Transient-absorption phases with strong probe and pump pulses

Vadim Becquet; Stefano M Cavaletto

doi:10.1088/1361-6455/aa8e6a

1. Introduction

Phases represent the essential feature of any wave-like phenomena, lying at the heart of coherence and interference effects in classical and quantum physics. In atoms and molecules, phases define the shape of a wave packet in a superposition of quantum states and hence determine its subsequent time evolution. Manipulating atomic and molecular dynamics with external electromagnetic fields [1–4], e.g., by using strong femto- or even attosecond pulses [5–9], thus requires full control of the generated quantum phases. However, already for relatively simple atomic systems, when a pulse is sufficiently intense to induce Rabi oscillations in the populations of the coupled states, a time-dependent field differently detuned from several excited states can induce nontrivial phase dynamics. Experimental methods for quantum-state holography [10, 11] in the presence of strong fields would enable one to extract information about the complex phases generated by such intense pulses directly from observable spectra. However, traditional spectroscopy methods usually do not provide access to the phase information: for instance, for nonautoionizing bound states, absorption spectra typically consist of Lorentzian lines, with spectral intensities quantifying the atomic populations.

The manipulation of absorption line shapes in transient-absorption-spectroscopy experiments [12–32] has been recently identified as a key mechanism to gain access to strong-field-generated phases. Absorption lines originate from the interference between a probe pulse transmitting through the medium and the field emitted by the system [33]. This interference pattern depends on the state of the system encountered by the probe pulse, whose absorption line shapes therefore encode atomic populations and phases. First experiments involving pico- to femtosecond pulses exploited a pump-probe setup [12, 13], where the dynamics triggered by the pump pulse are observed by measuring the absorption spectrum of the probe pulse at different time delays. Also attosecond extreme-ultraviolet pulses have been recently employed as probe pulses in attosecond transient-absorption spectroscopy, with their spectrum measured at different time delays in order to time-resolve the strong-field-ionization dynamics started by [15, 16] or in the presence of an intense infrared pump pulse [18–20].

Very recently, however, a complementary, probe-pump setup has received significant attention. In this case, a first-arriving probe pulse initiates the dynamics by exciting an initial state, which is nonlinearly modified by the subsequent action of a strong pump pulse [21–32]. Studies in helium have shown how the atomic-phase changes imposed by the action of the second-arriving pump pulse are encoded in the absorption line shapes of the probe pulse [23, 26, 27]. The line-shape-control concept thus provided has been generalized to additionally include pump-imposed amplitude changes [29, 34], also in more complex systems [30].

Related line-shape-manipulation experiments in gaseous rubidium have recently shown how this line-shape-control concept can be applied to more complex multilevel systems [35, 36] and used to understand their quantum dynamics. Femtosecond optical pump and probe pulses were used to excite the V-type three-level system formed by the $5s\,\,{}^{2}{S}_{1/2}\to 5p{}^{2}{P}_{1/2}$ ( $794.76\,\mathrm{nm}$ ) and $5s{}^{2}{S}_{1/2}\to 5p{}^{2}{P}_{3/2}$ ( $780.03\,\mathrm{nm}$ ) transitions, for a pump-probe and probe-pump setup and for different time delays. While the action of a weak pulse results in a small, well understood perturbation, Rabi flopping in the population of the coupled states is induced at higher intensities of $\sim {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . These intensities do not cause strong-field ionization, and the associated Rabi frequencies are still far smaller than the laser frequencies. Nevertheless, the presence of different detunings and of a time-dependent pulse envelope result in nontrivial population and phase dynamics of the multilevel system, whose prediction and understanding go beyond the linear regime of perturbation theory. These strong-field dynamics are imprinted into the line-shape of transient-absorption spectra, as shown in [35]. In particular, for both pump-probe and probe-pump setups, absorption line shapes display time-delay-dependent oscillations at the beating frequency between the two excited states. An increase in the intensity of the pump pulse, from the perturbative to the Rabi-flopping regime, causes a shift in the phase of these oscillating features, which can be associated with the transformation imposed by the pump pulse to the state of the system. In [35], in order to prove this connection, spectra were simulated with phases artificially added to the quantum state, effectively modeling the atomic-phase changes due to an intense pump pulse. Thereby, for both pump-probe and probe-pump setups, corresponding phase shifts in the absorption spectrum could be reproduced. In [36], an effective dipole-control model was used to fit absorption spectra from a probe-pump setup, providing a way to quantify the line-shape changes due to the action of an intense pump pulse. In both cases, however, the methods employed did not fully uncover the connection between absorption line shapes and strong-field-induced atomic population and phase changes. The V-type three-level system used to model Rb atoms, due to its simplicity, is especially suitable for this purpose, and can be utilized to understand the essential properties of the transient-absorption spectrum, which would also be exhibited in more complex multilevel systems excited by broadband pump and probe pulses.

Here, we develop an interpretation model to quantify the shifts in the time-delay-dependent oscillations featured by transient-absorption spectra. Due to recent experimental investigations, the model is developed for a V-type three-level system describing Rb atoms [35, 36]. However, our description can be generalized to more complex multilevel systems, whenever beating between several excited states results in time-delay-dependent oscillations of the absorption spectrum. Our model is based on recently introduced interaction operators [37], which provide a comprehensive formalism to describe the modification (in amplitude and phase) of the atomic system following the interaction with an intense short pulse. This is used to interpret the resulting absorption spectra and their intensity-dependent features. Compared to [35, 36], we explicitly show, in terms of interaction operators, which atomic population and phase changes are involved in the shifts exhibited by the time-delay-dependent oscillations of the spectra, and study them as a function of pulse parameters such as intensities and frequencies. Furthermore, and in contrast to previous studies only focusing on the intensity dependence of the pump pulse, here we fully account for the effect of a potentially intense probe pulse as well. On the one hand, this allows us to interpret transient-absorption spectra in terms of the pump and probe parameters of interest, without a priori assumptions, which may not correspond to the conditions featured in an experiment and, hence, could lead to an inappropriate or incomplete reconstruction of the strong-field dynamics of the system. On the other hand, by considering cases in which pump and probe pulses exhibit the same intensities, we can highlight the essential differences between spectra where the probe, i.e., measured, pulse either precedes or follows the pump pulse. A proper interpretation of transient-absorption spectra is crucial for the extraction of strong-field dynamical information from these spectra, and the implementation of recently suggested deterministic strong-field quantum-control methods [37].

The paper is structured as follows. In section 2, we present the theoretical model used to describe the evolution of the system and to predict the associated transient-absorption spectra. The numerical results are presented in section 3. In particular, time-delay-dependent transient-absorption spectra are shown in section 3.1 for different pump- and probe-pulse intensities. An analytical model based on recently introduced interaction operators [37] is used in section 3.2 to interpret the numerical results, focusing on the atomic-phase information which can be extracted from the spectra for different intensities and laser frequencies of the pump and probe pulses. Section 4 summarizes the results obtained. Atomic units are used throughout unless otherwise stated.

2. Theoretical model

2.1. Three-level model and equations of motion

We consider the V-type three-level system depicted in figure 1, modeling fine-structure-split $5s{}^{2}{S}_{1/2}\to 5p{}^{2}{P}_{1/2}$ and $5s{}^{2}{S}_{1/2}\to 5p{}^{2}{P}_{3/2}$ transitions in Rb atoms [38–40]. In particular, we introduce the state

$\begin{eqnarray}&&| \psi (t,\tau )\rangle =\displaystyle \sum _{i=1}^{3}{c}_{i}(t,\tau )| i\rangle ,\end{eqnarray} \tag{ 1 }$

written in terms of the ground state $| 1\rangle \equiv 5s{}^{2}{S}_{1/2}$ and the excited states $| 2\rangle \equiv 5p{}^{2}{P}_{1/2}$ and $| 3\rangle \equiv 5p{}^{2}{P}_{3/2}$ , with associated quantum amplitudes ${c}_{i}(t,\tau )$ and energies ${\omega }_{i}$ , $i\in \{1,2,3\}$ . The system interacts with a pump pulse, centered on t = 0 and modeled by the classical field

$\begin{eqnarray}&&{{\boldsymbol{ \mathcal E }}}_{\mathrm{pu}}(t)={{ \mathcal E }}_{\mathrm{pu},0}\,f(t)\cos ({\omega }_{{\rm{L}}}t){\hat{{\boldsymbol{e}}}}_{z},\end{eqnarray} \tag{ 2 }$

and a delayed probe pulse, centered on time delay $t=\tau$ and similarly described as

$\begin{eqnarray}&&{{\boldsymbol{ \mathcal E }}}_{\mathrm{pr}}(t)={{ \mathcal E }}_{\mathrm{pr},0}\,f(t-\tau )\cos [{\omega }_{{\rm{L}}}(t-\tau )]{\hat{{\boldsymbol{e}}}}_{z},\end{eqnarray} \tag{ 3 }$

as shown in figure 2. Both pulses are aligned along the polarization vector ${\hat{{\boldsymbol{e}}}}_{z}$ , have the same frequency ${\omega }_{{\rm{L}}}$ , vanishing carrier-envelope phases, and intensities ${I}_{\mathrm{pu}/\mathrm{pr}}={{ \mathcal E }}_{\mathrm{pu}/\mathrm{pr},0}^{2}/(8\pi \alpha )$ related to the peak field strengths ${{ \mathcal E }}_{\mathrm{pu}/\mathrm{pr},0}$ via the fine-structure constant α. We model their envelope functions as

$\begin{eqnarray}f(t)=\left\{\begin{array}{ll}{\cos }^{2}(\pi t/T) & \mathrm{if}\ | t| \leqslant T/2,\\ 0 & \mathrm{if}\ | t| \gt T/2,\end{array}\right.\end{eqnarray} \tag{ 4 }$

with $T=\pi {T}_{\mathrm{FWHM}}/(2\arccos \sqrt[4]{1/2})$ and ${T}_{\mathrm{FWHM}}=30\,\mathrm{fs}$ , defined as the full width at half maximum of $| f(t){| }^{2}$ [41]. Positive time delays correspond to a typical pump-probe setup, in which the system is first excited by the pump pulse and the resulting dynamics are measured by a probe pulse. In contrast, negative time delays describe experiments in which the dipole response generated by the first-arriving probe pulse is subsequently modified by the pump pulse, resulting in an intensity- and time-delay-dependent modulation of the line-shape of the absorption spectrum of the transmitted probe pulse.

Figure 1. Refer to the following caption and surrounding text. — **Figure 1.** V-type three-level scheme, with transition energies ${\omega }_{21}=1.56\,\mathrm{eV}$ and ${\omega }_{31}=1.59\,\mathrm{eV}$ and decay rates ${\gamma }_{2}={\gamma }_{3}=1/(500\,\mathrm{fs})$ , used to model Rb atoms interacting with broadband laser pulses of frequency ${\omega }_{{\rm{L}}}$ and spectral intensity ${S}_{\mathrm{in}}(\omega )$ .
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**Figure 1.** V-type three-level scheme, with transition energies ${\omega }_{21}=1.56\,\mathrm{eV}$ and ${\omega }_{31}=1.59\,\mathrm{eV}$ and decay rates ${\gamma }_{2}={\gamma }_{3}=1/(500\,\mathrm{fs})$ , used to model Rb atoms interacting with broadband laser pulses of frequency ${\omega }_{{\rm{L}}}$ and spectral intensity ${S}_{\mathrm{in}}(\omega )$ .
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Figure 2. Refer to the following caption and surrounding text. — **Figure 2.** Experimental setup for the detection of the optical-density transient-absorption spectrum of a transmitted probe pulse, delayed by τ with respect to a pump pulse, in a noncollinear geometry.
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The linearly polarized pulses excite electric-dipole-(E1-)allowed transitions $| 1\rangle \to | k\rangle$ , $k\in \{2,3\}$ , with equal magnetic quantum number, ${\rm{\Delta }}M=0$ , and dipole-moment matrix elements ${{\boldsymbol{D}}}_{1k}={D}_{1k}{\hat{{\boldsymbol{e}}}}_{z}$ . The formulas are written for general complex values of D_1k, although these are real and positive for our atomic implementation with Rb atoms, with ${D}_{12}=1.75\,{\rm{a}}.{\rm{u}}.$ and ${D}_{13}=2.47\,{\rm{a}}.{\rm{u}}.$ [39]. For the intensities considered here, we neglect the presence of higher excited states, to which states $| 2\rangle$ and $| 3\rangle$ could also be coupled. The total Hamiltonian of the system

$\begin{eqnarray}&&\hat{H}={\hat{H}}_{0}+{\hat{H}}_{\mathrm{int}}(t,\tau )\end{eqnarray} \tag{ 5 }$

then consists of the unperturbed atomic Hamiltonian

$\begin{eqnarray}&&{\hat{H}}_{0}=\displaystyle \sum _{i=1}^{3}({\omega }_{i}-{\rm{i}}{\gamma }_{i}/2)| i\rangle \langle i| \end{eqnarray} \tag{ 6 }$

and the E1 light–matter interaction Hamiltonian in the rotating-wave approximation [42–44]

$\begin{eqnarray}&&{\hat{H}}_{\mathrm{int}}=-\displaystyle \frac{1}{2}\displaystyle \sum _{k=2}^{3}{{\rm{\Omega }}}_{{\rm{R}}k}(t,\tau ){{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}t}\,| 1\rangle \langle k| +{\rm{h}}.{\rm{c}}.\end{eqnarray} \tag{ 7 }$

