Coherent strong-field control of multiple states by a single chirped femtosecond laser pulse

We present a joint experimental and theoretical study on strong-field photo-ionization of sodium atoms using chirped femtosecond laser pulses. By tuning the chirp parameter, selectivity among the population in the highly excited states 5p, 6p, 7p and 5f, 6f is achieved. Different excitation pathways enabling control are identified by simultaneous ionization and measurement of photoelectron angular distributions employing the velocity map imaging technique. Free electron wave packets at an energy of around 1 eV are observed. These photoelectrons originate from two channels. The predominant 2+1+1 Resonance Enhanced Multi-Photon Ionization (REMPI) proceeds via the strongly driven two-photon transition $4s\leftarrow\leftarrow3s$, and subsequent ionization from the states 5p, 6p and 7p whereas the second pathway involves 3+1 REMPI via the states 5f and 6f. In addition, electron wave packets from two-photon ionization of the non-resonant transiently populated state 3p are observed close to the ionization threshold. A mainly qualitative five-state model for the predominant excitation channel is studied theoretically to provide insights into the physical mechanisms at play. Our analysis shows that by tuning the chirp parameter the dynamics is effectively controlled by dynamic Stark-shifts and level crossings. In particular, we show that under the experimental conditions the passage through an uncommon three-state"bow-tie"level crossing allows the preparation of coherent superposition states.


Introduction
Selective excitation of preselected target states making use of shaped femtosecond laser pulses is at the heart of coherent quantum control [1,2,3,4,5,6,7,8,9,10]. Closedloop optimization strategies [4,11,12,13,14,15] have proven enormously successful in controlling a huge variety of quantum systems, however studies on model systems employing defined pulse shapes are the key to better understand the underlying physical mechanisms and to further develop quantum control concepts and techniques. This applies in particular to strong-field quantum control [16,17,18,19,20] characterized by non-perturbative interaction of a quantum system with intense shaped laser pulses. Strong-field physical mechanisms involve-besides the interference of multiple excitation pathways-adiabatic and non-adiabatic time evolution accompanied by Dynamic Stark-Shifts (DSSs) in the order of hundreds of meV. The latter is responsible for modification of the atomic states or molecular potential surfaces [21,22,23] such that new pathways become available and new target states-inaccessible in weak laser fields-open up. Recent studies of strong-field control on model systems devoted to the analysis of the basic physical mechanisms revealed that the concept of Selective Population of Dressed States (SPODS) [24] provides a natural description of controlled dynamics in intense shaped laser fields. For example, it was shown that ultrafast switching among different target channels by phase discontinuities within the pulse [16,24,25,26], Rapid Adiabatic Passage (RAP) by chirped pulses [27] and combinations thereof [28] are realizations of this general concept.
Chirped pulses are a well-established tool in quantum control because they usually serve as a prototype for shaped pulses with controllable envelope and timevarying instantaneous frequency. Therefore, they have played a prominent role in the development of quantum control concepts and techniques and are still the "workhorse" to test novel strategies in quantum control. Examples of quantum control with chirped pulses comprise studies of selective excitation and ionization of multi-level system in alkali atoms [19,27,29,30,31,32,33,34], control of molecular dynamics in diatomics and dyes [35,36,37,38,39,40,41,42], measurement of coherent transients [43] and the development of adiabatic passage techniques [44].
