# Magnetization induced dynamics of a Josephson junction coupled to a nanomagnet

## TL;DR

This paper analyzes how a nanomagnet coupled to a Josephson junction influences its superconducting current, revealing Shapiro-like steps in the I-V characteristics under various magnetic field conditions and extending the analysis to include dissipation effects.

## Contribution

The study provides an analytic perturbative solution for the coupled JJ-nanomagnet system under arbitrary magnetic fields and explores the impact of dissipation on magnetization-induced Shapiro steps.

## Key findings

- Existence of Shapiro-like steps in I-V characteristics under magnetic fields.
- Magnetic field rotation and AC drive affect the Shapiro steps.
- Dissipation modifies the magnetization-induced Shapiro steps.

## Abstract

We study the superconducting current of a Josephson junction (JJ) coupled to an external nanomagnet driven by a time dependent magnetic field both without and in the presence of an external AC drive. We provide an analytic, albeit perturbative, solution for the Landau-Lifshitz (LL) equations governing the coupled JJ-nanomagnet system in the presence of a magnetic field with arbitrary time-dependence oriented along the easy axis of the nanomagnet's magnetization and in the limit of weak dimensionless coupling $\epsilon_0$ between the JJ and the nanomagnet. We show the existence of Shapiro-like steps in the I-V characteristics of the JJ subjected to a voltage bias for a constant or periodically varying magnetic field and explore the effect of rotation of the magnetic field and the presence of an external AC drive on these steps. We support our analytic results with exact numerical solution of the LL equations. We also extend our results to dissipative nanomagnets by providing a perturbative solution to the Landau-Lifshitz-Gilbert (LLG) equations for weak dissipation. We study the fate of magnetization-induced Shapiro steps in the presence of dissipation both from our analytical results and via numerical solution of the coupled LLG equations. We discuss experiments which can test our theory.