Nonequilibrium and nonlinear defect states in microcavity-polariton condensates

The nonequilibrium and nonlinear defect modes (NNDMs), localized by a defect in a nonequilibrium microcavity-polariton condensate (MPC), are studied. There are three analytic solutions of NNDMs in a point defect: the bright NNDM, a bound state with two dark solitons for an attractive potential, and a gray soliton bound by a defect for a repulsive potential. We find that the stable NNDMs in a nonequilibrium MPC are the bright NNDM and gray soliton bound by a defect. The bright NNDM, which has the hyperbolic cotangent form, is a bright localized state existing in a uniform MPC. The bright NNDM is a unique state occurring in a nonequilibrium MPC that has pump-dissipation and repulsive-nonlinearity characters. No such state can exist in an equilibrium system with repulsive nonlinearity.

Figure 1

Wave function ${\psi }_0\left(x\right)$, squared wave function ${\left({\psi }_0\left(x\right)\right)}^2$ and reservoir density $n_0$ of the analytical localized states. Panels (a) and (d) show the bright NNDM localized by a point defect with $V_0=-1$; panels (b) and (e) show the bound state with two dark solitons localized by a point defect with $V_0=-1$; and panels (c) and (f) show the gray soliton localized by a point defect with $V_0=1$. The pump parameter is $\alpha =1$.

Figure 2

Absolute wave function $|{\psi }_0\left(x\right)|$, squared norm $|{\psi }_0{\left(x\right)|}^2$, and reservoir density $n_0$ of numerical solutions for $\alpha =1$ and $a=0.5$. Diagrams (a), (c), and (e) are the solutions of an attractive Gaussian defect ($V_d=-1$); panels (b), (d), and (f) are the solutions of a repulsive Gaussian defect ($V_d=1$). No solution can be found for an attractive Gaussian defect ($V_d=-1$) corresponding to the initial condition of Figs. 1, 1, and 1.

Figure 3

The Bogoliubov excitation spectra of the lowest excitation energies corresponding to the NNDMs shown in Figs. 1 and 2. Panels (a), (b), and (c) are excitation frequencies of the bright NNDM, the bound state with two dark solitons and the gray soliton bound by a point defect, respectively. Panels (d) and (e) are excitation frequencies of the bright NNDM and the gray soliton bound by a finite-size defect, respectively. The solid and empty circles represent the real and imaginary part of the eigenvalues, respectively. The parameters: $\sigma =5, \alpha =1, a=0.5, |V_0|=1$, and $|V_d|=1$.