Two-Dimensional Localization of Exciton Polaritons in Microcavities

We report two-dimensional localization of exciton polaritons in a coherently pumped planar semiconductor microcavity operating in the strong-coupling regime. Two-dimensional polariton solitons exist despite the opposite dispersion signs along the orthogonal in plane directions. Nonlinearities compensating the opposing dispersions have different physical origins and are due to the repulsion of polaritons on one side and due to parametric four-wave mixing on the other. Both of these nonlinearities can support their respective families of one-dimensional solitons, which coexist with each other and with the two-dimensional solitons.

Figure 1

Red (upper and lower) and blue (middle) lines in the ($\Omega ,k_x$) plane indicate the dispersion characteristics of the linear and nonlinear polaritons, respectively. $|{\Psi }_{\kappa }|$ is the spectrum corresponding to the 1D parametric soliton shown in Fig. 6c.

Figure 2

(a) Bistability loop of the homogeneous solution (HS). The dashed line corresponds to the unstable HS. The dotted line marks the maximum intensity of the 1D CPSs localized along $x$, as shown in (b). (b) Soliton stripe moving in the $x$ direction with velocity $V=0.56$. Parameters: $E_p=0.139$, $\Delta =-0.25$, $k_p=1.2$, ${\gamma }_{\mathrm{ph}}={\gamma }_{\mathrm{ex}}=0.1$.

Figure 3

Moving fronts connecting 1D single-hump (a) and double-hump (b) CPSs and HS background. (c) Velocities $V_f$ of the single-hump (solid line) and double-hump (dashed line) fronts. $V_f=0$ at the Maxwell points MP1 and MP2. Parameters as in Fig. 2.

Figure 4

(a) Shrinking of the initially localized excitation observed for $E_p=0.136$. (b) Spreading of the initial excitation observed for $E_p=0.141$. (c) Formation of a stable 2D CPS for $E_p=0.1378$. (d) Long-term dynamics showing the dynamical robustness and confirming the attractor properties of 2D CPSs, $E_p=0.1378$. Other parameters as in Fig. 2.

Figure 5

(a) Maxima of $|E|$ for different soliton solutions versus the pump $E_p$. CPS1 and CPS2 correspond to single- and double-hump solitons as in Figs. 6a, 6b, respectively. The dashed line marks a branch of the parametric CPSs; see Fig. 6c. Parameters: $\Delta =-0.25$, $k_p=1.2$. (b) $y$ profiles of the pump, signal, and idler components of the photonic component of the parametric CPS from Fig. 6c.

Figure 6

Profiles of the excitonic components of the stable bright polariton solitons in the coordinate (a)–(c) and momentum (d)–(f) space. Panels (a),(b),(d),(e) correspond to the single- and double-hump 2D solitons computed for $E_p=0.1378$. Panels (c),(f) correspond to the parametric soliton localized only along $y$ coordinate computed for $E_p=0.1375$. Parameters: $\Delta =-0.25$, $k_p=1.2$.