Open Access
Add to BibSonomy
Abstract
A non-linear spectroscopic study of the HDO molecule is performed in the wavelength range of 1.36–1.42 $\mathrm{\mu}$m using noise-immune cavity-enhanced optical-heterodyne molecular spectroscopy (NICE-OHMS). More than 100 rovibrational Lamb dips are recorded, with an experimental precision of 2–20 kHz, related to the first overtone of the O–H stretch fundamental of HD16O and HD18O. Significant perturbations, including distortions, shifts, and splittings, have been observed for a number of Lamb dips. These spectral perturbations are traced back to an AC-Stark effect, arising due to the strong laser field applied in all saturation-spectroscopy experiments. The AC-Stark effect mixes parity pairs, that is pairs of rovibrational states whose assignment differs solely in the Kc quantum number, where Kc is part of the standard $J_{K_a,K_c}$ asymmetric-top rotational label. Parity-pair mixing seems to be especially large for parity pairs with Ka ≥ 3, whereby their energy splittings become as small as a few MHz, resulting in multi-component asymmetric Lamb-dip profiles of gradually increasing complexity. These complex profiles often include crossover resonances. This effect is well known in saturation spectroscopy, but has not been reported in combination with parity-pair mixing. Parity-pair mixing is not seen in H$_2^{~16}$O and H$_2^{~18}$O, because their parity pairs correspond to ortho and para nuclear-spin isomers, whose interaction is prohibited. Despite the frequency shifts observed for HD16O and HD18O, the absolute accuracy of the detected transitions still exceeds that achievable by Doppler-limited techniques.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Most studies dealing with gas-phase rovibrational spectroscopy of small molecules utilize direct linear absorption and Fourier-transform spectroscopy [1]. In these measurements, the spectral resolution is severely limited by Doppler broadening. This situation can be improved significantly by employing saturation techniques of nonlinear spectroscopy [2]. The application of saturation techniques has the advantage that the linewidth, typically several hundred MHz under Doppler-broadened conditions, reduces to the sub-MHz regime. Saturation techniques combined with frequency-comb-based calibration, like noise-immune cavity-enhanced optical-heterodyne molecular spectroscopy (NICE-OHMS, see Refs. 3 and 4), can reach a spectral uncertainty below 10 kHz [5–15]. This accuracy represents an improvement over 2–3 orders of magnitude in the determination of line centers.
Based on the successful application of our NICE-OHMS apparatus [16,17] to study the rovibrational energy-level structure of two permutationally symmetric water isotopologues, H$_2^{~16}$O [9,13] and H$_2^{~18}$O [10], the principal objective of the present paper has been to conduct NICE-OHMS measurements on HD$^{16}$O. This investigation was designed to obtain accurate pure rotational energies for this permutationally nonsymmetric water isotopologue. Nevertheless, there is an important, symmetry-related difference between the spectroscopy of H$_2$O and HDO. Since H$_2$O isotopologues contain two equivalent H nuclei, these species have two nuclear-spin isomers, called ortho and para, and the interaction between the ortho and para states is symmetry forbidden. In contrast, HDO isotopologues have a single nuclear-spin isomer, allowing the interaction of close-lying rovibrational states of opposite parity, which may become mixed under the perturbing influence of external electric fields. The extent of such interactions under conditions of saturation spectroscopy has not been explored so far and thus its magnitude has remained unknown.
Our experimental campaign yielded more than 100 Lamb-dip features, involving states in the first overtone of the O–H stretch fundamental of HD$^{16}$O, with an experimental precision of 2–20 kHz. For transitions whose upper and/or lower states have $K_a>3$, multi-component Lamb-dip profiles have been identified ($K_a$ is part of the standard $J_{K_a,K_c}$ asymmetric-top rotational label [18]). Even for single Lamb dips, frequency shifts on the order of 100 kHz were detected through a comparison of the experimental (NICE-OHMS-based) energy splittings with their counterparts derived from an effective Hamiltonian (EH) model [19]. The complex Lamb-dip profiles and the frequency shifts are explained with mixing between nearby parity pairs, that is pairs of opposite-parity rovibrational states differing only in their $K_c$ values. To prove this interpretation, benchmark experiments have been performed for HD$^{18}$O, H$_2^{~16}$O, and H$_2^{~18}$O, as well, confirming that for transitions with the same lower and upper states (a) no spectral distortions are discernible in the Lamb-dip profiles of H$_2^{~16}$O and H$_2^{~18}$O, and (b) HD$^{18}$O exhibits similar complex Lamb-dip profiles as HD$^{16}$O.
2. Methodological details
2.1 NICE-OHMS measurements
A NICE-OHMS apparatus, shown schematically in Fig. 1 and described in Refs. 9 and 16, is used to record selected absorption lines of HD$^{16}$O and other water isotopologues, under saturation conditions. NICE-OHMS is a special form of cavity-enhanced absorption spectroscopy (CEAS, see Ref. 20), endowed with extreme sensitivity [3,4,21,22]. In our setup, a double modulation scheme is included via an electro-optic modulator (EOM) for imposing sideband modulations to the central carrier frequency. A low-frequency modulation at 20 MHz is employed for Pound–Drever–Hall (PDH) locking of the laser to the optical cavity with a finesse of 150 000 and a length of 51 cm. Higher-frequency sidebands, generated at 305 MHz and matched to the free spectral range of the cavity, are used to produce the NICE-OHMS signal [4]. For further noise reduction, a low-frequency (405 Hz) wavelength dither is applied on one of the cavity mirrors. For the demodulation steps and registration of the signal, an advanced high-frequency lock-in detector (Zurich Instruments, HF2LI) is utilized.
Fig. 1. Schematic layout of the NICE-OHMS setup employed during this study. NICE-OHMS = noise-immune cavity-enhanced optical-heterodyne-molecular spectroscopy, OFC = optical frequency comb, EOM = electro-optic modulator, FSR = free spectral range of the cavity, and PDH = Pound–Drever–Hall locking.
The frequency of the external-cavity diode laser (ECDL) is calibrated via a fiber link to an in-house metrology station, consisting of a frequency-comb laser, a Cs clock, and a device receiving signals from the global positioning system (GPS). Through this metrology station, long-term stabilization is ensured for the beatnote of the cavity-locked laser by a slow feedback on the cavity piezo for controlled acquisition and frequency scans. This construction allows the achievement of a frequency precision below 1 kHz.
Since the generic signal of NICE-OHMS is dispersive, the additional wavelength dither on the cavity produces a symmetric line shape, resembling the second derivative of a Voigt profile if $1f$ demodulation is employed. For the present measurement campaign, the relative absorption of the Lamb-dip features is typically on the order of $10^{-11}$ cm$^{-1}$, while the resonance widths are approximately 400–800 kHz (full width at half maximum, FWHM). In this study, intracavity powers of 5–100 W are utilized. Low powers are typically used to probe strong transitions, for which the Lamb-dips may suffer from power broadening. In a previous study [13], it was observed that strong lines eventually exhibit doubled Lamb-dip features at high power. This phenomenon, specifically associated with the NICE-OHMS detection technique, is avoided here to not confuse the new features observed in this study with doubled Lamb dips.
The wavenumber coverage of our NICE-OHMS spectrometer in its current setup is 7000–7350 cm$^{-1}$, limited by the tuning range of the ECDL, the highly-reflective cavity mirrors, as well as the transmission properties of certain electro-optical components. The accessible window of dipole moments, represented by the Einstein-$A$ coefficients, falls in the interval of $10^{-4}$–$10^2$ s$^{-1}$. Combined with the population distribution at the measurement temperature ($T=293$ K), the allowed range of intensities corresponds to $10^{-26}$–$10^{-20}$ cm molecule$^{-1}$.
