Attosecond Two-Photon Interferometry for Doubly Excited States of Helium

Attosecond two-photon interferometry for doubly excited states of helium

J. Feist,1, * S. Nagele,2, † C. Ticknor,3 B. I. Schneider,4 L. A. Collins,3 and J. Burgdörfer2 1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 2Institute for Theoretical Physics, Vienna University of Technology, 1040 Vienna, Austria, EU 3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 4Office of Cyberinfrastructure/Physics Division, National Science Foundation, Arlington, Virginia 22230, USA (Dated: November 20, 2018) We show that the correlation dynamics in coherently excited doubly excited resonances of helium can be followed in real time by two-photon interferometry. This approach promises to map the evolution of the two-electron wave packet onto experimentally easily accessible non-coincident single electron spectra. We analyze the interferometric signal in terms of a semi-analytical model which is validated by a numerical solution of the time-dependent two-electron Schrödinger equation in its full dimensionality.

PACS numbers: 32.80.Fb, 32.80.Rm, 42.50.Hz

Advances in optical technologies and laser sources in excitation by charged particles as pump and the velocity the past decade led to the production of extreme ultra- of post-collisional energy shifts as probe. From the veloc- violet (XUV) light pulses as short as 80 attoseconds (1 ity dependence of the angular differential autoionization attosecond = 10−18 seconds) [1–3]. Thus, the direct ex- spectra (“PCI effects”) the time evolution of collective ploration of the electronic dynamics in atoms, molecules two-electron variables such as the dipole, ⃗푟1 + ⃗푟2 (or ⟨ ⟩ and solids in the time domain came into reach. This Runge-Lenz vectors ⃗푎1 +⃗푎2 ), or the vibronic motion of ⟨ ⟩ advance initiated a whole new field, attosecond physics, the interelectronic angle ⃗푎1 ⃗푎2 could be identified and several pioneering experiments exploiting the novel [11]. For a XUV-XUV pump∼ ⟨ probe· ⟩ scenario a few the- technologies have already been performed (see [4–6] and oretical proposals to guide attosecond-pulse experiments references therein). Most measurement protocols ei- have been put forward. Hu and Collins [12] proposed ther realized or proposed rely up to now on an inter- to map out the wavepacket in coherently singly excited play of an attosecond XUV pulse and a few-cycle IR helium created by the pump pulse. They performed ab- pulse with durations 휏IR & 5 fs. Sub-fs time resolu- initio calculations for the double ionization by the probe tion is achieved through the exquisite sub-cycle control pulse as a function of delay time 휏 and showed that the over carrier-envelope phase (CEP) stabilized IR pulses total double ionization signal oscillation directly mirrors −2 with uncertainties as small as Δ휙 10 푇0 where 푇0 the radial breathing motion in the singly-excited state is the period of the IR oscillation.≈ However, the di- manifold. This scenario requires, however, a two-color rect analogue to femtosecond pump-probe spectroscopy XUV-XUV pump-probe sequence. Morishita et al. [13] in chemistry on the attosecond scale, i.e., excitation of showed, within lowest order perturbation theory, that the an electronic wavepacket by an attosecond pump pulse correlated motion of the two electrons in a wavepacket followed by an attosecond probe pulse to take snapshots among the doubly excited states (DES) of helium can be of the ensuing electronic motion remains to be accom- resolved by an XUV-XUV pump-probe scheme provided plished. One obvious difficulty is that current attosecond that the full six-dimensional two-electron momenta of the XUV pulses based on high-harmonic generation (HHG) ejected electrons are resolved in a kinematically complete had, up to now, insufficient intensity to efficiently realize experiment. multi-photon pump-probe protocols. Very recently, how- In this letter we present a novel single-color XUV-XUV arXiv:1104.3798v1 [quant-ph] 19 Apr 2011 ever, significant increases in HHG efficiency have been interferometric pump-probe protocol that allows to fol- reported [7,8]. Therefore, attosecond XUV-XUV pump- low the correlated two-electron motion in doubly excited probe experiments, which have been dubbed the “holy states in real time by observing only (relatively) easily ac- grail” of attosecond physics [5], will likely be realized in cessible integral and non-coincident experimental observ- the near future opening up a new stage of attosecond ables. To map out the electronic dynamics we exploit the science. interference between three two-photon double ionization This experimental perspective challenges theory to pathways (see Fig. 1) in a fashion which greatly enhances identify observables readily accessible in the experiment the observable signal. that map out non-trivial wavepacket dynamics of corre- We solve for the proposed scenario the time-dependent lated electronic motion. The paradigm system for corre- Schrödinger equation in its full dimensionality, includ- lated electron dynamics in real time are manifolds of co- ing electronic correlations without further approxima- herently excited doubly excited states (i.e., resonances) in tions (see [14]). The numerical parameters were chosen helium. Pioneering experiments [9, 10] utilized collision to ensure convergence. The XUV pulses have a sin2 enve- 2

