A Circuit-QED Lattice System with Flexible Connectivity and Gapped Flat Bands for Photon-Mediated Spin Models

TL;DR

This work demonstrates the first integration of superconducting transmon qubits into a coplanar-waveguide CPW lattice with non-trivial band structure, enabling photon-mediated spin interactions across flexible connectivities. By combining standard circuit-QED readout with a novel mode-mode spectroscopy technique, the authors map the photonic band structure—including gapped flat bands—and observe qubit–band hybridization and bound states in band gaps. They extract key parameters such as mode frequencies $\omega_\mu$ and hopping strengths $t_\mu$, and show tunable qubit–qubit interactions mediated by lattice bands, including two-photon transitions in the full-wave regime. The platform establishes a toolkit for realizing CPW lattices with diverse connectivities, potentially extending to two-dimensional and hyperbolic geometries and enabling driven-dissipative spin models with rich many-body physics.

Abstract

Quantum spin models are ubiquitous in solid-state physics, but classical simulation of them remains extremely challenging. Experimental testbed systems with a variety of spin-spin interactions and measurement channels are therefore needed. One promising potential route to such testbeds is provided by microwave-photon-mediated interactions between superconducting qubits, where native strong light-matter coupling enables significant interactions even for virtual-photon-mediated processes. In this approach, the spin-model connectivity is set by the photonic mode structure, rather than the spatial structure of the qubit. Lattices of coplanar-waveguide (CPW) resonators have been demonstrated to allow extremely flexible connectivities and can therefore host a huge variety of photon-mediated spin models. However, large-scale CPW lattices with non-trivial band structures have never before been successfully combined with superconducting qubits. Here we present the first such device featuring a quasi-1D CPW lattice with multiple transmon qubits. We demonstrate that superconducting-qubit readout and diagnostic techniques can be generalized to this highly multimode environment and observe the effective qubit-qubit interaction mediated by the bands of the resonator lattice. This device completes the toolkit needed to realize CPW lattices with qubits in one or two Euclidean dimensions, or negatively-curved hyperbolic space, and paves the way to driven-dissipative spin models with a large variety of connectivities.

Figures (12)

