Absorption and emission of single attosecond light pulses in an autoionizing gaseous medium dressed by a time-delayed control field Wei-Chun Chu1 and C. D. Lin1,2 1J. R. Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA 2Department of Physics, National Taiwan University, Taipei 10617, Taiwan (Dated: January 18, 2013) An extreme ultraviolet (EUV) single attosecond pulse passing through a laser-dressed dense gas 3 is studied theoretically. The weak EUV pulse pumps the helium gas from the ground state to the 1 2s2p(1P) autoionizing state, which is coupled to the 2s2(1S) autoionizing state by a femtosecond 0 infraredlaserwiththeintensityintheorderof1012W/cm2. Thesimulationshowshowthetransient 2 absorption and emission of the EUV are modified by the coupling laser. A simple analytical ex- n pression fortheatomicresponsederivedforδ-functionpulsesrevealsthestrongmodification ofthe a Fanolineshapeinthespectra,wherethesefeaturesarequiteuniversalandremainvalidforrealistic J pulseconditions. Wefurtheraccountforthepropagationofpulsesinthemediumandshowthatthe 6 EUVsignalattheatomicresonancecanbeenhancedinthegaseousmediumbymorethan50% for 1 specifically adjusted laser parameters, and that this enhancement persists as the EUV propagates in the gaseous medium. Our result demonstrates the high-level control of nonlinear optical effects ] that are achievable with attosecond pulses. h p PACSnumbers: 32.80.Qk,32.80.Zb,42.50.Gy - m o I. INTRODUCTION Very recently, with the emerging attosecond pulses, the t a timescale of the coupling has been pushed shorter than . the decay lifetime of an AIS [4, 5] for direct observation s The rapidlydevelopingtechnologiesofultrashortlight c sources have gained wide applications in the past 10 in the time domain. In the same scheme, numerous new i spectroscopicfeaturesweresuggestedbysimulations[16– s years [1, 2], most notably with the use of attosecond y 21]. Some of these features have been shown recently by pulses in time-resolvedspectroscopy. A typical measure- h experiments with improved energy resolution [22] of the ment scheme shines an extreme ultraviolet (EUV) light p spectrometers. Further technological developments will pulse, in the form of anattosecondpulse train (APT) or [ asingleattosecondpulse(SAP),togetherwithasynchro- likely bring more discoveries. 2 nizedinfrared(IR)laserpulse,onanatomicormolecular In this work, we focus on resonant laser coupling be- v target, and measures the photoions, photoelectrons, or tween two AISs, where an SAP passing through the 2 photoabsorption of the EUV. The adjustable time delay mediumisstronglyreshapedinitsspectralandtemporal 7 1 between the EUV and the IR gives information of the distributions by controlling the parameters of the laser 6 dynamicsofthetarget. FortheSAPinthex-rayregime, and the medium. This reshaping of the SAP, which be- . this technique has been used to study the emission of havesmuchbeyondwhatcanbe describedby the simple 1 Auger electrons [3] in the so-called “streaking” model. absorption rate or Beer’s law [23], has not been investi- 1 2 Down to the EUV energy, a similar scheme was used to gatedsofar. InRef.[14],theactualpropagationofEUV 1 time-resolve autoionization where the IR primarily just pulses was studied; however,those pulses with durations : depletedthe resonancesexcitedbythe EUV[4,5]. Fora of tens of femtoseconds are too narrow in bandwidth to v SAP at even lower energies, a series of bound states can demonstrateanymeaningfulmodificationintheresonant i X be excited at once. These states can then be ionized by spectrum. Inthisstudy, welookforhowanSAP evolves r theIRthroughdifferentquantumpathwaystoexhibitin- in the medium, which cannot be revealed by a single- a terference patterns [6], or dressed by the IR temporarily atom absorption cross section. to exhibit the ac Stark shift [7]. For demonstration at the single-atom model, a 200 as The studies of coupled autoionizing states (AIS) by EUVpulseandatime-delayed9-fslaserpulseareapplied lasers have been carriedout over the past 30 years theo- to the helium atom. The 2s2p(1P) and 2s2(1S) AISs retically [8–11] and experimentally [12]. In these earlier are coupled by a 540-nm laser with intensities between investigations, long pulses were used. For a typical AIS 1012 and1013 W/cm2,wheretheweakEUVexcites2s2p withalifetimeuptotenthsoffemtoseconds,thecoupling from the ground state. For clarity, the pulses are not lights were considered to be monochromatic, and the distinguished as the pump or the probe since they just measuredspectrumwasobtainedbyscanningthephoton couple different sets of states. In the single-atom calcu- energyoverthewidthsoftheresonances. Notuntilafew lation, the electronic wave function, photoelectron, and yearsagowasthe same coupling scheme extended to the EUV absorption spectra are carried out for a given set EUV energy range with femtosecondpulses, where tran- of pulses and atomic parameters. In orderto extractthe sient absorptions were measured or calculated [13–15]. universalfeaturesresultingfromthecoupling,wefurther 2 derivesimpleanalyticformsofthespectraforshortlight pulsesthatareapproximatedbyδfunctionsintime. This isavalidapproximationifthepulsedurationsareshorter b1 E than the atomic timescales, e.g., the 17 fs decay lifetime V1 1 of2s2p. TheFanoq parameter[24]preservedinthefinal Ω forms of the spectra indicates