# Disorder-Driven Density and Spin Self-Ordering of a Bose-Einstein Condensate in a Cavity

## TL;DR

This paper investigates how a two-component Bose-Einstein condensate in a cavity undergoes self-ordering driven by disorder, leading to simultaneous density and spin orderings, symmetry breaking, and characteristic excitation spectra at the superradiance transition.

## Contribution

It reveals the mechanism of disorder-driven spin and density self-ordering in a BEC within a cavity, highlighting symmetry breaking and the role of quantum fluctuations as a dynamical disorder source.

## Key findings

- Simultaneous density and spin order emerge at the superradiance transition.
- Spontaneous breaking of discrete and continuous symmetries occurs in the ordered phase.
- Elementary excitations show mode softening at the superradiance threshold.

## Abstract

We study spatial spin and density self-ordering of a two-component Bose-Einstein condensate via collective Raman scattering into a linear cavity mode. The onset of the Dicke superradiance phase transition is marked by a simultaneous appearance of a crystalline density order and a spin-wave order. The latter spontaneously breaks the discrete $\mathbf{Z}_2$ symmetry between even and odd sites of the cavity optical potential. Moreover, in the superradiant state the continuous $U(1)$ symmetry of the relative phase of the two condensate wavefunctions is explicitly broken by the cavity-induced position-dependent Raman coupling with a zero spatial average. Thus, the spatially-averaged relative condensate phase is locked at either $\pi/2$ or $-\pi/2$. This continuous symmetry breaking and relative condensate phase locking by a zero-average Raman field can be considered as a generic order-by-disorder process similar to the random-field-induced order in the two-dimensional classical ferromagnetic $XY$ spin model. However, the seed of the random field in our model stems from quantum fluctuations in the cavity field and is a dynamical entity affected by self-ordering. The spectra of elementary excitations exhibit the typical mode softening at the superradiance threshold.