Kibble-Zurek Mechanism in Topologically Nontrivial Zigzag Chains of Polariton Micropillars

Abstract

We consider a zigzag chain of coupled micropillar cavities, taking into account the polarization of polariton states. We show that the TE-TM splitting of photonic cavity modes yields topologically protected polariton edge states. During the strongly non-adiabatic process of polariton condensation, the Kibble-Zurek mechanism leads to a random choice of polarization, equivalent to dimerization of polymer chains. We show that dark-bright solitons appear as domain walls between polarization domains, analogous to the Su-Schrieffer-Heeger solitons in polymers. The soliton density scales as a power law with respect to the quenching parameter.As initially shown by Kibble [1] for the expansion and cooling of the early Universe, and then for liquid helium by Zurek [2], a system undergoing a second-order phase transition on a finite timescale develops domains with independent order parameters. The Kibble-Zurek mechanism (KZM) allows predicting the typical size of the domains and, therefore, the densities of the topological defects on their boundaries. Their scaling as a function of the quench rate is given by a power law, with the critical exponent of the transition being determined by its universality class [3].A very relevant system to study the KZM are the quantum fluids such as atomic Bose-Einstein condensates (BECs) formed by cooling. Indeed, quantum fluids support topological defects [4,5], their most famous example being a quantum vortex, which, contrary to a classical vortex, like a tornado, cannot disappear "by itself", via a continuous transformation. This property is due to the difference in vortex and ground state topologies, which is guaranteed by the irrotational nature of the fluid described by a complex wavefunction [6]. The nonadiabatic cooling of such a fluid allows the development of topological defects obeying the KZM scaling, as confirmed by experiments [7] and also predicted for multicomponent BECs [8,9]. However, solitons in 1D and half-vortices in 2D spinor BECs [4,5,10] are only quasitopological defects. Indeed, a dark soliton transforms into a grey and eventually disappears by simple acceleration, and a half-vortex can be unwound by a divergent magnetic field. Another system of interest to study the KZM are cavity exciton-polariton quantum fluids [11,12]. Because of the finite polariton lifetime, polariton condensation can be an out-of-equilibrium process driven by the condensation kinetics rather than by thermodynamics [12,13]. As previously pointed out [14,15], the establishment of a steady state by non-resonant pumping in an initially empty system cannot be an adiabatic process and is therefore equivalent to a quenching of the parameters of the system, leading to the appearance of topological defects.Another class of systems which possess topologically protected states are periodic lattices with topologically non-trivial band structures characterized by non-zero Chern numbers, or Zak phase, depending on their dimensionality. The most well-known examples of such systems are the topological insu...

Interaction and motion of vortices in nonequilibrium quantum fluids

“…The scenario of pulsed excitation is that of the Kibble-Zurek mechanism, with a fast ramp through the phase transition. So far, no experimental evidence of Kibble-Zurek scaling was found, but it has attracted theoretical attention in several works [7,8], and stimulated the theoretical study of the subsequent phase ordering kinetics that is governed by vortex-antivortex annihilation [9]. The motion of vortex pairs was analyzed numerically in [10] in order to interpret experimental measurements on polariton condensation.…”

Topological quantum paramagnet in a quantum spin ladder

“…Such bosonic topological bands have been observed not long ago for photons in dielectric superlattices [12]. Theoretical proposals of topological states in polaritonic systems have been made [13][14][15], some of which have been observed experimentally [16]. Besides these examples, bosonic band structures are also realized by elementary excitations of quantum spin systems, e.g., by magnons in (anti)ferromagnets or by triplons in dimerized quantum magnets.…”

Nonlinear gap modes and compactons in a lattice model for spin-orbit coupled exciton-polaritons in zigzag chains

“…As a specific example, a polariton BEC in a zigzag chain of polariton micropillars with photonic spin-orbit coupling, originating in the splitting of optical cavity modes with TE and TM polarization, was proposed in [6]. The simultaneous presence of zigzag geometry and polarization dependent tunneling was shown to yield topologically protected edge states, and in the presence of homogeneous pumping and nonlinear interactions the creation of polarization domain walls through the Kibble-Zurek mechanism, analogous to the Su-Schrieffer-Heeger solitons in polymers, was numerically observed [6]. Of crucial importance is the spin-orbit induced opening of a central gap in the linear dispersion relation.…”