how the population trans- bb 1 2 fer between the AISs is controlled by the coupling and how the wave packet evolves. E For the macroscopic model, the pulses are allowed to Ω ,Ω b2 V2 2 gb gE propagatethrougha 2mmheliumgaswithnumber den- 1 1 sity8×10−3cm−3 (orapressureof25Torratroomtem- perature). The detected EUV spectra as the pulses exit g themediumarecalculated. Wefoundthatatcertaincou- pling and gas conditions, an enhancement of more than 50%ofthe incident lightat the resonanceis producedin FIG. 1: (Color online) Coupling scheme used in the model. the transmission spectrum. This enhancement peak in The thick red arrow is the laser coupling. The thin purple arrow represents the EUV transition from the ground state the frequency domain persists through the propagation wherethesurroundingbackgrounddecaysalongthelight to the |b1i-|E1i resonance, where the bandwidth of the EUV pulse, indicated bythevertical purpleline, widely coversthe path. This demonstrates a remarkable extension of the wholelineshapeoftheresonance. Thegreenarrowswithhol- nonlinear optical control to the attosecond time regime. low heads are the configuration interactions responsible for theautoionization. II. MODEL hE |µ|b i = hb |µ|E i = 0. (2) The rotating wave ap- 2 1 2 1 proximation (RWA) is applied for the EUV coupling A. Single-atom wave function (but not the laser coupling) for its high frequency and low intensity. (3) The matrix elements of the Hamilto- In this section, we derive the time-dependent total nian involving |E i and |E i are independent of energy, 1 2 wave function of a three-level autoionizing system cou- whereweareonlyconcernedwiththeenergyrangesnear pled by an EUV pulse and a laser pulse. Two AISs, the resonances. This means that the target structure is composedbytheboundparts|b1iand|b2i,andtheasso- givenintermsoftheresonanceenergiesEb1 andEb2,the ciated background continua |E1i and |E2i, respectively, widths Γ1 and Γ2, and the q parameter of the |b1i-|E1i are coupled by the laser, while the |b1i-|E1i resonance resonance,whichtogetherconstitutethebasicFanoline- is coupled to the ground state |gi by the EUV, both shapes [24]. Note that, with our first approximation, q by dipole transitions. The coupling scheme can be ei- cannot be defined for the |b i-|E i resonance. 2 2 ther Ξ type or Λ type. A schematic of the system is Within the prescribed truncated space, the TDSE for in Fig. 1. The pulses are linearly polarized in the same the total wave function in Eq. (1) is reduced to the cou- direction and collinearly propagated. The EUV field is pledequationsofthecoefficientstherein,wheretheseco- expressed by EX(t) = FX(t)eiωXt +FX∗(t)e−iωXt where efficients as functions in time should be calculated ex- FX(t) is the envelope and ωX is the central frequency actly with the given initial conditions. However, the co- of the pulse. The pulse envelope is generally a complex efficients associatedwith the continua, E and E , make 1 2 function, whichincludes allthe phase factorsin addition the calculation less tractable. Thus, we adopt the adia- tothecarrier-frequencyterms. TheelectricfieldEL(t)of batic elimination of the continua by assuming that they the laser pulse, however, is kept in the exact form since change much more slowly than the bound states. Then, it may be a few-cycle pulse. The total wave function is the approximation c˙ (t) = c˙ (t) = 0 in the coupled E1 E2 in the general form of equations gives |Ψ(t)i=e−iEgtcg(t)|gi c (t)= 1 −F∗(t)D∗ c (t)+V c (t) (2) E1 δ −∆E X gE1 g 1 b1 +e−i(Eg+ωX)t(cid:20)cb1(t)|b1i+Z cE1(t)|E1idE1 c (t)= 1 1 1V(cid:2) c (t), (cid:3) (3) E2 δ −∆E 2 b2 2 2 +c (t)|b i+ c (t)|E idE . (1) b2 2 Z E2 2 2(cid:21) whereδ1 ≡EX−Eb1 andδ2 ≡EX−Eb2 arethedetuning of the EUV relative to the transitions from the ground In solving the time-dependent Schr¨odinger equation statetotheAISs,∆E ≡E −E and∆E ≡E −E are 1 1 a 2 2 b (TDSE), we make the following approximations: (1) thecontinuousenergiesrelativetotheresonanceenergies, All the second-order (two-electron) dipole matrix ele- the Ds are dipole matrix elements, and V ≡ hE |H|b i 1 1 1 ments, except the resonant excitation hb |µ|gi (e.g., the and V ≡ hE |H|b i are the strengths of the configura- 1 2 2 2 transition between 2s2p and 1s2), are neglected, i.e., tioninteraction. Inreturn,the coupledequationsforthe 3 bound state coefficients are reduced to satisfy 1. Photoelectrons ic˙ (t)=−F (t)D∗ λc (t)−i|F (t)|2j c (t) (4) In standard scattering theory, the atomic scattering g X gb1 b1 X gg g waves for momentum~k are given by ∗ ic˙ (t)=−F (t)D λc (t) b1 X gb1 g −(δ +iκ )c (t)−E (t)D c (t) (5) 2 1 ic˙ (t)=−(δ1+iκ1)cb1(t)−EL(t)Db∗1b2cb2(t), (6) ψ~k(~r)=rπkr ileiηlul(kr)Ylm(rˆ)Yl∗m(kˆ) (9) b2 2 2 b2 L b1b2 b1 Xlm in the energy-normalized form, where u are real stand- l where κ ≡ Γ /2 and κ ≡ Γ /2 are half widths of the 1 1 2 2 ingwaveradialfunctions,andη arethescatteringphase l resonances,λ≡1−i/qrepresentsthetransitionfromthe shifts. Both u and η are determined by the atomic po- groundstate to|b iand|E ijointly,andj ≡π|D |2 l l 1 1 gg gE1 tential. If one detects photoelectrons, the momentum is proportionalto the laser broadening. The dipoles (D) distributionwillbetheprojectionofthetotalwavefunc- in this work are all real numbers since real orbitals are tion at a large time onto the scattering waves,i.e., usedfortheboundstatesandstandingwavesareusedfor