Effect of $\alpha$ variation on the vibrational spectrum of Sr${}_2$

We consider the effect of $\alpha$ variation on the vibrational spectrum of Sr${}_2$ in the context of a planned experiment to test the stability of $\mu \equiv m_e/m_p$ using optically trapped Sr${}_2$ molecules [Zelevinsky et al., Phys. Rev. Lett. 100, 043201 (2008); Kotochigova et al., Phys. Rev. A 79, 012504 (2009)]. We find the prospective experiment to be 3 to 4 times less sensitive to fractional variation in $\alpha$ as it is to fractional variation in $\mu$. Depending on the precision ultimately achieved by the experiment, this result may give justification for the neglect of $\alpha$ variation or, alternatively, may call for its explicit consideration in the interpretation of experimental results.

Figure 1

Basic illustration of the experiment described by Zelevinsky et al.[7, 8] to test the stability of the electron-to-proton mass ratio using the vibrational spectrum of ${}^{88}$Sr${}_2$. The curve represents the $X{}^1{\Sigma }_g^{+}$ electronic potential, with horizontal dashed lines corresponding to select vibrational levels supported by this potential. Frequencies $\nu$ and ${\nu }^{\prime }$ are to be measured by an optical Ramsey scheme involving an intermediate excited electronic state (not shown). The dimensionless ratio $R$ is sensitive to variations in $\mu$ and is independent of any external reference (clock) frequency.

Figure 2

The function $f\left(x\right)=2[x-1+\sqrt{1-x}]$. For the Morse potential, this function modulates the level sensitivities to both $\mu$ and $\alpha$ variation across the vibrational spectrum, with the argument $x$ taken as ${\mathcal{E}}_n\equiv E_n/D$. The most deeply (${\mathcal{E}}_n\to 0$) and loosely (${\mathcal{E}}_n\to 1$) bound levels are insensitive to variations, while intermediate levels have much higher sensitivities.

Figure 3

Level sensitivities ${\partial }_{\mu }{\mathcal{E}}_n$ and ${\partial }_{\alpha }{\mathcal{E}}_n$ vs ${\mathcal{E}}_n$ for the $X{}^1{\Sigma }_g^{+}$ state of Sr${}_2$. Circles correspond to ab initio results for individual levels $n=0$ through $n=35$, with solid curves being fits to this data (fits are extended through ${\mathcal{E}}_n=1$).