High-order phase-dependent asymmetry in the above-threshold ionization plateau
M. Kübel, P. Wustelt, Y. Zhang, S. Skruszewicz, D. Hoff, D. Würzler, H. Kang, D. Zille, D. Adolph, A. M. Sayler, G. G. Paulus, M. Dumergue, A. Nayak, R. Flender, L. Haizer, M. Kurucz, B. Kiss, S. Kühn, B. Feti?, D. B. Miloševi?
HHigh-order phase-dependent asymmetry in the above-threshold ionization plateau
M. K¨ubel, ∗ P. Wustelt, Y. Zhang, S. Skruszewicz, D. Hoff, D. W¨urzler,H. Kang, D. Zille, D. Adolph, A. M. Sayler, † and G. G. Paulus Institute of Optics and Quantum Electronics, Max-Wien-Platz 1, D-07743 Jena, Germany andHelmholtz Institute Jena, Fr¨obelstieg 3, D-07743 Jena, Germany
M. Dumergue, A. Nayak, R. Flender, L. Haizer, M. Kurucz, B. Kiss, and S. K¨uhn
ELI-ALPS, ELI-HU Non-Profit Ltd., Wolfgang Sandner utca 3., Szeged, H-6728, Hungary
B. Feti´c and D. B. Miloˇsevi´c
Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina andAcademy of Sciences and Arts of Bosnia and Herzegovina,Bistrik 7, 71000 Sarajevo, Bosnia and Herzegovina (Dated: February 16, 2021)Above-threshold ionization spectra from cesium are measured as a function of the carrier-envelopephase (CEP) using laser pulses centered at 3.1 µ m wavelength. The directional asymmetry in theenergy spectra of backscattered electrons oscillates three times, rather than once, as the CEP ischanged from 0 to 2 π . Using the improved strong-field approximation, we show that the unusualbehavior arises from the interference of few quantum orbits. We discuss the conditions for observingthe high-order CEP dependence, and draw an analogy with time-domain holography with electronwave packets. Controlling electron motion using tailored laser fieldsis a central goal of strong-field and attosecond physics.This includes the motion of quasi-free electrons, underly-ing attosecond pulse generation [1] and encoding ultrafasttemporal information [2]; valence electrons in molecules[3], determining the outcome of chemical reactions; or ul-trafast currents in solids [4]. Various control techniqueshave been proposed and implemented, including attosec-ond pulses [5], multi-color pump-probe schemes [6, 7],and polarization-shaped laser pulses [8, 9].One of the most fundamental approaches, however,relies on the use of a few-cycle laser pulse, E ( t ; φ ) = E ( t ) cos ( ωt + φ ), with a controlled or known carrier-envelope phase (CEP) φ , and an envelope function whoseduration is comparable to an optical cycle T = 2 π/ω .This approach has been used to control various processesin atoms, molecules, and solids. The prototypical CEPeffect consists in an asymmetry in the above-thresholdionization (ATI) spectra of photoelectrons emitted intoopposite directions along the laser polarization [10]. Thesinusoidal oscillations of the CEP-dependent asymmetry, A ( φ ) ∝ sin( φ + φ ), represents the basis for measuringthe CEP using the stereo-ATI technique [11, 12], or otherapproaches [13, 14]. Here, φ is a phase offset, known asthe phase-of-the-phase [15], that generally depends onthe electron drift momentum.The CEP-dependence of ATI can be understood ona qualitative level by simple symmetry considerations:a CEP-stable few-cycle laser pulse has broken inversionsymmetry. Since the drift momenta of photoelectronsare determined directly by the electric field evolution,the asymmetry of the pulse is transferred onto the mo-tion of the field-driven electrons. As the field asymmetry is maximized for φ = 0 , π , and the field is symmetricfor φ = π/ , π/
2, one expects the aforementioned si-nusoidal oscillation for the photoelectron asymmetry, inaccordance with experimental results. CEP effects areparticularly pronounced for recollision processes [16], in-cluding high-order ATI [11], non-sequential double ion-ization [17, 18], and high-harmonic generation (HHG) [1].CEP effects with periodicity of π rather than 2 π , i.e.,with periodicity parameter m = 2 instead of m = 1 areobserved for the photon yields from HHG [1, 19] and fortotal yields of double ionization [20, 21] or fragmenta-tion [22]. The doubled periodicity is consistent with theabove symmetry considerations since total yields are in-sensitive to the direction of the field but are affected bythe modulation of the instantaneous intensity resultingfrom varying the CEP. The general theory of CEP ef-fects [23] predicts