In the present contribution we employ chirped ultrashort laser pulses resulting from phase modulation of the laser spectrum to study Resonance Enhanced Multi-Photon Ionization (REMPI) of a multi-level system in sodium atoms. We demonstrate experimentally, that different excitation pathways and, accordingly, different target channels can be addressed selectively by a single control parameter, i.e. the chirp. The nature of these pathways is unraveled by measurement of Photoelectron Angular Distributions (PADs) from Velocity Map Imaging (VMI) [45,46,47,48,49,50], yielding detailed information on the origin of the released photoelectron wave packets. Theoretical investigations of the light-atom interaction reveal an interplay of different physical mechanisms governing control. Analysis of the neutral excitation dynamics for a five-state model-atom (including the most relevant states 3s, 4s, 5p, 6p and 7p) under influence of a chirped ultrashort laser pulse highlights how physical mechanisms, such as RAP and DSS, act jointly to either address single states among the high lying sodium states 5p, 6p and 7p (cf. Fig. 1), or excite superpositions of any two neighboring states. We point out that the present paper extends two earlier techniques in several significant directions. The technique of Melinger et al. [29] uses a single chirped picosecond laser pulse to selectively excite the two fine-structure components 3p 1/2 and 3p 3/2 in sodium atoms. The present technique adds a DSS to the control tools, which enables the population of a third state, and also the creation of coherent superposition states. The technique of Clow et al. [34] makes use of a shaped femtosecond pulse to selectively populate a single highly excited state. The present technique is more flexible, since it allows to populate several different states by variation of a single parameter: the chirp.
The article is organized as follows. We start in Sec. 2 by introducing the excitation and ionization scheme of sodium atoms exposed to ultrashort near-infrared laser pulses, and subsequently describe the details of our experimental setup. The experimental results are presented in Sec. 3 along with a physical discussion of general features observed in the measured PADs supported by numerical simulations of the measurement results. Sec. 4 provides a detailed theoretical analysis of the strong-field induced chirped excitation dynamics in terms of adiabatic states, highlighting different physical mechanisms that govern the light-atom interaction. We conclude the paper with a brief summary and conclusions.

Experiment
In our experiment, we combine spectral phase shaping to produce chirped ultrashort laser pulses with the measurement of PADs resulting from REMPI of sodium atoms, employing the VMI technique. In this section, we first introduce the sodium excitation scheme with emphasis on the different accessible excitation and ionization pathways. Then we describe the experimental setup and layout of our photoelectron imaging spectrometer. Fig. 1 shows the excitation and ionization scheme of sodium atoms based on energy level information taken from the NIST-database [51]. Different multi-photon excitation pathways are accessible during the interaction of sodium atoms with intense ultrashort laser pulses (laser specifications are given in Sec. 2.2). The predominant excitation pathway is a 2+1+1 REMPI process via the two-photon transition 4s ←← 3s (red arrows in Fig. 1) which is nearly resonant with our laser spectrum [52]. Consequential population of states 5p, 6p and 7p gives rise to photoelectron wave packets in the ionization continuum having s or d-symmetry. The recorded PADs therefore exhibit a combined s and d-symmetry and are measured at the distinct kinetic energies 0.76 eV, 1.04 eV and 1.20 eV, corresponding to states 5p, 6p and 7p respectively. Alternatively, (Color online) excitation and ionization scheme of sodium atoms illustrating the excitation pathways that arise during the interaction with an intense 795 nm, 30 fs FWHM laser pulse. These pathways comprise a 2+1+1 REMPI (red arrows) and a 3+1 REMPI (green arrows) process from the 3s ground state as well as a two-photon ionization process from state 3p (blue arrows). Blurred red bars represent the one, two and three-photon spectrum of our laser respectively. Since state 4s lies within the bandwidth of the two-photon spectrum, the laser strongly drives the transition 4s ←← 3s. Once state 4s is populated, population flows to states 5p, 6p and 7p, giving rise to photoelectron wave packets with combined s and d-symmetry at characteristic kinetic energies 0.76 eV, 1.04 eV and 1.20 eV in the ionization continuum. A competing excitation pathway is opened up by three-photon absorption leading to population of states 5f and 6f in addition. Photoelectrons from this excitation channel are characterized by a combined d and g-symmetry of the measured PADs at kinetic energies 1.02 eV and 1.18 eV respectively. Two-photon