Multichromatic Polarization-Controlled Pulse Sequences for Coherent Control of Multiphoton Ionization
- Ultrafast Dynamics Group, Institut für Physik, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany
In this review, we report on recent progress in the generation and application of multichromatic polarization-tailored pulse sequences for the coherent control of multiphoton ionization (MPI) dynamics and present unpublished experimental results that complement our previous findings. Specifically, we utilize single-color, bichromatic, and trichromatic polarization-controlled pulse sequences generated by spectral amplitude, phase and polarization modulation of a carrier-envelope phase (CEP)-stable white light supercontinuum for MPI. The analysis of the number of ionization pathways and the number of distinct final free electron states shows that both increase significantly, but scale differently with the number of absorbed photons and the number of pulses in the sequence. In our experiments, ultrafast polarization shaping is combined with high-resolution photoelectron tomography to generate, control, and reconstruct three-dimensional photoelectron momentum distributions from atomic and molecular MPI. We discuss the use of polarization-controlled single-color and bichromatic pulse sequences in perturbative and non-perturbative coherent control of coupled electron-nuclear dynamics in molecules, atomic spin-orbit wave packet dynamics and the directional photoemission from atoms and chiral molecules. We compare the coherent control of CEP-insensitive intraband multipath interference in the MPI with a fixed number of photons with CEP-sensitive interband multipath interference in the ionization with a different number of photons. The generation and control of free electron vortices with even-numbered rotational symmetry by MPI with single-color pulse sequences is contrasted with the bichromatic control of CEP-sensitive electron vortices with odd-numbered rotational symmetry. To illustrate the potential of multichromatic pulse sequences for coherent control, we present a trichromatic scheme for shaper-based quantum state holography.
The basic principles of coherent control have been established more than 30 years ago [1–6]. Today many applications of coherent control have been demonstrated in various areas of physics and chemistry including multiphoton excitation [7–10] and ionization (MPI) of atoms [11, 12] and small molecules [13–15], electronic transitions in condensed matter , controlled molecular dissociation , spectroscopy [18, 19] and laser chemistry [20, 21]. Even in emerging fields such as high harmonic generation (HHG) [22, 23], material processing [24, 25], nonlinear microscopy , photoassociation , nanomaterial research  and quantum information  coherent control has proven its usefulness. A very effective method to control the dynamics of quantum systems is by specific manipulation of constructive and destructive quantum interferences  using ultrashort tailored laser pulses. Advanced techniques for generating tailored ultrashort laser pulses on the one hand and for highly differential detection on the other hand, have been key to experimental advances in coherent control. Since the implementation of the first single layer 128 pixel pulse shaping devices [30–40], remarkable advances have been achieved towards high precision pulse shaping using 640 pixel devices [41–50] and polarization shapers [51, 52], vector-field synthesizers [53–58] and supercontinuum pulse shaping [59–66]. Recently, we have introduced a scheme for polarization shaping of carrier-envelope phase (CEP)-stable over-octave-spanning white light supercontinua (WLS) to generate polarization-tailored bichromatic fields . The latter work resulted in the experimental demonstration of a new class of cycloidal pulse shapes, such as counterrotating circularly polarized (CRCP) and corotating circularly polarized (COCP) cycloidal fields. In general, polarization-tailored multicolor femtosecond laser fields have opened up new perspectives in numerous applications ranging from HHG [67–72, 74, 75] over the coherent control of ultrafast electron dynamics in atoms and molecules [73, 76–78] to the manipulation of coherent excitations in nanostructures [79, 80].
The use of advanced polarization-shaped pulses for the generation of free electron wave packets by MPI has necessitated 3D detection to characterize the full 3D photoelectron momentum distribution (PMD). While time-of-flight techniques enabled the kinetic energy-resolved detection of photoelectrons from ultrafast MPI dynamics [81, 82], their angular distribution became available through the use of velocity map imaging (VMI) techniques [83–85]. Currently, the most sophisticated detection method is based on COLTRIMS [86, 87]. Cylindrically symmetric PMDs can be retrieved from the measured 2D projections using the Abel-inversion technique . In order to image non-cylindrically symmetric PMDs, we have introduced a VMI-based photoelectron tomography technique . By now, photoelectron tomography has become an established method with versatile applications including 3D imaging of PMDs from atomic strong-field ionization , the reconstruction of molecular PMDs in the laboratory  and molecular  frame, the retrieval of circularly polarized XUV fields from HHG  or the reconstruction of PMDs from strong-field photodetachment of negatively charged ions . Applications of photoelectron tomography have ranged from the generation and characterization of designer electron wave packets [95, 96] and multiphoton photoelectron circular dichroism (PECD) measurements  to the extraction of transition matrix elements from tomographic data . More recently, we have used photoelectron tomography to study free electron angular momentum wave packets , spin-orbit  and Rydberg wave packet dynamics  and photoelectron vortices with even [102–104] and odd  rotational symmetries. An overview of our recent photoelectron tomography studies of coherent control of atomic MPI can be found in [105, 106].
In this contribution, we review the application of shaper-generated multichromatic polarization-controlled fields for coherent control of atomic and molecular MPI. We study the quantum dynamics of atomic and molecular model systems induced by absorption of
The MPI of atoms and molecules has been studied extensively both experimentally and theoretically. The fundamentals are documented in a vast body of literature, see for example [107–112]. In this section, we present the notation to describe multichromatic pulse sequences (Section 2.1) and summarize the theoretical concepts based on either non-perturbative description of the atomic and molecular neutral dynamics perturbatively coupled to the continuum (Section 2.2) or fully perturbative intraband and interband interference in MPI. The theoretical description is specifically adapted to reproduce the experimental results and discuss the relevant control aspects.
2.1 Ultrashort Polarization-Controlled Multichromatic Pulse Sequences
We start by describing the electric field of a pulse sequence consisting of
The total field is obtained by superposition of all
The physical field is given by Re[E(t)]. Schematic representations of single-color (
2.2 Simulation of Multiphoton Ionization
Many theoretical concepts and numerical methods are available to describe and simulate the quantum dynamics of atoms and molecules induced by the interaction with ultrashort laser pulses [4, 6, 114]. In the past two decades, we have used different methods to model ultrafast dynamics driven by shaped femtosecond pulses to model our experiments and interpret our observations. In parallel with our experimental studies, we have presented the corresponding theoretical descriptions and numerical simulations of the multilevel coherent control of atomic [115, 116] and molecular [117, 118] excitation. We have devised a resonant strong-field control mechanism based on the manipulation of dressed state populations and energies  and, in this context, developed a formalism to model photoelectron spectra from resonance-enhanced multiphoton ionization (REMPI) accounting for non-perturbative population dynamics and AC Stark shifts [118, 120]. In [48, 103], we provided a description for the calculation of sculpted 3D free electron wave packets from MPI, including the energy and angular distribution. Recently, we employed ab initio methods to numerically solve the time-dependent Schrödinger equation (TDSE) for a 2D model atom interacting with a polarization-tailored bichromatic femtosecond laser pulse sequence . The model was applied to reproduce and analyze our experimental results reported in [73, 103] taking into account phase fluctuations in the CEP and the Gouy phase.
2.2.1 Non-Perturbative Neutral Dynamics Coupled to the Continuum
Resonant strong-field control of MPI dynamics is based on the non-perturbative manipulation of resonant bound state dynamics and the simultaneous perturbative mapping of those dynamics into the ionization continuum by absorption of additional photons. To model this type of non-perturbative REMPI processes we adapt the concept from Meier et al.  and divide the calculation into two steps. In the first step, we calculate the non-perturbative neutral dynamics of the time-dependent amplitudes ci(t) by solving the corresponding TDSE numerically for a multistate quantum system, including all relevant bound states. This procedure yields the time-dependent bound state populations |ci(t)|2. In the second step, we compute the photoelectron kinetic energy spectrum using time-dependent perturbation theory by inserting the amplitudes determined in the first step. This strategy is illustrated in Figure 1A for a generic atomic system and in (B) for the potassium molecule.
