  We experimentally demonstrated that the stability of an atomic clock improves
at fastest rate $\tau^{-1}$ (where $\tau$ is the averaging time) when the phase
of a local oscillator is genuinely compared to the continuous phase of many
atoms in a single trap (atomic phase lock). For this demonstration, we
developed a simple method that repeatedly monitors the atomic phase while
retaining its coherence by observing only a portion of the whole ion cloud.
Using this new method, we measured the continuous phase over 3 measurement
cycles, and thereby improved the stability scaling from $\tau^{-1/2}$ to
$\tau^{-1}$ during the 3 measurement cycles.
  %Compared with the standard method that initialize phase during each
measurement cycle, the long term stability was improved by a factor of
$\sqrt{n_{cp}}$ (where $n_{cp}$ is the number of continuous phase
measurements).
  This simple method provides a path by which atomic clocks can approach a
quantum projection noise limit, even when the measurement noise is dominated by
the technical noise.