oscillations with periodicity parameter m to result from the interference of two pathways thatinvolve the absorption of n and n + m photons, respec-tively. However, experimental evidence for m > / cm , we ob-serve an unusual CEP-dependent asymmetry with m = 3in the ATI plateau region around 6 U p , where U p is theponderomotive potential. For lower intensity values, theCEP-dependent asymmetry exhibits the usual behaviorwith m = 1. The experimental results are interpreted us-ing the improved strong-field approximation (ISFA) andusing the saddle-point method [24]. We show that theobserved fast oscillations in the CEP-dependent asym-metry are due to the interference of a few quantum or- a r X i v : . [ phy s i c s . a t o m - ph ] F e b bits [25] which are modulated by the CEP. We discussthe conditions under which high-order CEP-dependentasymmetries are observable in ATI.The experiments are conducted using the mid-infrared(MIR) laser at the Extreme Light Infrastructure Attosec-ond Light Pulse Source (ELI-ALPS). It provides intenseultrashort ( ∼
42 fs) laser pulses centered at λ = 3100 nmat a repetition rate of 100 kHz [26]. The laser pulsesare post-compressed to a pulse duration (full width athalf-maximum of the intensity envelope) of τ p = 31 fs(corresponding to three optical cycles) using the methoddescribed in Ref. [27] [28]. The CEP of the laser pulseshas an excellent stability with a jitter of only 82 mradroot mean square [27]. In our experiments, the CEP iscontrolled with a pair of BaF wedges.The linearly polarized laser pulses are focused with athin CaF lens ( f = 200 mm) into a vacuum chamberwith base pressure of 10 − mbar, where they intersectan atomic beam of Cs, produced by evaporating Cs at ∼ ◦ C. The ionization potential of Cs (3 . , O ,H O). Hence, contamination of the measured photoelec-tron signal by ionization of background gas is negligi-ble at the relatively low intensities (
I < TW / cm )used in our experiments. The photoelectrons generatedin the laser focus are detected using two pairs of mul-tichannel plates detectors of 25 mm diameter, placed atthe ends of two 50 cm long drift tubes along the laserpolarization. The geometry supports a detection angleof ∼ ◦ , such that essentially only directly forward ordirectly backward scattered electrons are detected. Us-ing a pair of detectors on either side of the laser polar-ization (i.e., the stereo-ATI technique) helps suppressingthe effect of laser intensity fluctuations on the recordedCEP-dependent ATI spectra.Photoelectron spectra for ATI of Cs ionized by few-cycle pulses at 3100 nm wavelength are presented inFig. 1(a). It is interesting to compare our results to ear-lier ATI experiments. Despite the long laser wavelengthand owing to the relatively low intensity, the ponderomo-tive energy in our experiments amounts to only few eV.This is much less than in previous ATI experiments usingmid-IR light, where rare gases or molecules were used astargets, e.g. [29–31]. The low energy of recolliding elec-trons counteracts the infamous decay of the recollisionprobability [29] such that we observe a pronounced ATIplateau. Indeed, the situation in our experiment is rathercomparable to the typical case of xenon ionized by 800 nmlight, e.g. [11, 24].The high-energy cut-off of the ATI plateau at 10 U p is used to determine the laser intensity for each mea-surement. Using these values, we plot the intensitydependent electron yields in Fig. 1(d). The measureddata points agree well with predictions based on thePerelomov-Popov-Terent’ev (PPT) ionization rate [32].The significant flattening of the curve indicates that sat- FIG. 1. Experimental results for ATI of Cs by linearly polar-ized few-cycle mid-IR laser pulses ( τ p = 31 fs, λ = 3100 nm).(a) CEP-averaged photoelectron spectra measured for inten-sity values of approximately 0 . / cm (purple dotted line),1 . / cm (blue dashed line), 3 . / cm (green dashed-dotted line), and 4 . / cm (yellow solid line), respectively.