Matter-wave amplification in a seeded $^{23}$Na spinor Bose-Einstein condensate

In an $F=1$ spinor condensate, spin-changing interactions of atoms in the $|m_{F_A},m_{F_B}\rangle = |0,0\rangle$ state can only produce the $|0,0\rangle$ (unchanged) or $|1,-1\rangle$ states. Because of the ideally perfect correlation in the production of $m_F=-1$ and $m_F=+1$ atoms, the magnetization $m=n_{m_F=+1}-n_{m_F=-1}$ is a squeezed quadrature of the system. Here we use a microwave-dressed $^{23}$Na Bose-Einstein condensate to create a nonlinear matter-wave amplifier which can produce spin-squeezed states. We then use microwaves to transfer a fraction of the $m_F=0$ condensate into a coherent seed of $m_{F}=+1$ atoms. After some evolution time, we show that $n_{m_F=+1}$ can be used as a large amplitude measurement of only a few atoms initially in the $m_F=-1$ state. This kind of measurement may be important in achieving high phase sensitivity in Heisenberg-limited matter-wave interferometers.