Incoherent excitation and switching of spin states in exciton-polariton condensates

We investigate, theoretically and numerically, the spin dynamics of a two-component exciton-polariton condensate created and sustained by nonresonant spin-polarized optical pumping in a semiconductor microcavity. Using the open-dissipative mean-field model, we show that the existence of well defined phase-locked steady states of the condensate may lead to efficient switching and control of spin (polarization) states with a nonresonant excitation. Spatially inhomogeneous pulsed excitations can cause symmetry breaking in the pseudospin structure of the condensate and lead to formation of nontrivial spin textures. Our model is universally applicable to two linearly coupled polariton condensates, and therefore can also describe the behavior of condensate populations and phases in “double-well” type potentials.

Figure 1

(a)–(c) Stationary solutions of Eq. (4) for a linearly polarized pump. Branches 1 and 3 are elliptically polarized states; branches 2 and 4 are bonding ($s_1$) and antibonding ($s_2$) linearly polarized states, respectively. Only the solid branch is stable (see text). (d) Dynamical evolution of a spatially inhomogeneous state governed by Eq. (3) represented by integrated Stokes parameters (see Sec. 4). The pump is linearly polarized (${\overline{P}}_0=5, P_0^{-}/P_0=0.5$) and has a Gaussian profile with the widths $a_x=a_y=5$. Red dot: initial state of evolution; green rectangle: final state of evolution corresponding to the marked point in (a)–(c). Inset: evolution of the energy functional (7). Here $J=0.5$ and other parameters are given in Ref. [21].

Figure 2

Trajectories of the pseudospin vector obtained from numerical solutions of Eq. (4) for a linearly polarized, spatially homogeneous pump, ${\overline{P}}_{+}={\overline{P}}_{-}=1.2$ and $J=0.5$; the interaction strength $g_R$ between the condensate and the reservoir takes the same value as in Fig. 1 for (a) and (b); for (c) and (d) an artificially enlarged or reduced $g_R$ is used respectively. Red dots: initial states of evolution; green rectangles: final states of evolution. Initial conditions are arbitrary. Parameters are (a) $g_R=1.5\times {10}^{-2}, \theta \left(0\right)=0.5$; (b) $g_R=1.5\times {10}^{-2}, \theta \left(0\right)=2$; (c) artificially enlarged $g_R=1.5\times {10}^{-1}$; (d) artificially reduced $g_R=1.5\times {10}^{-3}$. Other parameters are given in Ref. [21].

Figure 3

(a)–(c) Selected stationary solutions of Eq. (4) with an elliptically polarized pump (${\overline{P}}_{+}=2.4$). Only the solid branches are stable. Red dot (green rectangle): the steady states before (during) addition of a pulsed perturbation. (d) Dynamics of spatially inhomogeneous states under pulsed perturbation represented by the integrated Stokes parameters (see Sec. 4). Steady states are supported by a Gaussian cw pump with ${\overline{P}}_0=4, P_0^{-}/P_0=0.4, a_x=a_y=5$, and perturbed by a pulse with ${\overline{P}}_0=0.12, P_0^{-}/P_0=1, a_{px}=a_{py}=5$. Red dot: initial state of evolution; green rectangle: final state of evolution. Inset: evolution of the energy functional (7). Here $J=0.5$ and other parameters are given in Ref. [21].

Figure 4

(a)–(c) Stationary, spatially homogeneous solutions of Eq. (4) with ${\overline{P}}_{-}=0$. Only the solid branch is stable (see text). (d) Dynamical evolution of a dynamically inhomogeneous state governed by Eq. (3) represented by integrated Stokes parameters (see Sec. 4). The pump is circularly polarized (${\overline{P}}_0=3, P_0^{-}=0$), and has a Gaussian profile with the widths $a_x=a_y=5$. Red dot: initial state of evolution; green rectangle: final state of evolution corresponding to the stable steady state indicated in (a)–(c). Inset: evolution of the energy functional (7). Here $J=0.5$ and other parameters are given in Ref. [21].

Figure 5

(a)–(c) Evolution towards a synchronized steady state with ${\overline{P}}_0=4, P_0^{-}/P_0=0.4, a_x=a_y=5$, and $J=0.5$: (a) density profiles (along $y=0$) and spatial distribution of $s_z$ for the steady state, (b) evolution of integrated Stokes parameters and energy functional (7) starting from an arbitrary small initial condition, and (c) evolution of integrated Stokes parameters on the Poincaré sphere from the initial (red dot) to the final (green rectangle) steady state. (d) Evolution of integrated Stokes parameters towards a desynchronized state with ${\overline{P}}_0=8, P_0^{-}/P_0=0.4, a_x=a_y=5$, and $J=0.5$. The initial state is marked by the red dot and the final state is a closed orbit. Inset: evolution of the mean-field energy functional (7) with the integration performed over the area where $n_c(x,y)>{10}^{-3}$. Other parameters are given in Ref. [21].

Figure 6

Steady state integrated quantities against the ratio $P_0^{-}/P_0$ for ${\overline{P}}_0=4, a_x=a_y=5, J=0.5$. Within the gap $|\delta P/P_0|<0.16$, one needs to increase ${\overline{P}}_0$ in order to reach steady states. Other parameters are given in Ref. [21].

Figure 7

Spin switching dynamics for the case of a Gaussian pump. (a)–(c) Energy of the initial and final states after (or during) the switching pulse. Dashed line: pulse switched on. Dash-dotted line: pulse switched off. (d)–(f) Integrated total density and integrated Stokes parameters during the switching process for switching between the steady states shown in (c). Parameters of the cw pump and pulsed perturbation in (c) are the same as in Fig. 3; parameters of the pulsed perturbation in (a) are ${\overline{P}}_0=0.4, P_0^{-}/P_0=0.65$ and in (b) ${\overline{P}}_0=0.9, P_0^{-}/P_0=0.65$. Other parameters are given in Ref. [21].

Figure 8

Excitation of vortex pairs for the case of an elongated elliptically polarized pump with $P_0^{-}/P_0=0.4$ and a tightly focused radially symmetric pulse with $P_0^{-}/P_0=0.65$ and different widths (see text). (a) Density profile of the ${\psi }_{+}$ component showing a density dip associated with a phase fold (b). (c) Cross section of density and phase across the fold. (d) Density profile of the ${\psi }_{-}$ component corresponding to (a). (e) Density profile of the ${\psi }_{+}$ component showing stationary configuration of vortex pairs connected by a phase fold (f). (g) Cross-section of density and phase across the phase fold. (h) Density profile of the ${\psi }_{-}$ component corresponding to (e). All density plots are logarithmic. Other parameters are given in Ref. [21].