the continua. With Eqs. (4)-(6), the bound state part of P(~k)= lim ψ |Ψ(t) 2. (10) the wave function is carriedout. The procedure so far is t→∞ ~k (cid:12)(cid:10) (cid:11)(cid:12) identical to what has been employedin the early studies (cid:12) (cid:12) Likewise, if one measures the energy of the pho- concerning coupled AISs [9–11], except that we do not toelectrons, the spectrum is given by P(E) = apply RWA on the laser pulse. limt→∞|hψE|Ψ(t)i|2,where|ψEiarethescatteringwaves Forpulsesthatarelongcomparedtotheresonancelife- for energy E. Remember that |E i and |E i are conven- 1 2 time, the pulse bandwidthis narrow,thus eachmeasure- tionally chosen as standing waves, which contains both ment gives the ionization yield at just one single photon the incoming and the outgoing components. At large energy in the spectrum. In this case, calculation of the times, since physically the AISs will decay to the con- ground-state population c (t) described above would be g tinuum, the wave packet will not have incoming com- enoughforcarryingoutthe ionizationspectra. However, ponents, i.e., the projection onto scattering waves is in the present study, the bandwidth of the attosecond the same as the projection onto standing waves. Thus, pulse is broad and can excite the whole AIS, including the photoelectron spectra corresponding to the two res- a significant fraction of the backgroundcontinuum. The onances are simply continuum part of the wave function is needed in de- termining the electron or absorption spectra in a single P (E )= lim |c (t)|2, measurement. Here, the coefficients of the continuum 1 1 t→∞ E1 part are retrieved from the original coupled equations P (E )= lim |c (t)|2, (11) 2 2 t→∞ E2 whichrequireustocalculatethetotalwavefunctionfora ic˙ (t)=(E −E )c (t)+V c (t) E1 1 X E1 1 b1 physicaltime muchlongerthanthe decaylifetimes, until ∗ ∗ −FX(t)DgE1cg(t) (7) cE1(t) and cE2(t) stop evolving other than an oscillating ic˙ (t)=(E −E )c (t)+V c (t). (8) phase. E2 2 X E2 2 b2 The retrieval of the continua can be viewed as a “cor- 2. Photoabsorption rection” after an iteration from the preliminary forms in Eqs. (2) and (3). Now, the time-dependent total wave In a given external field, the total energy absorbed by function in the form of Eq. (1) is completely solved. the atom is ∞ ∆U = ωS(ω)dω, (12) Z 0 B. Single-atom spectrum where the response function S(ω) for ω > 0 represents theabsorptionprobabilitydensity,andisproportionalto the pulse intensity in the frequency domain [14]. Thus, Inthefollowing,theelectronenergyspectrumandpho- P(E) and S(ω) have the same dimension and can be toabsorption spectrum for a single atom are defined by directly compared. The absorption cross section σ(ω) is the probability density per unit energy for emitting an related to S(ω) by electron and for absorbing a photon, respectively. In macroscopic cases where propagation is taken into ac- 4παωS(ω) σ(ω)= , (13) count,thespectroscopyisexpressedintermsofthetrans- 2 mitted light intensity versus the photon energy. E˜(ω) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 where α is the fine structure constant, and E˜(ω) is the system. In principle, both the electron and absorption Fourier transform of the electric field. By considering spectra carry the same essential information, depending dipoleinteractionbetweentheelectricfieldandtheatom, on the resonance structure of the states and the laser S(ω) is given by parameters. However, in electron measurements, high precision spectroscopy is usually harder to achieve since S(ω)=−2Im µ˜(ω)E˜∗(ω) , (14) the lineshape of most Fano resonances requires meV en- h i ergyresolution. Incontrast,suchresolutioniseasierwith where µ˜(ω) is the Fourier transform of the dipole mo- the transient absorption spectroscopy, as exemplified in ment. a recent report [22]. Thus, in the present work we will With the wave function in Eq.(1), the dipole moment put emphasis on absorption spectroscopy. is given by Inalinearmedium,theabsorptionoflightiscommonly described by Beer’s law as µ(t)=eiωXtu (t)+u (t)+c.c., (15) X L wherethecarrierfrequencyoftheEUVhasbeenfactored T(ω)=T0(ω)exp[−ρσ(ω)L], (18) out, and where T (ω) and T(ω) are light intensities at the en- uX(t)=Dgb1λ∗c∗b1(t)cg(t)−iFX(t)jgg|cg(t)|2, (16) trance an0d at the exit of the medium, respectively, ρ is u (t)=D c∗ (t)c (t). (17) the number density of the particles, σ(ω) is the absorp- L b1b2 b2 b1 tion cross section, andL is the traveldistance of light in Note that this model aims to deal with both short and the medium. This form requires that the cross section long pulse lasers. In the case of a few-cycle laser, the has no spatial and temporal dependence and is only a Fourier transformµ˜(ω) atvery low frequencies may con- function of energy, and the transmission of light simply tain the contributions from both uL(t) and its complex decays exponentially with its traveling distance and the conjugate, where the separation of the carrier frequency gas density. However, in the presence of intense ultra- andtheenvelopeisnotbeneficial;wethusdonotemploy shortIRpulses,theresponseofthemediumcanbemore this separation for the laser pulse in our study. complicated,whichshouldbecarriedoutbypropagating In a typical ultrafast EUV-plus-IR experiment, the the exact electric field in the medium. electron or absorption spectra