Modulation of attosecond beating by resonant two-photon transition
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p Modulation of Attosecond Beating by ResonantTwo-Photon Transition ´Alvaro Jim´enez Gal´an
Departamento de Qu´ımica, M´odulo 13, Universidad Aut´onoma de Madrid, 28049 Madrid,Spain, EUE-mail: [email protected]
Luca Argenti
Departamento de Qu´ımica, M´odulo 13, Universidad Aut´onoma de Madrid, 28049 Madrid,Spain, EU
Fernando Mart´ın
Departamento de Qu´ımica, M´odulo 13, Universidad Aut´onoma de Madrid, 28049 Madrid,Spain, EUInstituto Madrile˜no de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia),Cantoblanco, 28049 Madrid, Spain, EUCondensed Matter Physics Center (IFIMAC), Universidad Aut´onoma de Madrid, 28049Madrid, Spain, EU
Abstract.
We present an analytical model that characterizes two-photon transitions in thepresence of autoionising states. We applied this model to interpret resonant
RABITT spectra,and show that, as a harmonic traverses a resonance, the phase of the sideband beatingsignificantly varies with photon energy. This phase variation is generally very different fromthe π jump observed in previous works, in which the direct path contribution was negligible.We illustrate the possible phase profiles arising in resonant two-photon transitions with anintuitive geometrical representation.
1. Introduction
Autoionising states (AI) are hallmarks of electronic correlation, which shapes the reactivity ofall many-body systems. AI states have been the subject of extensive investigation since Maddenand Codling reported the asymmetric profile of helium doubly excited states in the first energy-resolved x-ray photoabsorption spectrum recorded using synchrotron radiation, a pioneeringexperiment which signed the birth of modern photoelectron spectroscopy [1]. Several yearsbefore, Fano had developed a model [2] in which he explained the asymmetric profiles in atomicphotoelectron spectra, such as those seen by Madden and Codling, as interferences between twoone-photon paths, a direct one, from the ground state to the continuum, and an indirect one,from the ground to the metastable state to the continuum. Synchrotron radiation gave access tothe study of one-electron processes with unprecedented detail; the width and energy of severalautoionising states have been accurately measured. Due to its typical properties (incoherentight pulses, with a duration of several picoseconds, that are highly monochromatized beforeimpinging on the sample), however, synchrotron radiation did not provide access to the energydependence of resonant transition phases. Hence, the dynamical information of the resonantprocess, which is encoded in such phases, was lost.The advent of attosecond light sources opened the way to monitor and control the electronmotion in atoms and molecules at their intrinsic timescale [3]. In particular, such coherent ultra-fast sources of light provide a new means to investigate the role of AI states in atomic transitionssince, first, the coherence of the pulses used permits to experimentally access the phases of thetransitions and, second, the attosecond resolution permits to follow the temporal evolution ofthe AI states on a time scale smaller than their lifetime. The technique of reconstruction ofattosecond beating in two-photon transitions (
RABITT ) [4], which has already been successfullyemployed to study resonant transitions [5,6], is particularly indicated to this task. So far,however, only transitions through either electronic bound states [5] or autoionising vibronicstates without any appreciable contribution from the intermediate continuum (direct path) [6]have been considered. In those conditions, the phase of the complex transition amplitude isexpected to undergo a change of π as a function of the detuning of the pump harmonic fromthe intermediate resonant state, and the measurements were indeed found to be compatiblewith this expectation. There remained a need to explore the case in which both the boundand the continuum intermediate states contributed to the two-photon transition. To do so, itwas necessary to extend Fano’s model to the multi-photon finite-pulse formulation required bymodern attosecond interferometric techniques. A two-photon finite-pulse resonant model, inparticular, serves as a framework to interpret the phases of resonant transition amplitudes incurrent experiments.In this paper, we provide a derivation of the latter theoretical model, which was originallypresented in [7], and which is able to reproduce to a great accuracy time-resolved resonant two-photon transitions. Indeed, the model has already been successfully used to interpret two recent RABITT experiments on resonant transitions in Helium [8] and Argon [9]. We also illustrate ageometrical construction that permits to interpret the phase profiles obtained from experiments.For a more complete set of results and a more detailed derivation, the reader is referred toRefs. [7,9,10].