FIGURE 1. Generic ionization schemes: (A) and (B) Non-perturbative neutral dynamics perturbatively coupled to the continuum . (A) Generic scheme for the selective population of dressed states . (B) Light-induced potentials of potassium dimers for ultrafast switching via the selective population of dressed states. (C) Scheme for perturbative control of intraband  and interband  MPI with polarization-shaped pulses along with multiphoton spectra. (D) Model potential for ab initio simulations of MPI with cycloidal laser pulses .
184.108.40.206 Atomic Multiphoton Ionization
To simulate our experiments, we have modeled atomic systems by multilevel schemes involving two [120, 123–125], three [48, 116] or more  bound states. As an example, we consider a two-level atom, consisting of a ground state |1⟩ and an excited state |2⟩ separated in energy by ℏω21 = ℏω2−ℏω1 and coupled by the transition dipole moment μ21. The non-perturbative dynamics of the atom interacting with an intense near-resnant femtosecond laser pulse
Herein, δ = ω0−ω21 is the detuning of the laser with respect to the atomic resonance. The TDSE in Eq. 5 is solved numerically on a discrete temporal grid using short-time propagator methods . This procedure yields the time evolution of the state vector
The notation was adapted from Dudovich et al. . The multiphoton transition dipole moment
220.127.116.11 Molecular Multiphoton Ionization
Molecular systems are modeled in the framework of the Born-Oppenheimer approximation by a set of N bound electronic states |ϕn⟩ characterized by the adiabatic potential energy curves Vn(R). For simplicity, we restrict ourselves to the diatomic case, where the nuclear coordinate R denotes the internuclear separation of the atoms and describes the vibration of the molecule. In the electronic basis, the TDSE for the state vector
The time-dependent probability amplitudes ψn(R, t) of the electronic states describe the nuclear wave packets in the electronic potentials Vn(R). Making use of the dipole approximation, the Born-Oppenheimer Hamiltonian is given by 
with the reduced mass mr of the molecule. Explicit expressions of the Hamiltonian matrix are provided in [118, 132]. The diagonal matrix
To calculate the photoelectron contribution from a bound electronic state Vn(R), produced by perturbative one-photon ionization with a probe pulse
If ΔVn(R) is non-monotonous within the ionization window, i.e. ΔVn′(R) vanishes locally, we decompose the wave packet into a set of Gaussian functions which are locally mapped into Gaussian-shaped photoelectron distributions with a narrow, but finite spectral width in the order of the experimental energy resolution. The second approach is based on a perturbative quantum mechanical treatment of the ionization process. Consider a transition from the bound vibrational eigenstate ν to an ionic vibrational eigenstate ν′ with the transition frequency ων′ν = ων′−ων. For a fixed spectral component ω of the probe pulse spectrum
The quantum mechanical description was used to reproduce and analyze the experimental results in [132, 137]. Using intense PLP single-color multipulse sequences followed by an ionizing probe pulse with a different color, we were able to demonstrate the resonant non-pertubative control scheme described above experimentally [131, 132, 137].
2.2.2 Perturbative Intraband and Interband Interference in Multiphoton Ionization
In this section we consider perturbative atomic MPI with a pulse train consisting of
the order of the MPI process is
which is written down explicitly for a tricromatic field (
Taking the Fourier transform of the temporal multiphoton field
Taking into consideration the selection rules for absorption of RCP (Δm = −1) and LCP (Δm = + 1) photons and ignoring for simplicity Δl = −1, the dipole moments
where we sum over all indices
FIGURE 2. Pathways for intraband MPI with
Control of multiphoton processes by multichromatic pulse sequences is based on the interference of multiple pathways leading to the same final state. As we increase the number of pulses, both the number of available quantum pathways and the number of final states increase significantly, opening up more versatile options for control. Figure 2 illustrates the pathways for absorption of
some of which may be degenerate. For each absorption of a photon, there are
An overview of the numbers of angular momentum states
TABLE 1. Overview of the total number of possible pathways for ionization with a given number of photons
In addition to the above discussed intraband interferences characterized by absorption of
If both pathways address overlapping final continuum states, i.e. the energy gained by absorption of
2.2.3 2D Ab Initio Simulation of Atomic Multiphoton Ionization in Cycloidal Laser Fields
Another efficient method to simulate atomic photoionization dynamics by polarization-shaped ultrashort laser fields is based on solving of the TDSE numerically for a 2D model system aligned in the laser polarization plane. In , we employed the 2D model potential shown in Figure 1D
with the error function erf and a soft-core parameter a, and solved the TDSE for the time-dependent Hamiltonian
on a 2D grid using the Fourier-based split-operator method . Specifically, atomic MPI with cycloidal, i.e. COCP and CRCP, femtosecond laser pulse sequences was investigated. Employing single-color pulse sequences, we were able to reproduce selected recent experimental results on the generation of photoelectron vortices [102, 103] and to analyze the transient vortex formation dynamics in different physical pictures, including the bound state population dynamics, the time evolution of the free electron wave function and its asymptotic topological charge. Employing cycloidal bichromatic fields, we reproduced the experimental results on the generation of free electron wave packets with odd rotational symmetries . We analyzed the final state wave function with respect to the relative orientation between field and PMD and extracted the azimuthal probability density current to validate the physical discussions in the experimental studies [73, 105, 106]. Furthermore, the simulations allowed us to examine the influence of experimental CEP and Gouy phase fluctuations on the measured PMD. Eventually, we applied the technique to investigate photoelectron vortices theoretically in so far unexplored intensity regimes.
In this section, we introduce our experimental scheme for the coherent control of ultrafast MPI dynamics in atoms and molecules using polarization-tailored single-color or multichromatic fields. In the experimental studies, we combine advanced supercontinuum pulse shaping techniques with the highly differential detection of photoelectrons using a VMI spectrometer. The experimental setup is portrayed in Figure 3. CEP-stable polarization-shaped laser pulse sequences are generated using a phase-stabilized white light polarization shaper in 4f geometry. Photoelectron wave packets, released by the interaction of the shaped pulses with the atomic or molecular sample in the VMI spectrometer, are observed with energy and angular resolution. By rotating the pulses about the laser beam axis using a wave plate, the VMI spectrometer is turned into a photoelectron tomography apparatus, allowing us to retrieve the full 3D PMD. In the following, we give an overview of white light polarization shaping in Section 3.1 and photoelectron tomography in Section 3.2. For a more detailed description of both techniques, we refer to .
FIGURE 3. Experimental setup for multichromatic coherent control of MPI utilizing a white light polarization pulse shaper (A) and a photoelectron VMI spectrometer (B). Multichromatic amplitude, phase and polarization tailoring of an input WLS is implemented by a specifically adapted polarization shaper in folded 4f geometry. The WLS input beam is spectrally dispersed at the input grating (G), collimated at the cylindrical mirror (CM) and spectrally modulated in the Fourier plane by a dual-layer LC-SLM and a custom composite polarizer (CP). Active CEP-stabilization is realized by an f-2f interferometer feeding the control loop of the laser system. A superachromatic λ/4 wave plate at the output allows to generate circularly polarized pulse sequences. Using an additional λ/2 wave plate, the shaped laser pulse is rotated about the beam axis, which is the basis for VMI-based photoelectron tomography. The VMI spectrometer on the right consists of an electrostatic lens system projecting the PMD onto an microchannel plate detector attached to a phosphor screen. Images of the resulting 2D projections are recorded with a charge-coupled device camera. Exemplarily, a CRCP (3ω:4ω) output pulse is depicted in magenta along with the tomographically reconstructed 3D PMD created by MPI of potassium atoms.
