Precision Measurement of Atomic Isotope Shifts Using a Two-Isotope Entangled State

Atomic isotope shifts (ISs) are the isotope-dependent energy differences between atomic electron energy levels. These shifts have an important role in atomic and nuclear physics, and have been recently suggested as unique probes of physics beyond the standard model under the condition that they are determined significantly more precisely than the current state of the art. In this Letter, we present a simple and robust method for measuring ISs by taking advantage of Hilbert subspaces that are insensitive to common-mode noise yet sensitive to the IS. Using this method we evaluate the IS of the $5S_{1/2}\leftrightarrow 4D_{5/2}$ transition between ${{}^{86}\mathrm{Sr}}^{+}$ and ${{}^{88}\mathrm{Sr}}^{+}$ with a $1.6\times {10}^{-11}$ relative uncertainty to be 570 264 063.435(5)(8) (statistical)(systematic) Hz. Furthermore, we detect a relative difference of $3.46(23)\times {10}^{-8}$ between the orbital $g$ factors of the electrons in the $4D_{5/2}$ level of the two isotopes. Our method is relatively easy to implement and is indifferent to element or isotope, paving the way for future tabletop searches for new physics, posing interesting prospects for testing quantum many-body calculations, and for the study of nuclear structure.

Figure 1

The oscillation in time of the parity signal $P_{+}\equiv P_{ee}+P_{gg}$ for the separable state and for the entangled state preparation sequence. The contrast of the entangled state sequence parity oscillation is almost twice that of the separable state sequence, due to the increased correlation of the entangled state. The oscillation frequency is determined by $\delta {\nu }_{88,86}^i-\delta f_{88,86}^i$, i.e., the difference between the isotope shift and the addressing fields frequency difference, which in this example is $\sim 26\mathrm{Hz}$. Decay is due to the finite lifetime of the excited state. Data are shown in circles, while lines show fit as a guide to the eye.

Figure 2

A 24-hour IS measurement. (a) The superposition phase is determined for short ($\sim 5\mathrm{ms}$) and long ($\sim 233\mathrm{ms}$) interrogation times by varying the phase difference of the addressing fields and measuring the parity signal $P_{+}=P_{SS}+P_{DD}$. From these the superposition oscillation frequency is extracted. The isotope crystal spatial configuration is alternated in order to average over external field gradients (color coded right and left). (b) Allan deviation analysis of data set shown in (c), consistent with shot-noise limit after 24 h of averaging. Mean values of every pair of alternate measurements is shown in blue. This measurement for $m_S=1/2$, $m_D=3/2$ with an external magnetic field of 5.17 G yields an oscillation frequency of 570264063.745(9) Hz, which differs from the reported mean value of 570 264 063.435(9) Hz due to the isotope shift of the orbital magnetic susceptibility.

Figure 3

Results summary: measuring $\delta {\nu }_{88,86}^{S{D}_{5/2}}$ and $\delta g_D^{88,86}$. (a) IS as a function of $B\times m_D$. Colors correspond to $B\approx 3.5\mathrm{G}$, $m_D=-3/2$; $B\approx 3.5\mathrm{G}$, $m_D=3/2$; and $B\approx 5.17\mathrm{G}$, $m_D=3/2$. Maximum likelihood linear fit shown in purple. The IS variation in $B\times m_D$ is due to $g_D$ dependence on isotope, of which the slope of $0.0388(26)\mathrm{Hz}/\mathrm{G}$ is a direct measurement. The black dot denotes the IS at null magnetic field: 570 264 063.435(9) Hz. (b) IS over time. Measurements taken several weeks apart remain within the statistical uncertainty of a single measurement. Error bars represent statistical uncertainty of one standard deviation for all plots.