path α 50 10

40 He++ 2 1 30 [eV]

β 2

path DES 20 0.1 + 1 He 10

0 0.01 path γ 0 10 20 30 40 50 1s2 1 [eV]

FIG. 2. Two-photon double ionization spectrum in the (휖1, 휖2) plane for a pump-probe sequence of two 1 fs sin2 20 nm pulses FIG. 1. Three-path interferometer for attosecond two-photon with a peak-to-peak delay of 휏 = 1500 as. The white lines de- double ionization probing the coherent dynamics in doubly limit the spectral window without contamination by sequen- excited states (DES). The three paths 훼, 훽, and 훾 are repre- tial contributions. The diagonal oscillations in 휖1 + 휖2 result sented by blue, green, and red arrows, respectively (see text). from the interference between pathways 훼 and 훽 + 훾. The Interference areas Δ퐸휏: Area 1 (light-green) is delineated complex interference pattern within the “sequential” peaks at (휔 − 퐼1, 휔 − 퐼2) is primarily due to interference between 훼, 훾, by (quasi-) bound states and is stable under average over 휖1 and the sequential pathway with one photon from each pulse (or 휖2). Area 2 (in light-blue) is delineated by the energy (see text). 퐸 = 휖1 + 휖2 of the two-electron continuum state and varies rapidly under variation of 휖1 (or 휖2).

gram (Fig. 1) between path 훼 and any other path rapidly lope with total duration of 1 fs, a FWHM of intensity of varies over the Fourier width of the total final energy, 390 as, and a central energy 휔 of 65.3 eV. All calculations 휖1 + 휖2, in the continuum (along the diagonal in Fig. 2). presented in the following were performed for peak in- Any partial trace over unobserved variables, e.g. the en- tensities of 1012 W/cm2 for rapid numerical convergence. ergy of one electron, will wipe out any interference fringes For the experiment, values close to 1015 W/cm2 would associated with path 훼 and will result in an incoherent be desired. We have explicitly checked that our results and 휏-independent background contribution to the ob- remain valid at such intensities, three-photon processes served electron spectra. and ground state depletion are still negligible. For energies close to (휖1, 휖2) = (휔 퐼1, 휔 퐼2) and its − − The present attosecond two-photon pump-probe se- exchange symmetric partner (휔 퐼2, 휔 퐼1) where 퐼1 and − − 2 + quence (Fig. 1) of DES can be viewed as a three-path 퐼2 are the ionization thresholds of He(1푠 ) and He (1푠), interferometer, with the time delay 휏 between the pulses the additional pathway of sequential two-photon ioniza- corresponding to the “arm length” of the interferometer. tion, creating first +He (1푠) by the pump and then He++ Path 훼 corresponds to two-photon double ionization by by the probe (omitted from Fig. 1 for clarity) gives rise the pump pulse which has been the subject of a large to additional rapidly oscillating fringes within the Fourier number of recent investigations (see e.g. [14] and ref- broadened “sequential peaks” (see Fig. 2). They can be erences therein). Path 훾 is its replica induced by the removed by choosing an appropriate spectral window for probe pulse delayed by a time interval 휏 relative to the the one-electron energies (휔 퐼2) < 휖 < (휔 퐼1) within pump pulse. The intermediate path 훽 represents a proper the “sequential” peaks. Focusing− in the following− on this pump-probe sequence where the first one-photon transi- energy window and integrating over the energy of the sec- tion coherently excites a wavepacket of an ensemble of ond electron leaves us with interference fringes that are 훽 훾 doubly excited states whose time evolution is then probed exclusively determined by the phases, 휑푚 = (퐸푚 퐸0 )휏. by double ionization by the second photon after the delay The enclosed area (Fig. 1) is delimited by the two− sharp time 휏. Two specific features of this three-path interfer- boundaries of the quasi-bound states of resonances (path- 2 ometer, which displays a complex fringe pattern in the way 훽) and by the ground state He(1푠 ) with energy 퐸0 (휖1, 휖2) plane of final energies of electron 1 and 2 (Fig. 2), (pathway 훾). Since the DES wavepacket encompasses 훽 are key to resolving the DES wavepacket dynamics. First, several resonances with energies 퐸푚 (푚 = 1,...), the path 훼 represents a “fuzzy” slit. The interference phase resulting interference fringes will display a fast oscilla- Δ퐸휏 represented by the area enclosed in the 퐸 푡 dia- tion on the attosecond scale given by the average phase − 3