measured over variable In the present work, by assuming that the laser is time delays between the two pulses are the most com- loosely focused, the electric field is a function of time mon. Photoions are also usually measured, but they do t and the spatialcoordinatein the propagationdirection not carry additional information. To study dynamic be- z only,wherethe dependence onthe transversedirection haviorofautoionizationdirectlyinthetimedomain,light is neglected. We assume that the propagation is only pulses shorter than the typical decay lifetime are neces- forward in z and approximately at the speed of light in sary. Bynowfew-cyclelasersofdurationsless than10fs vacuum, c. By expressing the time in the moving frame arequitecommon. Inthecasewherebothpulsesaresig- t′ ≡t−z/c, the Maxwell equation is reduced to nificantlyshorterthananyotheratomictimescale,major features of the dynamics can be recovered by assuming ′ ′ ∂E(z,t) ρ ∂µ(z,t) the pulses are δ-functions in time. With this approxima- =− , (19) ∂z cǫ ∂t′ tion, we are able to reduce the electron and absorption 0 spectratoverysimpleformsintermsofthepulseareasof where ρ is the gas density. Because the EUV and laser the EUV andthe laser,the time delay,andthe q param- frequencies are widely separate, they can be propagated eter, where the detailed derivations are shown in the ap- separately. For the EUV, the carrier oscillation terms in pendix. The modification brought by the laser will have field and in dipole are factored out, and the propagation maximumeffectswhenthelaserfollowstheEUVimmedi- for the field envelope is atelyanditspulseareais(4N+2)πforintegerN. Inthis case the photoelectron distribution flips about the reso- ′ ′ nance energy. For the light, the absorption line changes ∂FX(z,t) ρ ∂uX(z,t) ′ =− +iω u (z,t) . (20) to the emission line. These dramatic changes gradually ∂z cǫ0 (cid:20) ∂t′ X X (cid:21) die down when the time delay increases in terms of the decay lifetime of the observed resonance. This δ-pulse For the laser, the propagation simply follows Eq. (19) analysis will serve to qualitatively explain the result of where µ(z,t′) is replaced by u (t′) + c.c.. The dipole L realistic numerical calculation later in this article. oscillations corresponding to the EUV and laser fields are determined by Eqs. (16) and (17), respectively, and these dipoles determine how the pulses evolve along z C. Propagation of light by Eqs. (20) and (19). The TDSE for the single-atom wavefunction andthe Maxwellequationfor light propa- Wehaveshownhowtomodeltheresonanceshapeofan gationare executed sequentially until the pulses exit the AIS changed by the strong coupling laser in an atomic medium. 5 III. RESULTS AND ANALYSIS (10-6 a.u.) 40 8 Inthefollowing,energyandtimeareexpressedinelec- ((ba)) tron volts (eV) and femtoseconds (fs), respectively, and 30 6 the signal intensities in spectra are expressed in atomic units (a.u.), unless otherwise specified. y (fs) 20 4 a el d 2 e 10 A. Results for a single atom m Ti 0 0 In this study, the characteristics of two strongly cou- -2 pled AISs are demonstrated for neutral helium, the sim- -10 plest atomic system possessing electron correlation ef- u.) 35.2 35.4 35.6 35.8 fects. The lowest few AISs in helium are separated from a. 8 obnyelaasneorsthinerthoenntehaersIcRalteootfheeVvsisiabnled-laigrheteeanseilrygycoruapnlgeed. -6n (10 4 (b) tl0a s=e 3r-.f5re fes Hereweconsideralaserpulse[wavelengthλ =540nm, o fullwidthathalfmaximum(FWHM)duratioLnτL =9fs, ectr 0 fpoerakcoinnvteennsieitnyceI]Lt=o r2eIs0onwahnetrleyIc0o≡up1leTtWhe/2csm22p(h1ePre)aaftnedr hotoel 35.2 35.4 35.6 35.8 P 2s2(1S) resonances, and a weak EUV SAP (central pho- Electron energy (eV) ton energy ω = 60 eV, FWHM duration τ = 200 as, X X peak intensity 1010 W/cm2) to excite 2s2p from the FIG. 2: (Color online) Time-delayed photoelectron spectra ground state by one-photon transition. Both pulse en- near the 2s2p resonance excited by a 200 as EUV with a 9 velopes are in the sine-squared form. The Λ coupling fs laser coupling, calculated by the single-atom model. (a) scheme was chosen so that the binding energies of the The spectra between t0 = −10 and 40 fs. The spectra for two AISs are moderately high to prevent ionization by positive delays are to be compared with those in the δ-pulse the laser. The time delay t0 is adjustable, which is de- analysis in Fig. 7. (b) The spectrum for t0 = 3.5 fs, which fined by the time between the two pulse peaks, and it is correspondstothegreenlineinpanel(a). Atthistimedelay, positivewhentheEUVcomesfirst. ThebroadbandEUV theresonanceprofileflipshorizontallyfrom theoriginal Fano lineshape, which is shown by thegray curve. coversroughly from 50 to 70 eV and can in principle ex- cite many resonances at once, but due to the choice of thelaserwavelengthanditsrelativelynarrowbandwidth of about 200 meV, the coupling between the two AISs full cycle of Rabi oscillation. In Figs. 2(b) and 3(b), the specified above remains our focus, where it will only be spectrafort0 =3.5fsareplotted. TheflippedFanoline- slightly disturbed by the presence of other states if they shapes discussed in Sec. IIB2 and in the appendix are are not totally negligible. seen. In particular, the electron spectrum flips horizon- The time-delayed photoelectron and photoabsorption tally where the sign of q is changed, and the absorption spectra