Spin-orbit coupled interferometry with ring-trapped Bose--Einstein condensates

We propose a method of atom-interferometry using a spinor Bose-Einstein condensate (BEC) with a time-varying magnetic field acting as a coherent beam-splitter. Our protocol creates long-lived superpositional counterflow states, which are of fundamental interest and can be made sensitive to both the Sagnac effect and magnetic fields on the sub-micro-G scale. We split a ring-trapped condensate, initially in the $m_f=0$ hyperfine state, into superpositions of internal $m_f=\pm1$ states and condensate superflow, which are spin-orbit coupled. After interrogation, relative phase accumulation can be inferred from a population transfer to the $m_f=\pm1$ states. The counterflow generation protocol is adiabatically deterministic and does not rely on coupling to additional optical fields or mechanical stirring techniques. Our protocol can maximise the classical Fisher information for any rotation, magnetic field, or interrigation time, and so has the maximum sensitivity available to uncorrelated particles. Precision can increase with the interrogation time, and so is limited only by the lifetime of the condensate.