Observation of quantum interference of optical transition pathways in Doppler-free two-photon spectroscopy and implications for precision measurements

Doppler-free two-photon (DFTP) spectroscopy is a standard technique for precision measurement of transition frequencies of dipole-forbidden transitions. Here, we report the observation of quantum interference (QI) of optical transition pathways in DFTP spectroscopy of the cesium $6S\text{−}7D$ transitions chosen as a prototype system. The QI manifests itself as asymmetric line shapes of the hyperfine lines of the $7D$ states, observed through spontaneous emission following excitation by a narrow-linewidth cw laser. The interference persists despite the lines being spectrally well resolved. Ignoring the effect and fitting the spectrum to a Voigt profile causes large systematic shifts in the determination of the line centers, while accounting for QI resolves the apparent line shift and enables the precise determination of hyperfine splitting in the $7D$ states. We calculate the spectral line shape including the effect of QI and show that it agrees with the experimental observations. Our results are broadly applicable to other species and have implications for portable secondary optical clocks and precision measurements of hyperfine splittings, isotope shifts, and transition frequencies.

Figure 1

Schematic representation of the experiment. (a) Partial energy level diagram (not to scale) of ${}^{133}\mathrm{Cs}$. Numbers in italics are the HFS in MHz. Two-photon excitation of the $7d_{3/2}$ state is depicted with straight lines and the relevant spontaneous decay channels are depicted by wavy lines. (b) Schematic representation of interfering quantum pathways, analogous to a multislit interference experiment. (c) Sketch of the experimental setup. The laser beam with wave vector ${\vec{k}}_L$ propagates along the $\widehat{y}$ direction, is linearly polarized along ${\widehat{\epsilon }}_L$ in the $\widehat{x}-\widehat{z}$ plane, and makes an angle $\theta$ with the $\widehat{z}$ axis. The laser beam is then retroreflected. The detection is along the $\widehat{z}$ direction. The spectral bandwidth (10 nm) of the bandpass filter is orders of magnitude larger than the $7d_{\mathrm{J}}$ HFS, and cannot be used to distinguish between the decay channels; thus obscuring the knowledge of the intermediate state $|e\rangle$ and enabling the QI to survive.

Figure 2

Spectroscopy of the $7d_{3/2}$ state. (a),(b) Spectra for the $6s_{1/2}(F=3)\to 7d_{3/2}(F^{\prime }=2,3,4,5)$ and the $6s_{1/2}(F=4)\to 7d_{3/2}(F^{\prime }=2,3,4,5)$ transitions, respectively, at three different values of $\theta$ viz. $0^{\circ }\pm 2^{\circ }, {53}^{\circ }\pm 2^{\circ }$, and ${90}^{\circ }\pm 2^{\circ }$. Red lines are the measured data and the superimposed white lines are the fits to a combination of four independent Voigt profiles, each of which is symmetric around the line center. The residuals (blue dots) show the difference between the data points and the fitted function. For $\theta =0^{\circ }$, the asymmetry in the residuals on either side of the line center is most clearly visible in the $6s_{1/2}(F=3)\to 7d_{3/2}(F^{\prime }=5)$ and the $6s_{1/2}(F=4)\to 7d_{3/2}(F^{\prime }=2)$ lines. The asymmetry is reduced to the noise level of the experiment for $\theta ={53}^{\circ }$ and reappears for $\theta ={90}^{\circ }$. (c) The $7d_{3/2}$ state HFS ${\mathrm{\Delta }}_{23}, {\mathrm{\Delta }}_{34}$, and ${\mathrm{\Delta }}_{45}$ obtained from the Voigt fits depend on $\theta$ due to QI. Blue squares: excitation from the $6s_{1/2}(F=3)$ state, red circles: excitation from the $6s_{1/2}(F=4)$ state. The HFS for the two cases agree when $\theta \sim 54.7^{\circ }$ (and equivalently also at $\theta \sim 125.3^{\circ }$) where the QI effect vanishes.

Figure 3

Spectroscopy of the $7d_{5/2}$ state. The colour code and notations follow the conventions set in Fig. 2. (a),(b) Representative spectra at $\theta =0^{\circ }$ for the $6s_{1/2}(F=3)\to 7d_{5/2}(F^{\prime }=1,2,3,4,5)$ and the $6s_{1/2}(F=4)\to 7d_{5/2}(F^{\prime }=2,3,4,5,6)$ transitions. The asymmetry in the residuals is most clearly visible in the $6s_{1/2}(F=3)\to 7d_{5/2}(F^{\prime }=5)$ and the $6s_{1/2}(F=4)\to 7d_{5/2}(F^{\prime }=6)$ lines. (c) The $7d_{5/2}$ state HFS ${\mathrm{\Delta }}_{12}, {\mathrm{\Delta }}_{23}, {\mathrm{\Delta }}_{34}, {\mathrm{\Delta }}_{45}$, and ${\mathrm{\Delta }}_{56}$ obtained from the Voigt fits depend on $\theta$.

Figure 4

Simulated spectra of $7d_{3/2}$ state. (a),(b) The $6s_{1/2}(F=3)\to 7d_{3/2}(F^{\prime }=2-5)$ and the $6s_{1/2}(F=4)\to 7d_{3/2}(F^{\prime }=2-5)$ transitions, respectively. The red lines are the simulated data and the superimposed white lines are the fits to a combination of four independent Voigt profiles. The residuals (blue line) are asymmetric on either side of the line center except when $\theta ={53}^{\circ }$, where the residuals tend to zero. The residuals change sign at $\theta \sim 54.7^{\circ }$. (c) The $7d_{3/2}$ state HFSs obtained from the Voigt fits vary with $\theta$ due to QI. The HFS when exciting from the $6s_{1/2}(F=3)$ state (blue squares) and the $6s_{1/2}(F=4)$ state (red circles) agree at $\theta \sim 54.7^{\circ }$ (and equivalently also at $\theta \sim 125.3^{\circ }$) where the QI vanishes.

Figure 5

Simulated spectra of $7d_{5/2}$ state. (a),(b) The $6s_{1/2}(F=3)\to 7d_{5/2}(F^{\prime }=1-5)$ and the $6s_{1/2}(F=4)\to 7d_{5/2}(F^{\prime }=2-6)$ transitions, respectively. (c) The $7d_{5/2}$ state HFS obtained from the Voigt fits.

Figure 6

Schematic diagram of the experimental setup. AOM: acousto-optic modulator; PMT: photomultiplier tube; PD: photodiode; APD: avalanche PD; BS: beam sampler; PBS: polarizing beam splitter; BP: bandpass; SP: short pass; VCA: voltage controller attenuator.

Figure 7

Excitation and detection geometry.