Ramsey Spectroscopy with Displaced Frequency Jumps

Sophisticated Ramsey-based interrogation protocols using composite laser pulse sequences have been recently proposed to provide next-generation high-precision atomic clocks with a near perfect elimination of frequency shifts induced during the atom-probing field interaction. We propose here a simple alternative approach to the autobalanced Ramsey interrogation protocol and demonstrate its application to a cold-atom microwave clock based on coherent population trapping ($CPT$). The main originality of the method, based on two consecutive Ramsey sequences with different dark periods, is to sample the central Ramsey fringes with frequency jumps finely adjusted by an additional frequency-displacement concomitant parameter, scaling as the inverse of the dark period. The advantage of this displaced frequency-jump Ramsey method is that the local oscillator (LO) frequency is used as a single physical variable to control both servo loops of the sequence, simplifying its implementation and avoiding noise associated with controlling the LO phase. When tested using a $CPT$ cold-atom clock, the DFJR scheme reduces the sensitivity of the clock frequency to variations of the light shifts by more than an order of magnitude compared with the standard Ramsey interrogation. This simple method can be applied in a wide variety of Ramsey-spectroscopy based applications including frequency metrology with $CPT$-based and optical atomic clocks, mass spectrometry, and precision spectroscopy.

Figure 1

Illustration of light shifts and their mitigation using the DFJR method. The graphs show the central Ramsey-$CPT$ fringes obtained for two different dark periods ($T_L=16\mathrm{ms}$: blue line and $T_S=4\mathrm{ms}$: red dashed line). The center of each fringe is marked with a triangle. The steady-state value of the clock frequency is marked with a diamond symbol. Dotted lines show the frequency jumps applied to the sampling frequencies (marked with circles). (a) Light shift with standard Ramsey-$CPT$ interrogation ($T=16\mathrm{ms}$). The clock stabilizes to the shifted Ramsey fringe. (b) By displacing the frequency jumps—jumping a different amount to higher and lower frequencies—the clock frequency shifts can be mitigated. The challenge is then to find the suitable frequency displacement $\mathrm{\Delta }{f}_T$. (c) The DFJR method is used to find $\mathrm{\Delta }{f}_T$ and stabilize the clock frequency on the nonshifted resonance frequency ($f_c=0\mathrm{Hz}$). Using two different dark periods, $T_S=4\mathrm{ms}$ (lower part of the graph) and $T_L=16\mathrm{ms}$ (upper part of the graph) and adding the frequency displacement $\mathrm{\Delta }{f}_T$ as a $T$-dependent control parameter, the clock frequency is free from light shifts.

Figure 2

An illustration of the DFJR sequence, applied to a cold-atom $CPT$ clock. The sequence is composed of four consecutive Ramsey cycles—two with a long dark period, $T_L$ and two with a short dark period $T_S$. For each dark period, the two cycles include frequency jumps to the sides of the central fringe. However, the frequency jumps are corrected by a frequency displacement parameter $\mathrm{\Delta }{f}_T$, which scales as the inverse of the dark period [see Eq. (1)]. The frequency for each Ramsey cycle is noted on the left side of the figure. The error signal constructed from the long cycle pair, ${ϵ}_L$, is used to control the clock frequency $f_c$. The error signal derived from the short cycles, ${ϵ}_S$, is used to control the concomitant control parameter $\alpha$.

Figure 3

A time trace of the atomic clock operation based on DFJR. The upper pane shows the clock frequency shift $f_c-f_{Rb}$ and the lower pane shows the frequency displacement for the long cycle $\mathrm{\Delta }{f}_{T_L}=(\alpha /T_L)$. Each pane shows the original data (dots) and a moving average (solid line). During the clock run the $CPT$ intensity ratio is abruptly changed (switch times are shown in black dashed lines), changing the off-resonant light shift of the clock transition. It is evident that $\alpha$ adapts to accommodate the changing light shift leaving the clock frequency $f_c$ unaffected (at the value of the second order Zeeman shift).

Figure 4

Frequency shift $f_c-f_{Rb}$ of the clock, in Ramsey-$CPT$ (triangles, with $T=16\mathrm{ms}$) and DFJR (circles, with $T_S=4\mathrm{ms}$ and $T_L=16\mathrm{ms}$) methods, and frequency displacement for the long cycle $\alpha /T_L$ (squares) versus the $CPT$ intensity ratio. In the Ramsey-$CPT$ case, the clock frequency is shifted when the $CPT$ intensity ratio is changed, due to off-resonant light shift [32]. In the DFJR case, the clock frequency is nearly constant, and $\alpha$ changes to compensate for the light shift. The value of the single control parameter in the Ramsey-$CPT$ scheme (the clock frequency) is effectively split into the two control parameters with the DFJR method. The frequency displacement $\mathrm{\Delta }{f}_T$ is associated with the interrogation-related shifts. The clock frequency $f_c$ is free from interrogation related shifts, and is only shifted by the second-order Zeeman shift.

Figure 5

FFT power spectra of the clock frequency in Ramsey-$CPT$ (with $T=16\mathrm{ms}$) and DFJR ($T_S=4\mathrm{ms}$, $T_L=16\mathrm{ms}$) methods. Sinusoidal oscillations with an amplitude of $\pm 1\mathrm{MHz}$ and about 2 h period are applied to the OPD of the $CPT$ laser. An order-of-magnitude reduction of the oscillations amplitude is observed with the DFJR method.