Conditional Ramsey Spectroscopy with Synchronized Atoms

We investigate Ramsey spectroscopy performed on a synchronized ensemble of two-level atoms. The synchronization is induced by the collective coupling of the atoms to a heavily damped mode of an optical cavity. We show that, in principle, with this synchronized system it is possible to observe Ramsey fringes indefinitely, even in the presence of spontaneous emission and other sources of individual-atom dephasing. This could have important consequences for atomic clocks and a wide range of precision metrology applications.

Figure 1

(a) Conditional Ramsey spectroscopy where synchronized atoms are coupled collectively to a cavity and pumped individually with incoherent rate $w$ during the interrogation time. (b) Ramsey sequence showing initial preparation in state $|g⟩$ (pseudospins pointing down to the south pole of the Bloch sphere), the rotation to the equator by illuminating the atoms with a coherent near-resonant $\pi /2$ laser pulse, precession around the equator, and second $\pi /2$ $x$-axis rotation, after which the $z$-axis projection carries information about the cosine of the accumulated phase.

Figure 2

Calculations of Eq. (3) with $N=250$, ${\mathrm{\Gamma }}_C=0.2/T_1$, $T_2=T_1$, and $w=N{\mathrm{\Gamma }}_C/2$. (a) Ramsey fringes with synchronized atoms (red solid line) versus $T$. Conventional Ramsey fringes (blue dashed line) for the same $T_1$ and $T_2$. (b) During the interrogation time, the atomic inversion $⟨{\widehat{\sigma }}_j^z⟩$ (blue dashed line), spin-spin correlation $⟨{\widehat{\sigma }}_j^{+}{\widehat{\sigma }}_k^{-}⟩$ (red solid line), $⟨{\widehat{\sigma }}_j^{+}{\widehat{\sigma }}_k^{-}⟩\text{−}⟨{\widehat{\sigma }}_j^{+}⟩⟨{\widehat{\sigma }}_k^{-}⟩$ (red dotdashed line), and $⟨{\widehat{\sigma }}_j^{+}{\widehat{\sigma }}_k^z⟩/(⟨{\widehat{\sigma }}_j^{+}⟩⟨{\widehat{\sigma }}_k^z⟩)$ (green dotted line).

Figure 3

The decay rate of the visibility of Ramsey fringes at ${\mathrm{\Gamma }}_C=0.2/T_1$ and $T_2=T_1$ as a function of repumping for $N=200$ and as a function of $N$ for $w=N{\mathrm{\Gamma }}_C/2$ (Inset). The dots are numerical solutions of Eq. (3), and the solid blue line is the semiclassical approximation for comparison.

Figure 4

Quantum state diffusion calculations of conditional Ramsey fringes subject to continuous homodyne measurement of the cavity output field for $N=10$ and $w=N{\mathrm{\Gamma }}_C/2$. The blue dashed lines are the ensemble average for reference. (a) A typical Ramsey fringe for a single experimental trial (red solid line). (b) Histograms are the statistics of the positions of zero crossings of each fringe for 4000 trials. The blue solid lines are fitted Gaussian distributions with variance of $T{\mathrm{\Gamma }}_C$ centered on the zero crossing of the ensemble average.