(b,c) Measured CEP-dependent asymmetry parameter for 4.1and 3.2 TW / cm , respectively. The black box in (b) marksthe trifurcation of the asymmetry, which is absent in (c). (d)The average number of electrons detected per laser shot iscompared to predictions based on the PPT tunneling rate(solid line). The error bars depict an estimated 15% uncer-tainty in the intensity determination. Panel (e) presents theCEP-dependent asymmetries at electron energies of 6 U p , asmarked by the black arrows in (a). The high-order, m = 3,asymmetry oscillation in the case of the highest intensity isclearly visible and distinct from the usual m = 1 oscillationsobserved for the lower intensity values. uration takes place close to 2 TW / cm . In addition,the saturation intensity can be estimated using numer-ical solutions of the three-dimensional time-dependentSchr¨odinger equation (3D TDSE). To this end, we evalu-ate the survival probability at the end of the laser pulse,and find that saturation occurs between 2 TW / cm and3 TW / cm , in good agreement with Fig. 1(a), see Sup-plementary Material (SM) [33].CEP-dependent experimental results for ATI ofCs are presented in Figs. 1(b-e). In order toevaluate the CEP-dependence of the ATI spectrawe calculate the asymmetry parameter, A ( E, φ ) =[ R ( E, φ ) − L ( E, φ )] / [ R ( E, φ ) + L ( E, φ )]. Here, R ( E, φ )[ L ( E, φ )] are the yields of photoelectrons with energy E detected on the right (left) detector for a laser pulsewith CEP φ . The CEP-dependent asymmetry maps ofFigs. 1(b,c) reveal a clear CEP-dependence of the ATIspectrum, including features which are washed out whenthe data are averaged over the CEP. At low energieswhere the ATI spectrum is dominated by direct electronemission, the CEP dependence is rather weak and modu-lations associated with ATI peaks can be seen. At higherenergies, in the recollision plateau, significantly largerasymmetry values are observed and the regions of posi-tive/negative asymmetry are tilted, i.e. the phase-of-thephase depends on the electron energy in the characteris-tic fashion of the ATI plateau [11].The striking feature of our experimental results isshown in Fig. 1(b) and highlighted by the black box.At an electron energy of E ∼
20 eV, the CEP-dependentasymmetry trifurcates and exhibits a clear high-order os-cillation with m = 3. Around E ∼
30 eV, the asymme-try returns to the usual periodicity. This range corre-sponds to 5 U p (cid:46) E (cid:46) U p . The unusual behavior of theCEP-dependent asymmetry is only observed at the high-est intensity studied in our experiments, I = 4 TW / cm ,while it is absent at lower intensity values, as shown inFig. 1(c). The CEP-dependent asymmetry at E = 6 U p [black arrows in Fig. 1(a)] is quantified in Fig. 1(e).Clearly, only at the highest intensity value, high-orderoscillations are observed. We note that the intensity of I = 4 TW / cm is well beyond the saturation intensityof Cs, see Fig. 1(d). The role of saturation is addition-ally corroborated by unpublished experiments on ATI ofalkali atoms using 1800 nm few-cycle pulses [34].In order to interpret our experimental results and findthe origin of the asymmetry trifurcation, we employ threedifferent theoretical methods. The most accurate, andalso most computationally expensive method is the 3DTDSE, see Ref. [35] and SM for details on the imple-mentation used in the present work. An asymmetry mapcalculated with the TDSE is displayed in Fig. 2(a) andagrees very well with the experimental results shown inFig. 1(b), despite overestimating the asymmetry ampli-tude in the cut-off region above 35 eV. The phase depen-dence of the asymmetry varies throughout the electronspectrum, not only by phase but also by periodicity. Inparticular, the high-order asymmetry oscillations around20-25 eV can be clearly seen.For further analysis, we turn to the ISFA,[11, 24,36, 37], see SM for details. The ISFA is a quantum-mechanical theory which can be interpreted using thewell-known three-step model [16], where electrons firsttunnel from the atom around the field maxima, propa-gate in the continuum under the sole influence of the laserfield, and finally scatter off the parent ion upon recolli-sion. The electron acquires a phase given by the classicalaction S p ( t, t ; φ ). For a laser pulse with CEP φ , theyield y p of electrons with drift momentum p can be ob-tained by integrating over all possible ionization times t and rescattering times t , y p = p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dt (cid:90) dt V pk (cid:18) πiτ (cid:19) / e iS p ( t,t ; φ ) I k b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1)where τ = t − t is travel time, V pk is the Fourier trans-form of the