After two decades of development, cavity quantum electrodynamics with superconducting circuits has emerged as a rich platform for quantum computation and simulation. Lattices of coplanar waveguide ...resonators constitute artificial materials for microwave photons, in which interactions between photons can be incorporateded either through the use of nonlinear resonator materials or through coupling between qubits and resonators. Here we make use of the previously overlooked property that these lattice sites are deformable and permit tight-binding lattices that are unattainable even in solid-state systems. We show that networks of coplanar waveguide resonators can create a class of materials that constitute lattices in an effective hyperbolic space with constant negative curvature. We present numerical simulations of hyperbolic analogues of the kagome lattice that show unusual densities of states in which a macroscopic number of degenerate eigenstates comprise a spectrally isolated flat band. We present a proof-of-principle experimental realization of one such lattice. This paper represents a step towards on-chip quantum simulation of materials science and interacting particles in curved space.
Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, ...or non-interacting, band structure calculations, and in the limit of strong on-site confinement this can be reduced to graph-like tight-binding models. In this context, both mathematicians and physicists have developed largely independent methods for solving these models. In this paper we will combine and present results from both fields. In particular, we will discuss a class of lattices which can be realized as line graphs of other lattices, both in Euclidean and hyperbolic space. These lattices display highly unusual features including flat bands and localized eigenstates of compact support. We will use the methods of both fields to show how these properties arise and systems for classifying the phenomenology of these lattices, as well as criteria for maximizing the gaps. Furthermore, we will present a particular hardware implementation using superconducting coplanar waveguide resonators that can realize a wide variety of these lattices in both non-interacting and interacting form.
Photonic synthetic materials provide an opportunity to explore the role of microscopic quantum phenomena in determining macroscopic material properties. There are, however, fundamental obstacles to ...overcome — in vacuum, photons not only lack mass, but also do not naturally interact with one another. Here, we review how the superconducting quantum circuit platform has been harnessed in the last decade to make some of the first materials from light. We describe the structures that are used to imbue individual microwave photons with matter-like properties such as mass, the nonlinear elements that mediate interactions between these photons, and quantum dynamic/thermodynamic approaches that can be used to assemble and stabilize strongly correlated states of many photons. We then describe state-of-the-art techniques to generate synthetic magnetic fields, engineer topological and non-topological flat bands and explore the physics of quantum materials in non-Euclidean geometries — directions that we view as some of the most exciting for this burgeoning field. Finally, we discuss upcoming prospects, and in particular opportunities to probe novel aspects of quantum thermalization and detect quasi-particles with exotic anyonic statistics, as well as potential applications in quantum information science.This Review Article surveys the physics of many-body quantum states formed by microwave photons in circuit quantum electrodynamics environments.
Optical cavity QED provides a platform with which to explore quantum many-body physics in driven-dissipative systems. Single-mode cavities provide strong, infinite-range photon-mediated interactions ...among intracavity atoms. However, these global all-to-all couplings are limiting from the perspective of exploring quantum many-body physics beyond the mean-field approximation. The present work demonstrates that local couplings can be created using multimode cavity QED. This is established through measurements of the threshold of a superradiant, self-organization phase transition versus atomic position. Specifically, we experimentally show that the interference of near-degenerate cavity modes leads to both a strong and tunable-range interaction between Bose-Einstein condensates (BECs) trapped within the cavity. We exploit the symmetry of a confocal cavity to measure the interaction between real BECs and their virtual images without unwanted contributions arising from the merger of real BECs. Atom-atom coupling may be tuned from short range to long range. This capability paves the way toward future explorations of exotic, strongly correlated systems such as quantum liquid crystals and driven-dissipative spin glasses.
Phase transitions, where observable properties of a many-body system change discontinuously, can occur in both open and closed systems. By placing cold atoms in optical cavities and inducing strong ...coupling between light and excitations of the atoms, one can experimentally study phase transitions of open quantum systems. Here we observe and study a non-equilibrium phase transition, the condensation of supermode-density-wave polaritons. These polaritons are formed from a superposition of cavity photon eigenmodes (a supermode), coupled to atomic density waves of a quantum gas. As the cavity supports multiple photon spatial modes and because the light-matter coupling can be comparable to the energy splitting of these modes, the composition of the supermode polariton is changed by the light-matter coupling on condensation. By demonstrating the ability to observe and understand density-wave-polariton condensation in the few-mode-degenerate cavity regime, our results show the potential to study similar questions in fully multimode cavities.