near the 2s2p resonance energy, calculated by spectrum has a upside-down image, where the absorp- the single-atommodel, are shown in Figs. 2(a) and 3(a), tion peak in the laser-free spectrum points downward, respectively, for the time delay ranging from −10 fs to andthe signalsbelow the zeroline arefor the lightemis- 40 fs. Most informative spectral features are included in sion. These “flipped Fano lineshape” features have been this range. The resonant coupling laser with intensity shownin theoreticalstudies [16–18,20,21] andobserved 2I has a pulse area of 1.3×2π, which means that the in a recent experiment [22]. While some comprehensive 0 laser pulse, in its whole, is able to incur Rabi oscillation simulations were done previously, the analytical deriva- for 1.3 cycles. tion in this work would provide a simple yet effective The spectra in Figs. 2(a) and 3(a), calculated with re- explanation that could readily be applied. alistic pulse shapes, retain the main features shown in Note that the AISs decay continuously once they are Figs. 7 and 8. The latter two figures are derived analyt- populated,nomatterwhetherthe laserispresentornot. ically by assuming that the two pulses are δ-functions in For t = 3.5 fs, at the time of the laser peak, the 2s2p 0 time. The mostobviousdifferencesappearwhenthe two state has already partially autoionized, and the Rabi os- pulses overlap,where only the part of the laser after the cillation is able to affect only electrons that are not au- EUVpulse is responsiblefor the Rabioscillation. Unlike toionizedyet. Furthermore,thedecayof2s2 also“leaks” theδ-functionshortpulses,therealisticpulsesbringtheir some electrons resulting from Rabi oscillation, by emit- own timescales into account. Thus, we define an “effec- ting photoelectrons with its own decay lifetime of 5.3 fs. tivepulsearea”bycountingonlythe laserdurationafter These factorsreduce the distortionsthat the laser would the EUVpeak. Fort =3.5 fs, the effective pulse areais haveimposedontheoriginalFanolineshape,inboththe 0 2π, i.e., the laser at that time delay supports perfectly a electron and absorption spectra. As the time delay in- 6 (10-6 a.u.) (10-6 a.u.) 40 8 80 8 (b(a)) (b(a)) 60 30 6 6 s) s) 40 y (f 20 4 y (f 4 a a 20 el el d 2 d 2 e 10 e 0 m m Ti 0 Ti 0 -20 0 -2 -40 -2 -10 59.8 60 60.2 60.4 59.8 60 60.2 60.4 u.) 8 u.) 8 a. laser-free a. laser-free -610 4 (b) t0 = 3.5 fs -610 4 (b) t0 = 10 fs n ( n ( o o pti 0 pti 0 or or s s b b A 59.8 60 60.2 60.4 A 59.8 60 60.2 60.4 Photon energy (eV) Photon energy (eV) FIG. 3: (Color online) Same as Fig. 2 but for the photoab- FIG.4: (Coloronline)Time-delayedphotoabsorptionspectra sorption spectra. The spectra for positive delays are to be ofa200asEUVwitha50fsdressinglaserinthesingle-atom compared with the spectra in the δ-pulse analysis in Fig. 8. model. (a) The spectra between t0 = −50 fs and 80 fs. The The resonance profile for t0 = 3.5 fs flips vertically from the EIT condition is controlled by the time delay between the original Fano lineshape shown by the gray curve. Note that twopulses. (b)Thespectrumfort0 =10fsshowsthelargest thenegativesignalsatt0 =3.5fsaroundtheresonanceenergy separation in the Autler-Townes doublet, which matches the representtheemissionoflight. Incontrast,thephotoelectron Rabifrequency at thelaser peak. spectra in Fig. 2 donot havenegative signals. the pulses, the doublet can be “turned on” or “turned creases, more electrons are autoionized before the laser, off.” This is the theme of some recent studies applying and less are coupled by the laser to participate in the dressing lasers of tens to hundreds of femtoseconds in Rabi oscillation, and the resonance profile gradually ap- an EIT scheme while measuring the time-delayed EUV proaches the original Fano lineshape. transmission [13, 15, 28]. While similar three-level coupling schemes have been used for the electromagnetically induced transparency (EIT)effect[25,26],thespectralfeaturesinatypicalEIT B. Results for a gaseous medium setup, such as the transparency line or Autler-Townes doublet [27], are not recovered by our photoabsorption In Sec. IIIA, the photoabsorption spectrum reveals calculation in Fig. 3. Evidently, the approximation with the appearance of emission line at the resonance for δ-function pulses breaks down in the long dressing field. t = 3.5 fs. However, it is physically impossible for the 0 To differentiate our mechanism from EIT, we make an light at certain frequency to gain intensity indefinitely additionalcalculationbykeepingallthe parameters,but when passing through a medium. A realistic pulse has the laser duration is extended from 9 fs to 50 fs. The no singularity in the frequency domain. Thus, by intu- resultant spectra are plotted in Fig. 4. It shows that the ition,the single-atomresultdoes notrepresentwhatwill longlaserpulsesplitsthe resonancepeakintotwopeaks, actually be measured in a dense gaseous medium. With withthemaximumseparation0.45eVatt =10fs. The thesameEUVandlaserpulsesusedinSec.IIIA,wecon- 0 separationisexactlytheRabifrequencyatthelaserpeak, sider a gaseous medium made of non-interacting helium where the Autler-Townes doublet is reproduced. As the atoms with number density ρ = 8×1017 cm−3 (equiv- time delay changes from this optimal value, the separa- alent to pressure of 25 Torr at room temperature) and tion decreases, and finally drops to 0 for t < −40 fs thickness L=2 mm, and calculate the transmitted light 0 or t > 70 fs. It shows that contrary to short dress- spectra at the exit of the medium. 