rescattering potential V ( k is the intermediate FIG. 2. Calculated CEP-dependent asymmetry maps for (a)TDSE, (b) ISFA, (c) saddle-point calculations at intensity I = 4 TW / cm , averaged over the intensity distribution of aGaussian focal volume. While the TDSE results contain con-tributions from both direct and recollision electrons, the ISFAand saddle-point results are obtained for recolliding electronsonly. electron momentum), and I k b = (cid:104) k + A ( t ) | r · E ( t ) | ψ b (cid:105) is the ionization dipole matrix element ( ψ b is the boundground-state wave function).An asymmetry map calculated by numerically inte-grating Eq. (1) is displayed in Fig. 2(b). We focus onthe region of the rescattering plateau, where we observea pronounced oscillation with m = 3 in the energy rangeup to ∼
25 eV, where it returns to m = 1, in reasonableagreement with the experimental data. Since direct elec-trons are neglected in the ISFA results, the low-energypart cannot be directly compared to the experimentalresults and is omitted.It is insightful to evaluate the above integral [Eq. (1)]using the saddle-point approximation. The condition ofstationary action yields a discrete number of contribu-tions to the total integral. These contributions are anal-ogous to classical trajectories and referred to as quan-tum orbits [24, 38]. Each quantum orbit corresponds toionization and recollision at different times, t , t , respec-tively. The saddle-point results based on 8 pairs of quan-tum orbits are displayed in Fig. 2(c). They match theresults of the full numerical integration of Fig. 2(b) veryclosely, indicating that all significant contributions to therelevant photoelectron yield are included by consideringa limited number of quantum orbits. In the following,we will interpret the CEP-dependent asymmetry and itstrifurcation in terms of quantum orbits.Using the saddle-point method we obtain quantum or-bits that are associated with wavepackets that are cre-ated at specific half-cycle maxima of the electric field,at times t , and recollide at later times t . Based on the FIG. 3. Quantum orbit analysis. (a) The laser electric field ofa cos-like pulse ( φ = 0) is plotted along with the most relevantquantum orbits where the emission time t is connected withthe recollision time t . The solid (dashed) lines correspond toquantum orbits with electron emission in positive (negative)direction. Red integer (black half-integer) values correspondto long (short) orbits that lead to positive drift momenta. (b)For each quantum orbit, the electron yield is plotted againstthe electron kinetic energy. The sharp spikes at the cut-offenergies are an artefact of the saddle-point method. For theintensity of 4 TW / cm , the resulting CEP-dependent asym-metry map is plotted for (c) long orbits only, (d) short orbitsonly, and (e) all orbits. The black boxes mark the trifurcationregion. travel times τ = t − t , we distinguish between short or-bits with τ ≈ . T ( T = 10 . λ = 3100 nm) that rescatter on the first return, and longorbits with τ ≈ . T that rescatter on the second return.Orbits with longer travel times are not considered. InFig. 3(a), we plot t , and t for the most important orbitsin a cos-like pulse ( φ = 0). The time t can be the birthtime of short and long orbits, which produce electronwave packets with opposite drift momenta. For nomen-clature, we concentrate on electron emission with positivedrift momentum and use the emission time t at φ = 0 aslabels for different quantum orbits [see Fig. 3(a)]. Thus,half-integer values refer to short orbits, and full integersrefer to long orbits that produce electrons with positivedrift momentum. On the other hand, long orbits start-ing at half-integer times, and short orbits starting at fullinteger times result in negative drift momentum. Theseorbits are not labelled and are indicated by dashed linesin Fig. 3(a). When the CEP is scanned from 0 to 2 π ,the emission and rescattering times shift to earlier times,e.g. orbit 3.5 starts at t ≈ T for φ = π .Fig. 3(b) shows that the different orbits contribute todifferent parts of the spectrum. For example, the shortorbit 3 . E (cid:38) U p ) of theelectron spectrum. On the other hand, the long orbit 3 dominates the low energy part up to approximately 4 U p .Beyond 5 U p , it quickly drops below the yield from orbits3 . .