2.2 Notation
In this study, rovibrational states of water isotopologues are labelled as $(v_1\,v_2\,v_3)J_{K_a,K_c}$, where $(v_1\,v_2\,v_3)$ is the vibrational assignment based on the normal-mode picture, while $J_{K_a,K_c}$ stands for the standard asymmetric-top rotational quantum numbers [18]. The $v_1$, $v_2$, and $v_3$ quantum numbers correspond, respectively, to (a) the symmetric stretch, bend, and antisymmetric stretch normal modes for H$_2^{~16}$O and H$_2^{~18}$O, and (b) the O–D stretch, bend, and O–H stretch fundamentals for HD$^{16}$O and HD$^{18}$O. A rovibrational transition is denoted as $(v'_1\,v'_2\,v'_3)J'_{K'_a,K'_c} \leftarrow (v''_1\,v''_2\,v''_3)J''_{K''_a,K''_c}$, where $'$ and $''$ distinguish between the upper and lower states, respectively.
The parity of a $(v_1\,v_2\,v_3)J_{K_a,K_c}$ state is even/odd (e/o or $+/-$) if $K_c$ is even/odd. As the states $(v_1\,v_2\,v_3)J_{K_a,\dot {K}_c}$ and $(v_1\,v_2\,v_3)J_{K_a,\ddot {K}_c}$ with $\left | \dot {K}_c-\ddot {K}_c \right |=1$ practically differ only in their parity, they form what we call a parity pair. This parity pair is denoted as $(v_1\,v_2\,v_3)J_{K_a,\dot {K}_c/\ddot {K}_c}$ if $\dot {K}_c$ belongs to the higher-energy state. The energy difference of a parity pair is hereinafter called $K_c$ splitting. Clearly, the parity pairs of H$_2$O isotopologues are ortho–para nuclear-spin-isomer pairs. If $U_1$/$U_2$ and $L_1$/$L_2$ are parity pairs, then the transition doublet $(U_1 \leftarrow L_1, U_2 \leftarrow L_2)$ is named a parity doublet. By convention, $(v'_1\,v'_2\,v'_3)J'_{K'_a,\dot {K}'_c/\ddot {K}'_c} \leftarrow (v''_1\,v''_2\,v''_3)J''_{K''_a,\dot {K}''_c/\ddot {K}''_c}$ indicates a parity doublet whose higher- and lower-frequency lines are assigned as $(v'_1\,v'_2\,v'_3)J'_{K'_a,\dot {K}'_c} \leftarrow (v''_1\,v''_2\,v''_3)J''_{K''_a,\dot {K}''_c}$ and $(v'_1\,v'_2\,v'_3)J'_{K'_a,\ddot {K}'_c} \leftarrow (v''_1\,v''_2\,v''_3)J''_{K''_a,\ddot {K}''_c}$, respectively.
2.3 Effective Hamiltonian and variational computations
There are several investigations [19,23–28] which have helped to determine the empirical energy-level structure of HD$^{16}$O in its ground vibrational state, $(v_1\,v_2\,v_3)= (0\,0\,0)$. These studies involved more than 600 pure rotational Lamb-dip lines, often with resolved hyperfine structure. Based on these transitions, a global fit was performed in Ref. 19, yielding effective-Hamiltonian (EH) energies for the $(0\,0\,0)$ rotational states with an accuracy of $10^{-8}$–$10^{-6}$ cm$^{-1}$.
The empirical energies available for the $(0\,0\,2)$ rotational states are considerably less accurate, they are on the order of $10^{-3}$ cm$^{-1}$, as typically obtained from Doppler-limited rovibrational spectroscopy [28]. As shown in Refs. 9,10 and 13, small ($<0.01$ cm$^{-1}$) energy differences can be estimated much more accurately than $10^{-3}$ cm$^{-1}$ via first-principles nuclear-motion computations, provided that the underlying energy levels share the same vibrational state and $J$ value. Therefore, such computations have been carried out, up to $J = 8$, for HD$^{16}$O and HD$^{18}$O with the GENIUSH (GEneral rovibrational code with Numerical, Internal-coordinate, User-Specified Hamiltonians) code [29–31], using the potential energy surface of Ref. 32 and a sufficiently large vibrational basis set (as usual, the rotational basis is complete).
The computed $K_c$ splittings of HD$^{16}$O and HD$^{18}$O, as well as the EH-based estimates of Ref. 19, are given fully in Dataset 1 [33] and partly in Fig. 2. As apparent from panels (a) and (b) of Fig. 2, the variational $K_c$ splittings of HD$^{16}$O follow a monotonically decreasing trend as a function of $K_a$, for each $J$ value, in the $(0\,0\,0)$ and $(0\,0\,2)$ vibrational states, respectively. It must be noted that the computed $K_c$ splittings are in a remarkable agreement with their EH-based analogues in the $(0\,0\,0)$ state: their deviations are on the order of 0.1–5 % relative to the actual $K_c$ splitting values. In absolute terms, the discrepancies are well within 1 MHz when both $J$ and $K_a$ are larger than 2, showing the high absolute accuracy of the computed $K_c$ splittings, especially at higher $J$ and $K_a$ values.
Fig. 2. Variationally computed $K_c$ splittings of the HD$^{16}$O molecule up to $J=8$. Panels (a) and (b) correspond to the $(0\,0\,0)$ and $(0\,0\,2)$ vibrational states, respectively. The $K_c$ splittings are plotted, with various colors, for each $J$ value separately and connected with solid lines (the exceptions are the single $J=1$ points). The $K_c$ splittings are given both in cm$^{-1}$ (left vertical axes) and in Hz (right vertical axes).
3. Experimental results
3.1 Regular and perturbed Lamb-dip profiles
During the present NICE-OHMS study, Lamb dips were measured for more than 100 rovibrational transitions of HD$^{16}$O. A number of spectral lines recorded and analyzed exhibit a typical Lamb-dip profile, similar to the cases of the symmetric water isotopologues H$_2^{~16}$O [9,13] and H$_2^{~18}$O [10]. Figure 3 displays four symmetric Lamb dips of different strength for four transitions, exemplifying, at the same time, the remarkable signal-to-noise ratio obtained. In panels (b) and (d) of Fig. 3, some additional broadening is clearly visible, which is an evident proof of power broadening.
Fig. 3. Typical symmetric Lamb dips observed for a few transitions of HD$^{16}$O. The relative absorption strengths are indicated along the vertical axes. The spectra are plotted in the same detuning interval, showing significant differences in the line widths. The line centers, corresponding to zero detunings and given in kHz, are the following: 218 999 237 631 [panel (a)], 218 996 338 105 [panel (b)], 215 464 997 071 [panel (c)], and 217 318 701 228 [panel (d)].
For transitions with $\max (K'_a,K''_a)>3$, asymmetric Lamb-dip profiles have been found. To improve our understanding about the origin of this asymmetry, Lamb dips of a selected transition doublet, involving the $(0\,0\,0)4_{4,0/1}$ and $(0\,0\,2)4_{3,1/2}$ parity pairs, were recorded for four water isotopologues: HD$^{16}$O, HD$^{18}$O, H$_2^{~16}$O, and H$_2^{~18}$O. The experimental results of Fig. 4 show that asymmetry occurs only in the spectra of the two HDO species. This clearly suggests that the spectral distortion is due to the H$\rightarrow$D substitution and must be related to the different nuclear-spin symmetry of H$_2$O and HDO.
