Role of electronelectron interference in ultrafast timeresolved imaging of electronic wavepackets
Abstract
Ultrafast timeresolved xray scattering is an emerging approach to image the dynamical evolution of the electronic charge distribution during complex chemical and biological processes in realspace and realtime. Recently, the differences between semiclassical and quantumelectrodynamical (QED) theory of lightmatter interaction for scattering of ultrashort xray pulses from the electronic wavepacket were formally demonstrated and visually illustrated by scattering patterns calculated for an electronic wavepacket in atomic hydrogen [Proc. Natl. Acad. Sci. U.S.A., 109, 11636 (2012)]. In this work, we present a detailed analysis of timeresolved xray scattering from a sample containing a mixture of nonstationary and stationary electrons within both the theories. In a manyelectron system, the role of scattering interference between a nonstationary and several stationary electrons to the total scattering signal is investigated. In general, QED and semiclassical theory provide different results for the contribution from the scattering interference, which depends on the energy resolution of the detector and the xray pulse duration. The present findings are demonstrated by means of a numerical example of xray TRI for an electronic wavepacket in helium. It is shown that the timedependent scattering interference vanishes within semiclassical theory and the corresponding patterns are dominated by the scattering contribution from the timeindependent interference, whereas the timedependent scattering interference contribution does not vanish in the QED theory and the patterns are dominated by the scattering contribution from the nonstationary electron scattering.
I Introduction
To fully understand the functionality and dynamic behavior of molecules, solids and complex biological systems, it is important to image the motion of electrons in realtime and in realspace. The motion of atoms within molecules and solids that is associated with chemical transformations occurs on the femtosecond (1 fs = 10 s) timescale. The timescale of electronic motion, responsible for electronhole dynamics and electron transfer processes in molecules can be even faster, on the order of attoseconds (1 as = 10 s) (1); (2); (3); (4); (5). The ultimate goal of the emerging field of timeresolved imaging (TRI) is to visualize electronic motion on an ultrafast timescale as electrons move in atoms, complex molecules or solids, as occurring for instance in photoinduced exciton dynamics, during bond formation and breakage, conformational changes and charge migration (6); (7); (8); (9). With the tremendous advancement in technology for producing ultraintense and ultrashort xray pulses from novel light sources, it seems possible to obtain information about ultrafast dynamics of electrons. In extension of the concept of “molecular movies”, which track the motion of atoms on fs timescale (10); (11); (12); (13), ultrashort, tunable, and highenergy xray pulses from freeelectron lasers (FEL) (14); (15), laser plasmas (16) and highharmonic generation (17); (18) promise to provide “electronic movies” that take place on few fs to as timescale (19). Recent breakthroughs make it possible to generate hard xray pulses of a few fs (14); (20), and pulse duration of 100 as can in principle be realized (21); (22). Utilizing the remarkable properties of xrays from FEL, several new insights have been gained about systems ranging from atoms (23); (24), molecules (25); (26), clusters (27), complex biomolecules (28); (29), to matter in extreme conditions (30); (31).
Since the discovery of xrays (32), scattering of xrays from matter has been used to unveil the structure of molecules, solids and biomolecules with atomicscale spatial resolution (33); (34); (35); (36); (37). Also, scattering of xrays from atoms and molecules has been proposed to gain insight about the excited electronic states of atomic and molecular systems (38); (39). In order to image electronic motion in realtime and realspace with spatial and temporal resolutions of order 1 Å and 1 fs, respectively, one can perform scattering of ultrashort xray pulses from the dynamically evolving electronic charge distribution. Pumpprobe experiments are the most direct approach, where first a pump pulse induces the dynamics and then subsequently a probe pulse interrogates such induced dynamics. By varying the pumpprobe time delay, one obtains a series of scattering patterns that serve to image the electronic motion with atomicscale spatiotemporal resolution.
