Atomic clocks based on adaptive phase measurements with entangled atoms

We show that the frequency stability of atomic clocks limited by local oscillator frequency fluctuations [1] can be greatly improved by using an adaptive measurement strategy with entangled atoms. Our method uses multiple atomic sub-ensembles with various degrees of spin-squeezing and sequential adaptive measurements of the Ramsey phase. With properly optimized degree of squeezing, this method reaches the Heisenberg limit for phase measurements $\delta\phi\simeq 1/N$, where $N$ is the number of atoms. In addition, we show that multiple interrogation times for these sub-ensembles can be used to improve the long-term stability of the clock. This method allows one to use a very long interrogation time, limited only by environmental fluctuations. The combination of the above two methods leads to an ultimate long-term frequency stability of the clock scaling as $\sigma_y(\tau)=\frac{\langle\delta{\bar \omega}(\tau)^2\rangle^{1/2}}{\omega_0}\propto\frac{1}{N\tau}$, where $\tau$ is the averaging time, to be compared with the usual projection-noise limited clock stability scaling as $\sigma_y(\tau)\propto \frac{1}{\sqrt{N\tau}}$. [1] A. Andr\'{e}, A. S. S\o rensen, and M. D. Lukin, Phys. Rev. Lett. {\bf 92}, 230801 (2004).