Attosecond ionization dynamics of modulated, few-cycle XUV pulses

Few-cycle, attosecond extreme ultraviolet (XUV) pulses in the strong field regime are becoming experimentally feasible, prompting theoretical investigating of the ionization dynamics induced by such pulses. Here, we provide a systematic study of the atomic ionization dynamics beyond the regime of the slowly varying envelope approximation. We discuss the properties of such XUV pulses and report on temporal and spectral modulations unique to the attosecond nature of the pulse. By employing different levels of theory, namely the numerical solution to the time-dependent Schrödinger equation, perturbation theory and a semi-analytical approach, we investigate the ionization of atoms by modulated, few-cycle XUV pulses and distinguish first and higher order effects. In particular, we study attosecond ionization in different intensity regimes aided by a general wave function splitting algorithm. Our results show that polarization and interference effects in the continuum prominently drive ionization in the few-cycle regime and report on carrier-envelope phase (CEP)- and intensity-dependent asymmetries in the photoelectron spectra. The use of spectrally modulated attosecond pulses allows us to distinguish between temporal effects causing asymmetries and dynamic interference, and spectral effects inducing a redshift of the photoelectron spectrum.


Introduction
The attosecond timescale (1 as = 10 −18 s) is the natural time scale of electronic motion, and its study motivates the large current interest in attosecond science [1,2]. While in the beginning, the research focus of femtosecond dynamics was on the infrared regime, in recent years, research has been extended to the high frequency regime [1][2][3]. Due to the advances in light-source technology, such as free-electron lasers [4] and high-harmonic generation [5], short light pulses in the extreme ultraviolet (XUV) down to the soft x-ray are nowadays accessible not only in large facilities but also in table-top research labs [5][6][7][8]. XUV pulses of a few 100 as down to 50 as have been demonstrated [9]. Moreover, fewcycle XUV attosecond pulses with stable carrier-envelope phase (CEP) have been realized [10][11][12] and great effort is put into increasing the intensity of attosecond pulses [13,14].
These extremely short XUV pulses have been used to investigate light-matter interaction down to the attosecond time scale to probe, e.g. ultrafast electron dynamics and charge migration [3,[15][16][17][18][19]. Commonly, single ionization by XUV wave length is well described within linear response by singlephoton interaction. Therefore, for moderate and weak intensities, the impact of pulse shape and/or spectral phase variations on the ionization dynamics is negligible. This does not apply to higher intensities, when CEP-and other phase-modulations of the spectrally extremely broadband pulses are involved, as has been shown by several theoretical works. Firstly, in the area of double photoionization of the helium atom, an asymmetric photoelectron angular distribution has been reported that can be controlled by the CEP of the pulse [20][21][22][23][24]. Secondly, approaches to extend holographic mapping into the XUV regime using two-cycle pulses have been theoretically proposed [25][26][27]. Thirdly, molecular attosecond photoionization with few-cycle XUV pulses has been studied and, with the help of perturbative models, the redshift in the photoelectron spectra was rationalized [28].
In this work, we theoretically investigate the interaction of moderately to intense broadband XUV pulses with an atom. We emphasize that the common description of spectral phase modulation including chirping for femtosecond pulses with wavelengths in the visible spectral range, need to be adapted to account for the very broad spectral range of the attosecond XUV pulses as we will show below. We present atomic calculations on different levels of theory, specifically numerically exact solutions to the time-dependent Schrödinger equation (TDSE), perturbation theory (PT), and analytical approaches. We examine spectral shifts in the photoelectron spectra, analyze the interaction with spectrally modulated attosecond XUV pulses, distinguish temporal and spectral effects, and reveal a CEP-dependent asymmetry for intense few-cycle XUV pulses. The latter is related to constructive and destructive interference during the ionization process and strongly dependent on the light intensity. The results are analyzed by using a wave function splitting algorithm that allows a unique view on the ionization processes. We demonstrate its capabilities on the example of few-cycle effects in XUV ionization.
The paper is organized as follows: In the theory section, the properties of ultrashort XUV pulses and the employed model system are introduced. Special emphasis is put on temporal and spectral modulations in the few-cycle XUV regime. Furthermore, the different levels of theory are presented. Lastly, in section 3 the numerical results of the model system are discussed, with an emphasis on intense spectrally and CEP-modulated pulses. Finally, we summarize and conclude our findings. In the appendix, we present simulations on the 3D hydrogen atom and explain our wave function splitting algorithm in detail.