Circuit quantum electrodynamics is one of the most promising platforms for efficient quantum simulation and computation. In recent groundbreaking experiments, the immense flexibility of ...superconducting microwave resonators was utilized to realize hyperbolic lattices that emulate quantum physics in negatively curved space. Here we investigate experimentally feasible settings in which a few superconducting qubits are coupled to a bath of photons evolving on the hyperbolic lattice. We compare our numerical results for finite lattices with analytical results for continuous hyperbolic space on the Poincaré disk. We find good agreement between the two descriptions in the long-wavelength regime. We show that photon-qubit bound states have a curvature-limited size. We propose to use a qubit as a local probe of the hyperbolic bath, for example, by measuring the relaxation dynamics of the qubit. We find that, although the boundary effects strongly impact the photonic density of states, the spectral density is well described by the continuum theory. We show that interactions between qubits are mediated by photons propagating along geodesics. We demonstrate that the photonic bath can give rise to geometrically frustrated hyperbolic quantum spin models with finite-range or exponentially decaying interaction.
In 1826 Abel started the study of the polynomial Pell equation
x
2
−
g
(
u
)
y
2
= 1. Its solvability in polynomials
x
(
u
),
y
(
u
) depends on a certain torsion point on the Jacobian of the ...hyperelliptic curve
v
2
=
g
(
u
). In this paper we study the affine surfaces defined by the Pell equations in 3-space with coordinates
x, y, u
, and aim to describe all affine lines on it. These are polynomial solutions of the equation
x
(
t
)
2
−
g
(
u
(
t
))
y
(
t
)
2
= 1. Our results are rather complete when the degree of g is even but the odd degree cases are left completely open. For even degrees we also describe all curves on these Pell surfaces that have only 1 place at infinity.
Quantum simulators are a promising technology on the spectrum of quantum devices from specialized quantum experiments to universal quantum computers. These quantum devices utilize entanglement and ...many-particle behavior to explore and solve hard scientific, engineering, and computational problems. Rapid development over the last two decades has produced more than 300 quantum simulators in operation worldwide using a wide variety of experimental platforms. Recent advances in several physical architectures promise a golden age of quantum simulators ranging from highly optimized special purpose simulators to flexible programmable devices. These developments have enabled a convergence of ideas drawn from fundamental physics, computer science, and device engineering. They have strong potential to address problems of societal importance, ranging from understanding vital chemical processes, to enabling the design of new materials with enhanced performance, to solving complex computational problems. It is the position of the community, as represented by participants of the National Science Foundation workshop on “Programmable Quantum Simulators,” that investment in a national quantum simulator program is a high priority in order to accelerate the progress in this field and to result in the first practical applications of quantum machines. Such a program should address two areas of emphasis: (1) support for creating quantum simulator prototypes usable by the broader scientific community, complementary to the present universal quantum computer effort in industry; and (2) support for fundamental research carried out by a blend of multi-investigator, multidisciplinary collaborations with resources for quantum simulator software, hardware, and education.This document is a summary from a U.S. National Science Foundation supported workshop held on 16–17 September 2019 in Alexandria, VA. Attendees were charged to identify the scientific and community needs, opportunities, and significant challenges for quantum simulators over the next 2–5 years.
We show how quantum many-body systems on hyperbolic lattices with nearest-neighbor hopping and local interactions can be mapped onto quantum field theories in continuous negatively curved space. The ...underlying lattices have recently been realized experimentally with superconducting resonators and therefore allow for a table-top quantum simulation of quantum physics in curved background. Our mapping provides a computational tool to determine observables of the discrete system even for large lattices, where exact diagonalization fails. As an application and proof of principle we quantitatively reproduce the ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincaré disk, and show how conformal symmetry emerges for large lattices. This sets the stage for studying interactions and disorder on hyperbolic graphs in the future. Importantly, our analysis reveals that even relatively small discrete hyperbolic lattices emulate the continuous geometry of negatively curved space, and thus can be used to experimentally resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity.
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