Quantum Simulation of the Bosonic Creutz Ladder with a Parametric Cavity

There has been a growing interest in realizing quantum simulators for physical systems where perturbative methods are ineffective. The scalability and flexibility of circuit quantum electrodynamics make it a promising platform for implementing various types of simulators, including lattice models of strongly coupled field theories. Here, we use a multimode superconducting parametric cavity as a hardware-efficient analog quantum simulator, realizing a lattice in synthetic dimensions with complex hopping interactions. The coupling graph, i.e., the realized model, can be programmed in situ. The complex-valued hopping interaction further allows us to simulate, for instance, gauge potentials and topological models. As a demonstration, we simulate a plaquette of the bosonic Creutz ladder. We characterize the lattice with scattering measurements, reconstructing the experimental Hamiltonian and observing important precursors of topological features including nonreciprocal transport and Aharonov-Bohm caging. This platform can be easily extended to larger lattices and different models involving other interactions.

Figure 1

(a) Schematic representation of the Creutz ladder. The arrows indicate the sign of the hopping phase. (b) Device cartoon. We realize interactions between cavity modes by parametrically pumping the SQUID through a flux line. The system is then probed through the input capacitor by a coherent tone. (c) Synthetic lattice. We program a four-node lattice in synthetic dimensions using four pump tones, which have a well-controlled phase. We measure the scattering matrix of the system by probing near each node frequency and measuring the output at various nodes, which are separated in frequency space.

Figure 2

The scattering matrix. The magnitude of the experimental (left) and theoretical (right) scattering matrices as a function of lattice phase, $\varphi$, and frequency. The frequency axes give the detunings, ${\mathrm{\Delta }}_n$, from the uncoupled mode frequencies. The diagonal reflection coefficients, $\{S_{nn}\}$, provide the spectrum of the lattice eigenmodes. The off-diagonal elements, $\{S_{mn}\}$, are the magnitude of frequency-converting transport between the nodes. The $\{S_{mn}\}$ allows us to characterize the “spatial” support of each eigenmode over the lattice in the synthetic dimensions (see text). We see, clearly, that the transport is nonreciprocal with $\{S_{mn}\}$ and $\{S_{nm}\}$ often being complements of each other.

Figure 3

Topological precursors. (a) $\varphi =0$ line cuts of the measured (red) and theory (blue) scattering parameters in Fig. 2 when probing at node $a$. The vertical axes are normalized to the background. The measurements indicate the existence of an eigenmode at zero energy with significant support only in nodes $a$ and $b$, which are not directly connected. We infer this from the relatively high transmission amplitude from $a$ to $b$. We also observe a second zero mode localized on sites $c$ and $d$. (b) Twisted plaquette. We expect topological features of the Creutz ladder to appear at $\varphi =\pi$ and not $\varphi =0$. However, we note that, if we twist the plaquette as indicated, $\varphi =0$ regardless of the external flux. After twisting, the zero modes now appear at the two ends of the plaquette. These states are reminiscent of the predicted zero-mode end states [20]. (c) $\varphi =\pi$ line cuts when probing from node $a$. The scales of the vertical axes are the same as panel (a). The measurements indicate the existence of two pairs of degenerate eigenmodes, one at positive and one at negative detuning, that have support on all but one of the nodes. One of these four eigenmodes exists on each corner. (d) Caging. We can associate the corner eigenmodes with the soliton states in the Creutz ladder by identifying the lattice as the indicated trapezoidal path. Because of Aharonov-Bohm caging, an excitation at, e.g., node $a$ cannot propagate to node $b$.