Fig. 4. Lamb dips for a well-isolated parity doublet of various water isotopologues. Note the asymmetric broadening for the HD$^{16}$O and HD$^{18}$O isotopologues, in contrast to the fully symmetric resonances of the H$_2^{~16}$O and H$_2^{~18}$O species. The line centers, corresponding to zero detunings and given in kHz, are the following: $213\,591\,493\,210$ [panel (a)], $213\,600\,053\,170$ [panel (b)], $213\,012\,275\,678$ [panel (c)], $213\,021\,144\,460$ [panel (d)], $219\,363\,999\,176$ [panel (e)], $219\,416\,311\,674$ [panel (f)], $218\,657\,009\,781$ [panel (g)], and $218\,712\,072\,101$ [panel (h)].
For higher $J$ and $K_a$ values, the Lamb-dip profiles become gradually more complex, leading to multi-component spectral structures. Figure 5 gives an example for a parity doublet, $(0\,0\,2)7_{6,1/2} \leftarrow (0\,0\,0)6_{6,0/1}$, yielding very similar multi-component features for HD$^{16}$O and HD$^{18}$O. In H$_2^{~16}$O and H$_2^{~18}$O, this doublet, consisting of strongly forbidden ortho-para transitions, is not measured. Since the variationally computed line separations are 6.07 and 6.61 MHz for HD$^{16}$O and HD$^{18}$O, respectively, the resonances at detunings of $\pm 3.0$ (HD$^{16}$O) and $\pm 3.3$ MHz (HD$^{18}$O) are the main transition centers. Our variational computations also suggest that $(0\,0\,2)7_{6,2} \leftarrow (0\,0\,0)6_{6,1}$ is the lower-frequency line for both species. While the additional resonances are not analyzed in this extreme case, a semi-quantitative approach will be presented in Sec. 4 to explain the substructures of various multi-component HD$^{16}$O lines observed during this study.
Fig. 5. Multi-component saturation spectra detected for a parity doublet of HDO. Panels (a) and (b) show the recordings for the $(0\,0\,2)7_{6,1/2} \leftarrow (0\,0\,0)6_{6,0/1}$ doublet of HD$^{18}$O and HD$^{16}$O, respectively. In both spectra, the two outermost dips represent the main transition centers. The colors of the vertical bars are matched with the rovibrational assignments. Note that these transitions are forbidden in H$_2^{~16}$O and H$_2^{~18}$O due to ortho-para selection rules. Panel (c) exhibits the spectrum observed with the wavelength-modulated CEAS (WM-CEAS) technique. The symbols $1f$ and $2f$ denote the demodulation technique employed in the measurements. The detuning scale is relative to the absolute frequencies: 217 816 033 008 kHz for panel (a) and 218 444 739 748 kHz for panels (b) and (c).
To prove that the multi-component Lamb-dip profiles of HDO are not due to the special features of the NICE-OHMS method (i.e., the frequency sidebands shifted by $\pm 305$ MHz from the carrier frequency), the saturation spectrum of the HD$^{16}$O parity doublet depicted in Fig. 5(b) was also recorded with the wavelength-modulated CEAS (WM-CEAS) technique [20]. To obtain the best resemblance with the dispersive $1f$ NICE-OHMS signal, the WM-CEAS spectrum, resulting from direct absorption, was recorded at $2f$ demodulation. The WM-CEAS technique yielded a similarly complex Lamb-dip profile, see Fig. 5(c), albeit at lower signal-to-noise ratio. From this observation, one can conclude that these perturbations can be attributed to the use of saturation spectroscopy in general, and not to the special characteristics of the NICE-OHMS method.
3.2 Assessment of line-center uncertainties
A list of selected HD$^{16}$O transitions, whose frequencies could be extracted unambiguously, are presented in Table 1. The majority of the lines listed correspond to those cases for which single, symmetric Lamb dips were measured. For a few transitions, with frequencies set in bold face in Table 1, significant Lamb-dip asymmetry was observed. It must be stressed that no individual uncertainties are reported in Table 1, see the upcoming paragraphs for a detailed reasoning.
Table 1. A list of selected transitions observed during the present study for HD$^{16}$O.a
Under ideal circumstances, the uncertainty of the HDO line centers determined during this study should be similar to those obtained in previous studies on H$_2^{~16}$O [9,13] and H$_2^{~18}$O [10]. Beyond the statistical uncertainties (2–5 kHz), there is a minor systematic effect due to frequency calibration (1 kHz). To ascertain the magnitude of pressure shifts, five HD$^{16}$O transitions were studied over a set of pressure values. This analysis yielded an estimate of 4–8 kHz for the pressure-shift uncertainties of the remaining transitions measured at a pressure of 0.1–0.2 Pa. Taking all these uncertainty factors into account, the overall precision of the Lamb-dip centers reported in Table 1 is in the range of 2–20 kHz.
As noted in Sec. 3.1, those HDO lines with $\max {(K'_a,K''_a)} > 3$ produce distorted spectra even when saturation-spectroscopy techniques other than NICE-OHMS are employed. Thus, the possible frequency shifts due to the asymmetry effects observed cannot be uncovered through a direct comparison with results of independent nonlinear spectroscopic methods. For this reason, an indirect procedure is chosen to provide at least a partial assessment for the accuracy of our NICE-OHMS lines. This approach involves the comparison of the experimental $K_c$ splittings, related to the ground vibrational state and determined by our NICE-OHMS transition dataset, with their counterparts deduced from the pure rotational EH model of Ref. 19.
Numerical values of the $K_c$ splittings, together with the frequency combinations employed, are presented in Table 2. Table 2 shows that there are a number of deviations larger than our experimental precision, 2–20 kHz, but all deviations remain smaller than 0.5 MHz. (Note that in our earlier study [9] on H$_2^{~16}$O, no systematic deviations were observed during the EH analysis of the pure rotational energy levels.) When plotting the unsigned $K_c$-splitting deviations as a function of the EH-based $K_c$ splittings (see Fig. 6), it can be observed that these deviations are the largest where the $K_c$ splittings are the smallest. Although there is no asymmetry in the Lamb dips used to derive the $K_c$ splitting of the $(0\,0\,0)3_{3,0/1}$ pair, a relatively large deviation (91 kHz) can be seen for $(J,K_a)=(3,3)$. Apparently, the centers of symmetric Lamb dips may also become shifted in our nonlinear spectroscopic experiments. Based on the fourth column of Table 2, the underlying transitions may have uncertainties on the order of 100 kHz for $\max (K'_a,K''_a) \le 3$ and even significantly higher (probably less than 1 MHz) for $\max (K'_a,K''_a) > 3$.
Fig. 6. Unsigned $K_c$-splitting deviations as a function of the individual $K_c$-splittings. The $K_c$ splittings and the unsigned deviations are taken from the third and fourth columns of Table 2, respectively. The data point pertaining to the $(0\,0\,0)J_{K_a,J-K_a/ J-K_a+1}$ parity pair is labelled as $(J,K_a)$. The horizontal green strip at 20 kHz represents the typical experimental reproducibility of the NICE-OHMS lines.