2. Theory
The lifetimes of autoionising states are often comparable to the duration of the APT and of the IRpulses used in
RABITT spectroscopy. Furthermore, the central frequency of the harmonics maynot coincide with a nominal multiple of ω IR . For these reasons, to make quantitative predictionsit is necessary to use a finite-pulse formulation of the the perturbative transition amplitudes. Inthe present section, we provide a derivation for the time-dependent two-photon resonant model,in the simplified case of only one isolated intermediate resonance. The derivation accounts alsofor a direct dipolar coupling between the localised component of the intermediate metastablestate and the final continumm.The time-dependent lowest-order perturbative amplitude for the two-photon transition froman initial atomic bound state | g i to a final non-resonant continuum state | γE i , for a linearlypolarized field ~F ( t ) = ˆ ǫF ( t ), is A γE,g = − i Z dω ˜ F ( E − E g − ω ) ˜ F ( ω ) M γE,g ( ω ) , (1)where M γE,g ( ω ) = h γE |O G +0 ( ω g + ω ) O| g i is the two-photon dipole matrix element, the Fouriertransform (FT) is defined as ˜ F ( ω ) = (2 π ) − / R F ( t ) exp( iωt ) dt , G +0 ( E ) ≡ ( E − H + i + ) − is the retarded resolvent of the field-free hamiltonian H , O = ˆ ǫ · ~O is the dipole operator (inelocity gauge, ~F is the vector potential and ~O = α ~P , where α is the fine-structure constant).The index γ indentifies collectively all the quantum numbers, other than energy, that uniquelyidentify the final continuum, and which are needed to differentiate it from other continuumstates, such as those populated by one-photon transition from the ground state. Atomic unitsare used throughout, unless otherwise stated. We assume that the field of the impingingpulses can be expressed as a linear combination of Gaussian pulses, F ( t ) = F exp[ − σ ( t − t ) /
2] cos[ ω ( t − t ) + φ ], where F , ω , t , σ and φ are the amplitude, carrier frequency, center,spectral width and carrier-envelope phase of the pulse, respectively. The FT of such pulse is˜ F ( ω ) = ˜ F + ( ω )+ ˜ F − ( ω ), where ˜ F ± ( ω ) = F (2 σ ) − exp( iωt ) exp[ − ( ω ∓ ω ) / (2 σ )] exp( ∓ iφ ) arethe components responsible for photon absorption and emission, respectively. In a pump-probeexperiment with an XUV attosecond-pulse train (APT) in association with an isolated IR pulse,the APT center t XUV conventionally defines the time origin, t XUV = 0, while the center of theIR pulse coincides with the pump-probe time delay τ , t IR = τ . In the present context, the totalexternal field is conveniently expressed in terms of synchronized pulses for the odd harmonics,which give rise to the APT, and of a delayed pulse for the IR field, F ( t, τ ) = F APT ( t ) + F IR ( t − τ ),where F APT ( t ) = P n F H2 n +1 ( t ). In the RABITT scheme, where one XUV photon is absorbed andone IR photon is either absorbed or emitted, the transition amplitude corresponding to a givensideband SB n is given by the sum of four contributions, each associated to a time-orderedperturbative diagram. Since the contribution of the two diagrams in which the IR photon isexchanged first is generally small and can be neglected, so the transition amplitudes become A ± , H2 n ∓ γE,g = − i R dω ˜ F ± IR ( E − E g − ω ) ˜ F + H2 n ∓ ( ω ) M γE,g ( ω ). The energy-resolved intensity of thesideband, I SB , is computed from the transition amplitudes as I SB = (cid:12)(cid:12)(cid:12) A + , H2 