Driving Rabi oscillations at the giant dipole resonance in xenon
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Driving Rabi oscillations at the giant dipole resonance in xenon
Stefan Pabst,
1, 2, ∗ Daochen Wang,
1, 3 and Robin Santra
1, 4, † Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom Department of Physics, University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany (Dated: October 15, 2018)Free-electron lasers (FELs) produce short and very intense light pulses in the XUV and x-rayregimes. We investigate the possibility to drive Rabi oscillations in xenon with an intense FELpulse by using the unusually large dipole strength of the giant-dipole resonance (GDR). The GDRdecays within less than 30 as due to its position, which is above the 4 d ionization threshold. Wefind that intensities around 10 W/cm are required to induce Rabi oscillations with a periodcomparable to the lifetime. The pulse duration should not exceed 100 as because xenon will befully ionized within a few lifetimes. Rabi oscillations reveal themselves also in the photoelectronspectrum in form of Autler-Townes splittings extending over several tens of electronvolt. PACS numbers: 32.80.-t,31.15.A-
I. INTRODUCTION
In the last decade, with the emergence of free-electronlasers (FELs) [1–4], the door has been opened for study-ing multiphoton processes in the XUV and x-ray regimes.Complex ionization dynamics, resulting from a non-trivial interplay of photoabsorption and inner-shell pro-cesses such as Auger decay, have been experimentallyfound when exposing atoms [5–7], molecules [8, 9], andclusters [10, 11] to these high-fluence and high-intensityFEL pulses.From the extensive works in the last 40 years [12], itis well known that intense optical pulses with frequenciesresonant with internal transitions cause the system to be-have in completely new ways, resulting in novel effectssuch as Rabi oscillations, electromagnetically-inducedtransparency (EIT), lasing without inversion, and pop-ulation trapping. These processes have been primarilystudied in the outer-valence shells of atomic systems,which are accessible with optical frequencies and thephysical realization is quite close to ideal isolated two-,three-, and multi-level systems [13, 14]. With the arrivalof FELs, these processes have been extended to the XUVand x-ray regimes: namely stimulated emission [7], Rabioscillations [15, 16], and processes similar to EIT [17, 18].At XUV and x-ray photon energies ( ω >
10 eV),atomic bound-bound transitions must involve inner-valence and core shells. Once an electron is removedfrom a core orbital, the system wants to ’relax’ by fillingthis hole via spontaneous emission (more likely for heavyatoms) or Auger decay (more likely for light atoms) [19].In order to compete with the relaxation processes of thesystem, the time scale of the light-driven processes hasto be comparable or ideally faster than the time scale of ∗ [email protected] † [email protected] the relaxation. If this is the case, the light-driven processwill not just dominate over the relaxation processes butit also significantly alters the relaxation processes them-selves. The influence of light-driven processes on x-rayfluorescence [20] and Auger decay [16, 21–27] has beentheoretically studied in recent years.In this work, we investigate the possibility to induceRabi oscillations involving the giant-dipole resonance(GDR) in xenon. The GDR is located at around 100 eVabove the ground state and, therefore, also above the4 d ionization threshold. Even though the electron is ul-timately ionized at this energy, it is bound for a veryshort time in the vicinity of the atom leading to an en-hanced dipole transition strength which gave the GDR itsname [28, 29]. The unusually large dipole strength is ben-eficial in two ways. First, it induces fast Rabi oscillationsthat can compete with the short lifetimes ( <
