Dynamics of exciton-polaritons in discrete lattices under incoherent localized pumping

The paper deals with the spontaneous coherence building up between exciton-polaritons trapped in an array of deep potential wells in the presence of an incoherent pump. A theoretical approach based on a standard tight-binding mean-field approximation is used to reduce the continuous periodic problem to a discrete model. The typical dynamics of the nonlinear exciton-polariton system for the cases of spatially uniform and for localized pumps are discussed. Special attention is paid to the “staggered” coherent steady states with $\pi$ jumps in the phases between neighboring sites and to “uniform” states with a smooth phase distribution. It is shown that, apart from the states with a single frequency, mixed states with spectra with several harmonics can form in the system. The selection mechanism that controls the type of steady state growing from a weak noise is studied. It is found that in the case of localized pumps the decaying tails of the solutions play a crucial role in the dynamics of the polaritons. The applicability of the obtained theoretical results for a qualitative explanation of the complex phenomena observed in recent experiments is discussed.

Figure 1

(a) Polariton density ${\left|\psi \right|}^2$ vs pumping rate $P$ for a steady-state homogeneous solution with zero phase difference $Q=0$ between neighboring sites. The dashed line represents the modulationally unstable solution (MI). (b) The growth rate of the unstable perturbations on the parameter plane of the polariton density ${\left|\psi \right|}^2$ and phase difference between neighboring sites $q$. Parameters: $\sigma =0.2,\alpha =1.824,\beta =1.0965,{\gamma }_1=1$, and ${\gamma }_2=1.5$.

Figure 2

Stability diagram for staggered solution $\left(Q=\pi \right)$. The highest value of the growth rate of the unstable perturbations on the parameter plane of the pump, $P$, and phase difference between neighboring sites for perturbation, $Q-q$. Black curves correspond to the boundaries of the instability region. The panels are plotted for different values of coupling constant: (a) $\sigma =0.2$, (b) $\sigma =0.5$, and (c) $\sigma =2.0$. All other parameters are the same as in Fig. 1.

Figure 3

(a) The distribution of the amplitude $\left|\psi \right|$ (red circles) and the phase $Q$ (blue dots) of the stationary solution when only one site is pumped by the pump $P=2$. Pumped region is shaded. (b) The spatial spectrum of this solution. (c), (d) The same as (a) but for $P=4$ and $P=6$. (e) The numerically calculated dependency of the chemical potentials for the stationary solutions with one pumped site (blue circles) and two pumped sites. The red triangles correspond to uniform (U) solutions and the green triangles correspond to staggered (S) solutions. The dark solid line shows the analytical dependency (3) for the uniform solution, and the dashed line shows the same but for the staggered solution, Eq. (4). For all panels $\sigma =0.5$.

Figure 4

(a) The distribution of the amplitude $\left|\psi \right|$ (red circles) and the phase $Q$ (blue dots) of the stationary solution when only one site is pumped by the pump $P=2.2$. The pumped region is shaded. (b) The spatial spectrum of this solution. The blueshift is absent $\left(\alpha =0\right)$; all other parameters are the same as in Fig. 3.

Figure 5

(a) The distribution of the amplitude $\left|\psi \right|$ (red circles) and the phase $Q$ (blue dots) of the stationary (S) solution when two sites are pumped by the pump, $P=2$, with $\sigma =0.5$. (c) The corresponding spatial spectrum of the solution. (e) The stability diagram for S solutions obtained by numerical spectral analysis. The black curve corresponds to the boundary of the instability region. The dashed line corresponds to the condensation threshold. (b), (d), (f) The same but for $P=4$ when U solutions form.

Figure 6

The numerically calculated bifurcation diagrams. (a) The case when two sites are pumped. The area where an symmetric S state forms is shown in red. Green shows the area where the symmetric U state exists; it is marked by 0. The dark blue area marked as $d_1$ is the region of the existence of the asymmetric dynamical states. The area where the stationary solution can be either U or S depending on the initial conditions is shown in yellow and marked by $m$. The mixtures of yellow and green show the poorly defined border where a certain kind of solution dominates but the formation of the states of a different kind is sometimes observable in the numerical simulations. (b)–(d) The same but for three, four, and seven pumped sites, respectively. In (c) and (d) the light blue areas marked as $d_2$ show the regions of existence of symmetric dynamical states. In (d) small orange areas $a$ are the existence domains for the stationary asymmetric state. White regions correspond to a trivial solution.

Figure 7

The distributions of the absolute values of the fields and the spectra shown for relatively strong coupling, $\sigma =2$. The number of pumped sites is $N=7$. The states are grown from weak random noise. (a), (b) $P=1.8$ when according to Fig. 6 the state is in the domain where polaritons condensate into an S state. (c), (d) The state forming at $P=2.5$ and (e), (f) for the pump $P=3$. The formation of the U state is illustrated in (g) and (h) for the pump $P=5$.

Figure 8

The same as in Fig. 7 but for fewer pumped sites, $N=3$. The pump is (a), (b) $P=1.8$; (c), (d) $P=3$; and (e), (f) $P=4$. The coupling strength $\sigma =1.5$.

Figure 9

(a) The distribution of the absolute value of the complete dynamical state $\psi$ (black circles) and for the state with higher spatial harmonics with $Q>2$ filtered out, $\overline{\psi }$ (red circles). The open blue and green circles show the distribution of the real and imaginary part of the filtered state. (b) The same but for the case when low spatial harmonics, $Q<2$, are filtered out. (c), (d) Temporal dynamics of the asymmetry coefficients ${\mu }_a$ and ${\mu }_s$ for the complete states and for the state with filtered out (c) higher and (d) lower harmonics. The number of sites is $N=7$, the pump is $P=2.5$, and the coupling strength is $\sigma =2$.