Localizing High-Lying Rydberg Wave Packets with Two-Color Laser Fields
Seyedreza Larimian, Ji-Wei Geng, Stefan Roither, Daniil Kartashov, Li Zhang, Mu-Xue Wang, Qihuang Gong, Liang-You Peng, Christoph Lemell, Shuhei Yoshida, Joachim Burgdörfer, Andrius Baltuška, Markus Kitzler, Xinhua Xie
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r Localizing High-Lying Rydberg Wave Packets with Two-Color Laser Fields
Seyedreza Larimian, Ji-Wei Geng ( 耿 基 伟 ), Stefan Roither, Daniil Kartashov, Li Zhang ( 张 丽 ), Mu-Xue Wang ( 王 慕 雪 ), Qihuang Gong ( 龚 旗 煌 ),
2, 3
Liang-You Peng ( 彭 良 友 ),
2, 3, ∗ Christoph Lemell, Shuhei Yoshida, Joachim Burgd¨orfer, Andrius Baltuˇska, Markus Kitzler, and Xinhua Xie ( 谢 新 华 )
1, 5, † Photonics Institute, Technische Universit¨at Wien, A-1040 Vienna, Austria State Key Laboratory for Mesoscopic Physics and Collaborative Innovation Center of Quantum Matter,School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Extreme Optics,Shanxi University, Taiyuan, Shanxi 030006, China Institute for Theoretical Physics, Technische Universit¨at Wien, A-1040 Vienna, Austria Institute of Theoretical Chemistry, University of Vienna, A-1010 Vienna, Austria (Dated: August 31, 2018)We demonstrate control over the localization of high-lying Rydberg wave packets in argon atomswith phase-locked orthogonally polarized two-color (OTC) laser fields. With a reaction microscope,we measured ionization signals of high-lying Rydberg states induced by a weak dc field and black-body radiation as a function of the relative phase between the two-color fields. We find that thedc-field ionization yields of high-lying Rydberg argon atoms oscillate with the relative two-colorphase with a period of 2 π while the photoionization signal by black-body radiation shows a periodof π . These observations are a clear signature of the asymmetric localization of electrons recapturedinto high-lying Rydberg states after conclusion of the laser pulse and are supported by a semiclassicalsimulation of argon-OTC laser interaction. Our findings thus open an effective pathway to controlthe localization of high-lying Rydberg wave packets. PACS numbers: 33.80.Rv, 42.50.Hz, 82.50.Nd
Highly excited Rydberg atoms and molecules, in com-parison with ground-state atoms and molecules, haveunique properties [1]. Such atoms and molecules can beexploited in the studies of the quantum phenomena andthe transition from the quantum to the classical worldon a macroscopic length scale. They play an importantrole in chemistry and astrophysics and are considered tobe building blocks for future applications on quantum in-formation, chemistry and astrophysics [2]. Manipulatingelectrons in the ground and excited states of an atomor a molecule is of fundamental interest for physics andchemistry with a wide range of applications from highharmonic generation [3] to the control of chemical reac-tions [4]. Previous studies found that high-lying Rydbergstates can be steered by weak half-cycle pulses [5].In a strong laser pulse, valence electrons of an atomor a molecule can be detached through tunneling or bar-rier suppression ionization. After conclusion of the pulse,some of the released electrons may be recaptured by theionic Coulomb field and populate highly excited Ryd-berg states (“frustrated field ionization”) [6]. Recently,we have reported on the lifetime of such states mea-sured by electron-ion coincidence spectroscopy [7]. Ithas been demonstrated that electronically excited statesplay an important role in strong field phenomena, in-cluding ionization and molecular dissociation [8, 9], elec-tron wave packet interference, and high harmonic gen-eration [10–13]. Many strong field phenomena in atomsand molecules are governed by electronic dynamics thatare not only sensitive to the laser intensity but also tothe waveform of the laser field [14]. The latter can be controlled by varying the carrier-envelope phase of a few-cycle laser pulse or by the superposition of phase lockedpulses with different colors [14, 15].