Table 2. Experimental and effective-Hamiltonian $K_c$ splittings of HD$^{16}$O up to $J=5$.a
4. Phenomenological Lamb-dip-profile analysis
As mentioned in Sec. 3.1, during this study significant perturbations have been observed for the Lamb dips of HD$^{16}$O and HD$^{18}$O lines. For a parity doublet of the two HDO species, see panels (a)–(d) of Fig. 4, highly asymmetric Lamb-dip shapes were identified, which are similar to the typical profiles resulting from an (optical) AC-Stark effect [35,36]. In this section, an attempt is made to find a semi-quantitative explanation for the complex Lamb dips of HDO.
4.1 General framework: Stark-induced parity-pair mixing and cross-overs
We postulate that the cause of the significant perturbations in the Lamb dips of HDO originate in an interaction (mixing) of parity pairs. Since intramolecular perturbations occur only between quantum states of the same parity, the interaction mechanism of parity pairs must be related to an external cause. Such cause can be found in the intense laser field within the cavity, inducing an AC-Stark effect [37,38] between the two states of a parity pair. An argument supporting our hypothesis stems from perturbation theory, underpinning that interactions become larger when the interacting states, that is the two states of a parity pair in our case, are in proximity. The fact that spectral perturbations do not occur for H$_2$O, where parity pairs correspond to non-interacting ortho-para nuclear-spin-isomer pairs, further supports our assumption on parity-pair mixing in HDO.
Figure 7(a) displays how extra resonances can occur in the presence of close-lying parity pairs. The lines with frequencies $a_1$ and $a_2$ are dipole-allowed (a-type) transitions, following dipole-selection rules. With the decrease of the $K_c$ splittings, the AC-Stark mixing of parity pairs becomes more significant and thus makes the dipole-forbidden (parity-preserving, f-type) lines, with frequencies $f_1=a_2-{\varDelta }_U(K_c)$ and $f_2=a_1+ {\varDelta }_U(K_c)$ [see Fig. 7(a) and Supplement 1 [39] Sec. S1], quantum mechanically allowed in an intensity-borrowing process. Although the f-type transitions coincide with quadrupole lines on the frequency scale, they are are much stronger than generic quadrupole transitions. All the lines of Fig. 7(a) can be detected in saturation spectroscopy, based on the excitation of velocity components with ${v}_z=0$, where ${v}_z$ is the molecular velocity along the intra-cavity laser-beam propagation.
Fig. 7. Illustration of the extra resonances observed during the present study. The panels of this figure show (a) the transitions produced by parity-pair mixing, (b) the cross-over resonances in V (left) and $\Lambda$ (right) schemes, and (c) the nonuplet formed by the lines of panel (a) and their cross-overs. In the V ($\Lambda$) scheme, the two general (cyan) transitions share a common lower (upper) state. The parity pairs $U=(U_1,U_2)$ and $L=(L_1,L_2)$ belong to different vibrational states. The parity of a state is indicated within the ket symbol. The kets $|{l}\rangle$ and $|{u}\rangle$ of panel (b) stand for arbitrary lower and upper states, respectively. ${\varDelta }_X(K_c)$ and $V_X$ are the $K_c$ splitting and the (halfway) virtual state of $X=(X_1,X_2)$, respectively. The arrows in solid blue, dashed gray, and dotted red/green/yellow represent dipole-allowed (a-type), dipole-forbidden (f-type), and cross-over (c-type) lines, respectively. The labels on the arrows designate the resonance frequencies. Although $L_1$ and $U_1$ are chosen here to have opposite parity, the reverse case can also be realized (see Supplement 1 [39] Fig. S1). Both cases occurred during this study. The possible frequency orderings of the nine lines are presented in Supplement 1 [39] Table S1. The line frequencies relative to $c_0$ can be expressed as a combination of the two splitting values (see Supplement 1 [39] Sec. S1).
As explained in Ref. [40], the saturation conditions can induce additional lines, called cross-over (c-type) resonances, due to probing velocity classes different from those leading to regular Lamb dips. If two transition frequencies are close to each other in a V/$\Lambda$ scheme [see Fig. 7(b)], a c-type resonance appears at the average of the two frequencies. In fact, such a c-type resonance is delivered by the interaction of the counter-propagating carrier beams with Doppler-shifted ${v}_z \ne 0$ velocity classes. Therefore, cross-overs are only produced when the Doppler profiles overlap for the two transitions of a V/$\Lambda$ scheme. As obvious from Fig. 7(b), these c-type resonances can be treated as artificial transitions $V_U \leftarrow l$ and $u \leftarrow V_L$, where $u$/$l$ denotes an arbitrary upper/lower state and $V_X$ is a virtual state at halfway between the two states of the $X=(X_1,X_2)$ pair. Cross-over features were previously observed in (hyper)fine spectra [41,–43] and also in the Stark-perturbed spectrum of neon [44].
In our case, the c-type resonances are connected to parity pairs, as illustrated in Fig. 7(c). Considering all possible V and $\Lambda$ schemes between triplets $(L_1,V_L,L_2)$ and $(U_1,V_U,U_2)$, five distinct cross-overs with frequencies $c_0$, $c_1$, $c_2$, $C_1$, and $C_2$ can be specified. The $c_0$ line, which pertains to the V scheme of $(C_1,C_2)$ [and also to the $\Lambda$ scheme of $(c_1,c_2)$], can be viewed as a higher-order cross-over resonance. In total, the nine near-infrared resonances form a nonuplet, as shown in Fig. 7(c).
The intensities of the extra (f- and c-type) lines do not straightforwardly result from Wigner-$6J$ symbols, as is the case for hyperfine transitions. These intensities can be obtained from the numerical solution of the optical Bloch equations by integrating signals over the entire velocity distribution within the Doppler profile, as was shown in a recent study on HD [17]. As a result, the extra lines may acquire positive or negative excess intensity and thus generate Lamb dips or Lamb peaks, respectively [17,45]. In some cases, the c-type lines may be more intense than the a-type transitions.
4.2 Assignment of complex Lamb-dip spectra of HD$^{16}$O
Relying on the EH-based and variational $K_c$ splittings of the $(0\,0\,0)$ and $(0\,0\,2)$ vibrational states, respectively, our semi-quantitative approach is followed in this subsection to understand the relative positions of the observed extra lines at a sub-MHz level of accuracy, but no explanation is provided for the resonance intensities. Below, some typical multi-component Lamb-dip spectra of HD$^{16}$O are documented and analyzed. Furthermore, a complex saturation spectrum is presented around the $(0\,0\,2) 3_{2,1/2} \leftarrow (0\,0\,0)3_{3,0/1}$ doublet (see Supplement 1 [39] Sec. S2), where the a-type lines are characterized with well-isolated symmetric Lamb-dip profiles. This spectrum (see Fig. S2) shows two f-type resonances shifted from the a-type lines by $\pm 824.7$ MHz, corresponding to the $K_c$ splitting of the $(0\,0\,0)3_{3,0/1}$ parity pair.
