Robustness and observability of rotating vortex lattices in an exciton-polariton condensate

Exciton-polariton condensates display a variety of intriguing pattern-forming behaviors, particularly when confined in potential traps. It has previously been predicted that triangular lattices of vortices of the same sign will form spontaneously as the result of surface instabilities in a harmonic trap. However, natural disorder, deviation of the external potential from circular symmetry, or higher-order terms modifying the dynamical equations may all have detrimental effects and destabilize the circular trajectories of vortices. Here we address these issues by characterizing the robustness of the vortex lattice against disorder and deformations of the trapping potential. Since most experiments use time-integrated measurements, it would be hard to observe directly the rotating vortex lattices or distinguish them from vortex-free states. We suggest how these difficulties can be overcome and present an experimentally viable interference-imaging scheme that would allow the detection of rotating vortex lattices.

Figure 1

Density (grayscale) and superfluid streamlines (arrows) for different sizes of the pumping spot (red ellipse). (left) The surface instability leads to a spontaneously formed lattice with vortices moving on elliptical trajectories ($\delta =0.9$, $R=7$). (right) The pumping spot is smaller than $r_{\mathrm{TF}}$ and the instability does not appear ($\delta =0.9$, $R=4$).

Figure 2

(left) The averaged orbits of two vortices (green dashed and magenta dash-dotted lines, respectively). The red solid ellipse marks the pumping spot. (right) Snapshot of the vortex lattice, with the tracked vortices marked by green and magenta crosses. $\delta =0.9$.

Figure 3

(left) $\delta =0.85$; the regular vortex lattice is broken up, with some vortices still on regular elliptic orbits. (right) $\delta =0.80$; as $\delta$ decreases, the rotating lattice gives way to a fragmented state with vortices continually entering and leaving the cloud.

Figure 4

(top left) $\delta =0.25$. In a highly eccentric trap, the density tries to establish a profile of approximately uniform width along the length of the trap. Instability towards entrance of vortices leads to transverse oscillations. Note that the vertical axis has been rescaled so that the pumping spot appears circular in order to accommodate the extreme eccentricity. (top right) $\delta =0.1$. Extreme eccentricity suppresses fragmentation of the cloud. Transverse oscillations develop at the midpoint. (bottom left) Spontaneously entering vortices cause an effective pressure gradient, pushing the region of perturbations towards one end of the trap. The final position is shown. (bottom right) Vortices of both signs are present in the perturbed region. Arrows indicate circulation.

Figure 5

(top) The rotating vortex lattice forms also in the presence of weak disorder (a) but is destroyed by a disorder stronger than $V_{\mathrm{crit}}$, leading to a chaotic state (b). (bottom) The boundary between vortex-lattice and chaotic regimes in the $(V_0,\xi )$ parameter space. The choices for $(V_0,\xi )$ leading to states (a) and (b) are indicated in the diagram. The vortex lattice is strongly sensitive to the amount of disorder, whereas the correlation length is of only marginal importance.

Figure 6

Example of a vortex-lattice solution to Eq. (1). (top left) Density, showing the vortex lattice. (top right) The corresponding spectrum. Note the three bands corresponding to the regions of the condensate separated by the rings of vortices. (bottom left) Complex phase of the wave function and (bottom right) the time derivative of the phase. Note the separation into regions with an approximately constant time derivative corresponding to the bands in the spectrum.

Figure 7

Time-integrated interference image using the heterodyne scheme and the vortex lattice from Fig. 6. (left) ${\omega }_0=26.5$. (right) ${\omega }_0=14.5$.

Figure 8

Snapshot of (left) intensity and (right) phase after filtering $\psi$ from Fig. 6 to the band $26.25⩽\omega ⩽26.75$.

Figure 9

Frequency-filtered homodyne scheme applied to the solution in Fig. 6 (the result of the filtering shown in Fig 8). From top left to bottom right, the displacement [see Eq. (C2)] is $x_{\mathrm{d}}=0.2$, $x_{\mathrm{d}}=0.5$, $x_{\mathrm{d}}=1.0$, and $x_{\mathrm{d}}=2.0$. (Here $y_{\mathrm{d}}=0$.)

Figure 10

(top left) Snapshot of density, with crosses showing the points from which the reference is taken. (top right) Interference with the reference beam taken from the red cross, mapping out the interference fringes of all 18 vortices enclosed by that part of the condensate. (bottom left) Interference with light taken from the blue cross, giving six stationary interference fringes. (bottom right) Same as the bottom left panel, but with the image filtered to $14.15⩽\omega ⩽14.75$ before the reference is extracted and allowed to interfere with the filtered image.

Figure 11

Interference with light taken from the point indicated by the red cross in Fig. 10 after filtering to a narrow frequency band. The panels show the resulting interference image for different in-plane momenta in the $x$ direction. From left to right $k_x=1.0,2.0$, and $5.0$.

Figure 12

The top and bottom left panels show density and the corresponding spectrum. Crosses mark reference points for the filtered defocused homodyne scheme. The remaining panels show interference images corresponding to light taken from the red (top) and blue (bottom) crosses. From left to right in both rows, in-plane momentum in the $x$ direction $k_x=0$, $k_x=1$, and $k_x=2$.