Using the ground state of an antiferromagnetic spin-1 atomic condensate for Heisenberg-limited metrology

Abstract

We show that the ground state of a spin-1 atomic condensate with antiferromagnetic interactions constitutes a useful resource for quantum metrology upon approaching the Heisenberg limit. Unlike a ferromagnetic condensate state where individual atomic spins are aligned in the same direction, the antiferromagnetic ground-state condensate is a condensate of spin-singlet atom pairs. The inherent correlation between paired atoms allows for parameter estimation at precisions beyond the standard quantum limit (SQL) for uncorrelated atoms. The degree of improvement over the SQL is measured by the scaled quantum Fisher information (QFI), whose dependence on the ratio of linear Zeeman shift $p$ to spin-dependent atomic interaction $c$ is studied. At a typical value of $p=0.4c$, which corresponds to a magnetic field of $28.6\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{G}$ for $c=50\mathrm{h}$ Hz (for $^{23}\mathrm{Na}$ atom condensate in the $F=1$ state at a typical density of $\ensuremath{\sim}{10}^{14}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$), the scaled QFI can reach $\ensuremath{\sim}0.48N$, which approaches the limit of $0.5N$ for the twin-Fock state ${|N/2\ensuremath{\rangle}}_{+}{|N/2\ensuremath{\rangle}}_{\ensuremath{-}}$. Our work encourages experimental efforts to reach the ground state of an antiferromagnetic condensate at a extremely low magnetic field.