4.2.1 $(0\,0\,2)4_{3,1/2} \leftarrow (0\,0\,0)4_{4,0/1}$ doublet
As discussed in Sec. 3.1, highly asymmetric Lamb-dip profiles have been obtained for the $(0\,0\,2)4_{3,1/2} \leftarrow (0\,0\,0)4_{4,0/1}$ doublet. By performing a wider scan around these a-type transitions, four additional resonances have been identified. Figure 8 shows the saturation signal surrounding the $(0\,0\,2) 4_{3,1} \leftarrow (0\,0\,0)4_{4,0}$ transition, where the extra resonances are blue-shifted by 27.7 MHz [panel (b)] and 54.8 MHz [panel (c)]. These values coincide almost perfectly with the predicted positions of $f_2$ and $C_2$ relative to $a_2$, 27.3 and 54.6 MHz, respectively. The intensity of the $f_2$ transition, which is much smaller than that of the $a_2$ and $c_2$ lines, is strongly dependent on the intra-cavity circulating power [see panel (d)]. This intensity dependence provides further evidence for parity-pair mixing in HD$^{16}$O. The mirror image of Fig. 8(a)–(c), where the two additional resonances are red-shifted by 27.7 and 54.8 MHz from the $(0\,0\,2)4_{3,2} \leftarrow (0\,0\,0)4_{4,1}$ line, is not shown here. Since the $(0\,0\,2)4_{3,1/2}$ splitting, 8.6 GHz, is considerably larger than the Doppler width in the 7000–7350 cm$^{-1}$ region, the cross-overs probing the upper virtual state could not be observed.
Fig. 8. Saturation spectrum recorded for the $(0\,0\,2)4_{3,1} \leftarrow (0\,0\,0)4_{4,0}$ transition. Panels (a) to (c) contain the lines corresponding to the energy-level diagram of panel (e) with resonance frequencies $a_2$, $C_2$, and $f_2$, respectively. Panel (d) plots the peak intensity of the dipole-forbidden $f_2$ line as a function of the intra-cavity laser power, $P$. The conventions applied in panel (e) are described in the caption to Fig. 7. The MHz values of panel (e) denote the $K_c$ splittings of the upper and lower parity pairs. The parity of the individual states is indicated in square brackets. The vertical bars of panels (a)–(c) represent the theoretically predicted resonance positions relative to $a_2$, whose frequency is 213 600 053 170 kHz.
4.2.2 $(0\,0\,2)4_{4,0/1} \leftarrow (0\,0\,0)5_{5,1/0}$ doublet
Figure 9 displays the Lamb-dip spectra of the $(0\,0\,2)4_{4,0/1} \leftarrow (0\,0\,0)5_{5,1/0}$ doublet for HD$^{16}$O, H$_2^{~16}$O and H$_2^{~18}$O. This parity doublet, whose energy-level scheme [see Fig. 9(g)] is an example for the case reverse to that of Fig. 7(c), could not be measured in HD$^{18}$O, because its frequencies are not covered by our laser. For H$_2$O, single narrow Lamb dips are observed, once more demonstrating that the perturbative effects only occur in HDO, where the nuclear-spin distinction of H$_2$O species is lifted. In the complex Lamb-dip profile of HD$^{16}$O, the f-type transitions, $f_1$ and $f_2$, acquire almost no intensity. On the other hand, two strong c-type lines, $C_1$ and $C_2$, can be found near $a_1$ and $a_2$, respectively. The cross-overs linked with the upper virtual state, which should be in principle measurable, were not studied.
Fig. 9. Lamb dips for a close-lying parity doublet of different water isotopologues. For HD$^{16}$O, a multi-component Lamb-dip profile was detected, while the saturation spectra of H$_2^{~16}$O and H$_2^{~18}$O exhibit only a single resonance. The analogous HD$^{18}$O line, which should be very similar to that of HD$^{16}$O, falls outside the 7000–7350 cm$^{-1}$ region. The line centers, corresponding to zero detunings and given in kHz, are the following: 209 986 146 435 [panel (a)], 209 986 249 858 [panel (b)], 214 512 580 150 [panel (c)], 214 513 924 110 [panel (d)], 213 838 639 897 [panel (e)], and 213 840 105 039 [panel (f)]. Note the one-to-one correspondence between the vertical bars of the spectra and the arrows of the energy-level diagram of panel (g). The theoretical positions of the individual resonances are relative to $a_1$ [panel (a)] or $a_2$ [panel (b)]. For further details on the structure of the energy-level scheme, see the captions to Figs. 7 and 8.
4.2.3 $(0\,0\,2)6_{5,2/1} \leftarrow (0\,0\,0)6_{5,1/2}$ doublet
The first example for a complex Lamb-dip spectrum, where the nonuplet lines introduced in Fig. 7(c) could be unambiguously resolved, is shown in Fig. 10. In this case, the two $K_c$ splittings are small enough (<80 MHz) to keep the extra lines within the Doppler profiles of the a-type transitions. The nine resonances, whose relative positions agree reasonably well with their theoretical estimates, are contained in a single spectrum covering a frequency range of $\pm 60$ MHz. The presence of the (higher-order) $c_0$ resonance, which can be considered as a novel spectral feature, could be confirmed by studying the $\pm 10$ MHz detuning window at high intracavity power [see Fig. 10(b)].
Fig. 10. Saturation spectrum involving the $(0\,0\,2)6_{5,2/1} \leftarrow (0\,0\,0)6_{5,1/2}$ doublet. Panel (a) displays a scan of $\pm 60$ MHz around the central crossover, $c_0=215\,836\,700\,683$ kHz, corresponding to zero detuning. The environment of this $c_0$ line was remeasured at increased intra-cavity power, shown separately in panel (b). Note again the one-to-one correspondence between the vertical bars of the Lamb-dip spectrum [panel (a)] and the lines of the energy-level diagram [panel (c)]. The vertical bars indicate the theoretical positions of the nine lines relative to $c_0$. In this figure, as well as in subsequent figures, the scale of the vertical axis is not specified, because this information is not necessary to understand the spectral features observed.
4.2.4 $(0\,0\,2)5_{5,0/1} \leftarrow (0\,0\,0)6_{5,1/2}$ doublet
In the previous subsections, Lamb-dip spectra were analyzed whereby the $K_c$ splitting is smaller for the lower-energy parity pair than for the higher-energy one [i.e., ${\varDelta }_U(K_c) > {\varDelta }_L(K_c)$ in general terms]. The $(0\,0\,2)5_{5,0/1} \leftarrow (0\,0\,0)6_{5,1/2}$ doublet of Fig. 11 demonstrates the opposite case, with ${\varDelta }_U(K_c) < {\varDelta }_L(K_c)$, involving the inversions $a_1 \leftrightarrow a_2$, $C_1 \leftrightarrow c_2$, and $C_2 \leftrightarrow c_1$ in the ordering of the line positions. The a-type transitions of Fig. 11, from which the f-type lines are shifted by $\pm 7.2$ MHz, are characterized with a fairly small separation, 28.2 MHz. The $f_1$ and $c_2$ resonances are not plotted, they could not be measured due to an overlapping H$_2^{~16}$O line.
Fig. 11. Lamb-dip spectrum detected for the $(0\,0\,2)5_{5,0/1} \leftarrow (0\,0\,0) 6_{5,1/2}$ doublet. Note the one-to-one correspondence between the vertical bars of the spectrum [panel (a)] and the lines of the energy-level diagram [panel (b)]. The vertical bars provide the theoretical positions of the nine lines relative to the $c_0$ resonance coinciding with the line center at 213 065 604 238 kHz. The detuning region below $-13$ MHz, which is overlapped with the $(2\,0\,0)1_{1,0} \leftarrow (0\,0\,0)2_{2,1}$ transition of H$_2^{~16}$O, is not presented. Since the $K_c$ splitting is larger for the lower-energy parity pair, the ordering of the nine resonances is different from that of Fig. 10 (fact, the difference is connected to the $a_1 \leftrightarrow a_2$, $C_1 \leftrightarrow c_2$, and $C_2 \leftrightarrow c_1$ inversions).
