Spatial pattern formation and polarization dynamics of a nonequilibrium spinor polariton condensate

Quasiparticles in semiconductors—such as microcavity polaritons—can form condensates in which the steady-state density profile is set by the balance of pumping and decay. By taking account of the polarization degree of freedom for a polariton condensate, and considering the effects of an applied magnetic field, we theoretically discuss the interplay between polarization dynamics, and the spatial structure of the pumped decaying condensate. If spatial structure is neglected, this dynamics has attractors that are linearly polarized condensates (fixed points), and desynchronized solutions (limit cycles), with a range of bistability. Considering spatial fluctuations about the fixed point, the collective spin modes can either be diffusive, linearly dispersing, or gapped. Including spatial structure, interactions between the spin components can influence the dynamics of vortices; produce stable complexes of vortices and rarefaction pulses with both co- and counter-rotating polarizations; and increase the range of possible limit cycles for the polarization dynamics, with different attractors displaying different spatial structures.

Figure 1

(Color online) Basin of attraction of fixed point and limit cycle for full two-mode Josephson problem. Each panel is for a given value of $\Delta$, as indicated, and shows a color map according to the final state found by starting from the given initial conditions $\theta ,z$. Points that flow to the fixed point are colored gray, others are white. The fixed point is marked by a cross, and the limit cycle (when it exists) by a line.

Figure 2

(Color online) Fourier transform of ${\psi }_L\left(t\right)$, showing bifurcation of frequencies as $\Delta$ is increased, and illustrating critical behavior $\omega \sim 1/\text{ln}\left|\Delta -{\Delta }_c\right|$. Initial conditions for each $\Delta$ are chosen so that a limit cycle (i.e., desynchronized solution) is found for all $\Delta >{\Delta }_{c,\text{lower}}$. Inset: regions of stability of the synchronized and desynchronized solutions found from Eq. (3, 4, 5). The dotted line marks the approximate ${\Delta }_{c,\text{upper}}$ appropriate in the Josephson regime, Eq. (6), and the $\ast$ marks the known bifurcation point (Ref. 53) in that regime.

Figure 3

(Color online) Steady states, and damping of uniform fluctuations. All panels are plotted as a function of the phase difference, $\theta$, between the two circular polarization components. Top left: densities of components in steady-state solution. Top right: detuning corresponding to given solution. Bottom: real (left) and imaginary (right) parts of frequency for small fluctuations about the steady state. The gray shaded region is unstable to small fluctuations. The crosses mark the conditions used for the finite $k$ spectra in Fig. 3. Plotted for $\alpha =6.5,\sigma =0.3,J=1.0,u_a=1.1$.

Figure 4

(Color online) Spectra of normal modes in a uniform system; showing real (left) and imaginary (right) parts of each mode separately. The three rows correspond to the three points marked by crosses in Fig. 3. In all cases the total density modes are diffusive, while the spin mode changes from diffusive (top row, $\theta$ nearest $\pi /2$), to linearly dispersive (middle row) to underdamped spin oscillations (bottom row, $\theta$ nearest $\pi$).

Figure 5

(Color online) The outcome of instability of the vortex state $\left(+1,-1\right)$ for $\alpha =4.4$, $\sigma =0.3$, $r_0=3$, $\Delta =0$, and $J=0.5$ (top row), $J=1$ (second row), $J=1.5$ (left bottom) and $J=2$ (right bottom). The initial state is ${\psi }_{\pm }=\sqrt{\Theta \left(2{\mu }_{\text{TF}}-r^2\right)}\left(x\pm \text{i}y\right)/\sqrt{r^2+1/\left(2{\mu }_{\text{TF}}\delta \right)}$ where ${\psi }_R={\psi }_{-}$, ${\psi }_L={\psi }_{+}$, $2{\mu }_{\text{TF}}=3\alpha /2\sigma$ and $\delta$ is the vortex core parameter; see Appendix . Depicted are the density plots and streamlines of ${\psi }_L$ (left panels in first and second row, bottom row) and ${\psi }_R$ (right panels in first and second row). Luminosity of the background is proportional to the magnitude of the density, and luminosity of the streamlines is proportional to the velocity. The size of the pumping is shown as a circle of radius $r_0=3$. For $J=0.5$ the final state consists of two precessing vortices of opposite circulation. For $J=1$ the system evolves into a pair of two gray solitons. For $J=1.5$ the vortex of the negative circulation in ${\psi }_R$ leaves and a vortex of positive circulation enters the cloud. For $J=2$ both vortices leave the condensate and two pairs of vortices of opposite circulation enter. For $J=1.5$ and $J=2$ ${\psi }_L={\psi }_R$, so only ${\psi }_L$ is shown.

