Photonic Topological Spin Pump in Synthetic Frequency Dimensions

Reducing geometrical complexity while preserving desired wave properties is critical for proof-of-concept studies in wave physics, as evidenced by recent efforts to realize photonic synthetic dimensions, isospectrality, and hyperbolic lattices. Laughlin’s topological pump, which elucidates quantum Hall states in cylindrical geometry with a radial magnetic field and a time-varying axial magnetic flux, is a prime example of these efforts. Here we propose a two-dimensional dynamical photonic system for the topological pumping of pseudospin modes by exploiting synthetic frequency dimensions. The system provides the independent control of pseudomagnetic fields and electromotive forces achieved by the interplay between mode-dependent and mode-independent gauge fields. To address the axial open boundaries and azimuthal periodicity of the system, we define the adjusted local Chern marker with rotating azimuthal coordinates, proving the nontrivial topology of the system. We demonstrate the adiabatic pumping for crosstalk-free frequency conversion with wave front molding. Our approach allows for reproducing Laughlin’s thought experiment at room temperature with a scalable setup.

Figure 1

Photonic hardware for topological spin pumps. (a) Schematic of the pump. Each site for a resonance mode is labelled by the indices $m\in [1,M]$ and $n\in [1,N]$, where $M$ and $N$ are the numbers of the modes and resonators, respectively. Colored arrows indicate the coupling. (b) Coupled-resonator platform for realizing (a). Black circles and gray curved squares denote resonators and waveguide loops, respectively. The red and yellow regions are tunable phase shifters for the loops and resonators, respectively. Purple arrows show wave circulations. Blue regions indicate structural imbalance. (c) Layers of coupled resonators along the synthetic axis. ${\omega }_m$ is the $m$th-mode resonance frequency.

Figure 2

Band properties of pseudospin pumps with either $B_{\rho }$ or ${\partial }_t{\mathrm{\Phi }}_z(t)$. (a) Pump spectrum with $A_{\mathrm{D}}(t)=0$ for $B_{\rho }=\pi /8$, $M=32$, and $N=16$. (b) Eigenfunction intensities along the frequency ($m-$) axis. (c) Pump spectrum with $B_{\rho }=0$ for $A_{\mathrm{D}}(t)=0$. (d) Temporal evolution of the beam profiles on the cylinder according to linearly varying $A_{\mathrm{D}}(t)$. An initial wave packet is a superposition of two adjacent eigenfunctions in the lowest band [red dots in (c)]. $T_c=8/J$ is the characteristic time.

Figure 3

Real-space topological measures. (a),(b) Coordinate-indexing scheme and the lowest-band $C_{\mathrm{L}}({\mathbf{r}}_{mn})$ for (a) $M=32$ and (b) $M=16$. (c) $C_{\mathrm{L}}({\mathbf{r}}_{mn})$ as a function of $m/M$ for different $M$. (d) The averages and standard deviations of $C_{\mathrm{L}}({\mathbf{r}}_{mn})$ for 100 random realizations for each $W$, where $W$ denotes the degree of diagonal disorder uniformly distributed over $[-W,+W]$. The error bars are 10 times the standard deviations for clarity.

Figure 4

Adiabatic pumping. Pumping dynamics in (a),(b) band diagrams and (c)–(e) the corresponding temporal monitoring of beam profiles: (c),(d) for (a) and (e) for (b). In (a),(c),(d), initial states (black circles) undergo adiabatic intraband transition due to ${\mathrm{\Phi }}_z(t)$ and arrive at the final states (black triangles). In (b), black dots denote each excitation at the time $t$. The colored numbers of the lower (‘1,2’) and higher (‘1–4’) bands in (a) denote the sequences of the pumping in (c) and (d), respectively. In the left figures of (c)–(e), solid and dashed arrows represent ${\mathrm{\Phi }}_z(t)$ and the temporal range of wave excitations, respectively. (b) shows the lowest band occupation after the simultaneous excitation of the incidence and pumping, which leads to spatial ($n$) localization in (e). The displacement along the frequency ($m$) axis [center figures in (c)–(e)] is proportional to $\mathrm{\Delta }{\mathrm{\Phi }}_z$. Nodeless single rails in (c),(e) and double rails with a node in (d) originate from the ground and first excited state excitations in (a),(b).