• Figure 1: Target Lattice and its Band Structures. (a) The target quasi-1D lattice overlaid on top of the resonator chain which generates it. The lattice sites are indicated in gold with dark blue lines indicating nearest-neighbor connections. The CPW resonator that produces a lattice site is indicated by a light blue line in the background. (b)-(c) Tight-binding calculation of the photonic bands for the half-wave and full-wave modes of the lattice, respectively, without frequency-dependent corrections. The energy of an uncoupled resonator mode is set to zero, and the bands are plotted in units of the hopping strength $|t_{\mu}|$. Each set of modes features two sets of flat bands (one doubly degenerate), 3 bands with conventional quadratic band edges, and one set of linear band crossings (a one-dimensional version of a Dirac cone).
• Figure 2: Device Design. (a) Photograph of the device after the photolithography stage. Dark regions are tantalum, and white regions are the underlying $25.4 \times 25.4 \ \mathrm{mm}^2$ sapphire substrate. (b) CAD of the lattice device with color coding to illustrate different aspects of the design. The resonators forming the lattice are shown in black, with the exception of the central unit cell, which is highlighted in red. Input/output ports in the lower left and upper right of the device allow microwave transmission measurements. The locations of the three qubits are highlighted by the tan squares, and the associated on-chip flux bias lines, which allow independent frequency control of the qubits, are shown in orange. A fourth flux bias line (grey) is unused. The qubits are referred to as Q1, Q2, and Q3 from right to left. Transmon qubits, with the design shown in (d), are incorporated into the lattice using pockets opened in the side of the CPW shown in pink. To mitigate disorder, the capacitor paddles of a transmon are included in all 54 resonators, but the Josephson junctions needed to complete the transmons are only included at the three highlighted sites. (c) Effective tight-binding model describing photons in the device overlaid on the resonator network. (d) CAD design of the transmon qubit. The capacitor is digitated (yellow and dark blue regions), and the e-beam lithography pattern for the Josephson junctions is shown in red. The transmon is sandwiched between the center pin of the resonator just to its left (light blue) and its associated flux-bias line, the end of which can be seen in the lower right.
• Figure 3: Half-Wave Characterization. (a) Microwave transmission through the lattice as a function of flux applied to Q3, while the other two qubits are far detuned from the bands. The qubit Q3 is tuned through the half-wave modes to its frequency minimum at half flux ($\phi_0 = 0.5)$, creating a series of avoided crossings. (b) Non-linear mode-mode spectroscopy of the half-wave modes, taken with all three qubits far detuned from the bands. Because this measurement relies on coupling to the qubits, it is insensitive to any weakly coupled parasitic modes in the device packaging. As a result, the band gap and two sets of bands are more clearly visible here than in the transmission data. (c) Expected density of states from the band structure of an infinite lattice and empirically determined values of the hopping parameter ($\vert t_1\vert/ 2\pi = 40$ MHz) and band center ($\omega_1/ 2\pi = 4.889$ GHz). Dispersive bands are shown in light blue, and the flat bands in dark blue. The full height of the flat-band peaks is well above the scale shown here. The two regions in which avoided crossings are visible in (a) correspond to the two frequency regions with non-zero density of states, and the mode-mode spectroscopy shown in (b) is in good agreement with the theoretical density of states.
• Figure 4: Full-Wave Characterization. Combined characterization data for the full-wave modes of the device. (a) Microwave transmission through the lattice as function of flux applied to Q3, while the other two qubits are held at fixed detuning. Near integer flux, the qubit Q3 reaches its maximum frequency and tunes through the full-wave modes, creating a series of avoided crossings. In this frequency region, unwanted package modes are more significant than at the lower-frequency modes, resulting in a rolling background in transmission and causing many lattice modes to appear as Fano resonances. Also shown in (a), using a navy-gold color scale, is non-linear spectroscopy of the qubit transition itself. Two regions of this data are overlaid on top of the transmission, revealing qubit-like mid-gap states in which the qubit is heavily dressed by the the lattice modes. (b) Non-linear mode-mode spectroscopy of the full-wave modes, taken with all three qubits far detuned from the bands. The three sets of bands expected for this set of modes are easily distinguishable. Modes in the upper quadratic band and highly localized modes in the gapped flat bands that are difficult to observe in transmission are clearly visible in this mode-mode spectroscopy. (c) Expected density of states from the band structure of an infinite lattice using empirically determined values of the hopping parameter ($\vert t_2 \vert/ 2\pi = 82$ MHz) and band center ($\omega_2/ 2\pi = 9.726$ GHz). Dispersive bands are shown in light blue, and the flat bands in dark blue. The full height of the peaks due to the flat bands is well above the scale shown here.
• Figure 5: Photon-mediated avoided crossings. (a)-(f) Avoided crossings between Q1 and Q2 taken at six different frequencies. The flux applied to Q2 is held constant while Q1 is swept through resonance. The monitor mode of each plot is selected so that both qubits can be read out with the same mode. The two qubits couple differently to the selected monitor mode and can produce signals in different quadratures of the measurement signal. In order to combine both quadratures into one signal, we find the vector-valued change in the monitor signal and plot the magnitude of this change, known as the IQ distance. For each plot, the y-axis denotes the offset of Q1 from exact resonance with Q2 in units of milli-flux-quanta ($10^{-3}\Phi_0$). (a)-(c) show avoided crossings near the half-wave modes and (d)-(f) show avoided crossings near the full-wave modes. The strength of the interaction between the two qubit-photon bound states depends on their proximity to the closest band edge, with stronger interactions occurring as Q1 and Q2 are brought closer to a set of bands. Also visible in (d)-(f), but most prominently in (e), is the two-photon transition from $\vert 00 \rangle$ to $\vert 11 \rangle$, which lies between the two main branches of the avoided crossing.