Quantum interference in two-photon frequency-comb spectroscopy

Quantum interference arising from spontaneous emission, or cross-damping, is an important yet frequently overlooked systematic in precision spectroscopy experiments which aim to determine a transition frequency with an uncertainty smaller than the natural linewidth. Here, we calculate the effects of such interference in two-photon frequency-comb spectroscopy using a perturbative approach and by integration of the density matrix equations. We then apply these techniques to the two-photon spectroscopy of the hydrogen $1S-3S$ transition currently being performed in our group. Depending on the detection geometry, we find distortions of the line shapes which can lead to systematic errors of $\sim$1 kHz if such interference effects are ignored in the data analysis. This result is independent of whether a cw laser or frequency comb is used for the excitation. Finally, we propose a time-dependent detection scheme which, when used in conjunction with frequency-comb excitation, can mitigate the line distortions arising from such interference.

Figure 1

We consider two-photon excitation ($2\times \omega$) from initial states $|i\rangle$ through intermediate states $|p\rangle$ (far-off resonance) to excited levels $|e\rangle$. Excitation is detected through radiative decay at frequency ${\omega }_s$ to the final states $|f\rangle$. In principle, the manifold $|f\rangle$ may also decay back to the initial states, which is important to consider if optical pumping or saturation of the transition occurs.

Figure 2

Atomic trajectory passing through a Gaussian laser beam. We assume a linear trajectory which gives an atomic position of the form $\vec{r}\left(t\right)=[(x_0+v_{x}t),(y_0+v_{y}t),(z_0+v_{z}t)]$. We also assume the laser beam has a Rayleigh range much larger than the beam waist so that its $x$ dependence is negligible. A Doppler-free signal is obtained only from the pulse collision volume which is defined by the size of the beam waists in the transverse dimensions ($w_y$ and $w_z$) and the length of the transform-limited, colliding pulses along the $x$ axis ($\tau c$).

Figure 3

Relevant hydrogen levels with all hyperfine levels shown. The $m_F$ levels of the same $F$ are shown as horizontally displaced. We model the excitation of the allowed two-photon $1S-3S$ transitions ($F=0$ to $F=0$ and $F=1$ to $F=1$) using counterpropagating beams of $\pi$ polarization. To compare with Fig. 1, all $3S$ and $3D$ levels comprise the $|e\rangle$ manifold, the $2P$ levels comprise the $|f\rangle$ manifold, and the $1S$ levels comprise the $|i\rangle$ manifold. The $|p\rangle$ manifold (not shown) is all levels that are dipole allowed from the $1S$ level.

Figure 4

(a) Calculated fluorescence signal from the 3$S$ levels using Eq. (22) as the offset frequency ${\omega }_0$ is scanned with $t_{\delta }$= 2 $\mu$s and a repetition rate of ${\omega }_r=2\pi \times$ 348.16 MHz. The repetition rate is chosen so that only the 3$S$ levels are resonantly excited within the scan range shown. Since we show frequency-comb excitation, the $F=0$ and $F=1$ transitions appear closely spaced but are actually excited by separate comb modes. (b) Difference in the line shape between the full model and when only terms where $e=e^{\prime }$ are considered in Eq. (22) [note that the units in (b) are arbitrary, but the same as the units in (a)]. The errors depend on the detected dipole spherical components and average to zero if they are all detected with equal sensitivity. The ${\sigma }_{+}$ and ${\sigma }_{-}$ components are identical and are shown as the dashed line.

Figure 5

The line-center shift of the $1S$ $F=1$ to $3S$ $F=1$ transition which results from fitting the line profiles distorted by quantum interference with Lorentzian functions. We assume a small-area detector which detects a combination of the $\pi$ and ${\sigma }_{\pm }$ dipole spherical components with a laser polarization of $\mu =\nu =0$ (both beams linearly polarized along the $z$ axis). In this case, there is cylindrical symmetry, so we plot the result only as a function of the azimuthal angle $\theta$ (with respect to the $z$ axis). The solid and dashed lines represent the frequency-comb and cw excitation cases, respectively. Note that the shifts shown correspond to a $\sim$${10}^{-13}$ relative frequency error for the spectroscopic measurement.

Figure 6

Similar to Fig. 4 with only the $1S-3S$ transitions resonantly excited within the scan range shown. The transit time broadening is no longer negligible ($t_{\delta }$ = 200 ns), and therefore, the residual errors in (b) broaden and are proportionally larger than in Fig. 4.

Figure 7

Difference between the calculated line shapes when integrating the complete set of density matrix equations and when we set ${\gamma }_{ij}=0$ if $i\ne j$ (equivalent to ignoring the cross-damping). The dashed curves are reproduced from the perturbative calculations shown in Figs. 4 and 6 so that the two methods can be compared. In (a) $t_{\delta }=2$ $\mu$s, and in (b) $t_{\delta }=200$ ns.

Figure 8

Illustration of the interference between the time-dependent dipoles [$2\text{Re}\left\{d_S\left(t\right)d_D^{*}\left(t\right)\right\}$]. The rising and falling intensity of the pulses is because the atom has nonzero velocity and travels across a Gaussian laser beam. The additional signal which arises from this interference is proportional to the time integral of this interference [$2\int \text{Re}\left\{d_S\left(t\right)d_D^{*}\left(t\right)\right\}dt$]. This integral only accumulates when the pulse is at the position of the atom, and therefore, when applying an appropriate window function, $w\left(t\right)$, to the detected signal, the integral [$2\int w\left(t\right)\text{Re}\left\{d_S\left(t\right)d_D^{*}\left(t\right)\right\}dt$] is greatly reduced, mitigating the distortion.

Figure 9

Line center shift when the $1S$ $F=1$ to $3S$ $F=1$ transition is excited and detection of fluorescence is made through radiative decay to the $2P$ states after excitation by a 348.16-MHz repetition rate frequency comb with 1-ps pulses. A Gaussian window function is used to gate the detection between the pulses. (a) Line centers obtained from fitting a Lorentzian as a function of the timing of a Gaussian window function with a width of 500 ps. (b) Same as (a), but as a function of the width of the Gaussian window function.