In equation (6), the complex eigenvalues $({\omega }_{i}-{\rm{i}}{\gamma }_{i}/2)$ of ${\hat{H}}_{0}$ are given by the energies ${\omega }_{i}$ and the decay rates ${\gamma }_{i}$ , included in order to effectively account for broadening effects in the experiment and defining an effective time scale for the dipole decay [21, 26]. Transition energies ${\omega }_{{ij}}={\omega }_{i}-{\omega }_{j}$ are equal to ${\omega }_{21}=1.56\,\mathrm{eV}$ and ${\omega }_{31}=1.59\,\mathrm{eV}$ [38–40], whereas we set ${\gamma }_{1}=0$ and ${\gamma }_{2}={\gamma }_{3}=1/(500\,\mathrm{fs})$ . In equation (7), the time- and time-delay-dependent Rabi frequencies have been introduced [42]:

$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{R}}k}(t,\tau )={D}_{1k}[{{ \mathcal E }}_{\mathrm{pr},0}\,f(t-\tau ){{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}\tau }+{{ \mathcal E }}_{\mathrm{pu},0}\,f(t)].\end{eqnarray} \tag{ 8 }$

The equations of motion (EOMs) satisfied by the vector

$\begin{eqnarray}&&\vec{c}(t,\tau )={({c}_{1}(t,\tau ),{c}_{2}(t,\tau ),{c}_{3}(t,\tau ))}^{{\rm{T}}},\end{eqnarray} \tag{ 9 }$

of components given by the amplitudes of the state vector $| \psi (t,\tau )\rangle$ , are determined by the Schrödinger equation

$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{{\rm{d}}| \psi (t,\tau )\rangle }{{\rm{d}}t}=\hat{H}| \psi (t,\tau )\rangle ,\end{eqnarray} \tag{ 10 }$

which leads to

$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\vec{c}}{{\rm{d}}t}=\left(\begin{array}{ccc}0 & {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}2}}{2}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}t} & {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}3}}{2}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}t}\\ {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}2}^{* }}{2}\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}t} & -\tfrac{{\gamma }_{2}}{2}-{\rm{i}}{\omega }_{21} & 0\\ {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}3}^{* }}{2}\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}t} & 0 & -\tfrac{{\gamma }_{3}}{2}-{\rm{i}}{\omega }_{31}\end{array}\right)\vec{c}.\end{eqnarray} \tag{ 11 }$

The system is assumed to be initially in its ground state $| {\psi }_{0}\rangle =| 1\rangle$ , i.e., ${c}_{i,0}={\delta }_{i1}$ .

2.2. Transient-absorption spectrum

We solve the EOMs in equation (11) in order to simulate experimental optical-density absorption spectra

$\begin{eqnarray}&&{{ \mathcal S }}_{\exp }(\omega ,\tau )=-\mathrm{log}\left[\displaystyle \frac{{S}_{\mathrm{pr},\mathrm{out}}(\omega ,\tau )}{{S}_{\mathrm{pr},\mathrm{in}}(\omega )}\right],\end{eqnarray} \tag{ 12 }$

where ${S}_{\mathrm{pr},\mathrm{in}}(\omega ,\tau )$ is the spectral intensity of the incoming probe pulse, whereas ${S}_{\mathrm{pr},\mathrm{out}}(\omega ,\tau )$ is that of the transmitted probe pulse, explicitly dependent upon the time delay between pump and probe pulses. For low densities and small medium lengths, where propagation effects can be neglected, the time-delay-dependent absorption spectrum ${{ \mathcal S }}_{\exp }(\omega ,\tau )$ can be calculated in terms of the single-particle dipole response of the system [33]

$\begin{eqnarray}&&{{ \mathcal S }}_{1}(\omega ,\tau )\propto \\ &&\quad -\omega \,{\bf{Im}}\left[\displaystyle \frac{{\displaystyle \sum }_{k=2}^{3}{D}_{1k}^{* }{\displaystyle \int }_{-\infty }^{\infty }{c}_{1}(t,\tau )\,{c}_{k}^{* }(t,\tau ){{\rm{e}}}^{-{\rm{i}}\omega t}\,{\rm{d}}t}{{\displaystyle \int }_{-\infty }^{\infty }{{ \mathcal E }}_{\mathrm{pr}}^{-}(t){{\rm{e}}}^{-{\rm{i}}\omega t}\,{\rm{d}}t}\right],\end{eqnarray} \tag{ 13 }$

where

$\begin{eqnarray}&&{{ \mathcal E }}_{\mathrm{pr}}^{-}(t)=\displaystyle \frac{1}{2}{{ \mathcal E }}_{\mathrm{pr},0}\,f(t-\tau ){{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}(t-\tau )}\end{eqnarray} \tag{ 14 }$

is the negative-frequency complex electric field [41] and ${c}_{1}(t,\tau ){c}_{k}^{* }(t,\tau )$ here represents the dipole response of the kth transition. In the following calculations, the denominator in equation (16) is approximated by

$\begin{eqnarray}&&{\displaystyle \int }_{-\infty }^{\infty }{{ \mathcal E }}_{\mathrm{pr}}^{-}(t){{\rm{e}}}^{-{\rm{i}}\omega t}\,\,{\rm{d}}t\\ &&\quad =\,{{\rm{e}}}^{-{\rm{i}}\omega \tau }\,\displaystyle \frac{{{ \mathcal E }}_{\mathrm{pr},0}}{2}\,{\displaystyle \int }_{-\infty }^{\infty }f(t-\tau ){{\rm{e}}}^{-{\rm{i}}(\omega -{\omega }_{{\rm{L}}})(t-\tau )}{\rm{d}}t\\ &&\quad \approx \,{{\rm{e}}}^{-{\rm{i}}\omega \tau }\,\displaystyle \frac{{{ \mathcal E }}_{\mathrm{pr},0}}{2}{\displaystyle \int }_{-\infty }^{\infty }f(t)\,{\rm{d}}t={{\rm{e}}}^{-{\rm{i}}\omega \tau }\,{K}_{\mathrm{pr}},\end{eqnarray} \tag{ 15 }$

which is valid for an incoming probe pulse much broader than the transition energy between the two excited states, such that its spectral intensity can be approximately considered constant in the frequency range of interest. Spectra associated with different probe-pulse intensities, therefore, need to be properly normalized via the multiplication factor ${K}_{\mathrm{pr}}$ for comparison. Equation (16) can then be rewritten as

$\begin{eqnarray}&&{{ \mathcal S }}_{1}(\omega ,\tau )\,\propto \\ &&\quad -\omega \,{\bf{Im}}\left[\displaystyle \frac{{\displaystyle \sum }_{k=2}^{3}{D}_{1k}^{* }{\displaystyle \int }_{-\infty }^{\infty }{c}_{1}(t,\tau ){c}_{k}^{* }(t,\tau ){{\rm{e}}}^{-{\rm{i}}\omega (t-\tau )}\,{\rm{d}}t}{{K}_{\mathrm{pr}}}\right],\end{eqnarray} \tag{ 16 }$

with the Fourier transform in the numerator centered around the arrival time of the probe pulse.

Since both pump and probe pulses have the same laser frequency, the noncollinear geometry depicted in figure 2 is used in order to measure the spectrum of the probe pulse independent of the spectrum of the pump pulse. However, this also implies that any fast oscillations of the measured transient-absorption spectrum as a function of time delay τ cannot be distinguished [35, 36]. This can be explained as follows. If pump and probe fields propagate in nonidentical directions ${\hat{{\boldsymbol{e}}}}_{\mathrm{pu}}$ and ${\hat{{\boldsymbol{e}}}}_{\mathrm{pr}}$ , respectively, a particle at position ${\boldsymbol{r}}$ is excited with an effective delay $\tau ^{\prime} (\tau ,{\boldsymbol{r}})=\tau -({\hat{{\boldsymbol{e}}}}_{\mathrm{pr}}-{\hat{{\boldsymbol{e}}}}_{\mathrm{pu}})\cdot ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{0})/c$ , with ${{\boldsymbol{r}}}_{0}$ being the position of atoms which actually experience a time delay τ between pump and probe pulses. Since the experimental absorption spectrum results from contributions due to atoms at different positions ${\boldsymbol{r}}$ , i.e., of different effective time delays $\tau ^{\prime}$ , a finite interaction volume ${\rm{\Delta }}{\boldsymbol{r}}$ determines a finite time-delay window ${\rm{\Delta }}\tau$ over which time-delay-dependent features can be experimentally distinguished. As a result, fast changes of the spectrum as a function of time delay, which would characterize the absorption spectrum in a collinear geometry, are effectively averaged out if a noncollinear geometry is employed [35, 36]. Here, this is taken into account by convolving ${S}_{1}(\omega ,\tau )$ with a normalized Gaussian function $G(\tau ,{\rm{\Delta }}\tau )$ of width ${\rm{\Delta }}\tau \,=5\times 2\pi /{\omega }_{{\rm{L}}}$ , which leads to

$\begin{eqnarray}&&{ \mathcal S }(\omega ,\tau )={\int }_{-\infty }^{\infty }G(\tau -\tau ^{\prime} ,{\rm{\Delta }}\tau ){{ \mathcal S }}_{1}(\omega ,\tau ^{\prime} )\,{\rm{d}}\tau ^{\prime} .\end{eqnarray} \tag{ 17 }$

2.3. Analytical model in terms of interaction operators

In order to interpret numerical results from the simulation of ${ \mathcal S }(\omega ,\tau )$ , we employ the recently introduced strong-field interaction operators $\hat{U}(I)$ to model the effect of a pulse of intensity I on the atomic system [37].

The time evolution of the system $| \psi (t)\rangle$ from an initial time t₀, given by the solution of the EOMs (11), can be written in terms of the evolution operator $\hat{{ \mathcal U }}(t,{t}_{0})$ ,

$\begin{eqnarray}&&| \psi (t)\rangle =\hat{{ \mathcal U }}(t,{t}_{0})| \psi ({t}_{0})\rangle .\end{eqnarray} \tag{ 18 }$

In the absence of external fields, this reduces to the free-evolution operator

$\begin{eqnarray}&&\hat{V}(t)={{\rm{e}}}^{-{\rm{i}}{\hat{H}}_{0}t},\end{eqnarray} \tag{ 19 }$

which describes the dynamics of the unperturbed atomic system. The evolution of the system in the presence of a single pulse of intensity $I={{ \mathcal E }}_{0}^{2}/(8\pi \alpha )$ , peak field strength ${{ \mathcal E }}_{0}$ , centered around ${t}_{{\rm{c}}}=0$ and with the same envelope f(t) and pulse duration T we introduced in section 2.1, is then associated with the evolution operator ${\hat{{ \mathcal U }}}_{0}(t,{t}_{0})$ , solution of

$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\hat{{ \mathcal U }}}_{0}(t,{t}_{0})}{{\rm{d}}t}\\ \quad =\,\left(\begin{array}{ccc}0 & {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}2}}{2}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}t} & {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}3}}{2}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}t}\\ {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}2}^{* }}{2}\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}t} & -\tfrac{{\gamma }_{2}}{2}-{\rm{i}}{\omega }_{21} & 0\\ {\rm{i}}\tfrac{{{\rm{\Omega }}}_{{\rm{R}}3}^{* }}{2}\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}t} & 0 & -\tfrac{{\gamma }_{3}}{2}-{\rm{i}}{\omega }_{31}\end{array}\right){\hat{{ \mathcal U }}}_{0}(t,{t}_{0}),\end{eqnarray} \tag{ 20 }$

with initial conditions ${\hat{{ \mathcal U }}}_{0}({t}_{0},{t}_{0})=\hat{I}$ and the identity matrix $\hat{I}$ . In equation (20), the single-pulse Rabi frequencies ${{\rm{\Omega }}}_{{\rm{R}}k}(t)\,={D}_{1k}\,{{ \mathcal E }}_{0}\,f(t)$ are used. For the scheme discussed in this paper, where pump and probe pulses of equal femtosecond duration are employed, the time information related to the continuous evolution of the system in the presence of the pulse can be difficultly extracted. For our purposes, it is therefore beneficial to focus on the total action of the pulse, i.e., on the state reached by the system at the conclusion of the interaction with a pulse. Equation (20) can be used to calculate ${\hat{{ \mathcal U }}}_{0}(T/2,-T/2)$ and thus connect the initial state $| \psi (-T/2)\rangle$ with the final state $| \psi (T/2)\rangle$ at the end of the pulse:

$\begin{eqnarray}&&| \psi (T/2)\rangle ={\hat{{ \mathcal U }}}_{0}(T/2,-T/2)| \psi (-T/2)\rangle .\end{eqnarray} \tag{ 21 }$

However, one can also introduce effective initial ( $| {\psi }^{-}\rangle$ ) and final ( $| {\psi }^{+}\rangle$ ) states

$\begin{eqnarray}&&| {\psi }^{\pm }\rangle =\hat{V}(\mp T/2)| \psi (\pm T/2)\rangle ={{\rm{e}}}^{\pm {\rm{i}}{\hat{H}}_{0}T/2}| \psi (\pm T/2)\rangle \end{eqnarray} \tag{ 22 }$

and thus define the unique, intensity-dependent interaction operators

$\begin{eqnarray}&&\hat{U}(I)=\hat{V}(-T/2){\hat{{ \mathcal U }}}_{0}(T/2,-T/2)\hat{V}(-T/2)\end{eqnarray} \tag{ 23 }$

connecting them,

$\begin{eqnarray}&&| {\psi }^{+}\rangle =\hat{U}(I)| {\psi }^{-}\rangle ,\end{eqnarray} \tag{ 24 }$

thus capturing the essential features of the action of the pulse in terms of an effectively instantaneous interaction, as schematically represented in figure 3. An analytical model can then be derived to describe the associated ${ \mathcal S }(\omega ,\tau )$ , which enables one to quantify how pulse-induced changes in the population and phase of the atomic states are encoded in observable time-delay-dependent spectra.

