Polariton condensation threshold investigation through the numerical resolution of the generalized Gross-Pitaevskii equation

We present a numerical approach for the solution of the dissipative Gross-Pitaevskii equation coupled to the reservoir equation governing the exciton-polaritons Bose-Einstein condensation. It is based on the finite difference method applied to space variables and on the fourth order Range-Kutta algorithm applied to the time variable. Numerical tests illustrate the stability and accuracy of the proposed scheme. Then results on the behavior of the condensate under large Gaussian pumping and around the threshold are presented. We determine the threshold through the particular behavior of the self-energy and characterize it by tracking the establishment time of the steady state.

Figure 1

(a) Typical exciton-polariton microcavity. (b) Upper (UP) and lower (LP) polariton dispersion branches. Near $k=0$ where the condensation may occur, the exciton energy is almost constant. Created with optical pumping, a reservoir populates the condensate through relaxation mechanisms faster than radiative recombinations.

Figure 2

Spatial discretization grid.

Figure 3

(a) Temporal stability as function of the time step for a grid $400\times 400$ nodes. (b) Spatial convergence as function of the spatial step for a time step of ${\tilde{h}}_t=3\times {10}^{-4}$. Parameters: $m^{*}=7.44\times {10}^{-5}{m}_0$ ($m_0$ being the free electron mass); $g_R=0$; $\mathcal{G}=0.0175\mu {\text{m}}^2$; $g=0.015\text{meV}\mu {\text{m}}^2$; $\hslash R=0.05\text{meV}\mu {\text{m}}^2$; $\hslash {\gamma }_c=0.5\text{meV}$; $\hslash {\gamma }_R=2\text{meV}$; $V_{\mathrm{ext}}=0$; ${\sigma }_p=20\mu \text{m}$; $P_0=P=60.790\mu {\text{m}}^{-2}{\text{ps}}^{-1}$ [13] and $N_c^{(t=0)}=1$.

Figure 4

The evolution up to the steady state of the wave function of a nonequilibrium condensate in a harmonic potential. The circles indicate our numerical scheme results, while the dashed curves are reproduced with data extracted from [34] for $\tilde{t}=20$ (blue), $\tilde{t}=10$ (red), $\tilde{t}=5$ (green), and $\tilde{t}=1$ (black). Calculations were performed with a grid ($599\times 599$ points) in a domain with sides of ${\tilde{l}}_x={\tilde{l}}_y=40$ and with a time step of ${\tilde{h}}_t=0.001$. Parameters: $\tilde{g}=1.0,\tilde{P}=0.2,\tilde{\gamma }{}_c=\tilde{r}=0.1$.

Figure 5

Temporal evolution until the steady state of the condensate energies per particle. Parameters: $m^{*}=7.44\times {10}^{-5}{m}_0$; $g_R=0$; $\mathcal{G}=0.0175\mu {\text{m}}^2$; $g=0.015\text{meV}\mu {\text{m}}^2$; $\hslash R=0.05\text{meV}\mu {\text{m}}^2$; $\hslash {\gamma }_c=0.5\text{meV}$; $\hslash {\gamma }_R=2\text{meV}$; $V_{\mathrm{ext}}=0$; ${\sigma }_p=20\mu \text{m}$; $P_0=P=60.790\mu {\text{m}}^{-2}{\text{ps}}^{-1}$ [13]; and $N_c^{(t=0)}=1$.

Figure 6

The condensate local wave vector: comparison of our results (solid line) with the results in Ref. [13] (dashed line).

Figure 7

Temporal evolution of the condensate population by increasing the pump power from below (a) to above (c) threshold. Close to the threshold power (b), the steady state is unreachable. The circles mark the steady state beginning.

Figure 8

Steady state condensate and reservoir populations versus the initially input population. Zoom in the threshold vicinity is displayed in the inset.

Figure 9

Evolution of the different energies per particle in the condensate as a function of the pump power once the steady state is established. The inset displays a zoom on the self-energy per particle at the very threshold vicinity.

Figure 10

Steady state establishment time versus the pump power: the circles indicate the calculated values and the dashed curve is obtained with interpolation.