FIG. 1. (color online). Schematic view of the experiment. Anelectron from an argon atom released by the field of an OTClaser pulse may be recaptured into a high-lying Rydberg stateby frustrated field ionization. The Rydberg atom, in turn, issubsequently ionized either by the weak dc field in the targetregion or by photoionization by black-body radiation.
In this Letter, we report on the control of the formationof spatially localized high-lying Rydberg wave packetsby waveform controlled orthogonally polarized two-color(OTC) laser fields in argon atoms. With the help ofsemiclassical electron-trajectory simulations we analyzethe experimental observation and identify the underly-ing mechanism. In the present experiment we exploitthe relative phase of OTC laser fields to achieve tempo-ral and spatial shaping of the waveform of the laser field.Previously, OTC fields have been successfully proposedand applied to control electron rescattering and interfer-ence [12, 16–18], to image atomic wave functions basedon high harmonic generation [19], and to control electronemission and correlation in single and double ionizationof atoms [20]. Since controlling the wave form of an OTClaser field provides the capability of manipulating elec-tron trajectories in time and space, one may expect toachieve control over the formation of high-lying Rydbergstates [8, 21–23].In our experiment [Fig. 1], we use a reaction micro-scope to perform coincidence measurements of electronsand ions separated by the interaction of atoms with thelaser and the weak dc fields [24]. The ionization sig-nal of Rydberg states can be well distinguished of thatfrom prompt laser induced strong-field ionization and re-tains a very high signal-to-noise ratio. Details of theexperimental setup can be found in our previous pub-lications [11, 25]. Measurements were done with OTClaser fields formed by the superposition of a fundamentalpulse with a center wavelength of 800 nm and its sec-ond harmonic with pulse durations (FWHM) of 46 fs and48 fs, respectively. Temporal overlap of the two pulseswas ensured by compensating their different group veloc-ities with calcite plates and a pair of fused silica wedges.The electric field of the OTC pulses can be written as ~E ( t, ∆ ϕ ) = f x ( t ) cos( ωt ) ˆe x + f z ( t ) cos(2 ωt + ∆ ϕ ) ˆe z , with∆ ϕ the relative phase of the two colors and f x,z the pulseenvelopes. The waveform of the OTC pulse can be pre-cisely controlled on a sub-cycle time-scale via adjustingthe position of one of the wedges. The peak laser in-tensity was about 6 × W/cm (peak electric fieldon the order of 2 × V/cm) for each color. A weakhomogeneous dc field of 1.5 V/cm was applied in thetime-of-flight (TOF) spectrometer (along the polariza-tion direction of the 400 nm pulse) to accelerate chargedparticles towards the detectors. This field also inducesfield ionization of high-lying Rydberg states populatedduring the strong field-atom interaction [7]. A homo-geneous magnetic field of 12 gauss is applied to ensure4 π detection of electrons with velocities v < .