ionization from the non-resonant, transiently populated state 3p results in photoelectron wave packets at about 0.2 eV, having combined p and f -symmetry. For illustrative purposes, the relevant symmetries of the released photoelectron wave packets are visualized on top of the figure in red and blue, encoding the positive and negative sign of the electron wave function respectively. a 3+1 REMPI process (green arrows in Fig. 1) based on three-photon absorption from the 3s ground state with no intermediate resonances is taken into account, contributing also to the population of states 5p, 6p and 7p but, in addition, transferring population to states 5f and 6f . One-photon ionization of the latter results in photoelectron wave packets with d and g-symmetry at kinetic energies 1.02 eV and 1.18 eV respectively. These photoelectrons are distinguished from the p state contributions (at 1.04 eV and 1.20 eV) by the symmetry of their angular distributions. In the following we will refer to

Setup
In this section the experimental setup comprising the laser system and the photoelectron imaging spectrometer is described. Intense 795 nm, 30 fs FWHM (Full Width at Half Maximum) laser pulses provided by an amplified 1 kHz Ti:sapphire laser system (Femtolasers Femtopower Pro) were phase modulated in frequency domain by a homebuilt pulse shaper [53], applying quadratic phase masks of the form ϕ mod (ω) = ϕ 2 /2·(ω− ω 0 ) 2 , where ω 0 is the central frequency of our laser spectrum [27]. The chirp parameter ϕ 2 was varied in the range from −2000 fs 2 to +2000 fs 2 in steps of ∆ϕ 2 = 100 fs 2 . The chirped output pulses of 12 µJ energy were sent into a vacuum chamber and refocussed by a concave mirror (5 cm focal length; we estimated a peak intensity of about 10 13 W/cm 2 for the bandwidth-limited pulse) into sodium vapor supplied by an alkali metal dispenser source, as shown in Fig. 2. Photoelectrons released during the strong-field interaction of the shaped pulses with single atoms were detected by a photoelectron imaging spectrometer using the VMI method. In order to compensate the residual chirp of the unmodulated pulse, we performed an in situ adaptive optimization of the multi-photon ionization of water vapor background (about 4 × 10 −7 mbar) in the interaction region of the spectrometer. The resulting optimal compensation phase was additionally applied to the pulse shaper during the experiments, ensuring an error in the chirp parameter ϕ 2 of less than 150 fs 2 . The energy calibration of the imaging spectrometer was performed using a 3+1 REMPI of xenon atoms excited by a Nd:YAG ns laser system at 355 nm, achieving a spectrometer resolution of 60 meV at 0.5 eV. Employing the energy calibrated photoelectron imaging spectrometer we studied angular and energy resolved photoelectron spectra as a function of the chirp parameter ϕ 2 . Figure 3 (upper row) shows measured PADs from REMPI of sodium atoms with chirped fs laser pulses for three exemplary values of the chirp parameter ϕ 2 . The middle row displays the corresponding Abel-inverted (retrieved) PADs obtained by employing the pBasex-algorithm [50,54]. When PADs arise from ionization with polarization shaped pulses [55], direct tomography methods have been developed for three-dimensional reconstruction of ultrashort free photoelectron wave packets [56]. Angular sections through the retrieved PADs at kinetic energies 0.8 eV, 1.0 eV and 1.2 eV, as plotted in the lower row, serve to identify the symmetry of the different energy channels observed in the PADs. The PAD measured for the unmodulated, i.e., bandwidth-limited pulse is depicted in the central column. Three major contributions are observed at kinetic energies 0.8 eV, 1.0 eV and 1.2 eV, related to the energy channels discussed above (cf. Sect. 2.1). The angular section taken at 1.2 eV exhibits two minor nodes between 0 • and 180 • , i.e. d-symmetry. This channel is attributed mainly to ionization via state 7p (red excitation pathway in Fig. 1), though our numerical simulations (inset of Fig. 4) indicate, that also ionization via state 6f (green excitation pathway in Fig. 1) delivers a minor contribution. The contribution of an s-wave to this channel, as expected from the excitation scheme Fig. 1, is reflected in the weak equatorial signal: At an angle of 90 • s and d-wave have opposite sign and, thus, interfere destructively, whereas at the poles, i.e. at 0 • and 180 • , both waves add up constructively. The section taken at 1.0 eV exhibits 4 nodes between 0 • and 180 • , corresponding to g-symmetry. This contribution originates predominantly from ionization via state 5f . The observation that the lobe at 90 • (and 270 • respectively) is slightly lowered with respect to its two neighbors indicates a weak d-wave contribution interfering destructively with the g-wave in this angular segment. The contribution measured at 0.8 eV shows again combined s and d-symmetry and is ascribed to ionization via state 5p.