Properties of Ultrashort XUV Pulses
The electric field of a XUV pulse is described via its timedependent vector potential: where E 0 is the electric field strength, ω 0 is the field's carrier angular frequency, ϕ CE is the CEP, and β = 4 ln(2) τ −2 is the coefficient of the Gaussian envelope function, with τ being the full width at half maximum (FWHM) pulse duration. This corresponds to the electric field For comparatively long pulses with many optical cycles, the slowly varying envelope approximation (SVEA) is commonly assumed [29] and the first term in equation (2b) is negligible. However, for ultrashort, few-cycle pulses, the SVEA is generally not valid. The influence of the first term on the spectral properties of the pulse becomes clear by considering the following form: and is shown in figure 1 for a ultrashort, attosecond fewcycle XUV pulse (a), compared to a femtosecond XUV pulse (d). For ultrashort, few-cycle pulses (a), the rapidly varying envelope defines the pulses's spectral properties. The (dashed line). More importantly, the time-dependent phase ϕ(t) = arctan 2β ω0 t (dotted line) introduces an additional time-dependent envelope-induced frequency, ω ϕ (t) = d dt ϕ (t) (dashed-dotted line) and shifts the instantaneous frequency of the pulse asω 0 (t) = ω 0 + ω ϕ (t). At the maximum envelope intensity at t = 0, this results in a frequency ofω 0 (t = 0) = ω 0 + 2β ω −1 0 , giving rise to a blueshift with respect to the carrier frequency, ω 0 [30,31]. This blueshift is inversely proportional to the pulse length squared. Thus, it becomes particularly pronounced for attosecond pulses as used in the high frequency regime. In figure 1, for ω 0 = 15.5 eV and a pulse length τ = 140 as, the shift accounts for 5.8 eV. For long pulses (d), where the SVEA is applicable, the time-dependent phase ϕ(t) (dotted line) and its gradient are negligible compared to ultrashort, few-cycle pulses. Therefore, there is no envelopeinduced frequency (dashed-dotted line).
The spectral representation of the electric field is obtained via the (forward) Fourier transform (FT),Ẽ(ω) = FT {E(t)}. Alternatively, starting from the Euler decomposition of the vector field in equation (1b), we can writẽ Making use of the FT properties, the (negative) derivative in time corresponds to a multiplication by −i ω in spectral representation to obtain equation (6) using the identities β ω = (4β) −1 andẼ 0 = E 0 β ω /2, The linear factor in ω, which arises from the temporal derivative of the vector potential envelope function in equation (1a), guarantees a vanishing DC component,Ẽ(0) = 0, which is necessary to obtain a propagating pulse [32][33][34]. The linear frequency factor, ω, in the spectral representation (equation (6)) is another way to account for the effects of the envelope in ultrashort pulses, resulting in a blueshift of the central frequency,ω 0 , by scaling the analytical spec-trumẼ ± carrier (ω) = −Ẽ0 ω0 e −βω(ω∓ω0) 2 ±i ϕCE associated with the carrier frequency, ω 0 . Due to the ultrashort pulse duration, the positive and negative frequency components overlap at low energies. In figures 1(b) and (e), the two spectra, plotted for positive frequencies, are contrasted. The vertical dotted line indicates the blueshift of the central frequency,ω 0 , from the carrier frequency, ω 0 , (dashed line). Note that although the spectral phase is flat, there is a non-vanishing temporal phase, ϕ(t), due to the asymmetry of the spectrum.
In the following, we disentangle properties of the photoelectron spectra caused by the spectral width from temporal effects, i.e. effects that are explained by the attosecond nature and not the broad spectral width. To this end, we investigate ionization from spectrally modulated pulses with quadratic spectral modulation [35]. We introduce a quadratic spectral phase, ϕ 2 , in the frequency domain: The quadratic phase ϕ 2 causes a linear chirp in time domain and significantly elongates the pulse duration: where c.c. denotes the complex conjugate of the previous term, and β c = β/(1 + 4ϕ 2 2 β 2 ) defines the temporal profile with a FWHM of τ c = 4 ln 2/β c . Figures 2(a) and (c) display an example of a spectrally modulated few-cycle and long XUV pulse. The spectral intensity profile (power spectral density) of the unchirped (ϕ 2 = 0) and spectrally modulated pulses reads:Ĩ and consists of three terms: the term with positive and the term with negative frequency contributions, (solid red line). All three terms include a scaling by the frequency, ω 2 , that ensuresĨ(0) = 0, as seen in equation (6). Spectrally modulated and unchirped pulses are shown in figures 2(b), (d) and 1(c), (f), respectively.
The spectral intensity profile of unchirped and spectrally modulated pulse are identical in width but differ in the cross term,Ĩ cross (ω). The latter depends (on the CEP and) on the quadratic phase ϕ 2 that introduces amplitude modulations in the cross term, which are imprinted on the overall profile, I(ω). The amplitude modulated cross term is only relevant for attosecond pulses with a rapidly varying envelope, i.e. a broad spectral width (meaning a small value of β ω ). Another way of looking at this observation is this: The quadratic spectral phase, ϕ 2 , as introduced in equation (7), leads to different modulation of the positive and negative frequency components. For long XUV pulses, this solely results in a broadened temporal pulse with no effect on the spectral behaviour. However, for ultrashort XUV pulses, the positive and negative frequency components overlap (recall figure 1(b) dotted line). As a result, the spectra are amplitude modulated in the overlapping region giving rise to the cross term introduced above. Due to the amplitude modulations we refer to the spectrally modulated pulses as quasi-chirped.
For this study, we use an angular carrier frequency of ω 0 = 0.570 a.u. (15.5 eV) and different pulse intensities, represented by the electric field amplitudes E 0 = 1.69 × 10 −2 , 1.69 × 10 −1 , and 1.69 a.u. (corresponding to intensities of I = 10 13 , 10 15 , and 10 17 W cm −2 , respectively). If not stated otherwise, the intermediate value of E 0 = 1.69 × 10 −1 a.u. is employed. Different FWHM pulse lengths τ between 140 as and 10 fs, and different CEP values, ϕ CE ∈ {− π 2 , 0, π 2 , π}, are considered as well. We investigate unchirped, ϕ 2 = 0, and quasi-chirped, ϕ 2 = −0.125 fs 2 , pulses as displayed in figures 1 and 2. For the latter, the quasi-chirped vector field based on equation (8) is used: The full time-dependent Hamiltonian within the dipole approximation and employing the velocity gauge reads: where p is the electronic momentum. We numerically solve the time-dependent Schrödinger equation for atomic systems in 3D using QPROP 3.2 [36] and in 1D using a screened one-dimensional atomic model system with the potential [37][38][39][40] Here, x is the electron coordinate, erf denotes the error function and a = 2.83 a.u. is the truncation parameter, which results in an ionization potential of 0.320 a.u. (8.72 eV). Using a model system allows us to draw qualitative conclusions and compare to a variety of theory to draw intuitive pictures of the processes. It also reduces computational costs and numerical difficulties for the spectrally modulated pulses that are extremely broad in both time and energy. Thus, the results in the main text are obtained using aforementioned model system. Additionally, we compare and validate our findings to numerical solutions of the TDSE of the 3D hydrogen atom expanded in spherical harmonics in the appendix A. Atomic units (a.u.) are used throughout the manuscript if not specified otherwise. In the following, the terminology forward (backward) for x > 0 (x < 0) is used, in particular, to describe photoelectron emission directions and their asymmetries.

