Improved absolute clock stability by the joint interrogation of two atomic ensembles

Improving the clock stability is of fundamental importance for the development of quantum-enhanced metrology. One of the main limitations arises from the randomly fluctuating local oscillator (LO) frequency, which introduces “phase slips” for long interrogation times and hence the failure of the frequency-feedback loop. Here we propose a strategy to improve the stability of atomic clocks by interrogating two atomic ensembles sharing the same LO. The two ensembles are prepared in coherent spin states pointing along orthogonal directions in the Bloch sphere. While standard Ramsey interrogation can only determine phases unambiguously in the interval $[-\pi /2,\pi /2]$, the joint interrogation allows for an extension to $[-\pi ,\pi ]$, resulting in a relaxed restriction of the Ramsey time and improvement of absolute clock stability. Theoretical predictions are supported by ab initio numerical simulation for white and correlated LO noise. While our basic protocol uses uncorrelated atoms, we further extended it to include spin-squeezing and further improving the scaling of clock stability with the number of atoms. Our protocol can be readily tested in current state-of-the-art experiments.

Figure 1

(a) Scheme of the joint interrogation method. Two atomic ensembles are prepared in coherent spin states with mean spin direction pointing along orthogonal directions in the Bloch sphere ($x$ axis for the Ramsey interferometer $A$ and $y$ axis for $B$) and interrogate the same LO. The accumulated phase $\theta$ is the same in both interferometers. The insets show Husimi distributions during the different interferometer operations. (b) Ramsey signal $〈J_z\left(\theta \right){〉}_{\mathrm{out}}$ as a function of $\theta$ for the two interferometers. Combining the two out-of-phase signals (see text) it is possible to obtain an unbiased estimate of $\theta$ in the full $[-\pi ,\pi ]$ interval.

Figure 2

(a) Estimator bias as a function of $\theta$. The green dashed line is $|\theta -{\overline{\mathrm{\Theta }}}_A\left(\theta \right)|$ for the single-Ramsey interferometer with Eq. (2) as input, while the black solid line is $|\theta -{\overline{\mathrm{\Theta }}}_{AB}\left(\theta \right)|$ for the joint-Ramsey scheme. The horizontal dotted line is $1/\sqrt{N_t}$. (b) Mean squared error (MSE) as a function of $\theta$. The green dashed (black solid) line is obtained for the single- (joint-) Ramsey interferometer. The horizontal dotted line is $1/N_t$. The inset is a zoom showing the mean squared error multiplied by $N_t$ as a function of $\theta \sqrt{N_t}$ around $\theta =0$. In both panels, the single and joint protocols are compared by fixing the total number of particles $N_t=2000$, that is, $N_A=N_t$ for the single-Ramsey interferometer and $N_A=N_B=N_t/2$ for the joint-Ramsey scheme.

Figure 3

Allan variance as a function of $\tau /T$ for white LO noise and obtained with a clock based on a single Ramsey interferometer. Symbols are results of ab initio numerical simulation for different values of the Ramsey time: ${\gamma }_{\mathrm{LO}}T=0.25$ (blue circles), ${\gamma }_{\mathrm{LO}}T=0.17$ (green squares), and ${\gamma }_{\mathrm{LO}}T=0.12$ (red diamonds). Solid lines are Eq. (34), the dashed line is the SQL, Eq. (32), while the dotted lines are the asymptotic $\tau /T\to +\infty$ prediction of Eq. (35). Here $N_t=1000$

Figure 4

Allan variance (multiplied by ${\omega }_0^2\tau N_t/{\gamma }_{\mathrm{LO}}$) as a function of the Ramsey time ${\gamma }_{\mathrm{LO}}T$ for (a) white and (b) flicker LO noise, respectively. Symbols are results of ab initio numerical simulations: green diamonds are obtained for a single-Ramsey clock with $N_t=2000$ particles, while black circles are obtained for a joint-Ramsey clock with 1000 particles in each interferometer ($N_t=2000$ is the total number of particles used in both clocks). The dotted line in both panels is the SQL, ${\sigma }^2=1/\left({\omega }_0^{2}T\tau N_t\right)$. The solid and dashed lines in both panels are expected results. For the white-noise case, the dashed green line is Eq. (34), while the solid black line is Eq. (38). For the flicker noise, the lines are semi-analytical predictions given by Eq. (29) where the quantities $Q_T\left(\lfloor \tau /T\rfloor \right), P_T\left(n_c\right)$ and $c_T^2$ are obtained numerically and independently from the calculation of the Allan variance. The inset of panel (a) shows the gain $G$ defined in Eq. (39) for the white-noise LO case, as a function of the total averaging time ${\gamma }_{\mathrm{LO}}\tau$. The solid line is obtained analytically and shows that the gain in the noiseless case converges to $G=4$ (dashed line) for large $\tau$. Panels (c) and (d) show the probability of phase slip, Eq. (36), as a function of ${\gamma }_{\mathrm{LO}}T$, for white and flicker noise, respectively. The solid lines in panel (c) is Eq. (37). In all panels, ${\gamma }_{\mathrm{LO}}\tau =100$.

