Quantum engineering of atomic phase shifts in optical clocks

Quantum engineering of time-separated Raman laser pulses in three-level systems is presented to produce an ultranarrow optical transition in bosonic alkali-earth clocks free from light shifts and with a significantly reduced sensitivity to laser parameter fluctuations. Based on a quantum artificial complex wave-function analytical model and supported by a full density-matrix simulation including a possible residual effect of spontaneous emission from the intermediate state, atomic phase shifts associated with Ramsey and hyper-Ramsey two-photon spectroscopy in optical clocks are derived. Various common-mode Raman frequency detunings are found in which the frequency shifts from off-resonant states are canceled, while their uncertainties at the ${10}^{-18}$ level of accuracy are strongly reduced.

Figure 1

(a) Three-level atomic system configuration for spectroscopy of a forbidden atomic transition in ${}^{174}\mathrm{Yb}$ and ${}^{88}\mathrm{Sr}$ optical lattice clocks. Atomic parameters are described in text. (b) EIT-Raman laser pulses in a Ramsey-type (on, off, on) sequence with interrogation times $\tau$, ${\tau }_m^{\mathrm{echo}}$ (if inserting an echo pulse), and ${\tau }_m$, respectively, and free evolution at the clock frequency for a time $T$.

Figure 2

(left) ${}^{88}\mathrm{Sr}$ transition probabilities measured on the ${}^3{P}_0$ clock state and (right) $\delta \nu$ frequency shifts vs the common-mode detuning ${\Delta }_0$ for different operating parameters all satisfying the primary condition $\delta \nu \left({\Delta }_0^m\right)=0$. Rabi frequencies ${\Omega }_1=20\sqrt{\pi {\Delta }_0^m/3}$ and ${\Omega }_2={\Omega }_1/100$, free evolution time $T=3$ s, and pulse duration $\tau =0.1875=3/16$ s. In the right panels, the frequency shifts are shown with a relative $\pm 10\%$ pulse area variation (solid and dashed lines). All $\delta \nu \left({\Delta }_0\right)$ curves are computed from Eq. (14). (a) Ramsey spectroscopy based on the $\pi /2-T-\pi /2$ laser pulse sequence. (b) HRR scheme with the $\pi /2-T-3\pi /2$ pulse sequence. (c) HRR + phase and echo scheme with the $\pi /2-T-\pi -\pi /2$ pulse sequence including a phase reversal during the second pulse. In this case, a plateau is obtained around a magic common-mode Raman detuning ${\Delta }_0^m/2\pi \sim 11.2$ GHz, with all conditions of Eqs. (8) satisfied.

Figure 3

Fractional frequency shift (in ${10}^{-18}$ unit) based on a $\pi /2-T-3\pi /2$ laser pulse sequence in the presence of spontaneous emission relaxation. All conditions from Eqs. (8) are fulfilled at the millihertz level by adding, between the preparation and detection pulses, an echo pulse with a duration ${\tau }_m^{\mathrm{echo}}=2\tau$, reversing the sign of one Rabi frequency. (a) ${}^{88}\mathrm{Sr}$ clock frequency shift around ${\Delta }_0^m=11.2$ GHz with $\Gamma /2\pi =32$ MHz and (b) ${}^{174}\mathrm{Yb}$ clock frequency shift around ${\Delta }_0^m=326.7$ GHz with $\Gamma /2\pi =28.9$ MHz. The density matrix analyzes (solid dots and stars) are reproduced by the complex wave-function model (dashed and solid lines). Rabi frequencies are as in Fig. 2; free evolution time $T=5$ s.