Level of theory
To disentangle first and higher-order effects, and to identify specific quantum effects, such as interference and transient processes, different levels of theory are applied. In the following, the direct numerical solution of the TDSE will be contrasted against a first-order PT and a semi-analytical approach based on PT and continuum plane waves (PTPW).

Numerical details.
The time-dependent wave function |Ψ(t)⟩ is represented on a spatial grid of [−907,907] a.u. with 2048 grid points. The direct propagation is carried out numerically employing the short-time propagator with a time step size of ∆t = 1 as by the symmetric split operator technique [41] in coordinate and momentum space and using the FFTW3 library [42] for Fourier transform. Time propagation starts at t 0 = −5τ , if not indicated otherwise. Bound electronic eigenstates {|φ n ⟩} are obtained via the relaxation method [43] with the time-independent, field-free Hamiltonian yielding 37 bound states {|φ n ⟩}, n ∈ {0, . . . , 36}, with negative eigenenergy E n < 0. The electronic ground state is used as initial wave function, |Ψ(t 0 )⟩ = |φ 0 ⟩, in all propagation schemes.
To suppress reflection at the grid boundaries in forward (fwd) and backward (bwd) direction during propagation, cutoff functions are used in the asymptotic region of the atomic potential [44] with parameters ζ 1 = 0.025 a.u. and ζ 2 = 983 a.u. Parts of the wavefunction that reach the cut-off region, transformed into momentum space, and coherently added to ⟨p|Ψ out,ν (t)⟩ =Ψ out,ν (p, t) over time, which yields the photoelectron spectrum (PES) at the end of the propagation [44]: where only components in the direction of emission, i.e. p > 0 (p < 0) for ν =fwd (bwd), are considered. In our calculations, t → ∞ is set to t = 500 fs.

