Nonlinear matter-wave amplification in a $^{23}$Na spinor Bose-Einstein condensate

An $F=1$, $^{23}$Na Bose-Einstein condensate (BEC) in a far off-resonant optical trap has a spinor order parameter for the hyperfine sublevels $m_F=+1,0,-1$. At low magnetic fields pairs of atoms may undergo spin-changing collisions between the $|m_{F_A},m_{F_B}>=|0,0>$ and $|+1,-1>$ states. The ground state is a BEC in the $m_F=0$ state. However, by dressing the $F=1$ energy levels with a microwave field off-resonant from the $F=2$ state, the sign of the effective quadratic Zeeman energy is reversed and the $m_F=0$ BEC becomes metastable. Vacuum fluctuations in the initially empty $m_F=+1,-1$ states drive nonlinear amplification of $|+1,-1>$ atom pairs. When the energy difference is equal to an interaction energy, then the rate of emission of $|+1,-1>$ pairs is maximal. This realizes a phase-insensitive parametric amplifier for matter waves and is characterized by sub-shot-noise spin correlations between the $m_F=+1$ and $m_F=-1$ BECs. We discuss progress in realizing this amplifier in a Na BEC as well as possibilities for building a nonlinear matter-wave interferometer.