0 ing pulses, long dressing pulses with many Rabi cycles ThetransmittedEUVisplottedinFig.5forthreelaser creates the condition satisfying the “dressed state” pic- intensities, I = 1.1I , 4.5I , and 10I , where I is the L 0 0 0 L ture behind the EIT phenomenon, and the split states peak intensity. They safely fall into the intensity range appear as expected. By controlling the overlap between consideredbyourmodel. TheEUVspectrawithoverlap- 7 ping pulses fixed at t0 = 0 are plotted in Fig. 5(a). The 3.0 effective pulse areas for the three intensities are π, 2π, incident and 3π, which are responsible for the half, one, and one- a.u.) 2.5 (a) laser1-f.r1eIe0 and-half cycles of Rabi oscillation. For 1.1I0, the spec- -60 2.0 41.05II0 trumismostlyflat,indicatingthatwhilethe lasermoves 1 0 the electron population from 2s2p to 2s2, the 2s2p AIS sity ( 1.5 “disappears,”leavingonlythedirectlyionizedphotoelec- n e trons in the vicinity without the hint of autoionization. nt 1.0 V i For4.5I0,thelaserdrivestheelectronsbackto2s2pwith U 0.5 X a phase shift of π. The emission line that has been dis- cussedinthe single-atompictureis observedhere,where 0.0 59.8 60 60.2 60.4 the spectral peak appears to be more than 50% of the Photon energy (eV) signal intensity of the incident pulse, which is effectively a partial enhancement of the SAP passing through the 5 gas. When the laser intensifies further to 10I0, the Rabi (b) laisnecri-dfreenet oscillationdrivestheelectronsawayfrom2s2pagain;the 4 1.1I 0 ethnahnanacesmligehnttbiusmgopn.eT, haensdetthheresepleacsterruimntelonoskitsieflsadteomtohner- b. unit) 3 41.05II00 strate how the number of cycles in the Rabi oscillation ar controlsthephaseofthe2s2pAIS,whichthendetermines d ( 2 el the resonancelineshape atthe endofthe autoionization. Yi 1 The presentation in Fig. 5(a) is for the overlapping pulses, i.e., the Rabi oscillationhappens within the laser 0 duration at the beginning of the 17 fs decay of 2s2p. -10 0 10 20 30 40 It is a simplified case where autoionization occurs ap- Time delay (fs) proximately after the population transfer by the laser is done. However, for larger time delays, the laser comes FIG.5: (Coloronline)TransmissionofEUVthroughahelium after2s2pdecaysforsometimeandonlyaffectsthelater gaseousmediumdressedbya9fslaserofpeakintensities(IL) part of the decay. The total wave packet is the coherent of 0 (laser free), 1.1I0, 4.5I0, and 10I0. The incident light is sum of autoionizationbefore the laser,and the quantum plottedasareference. (a)TransmittedEUVprofilesforover- path going through the Rabi oscillation. In Fig. 5(b), lappingpulses(t0 =0), wheretheeffectivepulseareas ofthe the total signal yields in the 50 meV range around the laser are0, π,2π,and 3π. Theflippingoftheresonancepro- 2s2p resonance for different intensities and time delays file is predicted as the main feature by the single-atom case, are plotted. This plot is equivalent to a measurement although more structures are seen here. The signal at the of the EUV signal at the resonance energy with a spec- resonance for IL =4.5I0 is enhanced significantly, with more trometerof50meVresolutioninordertoinvestigatethe than 50% increase from the incident light. This measurable quantitycan only beobtained byapplying field propagation. enhancement. Foreachintensity,theyieldoscillateswith (b) Total photon signals (yield) gathered in the 50-meV en- the time delay. As seen in the figure, 4.5I is no longer 0 ergy range at the resonance [indicated in panel (a) by the thedefiniteoptimalintensitytoenhancethetransmitted yellowverticalstripe]asfunctionsofthetimedelay. Thisen- SAP resonance. For example, when the 10I pulse is at 0 ergy range complies with the spectrometer resolution under t =−2 fs, the enhancement is higher than what can be 0 currenttechnology. Theyieldforeachlaserintensityoscillates achieved by the 4.5I0 pulse. This is because even when with the time delay, which shows a high degree of control in both intensities have the same effective pulse area, the thetime domain. 10I pulse accomplishesthe Rabi oscillationin a slightly 0 earlier stage in the decay of 2s2p, where less electrons showthe transmitted EUV spectra atdifferent distances escape its influence by autoionization. It is very remark- alongthebeampathwiththe9-fs,4.5I dressingpulseat ablethatthispartialenhancementofaSAPiscontrolled 0 0 time delay. The transmitted EUV intensities and the within an energy scale of about 100 meV in the form of correspondingeffectiveabsorptioncrosssections,defined resonancelineshape,andwithinatimescaleofabout1fs by [see Eq. (18)] in the form of time delay. It has been demonstrated in Fig. 5(a) that while the 1 T(ω) σ (ω)≡− ln , (21) resonance lineshape of the EUV pulse is controlled by eff ρL T (ω) 0 thelasercoupling,thespectralprofileawayfromtheres- onance energy always drops in light transmission spec- are shown in Figs. 6(a) and (b), respectively. With such troscopy. Itisimaginablethattheoveralllightattenuates laser parameters, the “wings” on the two sides of the whenpropagatinginthegas. Itisthusveryintriguingto resonance profile descend, but the enhancement peak is knowhowtheexceptionalenhancedresonancepeakholds up to about 