High cooperativity coupling between a phosphorus donor spin ensemble and a superconducting microwave resonator

TL;DR

This paper demonstrates high cooperativity coupling between a phosphorus donor spin ensemble in isotopically purified silicon and a superconducting microwave resonator, showing strong coupling characteristics and temperature-dependent behavior.

Contribution

It introduces a statistical model describing how temperature affects the coupling strength in a spin ensemble-resonator system, highlighting the square root scaling with polarized spins.

Findings

Normal mode splitting indicates high cooperativity.

Coupling strength is comparable to spin system loss rate.

Temperature influences coupling, scaling with square root of polarized spins.

Abstract

We investigate the coupling of an ensemble of phosphorus donors in an isotopically purified $^{28}$Si host lattice interacting with a superconducting coplanar waveguide resonator. The microwave transmission spectrum of the resonator shows a normal mode splitting characteristic for high cooperativity. The evaluated collective coupling strength $g_{\mathrm{eff}}$ is of the same order as the loss rate of the spin system $\gamma$, indicating the onset of strong coupling. We develop a statistical model to describe the influence of temperature on the coupling strength from $50\,\mathrm{mK}$ to $3.5\,\mathrm{K}$ and find a scaling of the coupling strength with the square root of the number of thermally polarized spins.