Observing Dissipative Topological Defects with Coupled Lasers

Topological defects have been observed and studied in a wide range of systems, such as cosmology, spin systems, cold atoms, and optics, as they are quenched across a phase transition into an ordered state. These defects limit the coherence of the system and its ability to approach a fully ordered state, so revealing their origin and control is becoming an increasingly important field of research. We observe dissipative topological defects in a one-dimensional ring of phased-locked lasers, and show how their formation is related to the Kibble-Zurek mechanism and is governed in a universal manner by two competing time scales. The ratio between these two time scales depends on the system parameters, and thus offers the possibility of enabling the system to dissipate to a fully ordered, defect-free state that can be exploited for solving hard computational problems in various fields.

Figure 1

Experimental arrangement with representative results. (a) Experimental arrangement; $M_1$ (high reflectivity) and $M_2$ (93% reflectivity) are cavity mirrors, $L_1$ and $L_2$ are plano-convex lenses in a $4f$ telescope configuration, $L_3$ and $L_4$ are plano-convex lenses in a demagnifying telescope configuration, KTP is a nonlinear crystal for SHG, $L_5$ is an imaging lens. The filter transmits only the frequency-doubled wavelength. (b) Representative experimental results for a ring network of 10 lasers: ($b_1$) near-field intensity distribution, ($b_2$) first harmonic far-field intensity distribution, and ($b_3$) second harmonic far-field intensity distribution. (c) Calculated steady-state solutions of a ring network 10 lasers for different topological charges $q=-4$, $-3$, $-2$, $-1$, 0, $+1$, $+2$, $+3$, $+4$.

Figure 2

Topological defects in ring networks of 10 (top row) and 20 (bottom row) lasers. (a),(d) Experimental near-field intensity distributions. (b),(e) Experimental far-field intensity distributions after SHG. (c),(f) Experimental and calculated radial intensity profiles of far-field intensity distributions after SHG. The probability of topological defects is found to be 2% for 10 lasers, and 18% for 20 lasers.

Figure 3

Dissipative topological defects: (a) Probability of locally stable topological defects ($q\ne 0$) and globally stable in-phase solution ($q=0$) as a function of system size. Red circles are experimental results; blue triangles are numerical results with full laser rate equations showing good agreement with the experimental result; green squares are numerical results with the Kuramoto model indicating much higher defect probability than in the experiment. (b) For a ring network of 20 lasers, the probability of topological defects as a function of normalized pump strength $P_m/P_{\text{th}}$. The simulated results of the laser rate equations are in good agreement with the experimental results while the Kuramoto model results predict much higher defect probability. The coupling strength in the simulation was ${\kappa }_{mn}=0.01$.

Figure 4

Simulated dynamical behaviors of the laser amplitudes and phases at two different pump strengths, for a ring network of $N=20$ lasers. The amplitude spread and phase order parameter are shown as a function of time for pump strength $P_m/P_{\text{th}}=16.7$ in (a) and (b), and for $P_m/P_{\text{th}}=1.25$ in (c) and (d) with the same initial conditions and coupling strength $\kappa =0.001$. The arrows denote the amplitude synchronization time ($t_{\text{amp}}$) and phase locking time ($t_{\text{phase}}$). When $t_{\text{amp}}\ll t_{\text{phase}}$, the system is stuck in a locally stable state with $q\ne 0$ (top), and when $t_{\text{amp}}>t_{\text{phase}}$, the system reaches the globally stable state $q=0$ state (bottom).

Figure 5

Simulated probability of topological defects for a ring network of 20 lasers, using the laser rate equations, (a) as a function of synchronization time of the laser amplitude fluctuations at three different coupling strengths, (b) as a function of phase locking time at different coupling strengths, and (c) as a function of the ratio $⟨t_{\text{phase}}^{0.6}⟩/⟨t_{\text{amp}}⟩$. In all figures, for each coupling strength, every point on the curves corresponds to different pump strength. All trajectories collapse to a single curve, indicating universality in the system.