Spin oscillations of a single-mode polariton system driven by a plane wave

Theoretical study is performed of a single-mode polariton system with linear coupling of spin components. When combined with an ordinary two-particle interaction, the spin coupling involves a spontaneous symmetry breaking accompanied by a switch from linear to circular polarization under resonant driving. The asymmetric steady states can also lose stability, giving way to oscillatory and chaotic dynamics. Here, we explore a continuous transformation between the multistable regime, where the system is steady and locked in phase to the pump but has a broken spin symmetry, and full-span oscillations of the circular-polarization degree, owing to which the symmetry is effectively reestablished. Such oscillations are analogous to the intrinsic Josephson effect and prove to be robust against arbitrarily strong perturbations. Transitional phenomena include the Hopf bifurcation, spin bistability of limit cycles, and continuous transitions to and from dynamical chaos through series of period doubling and halving events.

Figure 1

Fixed-point states in case of a spin-symmetric excitation. (a) Symmetric states (15) with ${\overline{\psi }}_{+}={\overline{\psi }}_{-}$; (b) approximate asymmetric solution $f^2\left(u\right)=(2a-u){(D-u)}^2/V$; (c) exact asymmetric solution (12) in a region where it is noticeably different from (b); (d) spurious branch $(u<l_1)$. Parameters: $g=9\gamma , D=8.1\gamma$; $f_a=a{g}^2/4V$.

Figure 2

(a) Circular-polarization degree as a function of pump intensity. (b) Corresponding decay rates in the symmetric (dashed line) and main asymmetric (solid line) states of the system. $\lambda$ denotes the eigenvalues of (17). Parameters are the same as in Fig. 1.

Figure 3

Hybridization of elementary excitations. $\lambda$ denotes the eigenvalues of (17). Parameters in (b) correspond to the asymmetric solution in Fig. 2 at $f=f_a$. In (a), all parameters are the same except $g=0$. The dispersion law corresponds to a GaAs-based microcavity.

Figure 4

The Hopf bifurcation in the vicinity of $f=f_a$. The two lines represent the minimum and maximum of $|S_3|$ of an established solution depending on $f$. The initial conditions play no role.

Figure 5

Limit-cycle solutions at different ${(f/f_a)}^2$. Quantities $S_{1,2,3}$ are the Stokes vector components. The solutions are bistable for (a) and (b) and independent of the initial conditions for (c)–(f), so that the sole basin of attraction covers the entire phase space.

Figure 6

Bistable periodic solutions. The solid and dashed curves (also distinct in color) display alternative trajectories which are chosen depending on the initial conditions.

Figure 7

Transition to chaos through a cascade of the period-doubling bifurcations. Different colors in (a)–(c) represent alternative solutions. The time span in (c) and (d) is 10 ns.