Josephson junction-embedded transmission-line resonators: from Kerr medium to in-line transmon

TL;DR

This paper develops a method to derive the Hamiltonian of a nonlinear circuit with a Josephson junction embedded in a transmission-line resonator, revealing tunable Kerr effects and strong qubit-resonator coupling for quantum applications.

Contribution

It introduces a general approach to analyze nonlinear circuits with Josephson junctions, enabling precise control of Kerr effects and coupling strengths in microwave quantum circuits.

Findings

Kerr coefficient can be tuned from weak to very strong regimes.

The system can operate as an in-line transmon with adjustable nonlinearity.

Maximal qubit-resonator coupling strength is characterized.

Abstract

We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focussing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e. inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effect, as well as beam-splitter type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient K can be made to reach values that are weak (K < \kappa), strong (K > \kappa) or even very strong (K >> \kappa) with respect to the photon-loss rate \kappa. In the latter case, the resonator+junction circuit corresponds to an in-line version of the transmon. By replacing the single junction by a SQUID, the Kerr coefficient can be tuned in-situ, allowing for example the fast generation of Schr\"odinger cat states of microwave light. Finally, we explore the maximal strength of qubit-resonator coupling that can be reached in this setting.