Most experiments in recent years assume single-atom response in their transient absorption experiments. However, the experimental densities in these experiments often approach a limit where the simple Beer-Lambert law is no longer valid and macroscopic effects should be taken into account. We have explored this regime in our recent studies [

14,

28], where we show that the collective macroscopic XUV pulse propagation effect, also called the resonant pulse propagation (RPP) effect, should be considered. We also laid out the criteria for consideration of gas target as an optically thick medium. Here, we provide a simple overview of the complex interplay between LIP effect and RPP effect by summarizing it in a schematic of

Figure 4. The first column indicates experimental conditions, and the second column is the physical picture of the interaction, followed by its temporal description on the third column, and its corresponding spectral profile in the last column. In single-atom response picture—as explained in

Section 3 and shown in

Figure 4a—when XUV pulse alone (E

${}_{XUV}$) excites the dipole polarization (

P) of a resonant medium, the Fourier transform of this oscillating and decaying polarization shows a Lorentzian spectrum in the frequency domain. The introduction of strong, time delayed NIR pulse (E

${}_{NIR}$) (

Figure 4b) imparts an additional phase on the polarization, and this LIP effect results in a Fano spectral line shape. When an XUV propagation alone in a dense medium is considered (

Figure 4c), the polarization created by the incident XUV is strong enough to radiate XUV light, and this emission could further excite secondary polarization. In this self-consistent dipole-field interaction picture, the final polarization will be temporally reshaped and it can be described by a Bessel function of the first kind (J

${}_{1}$) [

29] as

${E}_{XUV}(z,t)\propto {J}_{1}\left(\mathrm{\Gamma}Pzt\right)/\sqrt{t}$, where Γ is the decay lifetime,

P is the gas pressure, and

z is the propagation distance. Therefore, the pressure-length product

$zt$ determines the reshaping of XUV pulse in a dense resonant medium. The temporal reshaping manifests as broadened spectral lineshape. An important feature of the reshaped XUV pulse profile is that the first sub-pulse will be out-of-phase compared to the original pulse, the second sub-pulse will be out of phase compared to the first sub-pulse, and so on. These temporal phase variations can be brought to light using the presence of delayed NIR pulse, where the NIR pulse samples the

π phase jumps between sub-pulses, and the LIP effect serves to broaden the fine spectral structure associated with these phase jumps. The non-linear interplay between RPP and LIP can thus be clearly seen through the appearance of new spectral features in the experimental transient absorption lineshape. It should be noted that, for this to happen, the duration of the first XUV sub-pulse has to be comparable or smaller than the NIR pulse duration, which means that pressure-length product has to be high enough to significantly reshape the XUV pulse through RPP effect.

In order to experimentally demonstrate the interplay between RPP and LIP effects, in

Figure 5a, we show 1s2p state evolution, at 400 Pa and 1200 Pa backing pressure for 786 nm laser wavelength, at –30 fs time delay, and ∼2 TW/cm

${}^{2}$ laser intensity. As gas pressure is increased, we clearly observe the appearances of new spectral features (indicated by vertical dashed line) near the line center of an otherwise simple Fano like profile. We also used an OPA to convert original NIR pulse to 1428 nm NIR pulse with similar pulse duration but weaker peak intensity at ∼0.2 TW/cm

${}^{2}$. The bottom curve in

Figure 5a shows the 1s2p line profile evolution under 1428 nm NIR imposed LIP effect at the same backing pressure (1200 Pa) and time delay (–30 fs). Although this intensity is an order of magnitude weaker, the LIP effect depends on the pondermotive energy shift that is proportional to

$I{\lambda}^{2}$, where

I is the peak intensity of the laser and

λ is the laser wavelength, so the LIP effect in the case of longer wavelength should only be three times smaller. We observe that the dispersive effect of 1428 nm light is actually similar compared to the 786 nm case, as seen from the overall broadening of the spectral lineshapes in the two cases. The quantitative comparison between these two cases requires calculation of the LIP effect by including one-photon couplings to nearby states. In our case, NIR couplings pathways to nearby dark 1sns states from the bright 1s2p state are shown in

Figure 5b. Depending on the detuning, the dipole strength of these transitions and the NIR intensity dependent Rabi Frequency, we will observe different AC Stark shift and line profiles in two cases. The calculations for 1s2p states reported in [

23] do show that it is possible to have wider lineshape for higher detuning cases. The overall line shape in 1428 nm case also shows an especially strong RPP peak when compared to the 786 nm NIR case. Detailed understanding of exact spectral features is not trivial, and it requires macroscopic calculations where a time-dependent Schrödinger equation calculation is coupled to the Maxwell wave equation based propagation.

If we continue looking at the long wavelength (1428 nm) case and further increasing helium gas backing pressure to 1600 Pa, we can enhance the main RPP peak so that it becomes comparable or greater in strength than the original Fano profile peaks as seen in

Figure 6a. Importantly, we can observe that, as the time delay is varied, the RPP strength changes monotonically; however, the dispersive Fano profile around the RPP peak changes its nature quite dramatically and going to positive to negative time delay results in complete reversal of the signal of OD. As for 1snp states in helium, including

n = 4, 5, 6, 7,… to ionization potential (IP), the transition strength is much smaller, and hence the RPP effect is not very significant; therefore, most states show Fano-like profiles even with dense gas as shown in

Figure 6b. Note that vertical gray dash lines are field-free energy level, and the ’real’ 1s2p and 1snp state under strong field dressing will be pondermotively shifted toward higher energy as shown in Ref. [

30].