93 a.u.( E kin < . µ s after thepulse.The strong laser fields induce not only tunneling ion-ization of argon atoms but may also excite them to long-lived high-lying Rydberg states through electron recap-ture [7, 21, 23]. These high-lying Rydberg states canbe ionized by a very weak dc field through over-the-barrier or tunneling ionization [26] or through photoion-ization by photons absorbed from black-body radiation(BBR) [27]. A typical photo-electron photo-ion coinci- dence (PEPICO) distribution for argon interacting withan intense OTC field is shown in Fig. 2(a). The delayed (cid:31)(cid:27)(cid:13)!(cid:9)(cid:14)(cid:6)(cid:8)(cid:11)%&’(cid:11)(cid:1)(cid:8)(cid:10)(cid:3)(cid:18)(cid:17)(cid:17)(cid:17) (cid:20)(cid:17)(cid:17)(cid:17) (cid:30)(cid:17)(cid:17)(cid:17) $ (cid:6)(cid:8) (cid:11) % & ’ (cid:11) (cid:1) (cid:8) (cid:10) (cid:3) (cid:29)(cid:17)(cid:17)(cid:17)(cid:17)(cid:29)(cid:18)(cid:17)(cid:17)(cid:17)(cid:29)(cid:20)(cid:17)(cid:17)(cid:17)(cid:29)(cid:30)(cid:17)(cid:17)(cid:17)(cid:29) (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3) (cid:5) (cid:6)(cid:7)(cid:8) (cid:9) (cid:10) (cid:11) (cid:12)(cid:13) (cid:14) (cid:11) (cid:4) (cid:15) (cid:8) (cid:16)(cid:17) (cid:18) (cid:16)(cid:17) (cid:19) (cid:16)(cid:17) (cid:20) (cid:21)(cid:21)(cid:22)(cid:11)(cid:12)(cid:23)(cid:6)(cid:9)(cid:6)(cid:15)(cid:6)(cid:8)(cid:15)(cid:24)(cid:2)(cid:9)(cid:15)(cid:6)(cid:8)(cid:25)(cid:5)(cid:11)(cid:26)(cid:15)(cid:13)(cid:27)(cid:28)(cid:11)(cid:15)(cid:6)(cid:8)(cid:15)(cid:24)(cid:2)(cid:9)(cid:15)(cid:6)(cid:8) (cid:16)(cid:17) (cid:16) (cid:16)(cid:17) (cid:18) (cid:16)(cid:17) (cid:19) (cid:16)(cid:17) (cid:20) (cid:16)(cid:17) (cid:29) (cid:16)(cid:17) (cid:30) (cid:25)(cid:5) (cid:21)(cid:21)(cid:22) (cid:31) (cid:15)(cid:10)(cid:10)(cid:15)(cid:6)(cid:8)(cid:11)(cid:9)(cid:15) (cid:13)(cid:11)(cid:1)(cid:8)(cid:10)(cid:3) (cid:1)!(cid:3) (cid:16)(cid:17) "(cid:20) (cid:16)(cid:17) "(cid:19) (cid:16)(cid:17) "(cid:18) (cid:17) (cid:20)(cid:17)(cid:17)(cid:17) $ (cid:6)(cid:8) (cid:15) (cid:24) (cid:2) (cid:9) (cid:15) (cid:6)(cid:8) (cid:11) (cid:14) (cid:2) (cid:9) (cid:13) (cid:11) (cid:1) (cid:8) (cid:10) " (cid:16) (cid:3) FIG. 2. (color online). (a) Photoelectron-photoion-coincidence (PEPICO) distribution for argon. The peak laserfield strengths of both colors are 2 × V/cm and the dcspectrometer field strength is 1.5 V/cm. (b) Signal of high-lying Rydberg states as a function of electron emission time[intensity distribution along diagonal in panel (a)]. (c) Ion-ization rate of Rydberg states derived from the data in panel(b). ionization signal from high-lying Rydberg atoms appearsalong the diagonal and can be easily separated from theprompt strong field ionization signal. From the corre-lated TOF signal between electrons and argon ions theemission time of the electrons from Rydberg atoms is ex-tracted. Fig. 2(b) shows emission times of up to 20 µ s.The ionization signal of high-lying Rydberg states con-tains two main contributions, one from field ionizationby the weak dc extraction field applied along the spec-trometer direction and the other one from photoioniza-tion induced by BBR at room temperature with a photonenergy of about 0.025 eV [28]. The (relatively) rapidlydecaying ( . µ s) component of the signal results fromRydberg states very close to the continuum thresholdand well above the potential barrier field ionized by theweak dc field of V dc = 1 . F dc = 1 / (9 n F ) yielding n F ≃ n > n BBR ≃ µ s. From the measured Rydberg signal, thecorresponding ionization rates Γ = − d ln[ I ( τ )] /dτ canbe derived [Fig. 2(c)]. The rate decreases from aboutΓ ≈ .
