Interacting quantum fluid in a polariton Chern insulator

Abstract

We study Bogoliubov excitations of a spinor Bose Einstein condensate in a honeycomb periodic potential, in the presence of a Zeeman field and of a spin-orbit coupling specific for photonic systems, which is due to the energy splitting between TE and TM polarized eigenstates. We also consider spin-anisotropic interactions typical for cavity polaritons. We show that the non-trivial topology of the single particle case is also present for the interacting system. At low condensate density, the topology of the single-particle bands is transferred to the bogolon dispersion. At a critical value, the self-induced Zeeman field at the Dirac points of the dispersion becomes equal to the real Zeeman field and then exceeds it. The gap is thus closed and then re-opened with inverted Chern numbers. This change of topology is accompanied by a change of the propagation directions of the one-way edge modes. This result demonstrates that the chirality of a topological insulator can be reversed by collective effects in a Bose-Einstein condensate. PACS numbers:The concept of topological insulators relies on the chirality of the Bloch bands. This concept was first introduced for electronic systems 1-5 and then extended to photonic 6-9 , and more generally to bosonic systems 10,11 . In this last case, the non-trivial topology of the system is addressed by either resonant excitation in photonics, or by driving an atomic gas strongly out of equilibrium 12 . Another field of research, extremely fruitful, is the one dealing with collective effects. In fermionic systems, the combination of non-trivial topology and many body effects led to the most intriguing phenomena in physics such as the fractional quantum Hall effect 13 , topological superconductors 5,14 , and a vast variety of other phenomena 15 . In bosonic systems, the most well-known collective effect is the Bose-Einstein Condensation 16,17 , leading to fascinating dynamical behaviour such as superfluidity 18 , and, more generally, to the physics of quantum fluids [19][20][21] . So far, the physics of bosonic quantum fluids in topologically non-trivial systems has been addressed only in a couple of works, essentially dealing with atomic condensates [22][23][24] .On the other hand, the mixed exciton-photon quasiparticles called exciton-polaritons represent a very good implementation of quantum fluids of light 21 in which a wide variety of phenomena such as polariton BEC 25 , superfluidity 26 , quantized vortices 27 have been observed. Polaritons possess the same type of spin-orbit coupling as other photonic systems, based on energy splitting between TE and TM polarized eigenmodes. (For a review on polariton spin effects see Ref.28 ). The specific relation between the polarization of the TE and TM modes and their propagation direction induces an intrinsic chirality for the photonic system which serves as a basis for several effects, such as the Hall effect for light 29 and optical spin Hall effect 30,31 . When this spin-orbit coupling (SOC) is combined with a finite Zeeman field, the...

Photonic versus electronic quantum anomalous Hall effect

“…This dependence has already been shown to lead to the inversion of the effective field sign (and thus the inversion of the topology) when both applied and SIZ fields are present in a Bose-Einstein condensate [51].…”

“…We choose this notation to make clearer the comparison between the two cases. Indeed, in our precedent works on polariton honeycomb lattices, we used the notation λ p = δJ [50,51,58]. Taking into account the TE-TM splitting the tunneling coefficients are defined in the circular-polarization basis as:…”

“…[51]. For this, we set a basis of Bogoliubov Bloch waves (u ± A/B,n , v ± A/B,n ) where n index numerates stripes, and k y is the quasi-wavevector in the zigzag direction.…”

“…It is at the origin of a very large variety of spin-related effects, such as the optical spin Hall effect [44,45], half-integer topological defects [46,47], Berry phase for photons [48], and the generation of topologically protected spin currents in polaritonic molecules [49]. The combination of a TE-TM SOC and a Zeeman field in a honeycomb lattice has indeed been found to yield a QAH phase [29,[50][51][52][53][54][55], and the related model represents a generalization of the seminal Haldane-Raghu proposal [56] of photonic topological insulator, recovered for large TE-TM SOC.In this manuscript, we demonstrate the role played by the winding number of the SOC on the QAH phases. We establish the complete phase diagram for both the photonic and electronic graphene.…”

Modulational instability and solitary waves in polariton topological insulators

“…The observed polariton effects with linear and nonlinear lattice potentials include one- [31] and twodimensional [32,33] gap polariton solitons, visualization of Dirac cones [34] and flat bands [35], and visualization of non-topological edge states [21]. Recently, it has been shown theoretically that attractive nonlinear interaction between polaritons with opposite spins can compensate and exceed Zeeman energy shifts due to magnetic field and thereby lead to the inversion of the propagation direction of the edge states [36]. Apart from this result the nonlinear effects with topological polariton edge states remain unexplored.…”