4.2.5 $(0\,0\,2)7_{6,1/2} \leftarrow (0\,0\,0)7_{6,2/1}$ doublet
As emphasized in Sec. 3.1, the saturation spectra of HD$^{16}$O reach a higher degree of complexity with increased $J$ and $K_a$ values. Figure 12 shows an interesting example, with an energy-level scheme similar to that of Fig. 10, but with significantly smaller ($<10$ MHz) $K_c$ splittings. In view of the dense structure of this multi-component spectrum, an additional measurement was performed with $3f$ demodulation of the NICE-OHMS signal, thereby obtaining a higher resolving power [46] as compared to the $1f$ detection scheme.
Fig. 12. Lamb-dip spectrum recorded for the $(0\,0\,2)7_{6,1/2} \leftarrow (0\,0\,0) 7_{6,1/2}$ doublet. Panels (a) and (b) are related to $1f$ and $3f$ demodulation signals, respectively. The nine resonances, which appear to be better resolved in the $3f$ spectrum, are associated with the lines of the energy-level scheme of panel (c). As in Figs. 10 and 11, the theoretical positions marked with vertical bars are relative to the central resonance, $c_0=215\,196\,861\,851$ kHz. As clear from panels (a) and (b), the Lamb dips of the a-type transitions are somewhat shifted from the theoretical predictions.
Although the strongest resonance is assigned to the $c_0$ cross-over in the $1f$ profile, the a-type transitions, separated by 8.6 MHz, are the most intense in the $3f$ recording. As in the preceding examples, the f-type components are extremely weak (in fact, they are barely detectable in the present case). The four c-type resonances are clearly visible in the $3f$ spectrum and agree particularly well with the theoretical positions relative to $c_0$. The two green cross-overs, $C_1$ and $C_2$, appear as Lamb-dips, while the red lines, $c_1$ and $c_2$, produce Lamb peaks, a typical sign-reversing effect in saturation spectroscopy [40]. In the $3f$ spectrum, further resonances are also observed, for which no assignment can be given based on our semi-quantitative model.
4.2.6 $(0\,0\,2)8_{7,1/2} \leftarrow (0\,0\,0)7_{6,2/1}$ and $(0\,0\,2)8_{7,1/2} \leftarrow (0\,0\,0)7_{7,0/1}$ doublets
During this study, a large number of Lamb dips were found mimicking a double-spike phenomenon, for which two examples are given in Fig. 13. For both examples, the upper states correspond to the $(0\,0\,2)8_{7,1/2}$ parity pair with a 0.6 MHz splitting. In one case [Fig. 13(a)], the lower-state $K_c$ splitting is 2.3 MHz, while the same splitting reduces to 10 kHz in the other example [Fig. 13(b)]. Due to their fairly high intensity, the $C_1$, $c_0$, and $C_2$ cross-overs can be easily recognized in Fig. 13(a), but no convincing assignment emerged for the remaining resonances.
Fig. 13. Typical double-spike Lamb-dip features identified during the present study. Panels (a) and (c) visualize the saturation spectra of the $(0\,0\,2)8_{7,1/2} \leftarrow (0\,0\,0)7_{7,0/1}$ and $(0\,0\,2)8_{7,1/2} \leftarrow (0\,0\,0)7_{6,1/2}$ doublets, respectively, together with the associated energy-level schemes in panels (b) and (d), respectively. Owing to the tiny $K_c$ splitting of the $(0\,0\,0)7_{7,0/1}$ pair, only the unresolved triplets of lines, sharing the same upper state and denoted with $t_1$, $t_2$, and $t_3$ [see also panel (d)], can be seen in panel (c).
As to the other doublet, the minuscule lower-state splitting leads to the coalescence of those lines with the same upper state at the current resolution of our NICE-OHMS spectrometer (200–300 kHz). Therefore, the prominent features of Fig. 13(b) can be ascribed to the $t_1$, $t_2$, and $t_3$ triplets defined in Fig. 13(d), which are located at detunings of –0.3, 0, and 0.3 MHz, respectively. In those cases when both splittings become smaller 200 kHz, only a single-spike feature, resembling a regular Lamb-dip with some spectral distortions, can be detected with our setup.
5. Conclusions
Motivated by our previous studies on H$_2^{~16}$O [9,13] and H$_2^{~18}$O [10], the original aim of this study was the derivation of kHz-accuracy energy levels within the ground vibrational state of HD$^{16}$O, by measuring near-infrared transitions via the NICE-OHMS saturation-spectroscopy technique. In principle, saturation spectroscopy enables the extraction of line frequencies with a few kHz uncertainty, corresponding to a 100–1000 times improvement in comparison to the MHz-level accuracy of Doppler-broadened approaches. Previously, over 50 000 rovibrational lines have been detected for HD$^{16}$O, linking some 10 000 of its quantum states in the electronic ground state [28,47].
In the present study, Lamb dips were recorded for more than 100 rovibrational transitions in the first overtone of the O–H stretch fundamental of HD$^{16}$O, yielding line positions with an experimental reproducibility of 2–20 kHz. For transitions with $\max (K'_a,K''_a) > 3$, multi-component Lamb-dip profiles with highly varying complexity were observed. Analyzing the experimental (NICE-OHMS-based) $K_c$ splittings of parity pairs (that is, pairs of opposite-parity energy levels differing only in their $K_c$ values) in the ground vibrational state, significant deviations were identified from predictions provided by an accurate effective-Hamiltonian (EH) model [19]. Even in view of the frequency shifts revealed, the line centers of symmetric Lamb dips still have an absolute accuracy on the order of 100 kHz, leading to a ten times improvement with respect to the uncertainties of Doppler-limited HD$^{16}$O lines.
The spectral perturbations discovered in the Lamb-dip profiles of HD$^{16}$O transitions can be semi-quantitatively described by assuming that an (optical) AC-Stark effect causes mixing between the states of parity pairs. This assumption was verified by measuring transition doublets with the same assignment for four water isotopologues: H$_2^{~16}$O, H$_2^{~18}$O, HD$^{16}$O, and HD$^{18}$O. The comparison of the results obtained for the different species suggests that (a) there is no perturbation in the near-infrared Lamb-dip spectra of H$_2^{~16}$O and H$_2^{~18}$O, and (b) the perturbative effects are equally present in HD$^{16}$O and HD$^{18}$O. The reason behind the dissimilar behavior of H$_2$O and HDO is that the parity pairs of H$_2$O are ortho–para nuclear-spin-isomer pairs, preventing their significant interaction. This means that the distortion effects are related to the reduction of the molecular symmetry under the H$\rightarrow$D replacement. Moreover, it is shown that the multi-component spectral features are not caused by the sidebands of the NICE-OHMS technique, but result from saturation spectroscopy as such.
To establish a semi-quantitative model, a general four-state scheme of two parity pairs is presented, explaining the appearance of extra resonances in the saturation spectra of HDO. In this scheme, four infrared lines (two dipole-allowed and two dipole-forbidden) are specified, where parity-pair mixing yields intensity for the two dipole-forbidden transitions, making them quantum mechanically allowed. Furthermore, cross-over resonances, previously observed in non-linear spectroscopy but not in connection to parity-pair mixing of polyatomic molecules, may also occur in the multi-component Lamb-dip profiles of HDO, leading to altogether nine near-infrared resonances in our four-state scheme.