Recently, the theory of TRI from a nonstationary electronic system, using both the semiclassical and the quantum electrodynamical (QED) treatment of lightmatter interaction, has been developed (40). In a semiclassical theory of lightmatter interaction, matter is treated quantum mechanically and light is treated classically. In such a situation, the timedependent Schrödinger equation is solved for the electrons together with Maxwell’s equations for the light. By solving Maxwell’s equations for given charge and current densities, the expression for the differential scattering probability (DSP) is obtained, which is a key quantity in xray scattering. According to the semiclassical theory, xray TRI would be expected to provide access to the instantaneous electron density of the nonstationary electronic system. On the other hand, in a consistent quantum theory of lightmatter interaction, where both matter and light are treated quantum mechanically, xray TRI encodes the information about spatiotemporal densitydensity correlation. Both the theories have been applied to an electronic wavepacket prepared as a coherent superposition of eigenstates of atomic hydrogen and it has been shown that the scattering patterns obtained using both the theories differ drastically from each other. Moreover, it was shown that the patterns obtained within QED theory follow the motion of the wavepacket providing the correct periodicity of the motion, which cannot be captured by the semiclassical theory. In that case, the notion of the instantaneous electron density as the key quantity being probed in xray TRI for a sufficiently short pulse completely breaks down (40). However, xray TRI from a single isolated hydrogen atom is not a realistic scenario. In practice, a sample contains several electrons and when a tunable pump pulse with broad energy bandwidth interacts with an electron system, one or few electrons participate in the formation of an electronic wavepacket and other electrons remain stationary. In such a situation, when an xray pulse scatters from a sample containing one or more nonstationary electrons and several stationary electrons, there is no way to know whether the scattering has taken place from the nonstationary electrons or from the stationary electrons and how the two scattering paths interfere with each other, i.e., interference between scattering from nonstationary and stationary electrons in the scattering process. Therefore, at this juncture it is important to analyze different types of contributions to the total scattering signal. The total signal can be decomposed into three main parts: first from stationary electrons, second from nonstationary electrons and third from the interference between nonstationary and stationary electrons. In the present work, we will analyze how these different contributions in an electron system contribute to the total scattering signal in both the theories (semiclassical and QED).
This paper is structured as follows. Section II discusses the formalism and results for xray TRI in the case of manyelectron systems, where only one electron forms an electronic wavepacket and other electrons serve as stationary reference scatterers in both the theories. Effects of different parameters such as energy resolution of the detector, pulse duration and spectral bandwidth of the xray pulse etc. for the electronelectron interference in the scattering process are discussed in detail. Section III presents a numerical example of xray TRI for an electronic wavepacket in helium, where one electron forms a coherent superposition of oneelectron eigenstates and the other electron remains stationary and serves as a reference scatterer. In this particular situation for helium, the role of the scattering interference is investigated. Conclusions and future outlook are presented in Sec. IV.
Ii Theory
Our investigations are based on the theory for xray TRI of electronic wavepacket motion as developed in Ref. (40). Our equations are expressed in atomic units (41). Under the assumptions that the probe pulse is centered at the energy of the incident pulse with very small energy width and the coherence length of the pulse is large in comparison to the size of the object, the expression for the DSP within the semiclassical theory is related to the Fourier transform of the instantaneous electron density, , as follows (40)
(1) 
Here, is the DSP for a free electron and is the photon momentum transfer. Here, we have assumed that the xray pulse duration is shorter than the dynamical timescale of the electronic wavepacket. According to Eq. (1), the measured scattering pattern provides access to the instantaneous electron density as a function of the pumpprobe delay time .
Let us consider a scenario for timeresolved scattering of ultrashort xrays in order to image the motion of a oneelectron wavepacket in the presence of stationary electrons. In such a situation, the stationary electrons serve as reference scatterers in the total scattering signal. In this case, using the language of second quantization (42), the total electronic wavepacket can be written as
(2) 
with
(3) 
Here, creates (annihilates) an electron in spin orbital and is the orbital energy corresponding to , i.e., . represents the electronic Hamiltonian at the meanfield level and is the unperturbed ground state of the electron system with the electrons filled to the Fermi level. Here and in the following, indices are used for general spin orbitals (occupied or unoccupied). Occupied orbitals in are presented by indices , whereas unoccupied (virtual) orbitals are symbolized by .