1푃 e symmetry is accessed, and we use the traditional but 3.2 numerical results imprecise labels 2푠푛푝± for brevity. Correlated two- semi-analytical model electron dynamics| unfolds⟩ in the quasi-bound part of modulation envelope 3.1 the resonances, the lifetime Γ−1 of which typically ex- ceeds 10 fs. To coherently excite a manifold of 1푃 e dou-

] 3 bly excited states in a one-photon transition from the 10 2 He(1푠 ) ground state, photon energies 휔XUV 60 eV (or 10 & × [ 2.9 wavelength . 20 nm) are required. The spectral width

M should be of the order of a few eV corresponding to P 2.8 an attosecond pulse with 푡XUV . 1 fs. Assuming that pump and probe pulses are temporally separated, the fi- 2.7 nal wavepacket can be written as

(2) −푖H^ ECS휏 (1) 휓푓 = U^ 푒 U^ 휓0 , (1) 2.6 | ⟩ | ⟩ 3 3.5 4 4.5 5 ^ ^ (푖) τ [fs] where HECS is the field-free ECS Hamiltonian, U is the time evolution operator associated with the 푖th pulse FIG. 3. Yield of restricted one-electron spectrum (16.3 eV < (1=pump, 2=probe), and 휏 is the duration of the field- 휖 < 28.6 eV) integrated over all emission angles resulting from free evolution between the pulses. We spectrally decom- ^ double ionization of He by a pump-probe sequence of a 20 nm pose the field-free propagation operator 푒−푖HECS휏 and 2 pump – 20 nm probe setup from the 1푠 singlet state as a retain only the relevant intermediate states, the initial function of delay time 휏 between pump and probe. Crosses: state 훾 휓0 (pathway 훾) and intermediate DES full numerical solution of the TDSE; blue line: semi-analytical 푚 | ⟩ ≡ |± ⟩ + 훽 2푠푛푝 (pathway 훽). Up to a global phase, model Eq. 2 including doubly excited resonances |2푠푛푝 ⟩, 푛 = | ⟩ ≡ | ⟩ 2−5, and |2푠3푝−⟩ as intermediate states; green line: envelope the double ionization amplitude at the conclusion of the of the modulation of the fast oscillation between pathways 훽 probe pulse is and 훾. −푖Δ퐸푚휏 푚 K 휓푓 = 훾K + 푒 훽K , (2) ⟨ | ⟩ 푚 ∑︁ 휑푚(휏) , and a slow modulation on a much longer time ⟨ ⟩ with K k k , , and scale, 휑 (휏) 휑 ′ (휏), an example of which is shown in ( 1 2) Δ퐸푚 = 퐸푚 퐸0 푔K = 푚 푚 ^ (2) ≡ ^ (1) − Fig. 3. This− interference with the reference wave (path- K U 푔 (푔 U 휓0 (for 푔 = 훽, 훾). In the intermediate ⟨ | | ⟩ | | ⟩ * way 훾) may appear as an unwanted background signal step, we use the modified inner product (푛 푚 = 푛 푚 , ^ | ⟩ ⟨ | ⟩ that overshadows the pump-probe pathway 훽 but, in- as HECS is complex symmetric. 퐸푚 (and thus Δ퐸푚) is stead, turns out to be the second key ingredient for complex, with the imaginary part describing the decay of improving the visibility of the coherent dynamics along the DES. * 푚 pathway 훽. Each of the mixed terms 훾K훽K in the probability 2 ∼ 푃K = K 휓푓 oscillates with frequencies Re(Δ퐸푚) The analysis of the interference signal is facilitated by |⟨ | ⟩| a simple semi-analytical model extending a similar treat- corresponding to periods of 70 as. The superposi- ≈ −푖Δ퐸푚휏 푚 ment for excited bound states to resonances [15, 16]. In tion of several terms, 훽K(휏) = 푚 푒 훽K , to this model for the two-photon interferometry we exploit which only resonances within the bandwidth of the pump ∑︀ the fact that only a limited number of states contributes pulse contribute, leads tô︀ a modulation with frequencies Re(퐸푚 퐸푚′ ) corresponding to periods on the (multi- to the (differential) double ionization signal within the − energy range of interest. These are the initial state (path )femtosecond scale (Fig. 3) given by the energy spacing 훾) and the intermediate DES within the pump pulse between the DES. Since 훽 2 is proportional to the prod- | K| bandwidth (path 훽). The latter are resonances embed- uct of XUV one-photon double excitation probabilities ded in the continuum of He+. While the full numeri- and the double ionization̂︀ probability from the weakly cal solution does not invoke an explicit representation of bound doubly excited states, it is three to four orders the DES, the model as well as the extraction of physical of magnitude smaller than the two-photon double ioniza- 2 observables of the correlated dynamics, 푂 DES, are fa- tion from the ground state 훾K . Consequently, the ⟨ ⟩ ∼* | | cilitated by the explicit calculation of the DES. They interferometric signal Re(훾K훽K) is enhanced by orders are determined by an exterior complex scaling (ECS) of magnitude compared∼ to the true pump-probe signal 2 transformation of the Hamiltonian (cf. [17] and refer- 훽K . It appears as the modulation̂︀ amplitude relative to | | 2 ences therein). DES can not generally be described by an approximately constant background 2 훾K (where ∼ 2 | | 2 independent-particle configurations, but require collec- thê︀ factor two takes the contribution 훼K 훾K into tive quantum numbers (cf. [18] and references therein). account). The latter can be independently| | ≈ determined | | In the current setup, only a restricted set of DES with from the measurement of the pump signal alone in the 4