Figure 6

(Color online) Stable state of cross-polarized vortices $\left(+1,-1\right)$ for $\alpha =4.4$, $\sigma =0.3$, $r_0=3$, $\Delta =8$, and $J=1$. Depicted are the density plots and streamlines of ${\psi }_L$ (left) and ${\psi }_R$ (right). Luminosity of the background is proportional to the magnitude of the density, and luminosity of the streamlines is proportional to the velocity. The size of the pumping is shown as a circle of radius $r_0=3$.

Figure 7

(Color online) Real (thick lines) and imaginary (thin lines) parts of ${\psi }_L$ (solid lines) and ${\psi }_R$ (dashed lines) of the rarefaction wave complex as a stationary solution of Eq. (2) with $v\left(r\right)=r^2$ for $\alpha =4.4$, $\sigma =0.3$, $J=1$ and $\Delta =0$ (left) and $\Delta =0.4$ (right) plotted along the axis of density symmetry. The functions are antisymmetric with respect to the origin ${\psi }_L\left(0,-y\right)={\psi }_R\left(0,y\right)$ for $\Delta =0$, but not for $\Delta \ne 0$.

Figure 8

(Color online) Density plots ${\left|{\psi }_L\right|}^2$ (top row) and ${\left|{\psi }_R\right|}^2$ (bottom row) for the outcome of instability of cross-polarized vortices obtained by numerical integration of Eq. (2) with $v\left(r\right)=r^2$, $\alpha =4.4$, $\sigma =0.3$, $J=1$ and various $\Delta$. The initial state is the same as in Fig. 5. Luminosity is proportional to density. The size of the pumping is shown as a circle of radius $r_0=3$. For $\Delta =0.4$ two rarefaction pulses form a stationary complex. For $\Delta =2$ a vortex of negative circulation in the $R$ component is coupled to a rarefaction pulse in the $L$ component. For $\Delta =4$ the two vortices of opposite circulation do not align their cores. For $\Delta =2$ and $\Delta =4$ the complexes precess around the center with the individual vortices moving along an epitrochoid; see also Fig. 9.

Figure 9

(Color online) Trajectories of vortices in the complex $\left(+1,-1\right)$ obtained by numerical integration of Eq. (2) with $v\left(r\right)=r^2$, $\alpha =4.4$, $\sigma =0.3$, $J=1$, and $\Delta =4$, and with pumping in a radius $r_0=3$. Lengths in units of $l=\sqrt{\hslash /m{\omega }_0}$. These parameters are the same as for the density profiles in the right hand column of Fig. 8. The trajectory of the vortex of positive (negative) circulation in the left (right) component is shown as thick black (thin blue) line. The start (end) points of the time interval plotted are shown as red/dark (green/light) filled circles. Note that the vortex trajectories are not closed.

Figure 10

(Color online) Chemical potentials of the two polarization components plotted against stepwise increased detuning $\Delta$. For solutions with time-dependent chemical potentials, a time average is shown.

Figure 11

(Color online) Josephson oscillations in the desynchronized solution at $\Delta =12.0,r_0=3.0,J=1.0$ in the limit of large time. Left: $n=\int {\left|\psi \right|}^2{d}^{2}r$ as a function of time. Right: The chemical potentials of the two polarization components. (Time given in units of $1/{\omega }_0$. Blue/dark and red/light color denote $L$ and $R$ components, respectively.)

Figure 12

(Color online) Phase portraits for the trapped problem with $J=1.0$ and $r_0=3.0$. Left: synchronized solution for small detuning. Right: desynchronized solution in the limit of large detuning (corrected for numerical precession).