Figure 3. Refer to the following caption and surrounding text. — **Figure 3.** Schematic illustration of the interaction operators $\hat{U}(I)$ , used to describe the action of an intense pulse on the state of the system, its amplitudes c_i, populations $| {c}_{i}{| }^{2}$ , and coherences ${c}_{i}{c}_{j}^{* }$ , in terms of an effectively instantaneous interaction. The product of matrix elements ${U}_{{ii}^{\prime} }{U}_{{jj}^{\prime} }^{* }$ describes how the action of the pulse connects the initial population/coherence ${c}_{i^{\prime} }{c}_{j^{\prime} }^{* }$ to the final one ${c}_{i}{c}_{j}^{* }$ .
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**Figure 3.** Schematic illustration of the interaction operators $\hat{U}(I)$ , used to describe the action of an intense pulse on the state of the system, its amplitudes c_i, populations $| {c}_{i}{| }^{2}$ , and coherences ${c}_{i}{c}_{j}^{* }$ , in terms of an effectively instantaneous interaction. The product of matrix elements ${U}_{{ii}^{\prime} }{U}_{{jj}^{\prime} }^{* }$ describes how the action of the pulse connects the initial population/coherence ${c}_{i^{\prime} }{c}_{j^{\prime} }^{* }$ to the final one ${c}_{i}{c}_{j}^{* }$ .
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For a weak and ultrashort pulse of peak field strength ${{ \mathcal E }}_{0}$ and envelope f(t), we can introduce approximated Rabi frequencies

$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{R}}k}(t)\approx {\vartheta }_{k}\,\delta (t),\end{eqnarray} \tag{ 25 }$

with the Dirac δ and the pulse areas

$\begin{eqnarray}&&{\vartheta }_{k}={\int }_{-\infty }^{\infty }{D}_{1k}\,{{ \mathcal E }}_{0}\,f(t)\,{\rm{d}}t.\end{eqnarray} \tag{ 26 }$

The solution of equation (20) and the use of the definition (23) allow one to calculate the associated interaction operator which, up to second order, reads

$\begin{eqnarray}{\hat{U}}_{\mathrm{weak}}=\left(\begin{array}{ccc}1-\tfrac{| {\vartheta }_{2}{| }^{2}+| {\vartheta }_{3}{| }^{2}}{8} & {\rm{i}}\tfrac{{\vartheta }_{2}}{2} & {\rm{i}}\tfrac{{\vartheta }_{3}}{2}\\ {\rm{i}}\tfrac{{\vartheta }_{2}^{* }}{2} & 1-\tfrac{| {\vartheta }_{2}{| }^{2}}{8} & -{\vartheta }_{2}^{* }{\vartheta }_{3}\\ {\rm{i}}\tfrac{{\vartheta }_{3}^{* }}{2} & -{\vartheta }_{2}{\vartheta }_{3}^{* } & 1-\tfrac{| {\vartheta }_{3}{| }^{2}}{8}\end{array}\right).\end{eqnarray} \tag{ 27 }$

For completeness, we also provide the solution of the set of differential equations (20) for single-pulse Rabi frequencies ${{\rm{\Omega }}}_{{\rm{R}}k}(t)={D}_{1k}\,{{ \mathcal E }}_{0}\,f(t)$ in the case of vanishing decay rates ${\gamma }_{k}$ and detunings ${{\rm{\Delta }}}_{k}={\omega }_{k1}-{\omega }_{{\rm{L}}}$ . In this case, one can introduce the effective Rabi frequency

$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{R}}}(t)=\sqrt{| {{\rm{\Omega }}}_{{\rm{R}}2}(t){| }^{2}+| {{\rm{\Omega }}}_{{\rm{R}}3}(t){| }^{2}},\end{eqnarray} \tag{ 28 }$

with the effective pulse area

$\begin{eqnarray}&&\vartheta =\sqrt{| {\vartheta }_{2}{| }^{2}+| {\vartheta }_{3}{| }^{2}}\end{eqnarray} \tag{ 29 }$

and the coefficients

$\begin{eqnarray}&&{x}_{k}=\displaystyle \frac{| {{\rm{\Omega }}}_{{\rm{R}}k}(t)| }{{{\rm{\Omega }}}_{{\rm{R}}}(t)}=\displaystyle \frac{| {\vartheta }_{k}| }{\vartheta }=\displaystyle \frac{{D}_{1k}}{\sqrt{{D}_{12}^{2}+{D}_{13}^{2}}},\end{eqnarray} \tag{ 30 }$

such that the associated interaction operator reads

$\begin{eqnarray}&&\hat{U}\\ =\,\left(\begin{array}{ccc}\cos \left(\tfrac{\vartheta }{2}\right) & {\rm{i}}{x}_{2}\sin \left(\tfrac{\vartheta }{2}\right) & {\rm{i}}{x}_{3}\sin \left(\tfrac{\vartheta }{2}\right)\\ {\rm{i}}{x}_{2}\sin \left(\tfrac{\vartheta }{2}\right) & {x}_{2}^{2}\cos \left(\tfrac{\vartheta }{2}\right)+{x}_{3}^{2} & {x}_{2}{x}_{3}\left[\cos \left(\tfrac{\vartheta }{2}\right)-1\right]\\ {\rm{i}}{x}_{3}\sin \left(\tfrac{\vartheta }{2}\right) & {x}_{2}{x}_{3}\left[\cos \left(\tfrac{\vartheta }{2}\right)-1\right] & {x}_{3}^{2}\cos \left(\tfrac{\vartheta }{2}\right)+{x}_{2}^{2}\end{array}\right),\end{eqnarray} \tag{ 31 }$

where we have used the fact that the dipole-moment matrix elements D_1k and hence the coefficients x_k are real in our case. Firstly, for $\vartheta \ll \pi$ , the weak-field solution ${\hat{U}}_{\mathrm{weak}}$ is recovered. Secondly, for increasing values of the pulse area ϑ, the matrix elements of the interaction operator display periodic oscillations, which are the consequence of Rabi flopping in the presence of the short pulse. Particularly relevant is the case of $\vartheta =\pi$ , where the system performs a full switch in the populations of ground and excited states. The solution (31), albeit obtained for vanishing detunings, is a useful starting point to investigate the dynamics of the three-level system in the presence of large-area pulses, and can guide one in the interpretation of the associated absorption line-shape changes.

In the following, we interpret intensity-dependent transient-absorption spectra in terms of the matrix elements of pump- and probe-pulse interaction operators for a probe-pump and pump-probe setup [37]. In contrast to previous results [35], population and phase changes due to the interaction with intense probe and pump pulses are both explicitly addressed. Since we are interested in atomic phases, and in particular in their connection with the phase of the time-delay-dependent oscillations displayed by transient-absorption spectra for positive and negative time delays, we do not focus on the case of overlapping pulses. We are therefore allowed to develop an analytical model in which the dynamics of the system are described in terms of well defined sequences of free evolution and interaction with a pump or a probe pulse of given intensity.

2.3.1. Probe-pump setup

In a probe-pump setup ( $\tau \lt 0$ ), for nonoverlapping pulses and neglecting the details of the continuous atomic dynamics in the presence of a pulse, the time evolution of the system can be written in terms of the state

$\begin{eqnarray}&&| \psi (t,\tau )\rangle \\ \quad =\,\left\{\begin{array}{ll}| {\psi }_{0}\rangle , & t\lt \tau ,\\ \hat{V}(t-\tau ){\hat{U}}_{\mathrm{pr}}({I}_{\mathrm{pr}})| {\psi }_{0}\rangle , & \tau \lt t\lt 0,\\ \hat{V}(t){\hat{U}}_{\mathrm{pu}}({I}_{\mathrm{pu}})\hat{V}(-\tau ){\hat{U}}_{\mathrm{pr}}({I}_{\mathrm{pr}})| {\psi }_{0}\rangle , & t\gt 0,\end{array}\right.\end{eqnarray} \tag{ 32 }$

with $| {\psi }_{0}\rangle =| 1\rangle$ and where we have introduced the pump- and probe-pulse interaction operators, ${\hat{U}}_{\mathrm{pu}}({I}_{\mathrm{pu}})$ and ${\hat{U}}_{\mathrm{pr}}({I}_{\mathrm{pr}})$ , dependent upon the respective pulse intensities. This can be included into equation (16) in order to model the probe-pump spectrum ${{ \mathcal S }}_{1}(\omega ,\tau )$ , $\tau \lt 0$ , in terms of interaction-operator matrix elements:

$\begin{eqnarray}&&{{ \mathcal S }}_{1}(\omega ,\tau )\propto -\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\,{\bf{Im}}\,\left\{\displaystyle \sum _{k=2}^{3}{D}_{1k}^{* }\right.\\ &&\quad \times \,\left[{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k1}^{* }{\displaystyle \int }_{\tau }^{0}{V}_{{kk}}^{* }(t-\tau ){{\rm{e}}}^{-{\rm{i}}\omega (t-\tau )}\,{\rm{d}}t\right.\\ &&\quad +\,\displaystyle \sum _{j,j^{\prime} =1}^{3}{U}_{\mathrm{pu},1j}{U}_{\mathrm{pu},{kj}^{\prime} }^{* }{V}_{{jj}}(-\tau ){V}_{j^{\prime} j^{\prime} }^{* }(-\tau ){U}_{\mathrm{pr},j1}{U}_{\mathrm{pr},j^{\prime} 1}^{* }\\ &&\quad \left.\left.\times \,{{\rm{e}}}^{{\rm{i}}\omega \tau }{\displaystyle \int }_{0}^{\infty }{V}_{{kk}}^{* }(t){{\rm{e}}}^{-{\rm{i}}\omega t}\,{\rm{d}}t\right]\right\},\end{eqnarray} \tag{ 33 }$

resulting in a sum of terms, each of which oscillates as a function of τ at a given frequency. Thereby, one can recognize, for the frequencies $\omega \approx {\omega }_{k1}$ in which we are interested, those terms responsible for fast oscillations of ${{ \mathcal S }}_{1}(\omega ,\tau )$ as a function of time delay which would not be exhibited by a spectrum measured in a noncollinear geometry. After neglecting these fast oscillating terms, the time-delay-average probe-pump spectrum reads

$\begin{eqnarray}{{ \mathcal S }}_{\mathrm{prpu}}(\omega ,\tau ) & \propto & -\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\,{\bf{Im}}\left\{\displaystyle \sum _{k=2}^{3}\displaystyle \frac{{D}_{1k}^{* }}{{\rm{i}}(\omega -{\omega }_{k1})+\tfrac{{\gamma }_{k}}{2}}\right.\\ & & \times [{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k1}^{* }(1-{{\rm{e}}}^{{\rm{i}}(\omega -{\omega }_{k1})\tau }{{\rm{e}}}^{\tfrac{{\gamma }_{k}}{2}\tau })\\ & & +{U}_{\mathrm{pu},11}{U}_{\mathrm{pu},k2}^{* }{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},21}^{* }{{\rm{e}}}^{{\rm{i}}(\omega -{\omega }_{21})\tau }{{\rm{e}}}^{\tfrac{{\gamma }_{2}}{2}\tau }\\ & & +{U}_{\mathrm{pu},11}{U}_{\mathrm{pu},k3}^{* }{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},31}^{* }{{\rm{e}}}^{{\rm{i}}(\omega -{\omega }_{31})\tau }{{\rm{e}}}^{\tfrac{{\gamma }_{3}}{2}\tau }]\phantom{\rule{}{3.9ex}}\}.\end{eqnarray} \tag{ 34 }$

This agrees with the dipole-control model used in [36] for a probe-pump setup, connecting the fit parameters there employed with interaction-operator matrix elements.

