Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models

Abstract

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.

Automorphic Bloch theorems for hyperbolic lattices

“…The above concrete realizations of hyperbolic lattices in the laboratory open up vistas for the exploration of quantum mechanics in (negatively) curved space, with possibly far-reaching implications for fundamental physics in the areas of string theory ( 10 – 12 ), quantum gravity ( 13 – 15 ), and quantum information ( 16 – 21 ). In the long-wavelength limit, the Hamiltonian of a quantum particle on a hyperbolic lattice reduces to the well-known Laplace–Beltrami operator on the Poincaré disk ( 22 , 23 ), whose spectrum is well understood. However, when the de Broglie wavelength approaches the lattice spacing, the geometry of the tessellation strongly affects both the spectrum and wave functions ( 24 – 28 ).…”

Selberg trace formula and hyperbolic band theory

“…With these definitions at hand, it now becomes clear that the second term in the Euclidean and hyperbolic trace formulas, Eqs. ( 13) and (22), are structurally very close.…”

“…Experimental realizations of hyperbolic lattices in both circuit quantum electrodynamics [1] and topoelectric circuits [2] recently resurged interest in the mathematical properties of hyperbolic space and physical systems in it [3][4][5][6]. A current and experimentally motivated focus of attention is on properties such as band structures [7][8][9][10][11][12][13][14] and interacting quantum systems [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Historically, however, hyperbolic space served as a crucial platform to investigate theories of both classical and quantum chaos, because key chaotic properties such as ergodicity can be proven mathematically for geodesic flow on hyperbolic surfaces [32][33][34][35][36][37].…”

Hyperbolic Lattice for Scalar Field Theory in AdS$_3$

“…One approach is to simulate this on a digital quantum computer with the standard Trotter expansion. An intriguing alternative is specific hardware being introduced [43][44][45][46][47] that purports to realize the discrete H 2 lattice. Finally, time evolution for gravity is an interesting and challenging problem.…”