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

We start from the SBE Hamiltonian expressed in second quantization

$$\begin{eqnarray}&&{\hat{H}}_{0}=\displaystyle \sum _{s}{E}_{s}\,{\hat{a}}_{s}^{\dagger }{\hat{a}}_{s}\end{eqnarray} \tag{ 1 }$$

in which the canonical operators satisfy the anticommutator relations $\left\{{\hat{a}}_{s},{\hat{a}}_{s^{\prime} }\right\}=\left\{{\hat{a}}_{s}^{\dagger },{\hat{a}}_{s^{\prime} }^{\dagger }\right\}=0$ and $\left\{{\hat{a}}_{s},{\hat{a}}_{s^{\prime} }^{\dagger }\right\}={\delta }_{{ss}^{\prime} }$, and E_s is the energy of an electron in the state s. The electric field coupling, in the length gauge, is written as

$$\begin{eqnarray}&&{\hat{V}}_{I}(t)={\boldsymbol{\varepsilon }}(t)\displaystyle \sum _{s^{\prime} ,s}{{\bf{d}}}_{{ss}^{\prime} }\,{\hat{a}}_{s}^{\dagger }{\hat{a}}_{s^{\prime} }\end{eqnarray} \tag{ 2 }$$

in which ${\boldsymbol{\varepsilon }}(t)$ is the electric field and ${{\bf{d}}}_{{ss}^{\prime} }$ stands for the dipole matrix element between states s and $s^{\prime}$. Our time-dependent Hamiltonian is $\hat{H}(t)={\hat{H}}_{0}+{\hat{V}}_{I}(t)$ and describes the dynamics of our many-body quantum state. By evolving the operators, within the Heisenberg picture, we find the equation

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{\partial }{\partial t}({\hat{a}}_{\lambda }^{\dagger }{\hat{a}}_{\lambda ^{\prime} })=({E}_{\lambda ^{\prime} }-{E}_{\lambda }){\hat{a}}_{\lambda }^{\dagger }{\hat{a}}_{\lambda ^{\prime} }-{\boldsymbol{\varepsilon }}(t)\displaystyle \sum _{s^{\prime} }{{\bf{d}}}_{\lambda s^{\prime} }^{* }{\hat{a}}_{s^{\prime} }^{\dagger }{\hat{a}}_{\lambda ^{\prime} }+{\boldsymbol{\varepsilon }}(t)\displaystyle \sum _{s^{\prime} }{{\bf{d}}}_{\lambda s^{\prime} }{\hat{a}}_{\lambda }^{\dagger }{\hat{a}}_{s^{\prime} }.\end{eqnarray} \tag{ 3 }$$

We define the time-dependent density matrix elements, evolving from an initial state $\hat{\rho }$, from the mean value of the canonical operator coherences [17]

$$\begin{eqnarray}&&{\rho }_{{\bf{k}}\lambda ,{\bf{k}}^{\prime} \lambda ^{\prime} }=\mathrm{Tr}[{\hat{a}}_{{\bf{k}}\lambda }^{\dagger }{\hat{a}}_{{\bf{k}}^{\prime} \lambda ^{\prime} }\hat{\rho }]\end{eqnarray} \tag{ 4 }$$

in which we explicitly separate the quasi-momentum k from other quantum numbers, such as the spin and the energy of the band. Using equation (4), the SBE is cast as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,{\dot{\rho }}_{\lambda \lambda ^{\prime} }({\bf{k}},t) & = & [{E}_{\lambda ^{\prime} }({\bf{k}})-{E}_{\lambda }({\bf{k}})]{\rho }_{\lambda \lambda ^{\prime} }({\bf{k}},t)+{\rm{i}}{\boldsymbol{\varepsilon }}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\rho }_{\lambda \lambda ^{\prime} }({\bf{k}},t)\\ & & +\displaystyle \sum _{s^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\lambda ^{\prime} s^{\prime} }({\bf{k}})\,{\rho }_{\lambda s^{\prime} }({\bf{k}},t)-\displaystyle \sum _{s^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\lambda s^{\prime} }^{* }({\bf{k}})\,{\rho }_{s^{\prime} \lambda ^{\prime} }({\bf{k}},t)\end{array}\end{eqnarray} \tag{ 5 }$$

