Measuring Many-body Coherence in Weakly Interacting Fermi Gases*

We study a coherently prepared, weakly interacting Fermi gas of $6\times10^4$ $^6$Li atoms, which offers a new foundation for understanding complex spin dynamics induced by the interplay between spin interactions and motion in many-body systems. Using phase-controlled radio-frequency (rf) pulses, we prepare an x-polarized collective spin state and implement a many-body echo: After an evolution time $\tau$, we apply a rotation by an angle $\phi$ about the x-axis. This is followed immediately by a $\pi$ rotation about the y-axis and inversion of the s-wave scattering length, which reverses the sign of the Hamiltonian. The rf detuning is sufficiently stable to measure the final collective spin vector as a function of $\phi$ for $2\tau$ up to 1 second. This system provides a powerful tool to study many-body coherence and entanglement spreading in quantum many-body systems.