Time- and Site-Resolved Dynamics in a Topological Circuit

From studies of exotic quantum many-body phenomena to applications in spintronics and quantum information processing, topological materials are poised to revolutionize the condensed-matter frontier and the landscape of modern materials science. Accordingly, there is a broad effort to realize topologically nontrivial electronic and photonic materials for fundamental science as well as practical applications. In this work, we demonstrate the first simultaneous site- and time-resolved measurements of a time-reversal-invariant topological band structure, which we realize in a radio-frequency photonic circuit. We control band-structure topology via local permutation of a traveling-wave capacitor-inductor network, increasing robustness by going beyond the tight-binding limit. We observe a gapped density of states consistent with a modified Hofstadter spectrum at a flux per plaquette of $\varphi =\pi /2$. In situ probes of the band gaps reveal spatially localized bulk states and delocalized edge states. Time-resolved measurements reveal dynamical separation of localized edge excitations into spin-polarized currents. The radio-frequency circuit paradigm is naturally compatible with nonlocal coupling schemes, allowing us to implement a Möbius strip topology inaccessible in conventional systems. This room-temperature experiment illuminates the origins of topology in band structure, and when combined with circuit quantum electrodynamics techniques, it provides a direct path to topologically ordered quantum matter.

Popular Summary

The interplay of quantum mechanics and topology is poised to revolutionize modern technology. By forcing particles to travel in chiral trajectories, it may be possible to produce materials capable of storing and transmitting quantum information that are not susceptible to losses. Such particles naturally bypass the disorder that produces resistance in conventional systems and, in the presence of strong interactions, these particles can be imbued with fractional statistics that have the potential to revolutionize modern computing. Understanding and controlling the properties of topological materials is thus a central goal of modern condensed-matter physics. Quantum circuitry provides a fascinating route to “blown-up” metamaterials that can be probed and manipulated in exquisite detail. We demonstrate the first such circuit with topological band structure, opening up a new frontier of materials with exotic geometries and topologies.

We conduct room-temperature experiments using a lattice that functions as a gauge field for radio-frequency photons. Since magnetic fields do not act on photons, we instead employ spin-orbit coupling such that opposing spin states experience opposite magnetic fields. We employ a metamaterial circuit consisting of $12\times 12$ sites, each composed of inductors and coupled by capacitors. We demonstrate time- and site-resolved measurements of topological band structure in our radio-frequency photonic circuit, observing spatially localized bulk states, and propagating edge states, within the bulk energy gaps.

The extreme scale of the system enables us to measure direct signatures of topological protection in both space and time and points the way to studies of many-body topological physics in superconducting quantum circuits.

Figure 1

(a) Circuit topological insulator schematic. The periodic structure is formed by on-site inductors and coupling capacitors (black) that are connected via a latticework of wires (light and dark blue lines). At each lattice site, the two inductors “A” and “B” correspond to right and left circularly polarized spins. When a photon traverses a single plaquette (indicated by orange), it accumulates a Berry phase of $\pi /2$. The phase is induced by braiding [indicated by the green boxes and specified in (b)] of the capacitive couplings. (b) Structure of the coupling elements between lattice sites. Each row shows one of the four rotation angles implemented by the capacitive coupling in the circuit. The rotation angle (left column) is induced by connecting inductors as shown (middle column). The corresponding rotation matrices (right column) indicate the inductors being coupled, as well as the signs of the couplings. (c) Photograph of circuit topological insulator. The inductors (black cylinders) are coupled via the capacitors (blue); circuit topology is determined by the trace layout on the printed circuit board (yellow). Inset: Zoom-in view of a single plaquette consisting of four adjacent lattice sites.

Figure 2

Site-resolved measurement of band structure and density of states (DOS). (a) The experimentally observed coupling (linear scale) between two points in the bulk of the circuit TI (blue line) with the bulk states indicated (orange bands). (b) Band structure of a circuit TI. A strip of circuit TI with fixed boundary conditions in the transverse direction is numerically diagonalized at finite longitudinal quasimomentum $q$, yielding four massive bulk bands (gray), and spin-orbit-locked edge states ($\text{red}=\uparrow$, $\text{blue}=\downarrow$) that reside in the bulk gap. The labels “top” and “bottom” denote the boundary that each edge mode propagates along (with opposing group velocities $d\omega /dk$). The spin-Chern numbers ($C^{\uparrow }$) of the spin-up bands are indicated next to each band ($C^{\downarrow }=-C^{\uparrow }$). The additional edge modes (indicated in purple) are not topologically protected. The higher-energy protected edge channel is localized to a single site along one direction, while the lower energy protected edge-channel is localized to two sites, respectively. (c) Experimentally observed coupling between two sites on the edge of the circuit, showing transmission through edge modes even within the bulk gaps. The structure within the gaps is to be expected, as the system is finite so the edge exhibits Fabry-Pérot resonances. (d) Experimental site-resolved response to excitation at a central lattice site, within the band gap at 160 kHz, showing bulk localization. (e) Experimentally observed, site-resolved edge-mode structure at 160 kHz. The arrows in (a) and (c) reflect the frequency of excitation in (b) and (d).

Figure 3

Time-resolved transport dynamics of the edge modes. (a) Spin-resolved detection of the splitting of a localized $\mathbf{A}\text{−}\text{site}=(\uparrow +\downarrow )/\sqrt{2}$ excitation in the protected gap (at 145 kHz). The $\uparrow$ (red) and $\downarrow$ (blue) signals are overlaid, demonstrating spin-momentum locking. Two round trips are visible; sites 0 and 42 are equivalent. (b) Spin-resolved excitation after exciting the same site in the unprotected upper edge mode at 225 kHz, where disorder immediately leads to backscattering for both spin states. Inset: Edge lattice-site numbering convention.

Figure 4

Topological insulator with Möbius global topology. (a) Schematic Möbius topological insulator. Top diagram: Möbius TI, with the arrow indicating the edge propagation direction, and color the spin state. Bottom diagram: Connectivity of the Mobius TI. PCB is shown in orange, with external connections generating the topology indicating spin on traversing the edge. This circuit is excited in the edge channel at 145 kHz, within the largest bulk gap, and the subsequent evolution of the wave packet around the full perimeter of the system is observed through time. (b) Spin-resolved detection of edge transport after excitation of $\uparrow$; $\uparrow$ and $\downarrow$ intensities are plotted in the red and blue channels, respectively, showing the conversion from $\uparrow$ to $\downarrow$ when the excitation moves from one edge to the other. Three round trips are visible.