First-order PT.
Within first-order PT, the timedependent wavefunction reads: with the time-dependent perturbation for a negatively charged particle The continuum propagation with the field-free Hamiltonian, H 0 (equation (14)) is carried out via the split operator method outlined above. This explicit propagation provides a description of the transition from the bound state to the continuum within first-order, i.e. without transient ionization, where the system does not necessarily remain ionized but may recombine within a very short time window [45][46][47]. The PES, σ PT ν (p), is obtained as in equation (16) replacingΨ out,ν (p, t) by its perturbative counterpart |Ψ PT(1) (t)⟩.

Perturbation theory with plane waves approach
(PTPW). As additional level of approximation, we represent the continuum by plane waves (PTPW). Starting from the PT approach (equation (17)), we replace the field-free propagator by a propagator with a (discretized) plane-wave basis (19) and the eigenvaluesẼ p = p 2 /2. We obtain The time integral can be rewritten aŝ In the momentum integral the transition dipole moment to the continuum (ionization dipole moment) occurs: In the femtosecond regime, the ionization transition dipole moment (equation (22)) is often assumed to be constant and set to one (Condon approximation) [48], which is a good approximation for the comparatively narrow spectral profile. However, in attosecond pulse interaction, the p-dependence becomes non-negligible. Thus, equation (20b) can be rewritten as: yielding the intuitive expression for the PES for t → ∞ ('Fermi's Golden Rule') with ν =fwd (bwd) for p > 0 (p < 0), as a product of the modulus squared of the transition dipole moment and the electric field in the frequency domain, shifted by the ground state energy, E 0 . The latter has been defined as the spectral intensity, see equation (9).

Pulse lengths dependence
In the long pulse limit, XUV ionization is intuitively described as a one photon process removing an electron from its bound state with energy E 0 and placing it into the continuum, with an excess kinetic energy, E, given by energy conservation  (25) (E 0 = −I P ). Recall, for long, spectrally narrow pulses, the carrier frequency ω 0 equals to the photon energy ω. In our model system employed throughout this chapter, this corresponds to E = 15.5 eV−8.7 eV = 6.8 eV. This is demonstrated in figure 3 for unchirped (ϕ 2 = 0) pulses of different pulse lengths τ . Apart from the obvious decrease in intensity due to the reduced total-i.e. time-integrated-pulse intensity, it is clear that equation (25) holds only for long XUV pulses comprised of many optical cycles. By reducing the pulse length, the PES not only broadens but substantially shifts its peak position towards lower kinetic energies, away from the expected one-photon long pulse limit (dashed vertical line).
This effect can be explained qualitatively by PTPW, which expresses the PES, equation (24), as the product of the transition dipole moment and the spectral intensity profile. Both quantities are shown in figure 4 panel (a) for three selected pulse lengths, τ = 5 fs, 560 as, and 140 as. The transition dipole moment, |µ(p)| 2 (orange curve) is largest for low energies. The spectral intensity profile of the XUV pulses, I(ω) (blue curves), does not only broaden with decreasing pulse duration, but also substantially blueshifts (vertical dotted lines), recall section 2.1. The overlap betweenĨ(ω) and |µ(p)| 2 is largest in the region of low kinetic photoelectron energies, thereby shifting the PES towards lower kinetic energies [28]. The transition dipole moment becomes the limiting contribution and we observe a dipole-dependent reshaping and imprint of µ(p) on the PES. In figure 4 panels (b)-(d), this effect is shown comparing results obtained by the TDSE, PT, and PTPW approach. All three levels of theory display the intensity shift towards low kinetic energies marking it as a linear property. While PT overall agrees well with the TDSE approach, the PTPW slightly underestimates the yield for low kinetic energies indicating the error induced by using plane waves in the latter. Not surprisingly, the deviations become larger, the smaller the kinetic energy of the emitted electron [49].

PES for a quasi-chirped pulse
Recent work on dynamic interference emphasizes the importance of the temporal envelope of pulses for XUV ionization [50][51][52] . In order to disentangle effects originating due to the temporal envelope of the ultrashort attosecond pulse from those originating from the extreme broad spectral profile, we next investigate attosecond few-cycle pulses with a spectral modulation [53][54][55]. As a consequence, while the spectral width remains broad, the temporal profile stretches (recall section 2.1). We employ the same parameters as in the previous section: a few-cycle pulse with pulse length of τ = 140 as and a spectral phase of ϕ 2 = −0.125 fs 2 , which linearly changes the instantaneous frequency and effectively elongates the pulse to τ c = 5.0 fs. The same analysis as for the unchirped pulses in the previous section is performed and shown in figure 5. The overall shift of the PES towards low energies remains unaffected by the quasi-chirp. This is also reproduced by the PT approaches (PT, PTPW). As before, the PES shift originates from the spectral profile of a pulse and not its temporal envelope function. Second, an oscillation in the spectral intensity can be observed, which is imprinted in the PES. This is reminiscent of the cosine cross term,Ĩ cross (ω), that becomes relevant for spectrally broad XUV pulses (recall equation (9) and figure 2(b)). The first-order PT approach with an exact continuum propagation is very similar to the TDSE results further underlining that higher order effects are negligible here and that we deal with the same linear effect as for unchirped pulses with the addition of oscillations due to the interplay of the spectral modulation with the broad spectrum of the attosecond pulse.

