Measurement of light shifts at two off-resonant wavelengths in a single trapped ${\mathrm{Ba}}^{+}$ ion and the determination of atomic dipole matrix elements

We have measured the ratio $R$ of the vector ac Stark effect (or light shift) in the $6S_{1∕2}$ and $5D_{3∕2}$ states of a single trapped barium ion to 0.2% accuracy at two different off-resonant wavelengths. We earlier found $R={\Delta }_S∕{\Delta }_D=-11.494\left(13\right)$ at $514.531\mathrm{nm}$ where ${\Delta }_{S,D}$ are the vector light shifts of the $6S_{1∕2}$ and $5D_{3∕2}$ $m=\pm 1∕2$ splittings due to circularly polarized light, and now we report the value at $1111.68\mathrm{nm}$, $R=+0.4176\left(8\right)$. These observations together yield a value of the $⟨5D\Vert er\Vert 4F⟩$ matrix element. Also, comparison of our results with an ab initio calculation of dynamic polarizability would yield a test of atomic theory and improve the understanding of atomic structure needed to interpret a proposed atomic parity violation experiment.

Figure 1

(Color online) This level diagram of ${\mathrm{Ba}}^{+}$ shows several relevant atomic states, transition wavelengths, lifetimes, and approximate decay branching ratios. The Zeeman structure of the $6S_{1∕2}$ and $5D_{3∕2}$ states is shown explicitly; an external magnetic field makes ${\omega }_S$ and ${\omega }_D$ a few MHz.

Figure 2

Structure of the multipole light shifts in the $6S_{1∕2}$ and $5D_{3∕2}$ states. In this schematic, we imagine a magnetic field splitting the $6S_{1∕2}$ and $5D_{3∕2}$ sublevels by ${\omega }_S$ and ${\omega }_D$ in the vector structure of the Zeeman interaction. Higher order terms in the ac Stark perturbation (assumed small) can mix the scalar, vector, and tensorlike shifts.

Figure 3

(Color online) (top) Estimated total light shifts ${\Delta }_S$ and ${\Delta }_D$ assuming a $20\mu \mathrm{m}$, circularly polarized laser beam. (middle) The fractional sizes and signs of contributions to the vector light shifts ${\Delta }_S$ and ${\Delta }_D$ due to various nearby states, shown on linear and (bottom) logarithmic scale plots. Notice, for instance, that in the visible spectrum, the dominant contribution to the $6S_{1∕2}$ vector shift ${\Delta }_S$ is due to the large and oppositely signed contributions of $6P_{1∕2}$ and $6P_{3∕2}$.

Figure 4

(Color online) Single ion spin resonances in the $6S_{1∕2}$ state as functions of frequency (top) and exposure time (bottom). See text for a description of the spin sensitive electron shelving technique. We found that the low contrast of the spin resonances (due to an unfavorable branching ratio of decays from $6P_{1∕2}$) could be improved by exposing the ion to several repeated rf spin flips and optical probe pulses as shown here.

Figure 5

(Color online) A rate equation model of optical pumping allows us to estimate the steady state population $a_{-1∕2}$ of the $5D_{3∕2}$, $m=-1∕2$ state when linearly polarized $493\mathrm{nm}$ and ${\sigma }^{-}$ polarized $650\mathrm{nm}$ light is incident on the single ion. We plot contours of $a_{-1∕2}$, an important parameter for a systematic line-shape effect, against the misalignment angle of the $650\mathrm{nm}$ beam to the magnetic field $\theta$, and the strength of circular polarization $\mid \sigma \mid$. Notice that $a_{-1∕2}$ is maximized with imperfect polarization.

Figure 6

(Color online) These data demonstrate theoretically (top) and experimentally (bottom) distortions in light shifted $5D_{3∕2}$ spin resonance curves resulting from using the same spin-flip exposure time for large and small light shifted resonances. At large light shifts ${\Delta }_D⪢{\Omega }_D$, the $5D_{3∕2}$, $m=\pm 1∕2$ levels are isolated into an effective two level system with a different Rabi frequency than the unshifted $J=3∕2$ system [18, 20]. At small light shifts $\Delta \sim {\Omega }_D$, the tensor light shift makes all the $5D_{3∕2}$ splittings quasiresonant, leading to complicated line shapes.

Figure 7

(Color online) In this example spin resonance data set, we find the Zeeman splittings ${\omega }_S$ and ${\omega }_D$ with and without application of an off-resonance light shift laser. Though coherent features are observed on the spectral lines, in practice we fit the resonance profiles to Gaussian line shapes since temporal magnetic drift and light shift laser intensity, pointing, and polarization noise tends to broaden the lines.

Figure 8

Light shift ratio data at 514 and $1111\mathrm{nm}$ plotted against the most important parameter for systematic effects, ${\Delta }_D∕{\Omega }_D$ (the light shift magnitude over the $5D_{3∕2}$ resonance linewidth). Data in the shaded regions are cut from the straight line fit because the size of the line-shape fitting error is expected to be $>{10}^{-3}$ while the sign is uncontrolled; this explains the large scatter at low ${\Delta }_D∕{\Omega }_D$. In each case, this data cut does not change the fitted values for $R$ beyond the quoted errors. No statistically significant slopes are derived from these plots which might otherwise imply an effect due to hyperpolarizability. To account for the larger than expected scatter in the $1111\mathrm{nm}$ data, we have ${\chi }^2$-corrected the reported value.

Figure 9

(Color online) Models of the $5D_{3∕2}$ spin resonance line-fitting error. The top graph compares the typical error magnitude for 514 and $1111\mathrm{nm}$ light shifts considered here. The bottom graph shows the error profiles for some other candidate light shift wavelengths. For each wavelength, the top curve assumes initial $5D_{3∕2}$ optical pumping fraction $a_{-1∕2}=0.8$, the maximum typically observed while the bottom curve assumes $a_{-1∕2}=0.51$ which minimizes line-shape error. Dashesd lines illustrate approximate bounds on ${\Omega }_D∕{\Delta }_D$ that keep the line fitting relative error below 0.1%.

Figure 10

(Color online) An estimate of the light shift ratio slope $\mid R^{\prime }\mid$ with respect to wavelength gives us an idea of how sensitive the measured light shift ratio will be to fluctuations in the laser frequency. The slope diverges near atomic dipole resonances. In regions with an expected relative error $\mid R^{\prime }\mid <0.1{\mathrm{nm}}^{-1}$, a laser with free running stability of $\Delta \lambda =0.01\mathrm{nm}$ contributes a relative systematic error to the light shift ratio of ${10}^{-3}$. The region with low $\mid R^{\prime }\mid$ around $470\mathrm{nm}$ is in-between the $6S_{1∕2}\leftrightarrow 6P_{1∕2}$ and $6S_{1∕2}\leftrightarrow 6P_{3∕2}$ transitions.

Figure 11

Data showing the measured $g$-factor ratio deviation with trap rf strength. An electrical model shows it is plausible for this shift to be due to the off-resonant ac Zeeman effect. In [20] we argue that the net effect on the measurements of $R$ reported here are at the $5\times {10}^{-4}$ level—far below the statistical sensitivity.