being ${\rho }_{\lambda \lambda ^{\prime} }({\bf{k}})\equiv {\rho }_{{\bf{k}}\lambda ,{\bf{k}}\lambda ^{\prime} }$. Here, we use the properties of the Bloch waves, $\langle {\bf{r}}| {\bf{k}},s\rangle ={{\rm{e}}}^{{\rm{i}}{\bf{k}}\cdot {\bf{r}}}{u}_{{\bf{k}},s}({\bf{r}})$, to separate the dipole matrix elements using the relation [29]

$$\begin{eqnarray}&&\langle {\bf{k}}^{\prime} ,s^{\prime} | {\bf{r}}| {\bf{k}},s\rangle =-{\rm{i}}\displaystyle \frac{\partial }{\partial {\bf{k}}}[\delta ({\bf{k}}-{\bf{k}}^{\prime} )\,{\delta }_{s^{\prime} s}]+\delta ({\bf{k}}-{\bf{k}}^{\prime} ){{\bf{d}}}_{s^{\prime} s}({\bf{k}})\end{eqnarray} \tag{ 6 }$$

where we define

$$\begin{eqnarray}&&{{\bf{d}}}_{s^{\prime} s}({\bf{k}})={\rm{i}}\,{\int }_{{{\rm{\Omega }}}_{{BZ}}}{\rm{d}}{{\bf{r}}}^{3}{u}_{{\bf{k}},s^{\prime} }^{* }({\bf{r}})\,\displaystyle \frac{\partial }{\partial {\bf{k}}}{u}_{{\bf{k}},s}({\bf{r}}).\end{eqnarray} \tag{ 7 }$$

The first term of the dipole matrix element (6), which emerges from intraband transitions, plays a significant role in the density matrix coherences of the system, especially when it is driven by mid-IR laser pulses as we will show in the following.