50% of the incident intensity, and roughly its shape and strength in the propagation. In Fig. 6, we maintains its height from 1 to 3 mm. The persistence of 8 theenhancementpeak,apartfromtheexponentialdecay 3.0 oafntihmeprreesstsiovfetnhoensliignneaalrsraelsopnogntsheeopfrtohpeaggaastioton,tdhiesplalasyesr u.) 2.5 (a) inc1id menmt a. 2 mm pulse. This violation to Beer’s law is in a very selective 6 3 mm -0 2.0 energy range and is sensitive to the parameters of the 1 dressinglaser. Toelucidatethe nonlinearityofthe dress- sity ( 1.5 ing laser, σ (ω) without the dressing field is plotted in n eff e Fig. 6(c). In this case, the propagation of the EUV ac- nt 1.0 V i tually follows Beer’s law stated in Eq. (18), where the U 0.5 X cross section determined by the original Fano lineshape remains the same over the distance. The pulse over the 0.0 59.8 60 60.2 60.4 whole energy range is absorbed indifferently. 1 mm (b) 2 mm 4.0 3 mm IV. CONCLUSION u.) 0.0 A three-level autoionizing system coupled by a SAP a. and a time-delayed intense femtosecond laser have been n ( o modeled. The photoelectron and photoabsorption spec- cti -4.0 e tra are responsive most obviously to the laser intensity s s 59.8 60 60.2 60.4 and its time delay. We analyze the spectra by assuming os both pulses are infinitely short, where the responses can e cr (c) 12 mmmm beexplainedbytheprecisecontroloftheRabioscillation ctiv 8.0 3 mm between the two AISs. The analysis also suggests that Effe the phase change in the wave packet during the Rabi 4.0 oscillationismappedoutintheflippingoftheFanoline- shape. The model calculation with realistic pulse pa- 0.0 rametersalsoproducesthesesignificantfeatures. Incon- trast, with long dressing fields, these spectral features 59.8 60 60.2 60.4 arereplacedby the Autler-Townesdoublet, whichis well Photon energy (eV) studiedinsystemscoupledbystablelasers. Thiscontrast demonstratesthattheultrashortcouplingdemolishesthe FIG.6: (Coloronline)(a)EUVtransmissionprofilesforprop- dressed-state picture, and the dynamics of the system is agation lengths L = 1, 2, and 3 mm in the gaseous medium better viewed directly in the time domain. with the overlapping 9 fs, 4.5I0 dressing laser pulse. When Focusing on the response of the SAP light in this the dressing laser is turned on, the enhanced peak persists coupled system, we further incorporate the single-atom while the surrounding background attenuates. Panels (b) response with the pulse propagation in the gaseous and (c) show effective absorption cross sections σeff(ω) for the same propagation lengths as in panel (a) with and with- medium. Themaincontrollablefeaturesoftheresonance out the dressing laser, respectively. With the dressing laser, profile shown by a single atom are well preserved in the σeff(ω) changes with L at the resonance but stays the same gaseousmedium. By applying alaserpulse witha 2π ef- intheoutsideregion. Withoutthedressingfield,σeff(ω)over fective pulse area to couple the 2s2p and 2s2 resonances thewholeenergyrangedoesnotchangewithL,whereBeer’s in the dense helium gas, an enhanced peak at the 2s2p lawapplies. Thecontrastbetweenthecaseswithandwithout resonance shows up in the transmitted SAP spectrum, thedressinglaserdemonstratesastrongnonlinearresponseof where the signal intensity retains more than 50% gain thegastothelaserfieldwhichisreflectedinthetransmission from the entrance to the exit of the medium. At the of the attosecond pulse. same time, the signals away from the resonance energy drops along the propagation as described by Beer’s law. It can be viewed as the pulse shaping of a SAP that is ergy Sciences, Office of Science, U.S. Department of En- measurable in the energy domain. This result illustrates ergy. C. D. L. would also like to acknowledge the par- a strong nonlinear manipulation of an SAP by an ultra- tial support of National Taiwan University (Grant No. fast dressing field, which would open new possibilities of 101R104021and No. 101R8700-2). control of attosecond dynamics. Appendix A: Atomic response in the short-pulse Acknowledgments limit This work is supported in part by Chemical Sciences, In the short-pulse limit, we assume that the field en- GeosciencesandBiosciencesDivision,OfficeofBasicEn- velopesarerectangularwithgivencarrierfrequencies. As 9 the durations approachzero, the envelopes are spectra for A = 2π are plotted in Figs. 7 and 8, re- L spectively. As seen in both figures, when τ is large, very A F (t)= XD δ(t) littlechangeisbroughtbythelaserontotheoriginalFano X 2 gb1 lineshape. However,intheτ →0limit,theelectronspec- A F (t)= LD δ(t−t ), (A1) trumflipsintheenergydimensionaroundǫ=0,andthe L 2 b1b2 0 absorptionspectrumdisplaysa strongemissionline near where A and A are equivalent to the pulse areas for ǫ=0. X L the EUV andlaser,respectively, andt is the time delay 0 betweenthepulsepeaks,whichispositivewhentheEUV comesfirst. Since the intensityofthe laserisnotenough to excite the atom from the ground state, we only con- 4 12 sider the time delay t ≥ 0. The infinitely short pulses 0 cutthe time intoregionswherethe TDSE ineachregion can be solved analytically. The EUV is weak so that 8 3 perturbation is applied on its transition. In order to ob- y servetheuniversalfeaturesoftheelectronandabsorption a 4 el spectra regardless of the atomic parameters, the energy d 2 isscaledbyǫ≡2(E−E )/Γ ,whereΓ istheresonance e b1 1 1 m 0 width (see Sec. II). Note that now Eb1 represents either Ti the electron energy or the photon energy depending on 1 -4 whether we look at the electron or absorption spectra. This scaling is the same as Fano’s characterization of resonances [24]. The time delay is scaled by τ ≡Γ t /2. 