Hyperbolic Lattices in Circuit Quantum Electrodynamics (1802.09549v2)

Abstract: After close to two decades of research and development, superconducting circuits have emerged as a rich platform for both quantum computation and quantum simulation. Lattices of superconducting coplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwave photons, where weak interactions can be introduced either via non-linear resonator materials or strong interactions via qubit-resonator coupling. Here, we highlight the previously-overlooked property that these lattice sites are deformable and allow the realization of tight-binding lattices which are unattainable, even in conventional solid-state systems. In particular, we show that networks of CPW resonators can create a new class of materials which constitute regular lattices in an effective hyperbolic space with constant negative curvature. We present numerical simulations of a series of hyperbolic analogs of the kagome lattice which show unusual densities of states with a spectrally-isolated degenerate flat band. We also present a proof-of-principle experimental realization of one of these lattices. This paper represents the first step towards on-chip quantum simulation of materials science and interacting particles in curved space.

• The paper demonstrates that superconducting CPW resonators can form tight-binding models of hyperbolic lattices, revealing isolated flat bands in non-Euclidean geometries.
• The paper employs numerical simulations and experimental proofs to realize hyperbolic analogs of kagome lattices with distinct spectral properties.
• The paper suggests that these hyperbolic lattice systems can advance quantum simulation techniques and inspire new quantum error correction codes.

Analysis of "Hyperbolic Lattices in Circuit Quantum Electrodynamics"

The paper "Hyperbolic Lattices in Circuit Quantum Electrodynamics" presents an exploration into the hitherto uncharted domain of utilizing superconducting circuits to simulate hyperbolic lattices—a concept that heralds a potential paradigm shift in materials science and quantum simulation. The thrust of the research hinges on the inherent flexibility allowed by circuit QED lattices, enabling the construction of two-dimensional coplanar waveguide (CPW) resonators configured in non-Euclidean geometries, specifically those with negative curvature.

Key Insights from the Study

Superconducting circuits have been extensively researched for their quantum computational capabilities; however, this paper emphasizes the potential of these systems for quantum simulation tasks, specifically for complex geometries that defy three-dimensional Euclidean constraints. The researchers harness the flexibility of CPW resonators in constructing artificial materials that embody hyperbolic geometries—spaces characterized by negative curvature and a distinctive non-Euclidean metric.

The paper underscores how lattice deformations within CPW resonators can lead to an unprecedented realization of tight-binding models that simulate hyperbolic spaces. Through numerical simulations, the authors succeed in demonstrating the emergence of hyperbolic analogs of well-known lattices like the kagome lattice. These lattices exhibit unique spectral properties, notably including a degenerate flat band that is spectrally isolated, a feature that is not inherently present in conventional Euclidean versions.

Numerical and Experimental Results

The core theoretical advancement lies in the numerical demonstrations of a hyperbolic kagome lattice, showing deviations in the density of states when compared to Euclidean counterparts. Experimentally, the researchers validate these findings by devising a proof-of-principle realization of a portion of a hyperbolic lattice via circuit QED, marking an initial but crucial step towards conducting quantum simulations that leverage curved spaces.

Implications and Future Directions

This exploration opens multiple avenues of inquiry, expanding the utility of superconducting resonators beyond traditional quantum computing, to include quantum simulations that require non-Euclidean topologies. The implications are vast, spanning the exploration of quantum systems under general relativistic conditions and providing insight into interacting quantum particles within such geometries.

The degenerate flat bands observed in the kagome analogs present unique opportunities for investigating strongly-correlated electronic states, such as fractional quantum Hall states, under artificial non-Euclidean conditions. The observed isolated flat bands may lead to a deeper understanding of electronic phases of matter that are heavily influenced by lattice geometry.

In future work, the integration of non-linear materials or qubit coupling may yield systems with enhanced interaction strengths—a promising domain for simulating physical processes that occur in curved spaces, which are otherwise non-trivial to realize experimentally. Continued research might also unravel more efficient quantum error correction codes rooted in hyperbolic surface codes, with enhanced connectivity properties crucial for quantum communication networks.

In summary, this paper delineates a novel approach to exploring hyperbolic geometries within a circuit quantum electrodynamics framework. The successful initial experimental results pave the way for more intricate studies on the interaction of quantum particles in simulated curved spaces, offering profound implications for theoretical physics, materials science, and quantum computing.