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

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.

Figure 1

(a) Single-excitation single-qubit bound state. Color indicates the amplitude of the wave function. Gray lines denote the connectivity of photonics sites. One site is coupled via $g$ to a qubit (gray circle). Inset: the photonic spectrum as a function of the eigenstate index $j$. All results are for $g=0.05$ and $\mathrm{\Delta }=-3.2$ below the band edge at $E_0\approx -2.9$. (b) Comparison of photonic amplitudes of the lowest eigenstate as a function of the distance $d(z_1,z_j)$ from the qubit at $z_1$—exact ${\psi }_j$ (diamonds), perturbative expressions from Eq. (4) based on discrete (squares) and continuous (dots) Green function. (c) and (d) The density of the photonic amplitude of a two-spin single-excitation bound state illustrating the photon-mediated interactions between spins on the graph (c) and in the continuum (d). We see that photons follow the shortest path between two spins [positioned at two black dots in (c)]—the geodesic shown as a (dashed) semicircle on the Poincaré disk.

Figure 2

Comparison of spectral properties from analytic limits to numerics of different-sized graphs in the long-wavelength limit near the LEBE. Graphs are characterized by the number $\ell$ of concentric rings of heptagons, where $\ell =1$ corresponds to a single heptagon. (a) The cumulative DOS $P(\omega )$ has data points nearly overlapping for different $\ell$. (b) The cumulative $J(\omega )$ is less smooth, but asymptotically ($\ell \to \infty$) approaches $J_{L=1}(\omega )$: The root-mean-square error with respect to $J_{L=1}$ for $\omega <-2$ is 0.0066, 0.0052, and 0.0044 for $\ell =5$, 6, and 7, respectively.

Figure 3

(a) Comparison of analytical expressions and numerical results (with the bin size $\mathrm{\Delta }\omega =0.15$) for $j$ and $\rho$. (b) Comparison of the numerical (with $\mathrm{\Delta }\omega =0.3$) $j$ and $\rho$ with the fitted $\mathrm{\Gamma }$ within time $\in [0,15]$ (error bars correspond to the standard deviation) for $g=0.3$, $\ell =7$.

Figure 4

Energy spectrum (a) as a function of $g$ for $\mathrm{\Delta }=-2.5$ and next-nearest neighbors separated by $z=0.5$, and (b) as a function of $\mathrm{\Delta }$ for $g=0.5$ and nearest neighbors separated by $z=h=0.276$. All lattice results are for $\ell =6$, which we compare with symmetric (lower black dashed curve) and antisymmetric (upper red dashed) 2-qubit bound states.

Figure 5

Spin model in the vicinity of the flatband for the heptagonal kagomelike line graph with $t=-1$ [14]. (a) The photonic spectrum as a function of the eigenstate index $j$. (b) We plot the interaction strength as a function of qubit position, assuming the position of the second qubit to be where the largest dot is. The dot color denotes the sign of the interactions [green (yellow) is $+(-)$], whereas its diameter is directly proportional to the interaction strength.