5. In the range around 6 U p , where the trifur-cation is observed, the orbits 3.5 and 4.5 yield approx-imately equal contributions to the photoelectron spec-trum.Figure 3(c) shows the asymmetry map calculated forthe contributions of the long orbits only. It exhibits theusual behavior with m = 1. This indicates that the longorbits alone are not responsible for the high-order oscil-lations of the CEP-dependent asymmetry.The asymmetry map calculated for the short orbitsonly is shown in Fig. 3(d) and exhibits clear high-orderoscillations up to approximately 30 eV, where it returnsto the usual behavior with m = 1. This transition co-incides with the cut-off energy of the orbit 4 . U p ).The absence of high-order oscillations above 7 U p impliesthat the high-order oscillations around 6 U p arise fromthe interference of the two short quantum orbits 3 . .
5. Even faster oscillation are present at energies below20 eV in Fig. 3(d). However, these cannot be directly ob-servered, since the dominant contribution is given by thelong quantum orbit 4 [see Fig. 3(b)], which exhibits onlythe fundamental oscillation, m = 1.The full asymmetry map composed of both short andlong orbit contributions is displayed in Fig. 3(e). It canbe seen that a pronounced high-order oscillation persistsin the range between 20 and 30 eV. At lower energies thebehavior of the asymmetry is dictated by the long quan-tum orbit. One might object that signatures of fast os-cillations are still observable at low energies in Fig. 3(e).However, the same can be said about the experimentaldata presented in Fig. 1(b), for example around 15 eV. Athigher energies ( E > U p ), on the other hand, the elec-tron spectrum is dominated by the contributions from asingle short quantum orbit 3.5, which also results in thefundamental oscillation period.The interference of quantum orbits underlying the ob-served trifurcation can be interpreted as holography inthe time domain. This differs from the position-spaceholography evoked in previous studies on photoelectronholography, where different trajectories originating in thesame laser half-cycle interfere [39–42]. The essential fea-ture of holography is the presence of a signal wave anda reference wave. For a useful reference, one of the in-terfering orbits in our experiment should exhibit a well-behaved phase evolution. For our case the orbit 3.5 isa good choice for the reference wave; it propagates inthe continuum around the center of the pulse, where theintensity variation are small compared to the situationof orbit 4.5, which propagates on the falling edge of thepulse envelope, where intensity variations are significant.For details, see the calculated phase evolution of orbits3.5 and 4.5 given in the SM.The question arises why high-order asymmetries havenot been reported in previous experiments. Part of theexplanation has been implicitly given above: in many ex-periments, the high-order oscillations are concealed bythe dominant contribution of a single quantum orbit.This is the case, in particular, for very short laser pulses,which are otherwise beneficial to observe pronouncedCEP effects. In our case, using three-cycle pulses, how-ever, two quantum orbits produce very similar contribu-tions in the energy range between 5 U p and 7 U p , whichcreates favorable conditions for the high-order oscilla-tions to become observable. In other words, for holo-graphic interferences, the pulse needs to be sufficientlylong to allow for one orbit to be considered a referencewave.The question remains why the effect is only observedat high intensity in our experiments. This can be ex-plained by the effect of ionization depletion during thelaser pulse, which suppresses the contribution of laterhalf-cycles with respect to earlier ones. In the highlightedregion of Fig. 3(b), the contribution from orbit 4.5 isstronger than that of orbit 3.5. For a certain degree of de-pletion the contributions of the two interfering orbits maybe equalized, thus maximizing the interference contrast.An additional factor might be that the measurement ofhigh-order oscillations requires both good statistics andexcellent CEP stability, which has been challenging toobtain. In the present experiment, these conditions havebeen met by using a highly CEP-stable high-repetitionrate laser [27].In conclusion, we have observed high-order CEP effectsin ATI of Cs driven by few-cycle MIR laser pulses. Ouranalysis based on quantum orbit theory shows that thefast oscillations of the CEP-dependent asymmetry canbe understood as the interference of two backscatteredquantum orbits. At this point it is unclear how this resultshould be interpreted in terms of the general theory ofCEP effects [23], given the large number of photons ( n >
50) involved. Future experiments could probe nuclear orelectronic dynamics by means of time-domain holographyof quantum orbits.
