Quantum Interferometry with Microwave-dressed F=1 Spinor Bose-Einstein Condensates: Role of Initial States and Long Time Evolution

We numerically investigate atom interferometry based on spin-exchange collisions in $F=1$ spinor Bose-Einstein condensates in the regime where both the truncated Wigner and the Bogoliubov approximations break down. The interferometer promises to beat the shot-noise limit even in the case of large atom-number population in the arms of the interferometer. Spin-exchange collisions in $F=1$ spinor Bose-Einstein condensates, where two atoms with magnetic quantum number $m_F=0$ collide and change into a pair with $m_F=\pm1$, are useful to implement matter-wave quantum optics in spin space, such as quantum-enhanced interferometry, because the collisions generate entanglement and they can be precisely controlled using microwave dressing. Here, we show numerically that the sensitivity of spin-mixing interferometry can be enhanced to go beyond the shot-noise limit even with a large population of $m_F=\pm 1$ states, $1\ll N_{m_F=\pm1}\ll N$, and after long evolution times. This is done by using classically seeded initial states with a small initial population in the $m_F=\pm1$ states and using long evolution times $t\gg h/c$, where c is the spin-dependent interaction energy.