We rewrite the key quantity in Eq. (1) in terms of the density operator
(4) 
with
(5) 
Here, is the electron density operator, and the field operator creates (annihilates) an electron at position . Using the expression for the wavepacket as introduced in Eq. (2), Eq. (4) simplifies as follows
(6c)  
with
(7) 
and
(8) 
Here, the righthand side of Eq. (6c) provides the timeindependent contributions due to scattering from the stationary electrons. The second term as shown in Eq. (6c) is due to the scattering interference between the stationary electrons and the nonstationary electron. The last timedependent term in Eq. (6c) is solely due to scattering from the nonstationary electron.
On the other hand, in the full quantum theory of xray TRI, both matter and xray pulse are treated quantum mechanically and firstorder timedependent perturbation theory is employed for the interaction between matter and x rays. Here we assume that the probe pulse has a small bandwidth and a small angular spread so that the pixel assignment is well defined in the momentum space, the coherence length of the pulse is large in comparison to the size of the object, and the pulse duration should be sufficiently short to freeze the dynamics of the electronic wavepacket. Under these assumptions, the resulting expression for the DSP from a coherent, Gaussian xray pulse is (40)
(9)  
Here, is a function of the pulse duration . and refer to the energy of the incident and scattered photon, respectively, while is a spectral window function centered at with a width . models the range of energies of the scattered photons accepted by the detector.
In the case of QED theory, an energyresolved scattering process is considered for xray TRI. Therefore, any inelastic (Compton) scattering contributions due to excitations from the stationary electrons can be easily distinguished by utilizing the energyresolving detector, if we assume that is small in comparison to the characteristic excitation energies of . Hence, excitations from the stationary electrons are not considered in the following. Similarly, on using the expression for the wavepacket as introduced in Eq. (2), the key expression of Eq. (9) is simplified as follows
(10c)  
Here, the first term, Eq. (10c), provides the timeindependent contribution due to scattering from the stationary electrons, which is identical to Eq. (6c). The second term in Eq. (10c) is due to the scattering interference between the stationary electrons and the nonstationary electron, which seems different to the one shown in Eq. (6c). The last timedependent term, Eq. (10c), is again solely due to scattering from the nonstationary electron. The scattering contributions from the nonstationary electron, Eqs. (6c) and (10c), are not identical and provide completely different information about the electronic motion as shown in the case of a oneelectron wavepacket in atomic hydrogen (40). It is evident from Eqs. (6) and (10), that the first term in both the theories, Eqs. (6c) and (10c), provides identical scattering contributions to the total signal. In many systems of interest, the number of stationary electrons is large and therefore, the timeindependent terms contribute a strong static background in the total signal. Due to the large number of the stationary electrons and one or few nonstationary electrons, the dominating timedependent contributions in the total scattering signal are due to the scattering interference between stationary and nonstationary electrons (unless one is considering for which is small). Therefore, it is crucial to analyze the scattering interference term, Eqs. (6c) and (10c), in both the theories.
On substituting Eqs. (6c) and (10c) in Eqs. (1) and (9), respectively, the expression for the scattering interference contribution to the DSP, , can be written as
(11) 
in the semiclassical theory, and in the QED formalism can be expressed as
(12)  
It is important to analyze several aspects of both expressions, Eqs. (11) and (12), under different circumstances and properties of the probe pulse.

The Gaussian distributions in Eq. (12) are centered at positions and , and the width of the distributions is determined by . Therefore, if is larger than the separation between the two distributions and the detector has a lower energy resolution than the energy spectral bandwidth of the xray pulse, then all the scattered photons are detected by the detector and is constant in the energy range contributing to the integral in Eq. (12). Hence, on performing the energy integral, Eq. (12) reduces to Eq. (11). However, when the energy resolution of the detector is poor, it is very difficult to assign a unique pixel in space to the scattered photon in the detector and it may no longer be possible to filter out Compton scattering from the stationary electrons.

