Emulation of lossless exciton-polariton condensates by dual-core optical waveguides: Stability, collective modes, and dark solitons

We propose a possibility to simulate the exciton-polariton (EP) system in the lossless limit, which is not currently available in semiconductor microcavities, by means of a simple optical dual-core waveguide, with one core carrying the nonlinearity and operating close to the zero-group-velocity-dispersion point, and the other core being linear and dispersive. Both two-dimensional (2D) and one-dimensional (1D) EP systems may be emulated by means of this optical setting. In the framework of this system, we find that, while the uniform state corresponding to the lower branch of the nonlinear dispersion relation is stable against small perturbations, the upper branch is always subject to the modulational instability. The stability and instability are verified by direct simulations too. We analyze collective excitations on top of the stable lower-branch state, which include a Bogoliubov-like gapless mode and a gapped one. Analytical results are obtained for the corresponding sound velocity and energy gap. The effect of a uniform phase gradient (superflow) on the stability is considered too, with a conclusion that the lower-branch state becomes unstable above a critical wave number of the flux. Finally, we demonstrate that the stable 1D state may carry robust dark solitons.

Figure 1

Scaled effective densities of the uniform exciton-polariton condensate versus the scaled chemical potential, $\mu /{\epsilon }_C$, for the effective (emulated) Rabi coupling $\Gamma =0.75{\epsilon }_C$. Dashed lines: effective exciton density ${\psi }_0^2$; dot-dashed lines: the respective cavity-photon density ${\varphi }_0^2$; solid lines: total density $n_0$. Other parameters are the effective exciton-exciton repulsion strength $g$, and the exciton and cavity-photon energies ${\epsilon }_X=0$ and ${\epsilon }_C$ (actually emulated by the propagating-constant shifts in the dual-core wave guide), at zero wave number. The curves below and above $\mu /{\epsilon }_C=1$ correspond, respectively, to the lower and upper branches.

Figure 2

Exciton fraction ${\psi }_0^2/n_0$ in the exciton-polariton condensate as a function of the scaled density of the cavity photons, $g{\varphi }_0^2/({\epsilon }_C-{\epsilon }_X)$. The four curves correspond to different values of the scaled Rabi coupling, $\Gamma /({\epsilon }_C-{\epsilon }_X)$.

Figure 3

Top panel: scaled energy $m_C{c}_s^2/({\epsilon }_C-{\epsilon }_X)$ of the first-sound mode, with the sound speed $c_s$, versus the scaled effective cavity-photon density, $g{\varphi }_0^2/({\epsilon }_C-{\epsilon }_X)$. Bottom panel: scaled energy gap ${\omega }_0^{+}/({\epsilon }_C-{\epsilon }_X)$ of the gapped branch versus the scaled cavity-photon density. In each panel, four curves correspond to different values of the scaled linear coupling, $\Gamma /({\epsilon }_C-{\epsilon }_X)$.

Figure 4

The evolution of the initial circular perturbation in the form of a small hole produced on top of the stable uniform state, per Eqs. (26) and (27). Each panel displays contour plots of the total density, $n_0(x,y,t)$, at fixed values of the propagation distance (effective time), $t.$ The top, middle, and bottom panels correspond to $t=9$, $t=15$, and $t=21$, respectively. Parameters are ${\epsilon }_X=0$, ${\epsilon }_C=1$, $g=1$, $\Gamma =0.75$. Initial conditions are taken as in Eqs. (26) and (27) with ${\psi }_0=1$, ${\varphi }_0=-1$, $A=0.1$ and $\sigma =2$.

Figure 5

Frequencies of small excitations ${\omega }_{(k,0)}$ above the stable uniform state with wave vector $\mathbf{q}=(q,0)$ of the phase flux. Parameters are ${\epsilon }_X=0$, ${\epsilon }_C=1$, $g=1$, $\Gamma =0.75$, $m_X=1000$, $m_C=1$, and $\mu =0.25$. Each line corresponds to a different solution (branch) of Eq. (32).

Figure 6

Real and imaginary parts, $\mathrm{Re}\left[{\omega }_k\right]$ and $\mathrm{Im}\left[{\omega }_k\right]$, of excitation frequencies ${\omega }_k$ on top of the unstable uniform state with wave vector $\mathbf{q}=(1.5,0)$ of the phase flux. Parameters are ${\epsilon }_X=0$, ${\epsilon }_C=1$, $\gamma =1$, $\Gamma =0.75$, $m_X=1000$, $m_C=1$, and $\mu =0.25$. Each line corresponds to a different solution (branch) of Eq. (32).

Figure 7

The evolution of a small hole produced, as a perturbation, on top of a stable uniform 1D state. In each panel, density profiles $n(x,t)$ are displayed. Parameters are ${\epsilon }_X=0$, ${\epsilon }_C=1$, $g=1$, and $\Gamma =0.75$. Initial conditions are given by Eqs. (26) and (27) with ${\psi }_0=1$, ${\varphi }_0=-1$, $A=0.1$, and $\sigma =2$.

Figure 8

The evolution of a small hole produced, as a perturbation, on top of an unstable 1D state. Parameters are ${\epsilon }_X=0$, ${\epsilon }_C=1$, $g=1$, and $\Gamma =0.75$. Initial conditions are given by Eqs. (26) and (27) with ${\psi }_0=1$, ${\varphi }_0=1$, $A=-0.1$, and $\sigma =2$.

Figure 9

The dark soliton for ${\epsilon }_X=\mu =0$, ${\epsilon }_C=1$, $\gamma =1$, and $\Gamma =0.75$. The exact solution (solid line) is produced by Eq. (48), and its approximate counterpart (dashed line) is given by Eq. (44). The top and bottom panels display the excitonic and photonic density profiles, ${\left|\Psi \left(x\right)\right|}^2$ and ${\left|\Phi \left(x\right)\right|}^2$, respectively.