30 as) ofthe GDR states. Second, it ensures that other ionizationpathways (out of other sub-shells), which do not involvethe GDR, are much weaker and are not of high relevance.In high-harmonic generation, this large dipole momentcan be used to significantly boost the high-harmonic yieldaround 100 eV [30–32].Theoretical studies [33, 34] have found that the GDRconsists of two sub-resonances. Recently, an experi-ment [35] has seen first indications of this sub-structurein the XUV two-photon above-threshold-ionization (ATI)spectrum of xenon. Another goal of this study is, there-fore, to investigate whether Rabi oscillations can uncoverthis sub-structure as well.In the following, we present in Sec. II our theoreticalmodel. In Sec. III we estimate which pulses are needed toinduce Rabi oscillations, study the population dynamics,and investigate how Rabi oscillations affect the 4 d holepopulation and the photoelectron spectrum. If not notedotherwise, atomic units are used throughout the paper. II. THEORY
We use our time-dependent configuration interactionsingles (TDCIS) approach [36], which we have success-fully applied in the strong-field [32, 37, 38], XUV [29,34, 35, 39, 40], and x-ray regimes [41, 42]. The N -bodywavefunction ansatz for TDCIS reads | Ψ( t ) i = α ( t ) | Φ ) + X ai α ai ( t ) | Φ ai ) , (1)where Φ is the Hartree-Fock (HF) ground state, and Φ ai is a one-particle-one-hole excitation where one electronis excited from orbital i into orbital a . The Hamiltonianis the exact non-relativistic N -body Hamiltonian,ˆ H ( t ) = ˆ H + ˆ H + A ( t ) ˆ p − E HF , (2)which is partitioned into four parts: (i) the Fock op-erator, ˆ H , describing non-interacting electrons in theHF mean-field potential plus a complex-absorbing po-tential, (ii) the residual Coulomb interaction, ˆ H , cap-turing the electron-electron interactions that go beyondthe HF mean-field picture, (iii) the light-matter interac-tion, A ( t ) ˆ p , in the velocity form of the dipole approxima-tion with A ( t ) being the vector potential and ˆ p being themomentum operator, and (iv) the Hartree-Fock energy, E HF , which shifts the spectrum such that the HF groundstate has the energy 0.The residual Coulomb interactions, ˆ H , can begrouped into two classes: intrachannel and interchannelcoupling. Intrachannel coupling, (Φ ai | ˆ H (cid:12)(cid:12) Φ bi (cid:1) , correctsthe mean-field potential due to the missing electron inthe atom, and leads to a long-range, − /r , potential forthe photoelectron. Interchannel coupling, (Φ ai | ˆ H (cid:12)(cid:12) Φ bj (cid:1) ( i = j ), describes the interaction where the excited elec-tron changes the ionic states i . This interaction leads cor-related electron dynamics that can significantly changethe overall response of the system [32] and has large ef-fects on coherence properties [39].Combining Eqs. (1) and (2), we find the equation ofmotion for the time-dependent CIS coefficients, i∂ t α ( t ) = A ( t ) X a,i (Φ | ˆ p | Φ ai ) α ai ( t ) , (3a) i∂ t α ai ( t ) = ( ε a − ε i ) α ai ( t ) + X b,j (Φ ai | ˆ H (cid:12)(cid:12) Φ bj (cid:1) α bj ( t )+ A ( t ) (cid:16) α ( t ) (Φ ai | ˆ p | Φ )+ X jb (Φ ai | ˆ p (cid:12)(cid:12) Φ bj (cid:1) α bj ( t ) (cid:17) , (3b)where | Φ ai ) and (Φ ai | are right and left eigenstates ofthe non-hermitian Fock operator ˆ H , respectively (seeRef. [36]). The energies of the occupied and virtual or-bitals are given by ε i and ε a , respectively.The existence of the GDR in xenon can be understoodin a single-particle picture with a central model poten-tial [28, 43]. CIS intrachannel already improves the de-scription of the GDR in terms of energy position and spectral width [36, 43]. However, many-body effects haveto be taken into account in order to reproduce the cor-rect position and width of the GDR. Interchannel in-teractions, (Φ ai | ˆ H (cid:12)(cid:12) Φ bj (cid:1) , within CIS improve greatly thedescription of the GDR in comparison to intrachannelCIS [29, 36, 43]. To obtain an even better description,electronic correlations of higher order, mostly double ex-citations, are needed [28].Recently, it has been shown that interchannel interac-tions lead to the emergence of a second dipole-allowedresonance state within the GDR [34]. This second res-onance is centered at 112 eV and lives only for 11 as(Γ = 58 . . III. RESULTS
We use the xcid program [44] (with the following nu-merical parameters in [45]) to verify that we can induceRabi oscillations using the GDR resonances. With TD-CIS, we do not simplify the problem to a two-level sys-tem, and we explicitly include all other ionization mech-anisms (that lead to singly ionized xenon). Furthermore,the GDR is properly described as a result of discretestates in the continuum [34] which can be modified bythe intense XUV pulse itself. This may lead to trendsthat deviate from the weak-field behavior.
