$N-\mathrm{atom}$ collective-state atomic clock with $\sqrt{N}-\mathrm{fold}$ increase in effective frequency and $\sqrt{N}-\mathrm{fold}$ reduction in fringe width

We describe a collective state atomic clock (COSAC) with Ramsey fringes narrowed by a factor of $\sqrt{N}$ compared to a conventional clock—$N$ being the number of noninteracting atoms—without violating the uncertainty relation. This narrowing is explained as being due to interferences among the collective states, representing an effective $\sqrt{N}$-fold increase in the clock frequency, without entanglement. We discuss the experimental inhomogeneities that affect the signal and show that experimental parameters can be adjusted to produce a near ideal signal. The detection process collects fluorescence through stimulated Raman scattering of Stokes photons, which emits photons predominantly in the direction of the probe beam for a high enough optical density. By using a null measurement scheme, in which detection of zero photons corresponds to the system being in a single collective state, we detect the population in a collective state of interest. The quantum and classical noise of the ideal COSAC is still limited by the standard quantum limit and performs only as well as the conventional clock. However, when detection efficiency and collection efficiency are taken into account, the detection scheme of the COSAC increases the quantum efficiency of detection significantly in comparison to a typical conventional clock employing fluorescence detection, yielding a net improvement in stability by as much as a factor of 10.

Figure 1

(a) Three level atomic system. (b) Population of $|E_N\rangle$ at the end of Ramsey pulse sequence as function of $\delta$.

Figure 2

Collective state signal at the end of the Ramsey field experiment for various numbers of atoms, with parameters $\mathrm{\Omega }=5\times {10}^5{\mathrm{s}}^{-1}$ and $T_2=3\times {10}^{-5}$ s. Plotted are the ideal signal (dashed line) $S_{\mathrm{col}}$ and the reduced signal (solid line) $S_{\mathrm{Dop}}$, where the effect of Doppler shift is taken into account.

Figure 3

Collective state signal at the end of the Ramsey field experiment for various Gaussian beam widths. $N=2\times {10}^6$ atoms; $\mathrm{\Omega }=5\times {10}^5{\mathrm{s}}^{-1}$; $T_2=3\times {10}^{-5}$ s. Plotted are the ideal signal (dashed line) $S_{\mathrm{col}}$ and the reduced signal (solid line) $S_{\mathrm{\Omega }}$, when the Gaussian nature of the beam is taken into account. The plots are for various ratios of the widths of the laser field to the atomic ensemble: ${\omega }_L/{\omega }_A$.

Figure 4

(left) Collective state signal (solid line) at the end of the Ramsey field experiment for $N=2\times {10}^6$ atoms; $\mathrm{\Omega }=5\times {10}^6{\mathrm{s}}^{-1}$; $T=3\times {10}^{-5}$ s. The dashed curves show the signal for $N+\mathrm{\Delta }N$ (narrower) and $N-\mathrm{\Delta }N$ (wider), where $\mathrm{\Delta }N/N=0.01$. (right) Plot of $\mathrm{\Delta }\mathrm{\Gamma }/\mathrm{\Gamma }$ as a function of $\mathrm{\Delta }N/N$.

Figure 5

Ramsey fringe experiment for an ensemble of $\mathrm{\Lambda }$-type atoms for the detection of collective state $|E_N\rangle$. Atoms are released from the trap, and the experiment is performed while they are free falling inside the vacuum chamber. They interact with two $\pi /2$ Ramsey pulses, which are separated in time by $T_2$, and are probed by a probe. The probe induces a unidirectional Raman transition in the atoms while producing Stokes photons in the direction of the detector, given high enough optical density. The combined signal from the probe and emitted Stokes photons are multiplied with the frequency produced by the FS in such a way that the resulting signal will be proportional to the number of Stokes photons detected. Determining the threshold of the zero emission signal, and counting how many trials result in zero emission, the histogram can be built to produce signal in Fig. 1.

Figure 6

Initialization of the system involves first optically pumping the atoms into $|F=1\rangle$ state by applying a laser field that is resonant with $5{}^2{S}_{1/2},|F=2\rangle \to 5{}^2{P}_{3/2},|F^{\prime }=2\rangle$ transition. Afterwards, as is depicted here, a $\pi$ polarized beam that is resonant with $5{}^2{S}_{1/2},|F=1\rangle \to 5{}^2{P}_{1/2},|F^{\prime }=1\rangle$ transition is applied. Because the $|F=1,m_F=0\rangle \to |F^{\prime }=1,m_{F^{\prime }}=0\rangle$ transition is forbidden for the $D1$ line, the atoms are eventually pumped into $|F=1,m_F=0\rangle$.

Figure 7

In the detection zone, we probe the population of state $|E_N\rangle$ by applying field ${\omega }_1$ and detecting Stokes photons produced during the Raman transition. In the bad cavity limit, the atomic system will not reabsorb the photon that has been emitted during the Raman process, such that the transition from $|E_k\rangle$ to $|E_{k+1}\rangle$ will occur, but not vice versa.

Figure 8

Left: Ratio of the QFF in the CC to the QFF in the COSAC, for $T_2=3\times {10}^{-5}$ s, $m=1000$, and $N=2\times {10}^6$. It should be noted that the fluctuation in the CC is independent of $f$ while that of the COSAC varies significantly with $f$. Right: Ratio of the SVS of the COSAC to the SVS of the CC for $T_2=3\times {10}^{-5}$ s, $m=1000$, and $N=2\times {10}^6$. The dashed vertical lines in the plots show where the ${\text{FWHM}}_{\mathrm{col}}$ are.

Figure 9

Plot of the ideal signal (solid line), the upper bound (dotted line), and the lower bound (dashed line) for $N=2\times {10}^6, T_2=3\times {10}^{-4}$ s, and ${\gamma }_{\mathrm{sa}}={10}^4{\mathrm{s}}^{-1}$. Note that in (c) and (d), the upper and lower bounds are virtually indistinguishable.

Figure 10

Collective state energy levels, separated by ${\omega }_0$, are excited by a field of frequency $\omega$. All the states from $|E_0\rangle$ to $|E_N\rangle$ are excited, and participate in producing an effective clock transition frequency proportional to $\sqrt{N}$.

Figure 11

Ramsey fringe experiment of a two level atom, in the Jaynes-Cummings model, involves states ${|1\rangle }_A{|m\rangle }_{\nu }$ and ${|2\rangle }_A{|m-1\rangle }_{\nu }$ where the state with subscript $A$ represents the atomic state, and subscript $\nu$ represents the Ramsey field. The phase difference of the two levels at the end of the experiment is $e^{i\delta T_2}$, and the signal produced would oscillate at frequency $\delta$; if a two level system existed in which the ground state were the collective state $|E_0{\rangle }_A{|m\rangle }_{\nu }$ and the excited state were the collective state $|E_N{\rangle }_A{|m-N\rangle }_{\nu }$, the phase accumulation between the two states at the end of the Ramsey fringe experiment would be $e^{iN\delta T_2}$, and the oscillation frequency would be $N\delta$.

Figure 12

In a two atom ensemble, each of the three collective states interfere with one another to produce different Ramsey fringes (a)–(c). The overall envelope is not drawn. The sum of these interferences gives the narrowing of the fringe linewidth as seen in (d). In (d), the dotted curve represents the signal from a single atom and the solid curve the signal from two atoms for comparison.