# Optimal synchronization deep in the quantum regime: resource and fundamental limit

## TL;DR

This paper develops an analytical framework for quantum synchronization in a spin-1 system, revealing fundamental limits, resource dependencies, and novel phenomena like interference blockade and extended Arnold tongues.

## Contribution

It introduces a unified analytical approach to quantum synchronization, demonstrating synchronization in a spin-1 system and uncovering new effects and fundamental limits.

## Key findings

- Quantum synchronization can be realized in a spin-1 system deep in the quantum regime.
- Identified interference-based quantum synchronization blockade.
- Derived the fundamental maximum synchronization achievable in the system.

## Abstract

We develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. Our unified description allows us to identify the resource on which quantum synchronization relies, and to compare quantitatively the synchronization behavior of different limit cycles and signals. We focus on the most elementary quantum system that is able to host a self-sustained oscillation, namely a single spin 1. Despite the spin having no classical analogue, we first show that it can realize the van der Pol limit cycle deep in the quantum regime, which allows us to provide an analytical understanding to recently reported numerical results. Moving on to the equatorial limit cycle, we then reveal the existence of an interference-based quantum synchronization blockade and extend the classical Arnold tongue to a snake-like split tongue. Finally, we derive the maximum synchronization that can be achieved in the spin-1 system, and construct a limit cycle that reaches this fundamental limit asymptotically.