If the probe pulse is very short in comparison to the dynamical timescale of the electronic motion, the dependent exponent in Eq. (10c) will reduce to unity. Therefore, if is smaller than the energy spectral width of the pulse and centered at the incident energy such that , the energy integral, Eq. (12) reduces to Eq. (11) and the scattering interference contributions from both the theories will be identical. However, an ultrashort pulse corresponds to a large energy spectral width due to the energytime uncertainty relation. Therefore, the unavoidable energy bandwidth of the pulse causes an uncertainty in the momentum distribution of the incoming photon. In such a situation uniqueness of a pixel in space is lost.

In general, however, the probe pulse is not very short in comparison to the dynamical timescale of the electronic motion. In such situations the dependent exponent in Eq. (10c) will not reduce to unity and both expressions for the interference will provide different contributions to the total signal.
In the following subsection, we present an example of oneelectron wavepacket motion in helium. In this example, we apply both the approaches to compute timeresolved scattering patterns and analyze the contributions from the scattering interference to the total scattering signal.
Iii Numerical Results and Discussion
A schematic scenario for probing an electronic wavepacket motion in helium is shown in Fig. 1. The ground state configuration for both the electrons is 1s. The ionization potential of the first electron is 24.59 eV and 54.42 eV for the second electron (43). A pump pulse with broad energy bandwidth excites one of the electrons from the ground state configuration and prepares a coherent superposition of the 1s3d and 1s4f configurations with the projection of orbital angular momentum being equal to zero. The energy difference between the 1s and 1s3d configurations is 23.07 eV and between the 1s3d and 1s4f configurations is 0.66 eV (see Fig. 1). Therefore, the dynamical timescale of the electronic wavepacket motion, which is inversely related to the energy spacing between the eigenstates participating in the wavepacket, is 6.25 fs. The spatial extension of the wavepacket is 14–17 Å along the axis and 7.5–9 Å along the and axes. It is known that for orbital angular momentum quantum number equal to or larger than two, the quantum defect is almost zero. Therefore, the electron in the orbital sees no shielding of the nuclear charge and the electron in the superposition of and orbitals sees complete shielding of the nuclear charge. Therefore, the wavefunction for the stationary electron in the orbital can be expressed in terms of the hydrogenic wavefunction with nuclear charge Z = 2. On the other hand, the wavefunction for the nonstationary electron in and orbitals as well as other higherlying orbitals can be written in terms of the corresponding hydrogenic wavefunction with Z = 1. This type of procedure for treating the twoelectron problem has already been used in the past and was successfully applied to describe different types of physical processes (44).
In order to compute timedependent scattering patterns of the electronic wavepacket (cf. Fig. 1) as a function of the delay time in helium, we employ both the semiclassical and the QED approaches, i.e., Eqs. (1) and (9). Since the nonstationary and stationary electrons are energetically distinguishable (energy difference is around 23 eV) and an energyresolved scattering process is considered with energy resolution at least equal to the unavoidable spectral bandwidth of the probe pulse, any excitation from the stationary electron can be easily filtered out and therefore is not considered in the present case. The scattering operator is expanded in terms of the spherical Bessel functions, , and spherical harmonics as
(13) 
where . After introducing hydrogenic wavefunctions, the expression in Eq. (9) factorizes into radial and angular parts. The angular part is given by
(14)  
whereas the radial part is numerically integrated. To calculate the patterns as a function of the delay times, we used a Gaussian pulse of duration 1 fs with 4 keV incoming photon energy, and assumed a Gaussian photon energy detection window of width 0.5 eV for the detector. The patterns are calculated for Å corresponding to a 3.14 Å spatial resolution and to a detection angle of scattered photons of up to 60. In order to compute the scattering contribution from the nonstationary electron to the total scattering signal, all transitions induced during the scattering process within the energy detection window are computed (40). Therefore, transition amplitudes from the eigenstates involved in the electronic wavepacket to all the electronic states within the detection range of are computed, which includes all types of multipole transitions allowed by the conservation of angular momentum and amounts to 22000 transition amplitudes. Also, the scattering patterns using semiclassical theory are calculated by convolution of the square of the Fourier transform of with an xray pulse of duration 1 fs.
