Amplification of the quantum superposition macroscopicity of a flux qubit by a magnetized Bose gas

We perform a microscopic derivation of the effective flux action of a hybrid quantum system consisting of a superconducting flux qubit (SFQ) in the presence of a spin-$F$ atomic Bose gas. For a magnetized Bose-Einstein condensed state (BEC) of $N_{B}\sim \mathcal{O}(10^6)$ atoms in an experimentally realistic geometry, we demonstrate a two- to five-fold amplification of the quantum superposition macroscopicity $\mathcal{M}$ of the ground state screening current superposition over the bare value without the BEC. Exploiting the connection between $\mathcal M$ and the maximal metrological usefulness of a multimode superposition state, we show that amplification of $\mathcal{M}$ in the ground state of the hybrid system is equivalent to a decrease in the quantum Cram\'{e}r-Rao bound for estimation of an externally-applied flux. Our result therefore demonstrates the increased usefulness of the BEC--SFQ hybrid system as a sensor of ultraweak magnetic fields below the standard quantum limit.