01 ns − at τ = 200 ns to about 2 × − ns − at τ = 5 µ s. For emission times longer than 6 µ s theionization rate becomes nearly constant with a value ofΓ ≈ × − ns − which agrees well with the simulatedphotonionization rate by BBR [7].To analyze the formation of high-lying Rydberg atomsin the presence of OTC fields, we performed semiclassicalelectron ensemble simulations. Briefly, in the model thetunneling rate for strong-field ionization is derived fromLandau’s effective potential theory. The tunneled elec-trons are assumed to have a Gaussian-like distributionover the transverse momentum perpendicular to the in-stantaneous laser field and zero longitudinal momentumalong the instantaneous laser field. Each launched elec-tron trajectory is weighted by the ADK ionization rate[29] and the initial lateral momentum distribution W ( v ⊥ i ) ∝ p I p | E ( t ) | exp " − p I p ( v ⊥ i ) | E ( t ) | . (1) E ( t ) is the laser electric field strength and I p is theionization potential. After tunneling and until the laserhas concluded the classical Newtonian equations of mo-tion ¨ ~r = − ~r/r − ~E ( t ) in the combined laser andCoulomb fields are solved numerically, where r is the dis-tance between the electron to the nucleus. For electronswith positive energy reaching the detector we calculatethe asymptotic final momentum using Kepler’s formula.Electrons with negative total energy are considered to betrapped in Rydberg states with large principal quantumnumbers n . 7 . × trajectories are simulated for Aratoms at an intensity of 6 × W/cm for each color.We use a trapezoidal envelope function with one opticalcycle ramping up and down with four optical cycles inthe plateau for the 800 nm laser pulse. The same enve-lope was used for the 400 nm laser pulse. For compara-bility with the experiment we consider only trajectorieswith final energies small enough to ensure 4 π detection( E kin < . + along the polarization axis of the 400 nm FIG. 3. (color online). (a) Measured momentum distribu-tion of Ar + along the polarization direction of the 400 nmpulse as a function of the relative phase ∆ ϕ between the twocolor fields. (b) The measured (blue circles) and simulated(red circles) average momentum of Ar + in the polarizationdirection of the 400 nm pulse as a function of the relativephase between the two color fields. (c) Normalized exper-imental strong field ionization yield (black squares) and dc-field ionization yield (blue squares with error bars) from high-lying Rydberg states as a function of the two-color phase.The red line is the projection of the dipole moment on inthe “downhill” direction on the polarization axis of the 400nm pulse. (d) Normalized BBR photoionization yield (greensquares) with a fitting curve (red solid line) with a function of0 . ± . . ± π ∆ ϕ − . π ( ± . < n < < n <
140 (blue solid line) as afunction of the relative phase. In all panels the absolute valueof the two-color phase has been determined by matching theoscillations of the average momentum in panel (b). pulse is shown as a function of the relative phase. Theclear periodic dependence of the momentum distributionon the relative phase indicates that a precise control ofthe cycle shape of the OTC field with the relative phaseis achieved in the experiments. The average momentumof Ar + ions along the polarization axis of the 400 nmpulse is plotted together with the simulated values inFig. 