Tunable coupling engineering between superconducting resonators: From sidebands to effective gauge fields

In this work we show that a tunable coupling between microwave resonators can be engineered by means of simple Josephson junctions circuits, such as dc and rf superconducting quantum interference devices. We show that by controlling the time dependence of the coupling it is possible to switch on and off and modulate the cross-talk and boost the interaction towards the ultrastrong regime, as well as to engineer red and blue sideband couplings, nonlinear photon hopping, and classical gauge fields. We discuss how these dynamically tunable superconducting circuits enable key applications in the fields of all-optical quantum computing, continuous-variable quantum information, and quantum simulation—all within the reach of the state of the art in circuit-QED experiments.

Figure 1

(a) Sketch of the geometrical arrangement of the two coupled superconducting stripline resonators of length $2l$. A finite interaction is present only in the coupling region of length $2l_c$ and is negligible elsewhere. (b) Schematics for the lumped circuit equivalent. We explicitly draw the mutual capacitances (dashed blue lines) and the mutual inductive coupling (red arrows). The node flux ${\varphi }_n$ is also indicated.

Figure 2

Architectures leading to a tunable microwave beam splitter. (a) A dc SQUID (superconducting loop intersected by two Josephson junctions) mediates the coupling between two stripline resonators (ground planes not shown). The established pointlike contact between the resonators takes the coupling to the ultrastrong domain. (b) A superconducting ring intersected by a Josephson junction now partially shares its branches with the cavities, improving the switching capability.

Figure 3

Coupling strength for differerent niobium transmission lines, as a function of the loop size. While homogeneous resonators (blue) hardly reach the strong-coupling regime, inhomogeneous ones (red dashed) do. The coupling strength can be further increased with a Josephson junction interrupting the center conductor, as shown for the homogeneous case (dotted) and the inhomogeneous one (dot dashed). We have considered for each resonator a frequency ${\omega }_0/2\pi =6.65\text{GHz}$.

Figure 4

(a) One-dimensional array of cavities coupled by means of a superconducting ring coupler. (b) Two-dimensional lattice of circular resonators, coupled by dc SQUIDs. Both lattices are bipartite. Using different resonator frequencies for each sublattice, ${\omega }_A\ne {\omega }_B$, we can use the techniques from Sec. 4a to engineer any sideband interaction between the arrays.

Figure 5

Energies of the full Hamiltonian of two $\mathrm{LC}$ resonators with frequency ${\omega }_1$ coupled by a SQUID (11), as a function of the effective Josephson energy $E_{\mathrm{eff}}$. We plot the transition frequencies relative to the ground state, ${\omega }_n=(E_n-E_0)/\hslash$. Note the splitting between energy levels which is linear with $E_{\mathrm{eff}}$. The inset shows the energy splitting ${\Delta }_n$, for $n=1$ (solid) and 2 (circles) photons, in perfect agreement with our linearized theory.