Quantum interference and light polarization effects in unresolvable atomic lines: Application to a precise measurement of the ${}^{6,7}$Li $D_2$ lines

We characterize the effect of quantum interference on the line shapes and measured line positions in atomic spectra. These effects, which occur when the excited-state splittings are of order of the natural line widths, represent an overlooked but significant systematic effect. We show that excited-state interference gives rise to non-Lorentzian line shapes that depend on excitation polarization, and we present expressions for the corrected line shapes. We present spectra of ${}^{6,7}$Li $D$ lines taken at multiple excitation laser polarizations and show that failure to account for interference changes the inferred line strengths and shifts the line centers by as much as 1 MHz. Using the correct line shape, we determine absolute optical transition frequencies with an uncertainty of $\le$25 kHz and provide an improved determination of the difference in mean-square nuclear charge radii between ${}^6$Li and ${}^7$Li. This analysis should be important for a number of high-resolution spectral measurements that include partially resolvable atomic lines.

Figure 1

Coordinate system: The excitation laser propagates along $\widehat{x}$ so that the linear polarization direction ${\widehat{\epsilon }}_{\mathrm{L}}$ lies in the $\widehat{y}$-$\widehat{z}$ plane, parameterized by ${\theta }_{\mathrm{L}}$. The detection direction $k_s$ lies in the $\widehat{x}$-$\widehat{z}$ plane and is parameterized by ${\theta }_{\mathrm{s}}$. In our apparatus, light collection is centered along $\widehat{z}$ with an angular spread determined by the numerical aperture of the imaging system. The atomic beam is along $\widehat{y}$.

Figure 2

The scattering rate (or intensity), $\frac{dR}{d\Omega }$ in arbitrary units, of the $F=2\to F^{\prime }=1,2,3$ Doppler-free feature in ${}^7$Li with ${\theta }_{\mathrm{L}}=0$. Red line shows the sum of Lorentzians with polarization independent weights, blue dashed line shows the sum of cross terms, black line shows the sum of Lorentzians and cross terms. The laser frequency ${\omega }_{\mathrm{L}}/\left(2\pi \right)$ is the offset from the $F=2\to F^{\prime }=2$ peak in units of MHz. The inset is the $F=2\to F^{\prime }=2$ peak enlarged to show the shift in the line center.

Figure 3

Error in the measured hyperfine splitting (in units of $\Gamma$) for a $F=1/2\to F^{\prime }=1/2,3/2$ transition as a function of the assumed separation when the theoretically calculated full line shape is fit as the sum of two Lorentzians. The dash dotted (solid) curve is for laser polarization ${\theta }_{\mathrm{L}}=\pi /2$ (0). The error in the hyperfine splitting is greatest where the assumed separation is about 1.3 times the natural line width. The red vertical line indicates the $x$ position of the actual hyperfine splitting for ${}^6$Li. Its vertical extent shows the range of errors that can occur when the laser polarization is varied between 0 and $\pi /2$.

Figure 4

Error in the measured center of gravity (in units of $\Gamma$) for a $F=2\to F^{\prime }=1,2,3$ transition as a function of the hyperfine $A$ constant when the theoretically calculated full line shape is fit as a sum of three Lorentzians. The dash dotted (solid) curve is for laser polarization ${\theta }_{\mathrm{L}}=\pi /2$ (0). The red vertical line indicates the $x$ position of the actual $A$ constant for ${}^7\mathrm{Li}$. Its vertical extent shows the range of errors that can occur when the laser polarization is varied between 0 and $\pi /2$.

Figure 5

Relevant ${}^{6,7}$Li level structure. The hyperfine components for the $D_2$ transition have natural widths of order of the hyperfine splitting. “IS” stands for isotope shift.

Figure 6

Simplified schematic of experimental apparatus. The interaction region is surrounded by three layers of mu-metal (not shown) to minimize the magnetic field. The coordinate system shown is consistent with Fig. 1.

Figure 7

Amplitude of scattered light, proportional to Eq. (7), as a function of laser frequency ${\omega }_{\mathrm{L}}$ and laser polarization angle ${\theta }_{\mathrm{L}}$. The laser frequency is offset from the ${}^6$Li $F=1/2$ ground state by 446 THz (see Table  for optical frequencies). The gray-scale surface is the complete theoretical line shape including cross terms and the red points are experimental data taken at ${\theta }_{\mathrm{L}}=0^{\circ },{25}^{\circ },{51}^{\circ },{75}^{\circ },\mathrm{and}{90}^{\circ }$. The central feature is the $F=1/2\to F^{\prime }=1/2,3/2$ transitions in ${}^6$Li while the two constant amplitude side peaks are the $F=1\to F^{\prime }=1,2$ $D_1$ lines of ${}^7$Li.

Figure 8

Line center of ${}^7$Li $F=1\to F^{\prime }=0,1,2$ feature fit from experimentally measured spectra as a function of laser polarization angle with respect to the collection direction. Transition frequencies are offset from the ${}^7$Li $F=1$ ground state by 446 THz (see Table  for optical frequencies). The black (red) data points were extracted by fitting the data to functions with (without) interference cross terms. Error bars represent the uncertainties given in Table .

Figure 9

${}^7$Li ground-state hyperfine interval ($F=1\to F=2$) as function of laser polarization angle. The measured GHI was determined by subtracting absolute measurements of the excited state $F=1\to F^{\prime }=0,1,2$ and $F=2\to F^{\prime }=1,2,3$ features including Doppler and intensity-dependent corrections (see the text). The red (black, blue) data are for an angular offset of $-3.7^{\circ }$ ($-0.7^{\circ }$, $2.3^{\circ }$). The red (black, blue) curve is of the form $A_{{\theta }_0}\sin \left(\theta \right)\cos \left(\theta \right)+{\mathrm{GHI}}_0$, where ${\mathrm{GHI}}_0$ is the value measured in Ref. [46] and $A_{{\theta }_0}$ is fit to the data. The triangular point is data from Ref. [6] reanalyzed using the procedure described in the present work. Error bars represent the uncertainties given in Table .

Figure 10

${}^6$Li ground-state hyperfine interval ($F=1/2\to F=3/2$) as a function of laser polarization angle. The measured GHI was determined by subtracting absolute measurements of the excited state $F=1/2\to F^{\prime }=1/2,3/2$ and $F=3/2\to F^{\prime }=1/2,3/2,5/2$ features. The solid line is the value measured in Ref. [46]. Error bars represent the uncertainties given in Table .

Figure 11

Measurements of difference in mean-square charge radius between ${}^7$Li and ${}^6$Li. The points are grouped by type of measurement and are then ordered chronologically within different types of measurement. The solid black line is the weighted average of the results of Refs. [36, 52, 53], along with the $D_1$ value from [6] and the $D_2$ value from this work. Error bars for the present work represent the uncertainties given in Table ; all other error bars represent the uncertainties given in the original references.