3(b). Both the measured and the simulated dataoscillate periodically with the relative phase. The ex-perimental curve, however, features a smaller oscillationamplitude and a different shape which could result fromaveraging effect due to the experimental pulse intensityprofile near the laser focus in the interaction region whichis not accounted for in our simulation.It is now instructive to analyze delayed dc field ion-ization and BBR photoionization separately. First, weselect the dc field ionization yield of high-lying Rydbergstates by integrating the signal over emission times inthe emission-time interval between 100 ns and 4 µ s afterthe laser pulse. This yield [blue squares with error barsin Fig. 3(c)] exhibits a clear 2 π -periodicity which is wellreproduced by our simulation (red solid line). It is im-portant to note that this 2 π -periodicity is neither relatedto the strong-field ionization yield (black dots) which isalmost constant for all relative phases nor to the elec-tron recapture rate [blue line in Fig. 3(d)] which oscil-lates with π -periodicity. A π -periodicity is also found forthe photoionization yield by BBR for electron emissiontimes longer than 4 µ s [green squares with error bars inFig. 3(d)]. The origin of the observed different periodic-ities lies in the localization of the Rydberg wave packetas revealed by our simulations.Starting point is the inversion symmetry of the atomand the π -periodicity of the laser intensity | ~E (∆ ϕ ) | = | ~E (∆ ϕ + π ) | . Therefore, all processes that do not dependon the directionality of the field, including the formationof high lying Rydberg states, will feature the same pe-riodicity as the intensity of the laser field. This is in-deed also observed in our simulation when counting thenumber of electrons with negative final energy [the bluesolid line and the black dashed line in Fig. 3(d)]. As alsoblack-body radiation is isotropic, the yield of post-pulsephotoionized Rydberg atoms is directly proportional tothe number of available highly excited atoms and, conse-quently, exhibits the same π -periodicity [Fig. 3 (d)]. FIG. 4. (color online) Atomic potential in a weak dc field withfield strength F dc = 1 . The situation is different for the post-pulse dc fieldionization where the weak extraction field present in the interaction region breaks the inversion symmetry of thesystem. As shown in Fig. 4, red-shifted Stark states withelongated orbitals pointing in the “downhill” direction,i.e., towards the potential barrier formed by the Coulomband the static fields have the highest probability to over-come the barrier even though blue-shifted states (“up-hill”) states are energetically higher [7]. We thus use theaverage dipole moment of the electrons or, equivalently,the spatial localization of the charge cloud, as an indi-cator for the probability of the ensemble to escape dueto the post-pulse interaction with the weak dc field. Tothis end we analyze the dipole moment of all states with(hydrogenic) primary quantum number 120 < n < π -periodicity, the orbit is elongated in down-hill direction only with 2 π -periodicity resulting in an in-creased ionization yield due to the presence of the weakdc field. The red line in Fig. 3(c) shows the projectionof the dipole moment in the “downhill” direction on thepolarization axis of the 400 nm pulse as a function of thetwo-color phase. Interestingly, the oscillation is not sym-metric for uphill and downhill directions but, instead,shows a “dip” in the uphill