In general, equation (5) accounts for all the energy bands of the system, although it is sufficient to truncate the sum to those bands in which the main dynamics is comprised. In simulating the dynamics in an attosecond x-ray transient absorption experiment, the coupling of the valence and conduction bands close to the Fermi level is driven by the IR pulse, while the coupling of core-electron bands and the bands close to the Fermi level is driven by the x-ray pulse. A certain numerical difficulty comes from the fact that both pulses represent two different timescales. However the interaction with the high-frequency nearly-resonant pulse can be described within the rotating-wave approximation, i.e. averaging the fast x-ray carrier oscillations. The electric field is composed by the sum of the IR, ${{\boldsymbol{\varepsilon }}}_{o}(t)$, and the x-ray pulse, ${{\boldsymbol{\varepsilon }}}_{x}(t)$. The x-ray pulse is described as ${{\boldsymbol{\varepsilon }}}_{x}(t)={{\bf{g}}}_{x}(t)\cos {\omega }_{x}t$, where g_x(t) is a slowly-variant envelope function and ω_x is the central frequency of the x rays. Defining the labels i, j, k, .. for energy bands close to the Fermi level and a, b, c, ... for core- and inner-shell bands, we obtain the core-state-resolved Bloch equations as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,{\dot{\rho }}_{{ij}}({\bf{k}},t) & = & \left[{E}_{j}({\bf{k}})-{E}_{{\rm{i}}}({\bf{k}})-{\rm{i}}\displaystyle \frac{{{\rm{\Gamma }}}_{{ij}}}{2}\right]{\rho }_{{ij}}({\bf{k}},t)+{\rm{i}}{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\rho }_{{ij}}({\bf{k}},t)\\ & & +\displaystyle \sum _{k^{\prime} }{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot {{\bf{d}}}_{{jk}^{\prime} }({\bf{k}})\,{\rho }_{{ik}^{\prime} }({\bf{k}},t)-\displaystyle \sum _{k^{\prime} }{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot {{\bf{d}}}_{{ik}^{\prime} }^{* }({\bf{k}})\,{\rho }_{k^{\prime} j}({\bf{k}},t)\\ & & +\displaystyle \sum _{c^{\prime} }\displaystyle \frac{{{\bf{g}}}_{x}(t)}{2}\cdot {{\bf{d}}}_{{jc}^{\prime} }({\bf{k}})\,{\tilde{\rho }}_{{ic}^{\prime} }({\bf{k}},t)-\displaystyle \sum _{c^{\prime} }\displaystyle \frac{{{\bf{g}}}_{x}(t)}{2}\cdot {{\bf{d}}}_{{ic}^{\prime} }^{* }({\bf{k}})\,{\tilde{\rho }}_{c^{\prime} j}({\bf{k}},t),\end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,{\dot{\tilde{\rho }}}_{{ib}}({\bf{k}},t) & = & \left[{E}_{b}({\bf{k}})-{E}_{{\rm{i}}}({\bf{k}})-{\rm{i}}\displaystyle \frac{{{\rm{\Gamma }}}_{{ib}}}{2}+{\omega }_{x}\right]{\tilde{\rho }}_{{ib}}({\bf{k}},t)+{\rm{i}}{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\tilde{\rho }}_{{ib}}({\bf{k}},t)\\ & & +\displaystyle \sum _{k^{\prime} }\displaystyle \frac{{{\bf{g}}}_{x}(t)}{2}\cdot {{\bf{d}}}_{{bk}^{\prime} }({\bf{k}})\,{\rho }_{{ik}^{\prime} }({\bf{k}},t)-\displaystyle \sum _{k^{\prime} }{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot {{\bf{d}}}_{{ik}^{\prime} }^{* }({\bf{k}})\,{\tilde{\rho }}_{k^{\prime} b}({\bf{k}},t)\\ & & +\displaystyle \sum _{c^{\prime} }{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot {{\bf{d}}}_{{bc}^{\prime} }({\bf{k}})\,{\tilde{\rho }}_{{ic}^{\prime} }({\bf{k}},t)-\displaystyle \sum _{c^{\prime} }\displaystyle \frac{{{\bf{g}}}_{x}(t)}{2}\cdot {{\bf{d}}}_{{ic}^{\prime} }^{* }({\bf{k}})\,{\rho }_{c^{\prime} b}({\bf{k}},t),\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,{\dot{\rho }}_{{ab}}({\bf{k}},t) & = & \left[{E}_{b}({\bf{k}})-{E}_{a}({\bf{k}})-{\rm{i}}\displaystyle \frac{{{\rm{\Gamma }}}_{{ab}}}{2}\right]{\rho }_{{ab}}({\bf{k}},t)+{\rm{i}}{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\rho }_{{ab}}({\bf{k}},t)\\ & & +\displaystyle \sum _{k^{\prime} }\displaystyle \frac{{{\bf{g}}}_{x}(t)}{2}\cdot {{\bf{d}}}_{{bk}^{\prime} }({\bf{k}})\,{\tilde{\rho }}_{{ak}^{\prime} }({\bf{k}},t)-\displaystyle \sum _{k^{\prime} }\displaystyle \frac{{{\bf{g}}}_{x}(t)}{2}\cdot {{\bf{d}}}_{{ak}^{\prime} }^{* }({\bf{k}})\,{\tilde{\rho }}_{k^{\prime} b}({\bf{k}},t)\\ & & +\displaystyle \sum _{c^{\prime} }{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot {{\bf{d}}}_{{bc}^{\prime} }({\bf{k}})\,{\rho }_{{ac}^{\prime} }({\bf{k}},t)-\displaystyle \sum _{c^{\prime} }{{\boldsymbol{\varepsilon }}}_{o}(t)\cdot {{\bf{d}}}_{{ac}^{\prime} }^{* }({\bf{k}})\,{\rho }_{c^{\prime} b}({\bf{k}},t),\end{array}\end{eqnarray} \tag{ 10 }$$