Intensity-dependent CEP effects
Results discussed above were obtained with a CEP of ϕ CE = 0 simulating the PES in fwd (x > 0) direction. For long and/or weak pulses, the CEP does not affect the PES. We now look into CEP effects of a few-cycle (τ = 140 as) pulse and its effect on the PES in fwd/bwd direction with different intensities. In figures 6(a)-(c), the PES in fwd and bwd direction is shown for CEP values of ϕ CE ∈ {−π/2, 0, π/2, π} and for different intensities of the original few-cycle XUV pulse (ϕ 2 = 0). We can distinguish three different regimes with unique CEPdependent asymmetries. For I = 10 13 W cm −2 , no CEP-dependent fwd/bwd asymmetry is present and the perturbative description applies. However, for half-integer CEPs, i.e. ϕ CE = ±π/2 (sine-like pulses), a higher signal intensity in the PES is obtained compared to the whole-integer cases, i.e. ϕ CE = 0 and π (cosinelike pulses). This effect is, again, traced back to the cosine cross term in the spectral intensity for attosecond pulses, equation (9), which is maximum for ϕ CE = ±π/2 and minimum for ϕ CE = 0, π. Consequently, the pulse's total intensity is enhanced for ϕ CE = 0, π leading to a higher yield in the PES according to equation (24). Since this CEP dependence is based on equation (9), it is a first order effect and, thus, fully reproduced applying the PT approach, figure 6(d).
For I = 10 17 W cm −2 , the asymmetry for cosine-like pulses is further enhanced, and additionally an asymmetry for the sine-like pulses appears.
To understand the CEP-dependent asymmetries in the two non-perturbative regimes (I = 10 15 W cm −2 and I = 10 17 W cm −2 ) in detail, we utilize our wave function splitting algorithm analysis (see appendix C). As outlined in the appendix, the strength of the splitting algorithm is that at any given time step t i , we dissect the full wave function |Ψ(t i )⟩ into the continuum partial wave function |Ψ cont (t i )⟩, the part of the wave function of instantaneously newly created continuum electrons within the previous time step, |Ψ new i ⟩, the part of the wave function describing the deexcitation from the continuum back to the bound state within the previous time step, |Ψ deex i ⟩, and the effect of interference of new born continuum electrons with the rest of the continuum, S(t i ). The latter is a scalar, quantifying constructive (S(t i ) > 0) and destructive (S(t i ) < 0) interference.
Our analysis shows that initially the light pulse interaction leads to a polarization of the bound wave function and, subsequently, a restoring force pointing to the center of the Coulomb potential. This translates to an initial net momentum for the newly created electrons in the continuum. All three effects are time-dependent and change during the light pulse interaction. This leads to newly created electrons in the continuum with varying net momentum and promotes strong interference effects between newly created and already present electrons in the continuum during the pulse interaction. Such continuum-continuum interferences represent the dominant part of the ionization process (see appendix C). The interference pattern in the continuum dictates the final emission asymmetry reported above. For I = 10 15 W cm −2 , we obtain a fwd (positive) asymmetry for ϕ CE = 0, while for ϕ CE = π 2 the interference contributions cancel out. For I = 10 17 W cm −2 and ϕ CE = π 2 , strong destructive interference of negative net momenta leads to the final fwd emission asymmetry.
For the quasi-chirped attosecond XUV pulse described in section 3.2, these non-perturbative CEP-and intensitydependent effects are not present. This is because-unlike the redshift of the PES reported above-these asymmetries are a temporal pulse effect originating from the time-dependent polarization and continuum-continuum interference. Since the quasi-chirped attosecond XUV pulse behaves in time like a femtosecond pulse, we do not observe these asymmetries.
Finally, we point out that for unchirped attosecond XUV pulses, no interference structure is found in the final PES neither for 1D (section 3.1) nor 3D appendix A simulations. We report-beyond the long pulse limit-a red-shift of the PES as a spectral effect, and a CEP-and intensity-dependent asymmetry as temporal interference effect. We did not find any indication for spatial or temporal interference leading to modulations/structures in the final PES for XUV ionization [25]. Modulations due to dynamic interference have been observed for longer high-frequency pulses [50][51][52] and such effects arise in our simulations only for a temporally elongated quasichirped attosecond XUV pulse with I = 10 17 W cm −2 .

