Experimental Observation of Aharonov-Bohm Cages in Photonic Lattices

We report on the experimental realization of a uniform synthetic magnetic flux and the observation of Aharonov-Bohm cages in photonic lattices. Considering a rhombic array of optical waveguides, we engineer modulation-assisted tunneling processes that effectively produce nonzero magnetic flux per plaquette. This synthetic magnetic field for light can be tuned at will by varying the phase of the modulation. In the regime where half a flux quantum is realized in each plaquette, all the energy bands dramatically collapse into nondispersive (flat) bands and all eigenstates are completely localized. We demonstrate this Aharonov-Bohm caging by studying the propagation of light in the bulk of the photonic lattice. Besides, we explore the dynamics on the edge of the lattice and discuss how the corresponding edge states can be continuously connected to the topological edge states of the Creutz ladder. Our photonic lattice constitutes an appealing platform where the interplay between engineered gauge fields, frustration, localization, and topological properties can be finely studied.

Figure 1

(a) Schematic diagram of a rhombic (or diamond) lattice with three sites ($A$, $B$, and $C$) per unit cell. (b) Complex-valued tunneling amplitudes are generated by applying a linear gradient of on-site energy $\mathrm{\Delta }$ along the lattice and then periodically modulating the on-site energies with frequency $\omega =\mathrm{\Delta }$ and phase $\theta$, see Fig. 2. (c) In this driving protocol, a nonzero synthetic magnetic flux ($\mathrm{\Phi }$) per plaquette is realized and it can be tuned by varying the phase of modulation $\theta$. (d) Simplified sketch illustrating the 3D waveguide paths of a driven photonic rhombic lattice with $\mathrm{\Phi }=\pi$ flux per plaquette. The linear gradient of on-site energy is realized by circularly curving the lattice. Here, the color contrast indicates a square wave modulation of the on-site energy.

Figure 2

(a) Variation of flux and the modulus of effective tunneling amplitude as a function of the phase of modulation ($\theta$) for $K/\omega =1.35$. (b) Quasienergy spectrum for a finite rhombic chain with $\theta =\pi /2$. Blue and red colors are associated with the bulk and edge modes, respectively. The dashed vertical lines indicate the values of $K/\omega$ (1.35 and 0.85) that were realized in the experiment. (c) A comparison of the Floquet quasienergy spectrum for $\omega /J=12.64$ and $\omega /J\to \infty$. Here, $K/\omega =1.35$. (d) Zoom of the energy spectrum of ${\widehat{H}}_{\text{eff}}^{(0)}$ with an additional detuning ${\delta }_A$ of the on-site energy at $A$ sites (full spectrum shown in the inset). The red dashed line is the energy ($E_{-}^{\text{Creutz}}=-4J_{\text{eff}}^2/{\delta }_A$) of the lowest flatband of the Creutz ladder into which we map ${\widehat{H}}_{\text{eff}}+{\widehat{H}}_{{\delta }_A}$ for ${\delta }_A\gg J_{\text{eff}}$. The upper flatband of the Creutz ladder is at energy $E_{+}^{\text{Creutz}}=0$. The black dotted line corresponds to the edge state, which becomes a topological midgap state (protected by an emergent chiral symmetry) in the Creutz-ladder limit ${\delta }_A\gg J_{\text{eff}}$. Parameters are chosen as in (c).

Figure 3

Observation of Aharonov-Bohm photonic caging in the presence of $\mathrm{\Phi }=\pi$ magnetic flux per plaquette. Experimentally measured output intensity distributions at five different propagation distances (shown on each image) for $K/\omega =1.35$ (a) and $K/\omega =0.85$ (b). The optical intensity exhibits a breathing motion whose frequency is determined by $K/\omega$. For all measurements, light was injected at the red circled $A$ site—this input state overlaps efficiently with the upper and lower flatbands and the frequency of the observed breathing motion is related to the energy gap between these two bands. Each image is normalized so that the total intensity is 1. The field of view is approximately $121\mu \mathrm{m}\times 61\mu \mathrm{m}$. The inset shows the spreading (delocalization) of the optical intensity in a straight lattice (i.e., $\mathrm{\Phi }=0$); see [36].

Figure 4

(a) Graphical representation of Fig. 3. The dashed and solid lines indicate the $z$ evolution of optical intensities (at the initially excited bulk $A$ site), which were numerically calculated solving the Schrödinger equation for $\omega /J=12.64$ associated with the Hamiltonians ${\widehat{H}}_{\text{eff}}$ and $\widehat{H}(t\leftrightarrow z)$, respectively. Small oscillations can be observed as a consequence of micromotion. The filled circles indicate the measured light intensities at the $A$ site. See also Fig. A3 in [36]. (b) Edge dynamics—when light is launched at the $A$ site on the edge, the input state overlaps with edge modes located in both band gaps. In this case, a breathing motion with relatively lower frequency (compared to the bulk excitation) was observed, as expected. Indeed, the frequency of this breathing motion is associated with the energy difference between the edge modes [see Fig. 2].