n − γE,g (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) A − , H2 n +1 γE,g (cid:12)(cid:12)(cid:12) + 2 ℜ h A + , H2 n − ∗ γE,g A − , H2 n +1 γE,g i (2)To evaluate the two-photon ionization matrix element M γE,g ( ω ), we assume that theintermediate eigenstates of H are well represented by a single resonant channel, H | ψ αE i = | ψ αE i E that can be expressed, following Fano’s formalism [2], in terms of known bound state, | a i , and featureless continuum states, | αε i , which are eigenstates of a reference hamiltonian H , H | a i = E a | a i , H | αE i = | αE i ε , h αE | αE ′ i = δ ( E − E ′ ), | ψ αE i = | αE i + (cid:18) | a i + Z dε | αε i V αε,a E − ε + i + (cid:19) V a,αE E − ˜ E a , h ψ αE | ψ αE ′ i = δ ( E − E ′ ) , (3)where V a,αE = h a | H − H | αE i , ˜ E a = ¯ E a − i Γ a / E a = E a + P R dε | V a,αε | / ( E − ε ), Γ a = 2 π | V a,αE | [2]. For simplicity, in the present derivation wedisregard the role played by virtual excitations of intermediate bound states. The transitionmatrix elements between a localised state and Fano continuum can be parametrized as h ψ αE |O| g i = O αE,g ( ǫ E + q )( ǫ E − i ), where ǫ E = 2( E − ¯ E a ) / Γ a and q = O ˜ ag / ( πV aE O Eg ) [2]. Thetwo-photon ionization matrix element, therefore, can be written as M γE,g ( ω ) = Z dε h γE |O| ψ αε i ω g + ω − ε + i + ǫ ε + qǫ ε − i O αε,g . (4)By applying the on shell approximation, h γE |O| αε i ≃ ¯ O γα ( E ) δ ( E − ε ), which, for singly-chargedparent ions, is quite accurate at energies of the order of 1 a.u. above the threshold or larger,and assuming that both ¯ O γα ( E ) and V a,αE are sufficiently slowly varying functions of E , it isimmediate to see that h γE |O| ψ αε i = ¯ O γα δ ( E − ε ) + ¯ O γα ε − E + i + π ( ǫ ε + i ) + O γE,a πV αE,a ( ǫ ε + i ) . (5)e can now insert this expression in the two-photon matrix element, M γE,g ( ω ) = ¯ O γα O αE,g ω g + ω − E + i + ǫ E + qǫ E − i + Z ( ǫ ε + q ) / ( ǫ ε + 1) ω g + ω − ε + i + (cid:18) ¯ O γα ε − E + i + + O γE,a V αE,a (cid:19) O αε,g π dε (6)The integral in this last expression can be easily computed closing the integration circuit in thelower half of the complex plane and applying Cauchy’s residual theorem, M γE,g ( ω ) = ǫ E + qǫ E + i ¯ O γα O αE,g ω g + ω − E + i + + (cid:18) β a − ǫ E + i (cid:19) ( q − i ) ¯ O γα O αE,g ω − ω ˜ ag , (7)where ω ij ≡ ω i − ω j and we introduced the parameter β a = π O γE,a V a,αE / ¯ O γα .If one is interested in the long-pulse limit only, for which the harmonic spectrum is stronglypeaked at ω H , the resonant and non-resonant two-photon transition matrix elements, for theabsorption of an XUV photon ω H followed by the absorption/emission of one IR photon, ω IR ,can be approximated with their value at ω = ω H , M ± γE,g ( ω ) ≃ M ± γE,g ǫ H + q ± ǫ H + i , q ± = q ∓ ( q − i ) ζ a , ζ a ≡ β a ω IR Γ a (8)where we used the energy-conservation principle E = E g + ω H ± ω IR and we introduced thenotation ǫ H = ǫ E g + ω H for the reduced detuning of the harmonic from the resonance. Noticethat when the intermediate autoionizing state | a i is not directly radiatively coupled to the finalcontinuum, i.e., β a = 0, we recover an expression equivalent to Eq. (4) in the main manuscript, M ± γE,g ( ω ) ≃ M ( α ) ± γE,g ǫ H + qǫ H + i + M (2) ± γE,g . (9)The dimensionless parameter ζ a = 2 ω IR β a / Γ a = O γE,a / ( ¯ O γα ω − IR V αa ) expresses the relativestrength of the direct | a i → | γE i radiative transition