Spontaneous formation and synchronization of vortex modes in optically induced traps for exciton-polariton condensates

We study the dynamics of exciton-polariton condensates in reconfigurable potentials created by an incoherent optical pump. The mode selection in the effective optically induced trap is shown to be determined by the largest growth rates of the linear eigenmodes of the effective complex potential created by the pump. We demonstrate that selection of higher order nonlinear modes with nonzero orbital angular momenta in such traps leads to formation of vortices in a single-component condensate and spin textures in exciton-polariton condensates consisting of two polarization components. We also show that cross-polarization coupling can lead to the synchronization of the modes belonging to different polarizations. We highlight clear signatures of the various spontaneously formed trapped states in the cavity photoluminescence spectrum, which should aid in their experimental observations.

Figure 1

Panel (a) shows the spatial distribution of the pump $p$, and the cross section of the pump is shown in panel (b). The dependencies of the real ${\omega }_r$ and imaginary parts $\nu$ of the eigenfrequencies of the linear modes are shown in panels (c) and (d). The numbers $n,m$ marking the curves are the indices of the modes; i.e., the curves show the dependencies ${\omega }_{rn,m}$ and ${\nu }_{rn,m}$ for the modes with the angular index $n$ and radial index $m$ as defined in (2). The rings and squares in panel (c) mark the frequencies of the stationary nonlinear states obtained by direct numerical simulations. The amplitude of the pump is $P=3.27$, the width of the pump ring is $w_p=1$, and the pump displacement ${\vec{r}}_0=0$.

Figure 2

(a) The polariton density and (b) phase distribution corresponding to a stationary vortex mode with $m=1$ supported by the pump ring with the width $r_p=1.7$. Panels (c) and (e) are the $k$-space density distributions of the stationary states with $m=0$ and $m=1$, respectively. The corresponding spectra are shown in panels (d) and (f). The width of the pump ring supporting the ground state shown in panel (c) is $r_p=1.15$.

Figure 3

Spectra of the nonsynchronized nonlinear states for $r_p=1.08$ are shown in panels (a) and (b) for the first ($+$) and the second ($-$) polarizations correspondingly. The relative displacement of the pumps is $r_0=0.3$. Panels (c) and (d) show the distributions of the densities of the coherent polaritons in the $(+)$ and $(-)$ polarizations, and the phase distributions of the coherent polaritons are shown in panels (e) and (f). The displacement of the pumps is $r_0=0.3$.

Figure 4

The dependence of the detuning of the $(+)$ polarization frequency on the frequency of the $(-)$ polarization is shown in panel (a) as a function of the radius of the $(+)$ pump. The detuning is defined as the difference between the frequencies of most intense spectral lines of the first and the second polarizations. The black circles are the measured detunings for $\sigma =0.15$. The detunings for the case $\sigma =0$ are shown by the blue squares. The lines are guides for the eye. Panel (b) shows the detuning of the frequencies of different polarizations as a function of the shift of the pump centers. The relative displacement of the pumps is $r_0=0.3$.

Figure 5

The spectra (a, b) and polariton density distributions (c, d) in the two polarization components under the same conditions as in Fig. 3, but for the radius of $(+)$-polarization pump $r_p=1.17$, allowing for the formation of a synchronized state.

Figure 6

(a) Vorticity of the $(-)$ polarization component as a function of time. The shift between the pump centers is $r_0=0.7$. The radius of the spatial area used for vorticity calculation is $r=3$. (b) The spectra of the two polarization components for the dynamical state with oscillating vorticity.

Figure 7

The phase (top row) and polariton density distributions (bottom row) in the $(-)$ spin component for different moments of time corresponding to the points marked (b, c, d) in Fig. 6.