Reply to Comment on 'Species-selective lattice launch for precision atom interferometry'

In our paper [1], we based the calculation of ${\lambda }_{{\rm{tune}} \mbox{-} {\rm{out}}}$ of ³⁹K on reference [2] and more specifically on the tabulated values in version 1.0, which reports the lifetimes as ${\tau }_{D1}={\tau }_{D2}=26.37(5)\,\mathrm{ns}$. This led to ${\lambda }_{{\rm{tune}} \mbox{-} {\rm{out}}}=768.959$ nm. Assuming the corrected values (table 1) in a more recent version of [2] (v1.02) or the ones in the original paper [3], we find the ${\lambda }_{{\rm{t}}{\rm{u}}{\rm{n}}{\rm{e}} \mbox{-} {\rm{o}}{\rm{u}}{\rm{t}}}=768.9691(20)\,{\rm{n}}{\rm{m}}$, which is the same result as Cronin et al [4] and explain the discrepancy of 9.4 nm. As suggested, we use in this reply a more accurate formula of the dipole potential accounting for the contribution of the off-resonant terms:

$$\begin{eqnarray*}&&{U}_{{\rm{dip}}}(\vec{r})=-\displaystyle \frac{\pi {c}^{2}}{2}\left[\displaystyle \frac{{{\rm{\Gamma }}}_{1}}{{\omega }_{{D}_{1}}^{3}}\left(\displaystyle \frac{1}{{\omega }_{{D}_{1}}-{\omega }_{L}}+\displaystyle \frac{1}{{\omega }_{{D}_{1}}+{\omega }_{L}}\right)+\displaystyle \frac{2{{\rm{\Gamma }}}_{2}}{{\omega }_{{D}_{2}}^{3}}\left(\displaystyle \frac{1}{{\omega }_{{D}_{2}}-{\omega }_{L}}+\displaystyle \frac{1}{{\omega }_{{D}_{2}}+{\omega }_{L}}\right)\right]I(\vec{r})\end{eqnarray*}$$

with the D₁, D₂ atomic transition lines ${\omega }_{{D}_{1}}$, ${\omega }_{{D}_{2}}$, the laser frequency ${\omega }_{L}$ and the speed of light c. This expression yields a small correction of 1.7 pm as pointed out by Cronin et al [4]. Contributions of core electrons and other atomic excitations to the residual static polarizability of alkalis can lead to further small corrections (less than 1 pm). We choose not to include them in this reply for the sake of simplicity. The interested reader is referred to table 5.5 in [5] for the considered isotopes.

Table 1.  Updated table 1 of [1], with the lifetime of the atomic excited level τ and the line transition frequency ${\omega }_{0}$. λ_tune-out is calculated according to the dipole potential expression U_dip stated above. The different lifetimes of K isotopes reflect different measurements and not an isotopic shift.

Isotopes	${\tau }_{{D}_{1}}$(ns)	${\tau }_{{D}_{2}}$(ns)	${\omega }_{0}^{{D}_{1}}/2\pi$(THz)	${\omega }_{0}^{{D}_{2}}/2\pi$(THz)	${\lambda }_{{\rm{tune}} \mbox{-} {\rm{out}}}$(nm)	References
Cs	34.750(70)	30.462(46)	335.116048807(41)	351.72571850(11)	880.1549(338)	[6–8]
85Rb	27.679(27)	26.2348(77)	377.107385690(46)	384.230406373(14)	790.0273(47)	[9]
87Rb	27.679(27)	26.2348(47)	377.107463380(11)	384.2304844685(62)	790.0272(47)	[10]
39K	26.72(5)	26.37(5)	389.286058716(62)	391.01617003(12)	768.9708(28)	[3, 11]
40K	26.79(7)	26.45(7)	389.286184353(73)	391.016296050(88)	768.9702(39)	[11, 12]
41K	26.79(7)	26.45(7)	389.286294205(62)	391.01640621(12)	768.9700(39)	[11, 12]
Na	16.299(21)	16.254(22)	508.3331958(13)	508.8487162(13)	589.55650(35)	[13]
6Li	27.102(9)	27.102(9)	446.789635(20)	446.799685(20)	670.987388(31)	[14, 15]

We recalculate the tune-out wavelengths in table 1 with updated and referenced D lines data and give uncertainties throughout.

In order to verify that the results of our proposal remain valid despite the change in the numerical values of ${\lambda }_{{\rm{tune}} \mbox{-} {\rm{out}}}$, we recalculate the acceleration sequence that we introduced in our proposal. The new potential expression led to the need to choose a new couple of integers ($N1=2028$ and $N2=982$) for the momentum kicks of the two species (to initially minimize the differential velocity to 77 µm s⁻¹). The required detuning, following equation (8) in our paper, is of 4.99 pm instead of 2.9 pm previously. This leads to a slightly larger value of the depth of the parasitic lattice for Rb (5.38 nK instead of 3.12 nK). Figure 1 illustrates the main result of our paper with these new parameters. As the reader can appreciate, no changes are to be reported. In the Rb isotope case, the effect of the parasitic lattice leads to a velocity offset of about 1 µm s⁻¹ consistent with the results of our papers ($P=4\,{\rm{W}}$ in figure 8). All conclusions drawn in our proposal remain, therefore, unaffected after the corrections proposed in the comment.

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