2.3.2. Pump-probe setup

When a pump-probe setup is utilized ( $\tau \gt 0$ ), for nonoverlapping pulses and neglecting the details of the continuous atomic dynamics in the presence of a pulse, the atomic state can be modeled as

$\begin{eqnarray}&&| \psi (t,\tau )\rangle \,\\ \quad =\,\left\{\begin{array}{ll}| {\psi }_{0}\rangle , & t\lt 0,\\ \hat{V}(t){\hat{U}}_{\mathrm{pu}}({I}_{\mathrm{pu}})| {\psi }_{0}\rangle , & 0\lt t\lt \tau ,\\ \hat{V}(t-\tau ){\hat{U}}_{\mathrm{pr}}({I}_{\mathrm{pr}})\hat{V}(\tau ){\hat{U}}_{\mathrm{pu}}({I}_{\mathrm{pu}})| {\psi }_{0}\rangle , & t\gt \tau ,\end{array}\right.\end{eqnarray} \tag{ 35 }$

with $| {\psi }_{0}\rangle =| 1\rangle$ . Also here, this can be included into equation (16), resulting in a model for the pump-probe spectrum ${{ \mathcal S }}_{1}(\omega ,\tau )$ , $\tau \gt 0$ , given by:

$\begin{eqnarray}&&{{ \mathcal S }}_{1}(\omega ,\tau )\propto -\,\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\,{\bf{Im}}\left\{\displaystyle \sum _{k=2}^{3}{D}_{1k}^{* }\right.\\ &&\quad \times \,\left[{U}_{\mathrm{pu},11}{U}_{\mathrm{pu},k1}^{* }\,{{\rm{e}}}^{{\rm{i}}\omega \tau }\,{\displaystyle \int }_{0}^{\tau }{V}_{{kk}}^{* }(t){{\rm{e}}}^{-{\rm{i}}\omega t}\,{\rm{d}}t\right.\\ &&\quad +\,\displaystyle \sum _{j,j^{\prime} =1}^{3}{U}_{\mathrm{pr},1j}{U}_{\mathrm{pr},{kj}^{\prime} }^{* }{V}_{{jj}}(\tau ){V}_{j^{\prime} j^{\prime} }^{* }(\tau ){U}_{\mathrm{pu},j1}{U}_{\mathrm{pu},j^{\prime} 1}^{* }\\ &&\quad \left.\left.\times \,{\displaystyle \int }_{\tau }^{\infty }{V}_{{kk}}^{* }(t-\tau ){{\rm{e}}}^{-{\rm{i}}\omega (t-\tau )}\,{\rm{d}}t\right]\right\}.\end{eqnarray} \tag{ 36 }$

By neglecting fast oscillating terms appearing in the resulting single-particle absorption spectrum at frequencies $\omega \approx {\omega }_{k1}$ , the time-delay-average pump-probe spectrum can be written in terms of the matrix elements of the interaction operators ${\hat{U}}_{\mathrm{pu}}({I}_{\mathrm{pu}})$ and ${\hat{U}}_{\mathrm{pr}}({I}_{\mathrm{pr}})$ as

$\begin{eqnarray}&&{{ \mathcal S }}_{\mathrm{pupr}}(\omega ,\tau )\propto -\,\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\,{\bf{Im}}\left[\displaystyle \sum _{k=2}^{3}\displaystyle \frac{{D}_{1k}^{* }}{{\rm{i}}(\omega -{\omega }_{k1})+\tfrac{{\gamma }_{k}}{2}}\right.\\ &&\quad \times \,({U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k1}^{* }| {U}_{\mathrm{pu},11}{| }^{2}\\ &&\quad +\,{U}_{\mathrm{pr},12}{U}_{\mathrm{pr},k2}^{* }| {U}_{\mathrm{pu},21}{| }^{2}\,{{\rm{e}}}^{-{\gamma }_{2}\tau }\\ &&\quad +\,{U}_{\mathrm{pr},12}{U}_{\mathrm{pr},k3}^{* }{U}_{\mathrm{pu},21}\,{U}_{\mathrm{pu},31}^{* }\,{{\rm{e}}}^{{\rm{i}}{\omega }_{32}\tau }\,{{\rm{e}}}^{-\tfrac{{\gamma }_{2}+{\gamma }_{3}}{2}\tau }\\ &&\quad +\,{U}_{\mathrm{pr},13}{U}_{\mathrm{pr},k2}^{* }{U}_{\mathrm{pu},31}\,{U}_{\mathrm{pu},21}^{* }\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{32}\tau }\,{{\rm{e}}}^{-\tfrac{{\gamma }_{2}+{\gamma }_{3}}{2}\tau }\\ &&\quad +\,{U}_{\mathrm{pr},13}{U}_{\mathrm{pr},k3}^{* }| {U}_{\mathrm{pu},31}{| }^{2}\,{{\rm{e}}}^{-{\gamma }_{3}\tau })\phantom{\rule{}{3.7ex}}].\end{eqnarray} \tag{ 37 }$

3. Results and discussion

3.1. Transient-absorption spectra for intense probe and pump pulses

Here, we apply our three-level model to study Rb atoms excited by intense femtosecond probe and pump pulses. Simulated time-delay-dependent transient-absorption spectra, obtained by numerically solving equation (11) and then using this solution in equations (16) and (17), are displayed in figure 4 for representative values of pump- and probe-pulse intensities and for a laser frequency of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ . The probe intensity is varied along the vertical axis, the pump intensity along the horizontal one. For both pump and probe pulses, we consider intensities which span from the weak-field limit ( $1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , $\vartheta =0.28\,\pi$ ) to the Rabi-flopping regime ( $1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , $\vartheta =0.87\,\pi ;$ and $2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , $\vartheta =1.46\,\pi$ ). In particular, for effective pulse areas of $\vartheta \sim \pi$ , nontrivial amplitude and phase changes are induced, whose signatures are expected to appear in the transient-absorption spectra. For all sets of intensities investigated, two absorption lines can be distinguished, respectively centered on the transition energies ${\omega }_{21}=1.56\,\mathrm{eV}$ and ${\omega }_{31}=1.59\,\mathrm{eV}$ . The shape and amplitude of these lines is modulated as a function of time delay, featuring oscillations whose period of $2\pi /{\omega }_{32}=140\,\mathrm{fs}$ is given by the beating frequency ${\omega }_{32}$ . This is stressed by the black lines, showing the spectra evaluated at the two transition energies ${\omega }_{21}$ and ${\omega }_{31}$ as a function of τ.

Figure 4. Refer to the following caption and surrounding text. — **Figure 4.** Absorption spectra for laser frequencies of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ , pump intensities of (a)–(c) ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , (d)–(f) ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and (g)–(i) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and probe intensities of (a), (d), (g) ${I}_{\mathrm{pr}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , (b), (e), (h) ${I}_{\mathrm{pr}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and (c), (f), (i) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . In each panel, the top (bottom) black lines represent the absorption spectra evaluated at the transition energy ${\omega }_{31}$ ( ${\omega }_{21}$ ) in arbitrary units. All black lines are on the same scale, with the 0 aligned on the corresponding transition energy. The blue continuous (red dashed) vertical lines correspond to local minima of the spectra evaluated at $\omega ={\omega }_{21}$ ( $\omega ={\omega }_{31}$ ).
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**Figure 4.** Absorption spectra for laser frequencies of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ , pump intensities of (a)–(c) ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , (d)–(f) ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and (g)–(i) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and probe intensities of (a), (d), (g) ${I}_{\mathrm{pr}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , (b), (e), (h) ${I}_{\mathrm{pr}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and (c), (f), (i) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . In each panel, the top (bottom) black lines represent the absorption spectra evaluated at the transition energy ${\omega }_{31}$ ( ${\omega }_{21}$ ) in arbitrary units. All black lines are on the same scale, with the 0 aligned on the corresponding transition energy. The blue continuous (red dashed) vertical lines correspond to local minima of the spectra evaluated at $\omega ={\omega }_{21}$ ( $\omega ={\omega }_{31}$ ).
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Figures 4(a)–(c) show transient-absorption spectra for a weak pump intensity of ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ ( $\vartheta =0.28\,\pi$ ) and three different values of probe intensity. Firstly, we notice that the amplitude of the time-delay-dependent oscillations displayed by the spectra is very small for these weak values of the pump intensity. The shape and amplitude of the absorption lines remain almost completely unchanged throughout the range of τ displayed, with no significant features distinguishing between positive and negative time delays. By modifying the probe intensity, we notice a variation in the strength of the lines, going from absorption for a weak intensity of ${I}_{\mathrm{pr}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ to emission at higher values of intensity.

When higher values of pump-pulse intensity are employed, clear time-delay-dependent features can be distinguished. The amplitude and the phase of these oscillations in τ vary differently, for positive and negative time delays, as a function of pump and probe intensities. Figures 4(a), (d), and (g) show spectra evaluated for a weak probe intensity of ${I}_{\mathrm{pr}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ and increasing values of ${I}_{\mathrm{pu}}$ . For intermediate values of the pump-pulse intensity ( ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ ) and for both positive and negative time delays, the phase of the exhibited time-delay-dependent spectra is the same for the two transition energies, as evinced by the vertical lines which highlight the position of the minima of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ . However, as already discussed in [35], a shift can be recognized for a higher pump intensity of ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ : while the spectra evaluated at ${\omega }_{21}$ and ${\omega }_{31}$ shift in opposite directions for $\tau \lt 0$ as a clear and distinguishable signature of the onset of strong-field effects, a common shift in the same direction takes place at $\tau \gt 0$ when the pump-pulse intensity is increased.

Recognizing these strong-field-induced features and understanding them in terms of intensity-dependent atomic phases becomes more complex when a probe pulse is used which is not sufficiently weak. This appears clearly when one compares figures 4(d)–(f), where results are shown for an intermediately strong pump pulse and different values of the probe intensity. Both at positive and negative time delays, absorption lines evaluated at ${\omega }_{21}$ and ${\omega }_{31}$ feature a shift in opposite directions, which becomes larger at high probe intensities. Similarly, spectra displayed in figures 4(g)–(i) for a pump intensity of ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ show that a probe-pulse-induced shift of the spectra evaluated at ${\omega }_{21}$ and ${\omega }_{31}$ arises for growing values of ${I}_{\mathrm{pr}}$ : at negative time delays, this enlarges the already existent shift due to the strong pump pulse; for positive time delays, where the increase in ${I}_{\mathrm{pu}}$ causes an aligned, common shift of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ , the presence of an intense probe pulse is reflected in additional shifts, analogous to those already recognized for ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ .

It should be noticed that the spectra in figures 4(a), (e), and (i) are calculated for equal pump- and probe-pulse intensities. The dynamics of the system are, therefore, perfectly symmetric with respect to τ, and the system features the same time evolution when equally delayed pump and probe pulses are used, independent of their arriving order. Nevertheless, the spectra exhibited in the above listed figures are clearly not symmetric with respect to τ, and different amplitudes and phases of the time-delay-dependent features of ${ \mathcal S }(\omega ,\tau )$ can be recognized at $\tau \gt 0$ or $\tau \lt 0$ , in spite of identical underlying dynamics. This can be understood by noticing that the spectrum arises from the interference between the electric dipole response of the atomic system with the probe pulse: even when the quantum dynamics are identical, the spectrum still reveals how these influence the first-(second-)arriving probe pulse for $\tau \lt 0$ ( $\tau \gt 0$ ). This is also evident from the definition of the absorption spectrum (16), where the Fourier transform is always centered on the central time τ of the probe pulse, and then from the analytic models in equations (34) and (37), respectively describing time-delay-averaged probe-pump and pump-probe spectra from a noncollinear geometry. Even when identical pump and probe pulses are used ( ${\hat{U}}_{\mathrm{pr}}={\hat{U}}_{\mathrm{pu}}$ ), the spectra evaluated at positive and negative time delays are determined by different interaction-operator matrix elements and hence differ.

In the previous discussion we have focused on the time-delay-dependent properties of the spectra ${ \mathcal S }({\omega }_{k1},\tau )$ , evaluated at the transition energies ${\omega }_{k1}$ . However, the identification of ${\omega }_{21}$ and ${\omega }_{31}$ may not be straightforward experimentally, affecting the properties of the observed time-delay-dependent features and the quantification of the associated phases. In order to better discuss this point and describe the line-shape changes ensuing from the presence of intense pump and probe pulses, in figures 5 and 6, for a probe-pump and pump-probe setup, respectively, we present transient-absorption spectra ${ \mathcal S }(\omega ,{\tau }_{k1})$ , $k\in \{2,3\}$ , evaluated as a function of frequency for fixed values of the time delay, ${\tau }_{21}$ and ${\tau }_{31}$ . Here, the time delay ${\tau }_{21}$ ( ${\tau }_{31}$ ) is the one for which ${ \mathcal S }({\omega }_{21},\tau )$ $[{ \mathcal S }({\omega }_{31},\tau )]$ has a local minimum, as identified in figure 4 by the vertical lines. The pictures show that the identified local-minimum points are not necessarily associated with emission peaks pointing downwards. Furthermore, for negative time delays, where additional frequency modulations appear as shown in figures 4 and 5, one has to disentangle the behavior of the peaks centered on ${\omega }_{k1}$ from the remaining modulations appearing as a function of frequency. Nevertheless, all panels confirm that it is possible to isolate the time-delay-dependent behavior of this central peak and, thereby, identify the particular time delay at which this is minimal.

