Isotope shifts of natural ${\mathrm{Sr}}^{+}$ measured by laser fluorescence in a sympathetically cooled Coulomb crystal

We measured by laser spectroscopy the isotope shifts between naturally occurring even isotopes of strontium ions for both the $5s{}^2{S}_{1/2}\to 5p{}^2{P}_{1/2}$ (violet) and the $4d{}^2{D}_{3/2}\to 5p{}^2{P}_{1/2}$ (infrared) dipole-allowed optical transitions. Fluorescence spectra were taken by simultaneous measurements on a two-component Coulomb crystal in a linear Paul trap containing ${10}^3$–${10}^4$ laser-cooled ${\mathrm{Sr}}^{+}$ ions. The isotope shifts are extracted from the experimental spectra by fitting the data with the analytical solution of the optical Bloch equations describing a three-level atom interacting with two laser beams. This technique allowed us to increase the precision with respect to previously reported data obtained by optogalvanic spectroscopy or fast atomic-beam techniques. The results for the $5s{}^2{S}_{1/2}\to 5p{}^2{P}_{1/2}$ transition are ${\nu }_{88}-{\nu }_{84}=+378\left(4\right)$ MHz and ${\nu }_{88}-{\nu }_{86}=+170\left(3\right)$ MHz, in agreement with previously reported measurements. In the case of the previously unexplored $4d{}^2{D}_{3/2}\to 5p{}^2{P}_{1/2}$ transition we find ${\nu }_{88}-{\nu }_{84}=-828\left(4\right)$ MHz and ${\nu }_{88}-{\nu }_{86}=-402\left(2\right)$ MHz. These results provide more data for stringent tests of theoretical calculations of the isotope shifts of alkali-metal-like atoms. Moreover, they simplify the identification and the addressing of ${\mathrm{Sr}}^{+}$ isotopes for ion frequency standards or quantum-information-processing applications in the case of multi-isotope ion strings.

Figure 1

Low-energy-level scheme for ${\mathrm{Sr}}^{+}$. In the present work we address the 711-THz violet transition (cooling) and the 275-THz infrared transition (“repumping”).

Figure 2

Schematic drawing of the isotope shifts of the even ${\mathrm{Sr}}^{+}$ isotopes studied in this paper. (a) $5s{}^2{S}_{1/2}\to 5p{}^2{P}_{1/2}$ violet transition [711 THz, blue (dark gray) arrows]; green (light gray) arrows represent the measured isotope shifts, while black arrows indicate the frequency shift with respect to the ${}^{85}$Rb $5s{}^2{S}_{1/2}(F=2)\to 6p{}^2{P}_{1/2}(F^{\prime }=3)$ reference transition. Note that we referenced the energies to the ground states. (b) $4d{}^2{D}_{3/2}\to 5p{}^2{P}_{1/2}$ infrared transition [275 THZ, red (medium gray) arrows]: thick green (light gray) arrows represent the measured isotope shifts with respect to the ${}^{88}$${\mathrm{Sr}}^{+}$ repumping transition. As both levels are isotope shifted in this diagram, we also show the isotope shift that affects the 711-THz transition with thin black arrows. The errors given are an estimate of the statistical variance in our measurements.

Figure 3

Schematic drawing of the atomic levels of ${\mathrm{Sr}}^{+}$ ($\Lambda$ configuration). One laser (cooling) addresses the $S\to P$ transition with a detuning ${\delta }_b$ and a Rabi frequency ${\Omega }_b$, while another laser (repumping) drives the $D\to P$ transition (detuning ${\delta }_r$ and Rabi frequency ${\Omega }_r$). The decoherence rate between the $D$ and $S$ levels is given by ${\gamma }_{DS}$.

Figure 4

Theoretical fluorescence spectrum as a function of ${\delta }_r$ (optical Bloch equations). The parameters of the plot are ${\Gamma }_b/2\pi =21.5$ MHz, branching ratio of 13.4 [29], ${\delta }_b/2\pi =-28$ MHz, ${\Omega }_b/2\pi =12.9$ MHz, ${\Omega }_r/2\pi =2$ MHz, and ${\gamma }_{DS}/2\pi =5.7$ MHz. These parameters correspond to the fit of the rightmost spectrum in Fig. 5.

Figure 5

Twin spectra obtained with a Coulomb crystal composed of equivalent proportions of ${}^{88}$${\mathrm{Sr}}^{+}$ and ${}^{86}$${\mathrm{Sr}}^{+}$ ions. The repumping laser detunings ${\delta }_r$ of the two isotopes are scanned simultaneously, but there is a constant frequency shift between the two beams imposed by fixed-frequency acousto-optic modulators ($\sim$40 MHz in this example). For each detuning of the master 275-THz laser two images are acquired. As an example we show two such images coded in false colors corresponding to the ${}^{88}$${\mathrm{Sr}}^{+}$ [blue (dark gray)] and ${}^{86}$${\mathrm{Sr}}^{+}$ [green (light gray)] isotopes. The effect of radial segregation in the trap pushes ${}^{88}$${\mathrm{Sr}}^{+}$ in the outer shells, with the core of the crystal being occupied by ${}^{86}$${\mathrm{Sr}}^{+}$. The total fluorescence signal plotted in the graph is obtained by integrating these images in the region of interest. The solid lines are the best fit of the spectra by an analytical model based on a three-level atom (optical Bloch equations). The fitting parameters for the ${}^{88}$${\mathrm{Sr}}^{+}$ spectrum are listed in the caption of Fig. 4; the fitting parameters of the ${}^{86}$${\mathrm{Sr}}^{+}$ spectrum are ${\Gamma }_b/2\pi =21.5$ MHz, ${\delta }_b/2\pi =-25.9$ MHz, ${\Omega }_b/2\pi =16.1$ MHz, ${\Omega }_r/2\pi =2.7$ MHz, and ${\gamma }_{DS}/2\pi =8.6$ MHz.