Scattering patterns in the  plane ( = 0) as a function of the delay time at times 0, T/4, T/2, 3T/4, and T are depicted in Fig. 2. The timedependent patterns shown in Fig. 2(a) are computed within QED theory using Eq. (9), whereas in Fig. 2(b) are computed within semiclassical theory using Eq. (1). It is evident from Fig. 2(a) that the patterns undergo spatial oscillation along in momentum space and reflect the motion of the wavepacket along in real space (see Fig. 1). One of the striking features of the patterns shown in Fig. 2(a) is that when the charge distributions are symmetric, and corresponding patterns are asymmetric and vice versa, which can be understood as follows. The charge distributions are identical at delay times T/4 and 3T/4, as may be seen in Fig. 1, while the electron clouds move in opposite directions at the two times. At time T/4, the flow of the electron cloud is downwards, whereas at time 3T/4 the flow is upwards. This is reflected by their corresponding patterns. Therefore, the patterns calculated within the QED theory capture the dynamics of the momentum distribution of the wavepacket. As a consequence, the apparent motions of the charge distributions and of the scattering patterns are shifted by 90. On the other hand, the patterns shown in Fig. 2(b) do not change significantly as a function of the delay time.
In order to understand the scattering from the nonstationary electron, the scattering patterns corresponding to the nonstationary electron are shown in Fig. 3 in the  plane ( = 0) as a function of the delay time. The patterns shown in Fig. 3(a) are obtained using Eq. (10c) and in Fig. 3(b) are obtained using Eq. (6c). It is evident from Fig. 3(a) that the scattering patterns undergo oscillations as a function of the delay time. The scattering patterns shown in Fig. 3(b) are localized in the low region, which reflects the spatial extension of the electronic charge distribution of the wavepacket and also undergo changes as a function of the delay time, but do not display oscillations as the charge distribution oscillates. Hence, patterns obtained using semiclassical theory provide half of the actual period of the motion as the patterns start repeating themselves in half of the actual time of the motion.
We now investigate the contribution from the scattering interference between the stationary and nonstationary electrons to the total scattering signal in both the theories. On rewriting Eqs. (7) and (8) in the case of helium, we find that
(15) 
is a purely real number, as is evident from Eqs. (13) and (14), whereas
(16) 
is a purely imaginary number, which also follows from Eqs. (13) and (14). Therefore,
(17) 
and
(18) 
In a similar way we can write , and , which are both real. On substituting the contributions of the ’s in Eq. (6c), the contribution from the scattering interference within the semiclassical theory can be written as
(19) 
which can be further decomposed into two parts: a timedependent scattering interference contribution
which is zero, and a timeindependent scattering interference contribution
(21) 
Similarly on substituting the contributions of the ’s in Eq. (10c), the contribution from the scattering interference within the QED theory can be written as the sum of two parts: a timedependent scattering interference contribution
(22b)  
which is nonzero, and a timeindependent scattering interference contribution
(23) 
which is identical to the one obtained within semiclassical theory. We rewrite the timedependent interference contribution, Eq (22), to the leading nonvanishing order in as
(24) 
Therefore, for the particular combination of orbitals involved in the oneelectron wavepacket and the orbital corresponding to the stationary electron in helium (see Fig. 1), the timedependent scattering interference between nonstationary and stationary orbitals is zero within the semiclassical theory and negligibly small in comparison to the total scattering signal in the QED theory. However, the timeindependent scattering interference between the orbitals, i.e., the interference between and orbitals and and orbitals, contributes equally to the total scattering signal in both the theories, as is evident from Eqs. (21) and (23). The reason why the timedependent scattering interference contribution to the total signal is so small in QED theory can be understood as follows: For probing the ultrafast motion, one has to satisfy , where is the characteristic energy scale of the electronic wavepacket, i.e., the pulse duration of the probe pulse should be much smaller than the characteristic timescale of the motion. Thus, the contribution from Eq. (24), which is proportional to , is suppressed.
