Kibble-Zurek Mechanism in Driven Dissipative Systems Crossing a Nonequilibrium Phase Transition

The Kibble-Zurek mechanism constitutes one of the most fascinating and universal phenomena in the physics of critical systems. It describes the formation of domains and the spontaneous nucleation of topological defects when a system is driven across a phase transition exhibiting spontaneous symmetry breaking. While a characteristic dependence of the defect density on the speed at which the transition is crossed was observed in a vast range of equilibrium condensed matter systems, its extension to intrinsically driven dissipative systems is a matter of ongoing research. In this Letter, we numerically confirm the Kibble-Zurek mechanism in a paradigmatic family of driven dissipative quantum systems, namely exciton-polaritons in microcavities. Our findings show how the concepts of universality and critical dynamics extend to driven dissipative systems that do not conserve energy or particle number nor satisfy a detailed balance condition.

Figure 1

Nonequilibrium phase transition in the polariton system. Top: OPO case. Steady-state, noise-averaged densities for the signal $|{\psi }_s{|}^2$ (blue solid line) and idler $|{\psi }_i{|}^2$ (blue dashed line) fields. Bottom: IP case. Steady-state, noise-averaged field density $|\psi {|}^2$ (blue solid line). In both panels the red curves indicate the steady-state, noise-averaged number of topological defects $⟨N_{\mathrm{v}}{⟩}^{\mathrm{ss}}$. All quantities are plotted as a function of the distance to criticality $\epsilon$. Insets: typical snapshots of the field profile in the initial and final states, as indicated by the thick green arrows at the top of the panels. Typical final state profiles are displayed for different ramp speeds of different timescale ${\tau }_Q$.

Figure 2

Average number of vortices as a function of the distance to criticality $\epsilon$ for different ramp speeds of characteristic time ${\tau }_Q$ (thin solid curves) and at steady state (red dashed curves) for OPO (top) and IP (bottom) pumping schemes. Chosen values for OPO (from right to left): ${\tau }_Q=0.3$ (blue curve), 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3, 6, 9, 12, 15, 18, 21, 24 ns (brown curve) (with corresponding IP values indicated within figure). For each value of ${\tau }_Q$, the dashed vertical lines indicate the point ${\widehat{\epsilon }}_{\mathrm{num}}({\tau }_Q)$ where the number of vortices in the dynamical evolution starts departing from the steady-state value (see Supplemental Material [49] for details). Insets: red solid curves show the characteristic relaxation time $\tau$ of the vortices as a function of the distance to criticality $\epsilon$. Dashed straight line indicates an example of the dependence of the characteristic time $\epsilon (t)/\dot{\epsilon }(t)$ on $\epsilon$ for a specific choice of ramp parameters, namely ${\tau }_Q=1.2\mathrm{ns}$ (OPO) and ${\tau }_Q=4.5\mathrm{ns}$ (IP) with $a_p=0.1942$ (OPO) and $a_p=0.2$ (IP).

Figure 3

Numerical prediction for the crossover time $|{\widehat{t}}_{\mathrm{num}}|$ (corresponding to ${\widehat{\epsilon }}_{\mathrm{num}}$ in Fig. 2) plotted with error bars [70] as a function of the theoretical crossover time $|{\widehat{t}}_{\mathrm{KZ}}|$ (see insets of Fig. 2) predicted by the Zurek relation Eq. (2) (with $A=1$) for the OPO (top) and IP (bottom) pumping schemes. The observed linear dependence between both variables is a clear indication of the applicability of Zurek’s relation Eq. (2) and of the KZ mechanism. We obtain a zero intercept (within the error bars) for the OPO polariton system and a small nonzero intercept for the IP case, which indicates nonuniversal subleading order corrections.