# Dynamics and thermodynamics of a central spin immersed in a spin bath

## TL;DR

This paper derives exact dynamical and thermodynamic descriptions of a central spin interacting with a thermal spin bath, revealing conditions for quantum coherence preservation and thermalization behavior.

## Contribution

It provides an exact reduced dynamical map, operator sum representation, and a time-local master equation for the central spin system, advancing understanding of its quantum and thermodynamic properties.

## Key findings

- Quantum coherence persists at low temperatures and finite bath sizes.
- High temperature and large bath size lead to Markovian, thermalizing dynamics.
- Long-time-averaged state can retain coherence under resonance conditions.

## Abstract

An exact reduced dynamical map along with its operator sum representation is derived for a central spin interacting with a thermal spin environment. The dynamics of the central spin shows high sustainability of quantum traits such as coherence and entanglement in the low-temperature regime. However, for sufficiently high temperature and when the number of bath particles approaches the thermodynamic limit, this feature vanishes and the dynamics closely mimics Markovian evolution. The properties of the long-time-averaged state and the trapped information of the initial state for the central qubit are also investigated in detail, confirming that the nonergodicity of the dynamics can be attributed to the finite temperature and finite size of the bath. It is shown that if a certain stringent resonance condition is satisfied, the long-time-averaged state retains quantum coherence, which can have far reaching technological implications in engineering quantum devices. An exact time-local master equation of the canonical form is derived. With the help of this master equation, the nonequilibrium properties of the central spin system are studied by investigating the detailed balance condition and irreversible entropy production rate. The result reveals that the central qubit thermalizes only in the limit of very high temperature and large number of bath spins.