3.1 Supercontinuum Polarization Pulse Shaping
Various experimental schemes for the generation of multicolor laser fields have been reported, including 4f pulse shapers adapted to an octave-spanning WLS [61, 64] and extensions to optical waveform synthesizers [139, 140] as well as four-wave mixing schemes  and molecular modulation [142–144]. Our method to generate multichromatic polarization-tailored pulse sequences is based on 4f-polarization pulse shaping of a white light supercontinuum. The primary light source of our experiments is an actively CEP-stabilized multipass chirped pulse amplifier system (Femtolasers Rainbow 500, CEP 4 module, Femtopower HR 3 kHz) providing 20 fs infrared pulses centered at 790 nm with an energy of 1.0 mJ. These pulses are used to seed a neon-filled hollow-core fiber (absolute pressure ∼ 2.0 bar) generating an over-octave spanning WLS which is amplitude- and phase-modulated in the spectral domain using a home-built 4f polarization pulse shaper [48, 50, 51, 145]. While traditional 4f pulse shaping allows for either independent amplitude and phase modulation or phase and polarization shaping [51, 145], recently we introduced a pulse shaping scheme specifically adapted for the independent shaping of amplitude, phase and polarization of bichromatic fields [65, 106, 146]. The bichromatic white light polarization shaper is shown on the lefthand side of Figure 3. Bichromatic amplitude profiles are sculptured from the WLS via combined amplitude and phase modulation using a traditional polarization shaping setup consisting of a 640 pixel dual-layer liquid crystal spatial light modulator (LC-SLM) in the Fourier plane of a folded 4f setup with holographic transmission gratings at the in- and output. A customized composite polarizer mounted directly behind the LC-SLM enables independent polarization control of both spectral bands (colors). Along the spectral axis, the polarizer is divided into two parts, each of which can be chosen in s- or p-configuration, resulting in the generation of either bichromatic PLP (both colors s- or p-polarized) or OLP (one color s-, the other one p-polarized) fields. The bichromatic pulse shaping approach allows us to control the center frequency ratio, spectral bandwidths and amplitude profiles of both spectral bands and, moreover, independently permits the application of arbitrary phase modulation functions. By use of a superachromatic quarter wave plate (λ/4) at the shaper output, we generate circularly polarized single-color or bichromatic pulse sequences. An additional half wave plate (λ/2) is used to rotate the polarization-shaped laser pulses about the propagation axis, which is the basis for the photoelectron tomography technique discussed in Section 3.2. To ensure the CEP-stability of the shaped pulses, we use the shaper to generate an additional (ω: 2ω) field which is split off the main beam by a dichroic mirror and sent to an f-2f interferometer. The interferometer feeds the active control loop of the amplifier, which stabilizes the CEP of the shaped pulses before they are focused into the VMI spectrometer. The root mean squared CEP stability after the shaper was measured over 11 h and is better than 215 mrad . Finally, dispersion control and pulse characterization are implemented in situ, i.e. in the interaction region of the VMI spectrometer, by using the shaper for adaptive optimization of the highly nonlinear MPI of xenon atoms and to perform shaper-based cross-correlation measurements .
3.2 Photoelectron Tomography
In order to study the full 3D PMD caused by the interaction of atoms and molecules with polarization-tailored ultrashort laser pulses, we employ the VMI technique [83, 103] for the energy- and angle-resolved measurement of photoelectron wave packets. The VMI spectrometer is shown on the righthand side of Figure 3. The tailored laser pulses are focused via a broadband silver mirror with a focal length of 250 mm into the interaction region of the VMI spectrometer. The background pressure is typically in the order of 5 × 10−7 mbar. All experiments were carried out in the gas phase. Alkali atomic samples are supplied by dispenser sources (working pressure: ∼10−7 mbar), whereas rare gases or molecular samples are injected using an effusive gas-inlet (working pressure: ∼10−6 mbar). An electrostatic lens setup consisting of a repeller, an extractor and an Einzellens, projects the 3D PMD created by MPI of the sample by the incident laser pulses onto a chevron-type microchannel plate detector stacked with a phosphor screen. The resulting 2D image of the projected PMD is recorded by a charge-coupled device camera. PMDs created by MPI with either linearly or circularly polarized laser pulses are, in general, cylindrically symmetric and can therefore be reconstructed from a single 2D image via Abel inversion, e.g. using the pBASEX algorithm . In contrast, PMDs created by polarization-shaped pulses typically exhibit no such symmetry. For the retrieval of 3D PMDs with arbitrary shape, we developed a tomographic reconstruction technique  based on the rotation of the laser pulse—and hence the PMD—using a superachromatic λ/2 wave plate. After acquiring numerous 2D projections of the PMD under various rotation angles, we retrieve the 3D PMD by application of tomographic techniques such as the Fourier slice algorithm [89, 147, 148] or the backprojection algorithm [90, 147, 149]. An advantage of the Fourier slice method is that the angular discretization inherent to the measurement appears only in the Fourier domain representation, whereas the retrieved real-space PMD is angularly smooth. For a typical tomographic measurement, we use about 45 different orientations of the λ/2 wave plate ranging from −90° to 86°. Energy calibration of the reconstructed 3D PMD is performed by taking 151 azimuthal 2D slices through the 3D data cube and convert the radial momentum distributions into corresponding energy distributions. We estimate an energy-resolution of better than 80 meV for photoelectrons with a kinetic energy of about 1 eV. Thus, our VMI-based photoelectron tomography technique is well suited to retrieve the highly structured 3D PMDs generally created by MPI with multichromatic polarization-shaped laser pulse sequences. Similar tomographic techniques have been applied, e.g., for the imaging of molecular orbitals , the discrimination of chiral molecules via the multiphoton PECD  and the time-resolved imaging of ultrafast laser-matter interactions inside transparent media .
4 Results and Discussion
In this section, we discuss our experimental results on photoelectron tomography of 3D PMDs obtained by atomic and molecular MPI using multichromatic polarization-controlled pulse sequences. We present our findings in the order of the type of pulse sequence: We start with experiments on MPI with single-color pulse sequences in Section 4.1 and focus on various coherent control scenarios by bichromatic pulse sequences in Section 4.2. New results on trichromatic pulse sequences are presented in Section 4.3.
4.1 Single-Color Pulse Sequences
Single-color pulses consist of a single connected spectral band with bandwidth Δω centered around a carrier frequency ω1. A set of single-color pulse sequences for different polarizations is depicted in Figure 4. These pulses are the starting point of our discussion.
FIGURE 4. Single-color pulse sequences. The projection in z-direction visualizes the time-dependent electric field vector at the position of the atom or molecule. (A) PLP, (B) OLP, (C) COCP and (D) CRCP single-color pulse sequences. Upper figures: Sequence with separated pulses τ > 2Δt, lower figures: Coinciding pulses at τ = 0.
4.1.1 Single-Color Linearly Polarized Pulse Sequences
Single-color PLP pulse sequences with variable time-delay τ, as shown in Figure 4A, are the standard tool in pump-probe experiments . Even in the overlapping region τ ≈ 0, such pulses remain linearly polarized. In contrast, OLP pulse sequences, as shown in Figure 4B, additionally address the spatiotemporal aspect of the light-matter interaction due to their time-dependent polarization. Their application ranges from the phase-control of currents in semiconductors  to the steering of molecular rotation [153, 154]. Crossed polarizations demand a vectorial description of the field and thus provide access to the angular properties of the interaction energy
18.104.22.168 Perturbative Multiphoton Ionization
In , Wollenhaupt et al. have used phase-locked pairs of linearly polarized single-color pulses to study the interference of ultrashort free electron wave packets generated by perturbative MPI of potassium atoms. By variation of the time delay in the PLP sequence, the interference pattern in the photoelectron kinetic energy spectrum from one-photon ionization of the excited 5p state was controlled. The observation of interferences in the photoelectron spectrum demonstrated that the coherence properties of the ultrashort laser pulses were transferred to the electrons. Using OLP sequences, i.e. crossed polarizations, no such interference fringes have been found in the threshold electrons. However, the quantum interferences reappeared at twice the modulation frequency in the above-threshold ionization (ATI).