Experimental results and discussion
Moreover, a weak contribution is observed at about 0.2 eV, a magnification of which is shown in the inset of Fig. 3(b). The nodal structure of this signal exhibits distinct fsymmetry. However, the pronounced poles of the PAD as well as the fact, that the nodes at 45 • and 135 • in the angular section are raised with respect to the node at 90 • give a hint on a p-wave contribution to the photoelectron signal. Observation of photoelectron wave packets with combined p and f -symmetry close to the ionization threshold is consistent with two-photon ionization from state 3p (blue pathway in Fig. 1). Note, that state 3p is-although non-resonant-transiently populated during the interaction, mediating the multi-photon processes to the state 4s and the high lying f states.
For large negative values of ϕ 2 (left column in Fig. 3), i.e. strongly down-chirped laser pulses, the outer channel at kinetic energy 1.2 eV is considerably enhanced in comparison to the bandwidth-limited case, whereas the intermediate channel at 1.0 eV is strongly reduced and the two innermost contributions have essentially vanished. Note the change in symmetry of the intermediate channel which exhibits combined s and d-symmetry in this case, indicating more efficient ionization from state 6p, while the 5f contribution is very small. Changing the sign of ϕ 2 , i.e. using strongly up-chirped laser pulses (right column in Fig. 3), suppresses the high energy channel in favor of the intermediate channel at 1.0 eV which dominates the PAD in this case. From its angular section at 1.0 eV we find a combined d and g-symmetry, as in the bandwidth-limited case. This contribution is therefore traced back mainly to state 5f . The finding that the symmetry of photoelectrons from the intermediate channel alters from d to g is rationalized by the change of the ordering of red and blue frequency components within the chirped pulse. For a down-chirped pulse, i.e. when the blue components arrive first, initially, the system is in resonance with the two-photon transition 4s ←← 3s implying efficient ionization via the p states (red pathway in Fig. 1). On the other hand, up-chirped pulses favor ionization via state 5f since at early times the system is in resonance with the three-photon transition 5f ←←← 3s (green pathway in Fig. 1). Such processes have also been observed in [39] under different excitation conditions.
In order to provide the full picture of the chirp dependent population flow to the different energy channels, we performed an angular integration of all 41 measured PADs and present the resulting energy-resolved photoelectron spectra in terms of a twodimensional map as a function of the kinetic energy and the chirp parameter ϕ 2 . The result obtained upon variation of ϕ 2 in the range from −2000 fs 2 to +2000 fs 2 is displayed in Fig. 4. The three major channels at 0.8 eV, 1.0 eV and 1.2 eV are clearly visible. Note, that for e.g. rare gas atoms under our experimental conditions ponderomotive shifts of more than 0.5 eV are calculated. No such shifts are observed in the experiment, since the high-frequency approximation [57,58] (necessary condition for the application of the ponderomotive energy concept) is not valid for alkalis excited by near infrared laser radiation. An analysis of the neutral excitation dynamics behind the observed contributions will be given in Sec. 4. The map illustrates the above statements, that for large negative values of ϕ 2 the high energy channel at 1.2 eV is addressed with high efficiency, i.e. a down-chirped pulse steers the population predominantly towards the high lying state 7p. For large positive chirp values the intermediate channel is selectively addressed, corresponding to predominant population of states 6p and 5f . The low energy channel is accessed most efficiently in the vicinity of ϕ 2 = 500 fs 2 . In fact, in the regime 0 ≤ ϕ 2 ≤ 1000 fs 2 the photoelectron spectrum is made up of contributions from states 5p, 6p and 5f . Because the excitation (and simultaneous ionization) takes place on an ultrashort time scale precluding decoherence processes, a coherent superposition of states 5p, 6p and 5f is excited in this chirp regime. Upon changing the sign of ϕ 2 , i.e. for −1000 fs 2 ≤ ϕ 2 ≤ 0, the laser pulse induces a coherent superposition of states 6p, 5f and 7p. Photoelectrons observed at about 0.2 eV for moderate positive chirps are attributed to two-photon ionization from state 3p.