On comparing the scattering contribution from the nonstationary electron with respect to total scattering signal, i.e., comparing Figs. 3(a) and 2(a), one can easily distinguish the scattering contributions from the stationary electron and the interferences to the total scattering patterns in the QED theory. In Fig. 2(a), the broadening of the scattering signal in the high region is the reflection of the contribution from the timeindependent scattering interference between orbitals, whereas the wing type structures along the diagonal, which change as a function of the delay time, are a reflection of the contribution from the timedependent scattering interference between orbitals, which is weak in comparison to the total scattering signal. On the other hand, on comparing the patterns shown in Figs. 2(b) and 3(b), one can only observe the broadening of scattering patterns, which reflects the contribution from the timeindependent scattering interference between orbitals.
Therefore, the patterns within the semiclassical theory are dominated by the scattering contribution from the timeindependent interference between orbitals, and are not changing significantly as a function of the delay time. In contrast, the patterns within the QED theory are dominated by the scattering contributions from the nonstationary electron due to Compton scattering within the finite energy detection range of the detector.
Iv Conclusions
This work is devoted to understanding ultrafast timeresolved xray scattering from a sample containing a mixture of a nonstationary electron in the form of a oneelectron wavepacket and one or more stationary electrons using the semiclassical theory and the QED theory of lightmatter interaction. The contributions of the scattering interference between the nonstationary and the stationary electrons to the total timedependent scattering signal are investigated in both the theories. Our investigations are based on the recent theory for timeresolved xray scattering to image the electronic wavepacket motion (40). First, we investigated different scattering contributions to the total scattering signal in both the theories and showed that the total signal can be decomposed into three main scattering contributions: first from the stationary electrons, second from the nonstationary electron and third from the interference between stationary and nonstationary electrons. In both the theories, the scattering contributions from the stationary electrons to the signal are identical, whereas scattering contributions from the nonstationary electron are completely different. In the QED theory, the scattering contributions from the interference depend on the energy resolution of the detector and the xray pulse duration. Therefore, in case of negligible energy resolution or extremely short pulses, QED theory provides identical contributions for the scattering interference as one obtaines in the semiclassical theory. On the other hand, if the pulse duration is not very short in comparison to the dynamical timescale of the motion and if the energy resolution is sufficiently high, the scattering interference in the QED theory does not provide identical result to the one obtaines in the semiclassical theory. It is important to note that most organic molecules and proteins contain mainly hydrogen, carbon, nitrogen and oxygen atoms and in such cases the scattering signal is dominated by the scattering from carbon, nitrogen and oxygen atoms. In these atoms, only two electrons are deeply bound core electrons, whereas other electrons are loosely bound valence electrons. When a pump pulse with broad bandwidth initiates excitation in such atoms it might be possible that more than one electron participates in the formation of the electronic wavepacket. In such situations, the timedependent scattering signal would be dominated by the scattering contributions from the nonstationary electrons rather than the stationary and the interference contributions.
Both the theories for lightmatter interaction are illustrated by means of calculating the timedependent scattering patterns for a oneelectron wavepacket in helium. In helium, the pump pulse excites one of the electrons from the ground state and prepares an electronic wavepacket as a coherent superposition of the 1s3d and 1s4f eigenstates. The scattering patterns are computed for the nonstationary electron in the presence of a stationary electron. The timedependent interference between the stationary and nonstationary electrons within the semiclassical theory is zero, and it is quite small in comparison to the total scattering signal in the QED theory. However, the timeindependent interference between the stationary and nonstationary electrons contributes identically to the total signal in both the theories. The patterns are dominated by the scattering contribution from the timeindependent interference within the semiclassical theory, whereas the patterns are dominated by the scattering contributions from the nonstationary electron due to Compton scattering within the QED theory. Henceforth, the dynamical features of the patterns cannot be captured within semiclassical theory. We expect that our present analysis of TRI using ultrafast xray scattering will find several important applications for exploring ultrafast dynamics in nature. With the recent advent of novel light sources, we also believe that our findings will shed light on ultrafast electronic motion, for example, in atoms, molecules and biological systems (45); (46); (47); (48); (49).
Acknowledgements.
We thank Jan Malte Slowik for careful reading of the manuscript.References
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