Figure 5. Refer to the following caption and surrounding text. — **Figure 5.** Probe-pump transient-absorption spectra evaluated as a function of frequency at two different time delays ${\tau }_{21}$ (blue, continuous) and ${\tau }_{31}$ (red, dashed), for laser frequencies of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ , pump intensities of (a)–(c) ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ and (d)–(f) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and probe intensities of (a), (d) ${I}_{\mathrm{pr}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , (b), (e) ${I}_{\mathrm{pr}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and (c), (f) ${I}_{\mathrm{pr}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . For each panel, the time delay ${\tau }_{21}$ ( ${\tau }_{31}$ ) at which the spectrum is evaluated is associated with the local-minimum point of ${ \mathcal S }({\omega }_{21},\tau )$ $[{ \mathcal S }({\omega }_{31},\tau )]$ highlighted in figure 4 by a corresponding vertical line for $\tau \lt 0$ .
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**Figure 5.** Probe-pump transient-absorption spectra evaluated as a function of frequency at two different time delays ${\tau }_{21}$ (blue, continuous) and ${\tau }_{31}$ (red, dashed), for laser frequencies of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ , pump intensities of (a)–(c) ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ and (d)–(f) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and probe intensities of (a), (d) ${I}_{\mathrm{pr}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , (b), (e) ${I}_{\mathrm{pr}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , and (c), (f) ${I}_{\mathrm{pr}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . For each panel, the time delay ${\tau }_{21}$ ( ${\tau }_{31}$ ) at which the spectrum is evaluated is associated with the local-minimum point of ${ \mathcal S }({\omega }_{21},\tau )$ $[{ \mathcal S }({\omega }_{31},\tau )]$ highlighted in figure 4 by a corresponding vertical line for $\tau \lt 0$ .
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Figure 6. Refer to the following caption and surrounding text. — **Figure 6.** Pump-probe transient-absorption spectra evaluated as a function of frequency at two different time delays ${\tau }_{21}$ (blue, continuous) and ${\tau }_{31}$ (red, dashed), for the same parameters used in figure 5. For each panel, the time delay ${\tau }_{21}$ ( ${\tau }_{31}$ ) at which the spectrum is evaluated is associated with the local-minimum point of ${ \mathcal S }({\omega }_{21},\tau )$ $[{ \mathcal S }({\omega }_{31},\tau )]$ highlighted in figure 4 by a corresponding vertical line for $\tau \gt 0$ .
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Encouraged by the results displayed in figures 5 and 6, in the following we focus on ${ \mathcal S }({\omega }_{k1},\tau )$ and the corresponding time-delay-dependent oscillations in order to draw conclusions about strong-field-induced atomic phases. Figure 7 shows the amplitude of the numerically calculated spectra ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ as a function of probe-pulse intensity for two different values of ${I}_{\mathrm{pu}}$ . The shifts in the phase of the time-delay-dependent spectra is here clearly apparent. For $\tau \gt 0$ or $\tau \lt 0$ , the effect of the intense pump and probe pulses appears in the spectrum as independent pump- and probe-induced phase shifts. In the following, in order to investigate this point further and identify how atomic-phase changes are encoded in transient-absorption spectra, we interpret our results in terms of the interaction operators introduced in section 2.3.

Figure 7. Refer to the following caption and surrounding text. — **Figure 7.** Absorption spectra evaluated at (a), (b) ${\omega }_{21}$ and (c), (d) ${\omega }_{31}$ as a function of probe intensity and time delay, for a laser frequency of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ and at fixed pump intensities of (a), (c) ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ and (b), (d) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . The blue continuous lines in panels (a) and (b) and the red dashed lines in panels (c) and (d) correspond to the local-minimum points (as a function of probe-pulse intensity) of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ , respectively.
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**Figure 7.** Absorption spectra evaluated at (a), (b) ${\omega }_{21}$ and (c), (d) ${\omega }_{31}$ as a function of probe intensity and time delay, for a laser frequency of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ and at fixed pump intensities of (a), (c) ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ and (b), (d) ${I}_{\mathrm{pu}}=2.8\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ . The blue continuous lines in panels (a) and (b) and the red dashed lines in panels (c) and (d) correspond to the local-minimum points (as a function of probe-pulse intensity) of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ , respectively.
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3.2. Interpretation of pump- and probe-pulse-induced phases in terms of interaction-operator matrix elements

Here, we use equations (34) and (37) in order to interpret the numerically calculated transient-absorption spectra presented in section 3.1 in terms of interaction-operator matrix elements. In particular, we focus on the phase of the time-delay-dependent oscillations exhibited by ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ (figure 7), and show how these can be understood via the strong-field-induced atomic phases quantified in ${\hat{U}}_{\mathrm{pu}}$ and ${\hat{U}}_{\mathrm{pr}}$ . For both a probe-pump and a pump-probe setup, we develop analytical interpretation models, calculate ${\hat{U}}_{\mathrm{pu}}$ and ${\hat{U}}_{\mathrm{pr}}$ with equations (20) and (23), and then use these interpretation models to understand the phase features displayed by the transient-absorption spectra in figures 4 and 7. Finally, we further investigate the dependence of the phases extractable from transient-absorption spectra upon the laser frequency of the pump and probe pulses.

3.2.1. Probe-pump setup

Firstly, we focus on the probe-pump interpretation model given by equation (34), aiming at better understanding the properties of the spectrum evaluated at $\omega ={\omega }_{k1}$ . For interpretation purposes, since ${\omega }_{32}\gg {\gamma }_{k}$ , we are allowed to neglect in first approximation the term proportional to ${D}_{1k^{\prime} }^{* }/({\rm{i}}\,{\omega }_{{kk}^{\prime} }+{\gamma }_{k}/2)$ , with $k^{\prime} \in \{2,3\}$ , $k^{\prime} \ne k$ , ${\omega }_{{kk}^{\prime} }=\pm {\omega }_{32}$ , thus obtaining

$\begin{eqnarray}&&{{ \mathcal S }}_{\mathrm{prpu}}({\omega }_{k1},\tau )\propto -\,\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\,\,{\bf{Im}}\left\{2\displaystyle \frac{{D}_{1k}^{* }}{{\gamma }_{k}}\right.\\ &&\quad \times \,[{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k1}^{* }(1-{{\rm{e}}}^{\tfrac{{\gamma }_{k}}{2}\tau })\\ &&\quad +\,{U}_{\mathrm{pu},11}{U}_{\mathrm{pu},{kk}}^{* }{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k1}^{* }{{\rm{e}}}^{\tfrac{{\gamma }_{k}}{2}\tau }\\ &&\quad +\,{U}_{\mathrm{pu},11}{U}_{\mathrm{pu},{kk}^{\prime} }^{* }{U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k^{\prime} 1}^{* }{{\rm{e}}}^{{\rm{i}}{\omega }_{{kk}^{\prime} }\tau }{{\rm{e}}}^{\tfrac{{\gamma }_{k^{\prime} }}{2}\tau }]\phantom{\rule{}{3.1ex}}\}.\end{eqnarray} \tag{ 38 }$

The only term which displays oscillations as a function of τ is given by

$\begin{eqnarray}&&{\tilde{{ \mathcal S }}}_{\mathrm{prpu}}({\omega }_{k1},\tau )\propto -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{1k}}{{\gamma }_{k}}{{\rm{e}}}^{\tfrac{{\gamma }_{k^{\prime} }}{2}\tau }\,{\bf{Im}}({Y}_{\mathrm{pu},k}\,{Y}_{\mathrm{pr},k}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{{kk}^{\prime} }\tau }),\end{eqnarray} \tag{ 39 }$

with

$\begin{eqnarray}{Y}_{\mathrm{pr},k} & = & {U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k^{\prime} 1}^{* },\\ {Y}_{\mathrm{pu},k} & = & {U}_{\mathrm{pu},11}{U}_{\mathrm{pu},{kk}^{\prime} }^{* },\end{eqnarray} \tag{ 40 }$

and where we have used explicitly the fact that, for our atomic implementation with Rb atoms, the projections D_1k of the dipole-moment matrix elements ${{\boldsymbol{D}}}_{1k}$ along the pulse polarization axis ${\hat{{\boldsymbol{e}}}}_{z}$ are real. We can more explicitly write

$\begin{eqnarray}{\tilde{{ \mathcal S }}}_{\mathrm{prpu}}({\omega }_{21},\tau ) & \propto & -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{12}}{{\gamma }_{2}}{{\rm{e}}}^{\tfrac{{\gamma }_{3}}{2}\tau }\,| {Y}_{\mathrm{pu},2}| \,| {Y}_{\mathrm{pr},2}| \\ & & \times \,{\bf{Im}}\{{{\rm{e}}}^{-{\rm{i}}[{\omega }_{32}\tau -\arg ({Y}_{\mathrm{pu},2})-\arg ({Y}_{\mathrm{pr},2})]}\}\\ & = & -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{12}}{{\gamma }_{2}}{{\rm{e}}}^{\tfrac{{\gamma }_{3}}{2}\tau }\,| {Y}_{\mathrm{pu},2}| \,| {Y}_{\mathrm{pr},2}| \\ & & \times \sin [{\omega }_{32}\tau -\arg ({Y}_{\mathrm{pu},2})-\pi -\arg ({Y}_{\mathrm{pr},2})]\end{eqnarray} \tag{ 41 }$

and

$\begin{eqnarray}{\tilde{{ \mathcal S }}}_{\mathrm{prpu}}({\omega }_{31},\tau ) & \propto & -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{13}}{{\gamma }_{3}}{{\rm{e}}}^{\tfrac{{\gamma }_{2}}{2}\tau }\,| {Y}_{\mathrm{pu},3}| \,| {Y}_{\mathrm{pr},3}| \\ & & \times \,{\bf{Im}}\{{{\rm{e}}}^{{\rm{i}}[{\omega }_{32}\tau +\arg ({Y}_{\mathrm{pu},3})+\arg ({Y}_{\mathrm{pr},3})]}\}\\ & = & -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{13}}{{\gamma }_{3}}{{\rm{e}}}^{\tfrac{{\gamma }_{2}}{2}\tau }\,| {Y}_{\mathrm{pu},3}| \,| {Y}_{\mathrm{pr},3}| \\ & & \times \sin [{\omega }_{32}\tau +\arg ({Y}_{\mathrm{pu},3})+\arg ({Y}_{\mathrm{pr},3})].\end{eqnarray} \tag{ 42 }$

With

$\begin{eqnarray}{Y}_{\mathrm{pr},2} & = & {U}_{\mathrm{pr},11}{U}_{\mathrm{pr},31}^{* },\\ {Y}_{\mathrm{pr},3} & = & {U}_{\mathrm{pr},11}{U}_{\mathrm{pr},21}^{* },\\ {Y}_{\mathrm{pu},2} & = & {U}_{\mathrm{pu},11}{U}_{\mathrm{pu},23}^{* },\\ {Y}_{\mathrm{pu},3} & = & {U}_{\mathrm{pu},11}{U}_{\mathrm{pu},32}^{* },\end{eqnarray} \tag{ 43 }$

and the phases

$\begin{eqnarray}{\varphi }_{\mathrm{pr},2} & = & -\pi -\arg ({U}_{\mathrm{pr},11}{U}_{\mathrm{pr},31}^{* }),\\ {\varphi }_{\mathrm{pr},3} & = & \arg ({U}_{\mathrm{pr},11}{U}_{\mathrm{pr},21}^{* }),\\ {\varphi }_{\mathrm{pu},2} & = & -\arg ({U}_{\mathrm{pu},11}{U}_{\mathrm{pu},23}^{* }),\\ {\varphi }_{\mathrm{pu},3} & = & \arg ({U}_{\mathrm{pu},11}{U}_{\mathrm{pu},32}^{* }),\end{eqnarray} \tag{ 44 }$

this reduces to

$\begin{eqnarray}{\tilde{{ \mathcal S }}}_{\mathrm{prpu}}({\omega }_{21},\tau ) & = & -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{12}}{{\gamma }_{2}}{{\rm{e}}}^{\tfrac{{\gamma }_{3}}{2}\tau }\,| {Y}_{\mathrm{pu},2}| \,| {Y}_{\mathrm{pr},2}| \\ & & \times \sin ({\omega }_{32}\tau +{\varphi }_{\mathrm{pr},2}+{\varphi }_{\mathrm{pu},2})\end{eqnarray} \tag{ 45 }$

and

$\begin{eqnarray}{\tilde{{ \mathcal S }}}_{\mathrm{prpu}}({\omega }_{31},\tau ) & = & -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{13}}{{\gamma }_{3}}{{\rm{e}}}^{\tfrac{{\gamma }_{2}}{2}\tau }\,| {Y}_{\mathrm{pu},3}| \,| {Y}_{\mathrm{pr},3}| \\ & & \times \sin ({\omega }_{32}\tau +{\varphi }_{\mathrm{pr},3}+{\varphi }_{\mathrm{pu},3}).\end{eqnarray} \tag{ 46 }$

The intensity-dependent position of the minima of ${ \mathcal S }({\omega }_{k1},\tau )$ for $\tau \lt 0$ , shown in figure 7 by the blue continuous and red dashed lines at negative time delays, can hence be quantified via equations (45) and (46) in terms of ${\varphi }_{\mathrm{pr},k}$ and ${\varphi }_{\mathrm{pu},k}$ . The sine functions appearing therein have local minima, respectively centered around

$\begin{eqnarray}{\tau }_{21}={\tau }_{0}-\displaystyle \frac{({\varphi }_{\mathrm{pr},2}+{\varphi }_{\mathrm{pu},2})}{{\omega }_{32}}, & & \mathrm{for}\ \omega ={\omega }_{21},\tau \lt 0,\\ {\tau }_{31}={\tau }_{0}-\displaystyle \frac{({\varphi }_{\mathrm{pr},3}+{\varphi }_{\mathrm{pu},3})}{{\omega }_{32}}, & & \mathrm{for}\ \omega ={\omega }_{31},\tau \lt 0,\end{eqnarray} \tag{ 47 }$

with an additive offset ${\tau }_{0}=(3\pi /2+2\pi m)/{\omega }_{32}$ , $m\in {\mathbb{Z}}$ . The positions of the local minima displayed in the previous figures by the blue continuous and red dashed lines correspond to ${\tau }_{0}=-9\pi /(2{\omega }_{32})$ . For real, positive dipole-moment matrix elements D_1k, and hence real positive pulse areas ${\vartheta }_{k}$ , the intensity-dependent variables ${Y}_{\mathrm{pr},k}$ and ${Y}_{\mathrm{pu},k}$ can be explicitly written in the case of weak pulses via equation (27) as

$\begin{eqnarray}{Y}_{\mathrm{pr},2}^{\mathrm{weak}} & = & -{\rm{i}}\displaystyle \frac{{\vartheta }_{3}}{2},\\ {Y}_{\mathrm{pr},3}^{\mathrm{weak}} & = & -{\rm{i}}\displaystyle \frac{{\vartheta }_{2}}{2},\\ {Y}_{\mathrm{pu},2}^{\mathrm{weak}} & = & -{\vartheta }_{2}{\vartheta }_{3},\\ {Y}_{\mathrm{pu},3}^{\mathrm{weak}} & = & -{\vartheta }_{2}{\vartheta }_{3},\end{eqnarray} \tag{ 48 }$

along with the associated phases

$\begin{eqnarray}{\varphi }_{\mathrm{pr},2}^{\mathrm{weak}} & = & -\pi /2,\\ {\varphi }_{\mathrm{pr},3}^{\mathrm{weak}} & = & -\pi /2,\\ {\varphi }_{\mathrm{pu},2}^{\mathrm{weak}} & = & \pm \pi ,\\ {\varphi }_{\mathrm{pu},3}^{\mathrm{weak}} & = & \mp \pi .\end{eqnarray} \tag{ 49 }$

For low intensities, the effect of the probe pulse is linearly proportional to the pulse areas ${\vartheta }_{k}$ and, therefore, of first order in the amplitude of the electric field, whereas the action of the pump pulse depends on the product of ${\vartheta }_{2}{\vartheta }_{3}$ and is hence of second order. This explains the small, almost vanishing amplitude of the time-delay-dependent oscillations displayed for $\tau \lt 0$ by the transient-absorption spectra in figures 4(a)–(c), for a small pump-pulse intensity of ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ .

