# Cross-Kerr nonlinearity for phonon counting

## TL;DR

This paper demonstrates a method for projectively measuring the quantum state of a trapped ion's motional mode using an effective cross-Kerr nonlinearity induced by Coulomb interactions, enabling phonon number reconstruction.

## Contribution

It introduces an experimental technique leveraging intrinsic Coulomb nonlinearity for phonon counting via cross-Kerr coupling in trapped ions.

## Key findings

- Resolved frequency shifts due to single phonons in the second mode.
- Successfully reconstructed phonon number distribution.
- Validated the effectiveness of cross-Kerr based phonon measurement.

## Abstract

State measurement of a quantum harmonic oscillator is essential in quantum optics and quantum information processing. In a system of trapped ions, we experimentally demonstrate the projective measurement of the state of the ions' motional mode via an effective cross-Kerr coupling to another motional mode. This coupling is induced by the intrinsic nonlinearity of the Coulomb interaction between the ions. We spectroscopically resolve the frequency shift of the motional sideband of the first mode due to presence of single phonons in the second mode and use it to reconstruct the phonon number distribution of the second mode.