Nonlinear symmetry breaking of Aharonov-Bohm cages

We study the influence of mean-field cubic nonlinearity on Aharonov-Bohm caging in a diamond lattice with synthetic magnetic flux. For sufficiently weak nonlinearities, the Aharonov-Bohm caging persists as periodic nonlinear breathing dynamics. Above a critical nonlinearity, symmetry breaking induces a sharp transition in the dynamics and enables stronger wave-packet spreading. This transition is distinct from other flatband networks, where continuous spreading is induced by effective nonlinear hopping or resonances with delocalized modes and is in contrast to the quantum limit, where two-particle hopping enables arbitrarily large spreading. This nonlinear symmetry-breaking transition is readily observable in femtosecond laser-written waveguide arrays.

Figure 1

(a) Schematic representation of a diamond lattice (characterized by bipartite symmetry) made of coupled optical waveguides. A synthetic magnetic flux $\mathrm{\Gamma }$ is applied in each plaquette. (b) Band structure for $\mathrm{\Gamma }=0,2\pi$ (dotted orange line), $\mathrm{\Gamma }=3\pi /4$ (dashed turquoise line), and $\mathrm{\Gamma }=\pi$ (blue solid lines). (c–e) Geometry-induced ${\beta }_{FB}=0$ flatband modes for different flux strengths (c, d) and additional flatband modes (${\beta }_{FB}=\pm 2$) originating due to the AB caging (e).

Figure 2

Linear instability spectrum of nonlinear CLS. (a, b) Positive frequency part of full linear perturbation eigenvalue (EV) spectrum for (a) $\mathrm{\Gamma }=3\pi /4$ and (b) $\mathrm{\Gamma }=\pi$. Pure real EVs (cyan – C) and nonzero real parts of complex EVs (black – B) mark regions of CLS linear instability. Pure imaginary EVs and imaginary parts of complex EVs are shown in green (G) and red (R), respectively. (c) Characteristic eigenmode profiles of perturbed CLS corresponding to three EV branches (1), (2), and (3) from (b). Size of the system is $N=21$.

Figure 3

Nonlinear delocalization and symmetry breaking of the CLS. (a–c) Intensity profiles $I_n=|a_n{|}^2+|b_n{|}^2+{\left|c_n\right|}^2$ at $z=10\pi$ for $\mathrm{\Gamma }=0$ (a), $\mathrm{\Gamma }=3\pi /4$ (b), and $\mathrm{\Gamma }=\pi$ (c). Cases when $g=0.5$ and $g=1$ are depicted with blue circle and red square symbol lines, respectively. Vertical dashed yellow lines mark the cell position of the input CLS. (d) Spreading of the CLS measured via the maximal value of the participation ratio $R$ within the propagation length $z=10\pi$. (e) $z$ average of the normalized leg imbalance $\eta$.

Figure 4

Periodic dynamics of $\mathrm{\Gamma }=\pi$ compact breathing modes in the weak-instability regime $g=0.5$. (a–c) Evolution of compact breathing modes when the initial excitation condition is (a) CLS from ${\beta }_{FB}=0$ submanifold, (b) single B site, and (c) single A site. Dashed purple lines mark edges of central unit cell, while yellow circles on the left denote excited sites at $z=0$. Vertical axes schematically represent waveguides of the array. Each cell contains three waveguides ($a, b$, and $c$) marked in plot (a–c) for the general case of the $n\mathrm{th}$ cell. All plots contain the same number of cells. (d–f) Power spectra of the breathers in (a–c) obtained from Fourier transform of the site intensities $|a_n{\left(z\right)|}^2$ (red/dark gray) and $|b_n{\left(a\right)|}^2$ (green/light gray) in the central unit cell.

Figure 5

Dynamics in the strong-instability regime $g=1.9$ for the three different excitation conditions of Fig. 4. (a–c) $z$ evolution exhibiting emergence of symmetry-broken profiles for the CLS and B excitations (a, b), while the A excitation remains symmetric. Vertical axes schematically represent waveguides of the array. Each cell contains three waveguides ($a, b$, and $c$) marked in plots (a–c) for the general case of the $n\mathrm{th}$ cell. (d–f) Corresponding power spectra of the intensities in the central unit cell.