Acknowledgements
We thank F. Kohl, A. Rose, T. Weber, F. Ronneberger,and the ELI-ALPS team for technical and logistical sup-port. The authors acknowledge funding by the DeutscheForschungsgemeinschaft (DFG) in the framework of theSchwerpunktprogramm (SPP) 1840, Quantum Dynam-ics in Tailored Intense Fields (project 281296000). MKacknowledges funding by the DFG under project no.437321733. The ELI-ALPS project (GINOP-2.3.6-15-2015-00001) is supported by the European Union and co-financed by the European Regional Development Fund.B.F. and D.B.M. acknowledge support by the Ministryfor Education, Science and Youth, Canton Sarajevo,Bosnia and Herzegovina. ∗ [email protected] † Present address: Benedictine College, Atchison, KS,USA[1] A. Baltuˇska, T. Udem, and M. Uiberacker, Nature (Lon-don) (2003).[2] R. Kienberger, E. Goulielmakis, M. Uiberacker,A. Baltuˇska, V. Yakovlev, F. Bammer, A. Scrinzi,T. Westerwalbesloh, U. Kleineberg, U. Heinzmann,M. Drescher, and F. Krausz, Nature , 817 (2004).[3] M. F. Kling, C. Siedschlag, A. J. Verhoef, J. I.Khan, M. Schultze, T. Uphues, Y. Ni, M. Uiberacker,M. Drescher, F. Krausz, and M. J. J. Vrakking, Science , 246 (2006).[4] A. Schiffrin, T. Paasch-Colberg, N. Karpowicz,V. Apalkov, D. Gerster, S. M¨uhlbrandt, M. Korb-man, J. Reichert, M. Schultze, S. Holzner, J. V. Barth,R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I.Stockman, and F. Krausz, Nature (London) , 70(2013).[5] E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Bal-tuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh,U. Kleineberg, U. Heinzmann, M. Drescher, andF. Krausz, Science , 1267 (2004).[6] M. K¨ubel, Z. Dube, A. Y. Naumov, M. Spanner, G. G.Paulus, M. F. Kling, D. M. Villeneuve, P. B. Corkum,and A. Staudte, Phys. Rev. Lett. , 183201 (2017).[7] M. K¨ubel, M. Spanner, Z. Dube, A. Y. Naumov,S. Chelkowski, A. D. Bandrauk, M. J. J. Vrakking, P. B.Corkum, D. M. Villeneuve, and A. Staudte, Nat. Com-mun. , 2596 (2020).[8] H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M.Shakya, E. Moon, and Z. Chang, Phys. Rev. Lett. ,103906 (2008).[9] O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin,D. Popmintchev, T. Popmintchev, H. Nembach, J. M.Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Co-hen, Nat Phot. , 99 (2015).[10] G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi,M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, Na-ture (London) , 182 (2001).[11] G. G. Paulus, F. Lindner, H. Walther, A. Baltuska,E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev.Lett. , 253004 (2003).[12] T. Rathje, N. G. Johnson, M. M¨oller, F. S¨ußmann,D. Adolph, M. K¨ubel, R. Kienberger, M. F. Kling, G. G.Paulus, and A. M. Sayler, J. Phys. B , 074003 (2012).[13] T. Paasch-Colberg, A. Schiffrin, N. Karpowicz, S. Kru-chinin, ¨O. Saˇglam, S. Keiber, O. Razskazovskaya,S. M¨uhlbrandt, A. Alnaser, M. K¨ubel, V. Apalkov,D. Gerster, J. Reichert, T. Wittmann, J. Barth,M. Stockman, R. Ernstorfer, V. Yakovlev, R. Kienberger,and F. Krausz, Nat. Photonics , 214 (2014).