compared to the indirect path for the sameprocess | a i → | αE i → | γE i , in which the system first decays non-radiatively to the continuumand subsequently exchanges a photon in a continuum-continuum transition. As ǫ H increasesfrom −∞ to + ∞ (i.e., from large negative to large positive detuning of the resonant harmonic)the resonant factor ( ǫ H + q ± ) / ( ǫ H + i ) describes, counterclockwise, a circular trajectory in thecomplex plane, which starts from (1 , − iq ± ) / | − iq ± | / ǫ H + q ± ǫ H + i = 12 (1 − iq ± ) + 12 (1 + iq ± ) e iφ H , φ H ≡ arctan( ǫ H ) + π/ . (10)The circle intercepts the origin only if ζ a = 0. If ζ a ≷
0, the origin falls outside the circle andhence the phase of the resonant matrix element M ± γE,g experiences a continuous excursion withno net variation, while if ζ a ≶
0, the circle encloses the origin, and the phase of M ± γE,g undergoesa smooth overall excursion of 2 π . In the long-pulse limit, therefore, when ζ a = 0, if the phaseof the resonant amplitude for the absorption of one IR photon performs a jump of 2 π , that forthe emission of an IR photon will experience no net variation, and viceversa.With finite pulses, the transition amplitude is given by the convolution (1) of M γE,g ( ω ) withthe FTs of the field. The calculation for Gaussian pulses is lengthy but straightforward, andthe result can be expressed in closed form in terms of the Faddeeva special function, w ( z ), heredefined as the analytic continuation of iπ − R ∞−∞ dt exp( − t ) / ( z − t ) ( ℑ z > ω H followed by the absorption (+) or emission ( − ) of an IRphoton ω IR , the result is A ± , H ( α ) γE,g = F ( τ ) e ± iω IR τ ¯ O γα ( E ) O αE,g (cid:20) ǫ E + qǫ E + i w ( z ± E ) + (cid:18) β a − ǫ E + i (cid:19) ( q − i ) w ( z ± ˜ E a ) (cid:21) , (11) igure 1. Trajectory of the resonant two-photon amplitude in the complex plane (upper panels)and corresponding phase (lower panels), as the reduced detuning ǫ = 2( E − ¯ E a ) / Γ a increasesfrom large negative to large positive values. In the left panels, the only contribution to thetransition comes from the intermediate bound component ( q = ∞ ). In the central panels, boththe intermediate continuum and bound components contribute to the dipolar transition from theground state ( q = 1), but the intermediate bound component is not radiatively coupled to thefinal continuum ζ a = 0. Finally, in the right panels, this final restriction is removed ( ζ a = 0 . π . See text for more details.where F ( τ ) is an inessential form factor of the field, and z ± ε ≡ σ t √ E − ( ε ± ω IR )] − [ E − ( E g + ω H ± ω IR )] /σ H + iτ √ σ t , σ t = q σ − H + σ − IR . (12)It is instructive to verify that, in the limit of long ( σ t Γ a ≫
1) overlapping ( τ ≪ σ t ) pulses,we recover expressions similar to the ones seen above, as expected. Indeed, in this limit,the argument of the Faddeeva function diverges, so one can use the asymptotic expansion w ( z ) ≃ iπ − / z − . At the nominal centre of the two-photon signal ( E = E g + ω H ± ω IR ),therefore, z ± ε ≃ σ t ( E g + ω H − ε ) / √
2. Using the energy-preserving condition, one can simplifythe transition amplitudes to A ± , H γE,g = √ iπσ t F ( τ ) e ± iω IR τ M ± γE,g ( ω ) . (13)With finite pulses, the resonant A ± , H ( α ) γE,g amplitudes describe contracted circles. For sufficientlysmall values of | ζ a | , therefore, both amplitudes would fail to enclose the origin, and hence thejump of 2 π wouldn’t be observable.