The inset to Fig. 4 shows results from a numerical simulation of the simultaneous multi-photon excitation and ionization process. The calculations are based on numerical integration of the time-dependent Schrödinger-equation for a neutral 20-state system (comprising those states labeled in Fig. 1 and taking the fine structure splitting into account) interacting with an intense chirped 795 nm, 30 fs FWHM Gaussian input pulse. One-photon ionization from the high lying p and f states is treated within a simplified model employing first order perturbation theory. We assume a flat continuum and unit coupling elements with no additional phases for all bound-free transitions. A more rigorous treatment of the ionization step involving the determination of radial coupling matrix elements also for the bound-free transitions is provided by, e.g., single-channel quantum defect theory [59] as reported for instance in [60,61]. In order to model the two-photon ionization from state 3p proceeding, for example, via state 3d as indicated by the blue pathway in Fig. 1, we employed second order perturbation theory. For a more detailed description of our method see [25,27,62]. The simulation of photoelectron spectra reproduces the main features of the experimental results very well. This allows us to look into the underlying neutral excitation dynamics and follow the population flow within the bound atomic system. We find that for large negative chirp ϕ 2 state 7p is addressed almost selectively, while for large positive ϕ 2 values both states 6p and 5f are populated efficiently in equal measure. The latter is in accordance with the experimental observation of the PAD with pronounced g-symmetry in the intermediate channel at 1.0 eV for large positive chirp (see Fig. 3(c)). The most efficient excitation of state 5p occurs for moderate positive chirp. However, in this chirp regime states 6p and 5f receive comparable population confirming the observation of a PAD with a contribution of gsymmetry at 1.0 eV and zero chirp. At moderate negative chirp, we obtain a coherent superposition of states 6p, 5f and 7p. Note that the weak contribution around 0.2 eV and small positive values of ϕ 2 observed in the experiment (shown in the inset to Fig. 3(b)) is also reproduced in the simulation. Within the framework of our simulation, these photoelectrons are ascribed to two-photon ionization from state 3p which receives nonperturbative transient population. We note, that in a perturbative regime, ionization from this transiently populated state could be interpreted as a transition from a virtual state.
In the next section, we will further investigate the neutral population dynamics by means of a reduced atomic system in order to rationalize the general features observed in the experiment in terms of physical mechanisms governing the excitation process.

Theoretical model
In this section we provide a mainly qualitative description of the system at hand. To this end, we assume that the photoelectron signal arises most significantly through the 2+1+1 REMPI channel (red pathway in Fig. 1), involving the five states 3s, 4s, 5p, 6p and 7p. The idea of this reduction is to demonstrate the basic principles influencing the dynamics of the whole system, which become more transparent in this simplified model, involving the most significant states for our experiment. In this approach, we adiabatically eliminated state 3p [44,63,64] because it is off resonance and receives smaller transient population than the other coupled states. Its presence, though, affects the population dynamics significantly for it induces strong dynamic Stark-shifts in the energies of states 3s and 4s, which substantially modify the energy diagram.