In figures 8(a) and (b), the total phases $[{\varphi }_{\mathrm{pr},2}+{\varphi }_{\mathrm{pu},2}\,-({\varphi }_{\mathrm{pr},2}^{\mathrm{weak}}+{\varphi }_{\mathrm{pu},2}^{\mathrm{weak}})]$ and $[{\varphi }_{\mathrm{pr},3}+{\varphi }_{\mathrm{pu},3}-({\varphi }_{\mathrm{pr},3}^{\mathrm{weak}}\,+{\varphi }_{\mathrm{pu},3}^{\mathrm{weak}})]$ (equations (44) and (49) after numerical calculation of ${\hat{U}}_{\mathrm{pr}}$ and ${\hat{U}}_{\mathrm{pu}}$ via equations (20) and (23)) are exhibited, as a function of ${I}_{\mathrm{pr}}$ and for a discrete set of values of ${I}_{\mathrm{pu}}$ . The very good agreement between the intensity dependence of these phases and the shift displayed by the time-delay-dependent features of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ (figure 7 and figures 8(c) and (d) at negative time delays) confirms the validity of our analytical interpretation model and in particular of equation (47). The shift in the phases (figures 8(a) and (b)) is reflected by an oppositely directed shift in the local-minimum points (figures 8(c) and (d)) as a function of ${I}_{\mathrm{pr}}$ and ${I}_{\mathrm{pu}}$ , as expected from the minus sign in equation (47).

Figure 8. Refer to the following caption and surrounding text. — **Figure 8.** Correspondence between strong-field-induced atomic phases and time-delay-dependent oscillations of the transient-absorption spectra in a probe-pump setup and for a laser frequency of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ . (a), (b) Total phases (a) $[{\varphi }_{\mathrm{pr},2}+{\varphi }_{\mathrm{pu},2}-({\varphi }_{\mathrm{pr},2}^{\mathrm{weak}}+{\varphi }_{\mathrm{pu},2}^{\mathrm{weak}})]$ and (b) $[{\varphi }_{\mathrm{pr},3}\,+{\varphi }_{\mathrm{pu},3}-({\varphi }_{\mathrm{pr},3}^{\mathrm{weak}}+{\varphi }_{\mathrm{pu},3}^{\mathrm{weak}})];$ and (c), (d) time delays associated with minima in the absorption spectra (c) ${ \mathcal S }({\omega }_{21},\tau )$ and (d) ${ \mathcal S }({\omega }_{31},\tau )$ for negative time delays. In all panels, curves are displayed as a function of probe intensity and for pump intensities of ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (blue continuous line), ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (orange dashed line), ${I}_{\mathrm{pu}}=1.9\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (green dotted line), and ${I}_{\mathrm{pu}}=2.8\,\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (red dashed–dotted line). Phases and time delays are shifted in opposite directions, as expected from the minus sign in equation (47).
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In order to understand the physics underlying the phase shifts ${\varphi }_{\mathrm{pr},k}$ appearing in the spectrum, we can use the schematic illustration of $\hat{U}(I)$ in figure 3 to clarify the meaning of the terms appearing in equations (43) and (44). The associated terms ${Y}_{\mathrm{pr},k}={U}_{\mathrm{pr},11}{U}_{\mathrm{pr},k^{\prime} 1}^{* }$ , $k^{\prime} \ne k$ , are the coherences (in amplitude and phase) generated by the first-arriving probe pulse acting on the ground state. The shift displayed by ${ \mathcal S }({\omega }_{k1},\tau )$ is therefore related to the phase of these strong-field-induced coherences. The different sign appearing in the definition of ${\varphi }_{\mathrm{pr},2}$ and ${\varphi }_{\mathrm{pr},3}$ also explains why the time-delay-dependent oscillations of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ shift in opposite directions for increasing probe-pulse intensities (figures 8(c) and (d)).

The explicit dependence of ${\varphi }_{\mathrm{pr},2}$ and ${\varphi }_{\mathrm{pr},3}$ upon the probe-pulse intensity was here calculated numerically and will be more thoroughly investigated in section 3.2.3, also as a function of the laser frequency. For interpretation purposes, these results can be compared with the analytical prediction provided by equation (31), where the interaction-operator matrix elements are calculated assuming vanishing detunings and decay rates. These formulas predict constant values of $\pm \pi /2$ for both ${\varphi }_{\mathrm{pr},2}$ and ${\varphi }_{\mathrm{pr},3}$ , with a sudden shift of ${\rm{\Delta }}{\varphi }_{\mathrm{pr}}=\pi$ when the effective pulse area is equal to $\vartheta =\pi$ ( ${I}_{\mathrm{pr}}=1.3\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ ). Clearly, these formulas cannot be rigorously applied in our case, where the detuning ${{\rm{\Delta }}}_{2}={\omega }_{21}-{\omega }_{{\rm{L}}}=-30\,\mathrm{meV}$ is comparable with the peak value of the effective Rabi frequency ${{\rm{\Omega }}}_{{\rm{R}}}(t)$ (ranging from 10 to $80\,\mathrm{meV}$ for the set of intensities considered in figure 8). The phases in figures 8(a) and (b) do not exhibit the sudden jump predicted by equation (31), but rather a smooth dependence on the probe-pulse intensity. Nevertheless, they feature a large change between $1\times {10}^{10}$ and $1.9\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ compared to the changes above and below these intensities, in an interval which is approximately centered around the intensity of a pulse with $\vartheta =\pi$ . This stresses the importance of strong-field-induced Rabi flopping in the nontrivial phase dynamics undergone by the system, also for the case of nonvanishing detunings, and the importance of experiment-based methods to extract these phase evolutions for the strong pulses considered here.

The second-arriving intense pump pulse nonlinearly modifies an already existent superposition of excited states, as previously investigated in [35]. The formalism developed here allows us to show how the shifts ${\varphi }_{\mathrm{pu},2}$ and ${\varphi }_{\mathrm{pu},3}$ in the oscillating features of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ , respectively, quantify the changes in the atomic phases induced by the pump pulse. This can be recognized via inspection of the associated interaction-operator matrix elements, ${Y}_{\mathrm{pu},k}={U}_{\mathrm{pu},11}{U}_{\mathrm{pu},{kk}^{\prime} }^{* }$ , $k^{\prime} \ne k$ (equations (43) and (44)), which describe how the pump pulse transforms an initial coherence between ground state and excited state $| k^{\prime} \rangle$ into a final coherence between ground state and excited state $| k\rangle$ (see also the schematic illustration in figure 3). The ensuing phase change determines the shift appearing in the oscillating features of the transient-absorption spectrum. Also in this case, the shift in opposite directions displayed by ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ for rising values of ${I}_{\mathrm{pu}}$ (figures 8(c) and (d)) is a consequence of the opposite sign with which ${\varphi }_{\mathrm{pu},2}$ and ${\varphi }_{\mathrm{pu},3}$ are related to the interaction-operator matrix elements (equation (44)).

3.2.2. Pump-probe setup

Here, we focus on the positive-time-delay part of the spectrum, and use the associated interpretation model given by equation (37) in order to better understand the properties of the spectra evaluated at $\omega ={\omega }_{k1}$ . For this purpose, as already performed in the previous part, we can neglect terms given by ${D}_{1k^{\prime} }^{* }/({\rm{i}}{\omega }_{{kk}^{\prime} }+{\gamma }_{k^{\prime} }/2)$ in equation (37), and thus identify those contributions which are responsible for the oscillations exhibited by the spectrum as a function of τ:

$\begin{eqnarray}&&{\tilde{{ \mathcal S }}}_{\mathrm{pupr}}({\omega }_{k1},\tau )\propto -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{1k}}{{\gamma }_{k}}\,{{\rm{e}}}^{-\tfrac{{\gamma }_{2}+{\gamma }_{3}}{2}\tau }\\ &&\quad \times \,{\bf{Im}}({U}_{\mathrm{pr},12}{U}_{\mathrm{pr},k3}^{* }{U}_{\mathrm{pu},21}{U}_{\mathrm{pu},31}^{* }\,{{\rm{e}}}^{{\rm{i}}{\omega }_{32}\tau }\\ &&\quad +\,{U}_{\mathrm{pr},13}{U}_{\mathrm{pr},k2}^{* }{U}_{\mathrm{pu},31}{U}_{\mathrm{pu},21}^{* }\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{32}\tau }).\end{eqnarray} \tag{ 50 }$

Also in this case, we have used explicitly the fact that the dipole-moment matrix elements D_1k are real. By introducing the intensity-dependent pump and probe variables

$\begin{eqnarray}{Z}_{\mathrm{pu}} & = & {U}_{\mathrm{pu},21}{U}_{\mathrm{pu},31}^{* },\\ {A}_{\mathrm{pr},k} & = & {U}_{\mathrm{pr},12}{U}_{\mathrm{pr},k3}^{* },\\ {B}_{\mathrm{pr},k} & = & {U}_{\mathrm{pr},13}{U}_{\mathrm{pr},k2}^{* },\end{eqnarray} \tag{ 51 }$

we can write equation (50) as

$\begin{eqnarray}&&{\tilde{{ \mathcal S }}}_{\mathrm{pupr}}({\omega }_{k1},\tau )\propto -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{1k}}{{\gamma }_{k}}\,{{\rm{e}}}^{-\tfrac{{\gamma }_{2}+{\gamma }_{3}}{2}\tau }\\ &&\quad \times \,{\bf{Im}}({A}_{\mathrm{pr},k}\,{Z}_{\mathrm{pu}}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{32}\tau }+{B}_{\mathrm{pr},k}\,{Z}_{\mathrm{pu}}^{* }\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{32}\tau }),\end{eqnarray} \tag{ 52 }$

and observe that the pump pulse equally acts on both terms of the above sums, resulting in a phase shift

$\begin{eqnarray}&&{\psi }_{\mathrm{pu}}=\arg ({Z}_{\mathrm{pu}})=\arg ({U}_{\mathrm{pu},21}{U}_{\mathrm{pu},31}^{* }).\end{eqnarray} \tag{ 53 }$

Furthermore, since ${\bf{Im}}(z)=-\,{\bf{Im}}({z}^{* })$ , we have that ${\bf{Im}}\{{B}_{\mathrm{pr},k}\,{Z}_{\mathrm{pu}}^{* }\,{{\rm{e}}}^{-{\rm{i}}{\omega }_{32}\tau }\}=-\,{\bf{Im}}\{{B}_{\mathrm{pr},k}^{* }\,{Z}_{\mathrm{pu}}\,{{\rm{e}}}^{{\rm{i}}{\omega }_{32}\tau }\}$ , and hence

$\begin{eqnarray}&&{\tilde{{ \mathcal S }}}_{\mathrm{pupr}}({\omega }_{k1},\tau )\propto -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{1k}}{{\gamma }_{k}}\,{{\rm{e}}}^{-\tfrac{{\gamma }_{2}+{\gamma }_{3}}{2}\tau }\,| {Z}_{\mathrm{pu}}| \\ &&\quad \times \,{\bf{Im}}[({A}_{\mathrm{pr},k}-{B}_{\mathrm{pr},k}^{* })\,{{\rm{e}}}^{{\rm{i}}({\omega }_{32}\tau +{\psi }_{\mathrm{pu}})}].\end{eqnarray} \tag{ 54 }$

