Topologically nontrivial Floquet band structure in a system undergoing photonic transitions in the ultrastrong-coupling regime

We consider a system of dynamically-modulated photonic resonator lattice undergoing photonic transition and show that in the ultrastrong-coupling regime such a lattice can exhibit nontrivial topological properties, including topologically nontrivial band gaps, and the associated topologically robust one-way edge states. Compared with the same system operating in the regime where the rotating wave approximation is valid, operating the system in the ultrastrong-coupling regime results in a one-way edge mode that has a larger bandwidth and is less susceptible to loss. Also, in the ultrastrong-coupling regime, the system undergoes a topological insulator-to-metal phase transition as one varies the modulation strength. This phase transition has no counterpart in systems satisfying the rotating wave approximation, and its nature is directly related to the nontrivial topology of the quasienergy space.

Figure 1

Lattice composed of two types of resonators, $A$ (red) and $B$ (blue). The lines represent coupling (or bonds) between nearest neighbors. All coupling strengths are modulated in time harmonically. For all bonds along the horizontal direction the modulation phase is zero. The bonds along the vertical direction have a spatial distribution of modulation phases as specified in the figure. The specified phases are for the hopping matrix elements of the Hamiltonian in Eq. (1) along the positive $y$ direction.

Figure 2

Floquet band structure for the Hamiltonian $\tilde{H}$ of Eq. (3): (a) with RWA from ${\tilde{H}}_{\mathrm{RWA}}$ (red solid line) and $V=0.02\mathrm{\Omega }$ (blue dashed line), (b) $V=0.5\mathrm{\Omega }$, (c) $V=1.1\mathrm{\Omega }$, and (d) $V=1.5\mathrm{\Omega }$.

Figure 3

The projected RWA band structure folded into the temporal first Brillouin zone as a function of $V$. The blue regions correspond to the bands. The white regions are the band gap regions.

Figure 4

(a) The bandwidth of the topologically nontrivial gap as a function of $V$. (b) The projected band structure at $V=0.5\mathrm{\Omega }$.

Figure 5

The simulation results with the Hamiltonian (1) in a $20\times 20$ resonator lattice $\left({\omega }_A=12\pi c/a,{\omega }_B=0\right)$. A point source is placed at the location (1,10). The propagation field profiles are plotted as follows: (a) at time $t=100a/c$ with $V=0.02\mathrm{\Omega }$ and the source frequency ${\omega }_s=V$; (b) at time $t=10a/c$ with $V=0.5\mathrm{\Omega }$ and ${\omega }_s=V$; (c) same as panel (a); and (d) same as panel (b) with a loss coefficient $\gamma =0.2c/a$ at steady state.