Two-dimensional topological effect in a transmon qubit array with tunable couplings

We investigate a square-lattice architecture of superconducting transmon qubits with interqubit interactions mediated by inductive couplers. Via periodically modulating the couplers, the Abelian gauge potential, termed effective magnetic flux, can be synthesized artificially, making the system an excellent platform for simulating two-dimensional topological physics. First, we focus on the three-leg ladder which only has three rows and investigate the chiral dynamics therein for the single-particle ground state when the effective magnetic flux varies. We find what we call the "staggered vortex-Meissner phase transition," where the vortex number can typically stagger a few times between one (defined as "vortex phase") and larger integers (defined as "Meissner phase") when the effective magnetic flux changes between -pi and pi. This phenomenon, actually not a phase transition by definition, is quite different from the vortex-Meissner phase transition in the two-leg ladder that, in contrast, possesses only two rows and is usually treated as the quasi-two-dimensional model. Also, we find that the chiral current relies on the effective magnetic flux according to a squeezed sinusoidal function. Both the staggering of the vortex number and squeezing of the chiral current can be controllable by the coupling ratio, which is defined by the coupling-strength ratio between the column direction and row direction. Second, we continue to increase the number of rows beyond three, and the topological band structure to be anticipated at an infinite number of rows begins to occur even for a relatively small number (10 or so for typical parameters) of rows. This heralds a small circuit scale enough to observe the topological band. The behavior of the edge state in the band gap can be interpreted by the topological Chern number, which can be calculated through integrating the Berry curvature with respect to the first Brillouin zone. Last, we present a systematic method on how to measure the topological band structure based on time- and space-domain Fourier transformation of the wave function after properly exciting the qubits, which should be helpful for comprehensively analyzing the topological physics since all the topological properties are mainly contained in the band structure. Our results offer an avenue for simulating two-dimensional topological physics on the state-of-the-art superconducting quantum chips.