direction due to the influenceof the static field on the orbit of the recaptured electron.The flat uphill part of the oscillation almost perfectlyreproduces the measured post-pulse dc field ionizationyield [blue open squares in Fig. 3 (c)]. The agreementbetween simulated and measured emission yields pointsto the high degree of spatial localization of states in onehemisphere which can be populated in laser-atom inter-actions by controlling the waveform of the exciting laserpulse.In conclusion, we have presented a joint experimentaland theoretical study on highly excited Rydberg statescreated during the interaction of argon atoms with wave-form controlled OTC laser fields. The ionization yieldsdue to weak dc field ionization and BBR photoionizationcould be measured separately as a function of the relativephase between the two colors. We found different oscilla-tion periods of 2 π and π , respectively. The measurementsand analysis of trajectories calculated in a semiclassicalsimulation suggest that Rydberg electrons recaptured bythe ion after conclusion of the pulse can be preferentiallysteered and localized into a hemisphere of the atom bythe shape of the laser field waveform. We have demon-strated this steering in our experiment by manipulatingthe relative phase between the two colors. For specifictwo-color phases, Rydberg wave packets with energiesabove the potential barrier are predominantly localizedon the downhill side of the atom. The ionization yield ofsuch electrons induced by a weak dc field is significantlyincreased and shows a modulation with a period of 2 π .The present study provides an effective way to controlthe population and the localization of high-lying Ryd-berg wave packets and may find potential applicationsin the manipulation of interacting Rydberg ensembles,Rydberg molecules, and chemistry.This work was financed by the Austrian Science Fund(FWF) under P25615-N27, P28475-N27, P21463-N22and P23359-N16, special research programmes SFB-041 ViCoM, SFB-049 NextLite, and doctoral collegeW1243, by the National Natural Science Foundation ofChina (NNSFC) under Grant No. 11574010, and by theNational Program on Key Basic Research Project (973Program) under Grant No. 2013CB922402. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] T. F. Gallagher,
Rydberg atoms , Vol. 3 (Cambridge Uni-versity Press, 2005).[2] M. Saffman, T. G. Walker, and K. Mølmer,Rev. Mod. Phys. , 2313 (2010); F. Merkt,Annual review of physical chemistry , 675 (1997);Y. Gnedin, A. Mihajlov, L. Ignjatovi´c, N. Sakan,V. Sre´ckovi´c, M. Zakharov, N. Bezuglov, andA. Klycharev, New Astronomy Reviews , 259 (2009),arXiv:1208.2516.[3] C. Winterfeldt, C. Spielmann, and G. Gerber,Rev. Mod. Phys. , 117 (2008).[4] R. E. Carley, E. Heesel, and H. H. Fielding,Chem. Soc. Rev. , 949 (2005).[5] F. B. Dunning, J. J. Mestayer, C. O.Reinhold, S. Yoshida, and J. Burgd¨orfer,Journal of Physics B: Atomic, Molecular and Optical Physics , 022001 (2009).[6] F. Krausz and M. Ivanov,Rev. Mod. Phys. , 163 (2009).[7] S. Larimian, S. Erattupuzha, C. Lemell, S. Yoshida,S. Nagele, R. Maurer, A. Baltuˇska, J. Burgd¨orfer, M. Kit-zler, and X. Xie, Phys. Rev. A , 033401 (2016).[8] B. Wolter, C. Lemell, M. Baudisch, M. G. Pullen,X.-M. Tong, M. Hemmer, A. Senftleben, C. D.Schr¨oter, J. Ullrich, R. Moshammer, J. Biegert, andJ. Burgd¨orfer, Phys. Rev. A , 063424 (2014); Q. Li,X.