Geometric frustration in polygons of polariton condensates creating vortices of varying topological charge

Vorticity is a key ingredient to a broad variety of fluid phenomena, and its quantised version is considered to be the hallmark of superfluidity. Circulating flows that correspond to vortices of a large topological charge, termed giant vortices, are notoriously difficult to realise and even when externally imprinted, they are unstable, breaking into many vortices of a single charge. In spite of many theoretical proposals on the formation and stabilisation of giant vortices in ultra-cold atomic Bose-Einstein condensates and other superfluid systems, their experimental realisation remains elusive. Polariton condensates stand out from other superfluid systems due to their particularly strong interparticle interactions combined with their non-equilibrium nature, and as such provide an alternative testbed for the study of vortices. Here, we non-resonantly excite an odd number of polariton condensates at the vertices of a regular polygon and we observe the formation of a stable discrete vortex state with a large topological charge as a consequence of antibonding frustration between nearest neighbouring condensates.

γ −1 = 5.5 ps. We choose values of interaction strengths typical of GaAs based systems: α = 3.3 µeV µm 2 and g = 2α. The redistribution rate of reservoir excitons is taken here as comparable to the condensate decay rate Γ −1 = 5 ps and the damping parameter is chosen small Λ = 0.05. The final two parameters are found by fitting numerical results to experiment which gives R = 33 µeV µm −2 , and G = 66 µeV µm −2 . The n-th pump element (vertex) is written as a Gaussian profile P n (r) = P 0 e −r 2 n /2w 2 RMS where r n = (x − x n ) 2 + (y − y n ) 2 , and (x n , y n ) denote the coordinates of the element. Here P 0 denotes the pump power density, and w RMS = 1.27 µm the RMS width (corresponds to a 3 µm full-width-half-maximum), slightly larger than the incident light beam width in order to account for the small diffusion of excitons from the pump spots. Condensate octagon and decagon geometries are displayed in Fig. S1, synchronised in in-phase (left hand side) and anti-phase (right hand side) configuration with respect to nearest-neighbour condensates. The in-phase states in Figs. S1(a,g) show the real-space configuration with an odd number of fringes between nearest neighbour condensates (in this case three fringes). In real-(Figs. S1(a,g)) and Fourier-space (Figs. S1(b,h)) both display a bright spot at the centre indicating that all spots are in-phase. The phase in Figs. S1(c,i) reflect the clear pattern in the real-space and it can be seen that the spot centres are all at the same phase. Conversely, the anti-phase configurations have a dark spot at the centre and display clear radial nodal lines in real-Figs. S1(d,j) and Fourier-space Figs. S1(e,k).
The corresponding polariton phase maps are shown in Figs. S1 (f,l), where neighbouring spots can be clearly observed in anti-phase.
Similarly, in the non-frustrated regime (J > 0) odd numbered pump polygons will form condensates in the in-phase configuration. The in-phase state has a bright fringe between adjacent condensates as well as the bright central fringe in both real-and Fourier-space images, as shown in Figs. S2(a,b,d,e,g,h) for a pentagon, heptagon and nonagon, respectively.
The corresponding polariton phase maps are shown in Figs. S2(c,f,i), where the neighbouring spots can be seen to be in-phase.

III. RADIUS SCAN OF HEPTAGON
A scan of the radius of a heptagon was performed whilst integrating over multiple instances of the condensate as the real-space shows in Fig. S3. The system was found to be quite robust and stable over the course of the measurement. The system is initially in the "fifth" in-phase configuration at 19.81 µm corresponding to five bright fringes observed between neighbouring condensates and a bright notch at the centre. Upon increasing the radius, the system alters whilst it does retain five fringes between vertices, the centre now has a small dark notch surrounded by a bright ring, most clearly seen at a radius of 21.12 µm corresponding to the first vortex state forming. Further increase of the radius caused the state to fracture one or more of the condensates similar to the vortex-antivortex state demonstrated in Fig. 5 (in the main text). Furthermore, at a radius of 22.53 µm, two of the condensates can be seen in the integrated real-space to be splitting (middle left-hand and bottom left-hand condensates), indicative that the frustration in the polygon has caused some of the condensates to split. A different number of fringes can be seen between different vertices (either five or six).
The next state to occur is seen at 23.29 µm with six fringes between vertices, where the real-space pattern has a heptagon at its centre surrounding a dark notch, (Fig. S3,  (cos (θ) − cos (θ n,n+1 )) 2 + (sin (θ) − sin (θ n,n+1 )) 2 . (S1) Hereθ is the expected phase configuration (e.g.,θ = 0 for in-phase configuration). At the end of each simulation we classify the formation of the expected state successful when ERR(θ) ≤ 0.05. The probability is then calculated as the number of successful formations over number of realisations. In Fig. S4(b) we show the probability of in-phase (blue whole line) and anti-phase (red dot-dashed line) configurations forming in a polygon of N = 6.
The results evidence that non-frustrated polygons have step-like domain walls separating the regimes of in-phase and anti-phase configurations. This is in contrast to the frustrated (e.g., N = 5) polygons shown in Fig. S4(a) where the vortex formation probability follows a more complex distribution. We also investigate the formation probability of a |m| = 2 vortex state for non-ideal pentagons. We introduce uncertainties to the coordinates of each pump vertex written (x n + dx, y n + dy). Here, dx, dy are normally distributed random variables with zero mean and standard deviation σ. In Fig. S4(c) we show the drop in probability of the |m| = 2 vortex forming for increasing standard deviation. These results carry an important message.
As the the number of pump spots increases the probability of maintaining a perfectly regular polygon drops since experimental uncertainties are never fully avoided, and therefore the probability of observing a vortex state diminishes. This challenge should then be overcome by either designing an alternative discrete rotational geometry which favours more strongly