A. Population dynamics
First, we have a look at the dynamics of the groundstate population (cf. Fig. 1a) and of the hole popula-tions of different sub-shells of xenon (cf. Fig. 1b-c) asit is exposed to a pulse with a center photon energy of109 eV (= 4 a.u.) and a FWHM duration (with respectto the intensity profile) of 36 as (= 1 . W/cm , we basically fully ionize xenon.As we increase the intensity, the ground state populationand the individual hole populations start to show oscil-latory behavior and do not monotonically increase as inthe “low” intensity limit (cf. 2 . · W/cm results inFig. 1). These are clear indications that we successfullyinduce Rabi oscillations. Rabi oscillations are stronglydamped due to the short lifetime of the GDR states lead-ing to a large amount of irreversible ionization per Rabicycle. Note that for relatively low intensities the holepopulations grow monotonically with time.In Fig. 1, we see that mainly the 4 d shell is depopu-lated as expected from the character of the GDR, whichmainly corresponds to a configuration where a 4 d elec-tron is excited into the l = 3 continuum. The ionizationof the 5 s and 5 p shells is ten times smaller but with 10% P opu l a t i on Time [as]00.20.40.60.8 (b) 4d hole P opu l a t i on P opu l a t i on I=1.2 10 +18
W/cm I=6.3 10 +17
W/cm I=2.3 10 +17
W/cm FIG. 1. (color online) The population of (a) the ground state,(b) a hole in the 4 d shell, and (c) a hole in the 5 s or 5 p shell inatomic xenon. The instantaneous peak intensity of the 36 aslong (FWHM) Gaussian pulse with a center photon energy of109 eV is varied from 2 . · W/cm to 1 . · W/cm . ionization probability it is large enough that it should betaken into account. B. Final hole population
Unfortunately, it is very hard to monitor the 4 d holepopulation as a function of time in an experiment. Withattosecond transient absorption spectroscopy [38, 46], itis in principle possible to do so. The pump-probe delaybetween the pulses can already be controlled on a fewattosecond scale, but the challenge lies in the generationof a probe pulse with a duration of a few attosecondsto achieve the required time resolution. This comes ontop of the already challenging requirements for the XUVpump pulse. It is, therefore, easier to probe Rabi oscil-lations by measuring the final 4 d hole population or thephotoelectron spectrum (see Sec. III C).The 4 d hole itself is not stable and will decay mainlyvia Auger decay. The lifetime of the 4 d hole is around6 fs. Therefore, we can safely neglect the hole decayduring the few attoseconds the XUV pulse drives Rabioscillations. The hole will, however, eventually decay.The Auger electron yield is, therefore, directly related tothe final 4 d hole population after the XUV pulse. Alsothe Auger electron spectrum should not be affected bythe XUV pulse as the 4 d hole decays predominantly afterthe pulse under field-free conditions.In Fig. 2, the final 4 d hole population (after the pulse) ω = 136 eVElectric field [V/a ]00.40.8 (b) ω = 109 eV ho l e popu l a t i on ω = 82 eVIntensity [10 +17 W/cm ]D=36 asD=73 as FIG. 2. (color online) The final 4 d hole population as a func-tion of pulse intensity for different center photon energies ω (a-c), and two pulse durations, 36 as (blue solid) and 73 as(red dashed). is shown as a function of the pulse intensity for theXUV photon energies, (a) ω = 82 eV [= 3 a.u.], (b) ω = 109 eV [= 4 a.u.], and (c) ω = 136 eV [= 5 a.u.],as well as for the pulse durations, (solid blue line) 36 as[= 1 . . (Ω eff T ), whereΩ eff = T − R ∞−∞ dt d E ( t ) = d E max p π/ (2 ln(2)) is thepulse-averaged Rabi frequency, E ( t ) is the pulse envelope, d is the dipole transition strength, and T is the durationof the pulse.Even though the variations are faster for longer pulses,the visibility decreases with pulse length. Due to theshort lifetimes of the GDR states, which are below 30 as,almost all electrons are irreversibly ionized after 30 as,and the hole population is always close to unity whetheror not Rabi oscillations were induced.The hole