The quantum dynamics of this five-state system obeys the time-dependent Schrödinger equation The Hamiltonian H(t) in the rotating-wave approximation, rotating with the instantaneous laser frequency ω(t) = ω 0 + 2at (see Eq. (A.5) in the appendix), is given by [44,64]: Here the explicit time dependence is dropped for ease of notation. The vector c(t) = [c 1 (t), c 2 (t), . . . , c 5 (t)] T consists of the amplitudes of the five states, ordered as shown above, which are obtained by numerical integration of the Schrödinger-equation (1), the respective populations are P n (t) = |c n (t)| 2 , ∆ n (t) = ω n − k ω(t) are the generally timedependent atom-laser detunings, where ω n are the atomic state eigenfrequencies, with ω 3s taken as zero, k is the transition order, Ω 2n = d 2n Ω 0 f (t) represent the one-photon couplings of state 2 to state n (n = 3, 4, 5), Ω 12 = q 12 Ω 2 0 f 2 (t) is the two-photon coupling between states 1 and 2, with f (t) being the chirped laser electric field envelope, d mn are the relevant transition dipole moments in atomic units, q 12 is the effective two-photon transition moment (cf. Eq. (A.1)) and S 1 and S 2 represent the DSS of states 1 and 2, respectively, The effect of the DSS due to state 3d is neglected for it is very weakly coupled to the states whose energies it might influence: the p states are coupled about 10 times stronger to state 4s as compared to state 3d; state 3d is not directly coupled to state 3s, but rather through a two-photon transition. In the first two diagonal elements of the Hamiltonian (associated with the energies of states 3s and 4s) the atom-laser detuning and the DSS add up to a time-dependent effective chirp: the former resulting from the time-dependent instantaneous laser frequency ω(t), and the latter deriving from the time-dependent shift of the level energies due to DSS.

Excitation regimes
In Fig. 5 we distinguish five different regimes in regard to the value of the chirp ϕ 2 . In all cases we plot the bare-state energies and analyze the dynamics by accounting for the presence of level crossings. Because it is the ionization signal that is observed in the experiment it is also important when a particular level crossing occurs: a level crossing at early time, and the ensuing adiabatic passage transition, would translate into a larger ionization signal than a late crossing, where even a significant population transfer to a certain discrete state would not be reflected in the ionization signal. Below we examine the dynamics of our system with particular interest in states 5p, 6p and 7p. In Fig. 5 we show the populations and the energies of the five bare states for  Fig. 5(a)) the laser field reaches resonances relative to the 7p ← 4s (one-photon) transition and the 4s ←← 3s (two-photon) transition in nearly the same instant, thus creating a "bow-tie" level crossing pattern [44, 65, 66, 67, 68, 69] which is of particular significance because it involves three rather than two states. This crossing results in efficient population transfer to states 4s and 7p and depopulation of state 3s. Because state 7p is populated at such early times, it is exposed to ionization for most of the interaction dynamics and hence has a dominant contribution in the photoelectron signal (see Fig. 4 at 1.2 eV and −2000 fs 2 ). Later on we observe almost adiabatic evolution and the population is shared mainly between states 4s and 7p in the form of Rabi-oscillations with fading amplitude [70]. State 6p acquires only marginal population mainly due to its crossing with state 3s (which is, however, already depleted due to the preceding "bow-tie" crossing) via a three-photon excitation through state 4s. The late crossings between states 3s and 5p, and also between states 4s and 6p are of no importance because they occur after the pulse intensity has essentially vanished. State 5p remains unpopulated since it is far off-resonant throughout the entire dynamics.  Fig. 5(e)) the energy diagram is mirrored compared to the one for large negative chirps ϕ 2 (Fig. 5(a)). Then initially the system evolves adiabatically, with minor (off-resonant) population transfer from state 3s to state 4s due to their strong mutual coupling. Around the time of the peak laser intensity, as state 3s sweeps across resonance with 6p, the latter starts to effectively populate through the three-photon 3s − 6p crossing. Because this crossing occurs approximately in the middle of the laser pulse the population of state 6p is exposed to ionization for a considerable time interval, which results in significant photoelectron signal from 6p (see Fig. 4 at 1.0 eV and +2000 fs 2 ). For the same reasonthe 3s − 6p crossing occurring near the laser pulse maximum-the population transfer from state 3s to state 6p is relatively efficient and only about half of the population is left in states 3s and 4s thereafter; then only a part of this already reduced population is transferred to state 7p at the subsequent "bow-tie" crossing 3s − 4s − 7p. Moreover, this crossing occurs at late times and hence state 7p is not visible in the photoelectron spectrum. State 5p remains unpopulated once again as it stays far off any resonance.