[14] M. Kubullek, Z. Wang, K. von der Brelje, D. Zimin,P. Rosenberger, J. Sch¨otz, M. Neuhaus, S. Sederberg,A. Staudte, N. Karpowicz, M. F. Kling, and B. Bergues,Optica , 35 (2020).[15] S. Skruszewicz, J. Tiggesb¨aumker, K.-H. Meiwes-Broer,M. Arbeiter, T. Fennel, and D. Bauer, Phys. Rev. Lett. , 43001 (2015).[16] P. B. Corkum, Phys. Rev. Lett. , 1994 (1993). [17] X. Liu, H. Rottke, E. Eremina, W. Sandner, E. Gouliel-makis, K. Keeffe, M. Lezius, F. Krausz, F. Lindner,M. Sch¨atzel, G. G. Paulus, and H. Walther, Phys. Rev.Lett. , 263001 (2004).[18] B. Bergues, M. K¨ubel, N. G. Kling, C. Burger, and M. F.Kling, IEEE J. Sel. Top. Quantum Electron. , 1 (2015).[19] R. Hollinger, D. Hoff, P. Wustelt, S. Skruszewicz,Y. Zhang, H. Kang, D. W¨urzler, T. Jungnickel,M. Dumergue, A. Nayak, R. Flender, L. Haizer, M. Ku-rucz, B. Kiss, S. K¨uhn, E. Cormier, C. Spielmann, G. G.Paulus, P. Tzallas, and M. K¨ubel, Opt. Exp. , 7314(2020).[20] N. G. Johnson, O. Herrwerth, A. Wirth, S. De, I. Ben-Itzhak, M. Lezius, B. Bergues, M. F. Kling, A. Sen-ftleben, C. D. Schr¨oter, R. Moshammer, J. Ullrich, K. J.Betsch, R. R. Jones, A. M. Sayler, T. Rathje, K. R¨uhle,W. M¨uller, and G. G. Paulus, Phys. Rev. A , 013412(2011).[21] M. K¨ubel, K. J. Betsch, N. G. Johnson, U. Kleineberg,R. Moshammer, J. Ullrich, G. G. Paulus, M. F. Kling,and B. Bergues, New J. Phys. , 093027 (2012).[22] X. Xie, K. Doblhoff-Dier, S. Roither, M. S. Sch¨offler,D. Kartashov, H. Xu, T. Rathje, G. G. Paulus,A. Baltuˇska, S. Gr¨afe, and M. Kitzler, Phys. Rev. Lett. , 243001 (2012).[23] V. Roudnev and B. D. Esry, Phys. Rev. Lett. , 220406(2007).[24] D. B. Miloˇsevi´c, G. Paulus, D. Bauer, and W. Becker, J.Phys. B , R203 (2006).[25] D. B. Miloˇsevi´c, G. G. Paulus, and W. Becker, Phys. Rev.A , 061404 (2005).[26] N. Thir´e, R. Maksimenka, B. Kiss, C. Ferchaud,G. Gitzinger, T. Pinoteau, H. Jousselin, S. Jarosch, P. Bi-zouard, V. D. Pietro, E. Cormier, K. Osvay, and N. For-get, Opt. Exp. , 26907 (2018).[27] M. Kurucz, R. Flender, L. Haizer, R. S. Nagymihaly,W. Cho, K. T. Kim, S. Toth, E. Cormier, and B. Kiss,Opt. Commun. , 126035 (2020).[28] The increased pulse duration with respect to the ∼
24 fsreported in Ref. [27] is attributed to the presence of high-order dispersion caused by using a 2 mm rather than a1 mm thick Si plate in the beam path.[29] P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler,C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March,G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di-Mauro, Nat. Phys. , 386 (2008).[30] B. Wolter, M. G. Pullen, M. Baudisch, M. Sclafani,M. Hemmer, A. Senftleben, C. D. Schr¨oter, J. Ullrich,R. Moshammer, and J. Biegert, Phys. Rev. X , 021034(2015).[31] H. Fuest, Y. H. Lai, C. I. Blaga, K. Suzuki, J. Xu,P. Rupp, H. Li, P. Wnuk, P. Agostini, K. Yamazaki,M. Kanno, H. Kono, M. F. Kling, and L. F. DiMauro,Phys. Rev. Lett. , 53002 (2019).[32] A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Sov.Phys. JETP , 924 (1966).[33] Supplementary Material is available at ¡url¿ and includesRefs. [23, 24, 35, 36, 43–62].[34] D. Zille, Scaling Ultrashort Light-Matter-Interaction tothe Short-Wave Infrared Regime and Beyond , Ph.D. the-sis, University of Jena (2018).