Precision measurement of the 5 ${}^2{S}_{1/2}$ – 4 ${}^2{D}_{5/2}$ quadrupole transition isotope shift between ${}^{88}\mathrm{Sr}$${}^{+}$ and ${}^{86}\mathrm{Sr}$${}^{+}$

We present a high-precision measurement of the isotope shift of the narrow quadrupole-allowed 5 ${}^2{S}_{1/2}$ – 4 ${}^2{D}_{5/2}$ transition in ${}^{86}\mathrm{Sr}$${}^{+}$ relative to the most abundant isotope ${}^{88}\mathrm{Sr}$${}^{+}$. This was accomplished using high-resolution laser spectroscopy of individual trapped ions, and the measured shift is $\Delta {\nu }_{\mathrm{meas}}^{88,86}=570.281(4)$ MHz. We also tested a recently developed and successful method for ab initio calculation of isotope shifts in alkali-metal-like atomic systems against this measurement, and our initial result of $\Delta {\nu }_{\mathrm{calc}}^{88,86}=457(28)$ MHz is also presented. While the measurement and the calculation are in broad agreement, there is a clear discrepancy between them, and we believe that the specific mass shift was underestimated in our calculation. Our measurement provides a stringent test for further refinements of theoretical isotope shift calculation methods for atomic systems with a single valence electron.

Figure 1

Level diagram for even isotopes of singly ionized strontium (i.e., Sr ii) with no net nuclear spin, ${}^{88}\mathrm{Sr}$${}^{+}$ and ${}^{86}\mathrm{Sr}$${}^{+}$ in this context. The $S_{1/2}$ level is the ground state of the single valence electron in these ions. The $P$ levels decay rapidly, having lifetimes of several to tens of nanoseconds, and the $D$ levels are metastable (lifetimes of a few tenths of a second). The double arrows indicate laser radiation used in the determination of the isotope shift in the 445 THz (red) quadrupole transition. The thick lines represent dipole-allowed ($E1$) transitions, while the thin line indicates the much weaker dipole-forbidden but quadrupole-allowed ($E2$) transition of interest. The natural lifetimes $\tau$ of the excited states are taken from [15]. The eigenstates of the optical qubit in the quantum-information-processing apparatus used to perform this measurement are $|0\rangle =|S_{1/2}\rangle$ and $|1\rangle =|D_{5/2}\rangle$.

Figure 2

Sample spectrum of the $S_{1/2}$ – $D_{5/2}$ transition in Sr ii acquired using the quantum shelving technique. The blue curve is a fit to eight Zeeman components of this quadrupole transition with the four most central Zeeman lines shown here. The dashed red line indicates the measured rf frequency delivered to the AOM at the transition center, which was determined as a parameter of the fit.

Figure 3

Measures of the AOM component $\Delta {\nu }_{\mathrm{AOM}}^{88,86}$ of the overall isotope shift. Black circles are measurements of the $S-D$ transition in ${}^{86}\mathrm{Sr}$${}^{+}$, and red squares are measurements of the same transition in ${}^{88}\mathrm{Sr}$${}^{+}$. The AOM frequency is the rf shift delivered to the probe light at the resonance line center not including the offset due to the free spectral range of the reference cavity that was used to lock and narrow the probe laser. The laser was locked one cavity FSR to the red when measuring the ${}^{86}\mathrm{Sr}$${}^{+}$ transition compared to when measurements of the ${}^{88}\mathrm{Sr}$${}^{+}$ transition were performed. The insets show the same points with zoomed ranges to clearly show the effect of cavity drift (note the different ordinate scales). An overall frequency offset of 200 MHz was subtracted from all rf values for readability.

Figure 4

Fits to determine AOM component of isotope shift. Black circles are measurements of the $S-D$ transition in ${}^{86}\mathrm{Sr}$${}^{+}$, and red squares are measurements of the same transition in ${}^{88}\mathrm{Sr}$${}^{+}$. The blue dashed line is a linear fit yielding an AOM shift of 29.020(2) MHz, and the green solid line is a quadratic fit, yielding an AOM shift of 29.019(2) MHz. The two panels are offset in the ordinate coordinate by 29.020 MHz, but are scaled identically (refer to left and right scales), as are the abscissas. An overall frequency offset of 200 MHz has been subtracted from all rf values for readability.