Creating lattice gauge potentials in circuit QED: The bosonic Creutz ladder

In this work we propose two protocols to make an effective gauge potential for microwave photons in circuit QED. The first scheme is based on coupled transmons whose on-site energies are harmonically modulated in time. We investigate the effect of various types of capacitive and inductive couplings, and the role of the phase difference between adjacent sites on creating a complex hopping rate between coupled qubits. The second method relies on the parametrically coupling the modes of a SQUID in a resonator and controlling the hopping phase via a coherent pump. Both proposals can be readily realized in a superconducting circuit with the existing technology and are suitable for scalable lattices. As an example benefiting from these complex-valued hopping terms, we simulated the behavior of a plaquette of bosonic Creutz ladder as one of the important models with interdisciplinary interest in various branches of physics. Our results clearly show the emergence of chiral edge modes and directional transport between lattice sites. Combined with intrinsic nonlinearity of the transmon qubits such lattices would be an ideal platform for simulating many different Hamiltonians such as the Bose–Hubbard model with nontrivial gauge fields. Important direct applications of the presented results span a broad range from signal processing in nonreciprocal transport to quantum simulation of gauge-invariant models in fundamental physics.

Figure 1

(a) A circuit of two transmons coupled with both inductive and capacitive elements. (b) Schematics of a plaquette from a bosonic Creutz ladder showing all the nodes and the coupling terms.

Figure 2

Time evolution of a two-site problem with one excitation driven by a modulated onsite frequency with frequency $\omega$ and amplitude of the modulation $\mathrm{\Omega }$. $\omega =\mathrm{\Omega }$ in all cases.

Figure 3

Temporal dependence of the local occupations of the modes involved in a plaquette configuration. This behavior has the signature of chirality. Starting with an initial state $|a_1\rangle$, the population of this state decreases while increases $|a_2\rangle +i|a_4\rangle$ population and finally the whole population appears in $|a_3\rangle$.

Figure 4

Time evolution of two bosonic excitations in a Creutz plaquette. The two limiting cases (no interaction $U=0$, and hard-core boson $U\to \infty$) can be characterized by a shift in the frequency of the time-dependent probability of finding the two excitations in the same place at a later time $t$.