Figure 5

Absolute stability gain $G$, Eq. (39), as a function of the parameter $s_{\epsilon }$ quantifying the imperfect alignment of the two probe states of the joint-Ramsey clock. Panel (a) is obtained for white LO noise, while panel (b) for flicker noise. In each panel, blue dots are results of numerical simulations, the dashed line is a guide to the eye. The inset of panel (b) shows the Allan variance as a function of the Ramsey time ${\gamma }_{\mathrm{LO}}T$. Different symbols are obtained for different values of $\epsilon$: $s_{\epsilon }=0$ (circles), $s_{\epsilon }=0.02$ (squares), $s_{\epsilon }=0.04$ (downward-pointing triangle), and $s_{\epsilon }=0.06$ (upward-pointing triangle). The solid line is Eq. (29), the dotted lines are a guide to the eye, and the dashed line is the SQL. In all panels ${\gamma }_{\mathrm{LO}}\tau =100$ and the total number of particles is $N_t=2000$.

Figure 6

(a) Stability gain $G$, Eq. (39), as a function of $N_A/N_t$. Symbols are results of numerical simulations: squares (dots) are obtained for white (flicker) LO noise. The lines are Eq. (41). Here $N_t=2000$. (b) Allan variance as a function of the Ramsey time in the presence of the number of particles fluctuations. Symbols are results of numerical simulations. Diamonds are obtained for a single Ramsey clock with $\overline{N}=2000$ and $\mathrm{\Delta }N=\sqrt{\overline{N}}$. Circles corresponds to a the joint clock scheme with ${\overline{N}}_A={\overline{N}}_B=1000$ and $\mathrm{\Delta }{N}_{A,B}=\sqrt{{\overline{N}}_{A,B}}$. The dotted line is the SQL, ${\sigma }^2=1/\left({\omega }_0^{2}T\tau {\overline{N}}_t\right)$ with ${\overline{N}}_t=2000$. Here ${\gamma }_{\mathrm{LO}}\tau =100$.

Figure 7

Allan variance as a function of the total cycle time time $T_C=T+T_D$ for (a) white and (b) flicker noise, respectively. Here the Ramsey time $T$ is set to that minimizing the Allan variance for $T_D=0$, see Fig. 4. In each panel, diamonds (circles) refer to results of numerical simulations for the single (joint) Ramsey clock scheme. Lines are the corresponding analytical behavior (see text). The inset shows the stability gain factor Eq. (39), calculated from the analytical functions. Here the total number of particles is $N_t=2000$, the averaging time is $\tau /T_C=100$.

Figure 8

Clock scheme combining joint-Ramsey interrogation and spin squeezing. It consists of three Ramsey interferometers. The first two interferometers (Ramsey A and Ramsey B) use the coherent spin states (2) and (7), respectively. The two estimates ${\mathrm{\Theta }}_A$ and ${\mathrm{\Theta }}_B$ are combined according to the discussion in Sec. 3c (green dotted square and lines). The joint estimate ${\mathrm{\Theta }}_{AB}$, Eq. (10) is used, via a phase feedback (red dashed square and lines), to bring the spin-squeezed state toward its optimal working point, on the equator of the Bloch sphere. The sum of ${\mathrm{\Theta }}_{AB}$ and ${\mathrm{\Theta }}_C$ giving ${\mathrm{\Theta }}_{ABC}$, Eq. (51), is used to steer the LO frequency via a frequency feedback (blue line).

Figure 9

(a) Bias $|\theta -{\overline{\mathrm{\Theta }}}_{ABC}\left(\theta \right)|$ and (b) mean squared error as a function of $\theta$ (solid red lines). Results are obtained for the squeezing parameter $s_{\mathrm{opt}}^2=1/{\left(2N^2\right)}^{1/3}$ and $N=1000$. The inset shows Eq. (56) as a function of the squeezing parameter. In panel (b) and in the inset, the dotted line is Eq. (58). In panel (a), the dotted line is the square root of Eq. (58).

Figure 10

Panel (a) shows the Allan variance for (i) a single Ramsey clock using a coherent state and $N_t=3000$ atoms (green diamonds); (ii) a joint Ramsey clocks using two coherent spin states with $N_t/2=1500$ atoms each (black circles); (iii) a hybrid clock using two coherent spin states and one optimized spin-squeezed state, each with $N_t/3=1000$ atoms each (red squares). Symbols are results of ab initio numerical simulations, while the solid lines are semi-analytical predictions (see text). The dashed line is ${\sigma }_{\mathrm{SQL}}^2=1/\left({\omega }_0^{2}T\tau N_t\right)$. The dotted line is Eq. (59). (b) Scaling of the Allan variance (minimized over the Ramsey time) as a function of $N_t$ for the different clock protocols (i) the single-Ramsey (green diamonds); (ii) the joint-Ramsey (black circles); (iii) the hybrid-Ramsey (red squares). Lines are fits: the dashed and dotted lines are ${\sigma }^2=O(1/N_t)$, while the solid line is ${\sigma }^2=O(1/N_t^{5/3})$. The numerical results in both panels are obtained for flicker LO noise and ${\gamma }_{\mathrm{LO}}\tau =100$.

Figure 11

(a) Phase variance as a function of ${\gamma }_{\mathrm{LO}}t$ (dots). The solid line is $v_1{\left(t\right)}^2={\left({\gamma }_{\mathrm{LO}}t\right)}^2$. (b) Power spectral density as a function of ${\gamma }_{\mathrm{LO}}t$ (dots). The solid line is a fit $S\left(f\right)=h_{\mathrm{LO}}/f$ giving $h_{\mathrm{LO}}=1/(2\chi log2)\times {({\gamma }_{\mathrm{LO}}/{\omega }_0)}^2$ with $\chi =1.4$. Panels (c) and (d) show $P_T\left(n_c\right)$ for ${\gamma }_{\mathrm{LO}}T=0.3$ and ${\gamma }_{\mathrm{LO}}T=0.45$, respectively.