Attosecond x-ray scattering from a particle-hole wave packet

We investigate the imaging of electron wave-packet dynamics by measuring the pattern of inelastically scattered photons. For this purpose, we develop a theory of time-resolved Compton scattering. We demonstrate that, by using a sufficiently short x-ray pulse, the scattering cross section directly reflects the instantaneous momentum density of the electron. Therefore, we propose time-resolved Compton scattering as a tool to image electron wave-packet dynamics in momentum space. To illustrate this, we simulate electron wave packets in argon by using the time-dependent configuration-interaction singles method. Specifically, we consider coherent particle-hole wave packets, where the hole is in either the $3p$ or the $3s$ shell, and the particle (the excited electron) is either in a Rydberg state or in the continuum. Our calculations confirm that the dynamics of electron wave packets can indeed be imaged by measuring the doubly differential Compton scattering cross section. When the x-ray detector has no energy resolution at all, the contribution of Compton scattering to the differential scattering cross section becomes stationary. In that case, the motion of the particle and the hole can no longer be inferred from the scattering pattern.

Figure 1

Elements of the ionic density matrix for atomic Ar driven by an XUV pump pulse. Panel (a) shows the populations of the Hartree–Fock ground state (solid black line), the $3s$ hole (dotted-dashed blue line), and the $3p_0$ hole (dashed red line), respectively. In panel (b) we show the degree of coherence between the $3s$ and the $3p_0$ holes (dotted-dashed blue line) and the pump electric field (solid gray line).

Figure 2

DDSCS for energy transfer $\omega =100$ eV and $T_R=23.5$ as at delay times (a) ${\tau }_{\text{delay}}=6.9$ a.u., (b) 10.9 a.u., (c) 13.9 a.u., and (d) 36.9 a.u. in units of $d{\sigma }_{\text{th}}/d\mathrm{\Omega }$. The $y$ component of the momentum-transfer vector is set to zero.

Figure 3

Contributions to the electron momentum-space density. Panels (a)–(d) show the momentum-space density of the excited electron $[p^{2}sin\left({\theta }_p\right){\rho }_{\text{elec}}(\mathbf{p},t)]$ at delay times (a) ${\tau }_{\text{delay}}=6.9$ a.u., (b) 10.9 a.u., (c) 13.9 a.u., and (d) 36.9 a.u. (d). In panels (e)–(h) we display the momentum-space density of the hole $[p^{2}sin\left({\theta }_p\right){\rho }_{\text{hole}}(\mathbf{p},t)]$ at the same delay times. Panels (i)–(l) contain the contribution of the interference term in Eq. (23) [multiplied by the Jacobian $p^{2}sin\left({\theta }_p\right)$], and panels (m)–(p) show the sum of all three contributions. Note that panels in the same column all refer to the same delay time.

Figure 4

Cuts through the DDSCS for $Q_x=0.07$ a.u. and $Q_y=0$ a.u. Energy transfer $\omega$ and pulse duration $T_R$ are the same as in Fig. 2. The delay times displayed are (a) ${\tau }_{\text{delay}}=6.9$ a.u, 10.9 a.u., 13.9 a.u., and (b) 45.9 a.u., 49.9 a.u., 66.9 a.u. The DDSCS is given in units of $d{\sigma }_{\text{th}}/d\mathrm{\Omega }$.

Figure 5

Real-space density $[r^{2}sin\left({\theta }_r\right)\rho \left(\mathbf{x}\right)]$ at delay times (a) ${\tau }_{\text{delay}}=6.9$ a.u., (b) 10.9 a.u., (c) 13.9 a.u., and (d) 36.9 a.u. The result for the Hartree–Fock ground state has been subtracted.

Figure 6

Elements of the ionic density matrix for atomic Ar excited by a sequence of two XUV pulses. Panel (a) shows the populations of the Hartree–Fock ground state (solid black line), the $3s$ hole (dotted-dashed blue line), and the $3p_0$ hole (dashed red line). In panel (b) we show the degree of coherence between the $3s$ and $3p_0$ hole (dotted-dashed blue line) and the XUV electric field (solid gray line).

Figure 7

Contributions to the electron momentum-space density. Panels (a)–(c) show the momentum-space density of the excited electron $[p^{2}sin\left({\theta }_p\right){\rho }_{\text{elec}}(\mathbf{p},t)]$ at delay times (a) ${\tau }_{\text{delay}}=51.2$ a.u., (b) 55.2 a.u., and (c) 58.2 a.u. In panels (d)–(f) we display the momentum-space density of the hole $[p^{2}sin\left({\theta }_p\right){\rho }_{\text{hole}}(\mathbf{p},t)]$ at the same delay times. Panels (g)–(i) contain the contribution of the interference term in Eq. (23) [multiplied by the Jacobian $p^{2}sin\left({\theta }_p\right)$], and panels (j)–(l) show the sum of all three contributions. Note that panels in the same column all refer to the same delay time.

Figure 8

DDSCS for an energy transfer $\omega =100$ eV and $T_R=23.5$ as at delay times (a) ${\tau }_{\text{delay}}=51.2$ a.u., (b) 55.2 a.u., and (c) 58.2 a.u. in units of $d{\sigma }_{\text{th}}/d\mathrm{\Omega }$. The $y$ component of the momentum-transfer vector is set to zero.