As illustrated for eight transition doublets, the experimental positions of the nine resonances, relative to the central cross-over, agree reasonably well with their theoretical (EH-predicted and/or variationally-computed) counterparts. Thus, the good matches between the experimental and theoretical relative positions confirm the assignments provided for the individual frequency components. The theoretical predictions also helped to find the weak forbidden lines for the $(0\,0\,2)3_{2,1/2}$ $\leftarrow$ $(0\,0\,0)3_{3,0/1}$ transition doublet (see Supplement 1 [39] Sec. S2), whose lines exhibit symmetric saturation spectra, but their frequency shifts are larger than 90 kHz. These weak signals demonstrate that the AC-Stark perturbations occur even in those cases when there is no asymmetry around the Lamb dips of the dipole-allowed transitions.
Although our four-state scheme provides unique assignments for the resonances of complex saturation spectra in HDO, a sophisticate quantum-chemical model is needed for a fully quantitative description of the multi-component Lamb-dip profiles and the frequency shifts observed. This model should consider (a) the usual conditions applied in saturation spectroscopy, (b) the dynamic coupling between lower and upper states by the laser field, (c) the AC-Stark interaction between the two states of parity pairs, and (d) the $2 J + 1$ Stark-induced “sub-levels” of the rovibrational states [36,48]. A set of optical Bloch equations, involving the possible sub-levels, might deal with all the coherences, also including cross-over resonances, as shown recently for HD [17] and for general multi-state interaction schemes [49,–51]. Since the cited papers investigate interactions between states with the same parity, their formulas cannot be employed directly for the complex Lamb-dip shapes of HDO, where the AC-Stark effect takes place between the two energy levels of parity pairs.
Funding
Nederlandse Organisatie voor Wetenschappelijk Onderzoek (16MYSTP); Nemzeti Kutatási Fejlesztési és Innovációs Hivatal (K138233); Horizon 2020 Framework Programme (654148).
Acknowledgments
The authors thank Prof. Ad van der Avoird (Radboud University Nijmegen) for insightful discussions on the symmetries of quantum states of water isotopologues.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Dataset 1 [33] and Dataset 2 [34].
Supplemental document
See Supplement 1 for supporting content.
References
1. M. Quack and F. Merkt, eds., Handbook of High-Resolution Spectroscopy (Wiley, 2011).
2. V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spectroscopy (Springer-Verlag, 1977).
3. L.-S. Ma, J. Ye, P. Dubé, and J. L. Hall, “Ultrasensitive frequency-modulation spectroscopy enhanced by a high-finesse optical cavity: theory and application to overtone transitions of C2H2 and C2HD,” J. Opt. Soc. Am. B 16(12), 2255–2268 (1999). [CrossRef]
4. A. Foltynowicz, F. M. Schmidt, W. Ma, and O. Axner, “Noise-immune cavity-enhanced optical heterodyne molecular spectroscopy: Current status and future potential,” Appl. Phys. B 92(3), 313–326 (2008). [CrossRef]
5. D. Gatti, R. Gotti, A. Gambetta, M. Belmonte, G. Galzerano, P. Laporta, and M. Marangoni, “Comb-locked Lamb-dip spectrometer,” Sci. Rep. 6(1), 27183 (2016). [CrossRef]
6. J. Wang, Y. R. Sun, L.-G. Tao, A.-W. Liu, T.-P. Hua, F. Meng, and S.-M. Hu, “Comb-locked cavity ring-down saturation spectroscopy,” Rev. Sci. Instrum. 88(4), 043108 (2017). [CrossRef]
7. H. Wu, C.-L. Hu, J. Wang, Y. R. Sun, Y. Tan, A.-W. Liu, and S.-M. Hu, “A well-isolated vibrational state of CO2 verified by near-infrared saturated spectroscopy with kHz accuracy,” Phys. Chem. Chem. Phys. 22(5), 2841–2848 (2020). [CrossRef]
8. Z. D. Reed, D. A. Long, H. Fleurbaey, and J. T. Hodges, “SI-traceable molecular transition frequency measurements at the 10−12 relative uncertainty level,” Optica 7(9), 1209–1220 (2020). [CrossRef]
9. R. Tóbiás, T. Furtenbacher, I. Simkó, A. G. Császár, M. L. Diouf, F. M. J. Cozijn, J. M. A. Staa, E. J. Salumbides, and W. Ubachs, “Spectroscopic-network-assisted precision spectroscopy and its application to water,” Nat. Commun. 11(1), 1708 (2020). [CrossRef]
10. M. L. Diouf, R. Tóbiás, I. Simkó, F. M. J. Cozijn, E. J. Salumbides, W. Ubachs, and A. G. Császár, “Network-based design of near-infrared Lamb-dip experiments and the determination of pure rotational energies of H218O at kHz accuracy,” J. Phys. Chem. Ref. Data 50(2), 023106 (2021). [CrossRef]
11. Z. Reed, B. Drouin, D. Long, and J. Hodges, “Molecular transition frequencies of CO2 near 1.6 µm with kHz-level uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 271, 107681 (2021). [CrossRef]
12. E. Fasci, S. Gravina, G. Porzio, A. Castrillo, and L. Gianfrani, “Lamb-dip cavity ring-down spectroscopy of acetylene at 1.4 µm,” New J. Phys. 23(12), 123023 (2021). [CrossRef]
13. M. L. Diouf, R. Tóbiás, T. S. van der Schaaf, F. M. Cozijn, E. J. Salumbides, A. G. Császár, and W. Ubachs, “Ultraprecise relative energies in the (2 0 0) vibrational band of H216O,” Mol. Phys. 120(15-16), e2050430 (2022). [CrossRef]
14. R. Aiello, V. Di Sarno, M. G. D. Santi, M. De Rosa, I. Ricciardi, G. Giusfredi, P. De Natale, L. Santamaria, and P. Maddaloni, “Lamb-dip saturated-absorption cavity ring-down rovibrational molecular spectroscopy in the near-infrared,” Photonics Res. 10(8), 1803–1809 (2022). [CrossRef]
15. M. Sinhal, A. Johnson, and S. Willitsch, “Frequency stabilisation and SI tracing of mid-infrared quantum-cascade lasers for precision molecular spectroscopy,” Mol. Phys., e2144519 (to be published). [CrossRef]
16. F. M. J. Cozijn, P. Dupré, E. J. Salumbides, K. S. E. Eikema, and W. Ubachs, “Sub-Doppler frequency metrology in HD for tests of fundamental physics,” Phys. Rev. Lett. 120(15), 153002 (2018). [CrossRef]
17. M. Diouf, F. Cozijn, B. Darquié, E. Salumbides, and W. Ubachs, “Lamb-dips and Lamb-peaks in the saturation spectrum of HD,” Opt. Lett. 44(19), 4733–4736 (2019). [CrossRef]
18. H. W. Kroto, Molecular Rotation Spectra (Dover, 1992).
19. G. Cazzoli, V. Lattanzi, J. L. Alonso, J. Gauss, and C. Puzzarini, “The hyperfine structure of the rotational spectrum of HDO and its extension to the THz region: Accurate rest frequencies and spectroscopic parameters for astrophysical observations,” Astrophys. J. 806(1), 100 (2015). [CrossRef]
20. D. Romanini, I. Ventrillard, G. Méjean, J. Morville, and E. Kerstel, “Introduction to cavity enhanced absorption spectroscopy,” in Cavity-Enhanced Spectroscopy and Sensing, (Springer, 2014), pp. 1–60.
