Curved-space topological phases in photonic lattices

We introduce topological phases in curved-space photonic lattices. In such systems, the interplay between the curvature of space and the topology of the system, as manifested in the topology of the band structure, gives rise to a wealth of new phenomena. We demonstrate the topological curved-space concepts in an experimentally realizable setting of a waveguiding layer covering the surface of a three-dimensional body, and show that the curvature of space can induce topological edge states, topological phase transitions, Thouless pumping, and localization effects. We also describe the analogy between our system and topological phases in dynamical curved space-time settings known from general relativity.

Figure 1

(a) A cylindrical waveguiding layer with an imprinted curved SSH lattice. The yellow arrow indicates the direction of light propagation. (b) Energy spectrum of the curved-space SSH as a function of the ratio of distances $u_d/v_d$; both $u_d$ and $v_d$ depend only on the metric. (c) Eigenmodes of the curved space SSH lattice. The color map represents the light intensity: Light propagating in a topological edge mode (upper) and in a nontopological mode when the system is topologically trivial (lower).

Figure 2

Lattices in dynamical curved space. (a), (b) Lattices of evanescently coupled waveguides (blue) on a light guiding cone (a) and on a periodic sinusoidal surface of revolution (b). (c) The flat-space potential obtained from (b) after mapping it to a flat plane, keeping the relative distances between waveguides the same (the outer waveguides in the lattice go through a longer optical path and stronger oscillations in amplitude than the waveguides in the middle of the lattice).

Figure 3

Topological phase transition induced by the curvature of space. (a) SSH lattice with a topological phase which is invariant to the expansion of space in $x$. The blue lines are waveguides embedded in the guiding SOR. (b) SSH lattice with a topological phase that depends on the expansion of space in $x$. The waveguides are placed in the inner and the outer side of the shell. If $usin\theta <v$, the uniaxial expansion induces a topological phase transition. (c) Realization of a SSH lattice on a SOR that is effectively the lattice of (b) on a small segment (in $z$) near the phase transformation. When $\mathrm{\Delta }z$ is small, the separation between waveguides changes approximately linearly with $z$. The yellow lines are the potential wells and the blue surface is the SOR. (d) Adiabatic propagation of light in the potential of (c) during a phase transition from topological (edge states exist) to trivial (no edge states). (e) Adiabatic propagation in an oscillatory SOR. The light alternates between bulk states and edge states as the topological phase changes due to periodic changes in the metric.

Figure 4

Thouless pumping induced by curved space. (a) The coupling coefficients ratio $u/v$ (red line) and the staggered potential $w$ (black line) in the lossy Rice-Mele pump as a function of $z$. (b) Propagation in an oscillating sinusoidal SOR with a staggered potential obtained by changing the width of the waveguides as a function of curvature according to the black line in (a). Most of the power is pumped from one side to the other periodically, with the same period as the curvature variation.

Figure 5

(a) Delocalization in a contracting Andre-Aubry-Harper lattice (AAH). Propagation in an AAH lattice on a cone, going from the region $t<p$, where all the modes are localized, to the $t>p$ region, where all the modes are extended. Light injected into a single waveguide remains localized until the duality point at $t=p$, after which the beam expands. (b) Energy spectrum of the AAH lattice as a function of the radius of the SOR. The arrow signifies that the cone crosses the duality point marked by the dashed line. (c) Inverse participation ratio of the model in (a) as a function of the radius.

Figure 6

Dynamic curvature-induced delocalization in an AAH lattice. (a) Propagation of an eigenstate in the AAH lattice under oscillating curvature in the $t<p$ regime. The light remains localized and behaves as if the system is adiabatic. (b) The same frequency and amplitude of the metric as in (a) but with average radius around the $t=p$ point. The light delocalizes in a nonadiabatic way. (c) Part of the energy spectrum as a function of the SOR radius; the arrows signify the oscillations of the models (a) and (b), the dashed line is the duality point, and only the green band's states couple in the settings of (a) and (b). (d) Projection of the beam at $z_{\mathrm{final}}$ on the eigenstates of the system (vertical axis) for each eigenstate (horizontal axis), after propagation in an oscillating SOR in the localized regime $(t<p$), for various frequencies ${\mathrm{\Omega }}_{\gamma }$ (second horizontal axis). There is a resonance at the metric frequency ${\mathrm{\Omega }}_{\gamma }=20\left[\frac{1}{m}\right]$.