Analysis of the coupling strength between transmons interacting via Left-Handed metamaterials

Left-Handed transmission lines can be designed to exhibit many modes near the typical transmon frequency. This multiplicity can be used to enhance qubit-qubit coupling, as opposed to just a single mode. We demonstrate that the strength of coupling between the transmon and the metamaterial modes depends on the qubit's frequency and its relative position with respect to the transmission line in a nontrivial manner, setting it apart from coupling via conventional (right-handed) transmission lines. Through simulations involving two transmons placed at different positions and operating at various flux-tuned frequencies, we can intentionally manipulate the coupling strength between the transmon and the resonator modes. Consequently, we study effective qubit-qubit interaction and its interesting features, which offer a high degree of flexibility in designing the desired interactions between the qubits, which can be activated or deactivated by making slight adjustments to the transmon frequency, typically within a range of just tens of megahertz.