21. H. Dinesan, E. Fasci, A. d’Addio, A. Castrillo, and L. Gianfrani, “Characterization of the frequency stability of an optical frequency standard at 1.39 µm based upon noise-immune cavity-enhanced optical heterodyne molecular spectroscopy,” Opt. Express 23(2), 1757–1766 (2015). [CrossRef]
22. S. Twagirayezu, M. J. Cich, T. J. Sears, C. P. McRaven, and G. E. Hall, “Frequency-comb referenced spectroscopy of ν4- and ν5-excited hot bands in the 1.5 µm spectrum of C2H2,” J. Mol. Spectrosc. 316, 64–71 (2015). [CrossRef]
23. E. B. Treacy and Y. Beers, “Hyperfine structure of the rotational spectrum of HDO,” J. Chem. Phys. 36(6), 1473–1480 (1962). [CrossRef]
24. P. Thaddeus, L. C. Krisher, and J. H. N. Loubser, “Hyperfine structure of the microwave spectrum of HDO, HDS, CH2O, and CHDO: beam-maser spectroscopy and asymmetric-top molecules,” J. Chem. Phys. 40(2), 257–273 (1964). [CrossRef]
25. H. Bluyssen, J. Verhoeven, and A. Dymanus, “Hyperfine structure of HDO and D2O by beam maser spectroscopy,” Phys. Lett. A 25(3), 214–215 (1967). [CrossRef]
26. S. G. Kukolich, “Beam maser double resonance measurements on HDO,” J. Chem. Phys. 66(10), 4345–4348 (1977). [CrossRef]
27. H. A. Fry and S. G. Kukolich, “Beam maser measurements of HDO hyperfine structure,” J. Chem. Phys. 76(9), 4387–4391 (1982). [CrossRef]
28. J. Tennyson, P. F. Bernath, L. R. Brown, et al., “IUPAC critical evaluation of the rotational-vibrational spectra of water vapor. Part II. Energy levels and transition wavenumbers for HD16O, HD17O, and HD18O,” J. Quant. Spectrosc. Radiat. Transfer 111(15), 2160–2184 (2010). [CrossRef]
29. E. Mátyus, G. Czakó, and A. G. Császár, “Toward black-box-type full- and reduced-dimensional variational (ro)vibrational computations,” J. Chem. Phys. 130(13), 134112 (2009). [CrossRef]
30. C. Fábri, E. Mátyus, and A. G. Császár, “Rotating full- and reduced-dimensional quantum chemical models of molecules,” J. Chem. Phys. 134(7), 074105 (2011). [CrossRef]
31. C. Fábri, M. Quack, and A. G. Császár, “On the use of nonrigid-molecular symmetry in nuclear-motion computations employing a discrete variable representation: a case study of the bending energy levels of CH5+,” J. Chem. Phys. 147(13), 134101 (2017). [CrossRef]
32. P. Barletta, S. V. Shirin, N. F. Zobov, O. L. Polyansky, J. Tennyson, E. F. Valeev, and A. G. Császár, “CVRQD ab initio ground-state adiabatic potential energy surfaces for the water molecule,” J. Chem. Phys. 125(20), 204307 (2006). [CrossRef]
33. M. L. Diouf, R. Tóbiás, F. M. J. Cozijn, E. J. Salumbides, C. Fábri, C. Puzzarini, A. G. Császár, and W. Ubachs, “Parity-pair-mixing effects in nonlinear spectroscopy of HDO,” figshare (2022), https://doi.org/10.6084/m9.figshare.20728444 Dataset 1.
34. M. L. Diouf, R. Tóbiás, F. M. J. Cozijn, E. J. Salumbides, C. Fábri, C. Puzzarini, A. G. Császár, and W. Ubachs, “Parity-pair-mixing effects in nonlinear spectroscopy of HDO,” figshare (2022), https://doi.org/10.6084/m9.figshare.20728552 Dataset 2.
35. C. E. Otis and P. M. Johnson, “The AC-Stark effect in molecular multiphoton ionization spectroscopy,” Chem. Phys. Lett. 83(1), 73–77 (1981). [CrossRef]
36. W. M. Huo, K. P. Gross, and R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54(10), 1012–1015 (1985). [CrossRef]
37. C. Wieman, M. Noecker, B. Masterson, and J. Cooper, “Asymmetric line shapes for weak transitions in strong standing-wave fields,” Phys. Rev. Lett. 58(17), 1738–1741 (1987). [CrossRef]
38. J. E. Stalnaker, D. Budker, S. Freedman, J. Guzman, S. Rochester, and V. Yashchuk, “Dynamic Stark effect and forbidden-transition spectral line shapes,” Phys. Rev. A 73(4), 043416 (2006). [CrossRef]
39. M. L. Diouf, R. Tóbiás, F. M. J. Cozijn, E. J. Salumbides, C. Fábri, C. Puzzarini, A. G. Császár, and W. Ubachs, “Parity-pair-mixing effects in nonlinear spectroscopy of HDO,” (2022), https://doi.org/10.6084/m9.figshare.21520698 Supplement 1.
40. W. Demtröder, Laser Spectroscopy: Vol. 2 Experimental Techniques (Springer, 2021).
41. H. J. Foth and F. Spieweck, “Hyperfine structure of the R(98), 58-1 line of 127I2 at 514.5 nm,” Chem. Phys. Lett. 65(2), 347–352 (1979). [CrossRef]
42. L. Julien, M. Pinard, and F. Laloë, “Hyperfine structure and isotope shift of the 640.2 and 626.6 nm lines of neon,” J. Phys., Lett. 41(20), 479–482 (1980). [CrossRef]
43. L. Bloomfield, H. Gerhardt, T. Hansch, and S. Rand, “Nonlinear UV-laser spectroscopy of the 23S–53P transition in 3He and 4He,” Opt. Commun. 42(4), 247–250 (1982). [CrossRef]
44. A. Schabert, R. Keil, and P. Toschek, “Dynamic Stark effect of an optical line observed by cross-saturated absorption,” Appl. Phys. 6(2), 181–184 (1975). [CrossRef]
45. C. Hertzler and H.-J. Foth, “Sub-Doppler polarization spectra of He, N2 and Ar+ recorded in discharges,” Chem. Phys. Lett. 166(5-6), 551–559 (1990). [CrossRef]
46. M. Melosso, M. L. Diouf, L. Bizzocchi, M. E. Harding, F. M. J. Cozijn, C. Puzzarini, and W. Ubachs, “Hyperfine-resolved near-infrared spectra of H217O,” J. Phys. Chem. A 125(36), 7884–7890 (2021). [CrossRef]
47. A. A. Kyuberis, N. F. Zobov, O. V. Naumenko, B. A. Voronin, O. L. Polyansky, L. Lodi, A. Liu, S.-M. Hu, and J. Tennyson, “Room temperature line lists for deuterated water,” J. Quant. Spectrosc. Radiat. Transfer 203, 175–185 (2017). [CrossRef]
48. J. S. Bakos, “AC Stark effect and multiphoton processes in atoms,” Phys. Rep. 31(3), 209–235 (1977). [CrossRef]
49. S. Mandal and P. N. Ghosh, “Line shape and frequency shift of Lamb dip and crossover-resonance dip in closely spaced transitions,” Phys. Rev. A 45(7), 4990–4997 (1992). [CrossRef]
50. S. Mandal and P. N. Ghosh, “Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate levels,” Phys. Rev. A 47(6), 4934–4945 (1993). [CrossRef]
51. S. Ghosh and S. Mandal, “Analytical studies on pump-induced optical resonances in an M-type six-level system,” J. Phys. B: At., Mol. Opt. Phys. 43(24), 245505 (2010). [CrossRef]