[35] B. Feti´c, W. Becker, and D. B. Miloˇsevi´c, Phys. Rev. A , 23101 (2020). [36] W. Becker, F. Grasbon, R. Kopold, D. B. Miloˇsevi´c,G. G. Paulus, and H. Walther, Adv. At. Mol. Opt. Phys. , 35 (2002).[37] D. B. Miloˇsevi´c, G. G. Paulus, and W. Becker, Opt. Exp. , 1418 (2003).[38] G. G. Paulus, F. Grasbon, H. Walther, R. Kopold, andW. Becker, Phys. Rev. A , 021401 (2001).[39] Y. Huismans, A. Rouz´ee, A. Gijsbertsen, J. H. Jung-mann, A. S. Smolkowska, P. S. W. M. Logman, F. L´epine,C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker,G. Berden, B. Redlich, A. F. G. van der Meer, H. G.Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y.Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, andM. J. J. Vrakking, Science , 61 (2011).[40] M. Haertelt, X.-B. Bian, M. Spanner, A. Staudte, andP. B. Corkum, Phys. Rev. Lett. , 133001 (2016).[41] S. G. Walt, N. Bhargava Ram, M. Atala, N. I. Shvetsov-Shilovski, A. von Conta, D. Baykusheva, M. Lein, andH. J. W¨orner, Nat. Commun. , 15651 (2017).[42] G. Porat, G. Alon, S. Rozen, O. Pedatzur, M. Kr¨uger,D. Azoury, A. Natan, G. Orenstein, B. D. Bruner,M. J. J. Vrakking, and N. Dudovich, Nat. Commun. ,2805 (2018).[43] D. B. Miloˇsevi´c, W. Becker, M. Okunishi, G. Pr¨umper,K. Shimada, and K. Ueda, J. Phys. B: At. Mol. Opt.Phys. , 015401 (2010).[44] M. D. Gregoire, I. Hromada, W. F. Holmgren, R. Trubko,and A. D. Cronin, Phys. Rev. A , 052513 (2015).[45] J. E. Sansonetti, Phys. Chem. Ref. Data , 761 (2009).[46] N. B. Delone and V. P. Kra˘ınov, Usp Fiz. Nauk [Sov.Phys. Usp.] , 753 (1999).[47] A. A. Radzig and B. M. Smirnov, Reference Data onAtoms, Molecules and Ions (Springer, Berlin, 1985).[48] C. F. Bunge, J. A. Barrientos, and A. V. Bunge, At. DataNucl. Data Tables , 113 (1993).[49] M. H. Mittleman and K. M. Watson, Phys. Rev. ,198 (1959).[50] A. E. S. Green, D. L. Sellin, and A. S. Zachor, Phys. Rev. , 1 (1969).[51] A. E. S. Green, D. E. Rio, and T. Ueda, Phys. Rev. A , 3010 (1981).[52] S. Augst, D. D. Meyerhofer, D. Strickland, and S. L.Chin, J. Opt. Soc. Am. B , 858 (1991).[53] G. N. Gibson, R. R. Freeman, T. J. McIlrath, and H. G.Muller, Phys. Rev. A , 3870 (1994).[54] M. A. Walker, P. Hansch, and L. D. V. Woerkom, Phys.Rev. A , R701 (1998).[55] R. Kopold, W. Becker, M. Kleber, and G. G. Paulus, J.Phys. B: At. Mol. Opt. Phys. , 217 (2002).[56] A. Gazibegovi´c-Busuladˇzi´c, D. B. Miloˇsevi´c, andW. Becker, Phys. Rev. A , 053403 (2004).[57] E. Hasovi´c, M. Busuladˇzi´c, A. Gazibegovi´c-Busuladˇzi´c,D. B. Miloˇsevi´c, and W. Becker, Laser Phys. , 376(2007).[58] A. ˇCerki´c, E. Hasovi´c, D. B. Miloˇsevi´c, and W. Becker,Phys. Rev. A , 033413 (2009).[59] D. B. Miloˇsevi´c, Phys. Rev. A , 063423 (2014).[60] N. J. van Druten, R. Trainham, and H. G. Muller, Phys.Rev. A , R898 (1995).[61] D. B. Miloˇsevi´c, E. Hasovi´c, M. Busuladˇzi´c, A. Gaz-ibegovi´c-Busuladˇzi´c, and W. Becker, Phys. Rev. A ,53410 (2007).[62] D. B. Miloˇsevi´c, in Computational strong-field quantumdynamics: Intense Light-Matter Interactions , edited by, edited by