Figure 13

(Color online) Progression from fully synchronized solution via formation of vortex lattice to fully desynchronized solution at $r_0=4.0$ and $J=1.0$. Top: phase portraits for the phase difference $\theta$ (a numerical precession has been removed from the $\Delta =20.0$ plot). Middle and bottom: total densities of the $L$ and $R$ components, respectively.

Figure 14

(Color online) Density profiles along the $x$ axis for representative examples of synchronized and desynchronized solutions. Left: synchronized solution without vortices $\left(r_0=3.0,\Delta =4.0\right)$. Middle: synchronized solution with vortex lattice $\left(r_0=6.0,\Delta =4.0\right)$. Right: desynchronized solution without vortices $\left(r_0=3.0,\Delta =12.0\right)$. Dashed and solid lines indicate left and right-polarized components, respectively. $J=1.0$ for all three examples.

Figure 15

(Color online) Top: highest $\Delta$ yielding synchronized solutions plotted as a function of $r_0$ for $J=0.5$, 1.0 and 2.0. Left: $\Delta$ increased stepwise. Right: resetting initial conditions for each $\Delta$. Bottom left: bifurcation of chemical potentials (plotted as in Fig. 10) for the two approaches, indicating the region of bistability for the example $J=1.0,r_0=3.0$. Bottom right: sketch of regions of stability of synchronized and desynchronized solutions for $r_0=3.0$. Compare with inset of Fig. 2.

Figure 16

(Color online) Spatially nonuniform interconversion rate ${\partial }_t{\rho }_1{\mid }_J$ at two different, but close, times. $J=1.0,r_0=6.0,\Delta =16.0$.

Figure 17

(Color online) Top: integrated (scaled) number density $n=\int {\left|\psi \right|}^2{d}^{2}r$ and chemical potentials for a desynchronized solution with counter-rotating components. Bottom from left to right: densities of the $L$ and $R$ polarization components, and interconversion rate ${\partial }_t{\rho }_1{\mid }_J$. (Note that the gray scale has been adjusted for each component separately to accentuate the density modulations.) $J=1.0,r_0=6.0,\Delta =32.0$.

Figure 18

(Color online) Top: Limit cycle with winding number 2 and retrograde loop. Right panel shows a blowup of the loop. A numerical precession has been removed from these plots. ($J=1.0,r_0=3.0,\Delta =6.4$.) Bottom left: possibly quasiperiodic behavior similar to a limit cycle with winding number 0 ($J=2.0,r_0=3.0,\Delta =14.0$ from stepwise increase). Bottom right: possible chaotic attractor ($J=2.0,r_0=3.0,\Delta =10.0$ from stepwise increase).

Figure 19

Spectral weight at $k_y=0$ for two vortex-lattice solutions. Left and right panels show left- and right-polarized components, respectively. Top: synchronized solution without a central vortex. ($J=0.5,\Delta =3.5,r_0=6.0$.) Bottom: desynchronized solution with a central vortex. ($J=0.5,\Delta =16.0,r_0=6.0$.)

Figure 20

(Color online) Radial velocity, $u\left(r\right)={\varphi }^{\prime }\left(r\right)$, of the vortex solutions of Eq. (A1) for $\alpha =1,\sigma =0.01$. The numbers next to the lines indicate the winding numbers of the vortices (1 and 2) and the ground state (0). The TF approximation of the gradient flow of the ground state for small $\alpha$ and $\sigma$ $u\left(r\right)=-r\sigma \left(2\mu -r^2\right)/6$, is given by the dashed line. The inset compares the density, $\rho \left(r\right)$, of a vortex solution for $\alpha =4,\sigma =0.27$ [$2\mu =30.28$, gray (red) line] with the density of the GPE vortex $\left(\alpha =0,\sigma =0\right)$ with the same number of particles ($2\pi \int {\left|\psi \right|}^{2}rdr=1000$, $2\mu =25.68$, black line). Both vortex solutions have topological charge 1.

Figure 21

Density of the left- and right-polarized components, respectively, showing an irregular vortex lattice. $J=1.0,r_0=6.0,\Delta =6.0$.