Chiral Bogoliubov excitations in nonlinear bosonic systems

We present a versatile scheme for creating topological Bogoliubov excitations in weakly interacting bosonic systems. Our proposal relies on a background stationary field that consists of a kagome vortex lattice, which breaks time-reversal symmetry and induces a periodic potential for Bogoliubov excitations. In analogy to the Haldane model, no external magnetic field or net flux is required. We construct a generic model based on the two-dimensional nonlinear Schrödinger equation and demonstrate the emergence of topological gaps crossed by chiral Bogoliubov edge modes. Our scheme can be realized in a wide variety of physical systems ranging from nonlinear optical systems to exciton-polariton condensates.

Figure 1

Top: Optical pumping scheme allowing one to create a suitable condensate for topological Bogoliubons, with incident field composed of six equal-frequency plane-wave components with wave vectors ${\mathbf{k}}_n$ (depicted by arrows) and phases chosen as ${\phi }_n=0$ except for ${\phi }_4=2\pi /3$ and ${\phi }_6=-2\pi /3$. Bottom: Intensity (left) and phase (right) pattern of the resulting field. Intensity maxima form a kagome structure (white lines) with vortex-antivortex pairs located such that each hexagonal plaquette is threaded by a flux $2{\mathrm{\Phi }}_0=4\pi$, with smaller triangular plaquettes threaded by $-{\mathrm{\Phi }}_0$. Although fluxes cancel out over the whole system, a well-defined chirality emerges, defined by the sign of ${\mathrm{\Phi }}_0$. Bogoliubov excitations of the condensate experience nontrivial fluxes due to interactions, leading to TR symmetry breaking and ultimately to topological states.

Figure 2

(a) Bogoliubov spectrum in the off-resonant tight-binding limit (in strip geometry): The positive part of the spectrum exhibits three bands with topological gaps crossed by pairs of counterpropagating chiral edge states, reminiscent of a kagome lattice model with staggered fluxes [40]. The red (blue) coloring indicates the degree of localization of states at the lower (upper) edge (as measured by their total intensity on the lower (upper) half of the system). Parameters were chosen as $\mathrm{\Omega }/t=6$ and $g/\mathrm{\Omega }=2/3$, in a strip geometry with periodic boundary conditions in the $x$ direction (see Supplemental Material [32] for details). (b) Unit cell used in the tight-binding model, with phases for the on-site coupling chosen as $(0,2\pi /3,-2\pi /3)$ in the basis $(A,B,C)$. (c) TR symmetry-breaking mechanism: The off-resonant coupling between $u$- and $v$-like excitations [see Eq. (3)] leads to virtual transitions which renormalize the hopping term $t$.

Figure 3

(a) Bogoliubov spectrum of exciton-polaritons (in strip geometry), exhibiting qualitatively similar low-energy features as the tight-binding spectrum of Fig. 2. Differences stem from longer-range hopping terms and weak repulsive interactions within the condensate which are neglected in tight-binding approximation (e.g., the upper topological gap vanishes because of the modified dispersion). Although $u$- and $v$-like excitations [see Eq. (3)] overlap in energy (around $\omega =0$), all eigenvalues are real, indicating that the system is stable. Parameters were taken from Ref. [53], giving a topological gap of about $0.06\mathrm{meV}$ [32]. (b) Practical realization in semiconductor microcavities composed of distributed Bragg reflectors (DBRs) [45], with periodic potential $V\left(\mathbf{x}\right)$ induced, e.g., via metal surface patterning (other mechanisms are discussed in the text). (c) Realization of the effective potential $V\left(\mathbf{x}\right)$ in a nonlinear optical system, using coupled waveguides [55].