# Revealing the nonlinear response of a tunneling two-level system ensemble using coupled modes

## TL;DR

This paper uses two-tone spectroscopy on coupled superconducting resonators to uncover the nonlinear response of a TLS ensemble, revealing average TLS coupling, nonlinear effects, and controllable resonance lifetimes.

## Contribution

It introduces a novel two-tone spectroscopy method on coupled resonators to analyze the nonlinear response of TLS ensembles and extract key parameters.

## Key findings

- Average single-photon Rabi frequency of TLSs is approximately 79 kHz.
- Nonlinear kinetic inductance causes enhanced frequency shifts at high photon numbers.
- Resonance lifetimes can be increased by pumping the coupled mode.

## Abstract

Atomic sized two-level systems (TLSs) in amorphous dielectrics are known as a major source of loss in superconducting devices. In addition, individual TLS are known to induce large frequency shifts due to strong coupling to the devices. However, in the presence of a broad ensemble of TLSs these shifts are symmetrically canceled out and not observed in a typical single-tone spectroscopy experiment. We introduce a two-tone spectroscopy on the normal modes of a pair of coupled superconducting coplanar waveguide resonators to reveal this effect. Together with an appropriate saturation model this enables us to extract the average single-photon Rabi frequency of dominant TLSs to be $\Omega_0/2\pi \approx 79 $ kHz. At high photon numbers we observe an enhanced frequency shift due to nonlinear kinetic inductance when using the two-tone method and estimate the value of the nonlinear coefficient as $K/2\pi \approx -1\times 10^{-4}$ Hz/photon. Furthermore, the life-time of each resonance can be controlled (increased) by pumping of the other mode as demonstrated both experimentally and theoretically.