Emergent quantum field theories on curved spacetimes in spinor Bose-Einstein condensates: from scalar to Proca fields

TL;DR

This work shows how excitations in a spin-1 Bose-Einstein condensate map onto emergent relativistic quantum field theories on curved acoustic spacetimes. Depending on the mean-field ground state (polar, ferromagnetic, or antiferromagnetic), density, spin-density, and spin-nematic fluctuations realize a massless scalar (phonon) and either massless vector (spin-nematic magnon) or massive Proca-type fields, with uni- or bi-/tri-metric spacetime structures. Finite Zeeman couplings and trap profiles enable explicit symmetry breaking, yielding controlled Proca masses and time-dependent FLRW-like metrics, which can drive cosmological particle production via quenches or ramps. The paper also outlines experimental pathways to observe spin-nematic squeezing, two-point correlators, and structure formation, establishing a concrete platform for analogue cosmology and vector-field quantum simulations in cold-atom systems.

Abstract

We consider excitations of a spin-1 Bose-Einstein-condensate (BEC) in the vicinity of different mean-field configurations and derive mappings to emergent relativistic quantum field theories minimally coupled to curved acoustic spacetimes. The quantum fields are typically identified with Nambu-Goldstone bosons, such that the structure of the analogue quantum field theories on curved spacetimes depends on the (spontaneous) symmetry breaking pattern of the respective ground-state. The emergent spacetime geometries are independent of each other and exhibit bi-metricity in the polar and antiferromagnetic phase, whereas one has tri-metricity in the ferromagnetic phase. Compared to scalar BECs, the spinor degrees of freedom allow to investigate massive vector and scalar fields where the former is a spin-nematic rotation mode in the polar phase which can be cast into a Proca field that is minimally coupled to a curved spacetime that emerges on length scales larger than the spin-healing length. Finally, we specify the Zeeman couplings and the condensate trap to be spacetime-dependent such that a cosmological FLRW-metric can be achieved. This work enables a pathway towards quantum-simulating cosmological particle production of Proca quanta via quenching the quadratic Zeeman-coefficient or via magnetic field ramps, which both result in the creation of spin-nematic squeezed states.

Figures (2)

• Figure 1: Emergent quantum field theories in curved spacetime (left image) associated to various mean-field ground states of a spin-one BEC with antiferromagnetic (center image) and ferromagnetic (right image) interactions. The insets show the dispersion relations in the infrared regime, leading either to Lorentz-invariant hyperbola or Galilei-invariant parabola (in an analogue sense), depending on whether the dispersion relation is quadratic or linear in the infrared regime. We notably find that transversal spin-nematic excitations of the polar state lead to an analogue vector QFTCS and a massive scalar QFTCS. Further details on the structure of the emergent are summarized in \ref{['tab:Summary']}. Solid (dashed) lines in the phase-diagram indicate first (second)-order quantum phase transitions where the derivative of $U_\mathrm{mag}$ with respect to $(p,q)$ changes continously (discontinously) Kawaguchi2012. Perceptually uniform colormaps designed by Crameri2023 are used.
• Figure 2: Dispersion relations $(\hbar \omega_\pm)^2$ of the exact eigenmodes formed by superpositions of phonons and magnons, resulting in an avoided crossing of pure density- and spin-modes with Bogoliubov dispersion $\hbar^2 \omega_\mathrm{D,S}^2$ for $c_0 / c_1 = 32$ which is representative of an antiferromagnetic $^{23}\mathrm{Na}$-BEC (cref. \ref{['tab:ExperimentalValues']}). Due to the high ratio $c_0/c_1$, an external magnetic field $f_z \neq 0$ only marginally influences the dispersion relations of the eigenmodes. Perceptually uniform colormaps designed by Crameri2023 are used.