Quarter-flux Hofstadter lattice in a qubit-compatible microwave cavity array

Topological and strongly correlated materials are exciting frontiers in condensed-matter physics, married prominently in studies of the fractional quantum Hall effect [H. L. Stormer et al., Rev. Mod. Phys. 71, S298 (1999)], There is an active effort to develop synthetic materials where the microscopic dynamics and ordering arising from the interplay of topology and interaction may be directly explored. In this work, we demonstrate an architecture for exploration of topological matter constructed from tunnel-coupled, time-reversal-broken microwave cavities that are both low loss and compatible with Josephson-junction-mediated interactions [A. Wallraff et al., Nature (London) 431, 162 (2004)]. Following our proposed protocol [B. M. Anderson et al., Phys. Rev. X 6, 041043 (2016)], we implement a square lattice Hofstadter model at a quarter flux per plaquette ($\alpha =1/4$), with time-reversal symmetry broken through the chiral Wannier orbital of resonators coupled to yttrium-iron-garnet spheres. We demonstrate site-resolved spectroscopy of the lattice, time-resolved dynamics of its edge channels, and a direct measurement of the dispersion of the edge channels. Finally, we demonstrate the flexibility of the approach by erecting a tunnel barrier and investigating dynamics across it. With the introduction of Josephson junctions to mediate interactions between photons, this platform is poised to explore strongly correlated topological quantum science in a synthetic system.

Figure 1

Schematic and photograph of microwave Chern insulator lattice. (a) Schematic of the cavity layout. The white circles denote cavities that do not shift the phase of a photon passing through them, while the cavities with the arrows shift the phase depending on the physical angle between the couplers. This layout guarantees phase of $\frac{\pi }{2}$ per plaquette. (b) Photograph of a $5\times 5$ section of the lattice measured in the data presented in the paper. The lattice sites are each tuned to 9.560 GHz $\pm 1$ MHz and evanescently tunnel coupled with hopping rate 30 MHz $\pm 1$ MHz. The typical resonator quality factor for fundamental cavities is $Q=3000$ and for the YIG cavities is $Q=1500$. The lattice spacing is 1.96 cm, resulting in a total edge-to-edge (including outer walls) lattice dimension of 24.0 cm. (c) Side profile of a $5\times 5$ lattice cut so that both types of cavities are visible. The second and fourth cavities are the phase-shifting YIG cavities, while the first, third, and fifth cavities are the fundamental cavities. The couplers are visible in between the cavities as gaps in the cavity walls. They are milled from the opposite side of the lattice as the cavities.

Figure 2

$S_{12}-S_{21}$ transmission between two antennas ${45}^{\circ }$ apart in a single YIG cavity as a function of magnetic field. There is stronger coupling between the cavity mode with the same chirality as the YIG sphere. The largest frequency splitting is 446 MHz. The color indicates the phase shift a photon acquires in transmission. After subtracting the two different directions of transmission to eliminate phase noise from cables and impedance mismatches, one chiral mode has a phase shift of ${90}^{\circ }$, while the other has a phase shift of $-{90}^{\circ }$, indicating the modes are of opposite chirality.

Figure 3

(a) Measured transmission spectrum between two bulk (purple) and two edge (orange) sites. The purple trace is the transmission between adjacent cavities in the bulk of the lattice [(5,6) and (6,6) defined from the upper-left lattice corner], while the orange trace is the transmission between cavities on the edge of the lattice [sites (1,1) and (1,11)]. The differential response between upper and lower gaps arises from the distance difference for clockwise and anticlockwise edge propagation between probe and measurement sites. (b) Projected band structure of both the bulk (blue/white density plot, with spatial Fourier limited resolution) and edge (red points) of the system, compared with theory for a $\alpha =1/4$ Hofstadter strip (purple/orange/gray-dashed curves). The bulk data result from a site-by-site measurement of the system response. The dispersion of the edge channel (explained in Appendix pp3) is extracted from the measured cavity-to-cavity phase shift. Within the bulk bands, this phase is sensitive to disorder and overall geometry, resulting in a near-random signal which we omit from the plot. The bulk-bulk transmission exhibits four distinct bands which arise from the four-site magnetic unit cell. Gaps are apparent between the first and second bands, as well as the third and fourth bands. The magnetic unit cell has two sites in each direction, compressing the Brillouin zone to $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$.

Figure 4

(a) The lattice's response to an edge excitation (red arrow), at a frequency of 9.622 GHz which is within the band gap. The presence of an edge channel at this frequency results in a delocalized chiral response along the system edge, decaying due to the finite resonator $Q$'s. (b) The response of the lattice when a bulk site (red arrow) is excited continuously at a frequency of 9.569 GHz within the upper band gap. The absence of bulk modes at the excitation frequency results in exponential localization of the response to the edge.

Figure 5

Spatiotemporal response of the edge cavities to a 50 ns pulse centered on 9.6 GHz. The $x$ axis is the indexed cavity number of each of the 40 edge sites and the $y$ axis is time. The brightness reflects the relative normalized transmission between the excited cavity and the measured cavity. The grid on the right illustrates the mapping used by the $x$ axis of the plot on the left. The solid cavities are the indexed sites, while the white bulk cavities are not indexed in the plot. The chirality of the edge channel is reflected in the unidirectional travel of the pulse. The weak stationary response results from a small Fourier-broadened excitation of bulk modes.

Figure 6

A wall is built into the lattice by detuning all but one of the cavities in the sixth row, separating the lattice into two $11\times 5$ lattices connected by one cavity. This effectively makes a beam splitter for running wave edge modes, as shown schematically at the right. When the lattice is pulsed at 9.6 Ghz (in the top band gap), the response is shown in the figure on the left. The pulse splits when it reaches the gap in the wall (end of the red), transmitting most of the pulse to the green part of the lattice and some of the pulse into the blue, unexcited $11\times 5$ sublattice. The grid on the right shows how the edge cavities are indexed in the plotted data, where the color indicates how the edge sites are indexed. Sites are counted first in the red region, then around the lattice in the blue region, and then from the blue region into the green region.

Figure 7

(a) Fundamental mode cavity electric field. The mode is localized at the bottom of the cavity. (b) Electric field simulation of one of the chiral YIG cavity modes. A video that shows the rotation as a function of time is available in the Supplemental Material [33]. (c) Magnetic field of one of the chiral YIG cavity modes viewed from the top. The field at the center of the cavity rotates in time, coupling to the magnetic moment of the YIG. A video is available in the Supplemental Material [33].

Figure 8

The lattice is shown on the right. A lid for the lattice is made so that one of the gold-colored antennas is connected to every lattice site. These antennas are connected through an rf switch to the network analyzer so that arbitrary transmission between pairs of sites can be measured.

Figure 9

We measure the difference in phase between an excited edge cavity and other edge cavities as a function of distance. The edge cavity number starts at 12 and ends at 18 so that only one edge is fit to a line, to avoid additional dispersion introduced by the corner. We choose the side furthest from the the excited cavity to minimize the effects of direct coupling. For the frequency shown here ($\omega =2\pi \times 9.61$ GHz), the lattice momentum is half the slope, or $-54deg$ per cavity.