3. Conclusions
We have derived a model which permits us to predict the phase variation of two-photon resonantamplitudes, extracted from current attosecond pump-probe experiments. In the limit of longpulses, we find an expression similar in form to that for the resonant one-photon transitions, butwhich requires an additional parameter ζ arising from the exchange of an IR photon with the igure 2. Sideband photoelectron signals in a
RABITT pump-probe photoionization of thehelium atom from the ground state, as a function of both the photoelectron energy (verticalaxis) and the pump-probe time delay (horizontal axis). Each panel corresponds to a differentreduced detuning ǫ H of harmonic H41 from the 2 s p intermediate P o doubly excited state( ¯ E = 35 .
56 eV, Γ = 0 .
037 eV, q = − . ζ/ω IR = 0 .
19 eV − ), whose energy is indicated bythe thin white dashed line: (a) ǫ H = −
43, (b) ǫ H = −
25, (c) ǫ H = 0 .
9, (d) ǫ H = 17, (e) ǫ H = 35.At large negative (a) or positive (f) detunings, the upper and lower sidebands are in phase toa very good approximation. As the harmonic traverses the resonance, on the other hand, thebeating of the upper and lower sidebands get clearly out of phase and the sideband profile itselfis distorted. These panels are computed with an extension of the model described in the text,which accounts for final resonant states. The thin horizontal feature above 37 eV, visible inpanels (c-e) is due to the final 2 p S e autoionising state.autoionising state. The model has a compact analytical finite-pulse formulation, which has beeninstrumental to interpret recent findings in resonant RABITT experiments [8,9]. With the useof the model and simple geometrical constructions, we have characterised the possible profilesof the amplitude phase that can be expected in resonant two-photon transitions as a functionof detuning. These can vary from a finite excursion with no net phase variation to a total jumpof 2 π , depending on the relative strength and phases of the atomic transitions involved, theduration of the pulses used and the number of intermediate open channels. Acknowledgments
We thank A. L’Huillier, A. Maquet, R. Ta¨ıeb, P. Sali`eres, E. Lindroth, M. Dahlstr¨om,M. Kotur, V. Gruson, T. Carette, D. Kroon, L. Barreau, J. Caillat, M. Gisselbrecht, andC. L. Arnold for useful discussions. We acknowledge computer time from the CCC-UAMand Marenostrum Supercomputer Centers and financial support from the European ResearchCouncil under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERCgrant agreement 290853 XCHEM, the MINECO project FIS2013-42002-R, the ERA-ChemistryProject PIM2010EEC-00751, the European COST Action XLIC CM1204, the Marie Curie ITNCORINF, and the CAM project NANOFRONTMAG.
References [1] R. Madden and K. Codling. Phys. Rev. Lett. , 516 (1963).[2] U. Fano, Phys. Rev. , 1866 (1961).[3] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163 (2009).[4] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Aug´e, Ph. Balcou, H. G. Muller and P.Agostini, Science , 1689 (2001).[5] M. Swoboda, T. Fordell, K. Kl¨under, J. M. Dahlstr¨om, M. Miranda, C. Buth, K. J. Schafer,J. Mauritsson, A. L’Huillier and M. Gisselbrecht, Phys. Rev. Lett. , 103003 (2010).[6] J. Caillat, A. Maquet, S. Haessler, B. Fabre, T. Ruchon, P. Sali`eres, Y. Mairesse, and R.Ta¨ıeb, Phys. Rev. Lett. , 093002 (2011).7] ´A. Jim´enez-Gal´an, L. Argenti, and F. Mart´ın, Phys. Rev. Lett. , 263001 (2014).[8] V. Gruson, L. Barreau, A. Jim´enez-Gal´an, F. Risoud, J. Caillat, A. Maquet, B. Carr´e, F.Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Ta¨ıeb, F. Mart´ın and P. Sali`eres, manuscriptin preparation (2015).[9] M. Kotur, D. Guenot, A. Jim´enez-Gal´an, D. Kroon, E. W. Larsen, M. Louisy, S. Bengtsson,M. Miranda, J. Mauritsson, C. L. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M.Dahlstr¨om, E. Lindroth, A. Maquet, L. Argenti, F. Mart´ın and A. L’Huillier, arXiv:1505.02024(2014).[10] ´A. Jim´enez-Gal´an, F. Mart´ın, and L. Argenti, manuscript in preparationmanuscript in preparation