To rationalize the experimental findings reported in , we apply the general formalism discussed in Section 2.2.2 to describe one- and two-photon ionization of a pre-excited atom with single-color PLP and OLP pulse sequences and analyze the full 3D PMD. Figure 5A shows a schematic energy level diagram of potassium atoms for the excitation of the 5p state and subsequent photoionization at the threshold (THR) and the ATI. The description in the spherical basis provides a consistent picture for excitation and ionization with both PLP and OLP sequences. Simulated PMDs at the ionization threshold and the ATI are depicted in Figure 5B. The left column shows the PMDs from ionization with either a single pulse or a PLP pulse sequence whose polarizations are parallel (p) to the initial blue excitation pulse. Ionization with a single orthogonal (o) polarized pulse or an OLP pulse sequence are depicted in the right column. To calculate the modulation of the interference in the photoelectron spectra, we first consider the wave function in the excited 5p state prepared by the initial, linearly polarized blue (405 nm) laser pulse. In the spherical basis, the p state is a superposition of the two ψ1,±1 states
Taking into account the transition dipole moments determined by the Wigner 3-j symbols, the absorption of another parallel or orthogonal polarized photon from the 5p state yields the wave functions for the photoelectrons at the ionization threshold
By calculating the electron density ρ(ɛ, θ, ϕ) = |Ψ(ɛ, θ, ϕ)|2, we see that ionization with a parallel polarized field yields the expected d-type electron density
characterized by the interference structures in the electron density
yielding the fully cosinusoidally modulated spectrum in agreement with the experimental results . In contrast, in the OLP case, the superposition of two different wave functions
results in the highly patterned electron density
In the next step, we apply the same procedure to analyze the PMDs at the ATI. To this end, we write down the corresponding wave functions and densities and calculate the photoelectron spectra to rationalize the experimental findings in the ATI spectrum. Absorption of another parallel (orthogonal) linearly polarized photon from the threshold states given by Eq. 22 yields the wave functions in the ATI
The corresponding densities
Again, in the PLP case, the PMDs given by
Using the above method to calculate the PMDs from two-photon ionization with an OLP sequence, given by
In contrast to the unmodulated photoelectron spectra from ionization with OLP pulses at the threshold (Eq. 26), the spectra in the ATI are indeed modulated—albeit more weakly—in the OLP case in agreement with the experimental work.
FIGURE 5. One- and two-photon ionization of potassium atoms with an infrared PLP or OLP pulse sequence after preparation of the atom in the excited 5p state by a blue prepulse. (A) Energy level diagram for excitation and ionization in the spherical basis. (B) Simulated electron densities of the excited 5p state (bottom) and the emitted photoelectron at the threshold (middle row) and ATI (top row). Left side of each column: Single pulse ionization at the threshold and the ATI for parallel
In another polarization-sensitive pump-probe experiment, single-color PLP and OLP pulse sequences have been used to control the perturbative MPI of potassium molecules . It was demonstrated that the ionization efficiency is higher for OLP pulses than for PLP pulses. This result was rationalized by the orthogonality of transition dipole moments encountered along the major MPI pathways in the potassium dimer. The experiment, which included the first demonstration of control of molecular dynamics via ultrafast polarization shaping, showed that tailored polarization-shaped pulses are better suited than the corresponding linearly polarized fields to manipulate the spatiotemporal evolution of molecular wave functions.
22.214.171.124 Non-perturbative Resonance-Enhanced Multiphoton Ionization
The basic concept behind the non-perturbative control of REMPI processes is depicted in Figure 1A. The scheme is based on the selective population of dressed states (SPODS), recently reviewed in , by a sequence of PLP single-color femtosecond laser pulses. A relatively weak prepulse with a pulse area of π/2  with respect to the resonant transition—typically between the ground and first excited state—is followed by an intense main pulse. Initially, the prepulse excites an electronic wave packet of maximum coherence, i.e., an efficient charge oscillation described by a time-dependent induced dipole moment
Resonant non-perturbative control via SPODS was first reported on 1 + 2 REMPI of potassium atoms . Using interferometric double pulse sequences, we demonstrated efficient ultrafast switching between different ionization target channels by controlling the dressed state populations of the resonant 4s−4p transition via the relative optical phase. The interpretation of the scheme in a dressed state picture, was presented in . On the same system, we also implemented SPODS with PLP single-color multipulse trains generated by sinusoidal spectral phase modulation . In that work, dressed state control by all sine-parameters was demonstrated.
Subsequently, we applied SPODS to the non-perturbative control of coupled electron-nuclear dynamics in molecules. In the molecular case, the charge dynamics induced by the prepulse is altered due to the coupling between electronic and nuclear degrees of freedom. For example, the amplitude of the dipole oscillation depends on the overlap of vibrational wave packets launched in the resonant electronic states. Also, as the vibrational wave packets propagate, the electronic resonance frequency varies changing the eigenfrequency of the dipole and introducing an additional phase drift which the main pulse needs to adapt to. Our proof-of-principle studies were performed on the potassium dimer serving as a molecular prototype system. The corresponding excitation scheme is depicted in Figure 1B. The prepulse resonantly couples the
4.1.2 Free Electron Vortices With Circularly Polarized Pulse Sequences
In this section we consider the creation of free electron wave packets by REMPI of potassium atoms with single-color circularly polarized pulse sequences (Figures 4C,D). First, in Section 126.96.36.199, we discuss the formation of a FEV by perturbative MPI at the ionization threshold with c6 rotational symmetry and in the ATI with c8 rotational symmetry. In addition, we study the evolution of the c6 FEV into an angular momentum eigenstate as the time-delay in the sequence vanishes. Then, non-perturbative REMPI resulting in a FEV with c4 rotational symmetry is examined in Section 188.8.131.52. We show, how the wave function of the latter FEV is reconstructed using a holographic method.
184.108.40.206 Perturbative Resonance-Enhanced Multiphoton Ionization
Recently, the generation of FEVs by photoionization with two time-delayed CRCP attosecond laser pulses has been theoretically predicted and termed “an unusual kind of Ramsey interference” . FEVs with
Note that, in the single-color case l1 = l2. Motivated by the helical interference structures in the electron density, Starace and coworkers  coined the term “electron vortex” for this type of PMD. In general, this notion of an electron vortex is not identical with a vortex state defined by its non-vanishing azimuthal probability density current
produces a vortex-shaped electron density distribution. In the same work, it was shown that changing from MPI with a time-delayed single-color CRCP sequence to the corresponding single-color COCP sequence (Figure 4C) results in the wave function
corresponding to a toroidal PMD with spectral interference fringes in the radial (energy) direction spaced by h/τ. Subsequently, Pengel et al.  reported on the formation of FEVs from polarization-shaped supercontinua and on the generation of an eight-armed FEV in the ATI with the wave function
The c8 rotationally symmetric FEV in the ATI is shown in Figure 6C. The time-dependent electric field of single-color CRCP sequences shown in Figure 4D reveals that, as the time delay vanishes, the sequence evolves into a linearly polarized field whose orientation is determined by the relative phase between the LCP and RCP components. In Figure 6D we present results from tomographic reconstruction of the PMDs as τ → 0. The initial c6 rotationally symmetric FEV (τ = 20 fs) evolves into a distorted vortex (14 fs) with more and more pronounced lobes in the vicinity of the x-axis at 10 fs and eventually turns into the rotated angular momentum state
where the amplitudes have been taken from Eq. 27. This result can be checked by decomposition of the rotated state into the unrotated basis
FIGURE 6. Tomographically reconstructed Archimedean spiral-shaped FEVs with c4, c6 and c8 rotational symmetry from 1 + 2 REMPI of potassium atoms using single-color CRCP pulses along with their projection in the z-direction . (A) Non-perturbative REMPI with a sequence of two π pulses creates a FEV with four arms. (B) Perturbative REMPI gives rise to a six-armed FEV. (C) Absorption of an additional photon in the continuum results in an eight-armed FEV in the ATI. (D) Evolution of the c6 FEV as the time delay decreases from τ = 20 fs to τ = 0 fs.