creation is most dominant at ω = 109 eVwhere the photoionization cross-section is largest (for thethree photon energies shown). At ω = 136 eV, the holecreation increases almost monotonically with the electricfield. Almost no oscillatory behavior is visible indicat-ing that at this photon energy we do not hit a reso-nance and ionization is predominantly irreversible. At ω = 82 eV, the first peak in the hole population appearsquite early followed by a relatively long plateau extend-ing up to 10 W/cm . This trend is not fully consistentwith the Rabi-oscillation picture of a two-level system.But xenon at these XUV photon energies cannot be de-scribed as a two-level system.As we see in Fig. 3, where the final 4 d hole population
1 2 3 4 5 6Electric field [V/a ]50100150 P ho t on ene r g y [ e V ] ho l e popu l a t i on (b) 1 2 3 5 7 10 15 20 2550100150 (a) Intensity [10 W/cm ] FIG. 3. (color online) The final 4 d hole population as a func-tion of pulse intensity and center pulse photon energy. Thepulse duration is (a) 36 as and (b) 73 as. is shown as a function of intensity and driving photonenergy, the photon energy where the 4 d shell is moststrongly ionized shifts to higher energies as the intensityincreases. This explains why for ω = 82 eV in Fig. 2a) aplateau appears. At low intensities, it probes the lowerend of the GDR. At higher intensities, the GDR moves tohigher energies and the dipole transition strength drops,counter-balancing the increase in field strength.The energy shift of the GDR is a result of the veryintense XUV pulse, which dresses the excited and thecontinuum states. The polarizability of a flat contin-uum is given by the ponderomotive potential U p = E ω and shifts the continuum to higher energies [47]. At E = 2 a.u. and ω = 109 eV, this yields an energy shiftof less than 2 eV. The observed energy shift of around20 eV is much larger. Furthermore, the energy shiftsdepend rather linearly than quadratically on E . As wewill see in Sec. III C, the kinetic energy of the photoelec-tron even increases or decreases with intensity depend-ing on whether the photon energy is above or below theGDR, respectively. This clearly shows that the contin-uum in xenon is structured around the GDR and is notflat. Furthermore, when z E ≫ ω linear energy shiftsare expected [47]. In Fig. 3, we find the linear energyshifts especially at intensities above 10 W/cm where z E > ω .For the short 36 as pulse, the effect of Rabi oscillationsis nicely visible. At ω ∼ d hole popula-tion shows a local minimum around 7 − · W/cm and an island of enhanced ionization emerges at lowerintensities. For the longer pulse, more Rabi oscillationoccur as the intensity increases and, therefore, two is-lands of enhanced ionization occur. Again, the visibilityis greatly reduced for longer pulses as explained above.At photon energies below 100 eV, no signs of Rabioscillations can be seen. Even though the dipole strengthfalls off quite symmetrically around 100 eV in the weak- field regime, the dressing of the continuum states createsa clear preference for higher energies as the field strengthincreases. Furthermore, we see no indication of the twoGDR subresonances. Rabi oscillations seem to be onlysensitive to the overall dipole strength. Note that at eachphoton energies the ground state is coupled resonantlyto an excited/continuum state in contrast to a two-levelsystem where only one excited state at a specific energyexists.Another general trend that we can observe in Fig. 3is the broadening of the spectral feature with intensity.This indicates, on the one hand, that the lifetimes ofthe excited states are decreasing. On the other hand,the spectral broadening is also a direct consequence ofsaturation of ionization. We already saw in Figs. 1 and2 that at these intensities we basically fully ionize thesystem.Also new multiphoton ionization pathways start toemerge for photon energies below the 4 d ionizationthreshold and at intensities above 2 · W/cm ( E ≥ . C. Photoelectron spectrum
