# Magnon Condensation in a Dense Nitrogen-Vacancy Spin Ensemble

## TL;DR

This paper explores the potential for Bose-Einstein condensation of magnons in dense nitrogen-vacancy spin ensembles in diamond, estimating critical concentrations and temperatures for magnetic self-ordering and magnon condensation.

## Contribution

It provides a theoretical analysis of magnon condensation feasibility in NV ensembles, including critical concentration and temperature estimates using the Sherrington-Kirkpatrick model.

## Key findings

- Magnetic self-ordering occurs above approximately 70 ppm concentration.
- Magnon condensation may be detectable in thin films at concentrations greater than 90 ppm.
- Estimated transition temperature exceeds 10 mK at 90 ppm and reaches room temperature at around 450 ppm.

## Abstract

The feasibility of creating a Bose-Einstein condensate of magnons using a dense ensemble of nitrogen-vacancy spin defects in diamond is investigated. Through assessing a density-dependent spin exchange interaction strength and the magnetic phase transition temperature ($T_c$) using the Sherrington-Kirkpatrick model, the minimum temperature-dependent concentration for magnetic self-ordering is estimated. For a randomly dispersed spin ensemble, the calculated average exchange constant exceeds the average dipole interaction strengths for concentrations approximately greater than 70 ppm, while $T_c$ is estimated to exceed 10 mK beyond 90 ppm, reaching 300 K at a concentration of approximately 450 ppm. On this basis, the existence of dipole-exchange spin waves and their plane-wave dispersion is postulated and estimated using a semiclassical magnetostatic description. This is discussed along with a $T_c$-based estimate of the four-magnon scattering rate, which indicates magnons and their condensation may be detectable in thin films for concentrations greater than 90 ppm.