220.127.116.11 Non-Perturbative Resonance-Enhanced Multiphoton Ionization
So far, we discussed c6 and c8 rotationally symmetric FEVs generated by perturbative three- and four-photon ionization, respectively, with the intermediate resonant 4p state being only weakly populated and not contributing significantly to the vortex formation. In contrast, it was demonstrated that FEVs with c4 rotational symmetry can be generated by non-perturbative REMPI . By using CRCP pulse sequences with a pulse area of π , the 4s state was fully depleted and the 4p state fully populated. Hence, ionization with the second pulse started from the excited 4p state resulting in a wave function characterized by the superposition of angular momentum states with different values of |m|
The tomographic reconstruction of the corresponding PMD depicted in Figure 6A shows a FEV with four arms. Using the same REMPI scheme, Pengel et al.  demonstrated a holographic method for direct measurement of the wave function of the non-perturbative c4 vortex by measuring a π shift in the azimuthal interferences in the polar direction of the vortex due to the polar nodal line of the ψ3,1 angular momentum state.
4.2 Bichromatic Pulse Sequences
In this section, we discuss a number of applications of bichromatic polarization-shaped pulse sequences for femtosecond spectroscopy and coherent control of ultrafast quantum dynamics [105, 106]. Bichromatic fields are characterized by two spectral bands with individual bandwidths of Δω1 and Δω2 centered around two different carrier frequencies ω1 and ω2, as detailed in . The spectral band with the lower (higher) carrier frequency is typically referred to as red (blue) component. Our shaper-based approach for the generation of bichromatic fields offers independent control over the amplitude, phase and polarization profile of the field [65, 146]. Shaper-generated polarization-tailored bichromatic fields are therefore very versatile, not only regarding their use in bichromatic coherent control schemes but also in terms of their application in time-resolved studies of ultrafast dynamics. For example, employing a linear spectral phase function (ω−ωi) ⋅ τ allows us to introduce a variable time delay τ between the two colors to generate bichromatic pulse sequences for polarization-sensitive two-color pump-probe experiments . Several examples of polarization-controlled bichromatic pulse sequences, with both colors being fully separated in time, are depicted in the top row of Figure 7, including the prototypical PLP, OLP, COCP and CRCP fields. The bottom row illustrates the same sequences for τ = 0. In contrast to the single-color case, temporally overlapping bichromatic fields occur in a plethora of shapes and symmetries, depending on the frequency ratio, polarization, amplitude and relative phase of the two colors. For example, in the OLP case shown in Figure 7B, the bichromatic polarization profile exhibits a Lissajous-type shape, while in the COCP and CRCP cases, shown in (C) and (D), the polarization profiles are cycloidally shaped. The applications of polarization-tailored bichromatic fields are as versatile as their shapes. For example, PLP bichromatic fields were applied to control plasmon-enhanced photoemission from silver clusters  and strong-field photoemission from gold nanotips . OLP bichromatic fields with commensurable frequencies, characterized by Lissajous-type polarization profiles which exhibit a time-varying optical chirality, have been used to investigate sub-cycle variations in the PECD of chiral molecules [179, 180]. Commensurable CRCP bichromatic fields exhibit unique propeller shapes with distinct rotational symmetry. Currently, such fields play a prominent role in HHG [67, 68, 70–72, 181, 182], strong-field ionization [158, 183–186], and were employed for the time-resolved probing of molecular chirality . In the perturbative regime, coherent control of bichromatic MPI is governed by the manipulation of intra- and interband interferences  (Section 2.2.2). In the following, we discuss examples for the bichromatic control of MPI in atoms (Section 4.2.1) and molecules (Section 4.2.2).
FIGURE 7. Bichromatic pulse sequences. The projection in z-direction visualized the time-dependent electric field vector at the position of the atom or molecule. (A) PLP, (B) OLP, (C) COCP and (D) CRCP bichromatic pulse sequences. Upper figures: Sequence with separated pulses τ > 2Δt, lower figures: Coinciding pulses at τ = 0.
4.2.1 Atomic Multiphoton Ionization
Atomic model systems interacting with multichromatic pulse sequences are ideally suited to elucidate the physical mechanisms underlying the coherent control. In this section, we study two mechanisms of coherent control of multipath interference in the atomic MPI by polarization-tailored bichromatic fields. We compare the CEP-insensitive intraband MPI with
18.104.22.168 Linearly Polarized Bichromatic Pulse Sequences
The two prototypes of linearly polarized bichromatic pulse sequences, PLP and OLP fields, are illustrated in Figures 7A,B, respectively. To illustrate the difference between interband and intraband interferences in optical phase-control, we discuss below the results from bichromatic MPI of alkali atoms using CEP-stable commensurable PLP (3ω:4ω) fields  on the one hand and incommensurable OLP bichromatic fields [99, 100] on the other hand. We start by investigating phase-sensitive interband interferences arising from the superposition of continuum states with opposite parity [73, 188]. MPI of sodium atoms with a PLP (3ω:4ω) field creates an f-type photoelectron wave packet (ψ3,0) via 3-photon ionization by the blue pulse (ω2) and a g-type wave packet (ψ4,0) via 4-photon ionization in the continuum by the red pulse (ω1). Both wave packets overlap at a kinetic energy of ɛ = 4ℏω1 = 3ℏω2 resulting in interband interferences. Because the difference between the number of photons absorbed by each partial wave is odd, the resulting photoelectron wave function is a superposition of two states with opposite parity
As illustrated in Figure 8A, this wave function exhibits a directional asymmetry along the laser polarization axis (y-axis), which is sensitive to the CEP. To study the CEP-dependence of the asymmetry, we measured 2D projections of the PMD for φce = 0 and φce = π. The recorded projections were Abel-inverted using the pBASEX algorithm . The retrieved 2D sections of the PMD through the (x, y) polarization plane are shown in the left and middle frames of Figure 8A. The sections display a pronounced left/right asymmetry which is inverted by switching the CEP from 0 to π. To quantify the asymmetry, we derived the asymmetry contrast [105, 189] directly from the measured projections. The result shown in the right frame of Figure 8A reveals a CEP-induced directional asymmetry of up to ±28%. A similar scheme for the control of directional photoemission from xenon atoms by bichromatic MPI with phase-modulated PLP (7ω:8ω) fields was recently reported in [189, 190]. In that contribution, the bichromatic pulses were specifically tailored to induce phase-controlled 7- vs. 8-photon interband interference in the 5P3/2 ionization continuum of xenon. An alternative strategy to steer the directionality of ultrafast electronic processes in atoms [188, 189, 191, 192], molecules [193–195] or solids [196–201] is based on the use of CEP-stable few-cycle femtosecond laser pulses. In this case, however, the interpretation of the interference mechanism is generally complicated by the multitude of MPI pathways connecting the ground state to a specific energy target state. Using shaper-generated commensurable bichromatic fields tailored to the MPI process, has enabled us to excite specific asymmetric target states in the continuum by two-path interband interference, fully controllable via the CEP and the relative phase of the two colors. In contrast to the few-cycle approach, we were thus able to localize phase-controlled asymmetries in a preselected photoelectron energy window . To demonstrate the concept of intraband interference, we employed temporally overlapping (τ < Δt) OLP bichromatic fields with incommensurable frequencies for 3-photon ionization of potassium atoms. While both the blue and the red component produce individual single-color signals centered at ɛ0 = 3ℏω1−ℏωIP and ɛ3 = 3ℏω2−ℏωIP, third-order intrapulse frequency mixing leads to two additional contributions centered at ɛ1 = 2ℏω1 + ℏω2 − ℏωIP and ɛ2 = ℏω1 + 2ℏω2 − ℏωIP, which are energetically disentangled from the single-color signals (cf. Eq. 15) [99, 100]. The measured and tomographically reconstructed 3D PMD is shown in the main frame of Figure 8B. The different above mentioned contributions are color-coded. Since each of the frequency mixing pathways is composed of three photons—albeit with different combinations of red and blue photons—the accumulated optical phase in each target state is identical, rendering the resulting intraband interference of all target states CEP-insensitive. Each of the photoelectron angular distributions within a given energy interval is determined by a specific coherent superposition of angular momentum states as described in Eq. 15. For example, the two single-color signals are described by rotated f-type wave packets perpendicularly to one another along the respective laser polarization axis, i.e. the x- and y-axis for the OLP pulse shown in Figure 7B. The two inner partial wave packets in the energy interval around ɛ1 (magenta) and ɛ2 (purple) resulting from intraband interference read
analogously to Eq. 27, where the upper and lower signs in the sum correspond to ɛ1 and ɛ2, respectively. Both reconstructed partial wave packets are shown separately in Figure 8C. They are also aligned perpendicularly with respect to one another and exhibit a pseudo c6 rotational symmetry. Using (ω:2ω) bichromatic OLP fields, we recently demonstrated the generation of a c4 rotationally symmetric PMD in the 2-photon ionization of potassium atoms  similar to the dx,y state shown in Figure 5B. Again, the mechanism was based on the cancellation of specific quantum pathways by intraband interference. Due to the energetic disentanglement of the different photoelectron contributions from intraband interference, this scheme is suitable for pump-probe experiments featuring background-free detection of ultrafast dynamics. For example, employing shaper-generated linearly polarized bichromatic pump-probe pulse sequences, we investigated Rydberg  and spin-orbit  wave packet dynamics. The full potential of polarization-shaped bichromatic fields to generate and manipulate angular momentum superposition states via frequency mixing has been studied in . Very recently, we introduced a shaper-based quantum state holography (SQuaSH) method by combining intra- and interband interferences using commensurable CEP-stable bichromatic fields for phase-sensitive pump-probe measurements .