We now turn our attention to the regimes of a moderately large chirp ϕ 2 , where the photoelectron spectrum changes from a single-state feature to one displaying double features.

Moderate negative chirp
For a moderate negative chirp (ϕ 2 = −500 fs 2 , Fig. 5(b)) an early crossing occurs between states 3s, 4s and 7p in the rising edge of the pulse, which leads to a partial population transfer from state 3s to states 4s and 7p, because the laser intensity is not strong enough to enforce adiabatic evolution. The population in state 7p is exposed to ionization for the rest of the pulse, whereas the population in state 4s proceeds until the subsequent 4s − 6p crossing where it is partially transferred to state 6p. The leftover temporary flows into state 5p, which starts to emerge in the photoelectron spectrum, and is finally driven back into state 3s. In result, all states 5p, 6p and 7p are visible in the photoelectron signal, which is an indication for the creation of a coherent superposition of these (see Fig. 4 at about −500 fs 2 ).

Moderate positive chirp
For moderate positive chirps (ϕ 2 = 500 fs 2 , Fig. 5(d)) state 3s first comes very close to state 5p at times of the laser pulse maximum; during this proximity the population undergoes Rabi-type oscillations between states 3s and 5p and is exposed to ionization from state 5p. The signature of state 5p is clearly visible and indeed, this is the regime where this state indisputably dominates in the photoelectron signal (see Fig. 4 at 0.8 eV and +500 fs 2 ). In other words, it is the DSS induced by the two-photon transition 4s ←← 3s, which makes the population of the far-off-resonant state 5p possible [71]. If this Stark-shift were absent (e.g. if the two-photon transition 4s ←← 3s were instead a single-photon one in a gedanken scenario) state 5p would never receive sizeable population. As we proceed beyond the pulse maximum state 3s crosses state 6p and the population is partially transferred to the latter. Hence state 6p emerges in the photoelectron signal due to the ensuing ionization, whereas state 7p is invisible in this regime because all population left flows into state 4s.

Zero chirp.
In this regime the laser pulse is unchirped, ϕ 2 = 0. Therefore, the effective chirp is entirely due to ac Stark-shift. The latter is symmetric to the pulse because it is induced by the same pulse. Moreover, because state 3s crosses states 6p and 5p (Fig. 5(c)), sizeable population will visit these two states through the respective first crossings 3s − 5p and 3s − 6p. A second pair of crossings in the falling edge of the pulse will induce additional transitions 5p ←←← 3s and 6p ←←← 3s. The implication is that states 5p and 6p will contribute significantly to the photoelectron signal (see Fig. 4 around ϕ 2 = 0). State 7p, on the other hand, remains well off resonance throughout and receives only a small population due to (weak) non-resonant interaction. Its contribution to the photoelectron signal should be therefore more muted than these from states 5p and 6p.

Discussion
Below we discuss the five excitation regimes in the dressed state (adiabatic) context. When adiabatic, which demands large couplings and low chirp rates for the avoided crossings in question, starting in state 3s we end up in state 7p for ϕ 2 < 0 or in state 6p for ϕ 2 > 0 (Fig. 5, middle frames; in the latter case a fully non-adiabatic passage across state 5p occurs, since the pulse intensity is negligible for the 3s − 5p resonance). Therefore, clearly from Fig. 5, our system exhibits a somewhat adiabatic behavior for chirp ϕ 2 away from the origin. As we get closer, the crossings shift towards the pulse wings, whereas the pulse gets narrower in time, which in combination results in breaking adiabaticity. The latter is further hindered by the increased DSS, which effectively enhances the chirp rate.
We expect adiabaticity to remain almost unaffected for large negative values of the chirp ϕ 2 , since the chirp rate a ∝ 1/ϕ 2 and Ω ∝ 1/ √ ϕ 2 , and to break down for large positive values, for it relies on the three-photon transition 6p ←←← 3s, which gets weaker, as the resonances relative to 3s − 4s and 4s − 6p further separate in time.