By further introducing the phases

$\begin{eqnarray}{\psi }_{\mathrm{pr},2} & = & \arg ({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })\\ & = & \arg ({U}_{\mathrm{pr},12}{U}_{\mathrm{pr},23}^{* }-{U}_{\mathrm{pr},22}{U}_{\mathrm{pr},13}^{* }),\\ {\psi }_{\mathrm{pr},3} & = & \arg ({A}_{\mathrm{pr},3}-{B}_{\mathrm{pr},3}^{* })\\ & = & \arg ({U}_{\mathrm{pr},12}{U}_{\mathrm{pr},33}^{* }-{U}_{\mathrm{pr},32}{U}_{\mathrm{pr},13}^{* }),\end{eqnarray} \tag{ 55 }$

the spectrum can be written as

$\begin{eqnarray}&&{\tilde{{ \mathcal S }}}_{\mathrm{pupr}}({\omega }_{k1},\tau )\propto -2\displaystyle \frac{\omega }{{K}_{\mathrm{pr}}}\displaystyle \frac{{D}_{1k}}{{\gamma }_{k}}\,{{\rm{e}}}^{-\tfrac{{\gamma }_{2}+{\gamma }_{3}}{2}\tau }\,| {Z}_{\mathrm{pu}}| \,| {A}_{\mathrm{pr},k}-{B}_{\mathrm{pr},k}^{* }| \\ &&\quad \times \sin ({\omega }_{32}\tau +{\psi }_{\mathrm{pu}}+{\psi }_{\mathrm{pr},k}).\end{eqnarray} \tag{ 56 }$

This implies that the intensity-dependent positions of the minima of ${ \mathcal S }({\omega }_{k1},\tau )$ , shown in figure 7 by the blue continuous and red dashed lines at positive time delays, can be quantified via equation (56) in terms of ${\psi }_{\mathrm{pu}}$ and ${\psi }_{\mathrm{pr},k}$ . The sine functions appearing therein have local minima respectively centered around

$\begin{eqnarray}{\tau }_{21} & = & {\tau }_{0}-\displaystyle \frac{({\psi }_{\mathrm{pu}}+{\psi }_{\mathrm{pr},2})}{{\omega }_{32}},\ \mathrm{for}\ \omega ={\omega }_{21},\tau \gt 0,\\ {\tau }_{31} & = & {\tau }_{0}-\displaystyle \frac{({\psi }_{\mathrm{pu}}+{\psi }_{\mathrm{pr},3})}{{\omega }_{32}},\ \mathrm{for}\ \omega ={\omega }_{31},\tau \gt 0,\end{eqnarray} \tag{ 57 }$

with an additive offset ${\tau }_{0}=(3\pi /2+2\pi m)/{\omega }_{32}$ , $m\in {\mathbb{Z}}$ . The positions of the local minima displayed in figures 4 and 7 by the blue continuous and red dashed lines correspond to ${\tau }_{0}=7\pi /(2{\omega }_{32})$ . For real, positive dipole-moment matrix elements D_1k, and hence real positive pulse areas ${\vartheta }_{k}$ , the intensity-dependent variables ${Z}_{\mathrm{pu}}$ and $({A}_{\mathrm{pr},k}-{B}_{\mathrm{pr},k}^{* })$ can be explicitly written in the case of weak pulses via equation (27) as

$\begin{eqnarray}{Z}_{\mathrm{pu}}^{\mathrm{weak}} & = & \displaystyle \frac{{\vartheta }_{2}{\vartheta }_{3}}{4},\\ {A}_{\mathrm{pr},2}^{\mathrm{weak}}-{({B}_{\mathrm{pr},2}^{\mathrm{weak}})}^{* } & = & {\rm{i}}\displaystyle \frac{{\vartheta }_{3}}{2},\\ {A}_{\mathrm{pr},3}^{\mathrm{weak}}-{({B}_{\mathrm{pr},3}^{\mathrm{weak}})}^{* } & = & {\rm{i}}\displaystyle \frac{{\vartheta }_{2}}{2},\end{eqnarray} \tag{ 58 }$

along with the associated phases

$\begin{eqnarray}{\psi }_{\mathrm{pu}}^{\mathrm{weak}} & = & 0,\\ {\psi }_{\mathrm{pr},2}^{\mathrm{weak}} & = & \pi /2,\\ {\psi }_{\mathrm{pr},3}^{\mathrm{weak}} & = & \pi /2.\end{eqnarray} \tag{ 59 }$

Also in a pump-probe setup, the effect of a weak probe pulse is linearly proportional to the pulse areas ${\vartheta }_{k}$ and, therefore, of first order in the amplitude of the electric field. The action of a weak pump pulse depends on the product of ${\vartheta }_{2}{\vartheta }_{3}$ and is hence of second order. Also in this case, this explains the small, almost vanishing amplitude of the time-delay-dependent oscillations displayed for $\tau \gt 0$ by the transient-absorption spectra in figures 4(a)–(c), for a small pump-pulse intensity of ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ .

Figures 9(a) and (b) display the total phases $[{\psi }_{\mathrm{pu}}+{\psi }_{\mathrm{pr},2}-({\psi }_{\mathrm{pu}}^{\mathrm{weak}}+{\psi }_{\mathrm{pr},2}^{\mathrm{weak}})]$ and $[{\psi }_{\mathrm{pu}}+{\psi }_{\mathrm{pr},3}-({\psi }_{\mathrm{pu}}^{\mathrm{weak}}\,+{\psi }_{\mathrm{pr},3}^{\mathrm{weak}})]$ (equations (53), (55), and (59), after numerical calculation of ${\hat{U}}_{\mathrm{pr}}$ and ${\hat{U}}_{\mathrm{pu}}$ via equations (20) and (23)) as a function of ${I}_{\mathrm{pr}}$ and for different values of the pump-pulse intensity ${I}_{\mathrm{pu}}$ . The dependence of these phases on pulse intensities matches that exhibited by the time-delay-dependent features of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ in figure 7 and in figures 9(c) and (d) at positive time delays, confirming the validity of equation (57) for the interpretation of the phase of the oscillating features displayed by transient-absorption spectra. Also in this case, the minus sign in equation (57) results in a shift in the local-minimum points in figures 9(c) and (d) in a direction which is opposite to the change in phase exhibited by figures 9(a) and (b).

Figure 9. Refer to the following caption and surrounding text. — **Figure 9.** Correspondence between strong-field-induced atomic phases and time-delay-dependent oscillations of the transient-absorption spectra in a pump-probe setup and for a laser frequency of ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ . (a), (b) Total phases (a) $[{\psi }_{\mathrm{pr},2}+{\psi }_{\mathrm{pu}}\,-({\psi }_{\mathrm{pr},2}^{\mathrm{weak}}+{\psi }_{\mathrm{pu}}^{\mathrm{weak}})]$ and (b) $[{\psi }_{\mathrm{pr},3}\,+{\psi }_{\mathrm{pu}}-({\psi }_{\mathrm{pr},3}^{\mathrm{weak}}+{\psi }_{\mathrm{pu}}^{\mathrm{weak}})];$ and (c), (d) time delays associated with minima in the absorption spectra (c) ${ \mathcal S }({\omega }_{21},\tau )$ and (d) ${ \mathcal S }({\omega }_{31},\tau )$ for positive time delays. In all panels, curves are displayed as a function of probe intensity and for pump intensities of ${I}_{\mathrm{pu}}=1\times {10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (blue continuous line), ${I}_{\mathrm{pu}}=1\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (orange dashed line), ${I}_{\mathrm{pu}}=1.9\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (green dotted line), and ${I}_{\mathrm{pu}}=2.8\,\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (red dashed–dotted line). Phases and time delays are shifted in opposite directions, as expected from the minus sign in equation (57).
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In contrast to the previously discussed probe-pump case, here, the first-arriving pump pulse equally influences the shift in the spectra evaluated at ${\omega }_{21}$ and ${\omega }_{31}$ . This was already observed in [35], and can now be understood in terms of the atomic-phase changes imposed by the pump pulse via equation (57). The same phase ${\psi }_{\mathrm{pu}}$ equally affects both spectra, with a common shift which quantifies the phase difference between excited states generated by the first-arriving pump pulse. This is apparent by the definition of ${\psi }_{\mathrm{pu}}$ (equation (53)) and of the associated term ${Z}_{\mathrm{pu}}={U}_{\mathrm{pu},21}{U}_{\mathrm{pu},31}^{* }$ (equation (51)), which represents the coherence between excited states $| 2\rangle$ and $| 3\rangle$ resulting from the interaction with the pump pulse, as schematically illustrated in figure 3.

Quantifying the shift in the spectra induced by the second-arriving probe pulse is more complex. In a pump-probe setup, the probe pulse modifies the state excited by the first-arriving pump pulse, inducing atomic-phase changes which are encoded in the spectrum. However, in this case, the phases ${\psi }_{\mathrm{pr},k}$ (equation (55)) of the time-delay-dependent oscillations of ${ \mathcal S }({\omega }_{k1},\tau )$ are due to a sum of terms ( $({A}_{\mathrm{pr},k}-{B}_{\mathrm{pr},k}^{* })$ from equation (51)). As a result, the phases ${\psi }_{\mathrm{pr},k}$ , and hence the corresponding phase shifts featured by the spectra, are not only determined by the phases of the corresponding interaction-operator matrix elements ( ${A}_{\mathrm{pr},k}={U}_{\mathrm{pr},12}{U}_{\mathrm{pr},k3}^{* }$ and ${B}_{\mathrm{pr},k}={U}_{\mathrm{pr},13}{U}_{\mathrm{pr},k2}^{* }$ ), but also by their amplitudes. The definition of the interaction operator $\hat{U}(I)$ allows one to see that ${A}_{\mathrm{pr},k}$ and ${B}_{\mathrm{pr},k}$ describe how the probe pulse transforms an initial coherence between the excited states $| 2\rangle$ and $| 3\rangle$ into a coherence between ground state and excited state $| k\rangle$ (see also the schematic illustration in figure 3). Amplitude and phase of these interaction-operator matrix elements both enter the definition of ${\psi }_{\mathrm{pr},k}$ and are hence encoded in the absorption spectra.

3.2.3. Dependence on laser frequency

Since we confirmed in the previous subsections the validity of equations (47) and (57) for the interpretation of transient-absorption spectra in terms of pump- and probe-pulse-generated phases, here we focus on the previously introduced phases ${\varphi }_{\mathrm{pr},k}$ , ${\varphi }_{\mathrm{pu},k}$ , ${\psi }_{\mathrm{pr},k}$ , and ${\psi }_{\mathrm{pu}}$ , and investigate their dependence upon the frequency of the laser. Also in this case, this is achieved by using equations (44), (49), (53), (55), and (59), after having numerically calculated ${\hat{U}}_{\mathrm{pr}}$ and ${\hat{U}}_{\mathrm{pu}}$ via equations (20) and (23). However, while we assumed in the previous sections that both pump and probe pulses were characterized by a laser frequency ${\omega }_{{\rm{L}}}=1.59\,\mathrm{eV}$ , we display here intensity-dependent results for 5 discrete values of laser frequency, equally spaced between ${\omega }_{{\rm{L}}}={\omega }_{21}=1.56\,\mathrm{eV}$ and ${\omega }_{{\rm{L}}}\,={\omega }_{31}=1.59\,\mathrm{eV}$ .

In figures 10 and 11, we focus on the probe-pump setup discussed in section 3.2.1. In particular, figure 10 exhibits the phases induced by the probe pulse, as a function of its intensity and for different values of the laser frequency. The phases presented in this figure are independent of the pump-pulse properties. In figure 10(a), the phase $({\varphi }_{\mathrm{pr},2}-{\varphi }_{\mathrm{pr},2}^{\mathrm{weak}})$ is shown, which determines the probe-intensity-dependent shift featured by the absorption spectra ${ \mathcal S }({\omega }_{21},\tau )$ evaluated at ${\omega }_{21}$ . These phases are related to the argument of ${Y}_{\mathrm{pr},2}$ (figure 10(b) and equation (43)), which represents the coherence between states $| 1\rangle$ and $| 3\rangle$ generated by the first-arriving probe pulse. At low intensities, all curves are characterized by negative, purely imaginary values of ${Y}_{\mathrm{pr},2}$ , in agreement with equation (48). The laser frequency influences the path followed by ${Y}_{\mathrm{pr},2}$ at increasing intensities, and whether this will move towards regions characterized by positive or negative real parts. This influences the behavior of the phases in figure 10(a) as well, deciding whether the shift is towards values of ${\varphi }_{\mathrm{pr},2}$ larger or smaller than the weak-limit value. Similarly, the behavior of ${Y}_{\mathrm{pr},3}$ displayed in figure 10(d) determines the intensity-dependent shift $({\varphi }_{\mathrm{pr},3}-{\varphi }_{\mathrm{pr},3}^{\mathrm{weak}})$ featured by the absorption spectra ${ \mathcal S }({\omega }_{31},\tau )$ evaluated at ${\omega }_{31}$ . Here, ${Y}_{\mathrm{pr},3}$ is the coherence between states $| 1\rangle$ and $| 2\rangle$ generated by the first-arriving probe pulse. Also in this case, weak intensities correspond to negative, purely imaginary values of ${Y}_{\mathrm{pr},3}$ , agreeing with equation (43). A different dependence of ${Y}_{\mathrm{pr},3}$ on probe-pulse intensities is featured for different values of the laser frequency, analogously influencing the intensity-dependent shift ${\varphi }_{\mathrm{pr},3}$ exhibited by figure 10(c).

