Topological States Enabled by Non-local Nonlinearity in Synthetic Dimensions

TL;DR

The work investigates topology in a synthetic SSH lattice with all-to-all nonlocal nonlinearity, developing a Bogoliubov nonlinear adiabatic theory that yields a quantized nonlinear winding and Berry phase protected by an effective chiral symmetry. As nonlinearity increases, swallowtail band structures and band swapping emerge, producing fractional windings and period-doubled Bloch oscillations that connect to discrete time-crystal physics. Remarkably, strong nonlocal nonlinearity can induce emergent topological phases even when the linear system is trivial, with nonlinear bulk-edge phenomena shaped by the boundary and nonlocal coupling. The proposed photonic realization with degenerate cavities and Rydberg-mediated interactions offers a feasible platform to explore nonlinear topological dynamics in synthetic dimensions and to realize robust time-multiplexed topological states.

Abstract

The interplay between topology and nonlinearity represents a central challenge in modern physics. Here, we investigate this interplay by considering a synthetic Su-Schrieffer-Heeger lattice with all-to-all nonlocal interactions. We find that the distinctive nonlinearity maintains an effective chiral symmetry and leads to a quantized nonlinear winding and Berry phase, as corroborated by the developed Bogoliubov nonlinear adiabatic theory. Increasing nonlinearity drives a sequence of topological transitions signaled by the appearance of characteristic swallowtail band structures at intermediate interaction strengths and band swapping in the strong nonlinear regime. The band swapping results in quantized fractional windings and double-period Bloch oscillations that are closely related to discrete time crystals. Remarkably, even starting from a topologically trivial linear system, nonlocal nonlinearity can induce an emergent topological phase with fractional windings. Experimentally, our model can be realized using photons in a degenerate optical cavity with Rydberg-mediated interactions. Our results establish a rigorous framework and pave the way for exploring nonlinear topological phenomena and their applications in synthetic quantum platforms.

Figures (10)

• Figure 1: Nonlinear bands and winding properties. With increasing nonlinearity: (a) Two well-defined bands. (b) Swallowtail emerges in the lower band. (c) Swallowtail forms in both bands. (d) Swallowtails connect into a four-band structure. Color bar indicates the phase $\varphi_{m,k}$, whose integral yields the winding number. $J=1$ and $\delta J=0.3$ in all plots.
• Figure 2: Phase diagram and edge states. (a) The winding number of each band are presented for different phases, with $\varnothing$ denoting the ill-defined winding number. Phases $\text{T}_4$ and $\text{T}_4'$ have four well-defined bands (i.e., four winding numbers). (b) The synthetic lattice with open boundary at $l = 0$ and $l = 1$. (c, d) The anti-symmetric and symmetric edge states for the lattice geometry in (b), with $\delta J=0.3$, $U=gN_{\rm edge}=0.2$. $\bar{\psi}=\psi/\sqrt{N_{\rm edge}}$ is the normalized wave function. For $\delta J>0$, two edge-state solutions exist at small $U$ while only the anti-symmetric one persists into the large-$U$ regime, as delineated by the red-dotted lines in (a). $J=1$ in all plots.
• Figure 3: Nonlinearity induced period multiplexing. (a) Overlap of low-energy eigenstates between $k\ne0$ and $k=0$. It recovers the initial value after $k$ traverses the BZ once (twice) for weak (strong) nonlinearity. (b) Bloch oscillation of two wave packets initialized in the instantaneous eigenstate at $k=0$. Color bar shows the phase evolution $\arg[a_k^*(t)b_k(t)]$ during propagation along $k=\epsilon t$ with $\epsilon=0.01$ and $U=1.5$. The large-amplitude wave packet exhibits period-doubling, compared with weak-amplitude ordinary Bloch oscillation, as marked by the vertical arrows. (c) Full quantum energy bands (colored solid lines) with $\hat{n}_k=27$ and $U=g\hat{n}_k$ , comparing to the mean-field results (black dots). Color bar indicates the phase $\varphi_k=\arg[\langle \hat{a}^\dagger_k \hat{b}_k\rangle]$. $J=1$, $\delta J=0.3$ in all plots.
• Figure 4: Schematic of the proposed experimental setup. (a) OAM modes in the main cavity are coupled by coupler cavities consisting spatial light modulators (SLMs). The beam rotators (BRs) would induce a tunneling phase $e^{il\theta_R}$Luo2018TopologicalPhotonic. (b) Nonlinearity is introduced by coupling the cavity photon with atomic Rydberg states through Raman process, with atom-cavity coupling $g_\text{a}$ and external pump $\Omega$.
• Figure 5: Adiabatic evolution in $\theta$-space. (a) The wave packet eventually converts from $\theta$ to $\theta+\pi$ during the propagation for weak nonlinearity, restoring its initial state with a period of $\pi$. (b) Strong nonlinearity suppresses the conversion, doubling the period to $2\pi$.