Probe light-shift elimination in generalized hyper-Ramsey quantum clocks

We present an interrogation scheme for the next generation of quantum clocks to suppress frequency shifts induced by laser probing fields that are themselves based on generalized hyper-Ramsey resonances. Sequences of composite laser pulses with a specific selection of phases, frequency detunings, and durations are combined to generate a very efficient and robust frequency locking signal with an almost perfect elimination of the light shift from off-resonant states and to decouple the unperturbed frequency measurement from the laser's intensity. The frequency lock point generated from synthesized error signals using either $\pi /4$ or $3\pi /4$ laser phase steps during the intermediate pulse is tightly protected against large laser-pulse area variations and errors in potentially applied frequency shift compensations. Quantum clocks based on weakly allowed or completely forbidden optical transitions in atoms, ions, molecules, and nuclei will benefit from these hyperstable laser frequency stabilization schemes to reach relative accuracies below the ${10}^{-18}$ level.

Figure 1

Time sequence of a composite three-pulse Ramsey interrogation. The parameters of the composite pulse sequence are detuning ${\delta }_l$, complex field amplitude ${\mathrm{\Omega }}_l{e}^{i{\phi }_l}$, pulse duration ${\tau }_l$, where $l=\text{i,j,k}$ and the two different free-evolution times $T_1$ and $T_2$ between pulses. Phase-step modulations ${\phi }_l$ of the laser fields are applied to generate a specific and robust error signal.

Figure 2

Generalized hyper-Ramsey dispersive error signals vs the frequency clock detuning $\delta$ with $\mathrm{\Delta }=0$. (a) $\mathrm{\Delta }{E}_{\text{GHR}(\pi /4)}$ and (b) $\mathrm{\Delta }{E}_{\text{GHR}(3\pi /4)}$ are calculated with Eqs. (4) and (5a), respectively. The pulse duration is $\tau =0.1875\mathrm{s}$, the Rabi frequency is fixed to $\mathrm{\Omega }=\pi /2\tau$, and the free-evolution time is $T=2\mathrm{s}$.

Figure 3

Error signal $\mathrm{\Delta }E$ vs $\delta$ in the presence of residual light-shift perturbations $\mathrm{\Delta }\ne 0$ (we use ${\mathrm{\Delta }}_{\text{i,j,k}}=\mathrm{\Delta }$). In (a)–(c), the three phase-steps protocols, $\mathrm{\Delta }{E}_{\text{HR}}, \mathrm{\Delta }{E}_{\text{GHR}(\pi /4)}$, and $\mathrm{\Delta }{E}_{\text{GHR}(3\pi /4)}$, respectively. Same conditions as in Fig. 2.

Figure 4

Normalized slopes of error signals (a) $\mathrm{GHR}(\pi /4)$ and (b) $\mathrm{GHR}(3\pi /4)$ vs the uncompensated residual offset. The slope predicted by Eqs. (6) and (7) is plotted for a Rabi frequency increased or decreased by $10\%$ with a fixed pulse duration $\tau$. All others conditions as in Fig. 2.

Figure 5

Sensitivity of the clock frequency shift $\delta \nu$ for HR and GHR protocols to some pulse area variations of $\mathrm{\Delta }\theta /\theta =\pm 10\%$ during laser pulses vs uncompensated residual offsets. All other conditions as in Fig. 2.