(b) 1 (a) 7 110 P β /P αγ × M M FIG. 4. (푎) Yield 푃 훽 from DES and modula- 0.5 6 A /P αγ 푀 M M tion 퐴 , shown as ratios to the background 12 2 2 푀 θ µˆ r 훼훾 0 12− β ∫︀ 2 5 yield 푃푀 = 2 푀 |훾K| dK from paths 훼 cos and 훾, for the restricted one-electron spec- -0.5 4 trum (17.7 eV<휖<30.0 eV) from double ion- -11 ization integrated over all emission angles, (c) [%] 3 compared with the DES expectation value 0.5 2 −2 2 ⟨휇^ 푟12 ⟩훽 . The pulses (sin shape with 2 fs 12

θ 2 0 total duration, central wavelength 19 nm) co- +

cos herently excite |2푠푛푝 ⟩ (푛 = 3 − 8) with ap- -0.5 1 preciable probability. Two-dimensional pro- jections of the two-electron wavepacket on -1 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 the (푟12, cos 휃12) plane at the maximum (푏) and minimum (푐) of the modulation 퐴 . r12 [a.u.] τ [fs] 푀

absence of a probe pulse. In turn, the modulation ampli- Invoking the closure approximation K0 K0 dK 1, 푀 | ⟩⟨ | ≈ tude 퐴푀 follows from Eq. 4 reduces to the expectation value ∫︀ 훽 2 −2 * 푃푀 (휏) 휓훽 휇^ 푟12 휓훽 , (5) 퐴푀 (휏) = 4 훾K훽KdK (3) ∝ ⟨ | | ⟩ ⃒∫︁푀 ⃒ i.e., the dipole-weighted square of the electron-electron ⃒ ⃒ where 푀 is the region of final-state⃒ ̂︀ electron⃒ momenta in- interaction. Eq. 5 agrees remarkably and, in view of the a ⃒ ⃒ tegrated over: an energy window for electron 1 (Fig. 2), priori poorly justified closure approximation, surprisingly all emission angles of electron 1 and all vectorial mo- well with the simulated modulation signal 퐴푀 (Fig. 4) for menta of electron 2. The modulation 퐴푀 (휏) is the ex- the complex modulation pattern resulting from a pump- perimentally accessible signal monitoring the wavepacket probe sequence with a central wavelength of 19 nm. We dynamics in the collectively excited DES manifold, and note that leaving out the dipole operators, i.e., using the −2 agrees remarkably well with the (experimentally inacces- expectation value 휓훽 푟12 휓훽 works equally well. Fig. 4 sible) direct contribution from the DES pump-probe path clearly represents⟨ signatures| | ⟩ of the time-resolved corre- 훽 2 훽, 푃푀 = 푀 훽K dK (Fig. 4). This good agreement relation dynamics appearing in the non-coincident single- sults from the| fact| that two-photon double ionization in electron spectrum. ∫︀ a single pulsê︀ produces a “well-behaved” reference wave. The numerical simulation allows to explore the cor- It is now of crucial importance to identify the expec- related two-electron dynamics, the projection of which tation values of observables within the DES manifold onto the single-electron spectrum is monitored by 퐴푀 . with which the probe signal 퐴푀 approximately corre- Snapshots in the (푟12, cos 휃12) plane reveal that maxima lates. Clearly, because of the dipole selection rules the (minima) in 퐴푀 are associated with minima (maxima) two-photon XUV pump-probe scenario will give access to in the inter-electronic separation rather than with the observables differently from those monitored by charged- one-electron distance from the nucleus. The latter would particle collisions [9–11]. Key is the observation that be the hallmark of mean-field (or independent particle) double ionization of DES by absorption of a single pho- processes. ton from the probe pulse is mediated by final state cor- In summary, we have shown how correlated dynam- relation. To lowest order perturbation theory, this is ics in doubly excited states of helium can be accessed the well-known two-step-one (TS1) process frequently in- by two-photon interferometry with identical attosecond voked for both photoionization and charged particle ion- pulses. Supported by a full numerical solution of the ization [19, 20]. Accordingly, one electron absorbs the Schrödinger equation, we have shown that contributions photon energy and ejects the second electron by a colli- from two-photon absorption within a single pulse pro- sional Coulomb interaction in the exit channel. The am- vide a reference wave that the signal of interest inter- −1 plitude of this process is proportional to K 푟 휇^ 휓훽 , feres with and that greatly enhances the observable sig- ⟨ (0)| 12 | ⟩ where K(0) represents the uncorrelated final two-electron nal. The present protocol may provide an avenue for continuum state, 휇^ = 푝푧,1 + 푝푧,2 is the dipole transition directly observing correlation dynamics with attosecond operator and 휓훽 is the DES part of the intermediate pulses available presently or in the near future without wave packet. The| ⟩ probability for one-photon double ion- coincidence requirements. ization of DES with final momenta in the restricted re- The authors wish to thank Renate Pazourek for valu- gion is therefore able discussions. We acknowledge support by the FWF- Austria, grants No. SFB016 and P21141-N16 (S.N. & 훽 −1 −1 푃 (휏) dK 휓훽 휇푟^ K K 푟 휇^ 휓훽 . (4) J.B.) and by the NSF through a grant to ITAMP (J.F.). 푀 ∝ ⟨ | 12 | (0)⟩⟨ (0)| 12 | ⟩ 푀∫︁ C.T. and L.A.C. acknowledge support from LANL, which 5 is operated by LANS, LLC for the NNSA of the U.S. DOE G. Sansone, and M. Nisoli, Nat. Phot. 4, 875 (2010). under Contract No. DE-AC52-06NA25396. The com- [8] T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, putational results have been achieved using the Vienna and H. C. Kapteyn, Nat. Phot. 4, 822 (2010). Scientific Cluster, Institutional Computing resources at [9] P. van der Straten and R. Morgenstern, Phys. Rev. A 34, 4482 (1986). Los Alamos National Laboratory, and NSF TeraGrid re- [10] Y. 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