# Phase space curvature in spin-orbit coupled ultracold atom systems

## TL;DR

This paper derives and analyzes semiclassical equations of motion for spin-orbit coupled ultracold atoms, highlighting the role of Berry curvature in particle dynamics and effective mass, with implications for understanding quantum transport.

## Contribution

It introduces a comprehensive derivation of phase space equations of motion incorporating Berry curvature effects in spin-orbit coupled ultracold atoms, linking curvature to effective mass.

## Key findings

- Berry curvature influences particle trajectories in phase space.
- Effective mass is directly related to phase-space Berry curvature.
- Semiclassical equations recover known results in low-curvature limit.

## Abstract

We consider a system with spin-orbit coupling and derive equations of motion which include the effects of Berry curvatures. We apply these equations to investigate the dynamics of particles with equal Rashba-Dresselhaus spin-orbit coupling in one dimension. In our derivation, the adiabatic transformation is performed first and leads to quantum Heisenberg equations of motion for momentum and position operators. These equations explicitly contain position-space, momentum-space, and phase-space Berry curvature terms. Subsequently, we perform the semiclassical approximation, and obtain the semiclassical equations of motion. Taking the low-Berry-curvature limit results in equations that can be directly compared to previous results for the motion of wavepackets. Finally, we show that in the semiclassical regime, the effective mass of the equal Rashba-Dresselhaus spin-orbit coupled system can be viewed as a direct effect of the phase-space Berry curvature.