Larger peak intensities Ω 0 strengthen adiabaticity for the transition 7p ←←← 3s and make complete population transfer possible, as also indicated in [34], whereas for the transition 6p ←←← 3s due to the unfavorable influence of the increased DSS we predict the contrary.

Summary and Conclusion
In this contribution we presented a joint experimental and theoretical study on strongfield Resonance Enhanced Multi-Photon Ionization (REMPI) of sodium atoms using chirped femtosecond laser pulses. Experimentally, Photoelectron Angular Distributions (PADs) have proven the essential tool to identify the different excitation and ionization pathways.
We observed three distinct ionization pathways contributing to the measured PADs. The predominant contribution with combined s and d-symmetry is due to a 2+1+1 REMPI processes involving the strongly driven two-photon transition 4s ←← 3s, and subsequent ionization from the states 5p, 6p and 7p. Photoelectrons with combined d and g-symmetry originated from 3+1 REMPI via states 5f and 6f . A weak contribution with combined p and f -symmetry close to the ionization threshold is attributed to the third channel, that is two-photon ionization of the non-resonant transiently populated state 3p.
Selective population of the highly excited states 5p, 6p, 7p and 5f , 6f was achieved by controlling a single pulse parameter, i.e. the chirp parameter ϕ 2 . In particular, we observed highly selective population of state 7p using strongly down-chirped laser pulses. For strongly up-chirped laser pulses states 6p and 5f were populated with high efficiency and a dominant signal from state 5p was obtained for moderately up-chirped laser pulses. Moreover, in the intermediate chirp regions coherent superpositions of neighboring states have been excited.
Simulations based on numerical integration of the time-dependent Schrödingerequation for a neutral 20-state system are in agreement with our experimental findings. In addition, a five-state model was developed in order to provide insights into the physical mechanisms at play. Our analysis of the time-dependent populations showed that by tuning the chirp parameter distinct physical mechanisms have been addressed, involving adiabatic and non-adiabatic time evolution along with Dynamic Stark-Shifts (DSSs) and (multiple) level crossings. It was pointed out that the occurrence of an uncommon "bow-tie" level crossing is responsible for the excitation of coherent superposition states as observed in the experiment. The strong DSS of the two-photon transition 4s ←← 3s turned out to be of particular significance for populating state 5p being inaccessible in weak laser fields.
Our results highlight the importance of studying model systems experimentally and theoretically to better understand the physical mechanisms of strong-field coherent control. Our findings demonstrate that, in general, in strong-field control multiple pathways involving different physical mechanisms are at play simultaneously.

Appendix A. Details of calculations
Each p state consists of p 1/2 and p 3/2 substates, coupled by Ω 1/2 and Ω 3/2 , respectively, to a relevant s state. Therefore initially our system comprises overall 10 states (prior to eliminating state 3p). To simplify our approach we perform a transformation to a darkbright basis for each of the p states and thus eliminate half of the p substates as dark (uncoupled) states, and keep the rest, which become coupled by the root mean square of the relevant Ω 1/2 and Ω 3/2 and are the ones to be referred to as p states throughout the theoretical part of the paper. The effective two-photon transition moment between states 3s and 4s is where d a,c and d b,d are the dipole moments for the transitions 3p 1/2,3/2 ← 3s 1/2 and 4s 1/2 ← 3p 1/2,3/2 , respectively. The effect of a quadratic phase modulation in frequency domain of the form is described in time domain by a modulated linearly polarized laser electric field E(t) given as [72] E (t) = 2Re E + (t) , Here ∆t denotes the FWHM of the intensity I(t) of the unmodulated pulse, ω 0 is the laser carrier frequency and ϕ 2 is the chirp parameter to be varied.
We define a reference Rabi-frequency Ω(t) = Ω 0 f (t), where f (t) is the laser electric field envelope