Attosecond x-ray transient absorption in condensed-matter: a core-state-resolved Bloch model

Abstract

Attosecond transient absorption is an ultrafast technique that has opened the possibility to study electron dynamics in condensed matter systems at its natural timescale. The extension to the x-ray regime permits one to use this powerful technique in combination with the characteristic element specificity of x-ray spectroscopy. At these timescales, the coherent effects of the electron transport are essential and have a relevant signature on the absorption spectrum. Typically, the complex lightdriven dynamics requires a theoretical modeling for shedding light on the time-dependent changes in the spectrum. Here we construct a semiconductor Bloch equation model for resolving the lightinduced and core-electron dynamics simultaneously, which enables to easily disentangle the interband and intraband contributions. By using the Bloch model, we demonstrate a universal feature on attosecond x-ray transient absorption spectra that emerges from the light-induced coherent intraband dynamics. This feature is linked to previous studies of light-induced Fano resonances in atomic systems.relied on time-dependent density functional theory approaches [13], which has provided good results in comparison with experiments [5,8]. However, the disentanglement of intraband and interband dynamics is not straightforward and adds an extra difficulty in the interpretation of the numerical results.On the other hand, semiconductor Bloch equations (SBE) [14-17] have been successfully applied to describe the nonlinear response of semiconductors and two-dimensional materials under the interaction of intense IR and mid-IR laser pulses [18][19][20][21][22][23], and also to model ultrafast studies [6]. This many-body theory offers clear advantages for understanding and interpreting the induced complex dynamics, such as separating the intraband and interband contribution as well as reducing the numerical effort. There is still an ongoing debate about the underlying mechanisms of high-harmonic generation in solids and their intrinsic interband and intraband nature [24][25][26][27]. In this context, theoretical approaches that allow to disentangle both contributions are desirable.In this work we present an extension of the standard SBE theory to the x-ray regime, named core-stateresolved Bloch equations (cBE), by including core-hole states. Using cBE, the induced dynamics driven by the IR pump and the x-ray probe pulse can be efficiently resolved. We demonstrate a universal feature of the pumpinduced intraband dynamics on the attosecond x-ray transient absorption spectra in condensed-matter systems. This intraband effect is originated from the coherent lightwave electron transport prior to decoherence by scattering interactions or core-hole decay, and it can be connected to previous works on light-induced Fano spectral lineshapes in atomic systems [28]. This effect is important when the time of the attosecond excitation or the core-hole decay overlaps with the IR laser pulse. We also discuss the importance of this effect on ultrafast studies with mid-I...

Attosecond state-resolved carrier motion in quantum materials probed by soft x-ray XANES

“…We performed a firstprinciples electron dynamics simulation of the full pump-probe experiment based on real-time time-dependent density functional theory (RT-TDDFT). 99 In addition to the ab initio model, we developed a core-state-resolved Bloch equation model (cBE) 100 to disentangle the various concurrent inter-and intra-band contributions in response to the control field. The cBE model retains the relevant seven bands in three dimensions; it includes the three highest-occupied valence, two lowest-unoccupied conduction band states together with the Ti-2p core states, and bandgap renormalization (further details in supplementary material).…”

Theoretical Approach for Electron Dynamics and Ultrafast Spectroscopy (EDUS)

“…The induced dipole density of the system, or polarization of the system, is given by ⟨ P̂ ⟩ = prefix− false| e false| V ⟨ r̂ ⟩ where −| e |⟨ r̂ ⟩ represents the dipole of all electrons and V represents the volume of the quantum system. By analyzing the gain and loss of a quantum system under a laser pulse, one can relate the Fourier transform of the polarization with the absorption coefficient as , α ( ω ) = ω n b c ϵ 0 Im false[ boldP ( ω ) · bold-italicε * ( ω ) false] | ε false( ω false) | 2 where ϵ 0 is the dielectric permittivity of the vacuum, n b is the background refractive index, ε(ω) is the Fourier transform of the external electric field ε( t ), and P (ω) ≡ ∫d t e –iω t ⟨ P̂ ⟩( t ). Note that defining the Fourier transform of the current as J (ω) ≡ ∫d t e –iω t ⟨ Ĵ ⟩( t ), it is easy to check that i ω P (ω) = J (ω).…”

“…Writing the off-diagonal term of the density matrix as ρ nn c = e –iω x t ρ̃ nn c , where n c and n labels refer to the core band and a valence/conduction band, respectively, enables us to separate the fast- from the slowly variant changes in time. Including this transformation in the EOM, eq , and performing the RWA, i.e., neglecting the fast terms in time, one obtains normali ℏ ∂ ρ italicnm false( R false) ( k , t ) ∂ t = normali ℏ ∂ ρ nm ( R ) false( boldk , t false) ∂ t | deph + S italicnm ρ italicnm false( R false) ( k , t ) + false[ H 0 false( R false) ( k ) + H normale −e false( R false) ( k ) + | e false| bold-italicε false( R false) ( t ) · bold-italicξ ( k ) , ρ false( R false) …”

Simulation of attosecond transient soft X-ray absorption in solids using generalized Kohn-Sham real-time TDDFT

“…Attosecond Transient Absorption Spectroscopy (ATAS) [9,7,64,65,23,27] typically involves a pump-probe protocol wherein the absorption of an ultrashort probe pulse in a material excited by a pump pulse is measured. The intensity, duration and shape of the pump pulse as well as the pump-probe delay provide external control parameters that can be tuned.…”

“…The simulation, within a TDDFT framework, of ATAS employing low energy IR-XUV probe pulses in molecules [23,25] and solids [66,27] has been discussed in the literature. Soft X-ray transient absorption (XTAS) [9,7,65] is similar in principle except that higher frequency (100 -1000 eV) wide band-width X-ray probe pulses tuned to core-excitation edges are typically employed yielding an element specific perspective of transient electron dynamics with attosecond time-resolution Within the real-time velocity gauge formulation of TDDFT [29], one has access to the time-dependent macroscopic current − → I (t) (Eqn. 3) within a material that is under the influence of external laser fields described by the vector potential…”