Figure 10. Refer to the following caption and surrounding text. — **Figure 10.** Probe-pulse-induced phases in a probe-pump setup as a function of probe-pulse intensity and for ${\omega }_{{\rm{L}}}=1.560\,\mathrm{eV}$ (blue continuous line), ${\omega }_{{\rm{L}}}=1.567\,\mathrm{eV}$ (orange dashed line), ${\omega }_{{\rm{L}}}=1.575\,\mathrm{eV}$ (green dotted line), ${\omega }_{{\rm{L}}}=1.582\,\mathrm{eV}$ (red dashed–dotted line), ${\omega }_{{\rm{L}}}=1.590\,\mathrm{eV}$ (brown dashed-double-dotted line). The panels display (a) $({\varphi }_{\mathrm{pr},2}-{\varphi }_{\mathrm{pr},2}^{\mathrm{weak}})$ and (c) $({\varphi }_{\mathrm{pr},3}-{\varphi }_{\mathrm{pr},3}^{\mathrm{weak}})$ , with the real and imaginary parts of the corresponding complex numbers (b) ${Y}_{\mathrm{pr},2}$ and (d) ${Y}_{\mathrm{pr},3}$ .
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**Figure 10.** Probe-pulse-induced phases in a probe-pump setup as a function of probe-pulse intensity and for ${\omega }_{{\rm{L}}}=1.560\,\mathrm{eV}$ (blue continuous line), ${\omega }_{{\rm{L}}}=1.567\,\mathrm{eV}$ (orange dashed line), ${\omega }_{{\rm{L}}}=1.575\,\mathrm{eV}$ (green dotted line), ${\omega }_{{\rm{L}}}=1.582\,\mathrm{eV}$ (red dashed–dotted line), ${\omega }_{{\rm{L}}}=1.590\,\mathrm{eV}$ (brown dashed-double-dotted line). The panels display (a) $({\varphi }_{\mathrm{pr},2}-{\varphi }_{\mathrm{pr},2}^{\mathrm{weak}})$ and (c) $({\varphi }_{\mathrm{pr},3}-{\varphi }_{\mathrm{pr},3}^{\mathrm{weak}})$ , with the real and imaginary parts of the corresponding complex numbers (b) ${Y}_{\mathrm{pr},2}$ and (d) ${Y}_{\mathrm{pr},3}$ .
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Figure 11. Refer to the following caption and surrounding text. — **Figure 11.** Pump-pulse-induced phases in a probe-pump setup as a function of pump-pulse intensity and for the same laser frequencies employed in figure 10. The panels display (a) $({\varphi }_{\mathrm{pu},2}-{\varphi }_{\mathrm{pu},2}^{\mathrm{weak}})$ and (c) $({\varphi }_{\mathrm{pu},3}-{\varphi }_{\mathrm{pu},3}^{\mathrm{weak}})$ , with the real and imaginary parts of the corresponding complex numbers (b) ${Y}_{\mathrm{pu},2}$ and (d) ${Y}_{\mathrm{pu},3}$ .
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**Figure 11.** Pump-pulse-induced phases in a probe-pump setup as a function of pump-pulse intensity and for the same laser frequencies employed in figure 10. The panels display (a) $({\varphi }_{\mathrm{pu},2}-{\varphi }_{\mathrm{pu},2}^{\mathrm{weak}})$ and (c) $({\varphi }_{\mathrm{pu},3}-{\varphi }_{\mathrm{pu},3}^{\mathrm{weak}})$ , with the real and imaginary parts of the corresponding complex numbers (b) ${Y}_{\mathrm{pu},2}$ and (d) ${Y}_{\mathrm{pu},3}$ .
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Figure 11 shows the additional phase shift owing to a strong pump pulse as a function of its intensity. The (amplitude and phase) changes resulting from the interaction with the pump pulse are encoded in the complex numbers ${Y}_{\mathrm{pu},2}$ and ${Y}_{\mathrm{pu},3}$ , whose dependence on intensity and laser frequency is shown in figures 11(b) and (d), respectively. As noticed in equation (48), ${Y}_{\mathrm{pu},k}$ are of second order in the effective pulse area ϑ for weak values of the pulse intensity. As a result, for small pulse intensities, ${Y}_{\mathrm{pu},2}$ and ${Y}_{\mathrm{pu},3}$ tend to vanishing values for all considered laser frequencies. The associated atomic-phase change results in the phase shifts displayed in figures 11(a) and (c). For all considered laser frequencies, ${\varphi }_{\mathrm{pu},2}$ and ${\varphi }_{\mathrm{pu},3}$ evolve in opposite directions for increasing values of the pump-pulse intensity.

In figures 12 and 13 we present laser-frequency-dependent results for the pump-probe setup discussed in section 3.2.2. Figure 12 focuses on the phases induced only by the probe pulse, as a function of its intensity and for different values of the laser frequency. The phase $({\psi }_{\mathrm{pr},2}-{\psi }_{\mathrm{pr},2}^{\mathrm{weak}})$ , defining the intensity-dependent shift of ${ \mathcal S }({\omega }_{21},\tau )$ , is shown in figure 12(a). Also in this case, the dependence upon intensity and laser frequency can be better understood by referring to the complex numbers $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ (equation (51)), displayed in figure 12(b). As discussed previously, these complex numbers are related to the transformation induced by the second-arriving probe pulse, quantifying how an initial coherence between states $| 2\rangle$ and $| 3\rangle$ is transformed into coherence between $| 1\rangle$ and $| 2\rangle$ . At low intensities, all curves tend to positive, purely imaginary values of $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ , in agreement with equation (58). The path followed by $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ at increasing intensities depends on the laser frequency, and reveals interesting features about the intensity dependence of $({\psi }_{\mathrm{pr},2}-{\psi }_{\mathrm{pr},2}^{\mathrm{weak}})$ shown in figure 12(a). For example, one can notice how relatively similar values of $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ , such as those displayed by the green, red, and brown curves in figure 12(b), can lead to a very different behavior of the corresponding phases (figure 12(a)). This is due to the fact that the amplitude of $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ is very close to vanish for all three considered curves. A small change in the actually followed path can therefore lead to a completely oppositely directed shift in the corresponding phase. The phases $({\psi }_{\mathrm{pr},3}-{\psi }_{\mathrm{pr},3}^{\mathrm{weak}})$ shown in figure 12(c), determining the intensity-dependent shift of ${ \mathcal S }({\omega }_{31},\tau )$ , display a more regular dependence upon intensity and laser frequency. This is essentially related to the fact that the corresponding complex numbers $({A}_{\mathrm{pr},3}-{B}_{\mathrm{pr},3}^{* })$ do not approach vanishing values for the range of intensities and laser frequencies considered, as exhibited by figure 12(d). The complex numbers $({A}_{\mathrm{pr},3}-{B}_{\mathrm{pr},3}^{* })$ (equation (51)) quantify how an initial coherence between states $| 2\rangle$ and $| 3\rangle$ is transformed into coherence between $| 1\rangle$ and $| 3\rangle$ and, at low intensities, tend to positive, purely imaginary values (figure 12(d)), in agreement with equation (58).

Figure 12. Refer to the following caption and surrounding text. — **Figure 12.** Probe-pulse-induced phases in a pump-probe setup as a function of probe-pulse intensity and for the same laser frequencies employed in figure 10. The panels display (a) $({\psi }_{\mathrm{pr},2}-{\psi }_{\mathrm{pr},2}^{\mathrm{weak}})$ and (c) $({\psi }_{\mathrm{pr},3}-{\psi }_{\mathrm{pr},3}^{\mathrm{weak}})$ , with the real and imaginary parts of the corresponding complex numbers (b) $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ and (d) $({A}_{\mathrm{pr},3}-{B}_{\mathrm{pr},3}^{* })$ .
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**Figure 12.** Probe-pulse-induced phases in a pump-probe setup as a function of probe-pulse intensity and for the same laser frequencies employed in figure 10. The panels display (a) $({\psi }_{\mathrm{pr},2}-{\psi }_{\mathrm{pr},2}^{\mathrm{weak}})$ and (c) $({\psi }_{\mathrm{pr},3}-{\psi }_{\mathrm{pr},3}^{\mathrm{weak}})$ , with the real and imaginary parts of the corresponding complex numbers (b) $({A}_{\mathrm{pr},2}-{B}_{\mathrm{pr},2}^{* })$ and (d) $({A}_{\mathrm{pr},3}-{B}_{\mathrm{pr},3}^{* })$ .
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Figure 13. Refer to the following caption and surrounding text. — **Figure 13.** Pump-pulse-induced phases in a pump-probe setup as a function of pump-pulse intensity and for the same laser frequencies employed in figure 10. The panels display (a) $({\psi }_{\mathrm{pu}}-{\psi }_{\mathrm{pu}}^{\mathrm{weak}})$ and (b) the real and imaginary parts of the corresponding complex number ${Z}_{\mathrm{pu}}$ .
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Figure 13 shows the additional phase shift $({\psi }_{\mathrm{pu}}-{\psi }_{\mathrm{pu}}^{\mathrm{weak}})$ induced by the pump pulse. In a pump-probe setup ( $\tau \gt 0$ ), this shift equally affects the oscillations of ${ \mathcal S }({\omega }_{21},\tau )$ and ${ \mathcal S }({\omega }_{31},\tau )$ , as described in equation (57). The associated complex numbers ${Z}_{\mathrm{pu}}$ , quantifying the coherence between excited states generated by the first-arriving pump pulse, are exhibited in figure 13(b), displaying a small dependence on the laser frequency ${\omega }_{{\rm{L}}}$ . We notice that ${Z}_{\mathrm{pu}}$ tends to 0 for small intensities, being of second order in the effective pulse area ϑ as predicted by equation (58). The dependence of ${Z}_{\mathrm{pu}}$ on pump-pulse intensity and laser frequency is reflected in the associated phases $({\psi }_{\mathrm{pu}}-{\psi }_{\mathrm{pu}}^{\mathrm{weak}})$ , as shown in figure 13(a).

4. Conclusion

In summary, we have investigated the interaction of a sample of Rb atoms, modeled as a V-type three-level system, with intense probe and pump pulses separated by a positive or negative time delay in a transient-absorption-spectroscopy setup. The choice of this model system was motivated by its experimental accessibility. Furthermore, this simple system, with two excited states optically coupled to the same ground state by broadband femtosecond pulses, contains already the main features characterizing transient-absorption spectra of a multilevel system excited by pump and probe pulses. This renders it particularly suitable for the development of interpretation models. We considered pulses of intensities ranging from $1\times {10}^{9}\,\,{{\rm{W}}{\rm{c}}{\rm{m}}}^{-2}$ to $3.5\times {10}^{10}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ , spanning from the perturbative weak-field regime to the appearance of Rabi oscillations in the population of the excited states. For these intensities, and in particular for effective pulse areas of $\vartheta \sim \pi$ , the broadband pulse, with a central frequency which is differently detuned from the two excited transitions, induces nontrivial amplitude and phase dynamics. We investigated how these dynamics are imprinted into the transient-absorption spectrum, focusing on the time-delay-dependent oscillations displayed by the spectral line shapes.

The three-level model was used to describe the evolution of the atomic system and, thereby, to numerically simulate experimental time-delay- and pulse-intensity-dependent spectra for a noncollinear geometry. The absorption spectra exhibited oscillations in their line shapes as a function of time delay, whose phases are directly linked to the intensity of the pump and probe pulses employed. In order to interpret these numerical results, and in particular the intensity-dependent features for positive and negative time delays, an analytical interpretation model was developed, directly connecting the time-delay-dependent oscillations featured by the spectra with the atomic amplitude and phase changes induced by the pump and probe pulses, as quantified by the matrix elements of the interaction operators $\hat{U}$ . Specifically, this interpretation model was used to quantify the phases of the time-delay-dependent oscillations featured by the spectra, which were systematically investigated as a function of pump- and probe-pulse intensity and laser frequency.

Our study shows how strong-field quantum-dynamics information is encoded in transient-absorption spectra for the three-level scheme modeling Rb. Hence, it represents an important step towards experimental methods of quantum-state holography, which aim at the extraction of quantum-dynamics information from time-dependent spectra in more complex, atomic and molecular multilevel systems. This full reconstruction can have significant applications as for, e.g., coherent quantum control with intense ultrashort pulses, and for the experimental characterization of the pulses used, as required at higher frequencies for the x-ray pulses provided by recently established x-ray free-electron lasers.

Further studies could include a more thorough analytical and theoretical description of the frequency dependence of the phases, as well as an atomic-system description going beyond the three-level model employed here. For high densities or long media, it could be important to further investigate how propagation effects can be included in our interpretation models.

Acknowledgments

The authors acknowledge valuable discussions with Zoltán Harman, Christoph H Keitel, and Thomas Pfeifer. The work of VB has been carried out thanks to the support of the A*MIDEX grant (No. ANR-11-IDEX-0001-02) funded by the French Government 'Investissements d'Avenir' program.

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2.3.1. Probe-pump setup

2.3.2. Pump-probe setup

3.2.1. Probe-pump setup

3.2.2. Pump-probe setup

3.2.3. Dependence on laser frequency

Transient-absorption phases with strong probe and pump pulses

Author notes

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Abstract

1. Introduction

2. Theoretical model

2.1. Three-level model and equations of motion

2.2. Transient-absorption spectrum

2.3. Analytical model in terms of interaction operators

2.3.1. Probe-pump setup

2.3.2. Pump-probe setup

3. Results and discussion

3.1. Transient-absorption spectra for intense probe and pump pulses

3.2. Interpretation of pump- and probe-pulse-induced phases in terms of interaction-operator matrix elements

3.2.1. Probe-pump setup

3.2.2. Pump-probe setup

3.2.3. Dependence on laser frequency

4. Conclusion

Acknowledgments