Summary
In this work, we have addressed systematically several quantum effects occurring in few-cycle XUV attosecond ionization. To this end, we thoroughly introduced the properties and modulations unique to attosecond, few-cycle XUV pulses beyond the SVEA.
Apart from the trivial effect of a broad spectral bandwidth, envelope-induced contributions lead to a temporal modulation. In the frequency realm this results in a strong blueshift and a cross term in the spectral intensity profile depending on the CEP and spectral modulations.
For weak to moderate intensities, the light-atom interaction can be well described within linear response: despite the blueshift of an attosecond XUV pulse, the resulting PES are shifted towards lower kinetic energies. This is reasoned by the two contributions, the p-dependence of the ionization matrix element, and the broad spectral profile of the pulse. Since this shift is a linear effect in the spectral domain, a quasi-chirped XUV attosecond pulse, which stretches the pulse in the time domain, while only marginally affecting its spectral intensity width, results in the same red shift.
These cross terms of the spectral intensity, that is unique to attosecond pulses beyond the SVEA, vary the total pulse intensity, rendering the PES sensitive to the CEP of an ultrashort pulse even within first-order effects. For half-integer CEPs, the yield in the PES is maximum, whereas for wholeinteger CEPs, a minimum signal follows.
As a non-perturbative effect, we report on fwd/bwd asymmetries in the PES for unchirped few-cycle XUV pulses, which depend on the field strength and the CEP of the pulse. This effect is defined by the temporal property of the pulse and is not present for temporally elongated, quasi-chirped XUV pulses. By applying a wave function splitting algorithm, we traced the origin of the asymmetries back to an initial bound state polarization and CEP-and intensity-specific continuumcontinuum interferences that drive the ionization process.
Future questions in this field could be investigating dynamical interference of strong, (un)chirped, few-cycle XUV pulses and hereby extending the works of long, high frequency pulses [50][51][52]. It is yet to be addressed how temporal dynamic interference effects are present for these kind of pulses and have to be thoughtfully dissected from the CEP-and intensitydependent effects unraveled in this work. Moreover, it has been shown recently that electronic and nuclear correlation lead to asymmetries in the PES during XUV ionization [56], therefore, the interplay with the asymmetry from attosecond pulses reported here would be of future interest.

Data availability statement
All data that support the findings of this study are included within the article and appendix.

Appendix A. Calculation on 3D hydrogen
In order to validate and quantify the results of the text, we performed TDSE calculations on the hydrogen atom in 3D using an expansion in spherical harmonics. To this end, we utilized QPROP 3.2 [36] modified to allow for our pulse definitions in section 2.1.
The electronic ground state was obtained by imaginary time propagation. The light pulse was linearly polarized and added in the dipole approximation using velocity gauge with the vector potentials shown in equation (1a) (unchirped) and equation (10) (quasi-chirped). The PES was calculated via t-SURFF [57] during the pulse interaction time, while the propagation of the wavefunction from the end of the pulse to infinity used i-SURFF [58]. For the expansion in spherical harmonics, L max = 15 was used and m = m 0 = 0 since the magnetic quantum number of the initial state is conserved. The grid, time propagation, and t/i-SURFF parameter had to be chosen carefully to encapsulate the high frequency, broad spectrum with very slow and fast electrons and (temporally elongated) time span of the (modulated) light pulses. We used a time step of ∆t = 0.02, grid spacing of ∆r = 0.2, and a grid size of R grid = 1600 with cut-off of the binding potential to a linear function becoming zero at the grid boundaries at R CO = 400. The t-SURFF flux is collected at the grid boundary. The imaginary potential to suppress grid reflection used the parameters V im,max = 100 and W im = 400 with a factor of 10 for the i-SURFF propagation. For further detail on the implementation of the parameters, we refer to the original QPROP and t/i-SURFF publications [36,57,58].
In the simulations, we used ω = 0.749, E 0 = 0.0169 and three pulses identical to the ones used in the text for the model Figure S1. Normalized PES of hydrogen in 3D using an expansion in spherical harmonics and obtained by numerical solution to the TDSE. Calculations with the same properties as in the text were performed validating the model system approach. A long, femtosecond XUV pulse (blue dotted line) leads to a spectrum centered around the one-photon limit (vertical grey dashed line). The attosecond pulse (dashed orange line) exhibits a red shift. The quasi-chirp of the attosecond pulse (green line) leads to an imprint of the spectral modulation parameter due to the unique cross term for attosecond pulses discussed in the text.