FIGURE 8. Measured PMDs from bichromatic MPI of sodium and potassium atoms. (A) Retrieved (x, y)-sections from the photoemission of sodium atoms using CEP-stable (3ω:4ω) PLP fields. The left and middle frames show the Abel-inverted sections for φce = 0 and φce = π, respectively. The top insets show the corresponding generic 3D PMDs of the wave packets. The calculated asymmetry contrast, depicted in the right frame, shows pronounced asymmetries in the laser polarization plane originating from CEP-sensitive interband interference. (B) Measured and tomographically reconstructed 3D PMD from three-photon ionization of potassium atoms using bichromatic fields with incommensurable frequencies. The PMD shows four energetically separated shells originating from single-color and frequency mixing contributions, equidistantly centered around energies ɛi (i = 0, 1, 2, 3). (C) The partial wave packets from the intermediate energy channels ɛ1 and ɛ2, exhibit pseudo c6 rotational symmetry resulting from intraband interference.
22.214.171.124 Circularly Polarized Bichromatic Pulse Sequences
Circularly polarized bichromatic fields, with their cycloidal polarization profiles (Figures 7C,D), offer fascinating possibilities for the generation and control of electronic angular momentum wave packets with unusual physical properties, whether in the neutral system as in the case of spin-orbit wave packets (SOWPs) or in the ionization continuum as in the case of free electron wave packets. Similar to the linearly polarized case discussed in Section 126.96.36.199, the mechanism to control the spectrum, the symmetry and the rotation of the PMD is based on the manipulation of intraband and interband interference by the polarization state of the sequence (CRCP or COCP) and the optical phases. We illustrate the basic concepts with two examples. In the first example, we use bichromatic CRCP pulse sequences with incommensurable frequencies for MPI of potassium atoms and make use of the energetic disentanglement of the angular momentum states in the continuum. This ionization scheme is utilized for background-free observation of SOWP dynamics in the bound state. The second example deals with the creation and coherent control of FEVs with unusual rotational symmetry by interband interference in the MPI of sodium atoms with CEP-stable commensurable cycloidal bichromatic pulse sequences.
Recently, we demonstrated intraband frequency mixing by shaper-generated CRCP bichromatic fields in the three-photon ionization of potassium . Using temporally overlapping (τ < Δt) sequences of two colors with incommensurable frequencies, we showed that in the CRCP case, frequency mixing results in a complete energetic disentanglement of the different angular momentum target states in the ionization continuum. The reconstructed 3D PMD is depicted in Figure 9A. The two single-color contributions in the inner- (red) and outermost (blue) energy channels correspond to the torus-shaped partial waves ψ3,±3. The two frequency mixing contributions in the intermediate energy channels ɛ1 and ɛ2 correspond to the partial waves
displayed in the insets. This result shows that in the CRCP case, bichromatic frequency mixing provides a unique mapping between the number of absorbed red and blue photons and the different continuum target states, enabling the selective excitation of individual angular momentum states which otherwise (e.g. in the single-color case) overlap inseparably in the energy spectrum (Eq. 36). As pointed out in Section 188.8.131.52, if the two colors are spectrally disjoint, the frequency mixing signals are insensitive to the optical phases implying that frequency mixing signals are inherently robust with respect to phase fluctuations of the CEP, the relative phase between the colors or the Gouy-phase. Moreover, the process of frequency mixing is very efficient due to the large number of pathways leading to the same final state (see Figure 2). These properties, i.e. efficiency, robustness and the disentanglement of target states, make frequency mixing a well-suited test-bed for bichromatic pump-probe studies. The potential of shaper-based bichromatic pump-probe spectroscopy was recently demonstrated on the examples of time- and angular-resolved measurement  and holographic observation  of ultrafast Rydberg dynamics and the time-resolved 3D imaging of ultrafast SOWP dynamics in potassium atoms .
FIGURE 9. Tomographically reconstructed 3D PMDs of free electron wave packets created by MPI of potassium and sodium atoms with circularly polarized bichromatic pulses. (A) Intraband interferences and single-color contributions from MPI of potassium using CRCP bichromatic fields with incommensurable frequencies. (B) FEV with c1 rotational symmetry from interband interference in the MPI of sodium using CEP-stable COCP (3ω:4ω) bichromatic fields. (C) FEV with c7 rotational symmetry created by MPI of sodium using CRCP (3ω:4ω) fields.
In the latter experiment, the LCP blue pulse (pump) was tuned on resonance with the two 4p fine structure states 4p1/2 and 4p3/2. These states are separated energetically by Δɛ = 7 meV which corresponds to a SOWP oscillation period of T = h/Δɛ = 580 fs. The bandwidth of the pump was chosen sufficiently large to excite both states coherently and, thus, launch the SOWP in the neutral system. After a variable time delay τ, introduced experimentally by linear spectral phase modulation via the shaper, the RCP red pulse (probe) ionized the atom from the 4p states in a two-photon process, mapping the SOWP into the ionization continuum as depicted in Figure 10A. More specifically, the electron dynamics in the neutral atom were mapped into the ɛ1-channel, where—due to the energetic disentanglement from other contributions—the time evolution of the SOWP could be observed without any interfering background. By tomographic reconstruction of the PMD as a function of time delay τ, we obtained 3D images of the spatiotemporal SOWP dynamics. The images are shown in Figure 10B. Initially, at τ = 0, the partial wave from the ɛ1-channel displays ψ3,−1 symmetry. This symmetry maps the torus-shaped ψ1,1 state (4p (m = 1)) in the neutral system, which is consistent with the impulsive excitation of the 4p state by the circularly polarized pump pulse. At half-period τ ≈ T/2, the symmetry of the photoelectron angular distribution in the ɛ1-channel evolves into that of a ψ3,−2 state, mapping the dumbbell-shaped ψ1,0 state [4p (m = 0)]. The time evolution of the SOWP between these two states is driven by the spin-orbit interaction which, in a semi-classical picture, is interpreted as the precession of the coupled spin and orbital angular momenta around the total angular momentum. After a full period T, the distribution in the ɛ1-channel returns to ψ3,−1 symmetry indicating the completion of the first SOWP oscillation cycle. These findings, along with additional results obtained for OLP bichromatic pump-probe sequences [99, 100], showcase the capabilities of bichromatic pump-probe spectroscopy, based on shaper-generated polarization-shaped tailored bichromatic pulse sequences in combination with photoelectron tomography, for the background-free 3D imaging of ultrafast spatiotemporal quantum dynamics.