0 -8 1 0 With some algebra, the electron spectrum is found to -8 -4 0 4 8 be given by Energy |A |2 P(ǫ)= X (q+ǫ)2−ce−τf (ǫ) , (A2) 1+ǫ2 P (cid:2) (cid:3) FIG.7: Photoelectron spectrawithinfinitelyshort pulses,as where described by Eq. (A2). The energy is scaled by Γ1/2, and the time delay is scaled by 2/Γ1. The pulse area of laser is c≡1−cos(AL/2) (A3) AL = 2π, which indicates a full cycle of Rabi oscillation by thelasercoupling. Whenthetimedelayislarge,thespectrum and approachestheoriginalFanolineshape. Whenthetimedelay f (ǫ)≡2(q+ǫ)[qcos(ǫτ)+sin(ǫτ)]+ce−τ(q2+1). (A4) is 0, the spectrum is close to a horizontally flipped image of P theoriginal Fanolineshape. Similarly, the absorption spectrum is given by |A |2 The significant change of spectral features at τ = 0 S(ǫ)= 1+Xǫ2 (q+ǫ)2−ce−τfS(ǫ) , (A5) can be expressedanalytically by reducing Eqs.(A2) and (cid:2) (cid:3) (A5) to where fS(ǫ)≡(q2−1+2qǫ)cos(ǫτ)+(cid:2)2q+(1−q2)ǫ(cid:3)sin((ǫAτ6).) P(ǫ)=|AX|2 (−q1++ǫ)ǫ22+4, (A7) Both spectra are composed of two terms. The first term S(ǫ)=|A |2 −(q+ǫ)2 +2 . (A8) istheoriginalFanolineshapedefinedbythe|b1i-|E1ires- X (cid:20) 1+ǫ2 (cid:21) onance alone, without any influence from the laser. The second term decays exponentially with the time delay τ, In the electron spectrum, the mirror image of Fano res- andisproportionaltothecoefficientcdeterminedbythe onance, indicated by the minus sign of the q parame- laserpulseareaALasdefinedbyEq.(A3). Theminimum ter, is added onto a Lorentzianshape. In the absorption value of c is 0, which corresponds to AL = 0,4π,8π,..., spectrum,besideanadditionalconstantbackground,the where the Rabi oscillationhas even number ofcycles. In wholeresonancefunctionflipsupside-down,whichmeans thiscondition,theelectronscomingbackto|b1icarryno thatthe absorptionline becomes the emissionline. Note additional phase, and the laser effectively does nothing. that Eqs. (A7) and (A8) are the results for a full Rabi Onthecontrary,themaximumvalueofc,2,corresponds cycle and represent the extreme changes that the strong to AL = 2π,6π,10π,... and odd number of cycles. The coupling can bring. For non-integer number of cycles, returning electrons have additional phase of π. among the two resonances, some electrons will stay at In orderto demonstrate the influence of laserstrength |b i at the end of the laser, and will not be combined 2 more clearly, the electron and absorption time-delayed with the photoelectrons in the neighborhood of |b i. 1 10 4 12 8 3 y a 4 el d 2 e m 0 Ti 1 -4 0 -8 -8 -4 0 4 8 Energy FIG. 8: Same as Fig. 7 but for the photoabsorption, as de- scribed byEq.(A5). Whenthetimedelayis0, thespectrum isclosetoavertically flippedimageoftheoriginal Fanoline- shape. [1] F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 [14] M. B. Gaarde, C. Buth, J. L. Tate, and K. J. Schafer, (2009). Phys. Rev.A 83, 013419 (2011). [2] G. Sanone, L. Poletto, and M. Nisoli, Nature Photonics [15] M. Tarana and C. H. Greene, Phys. Rev. A 85, 013411 5, 655 (2011). (2012). [3] M. Drescher, M. Hentschel, R. Kienberger, M. Uiber- [16] W.-C.Chu,S.-F.Zhao,andC.D.Lin,Phys.Rev.A84, acker, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. 033426 (2011). Kleineberg, U. Heinzmann, and F. Krausz, Nature 419, [17] W.-C. Chu and C. D. Lin, Phys. Rev. A 85, 013409 803 (2002). (2012). [4] H. Wang, M. Chini, S. Chen, C.-H. Zhang, F. He, Y. [18] J. Zhao and M. Lein, NewJ. Phys.14, 065003 (2012). Cheng, Y. Wu, U. Thumm, and Z. Chang, Phys. Rev. [19] A.N.Pfeiffer andS.R.Leone, Phys.Rev.A85, 053422 Lett.105, 143002 (2010). (2012). [5] S. Gilbertson, M. Chini, X. Feng, S. Khan, Y. Wu, and [20] W.-C. Chu and C. D. Lin, J. Phys. B: At. Mol. Opt. Z. Chang, Phys. Rev.Lett. 105, 263003 (2010). Phys. 45, 201002 (2012). [6] J. Mauritsson et. al., Phys. Rev. Lett. 105, 053001 [21] L. Argenti, C. Ott, T. Pfeifer, and F. Mart´ın, (2010). arXiv:1211.2526v1 (2012). [7] M. Chini, B. Zhao, H. Wang, Y. Cheng, S. X. Hu, and [22] C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, Z. Chang, Phys. Rev.Lett. 109, 073601 (2012). Y.Zhang,S.Hagstotz,T.Ding,R.Heck,andT.Pfeifer, [8] P. Lambropoulos and P. Zoller, Phys. Rev. A 24, 379 arXiv:1205.0519v1 (2012). (1981). [23] A. Ishimaru, Wave Propagation and Scattering in Ran- [9] H. Bachau, P. Lambropoulos, and R. Shakeshaft, Phy. dom Media, Chap. 6 (New York: IEEE, 1997). Rev.A 34, 4785 (1986). [24] U. Fano, Phys. Rev.124, 1866 (1961). [10] L. B. Madsen, P. Schlagheck, and P. Lambropoulos, [25] S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Phys.Rev.Lett. 85, 42 (2000). Lett. 64, 1107 (1990). [11] S.I.Themelis,P.Lambropoulos,andM.Meyer,J.Phys. [26] M.Fleischhauer,A.Imamoglu,andJ.P.Marangos,Rev. B: At.Mol. Opt.Phys. 37, 4281 (2004). Mod. Phys.77, 633 (2005). [12] N. E. Karapanagioti, O. Faucher, Y. L. Shao, D. Char- [27] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 alambidis, H. Bachau, and E. Cormier, Phys. Rev.Lett. (1955). 74, 2431 (1995). [28] T. E. Glover et al., NaturePhys.6, 69 (2010). [13] Z. H. Loh, C. H. Greene, and S. R. Leone, Chem. Phys. 350, 7 (2008).