Appendix B. Wave function splitting algorithm
In order to understand and quantify different effects contributing to attosecond pulse interaction, such as direct ionization, deexcitation and continuum interference, a wave function splitting algorithm is derived. To that end, a projector P = n |φ n ⟩⟨φ n | (B1) and its complement Q = 1 − P are introduced, dissecting a wavefunction into its 'bound' and 'unbound' component, respectively.
Applying the projectors to the full wave function, , at a certain point in time t i yields two partial wave functions, indicating which components can be considered to be bound at time step t i and which components are unbound, i.e. in a continuum state (cont): Consequently, the instantaneous new creation of unbound continuum electrons from the bound components of the wavefunction during a single propagation time step ∆t at time t i can be obtained via whereÛ(t i , t i−1 ) is the short-time propagator with the full Hamiltonian and the wavefunction index i indicates the time of birth at timestep t i of the newly created unbound electron. Further propagation of the newly generated part of the continuum electron's wave function, |Ψ new i ⟩, from its time of birth t i to a later time t j , j ⩾ i, and again dissecting the wavefunction, allows monitoring the influence of the Coulomb potential and light field on the continuum electrons: |Ψ free i (t j )⟩ represents those parts of the wave function that were entering the continuum at time t i and are still found to be unbound at time t j . This will be especially interesting for the final time step, j = N t , i.e. when the light pulse is switched off. Note that the coherent sum of all continuum electrons born at different times t i (and propagated to t j ) yields the unbound part of the total wavefunction: Analogously to equation (B4), the deexcitation from the transiently populated continuum into the bound states within a single time step ∆t is calculated as: Moreover, the splitting approach allows to isolate the influence of interference between newly generated continuum electrons at time t j , |Ψ new j ⟩, and continuum electrons generated at earlier points in time i<j |Ψ free i (t j )⟩ on the total amount of the photoionization probability. To this end, we start from equation (B6), write the last time step explicitly, and use the identities equations (B4) and (B7): This expression correspond to the coherently propagated wavefunction of the unbound electrons, which is extended by the newly generated photoelectrons at time step t j and reduced by the electrons that return into a bound state at the same time.
The total photoionization probability is therefore where we used ⟨Ψ new j |Ψ deex j ⟩ = 0. Using equation (B7) and the projector identity P = P 2 , the second to last term becomes The last term in equation (B9) represents the interference S(t j ) of newly generated photoelectrons at timestep t j with previously generated photoelectrons, which can be positive, in case of constructive interference, or negative in case of destructive interference. Using equations (B4) and (B6), and Q = Q 2 , it can be written as The total photoionization probability at time t j is then given by: where we have applied the decomposition of |Ψ cont (t j )⟩ recursively to all previous time steps t i . Thus, we interpret the total photoionization probability as the sum of all contributions being excited into the continuum states over all points in time minus the probability of its deexcitation back into a bound state plus the interference term, which is further examined in section 3.3. Figure S2 visualizes the wave function splitting algorithm and its contributions outlined above for three explicit time steps. The wave function splitting algorithm is mathematically not depending on the dimensionality of the wave function and can be applied as long as the basis of the projection operators can be calculated. For higher dimensions, obtaining this basis is computationally demanding. Figure S2. Graphical depiction of wave function splitting algorithm for three example time steps. All wave functions and operators as described in the text. The top, blue row shows the propagation of the full wave function. In each time step, t i , the wave function is dissected into a bound (brown) and continuum (red) part, |Ψ bound (t i )⟩ and |Ψ cont (t i )⟩, via the respective projection operators, P and Q. Further propagation and projection of those by one time step gives the partial wave function of newly created continuum electrons (green), |Ψ new i ⟩, and deexcitation into bound states (violet), |Ψ deex i ⟩. The interference term (red), S(t i ) describes interference between newly created continuum electrons and continuum electrons created at previous times.

