Kirrander, A.

Suominen, H. J.

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     Edinburgh Research Explorer                                      Comment on "How to observe coherent electron dynamicsdirectly" ReplyCitation for published version:Suominen, HJ & Kirrander, A 2014, 'Comment on "How to observe coherent electron dynamics directly"Reply' Physical Review Letters, vol. 113, no. 18, 189302. DOI: 10.1103/PhysRevLett.113.189302Digital Object Identifier (DOI):10.1103/PhysRevLett.113.189302Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordPublished In:Physical Review LettersPublisher Rights Statement:Copyright © 2014 American Physical Society. This article may be downloaded for personal use only. Any otheruse requires prior permission of the author and the American Physical Society.General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019Comment on “How to observe coherent electrondynamics directly”The main results of Ref. [1] rely on the assumption of thevalidity of Eq. (1) in Ref. [1]. In essence, that equation ismeant to establish a connection between the time-dependentelectron density of a nonstationary electronic system and theobservable x-ray scattering pattern associated with thatsystem. The authors of Ref. [1] claim that their Eq. (1)rests exclusively on the assumption that the electronicdynamics to be imaged are much slower than the durationof the x-ray pulse employed to probe those dynamics. (Thisis in addition to the assumption of nonresonant x-ray probeconditions.) The purpose of this Comment is to point out thatEq. (1) in Ref. [1] is generally invalid; a short pulse durationis by no means sufficient to guarantee the validity of thatequation. An example demonstrating the failure of Eq. (1) inRef. [1] for an x-ray pulse shorter than the electron dynamicsto be imaged may be found in Ref. [2]. Therefore, weconsider the validity of the results of Ref. [1] questionable.Particularly, it must be expected that the patterns in Fig. 5 ofRef. [1] differ qualitatively from what would be found inexperiment.Let us assume that the electronic wave packet of interestevolves freely, i.e.,jΨ; ti ¼XIαIe−iEItjΨIi; ð1Þwhere jΨIi is an eigenstate of the electronic Hamiltonian, EIis the associated eigenenergy, and αI is a time-independent,generally complex expansion coefficient. In order to makethings particularly transparent, let us assume that only twoelectronic eigenstates—jΨ1i and jΨ2i—contribute to thewave packet. Then, the expectation value of any observablewill oscillate periodically with the period T ¼ 2π=jE2 − E1j.A probe pulse that can resolve these oscillations willnecessarily be shorter than T; equally necessarily, such apulse has a spectral bandwidth that exceeds the energysplitting between jΨ1i and jΨ2i. This is obvious fromFourier considerations, and it holds irrespective of whether Tis a femtosecond or much longer. It is fundamentallyimpossible to make the spectral bandwidth of the incomingx-ray beam small in comparison to the energy splittingbetween jΨ1i and jΨ2i if one is using an x-ray pulse that isshort enough to resolve the dynamics associated withcoherent superpositions of those electronic eigenstates. Itmust therefore be expected that even if the x-ray scatteringdetector has perfect energy resolution (a most optimisticassumption), the scattering signal will involve an incoherentsum over all final states that are energetically accessiblewithin the spectral bandwidth of the incoming x-ray beam. Acareful analysis within the framework of quantum electro-dynamics demonstrates that this is indeed the case [2],leading, in general, to a failure of Eq. (1) in Ref. [1].This failure is particularly easy to see if we assume, forsimplicity, that the only final states energetically accessibleare jΨ1i and jΨ2i themselves (a gross oversimplification inview of the nonuniform energy level structure in theCoulomb problem). Then, for a probe pulse much shorterthan T, the differential scattering probability per x-raypulse, at high photon energy, isdPdΩ¼ ζZd3xZd3x0hΨ; tdjnˆðxÞ× fjΨ1ihΨ1j þ jΨ2ihΨ2jgnˆðx0ÞjΨ; tdieiQ·ðx−x0Þ: ð2ÞHere, nˆðxÞ is the electron density operator, Q is the photonmomentum transfer, td is the time at which the x-ray probepulse scatters from the electronic wave packet (the pump-probe time delay), and the constant ζ depends, among otherthings, on the spectrum of the incoming x-ray beam and onthe spectral response of the x-ray scattering detector.Equation (2) may be easily verified by using the resultsof Ref. [2]. (One may arrive at the same conclusion byapplying the analyses of Refs. [3] and [4].) The key pointhere is that the right-hand side of Eq. (2) cannot be writtenin the formZd3xZd3x0hΨ; tdjnˆðxÞjΨ; tdihΨ; tdjnˆðx0ÞjΨ; tdieiQ·ðx−x0Þ;which, up to a prefactor, is Eq. (1) from Ref. [1] in thenotation employed here. In other words, the requirement ofa short pulse—or slow electronic dynamics—does notensure that the final state reached in the photon collisionprocess equals the electronic wave-packet state right beforethe collision. Finally, we would like to mention thatanalogous considerations have been shown to apply totime-resolved electron scattering [5].R. Santra,1,2 G. Dixit3 and J. M. Slowik1,21Center for Free-Electron Laser ScienceDESY, Notkestraße 85, 22607 Hamburg, Germany2Department of Physics, University of HamburgJungiusstraße 9, 20355 Hamburg, Germany3Max Born InstituteMax-Born-Straße 2 A, 12489 Berlin, GermanyReceived 4 March 2014; published 28 October 2014DOI: 10.1103/PhysRevLett.113.189301PACS numbers: 32.30.-r, 32.80.Ee, 34.80.Qb, 82.53.Hn[1] H. J. Suominen and A. Kirrander, Phys. Rev. Lett. 112,043002 (2014).[2] G. Dixit, O. Vendrell, and R. Santra, Proc. Natl. Acad. Sci.U.S.A. 109, 11636 (2012).[3] S. Tanaka, V. Chernyak, and S. Mukamel, Phys. Rev. A 63,063405 (2001).[4] N. E. Henriksen and K. B. Møller, J. Phys. Chem. B 112, 558(2008).[5] H.-C. Shao and A. F. Starace, Phys. Rev. A 88, 062711(2013).PRL 113, 189301 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending31 OCTOBER 20140031-9007=14=113(18)=189301(1) 189301-1 © 2014 American Physical Society

Comment on "How to observe coherent electron dynamics directly" Reply

Henri J. Suominen

Adam Kirrander

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