Quantized Hall drift in a frequency-encoded photonic Chern insulator

TL;DR

The paper demonstrates a frequency-encoded photonic Chern insulator by embedding a Haldane-like honeycomb lattice in the synthetic frequency dimension of a fiber-loop system. It achieves bulk-band topology mapping through Bloch-state tomography, extracting Berry curvature across the Brillouin zone and confirming Chern numbers $\mathcal{C}=0,\pm1$ for different topological phases. A driven-dissipative analogue of the quantum Hall effect is observed as a quantized Hall drift in frequency space, with an experimental procedure that cancels non-Berry contributions and yields $\mathcal{C}$ via integration over detuned drive spectra. The results establish a versatile platform for robust photonic transport in frequency-multiplexed systems and open avenues for metrology and photonic quantum information processing using topological light. The approach supports tunable, chip-relevant implementations and motivates future exploration of driven-dissipative topological photonics in higher dimensions or with non-Abelian gauge structures.

Abstract

The quantization of transport and its resilience to backscattering are key features for leveraging topological matter in applications that demand stringent noise mitigation, such as metrology and quantum information processing. Due to the bosonic nature of light, engineering such robust, ``one-way'' channels in synthetic photonic systems imposes the implementation of topological models with broken time-reversal symmetry; this is challenging since photons possess neither an electric charge nor a magnetic moment. Here, we propose and demonstrate a novel approach to realizing photonic Chern insulators - topological insulators with broken time-reversal symmetry - by encoding a Haldane-like model in the synthetic frequency dimension of an optical fiber loop platform. The bands' topology is assessed by reconstructing the Bloch states geometry across the Brillouin zone. We further highlight its consequences by measuring a driven-dissipative analogue of the quantized transverse Hall conductivity. Our results open new avenues for harnessing topologically protected light propagation in frequency-multiplexed photonic systems, with applications ranging from precision metrology to photonic quantum processors.

Figures (17)

• Figure 1: Implementation of the Haldane-like model in the synthetic frequency dimension. (a) Schematic representation of our simplified Haldane model with NN couplings (blue lines) and a single pair of NNN couplings (red and green lines) of opposite phase ($\pm\phi_h$) for each sub-lattice. (b) Topological phase diagram, as a function of $\phi_{h}$ and $\Delta/J'$, exhibiting three distinct phases with Chern number $\mathcal{C}=0,\pm1$. (c) Schematic representation of the experimental setup depicting a single loop whose CCW (red arrows) and CW (blue arrows) circulating modes are coupled with a 25/75 fiber coupler (FC). On the right side, a pair of optical circulators allows modulating CW and CCW modes individually. The loop is probed with a photodiode (PD), using a transmission line coupled to the main loop with a $99.9\%$ transmission FC. (d) Optical mode structure for two pairs of supermodes $\ket{i,a}$, $\ket{i,b}$ with $i=m, m+1$, formed from the coupling of CW and CCW modes split by a frequency $\delta_0$. (e) Representation of the brick-wall Haldane-like Hamiltonian with a folding period of $M\Omega$ - each site represents a supermode of the fiber loop. A unit cell is depicted in orange. Blue lines indicate NN couplings with dotted lines indicating the twisted boundary conditions. NNN couplings are indicated with red and green lines for the $\ket{a}$ and $\ket{b}$ sublattices, respectively (for the sake of clarity, boundary conditions are not depicted for NNN couplings). The insets above depicts the associated reciprocal space (left) and a zoom on the first BZ.
• Figure 2: Band structure measurements for graphene-like, hBN-like and Haldane-like models. (a),(e),(g) Band structure measurements for configurations emulating graphene (a), hexagonal boron nitride (e) and the Haldane model with $\phi_h=\pi/2$ (g). The band dispersion is obtained by measuring the time-resolved transmission of the loop as a function of time and laser detuning. (b) The simulation of the graphene-like configuration agrees with the experimental results in (a). (c) Lower band dispersion as a function of $(k_x,k_y)$; a fine coverage of the BZ is obtained by scanning the phase of one driving component. (d),(f),(h) Band dispersion along a trajectory through high-symmetry points $M-K-\Gamma-M$ indicated in (c) for graphene-like (d), hBN-like (f) and the Haldane-like model (h). The latter two clearly show a bandgap, indicated by dashed horizontal lines.
• Figure 3: Extraction of the Berry curvature and Chern number. (a)-(e) Tomographic measurement of the lower band's eigenstates $\ket{\psi^{(+)}_{\textbf{k}}}$ across the full BZ for the cases of graphene-like (a) and hBN-like (b) lattices, for topologically non-trivial Haldane-like model with $\phi_h=+\pi/2$ (c) and $\phi_h=-\pi/2$ (d), and topologically trivial Haldane-like model with $\phi_h=+\pi/2$ and $\Delta>\sqrt{3}J'$(e). The north and south poles of the Bloch spheres correspond to the sub-lattice basis states $\ket{a}$ and $\ket{b}$ respectively. The color code refers to the time bin of each $k$-point. (f)-(j) The Berry curvature, extracted using the Fukui method from the momentum-resolved eigenstates shown in Panels (a)-(e), exhibit clear peaks at the positions of the Dirac point $K$ and $K'$. The value of the Chern number obtained by integrating the Berry curvature over the BZ is indicated in the corner of each panel.
• Figure 4: Photonic analogue of the transverse Hall conductivity in frequency space. (a) Schematic representation of the experimental protocol for measuring the transverse displacement. The effective scalar potential induced by the detuning along $\hat{y}$ is depicted by the blue colour gradient. The driven site is indicated by the red spot and the displacement by the red arrow. (b) Experimental setup for the heterodyne measurement. The local oscillator consists of the laser signal shifted by $200MHz$ and split in two to beat with both the CCW and CW modes, respectively measured with photodiodes PD2 and PD3. (c) Example of a heterodyne spectrum exhibiting broad regions separated by $10\Omega$ with a high density of narrowly-spaced peaks. The inset shows a zoom-in on the central region exhibiting pairs of peaks separated by $\Omega$. (d)-(e) Anomalous transverse displacement in units of the unit cell ($\Omega$) as a function of the laser detuning for a graphene-like lattice (d) and two topologically distinct phases of the Haldane-like model (e) . For the graphene-like lattice, we indicate the Chern number extracted by integrating the displacement over the laser detuning bandwidth corresponding to the lower and upper band. (f) Summary of the extracted Chern numbers extracted for the Haldane-like model where 3 measurements are taken with $\phi_h=+\frac{\pi}{2}$ (blue) and 4 measurements taken with $\phi_h=-\frac{\pi}{2}$ (red). Circles (diamonds) correspond to integrating over a bandwidth corresponding to the lower (upper) band. Taking the mean of the absolute value of all these points, we obtain $\abs{\mathcal{C}}_{mean}=0.95$ with a standard deviation of $\sigma=0.14$. The error bars reflect an experimental uncertainty of $\pm 0.10$ on the extracted Chern numbers, obtained by propagating the variance of the graphene-like measurement through Eq. (\ref{['eq:Chern_extraction_main']}) (see Appendix C).
• Figure 5: Schematic representation of the experimental setup. (a) Optical setup, including the cavity and the heterodyne measurement and (b) electronic setup, including electrical controllers and digitizers.