FIGURE 10. Applications of intra- and interband interference in the MPI of potassium and sodium atoms using circularly polarized bichromatic pulse sequences. (A) Excitation scheme for the time-resolved 3D imaging of ultrafast SOWP dynamics in potassium using shaper-generated bichromatic pump-probe pulse sequences. (B) Reconstructed 3D PMDs for different stages of the SOWP cycle, revealing the orbital realignment dynamics in the excited 4p state due to the spin-orbit interaction. (C) and (D) Coherent control of the azimuthal rotation of the c1 and c7 rotationally symmetric FEVs from Figures 9B,C, respectively. (E) Reconstructed 3D PMD of a seven-arm Archimedean spiral-shaped FEV created by MPI of sodium using a time-delayed (τ1 = −20 fs) CRCP (3ω:4ω) bichromatic pulse sequence.
Next, we discuss the application of circularly polarized bichromatic pulse sequences with commensurable frequencies for the coherent control of interband
Here, the upper and lower sign correspond to CRCP and COCP pulses, respectively, and gcd denotes the greatest common divisor. For the sake of clarity, we discuss below the case of
The phase term Δφ = 4φ1 − 3φ2 + φce describes the optically induced relative phase between both partial waves. The phase term ɛτ1/ℏ accounts for an additional energy-dependent propagation phase accumulated if the red pulse is time-delayed by τ1 relative to the blue pulse. The FEV described by Ψ1, created by COCP ionization, exhibits a single azimuthal lobe reminiscent of a crescent, while the FEV with the wave function Ψ7, created by CRCP ionization, displays seven lobes in the polarization plane. The corresponding tomographically reconstructed 3D PMDs, measured at the parameters Δφ = 0 and τ1 = 0, are shown in Figures 9B,C, respectively. To demonstrate the rotational control of both FEVs, we varied the CEP from φce = 0 to φce = π, analogously to the directional control discussed in Section 184.108.40.206 (Figure 8). The resulting rotation of the PMD is depicted in Figures 10C,D, indicated by the green and blue angle segments, respectively. While the c1 FEV is rotated by αCOCP = π, the c7 FEV is rotated by an angle of
4.2.2 Molecular Multiphoton Ionization
Two-color femtosecond pump-probe experiments are ideally suited to study ultrafast dynamics in molecules driven by femtosecond laser pulses . In the following Section 220.127.116.11 and Section 18.104.22.168, we discuss the application of this concept to the observation of non-Born-Oppenheimer dynamics in sodium dimers and the mapping of non-perturbative control of concerted electron-nuclear dynamics in potassium dimers. In Section 22.214.171.124, we present new results demonstrating axial and lateral asymmetries in the photoemission from chiral molecules induced by MPI with CEP-stable circularly polarized few-cycle pulses from a WLS.
126.96.36.199 Non-adiabatic Molecular Dynamics
In , we used a PLP two-color pump-probe sequence to study the effect of adiabatic (non-Born-Oppenheimer) dynamics in the
188.8.131.52 Non-perturbative Electron-Nuclear Dynamics
In the experiments reported in [132, 137], we applied a two-color pump-probe scheme to demonstrate efficient ultrafast switching between bound electronic target states in the potassium dimer, based on a detailed understanding of the underlying strong-field control mechanism. Intense femtosecond pulse sequences, shaped with attosecond precision , were used to switch the neutral excitation selectively between the 21Πg and the
184.108.40.206 Asymmetries in the Photoemission From Chiral Molecules
Polarization-shaped bichromatic fields are particularly suitable to chiral applications. Recent examples comprise the control of optical chirality , fundamental investigations on the PECD of chiral molecules [179, 180] and time-resolved studies of molecular chirality [187, 207]. Motivated by our experimental findings on the CEP-control of lateral asymmetries in the atomic MPI of sodium atoms by (3ω: 4ω)  and xenon atoms by (7ω: 8ω)  bichromatic PLP pulses (cf. Section 220.127.116.11), we investigated asymmetries in the PMD from MPI of the chiral fenchone molecule using CEP-stable few-cycle pulses. The manipulation of lateral asymmetries in chiral molecules is particularly interesting because these molecules also exhibit a pronounced axial, i.e., forward/backward asymmetry in the photoelectron emission due to the PECD . Recent measurements of the PECD of (-)-fenchone molecules from MPI with circularly polarized 400 nm pulses, depicted in Figure 11A, reproduce our previous results [97, 150]. Displayed is the antisymmetric part of the PMD with respect to the polarization plane, showing the well-known asymmetries of up to ±12%. Using a CEP-stable circularly polarized bandwidth-limited white light pulse centered at 800 nm, we observed CEP-sensitive lateral, i.e., left/right asymmetries in the photoelectron emission of (-)-fenchone. In Figure 11B we plot the difference between the PMDs measured with a CEP of φce = 0 and φce = π. This representation yields the antisymmetric part of the spectrum with respect to the propagation direction (z-direction), revealing pronounced lateral asymmetries of up to ±15%. The CEP-dependence of this asymmetry is shown in Figure 11C in the form of an energy-resolved asymmetry map obtained by integrating the spectra over the angular coordinate in one half-plane. These results demonstrate that MPI of chiral molecules with circularly polarized few-cycle pulses can be used to control the axial asymmetry via the PECD and, moreover, the lateral asymmetry via the CEP. Very recently, fully 3D polarization-tailored laser fields, generated by superposition of two polarization-controlled multichromatic laser pulses propagating non-collinearly in different directions, have been proposed as novel tools for the investigation of chiral light-matter interactions [209–212]. These so-called superchiral light fields  promise an unprecedented enantio-sensitivity which makes them highly attractive for chiral discrimination applications.
FIGURE 11. PECD and lateral photoelectron asymmetries in the MPI of (-)-fenchone molecules using circularly polarized femtosecond laser pulses. (A) PECD observed at 400 nm. (B) Previously unpublished results on lateral asymmetries in the photoelectron emission using LCP few-cycle pulses with a duration of Δt ≈ 5 fs centred at λ0 = 800 nm. (C) CEP-dependence of the lateral asymmetry displayed as an energy-resolved asymmetry map.
4.3 Multichromatic Pulse Sequences
A natural extension of bichromatic white-light polarization pulse shaping (cf. Section 3.1) is to generate multichromatic polarization-controlled pulse sequences. As a first step towards more complex pulses, we consider a trichromatic application. Trichromatic fields offer enhanced possibilities for ultrafast applications such as multicolor spectroscopy , strong-field physics [215, 216], the generation of terahertz radiation [217, 218] and HHG . Further examples of trichromatic pulse sequences are illustrated in Figure 12. Each sequence consists of a temporally overlapping polarization-tailored bichromatic pulse which is either preceded by a circularly polarized [(A) and (B)] or followed by a linearly polarized [(C) and (D)] single pulse of a third color. The use of an additional color allows us to combine the bichromatic mechanism of intraband interference, e.g. for the background-free observation of ultrafast dynamics, with phase-sensitive interband interferences for the coherent control of the underlying multiphoton processes. For example, an auxiliary pulse can be used to provide a separate reference wave packet for an experimental implementation of wave packet holography. This extension opens up a manifold of coherent control scenarios ranging from multichromatic SQuaSH via the creation and manipulation of multiple FEVs to multichannel quantum-phase clocks [101, 220]. However, trichromatic MPI is not only an extension of bichromatic MPI but also changes the physical picture, since the number of available pathways and the number of target states scales differently with the number of pulses, as discussed in Section 2.2.2. So far, we have distinguished between intraband and interband interference in bichromatic MPI scenarios. Phase-sensitive interband interference was characterized by interfering pathways involving the absorption of a different number of photons per color, i.e.
involving two distinct pathways, e.g., with three pairwise different numbers of photons
FIGURE 12. Trichromatic pulse sequences. The projection in z-direction visualizes the time-dependent electric field vector at the position of the atom or molecule. (A) Circularly polarized single pulse followed by a CRCP pulse sequence, (B) Circularly polarized single pulse followed by an OLP pulse sequence, (C) CRCP pulse sequence followed by a linearly polarized (s) single pulse and (D) COCP pulse sequence followed by a linearly polarized single pulse tilted by 45°. In all figures the single pulse is separated by a delay τ > 2Δt from the coinciding pulse sequences.