Appendix C. Detailed analysis of Intensity-dependent CEP effects with a wave function splitting algorithm
Here, we present a detailed analysis of the effects described in section 3.3 using the wave function splitting algorithm of appendix B.
All expectation values to an operator O shown in the following are calculated via ⟨⟨Φ | O | Φ⟩⟩ = ⟨Φ | O | Φ⟩/⟨Φ | Φ⟩, where Φ represents either the full wavefunction or one of its projections.
The results of the wave function splitting algorithm are shown in figure S3. The first row a) shows the electric field and vector potential, followed by the momentum and coordinate expectation value of the bound wave function, ⟨⟨Ψ bound (t j )|p|Ψ bound (t j )⟩⟩ and ⟨⟨Ψ bound (t j )|x|Ψ bound (t j )⟩⟩, in the second row b). The third row c) depicts the momentum expectation values of the continuum wave function, ⟨⟨Ψ cont (t j )|p|Ψ cont (t j )⟩⟩, the momentum of newly created continuum electrons, ⟨⟨Ψ new j |p|Ψ new j ⟩⟩ and the change of this momentum after the light pulse has vanished at t end = N t ∆t, ⟨⟨Ψ free j (N t )|p|Ψ free j (N t )⟩⟩. The interference term, S(t j ) (equation (B10c)), can be rewritten to extract the momentum influenced by interference: The (norm-weighted) momentum influenced by interference, S p (t j )/S(t j ), follows for all our examples the momentum of newly created electrons, ⟨⟨Ψ new j |p|Ψ new j ⟩⟩. The cumulative sum yields the continuum momentum at time t j purely defined by interference effects: i⩽j S p (t i ). The last row d) in figure S3 shows the norm of all contributing processes according to equation (B11b). To improve readability, we will treat time as a continuous quantity below, despite its numerical representation via a discretized grid.
C.1. Asymmetry at I = 10 15 W cm −2 Starting with the cosine-like pulse of ϕ CE = 0 (column I in figure S3), we see in I b) that the initial negative vector potential leads to a polarization of the wave function's average position, ⟨⟨Ψ bound (t)|x|Ψ bound (t)⟩⟩ (I b dotted line), to negative coordinate values following´A(t) dt. This leads to a positive restoring force due to the Coulomb potential and, thus, the positive momentum expectation value ⟨⟨Ψ bound (t)|p|Ψ bound (t)⟩⟩ (I b solid line). The positive momentum in the bound state is Figure S3. Results of the wave function splitting algorithm for column I: ϕ CE = 0 and I = 10 15 W cm −2 , column II: ϕ CE = π leading to a change in restoring force (II b solid line) imprinted on the continuum electrons (II c dashed line) and, secondly, an interference that compensates constructive and destructive parts after t > 0.07 fs.
C.2. Asymmetries at I = 10 17 W cm −2 The latter point of constructive and destructive compensation in interference contribution is the essential change when going to I = 10 17 W cm −2 that leads to an asymmetry for ϕ CE = π 2 (column III in figure S3): For the increased intensity, the bound state polarization (III b dotted line) scales by a factor of ten (proportional to the electric field strength) compared to I = 10 15 W cm −2 , which leads to larger restoring force (III b solid line) and, subsequently, momentum expectation values (III c). However, their qualitative behavior is identical for both intensities. The momentum modulation in the continuum, ⟨⟨Ψ cont (t)|p|Ψ cont (t)⟩⟩ (III c solid line), is again interferencedriven and can be explained by the change in positive and negative values of S(t) (III d dashed-dotted line). Here lies the difference to the asymmetry-free sine-like pulse at I = 10 15 W cm −2 . After t > 0.07 fs the interference, S(t), is strictly negative for I = 10 17 W cm −2 (III d dashed-dotted line), while it is equally positive and negative for I = 10 15 W cm −2 (II d dashed-dotted line). This difference leads to a final positive asymmetry for I = 10 17 W cm −2 , that is absent for lower intensities. The negative (destructive) interference, S(t) (III d dashed-dotted line), of negative momentum components, ⟨⟨Ψ new t |p|Ψ new t ⟩⟩ (III c dashed line), leads to a net positive momentum imprinted in the final continuum wave packet after ionization, ⟨⟨Ψ cont (t end )|p|Ψ cont (t end )⟩⟩ (III c solid line).
Moreover, row III d) shows a change in relative contribution of the different processes involved. The deexcitation term is increased compared to lower intensity, but still overall negligible (factor ten compared to contribution of newly created electrons). In general, the overall ionization magnitude and the norm of the processes involved follows the vector potential (row d), i.e. are zero if the vector potential is zero.
To summarize, while the polarization of the bound wave function is responsible for the initial momentum of newly created electrons in the continuum, the important term driving the ionization is interference of newly created continuum electrons with the already present continuum electrons (continuum-continuum interference). This term explains the temporal momentum modulations and is responsible for the final emission asymmetry. Moreover, the CEP-and intensitydependent effect reported in this section is-unlike the redshift of the PES reported in section 3.1-a temporal pulse effect, which is not present in our simulations for a temporally elongated, quasi-chirped attosecond XUV pulse with a comparable spectral intensity.
We are confident that the wave function splitting algorithm can be applied to a plethora of problems and systems. In this first example, interference turned out to be the driving force, but for other setups deexcitation might become relevant and can be monitored in detail using this algorithm. We also want to point out that the partitioning by bound and continuum projection operators is just one example. In principle, the algorithm can be applied to any kind of Hilbert space partitioning, as long as the eigenfunctions of the projection operators are computationally accessible.