-M. Tong, T. Morishita, C. Jin, H. Wei, and C. D. Lin,Journal of Physics B: Atomic, Molecular and Optical Physics , 204019 (2014);H. Liu, Y. Liu, L. Fu, G. Xin, D. Ye, J. Liu, X. T.He, Y. Yang, X. Liu, Y. Deng, C. Wu, and Q. Gong,Phys. Rev. Lett. , 093001 (2012).[9] R. Minns, D. Lazenby, F. Hall, N. Jones, R. Patel, andH. Fielding, Molecular Physics , 1808 (2014).[10] M. Chini, X. Wang, Y. Cheng, H. Wang, Y. Wu, E. Cun-ningham, P.-C. Li, J. Heslar, D. A. Telnov, S.-I. Chu,and Others, Nature Photonics , 437 (2014).[11] X. Xie, S. Roither, D. Kartashov, E. Persson, D. G.Arb´o, L. Zhang, S. Gr¨afe, M. S. Sch¨offler, J. Burgd¨orfer,A. Baltuˇska, and M. Kitzler, Phys. Rev. Lett. ,193004 (2012).[12] X. Xie, Phys. Rev. Lett. , 173003 (2015); Y. Dengand X. Xie, Phys. Rev. A , 043414 (2015).[13] D. G. Arb´o, S. Nagele, X.-M. Tong, X. Xie, M. Kitzler,and J. Burgd¨orfer, Phys. Rev. A , 043414 (2014). [14] A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel,E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev,A. Scrinzi, T. W. H¨ansch, F. Krausz, and A. Baltuˇska,Nature , 611 (2003).[15] H.-S. Chan, Z.-M. Hsieh, W.-H. Liang, a. H. Kung,C.-K. Lee, C.-J. Lai, R.-P. Pan, and L.-H. Peng,Science (New York, N.Y.) , 1165 (2011).[16] M. Kitzler and M. Lezius,Phys. Rev. Lett. , 253001 (2005); M. Kit-zler, X. Xie, A. Scrinzi, and A. Baltuska,Phys. Rev. A , 011801 (2007).[17] M. Richter, M. Kunitski, M. Sch¨offler, T. Jahnke,L. P. H. Schmidt, M. Li, Y. Liu, and R. D¨orner,Phys. Rev. Lett. , 143001 (2015).[18] J.-W. Geng, W.-H. Xiong, X.-R. Xiao, L.-Y. Peng, andQ. Gong, Phys. Rev. Lett. , 193001 (2015).[19] M. Kitzler, X. Xie, S. Roither, A. Scrinzi, andA. Baltuska, New Journal of Physics , 025029 (2008);D. Shafir, Y. Mairesse, D. Villeneuve, P. Corkum, andN. Dudovich, Nature Physics , 412 (2009).[20] L. Zhang, X. Xie, S. Roither, Y. Zhou, P. Lu,D. Kartashov, M. Sch¨offler, D. Shafir, P. B.Corkum, A. Baltuˇska, A. Staudte, and M. Kit-zler, Phys. Rev. Lett. , 193002 (2014); Y. Zhou,C. Huang, A. Tong, Q. Liao, and P. Lu,Opt. Express , 2301 (2011); L. Zhang, X. Xie,S. Roither, D. Kartashov, Y. Wang, C. Wang,M. Sch¨offler, D. Shafir, P. B. Corkum, A. Baltuˇska,I. Ivanov, A. Kheifets, X. Liu, A. Staudte, and M. Kit-zler, Phys. Rev. A , 061401 (2014).[21] T. Nubbemeyer, K. Gorling, A. Saenz, U. Eichmann,and W. Sandner, Phys. Rev. Lett. , 233001 (2008);U. Eichmann, T. Nubbemeyer, H. Rottke, andW. Sandner, Nature , 1261 (2009); U. Eichmann,A. Saenz, S. Eilzer, T. Nubbemeyer, and W. Sandner,Phys. Rev. Lett. , 203002 (2013).[22] A. S. Landsman, A. N. Pfeiffer, C. Hofmann,M. Smolarski, C. Cirelli, and U. Keller,New Journal of Physics , 013001 (2013).[23] E. Diesen, U. Saalmann, M. Richter, M. Ku-nitski, R. D¨orner, and J. M. Rost,Phys. Rev. Lett. , 143006 (2016).[24] J. Ullrich, R. Moshammer, A. Dorn, R. Doerner,L. P. H. Schmidt, and H. Schmidt-Boecking,Reports on Progress in Physics , 1463 (2003);R. D¨orner, V. Mergel, O. Jagutzki, J. U. L. Spiel-berger, R. Moshammer, and H. Schmidt-B¨ocking,Physics Reports , 95 (2000).[25] 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).[26] T. Morishita and C. D. Lin,Phys. Rev. A , 063405 (2013).[27] T. F. Gallagher and W. E. Cooke,Phys. Rev. Lett. , 835 (1979).[28] W. P. Spencer, A. G. Vaidyanathan, D. Klepp-ner, and T. W. Ducas, Phys. Rev. A , 1490 (1982);I. I. Beterov, D. B. Tretyakov, I. I. Ryabtsev,V. M. Entin, A. Ekers, and N. N. Bezuglov,New Journal of Physics , 013052 (2009).[29] M.-V. Ammosov, N.-B. Delone, and V.-P. Krainov,Soviet-Physics-JETP64