Hyperbolic lattices in circuit quantum electrodynamics

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 previouslyoverlooked 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. arXiv:1802.09549v2 [quant-ph]

Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

“…Adding non-linearity and effective photon-photon interactions to such isolated flat bands is an ideal starting point for quantum simulation of strongly correlated many body physics with photons and will be the focus of future experimental and theoretical work. [6,8] For regular layouts, σ(H α (X)) can be completely understood by examining σ(A X ). Because of the minus sign, a spectral gap at −2 forH a (X) arises only due to an expander gap in the spectrum of A X .…”

“…We now consider the implementation of lattices using distributed elements, most commonly microwave CPW resonators in superconducting circuits. [4][5][6]13] Coplanar waveguides are effectively a two-dimensional analog of cylindrical coaxial cables fabricated from metallic films, whose modes are described by the transmission line Lagrangian. [13][14][15] The classical literature [14] gives the mode functions in terms of the voltage along the transmission line, whereas the quantum circuits literature [13,15] typically uses the closely related generalized flux Φ(x).…”

“…Networks of coupled CPW resonators can be regarded as artificial materials in the tight-binding approximation where the individual resonators replace the on-site potential V 0 ( x), the single resonator eigenmodes take the place of the bound state wavefunction, and microwave photons replace carrier electrons. [4,6,13] The appropriate description of kinetic-energy-like terms due to movement of photons between resonators can be derived through careful analysis of the transmission line Lagrangian, and gives rise to a new term in the total Hamiltonian H = resonators ω 0 a † n a n + resonator ends…”

“…The purpose of this work is to collect and combine complementary results from both solid-state physics and mathematics and thereby gain new insight into the properties of H. In particular, we will apply these insights to the circuit QED lattices introduced in Refs. [4][5][6]. To this end, the remainder of paper is organized as follows.…”

Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

“…ion traps [39,42], modular architectures of superconducting cavity networks [1,18,19,37], Rydberg atoms [20,43,49,50] and silicon photonics [30]), and a necessary requirement for any QECC with constant space overhead. In particular, a recent experimental realization of hyperbolic circuit QED lattices [35] is achieved by utilizing the feature that the cavity quantum bus (used to connect superconducting qubits) can have a wide range of length scales.…”