Observation of Unidirectional s-p Orbital Topological Edge States in Driven Photonic Lattices

Abstract

Time-periodic modulation of a static system is a powerful method for realizing robust unidirectional topological states. So far, all such realizations have been based on interactions among $s$ orbitals, without incorporating inter-orbital couplings. Here, we demonstrate higher-orbital Floquet topological insulators by introducing periodically modulated couplings between the optical $s$ and $p$ orbitals in a square lattice. The staggered phase of the $s$-$p$ couplings gives rise to a synthetic uniform $π$ magnetic flux per plaquette of the lattice, and periodic driving of the couplings opens a topological bandgap, characterized by the Floquet winding number. We image topological edge modes of $s$-$p$ orbitals traveling unidirectionally around a corner. Here, the topological phases are realized by a combined effect of the periodic driving and synthetic magnetic flux. Consequently, when the synthetic flux is turned off, the system becomes trivial over a range of driving parameters. Our results open a promising pathway for exploring topological phenomena by introducing the orbital degree of freedom.

Figures (9)

• Figure 1: Photonic Floquet topological insulator of $s$-$p$ orbitals. (a) Schematic of a square lattice of $s$-$p$ orbitals with two sites ($\text{A}$ and $\text{B}$) per unit cell and nearest-neighbor couplings $J_{1-4}^{sp}$. Note that $J_1^{sp}$ is negative, realizing a uniform synthetic $\pi$ magnetic flux per plaquette. The inset on the top shows the refractive index profile and the resonance condition, $\beta_s^\text{A}\!=\!\beta_p^\text{B}$. (b) Photonic bands of propagation constant $\beta(k_x, k_y)$ (analogous 'energy') for homogeneous coupling strengths $|J_{1-4}^{sp}|=J$, exhibiting two Dirac cones at $(k_xa,k_ya)\!=\!(\pm\pi/2, 0)$ with linear dispersion. The system in (a) is made topological by periodically switching on and off the couplings (one at a time) in a clockwise manner along the propagation distance, $z$. (c) Quasienergy spectrum for a strip geometry along the $y$ direction of the Floquet square lattice with experimentally realized parameter $\Lambda_{1-4}\!=\! 0.76\,\pi/2$. (d-f) Intensity distributions of bulk and edge modes (on the top and bottom edges) of $s$-$p$ orbitals, respectively. (g) Simplified sketch showing the photonic realization of the Floquet topological insulator of $s$-$p$ orbitals. The coupling between any two sites is switched on and off by synchronously bending the waveguide paths along $z$. Non-zero horizontal couplings are realized by changing the waveguide positions as indicated in the inset. (h) White-light transmission micrograph (cross-section) of a femtosecond laser fabricated $s$-$p$ orbital square lattice.
• Figure 2: Probing unidirectional $s$-$p$ orbital topological edge states. Experimentally measured output intensity distributions at $z\!=\!2z_0$. Here, $\Lambda =(0.76\pm 0.02)\pi/2$. The red circle marks the input coupling position. In (a) and (b), the light is coupled to the $s$ mode of A sites at the input, whereas in (c) and (d), it is mostly ($\sim \!81\%$) coupled to the $p$ mode of B sites. The topological edge modes propagate unidirectionally and are not scattered by corners. Each measured intensity pattern is normalized to $1$.
• Figure 3: Topological invariants. (a) Bulk quasienergy spectrum of the $s$-$p$ lattice as a function of $\Lambda$ showing the bandgap closing and opening, which indicates topological phase transitions. (b) Winding numbers W$_\varepsilon$ for the gaps centered around quasienergy $\varepsilon\!=\!0$ and $\pm \Omega/2$ for the $s$-$p$ lattice. The solid lines are guides to the eye. (c, d) Similar calculations for a lattice with $s$-$s$ couplings with zero flux. The quasienergy gap $\varepsilon\!=\!\pm \Omega/2$ is topological for $\pi/4 < \Lambda < 3\pi/4$. The dashed vertical lines in (b, d) indicate $\Lambda$ values used in Figs. \ref{['Fig_SP_SS_edge_excitation']} (a, b).
• Figure 4: Importance of synthetic $\pi$ flux. (a) Experimental signature of Floquet topological edge modes in a $s$-$p$ orbital lattice with $\Lambda\!=\!(0.3\pm0.04)\pi/2$. (b) Same as (a) but in a lattice consisting of $s$-orbitals only with $\Lambda\!=\!(0.3\pm0.03) \pi/2$. In this case, the lattice is trivial as indicated by the near-symmetric (i.e., not unidirectional) spreading of the output intensity pattern along the edge. Each measured intensity pattern is normalized to $1$. (c, d) Quasienergy spectrum associated with (a, b), respectively, further highlighting the topological characteristics of the lattices
• Figure S1: (a) Schematic of femtosecond laser writing. (b) Sketch showing the bending profiles of the A and B waveguides along the propagation direction. (c, d) Schematics of the cross-sections of relative positions of the $s$ and $p$ orbitals at three different $z$ values (i.e., $z_1$, $z_2$, and $z_3$) for the $J_3^{sp}$ and $J_4^{sp}$ couplings, respectively.