Exponential decrease of Ramsey linewidth via coherent atomic phase tracking

We propose an effective method to decrease the Ramsey linewidth of microwave atomic clock by tracking the coherent atomic phase through nondestructive measurement. The free evolution time T between two Ramsey pulses is divided into N subsections. In each subsection, the coherent phase of the atomic ensemble is probed by an additional large-detuning laser. The probed phase in the previous subsection is used to change the phase of the microwave pulse in the following subsection. After tracking the coherent atomic phase for N-1 times, the final linewidth of Ramsey fringe decreases quickly as N/(2N-1). We then demonstrate this proposal in a vapor cell atomic clock. The linewidth of the Ramsey fringe is decreased down to 27 Hz after 5 successive sub-interrogations, which is much narrower than the linewidth of 142 Hz obtained by normal Ramsey spectroscopy.