] for at the left edge; (e, f) Reconstructed solitons at the right edge of the soliton branch for $E_p=0.067$; (g,h) Full-model 2D solitons at the right edge of the soliton branch.

Figure 11

Reconstructed $Q_0$, $Q_2$, and $Q_4$ soliton components from final evolved full-model ($E,\mathrm{\Psi }$) solutions at $\mathrm{\Delta }=0,{\gamma }_0=0.04$, using inverse transform [Eq. (4)]: upper row—$E_p=0.0672$, corresponding to a single-hump soliton; lower row—$E_p=0.0732$, corresponding to a double-hump soliton.

Figure 12

Soliton branches of type 1 stable and type 2 unstable solitons at $\mathrm{\Delta }=0,{\gamma }_0=0.04$ superimposed on the coupled multistability ($Q_0,Q_2$) curves of the reduced model and the multistability curve ($P_{\mathrm{\Psi }}=\int {\left|\mathrm{\Psi }\right|}^{2}dy$ vs $E_p$) full model (black dash-dotted curve). The domain of stable soliton existence is indicated by a rectangle. The soliton branch ($max\left|E\right|$) computed by 2D Newton-Raphson method for the full model is shown for comparison. The points correspond to soliton solutions that cannot be connected to the stable branch, since they represent multihump soliton solutions.

Figure 13

Stability analysis of the homogeneous solution for (a) $\mathrm{\Delta }=-0.1,{\gamma }_0=0.04$; (b) $\mathrm{\Delta }=0,{\gamma }_0=0.04$. Solid blue, dashed red, and solid green curves correspond to stable, unstable, and modulationally unstable branches, respectively.