# Spinor Bose-Einstein Condensate Interferometer within the Undepleted Pump Approximation: Role of the Initial State

## TL;DR

This paper derives analytical expressions for phase sensitivity in spinor Bose-Einstein condensate interferometers within the undepleted pump approximation, analyzing different initial states and seeding conditions to guide experimental and numerical studies.

## Contribution

It provides the first analytical treatment of SU(1,1) interferometers with spinor BECs under the undepleted pump approximation for various initial states and seeding configurations.

## Key findings

- Analytical phase sensitivity expressions derived for different initial states.
- Assessment of the undepleted pump approximation's validity through quantum calculations.
- Guidance for experiments and numerical studies exploring beyond the approximation.

## Abstract

Most interferometers operate with photons or dilute, non-condensed cold atom clouds in which collisions are strongly suppressed. Spinor Bose-Einstein condensates (BECs) provide an alternative route toward realizing three-mode interferometers; in this realization, spin-changing collisions provide a resource that generates mode entanglement. Working in the regime where the pump mode, i.e., the m=0 hyperfine state, has a much larger population than the side or probe modes (m=+1 and m=-1 hyperfine states), f=1 spinor BECs approximate SU(1,1) interferometers. We derive analytical expressions within the undepleted pump approximation for the phase sensitivity of such an SU(1,1) interferometer for two classes of initial states: pure Fock states and coherent spin states. The interferometer performance is analyzed for initial states without seeding, with single-sided seeding, and with double-sided seeding. The validity regime of the undepleted pump approximation is assessed by performing quantum calculations for the full spin Hamiltonian. Our analytical results and the associated dynamics are expected to guide experiments as well as numerical studies that explore regimes where the undepleted pump approximation makes quantitatively or qualitatively incorrect predictions.