Comment on “Self-Referenced Coherent Diffraction X-Ray Movie of Ångstrom- and Femtosecond-Scale Atomic Motion”

Abstract

In this comment, we challenge the interpretation of ultrafast optical pump X-ray probe diffraction experiments on gas phase I 2 put forth recently by Glownia et al. [1]. In that Letter, the x-ray diffraction from a sample perturbatively prepared with excited state population a is given asS = N |af e (q) + (1 − a)f g (q)| 2( 1) where N is the number of molecules in the gas and f g/e (q) = g/e|σ(q)|g/e is the ground/excited state elastic scattering amplitude (related to the Fourier transform of the electronic charge densityσ operator). Reference [1] assumed "incoherent mixtures of ground and excited electronic states", neglecting electronic coherences from the onset. We thus consider only a diagonal electronic density matrix with elements a and 1 − a and restrict attention to elastic scattering. Importantly, the cross term (f g (q)f e (q)) resulting from the squaring in Eq. (1) amounts to heterodyne detection, the interference of a weak signal field (f e ) with a strong reference (f g ). Such holographic detection has been reported in transient X-ray diffraction in crystals [2]. For weak excitations, where only a small fraction of the molecules are excited (a ≪ 1), the ground-state signal serves as an in situ local oscillator for the weaker excited-state signal.In Ref.[1], it was argued that the linearity in the excitation fraction a of the cross term in Eq. (1) renders detection feasible in a heterodyne fashion, while the pure excited-state diffraction scales quadratically in a and is negligible. While we agree that "This signal is an incoherent sum of the coherent diffraction from each molecule", we point out that the correct expression [3,4] for such a signal iswhere the expectation value . . . = Tr [. . . ρ], can be evaluated via a trace over the density matrix. The excited-state diffraction from a gas thus comes linear in the excitation fraction and the amplitude boost from heterodyne detection is neither necessary nor possible. Equation (2) and equivalents obtained from the independent atom approximation and rotational averaging have been known in the literature on time-resolved X-ray scattering for many years and appear also in electron diffraction [3,5,6].The possibility of heterodyne-detected diffraction in crystals (and other systems with long-range order) can be seen by partitioning the total charge density as a sum of molecular charge densitiesσ gas = ασ α in S = σ * (q)σ(q) . The diagonal terms in this doublesum generate Eq. (2) while the remaining, two-molecule terms arewhere we have assumed identical molecules located at positions r α . This amounts to the observation that the electronic charge densities of distinct molecules are uncorrelated so that, for α = β, we have σ *The double-summation pre-factor in Eq. (3) encodes the long-range structure of the sample and, in crystals, results in bragg peaks at the reciprocal lattice vectors q Bragg [2]. It is well-established that S 2 averages out in the gas phase due to random molecule positions and signals are given by S 1 whereas in crystals S 2 which ...

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“…[12]; our present result is thus consistent with the brief remark made by Mukamel and co-workers [14] (see also Refs. [18,19]).…”

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