Rabi oscillations should be also visible in the photo-electron spectrum. In a time domain picture, the field-driven coupling between two states leads to Rabi oscil-lations. In an energy domain picture, the excited state(also the ground state) splits into two states, known asthe Autler-Townes doublet, which are separated by theRabi frequency Ω [49]. In our case, the excited stateis in the continuum, and the energy splitting should bedirectly imprinted in the kinetic energy of the photoelec-tron.In Fig. 4, the total photoelectron spectrum is shown asa function of intensity for (a) a 36 as and (b) a 73 as pulsewith ω = 109 eV. We clearly see the Autler-Townes dou-blet for both pulse durations. The lower energy branchof the Autler-Townes doublet approaches 0 eV and doesnot survive at high intensities as the kinetic energy ofthe electron has to be positive. At higher intensities, theionization dynamics become less periodic and additionalpeaks in the photoelectron spectrum occur [21, 50].Once the driving photon energy moves away from110 eV, no indication of Rabi oscillations are seen in the4 d hole population (see Fig. 3). The same is true forthe photoelectron spectrum. In Fig. 5, the photoelectronspectrum is shown for (a) ω = 136 eV and (b) ω = 82 eVas the pulse intensity is varied. For ω = 136 eV, thekinetic energy of the electron slightly increases with in- ]104070100 E l e c t r on ene r g y [ e V ] P ho t oe l e c t r on s pe c t r u m d P / d E (b) 1 2 3 5 7 10104070100 (a) Intensity [10 W/cm ] FIG. 4. (color online) The photoelectron spectrum as a func-tion of the pulse intensity. The photon energy is centered at109 eV and the pulse duration is (a) 36 as and (b) 73 as. ]104070100 E l e c t r on ene r g y [ e V ] P ho t oe l e c t r on s pe c t r u m d P / d E (b) Intensity [10 W/cm ] 1 2 3 5 7 10104070100 (a) Intensity [10 W/cm ] FIG. 5. (color online) The photoelectron spectrum as a func-tion of the pulse intensity. The pulse has a duration of 73 asand the photon energy is centered (a) at 136 eV and (b) at82 eV. tensity. The coupling to the lower lying GDR resonancepushes the continuum states above the GDR to lowerenergies. At ω = 163 eV (= 6 a.u.)—these results are not shown—the energy separation to the GDR is largeand the coupling can be neglected and the kinetic energyof the photoelectron does not significantly change withintensity.For photon energies below the GDR (see Fig. 5b), thetrend is reversed and the kinetic energy of the electron de-creases with intensity. This clearly shows that the polar-ization of the continuum at these frequencies is stronglyaffected by the GDR and cannot be considered flat. Fora flat continuum, the kinetic energy of the photoelectronspectrum would always decrease by the ponderomotivepotential ( ∝ ω − ) and would never increase as ω increase. IV. CONCLUSION
We have investigated to which extent intense FELpulses could be used to drive Rabi oscillations betweenthe neutral ground state and the GDR in xenon. Wefound that intensities around 10 W/cm are needed tosee an impact of Rabi oscillations on the 4 d hole popula-tion. We could find indications that Rabi oscillation canbe used to uncover the substructure of the GDR, i.e., thetwo dipole-allowed resonances at 73 eV and 112 eV [34].There are two ways how Rabi oscillations can be ob-served: (1) by measuring the 4 d hole population via theAuger electron that gets emitted when the 4 d hole de-cays, or (2) via the photoelectron spectrum in the formof Autler-Townes splitting.To see Rabi oscillations in the 4 d hole population, ashort pulse duration comparable to the lifetimes of theGDR states is needed. For the photoelectron spectrum, alonger pulse seems to be beneficial as the Autler-Townessplitting is spectrally better visible. The energy shiftsof the photoelectron show clearly that the continuumaround the GDR is structured and highly polarizable andcannot be assumed flat. With seeded FELs, pulse inten-sities and pulse durations are in reach to induce Rabioscillations that are driven by XUV light. ACKNOWLEDGMENTS
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