Dynamics of a Vortex Lattice in an Expanding Polariton Quantum Fluid

If a quantum fluid is driven with enough angular momentum, at equilibrium the ground state of the system is given by a lattice of quantized vortices whose density is prescribed by the quantization of circulation. We report on the first experimental study of the Feynman-Onsager relation in a nonequilibrium polariton fluid, free to expand and rotate. Upon initially imprinting a lattice of vortices in the quantum fluid, we track the vortex core positions on picosecond timescales. We observe an accelerated stretching of the lattice and an outward bending of the linear trajectories of the vortices, due to the repulsive polariton interactions. Access to the full density and phase fields allows us to detect a small deviation from the Feynman-Onsager rule in terms of a transverse velocity component, due to the density gradient of the fluid envelope acting on the vortex lattice.

Figure 1

Schematic representation of the experimental setup. A laser beam is divided in two paths by a beam splitter (BS1) and one beam is diffracted by a SLM, where the phase pattern of a lattice of vortices is displayed. The beam with the imprinted phase profile is imaged on the sample surface by a system of lenses. The signal is made to interfere with a time-delayed reference beam. The interferogram is acquired by a CCD camera, and both density and phase are reconstructed in space and time by digital off-axis holography.

Figure 2

(a) Phase map $\phi (\mathbf{r})$ of the initial state of a $7\times 7$ square lattice of vortices (left) with momentum $\mathbf{k}(\mathbf{r})=\mathrm{\nabla }\phi$ representing the velocity field (right). The color scale of the momentum map is bounded at $|k|=1.6\mu {\mathrm{m}}^{-1}$ to avoid saturation inside the cores. The regular lattice builds up a continuous increase of azimuthal momentum reaching a maximum at its boundary of $\approx 0.42{\mu \mathrm{m}}^{-1}$ corresponding to a fluid velocity of $1.2\mu \mathrm{m}{\mathrm{ps}}^{-1}$. Overlapped streamlines (black lines) display the velocity direction, indicating the rotating motion. (b) The normalized amplitude maps over a span of 45 ps. The outer boundary of the lattice is shown by a square (thin line).

Figure 3

(a) Intervortex distances over time for three cases with total charge equal to $Q=\pm 49$ (red and blue dots, respectively), and $Q\simeq 0$ (green dots). Error bars come from the estimate of the vortex positions. (b) Mean intervortex distances for three different initial separations and $Q=-49$. (c) The orientation angle of the whole lattice for the same cases as in (b). In panels (b) and (c), solid lines are the best fits of Eqs. (2) and (3), with $a=11.6\pm 0.9{\mu \mathrm{m}}^2{\mathrm{ps}}^{-1}$ and $b=0.99\pm 0.09$. The small contraction of the lattice at short time lags, due to a residual curvature of the phase profile of the exciting beam, is taken into account by introducing a time offset $t_0$ in the fitting functions used in panels (b),(c): $t_0=7$, 11, and 17 ps for $d_0=21$, 26.5, and $30\mu \mathrm{m}$, respectively. (d) Rotation angle as a function of the vortex distance during the expansion of the lattice for the three cases shown in panels (b) and (c). Only positive angles are shown in panel (d), i.e., for $t>t_0$, to allow the comparison between different $d_0$. Solid lines correspond to the first 150 ps of expansion and rotation. Inset: the value of $ab/a_0$ as extracted from individual fit [same color legend as in panels (b) and (c)], and for a global fit (yellow point). The deviation from the Feynman-Onsager relation is quantified by the distance from the dashed line, with $a_0=9.85{\mu \mathrm{m}}^2{\mathrm{ps}}^{-1}$ and $b=1$ of the linear case.

Figure 4

(a) Graphical representation of the effect of nonlinear interactions producing a bending of the trajectories from the straight lines. (b) The time evolution of the intervortex separation $d$ for a lattice of $5\times 5$ vortices. Red points with error bars are experimental data at chemical potential $\mu =0.12\mathrm{meV}$ (corresponding to a polariton density $n\sim 100{\mu \mathrm{m}}^{-2}$, with an interaction strength $g={10}^{-3}\mathrm{meV}\mu {\mathrm{m}}^2$), and the red solid line is the result of a numerical simulation at equal $\mu$; from the best fit, the corresponding nonlinear expansion factor is $a=1.38a_0$. The black dashed line is the evolution in the linear case corresponding to $\mu \simeq 0\mathrm{meV}$, and $a=a_0=5.04{\mu \mathrm{m}}^2{\mathrm{ps}}^{-1}$. (c) Rotation angles as a function of time corresponding to the lattice expansions shown in (b). Red line is the best fit of Eq. (3) with $a=1.38a_0$ and $b=0.8$; the black dashed line is the angle evolution in the linear case ($a=a_0$, $b=1$). In the inset, an enlargement at small time lag shows the experimental data (red points) superposed to the best fit (red line). (d) Parameters $(a,b)$ extracted from experiments (crosses) and simulations (circles, squares) at different values of $\mu$. The red line is a polynomial fit to $a$ values; the blue line is the expected behavior for $b$ in the absence of the Magnus effect, $a_0/a$.