As shown in the excitation scheme in Figure 13A a photoelectron wave packet is created by resonant two-photon excitation of the 3d state by the red pump and subsequent one-photon ionization by the blue probe pulse. This wave packet interferes with the reference wave packet originating from off-resonant 3-photon ionization of the ground state by the green reference pulse to generate a phase-sensitive photoelectron hologram. We expect the hologram to be sensitive to the relative phases between the pump, the probe and the reference, but, because the same number of photons are absorbed in each pathway, the interference pattern should not depend on the CEP. To investigate this scheme experimentally, we apply spectral amplitude modulation of the WLS (cf. Figure 13B) to generate a tailored PLP trichromatic pulse sequence composed of a λpu = 928 nm pump pulse (red band), followed by a λpr = 722 nm probe (blue band) and λref = 848 nm reference pulse (green band). The time delays between the pulses are introduced via additional linear spectral phase modulation of the pump and probe band with τpu = −215 fs and τpr = −15 fs relative to the reference pulse. The wave function of the resulting photoelectron hologram reads
where cpu, cpr and cref denote the complex valued amplitudes including delay-dependent phases and φpu, φpr and φref the relative optical phases corresponding to the pump-the probe- and the reference pulse, respectively. A tomographically reconstructed 3D PMD is depicted in Figure 13C. To evaluate the phase-dependence of the hologram, we measure the modulation of the integral cross-section in the SQuaSH channel of the photoelectron spectrum by variation of the relative optical phase φref ∈ [−π, π]. To this end, the measured PMDs are energy-calibrated  and integrated over the angular coordinates to yield the photoelectron spectra. In Figures 13D,E the resulting energy-resolved photoelectron spectra I (ɛ; φref) with ɛ ∈ [0.01, 0.1] eV are plotted as a function of the relative phase of the reference pulse. The insets in the top frame show the measured projections for constructive (φref = −1.64 rad) and destructive interference (−0.77 rad). In agreement with Eq. 46, the interference signal shows a
FIGURE 13. SQuaSH on potassium atoms using PLP trichromatic pulse sequences. (A) Excitation scheme for the two-photon excitation (pump) and time-delayed one-photon ionization (probe) of the potassium 3d state. Interference of the resulting free electron wave packet with a reference wave packet created by three-photon ionization of the ground state by the reference pulse provides a phase-sensitive photoelectron hologram in the continuum. (B) The triple pulse sequence is generated by trichromatic WLS shaping applying linear spectral phases to introduce the time delays of pump and probe relative to the reference pulse. (C) Tomographically reconstructed 3D PMD from SQuaSH with PLP trichromatic pulses. (D) Angularly integrated photoelectron spectra I (ɛ; φref) versus the applied optical phase φref of the reference pulse for φpu = 0 showing a pronounced
5 Summary and Outlook
In this review, we have reported on coherent control of MPI dynamics of atomic and molecular model systems using fully coherent polarization-tailored single-color, bichromatic and trichromatic pulse sequences. Building on previous experiments with single-color pulse sequences, we have shown that the advanced polarization shaping of CEP-stable supercontinua enables the generation of unprecedented bichromatic laser fields such as propeller-shaped CRCP pulses or Lissajous-like OLP pulses. Even more flexibility is achieved in trichromatic polarization shaping by the many possible combinations of linearly or circularly polarized pulses, resulting in an increasing number of final states and interfering pathways. These new pulse shapes enable the targeted use of CEP-insensitive intraband and CEP-sensitive interband multipath interference for quantum control. The experimental results on atomic and molecular MPI by multichromatic pulse sequences reviewed here, serve to illustrate the new options for coherent control. We have summarized the theoretical methods used to reproduce and analyze our experimental results and presented a powerful experimental technique combining supercontinuum polarization shaping and VMI-based high-resolution photoelectron tomography to reconstruct the full 3D PMD. The presented applications of single-color pulse sequences included perturbative and non-perturbative control of atomic and molecular MPI. We have analyzed the 3D PMD from a previous experiment on the interference of free electron wave packets created by PLP sequences and reported on free electron vortices with even-numbered rotational symmetry originating from MPI with single-color CRCP sequences. We have reviewed results on bichromatic coherent control by intraband and interband multipath interference on various examples including control of the directional photoemission in the MPI of atoms and chiral molecules by PLP and COCP fields, the use of COCP and CRCP cycloidal fields to generate and manipulate free electron vortices with odd-numbered rotational symmetry, and spin-orbit wave packets controlled by PLP pulses. Finally, we have introduced trichromatic shaper-based quantum state holography using a trichromatic PLP pulse sequence by adding a reference pulse to a bichromatic pump-probe sequence. As an example for the new options of trichromatic pulse shaping, we have devised an MPI scheme based on a polarization-tailored trichromatic CRCP pulse sequence which results in a CEP-insensitive c6 rotationally symmetric inner vortex surrounded by a CEP-sensitive crescent-shaped wave-packet (Figure 14). Extensions of our shaping scheme towards the generation of independently polarization-controlled multichromatic multipulse sequences are currently explored in our labs.
FIGURE 14. (A) Trichromatic pulse sequence consisting of a circularly polarized single pulse temporally overlapping with a CRCP pulse sequence. While the pump-probe pulse sequence is given by a (3ω:4ω) bichromatic field, the additional green component exhibits a frequency ratio of 5:6 with respect to the probe field. (B–D) show the resulting PMDs displaying a CEP-insensitive 6-armed inner vortex surrounded by a CEP-sensitive 1-armed outer vortex, for selected values of the CEP and the relative optical phase φref.
Shaper-based multi-color 2D spectroscopy using fully coherent multichromatic CEP-stable pulse sequences is another emerging application of our technique. Taking advantage of the shaper’s capability to additionally introduce higher order spectral phase modulation, the trichromatic scheme can be further developed into a powerful tool for coherent control spectroscopy by using sequences of tailored pulses, thus generalizing the concept of 2D spectroscopy. In addition, the tunability of the shaper-generated polarization-controlled multi-color fields makes them attractive for applications in quantum information , quantum metrology [109, 110, 223] and ultrafast nanotechnology [79, 178, 214, 224, 225]. In all the experiments reported above, the tailored polarization profile was confined to the (x, y)-plane perpendicular to the propagation direction of the pulse. Building on recent theoretical proposals [209–212, 226], we envision the next challenge in experimental coherent control to be the generation and tailoring of full 3D light fields by superimposing two polarization-shaped laser pulses propagating non-collinearly in different directions.
KE and MW developed the topic and idea, collected the data, and prepared the original manuscript. MW coordinated this review and managed the project. KE, LE, TB, and MW contributed to the conception and development of the narrative. KE, TB, and MW were responsible for data analysis and simulations. LE contributed to the drafting of the manuscript and prepared the illustrations. All authors contributed to manuscript revision, read, and approved the submitted version.
This research was funded by the Deutsche Forschungsgemeinschaft (DFG) via the priority program SPP1840 QUTIF. Open access publication fees are granted by the University Oldenburg Open-Access publication fond supported by the DFG program Wissenschaftliche Literaturversorgungs-und Informationssysteme.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Keywords: coherent control, bichromatic polarization shaping, multiphoton ionization, photoelectron tomography, chiral molecules
Citation: Eickhoff K, Englert L, Bayer T and Wollenhaupt M (2021) Multichromatic Polarization-Controlled Pulse Sequences for Coherent Control of Multiphoton Ionization. Front. Phys. 9:675258. doi: 10.3389/fphy.2021.675258
Received: 02 March 2021; Accepted: 23 July 2021;
Published: 20 August 2021.
Edited by:Robert Gordon, University of Illinois at Chicago, United States
Reviewed by:Jianming Wen, Kennesaw State University, United States
David Ayuso, Imperial College London, United Kingdom
Nicolas Douguet, Kennesaw State University, United States
Copyright © 2021 Eickhoff , Englert , Bayer and Wollenhaupt. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Matthias Wollenhaupt, email@example.com