where we perform the phase transformation ${\rho }_{{ib}}={{\rm{e}}}^{{\rm{i}}{\omega }_{x}t}{\tilde{\rho }}_{{ib}}$ and introduce the dephasing and relaxation terms Γ. Within the density matrix formalism, scattering effects accounting for electron–electron and electron–phonon interactions are included by dephasing or relaxation terms [16]. In ATA studies, the explored time scales are at most few femtoseconds and most scattering effects do not play a major role in the dynamics. For example, in non-equilibrium photo-carriers excitation studies, electron–electron scattering effects in graphite are found to be around 30 fs [30] and in monolayer MoSe₂ are found to be around hundreds of femtoseconds [31]. On the other hand, relaxation terms arisen from core-hole decays must be included. Core-hole relaxations are in the order of hundreds of attoseconds to few femtoseconds, mainly triggered by Auger or fluorescence transitions.

ATA is based on a time-resolved scheme. A pump IR pulse excites the system inducing the dynamics to be investigated. Then, with a controlled time delay with respect to the pump pulse, a second XUV/x-ray pulse, of only several to hundreds of attoseconds, propagates through the system. By measuring the absorption of the probe pulse with and without the pump pulse, one is able to retrieve information of pump-induced dynamics. It is important to take into account the broad bandwidth of the probe pulse and to model the response of the system at such short timescales for calculating the ultrafast absorption.

In the recent years, the theoretical modeling for calculating ATA spectra has been developed, first in atomic systems [32], and then extended to condensed-matter systems [4]. The ultrafast absorption can be calculated via the so-called response function

$$\begin{eqnarray}&&S(\omega )=2\,\mathrm{Im}[{\boldsymbol{\mu }}(\omega )\cdot {{\bf{E}}}^{* }(\omega )]\end{eqnarray} \tag{ 11 }$$

for positive frequencies ω > 0, where E and ${\boldsymbol{\mu }}$ stands for the Fourier transform of the electric field and dipole response of the system, respectively. The dipole response of the system is defined as

$$\begin{eqnarray}&&{\boldsymbol{\mu }}(t)=q\,\mathrm{Tr}[\displaystyle \sum _{{\bf{k}},{\bf{k}}^{\prime} }\displaystyle \sum _{\lambda ,\lambda ^{\prime} }{{\bf{d}}}_{{\bf{k}}\lambda ,{\bf{k}}^{\prime} \lambda ^{\prime} }\,{\hat{a}}_{{\bf{k}}\lambda }^{\dagger }{\hat{a}}_{{\bf{k}}^{\prime} \lambda ^{\prime} }\,\hat{\rho }]\end{eqnarray} \tag{ 12 }$$

q is the charge of the electron. Since we are interested in the absorption of the XUV/x-ray pulse, the main contribution in the high-frequency region of the dipole response involves transitions between core sates and states near the Fermi level, and is given by

$$\begin{eqnarray}&&{\boldsymbol{\mu }}(t)\approx q\,\displaystyle \sum _{{\bf{k}}}\displaystyle \sum _{{i},b}[{{\bf{d}}}_{{ib}}({\bf{k}})\,{\tilde{\rho }}_{{ib}}({\bf{k}},t)\,{{\rm{e}}}^{{\rm{i}}{\omega }_{x}t}+{\rm{c}}.{\rm{c}}.].\end{eqnarray} \tag{ 13 }$$

By integrating the cBE equations (8)–(10), we compute the response function given by equation (11) at the end of the simulation, and from the response function we calculate the absorption of the system using the relation [32]

$$\begin{eqnarray}&&\sigma (\omega )=\displaystyle \frac{4\pi \alpha \omega \,S(\omega )}{| E(\omega ){| }^{2}},\end{eqnarray} \tag{ 14 }$$

where α is the fine-structure constant.

The derivation of the SBE and cBE theory is based on creation and annihilation operators and is different from other many-body approaches used in condensed-matter systems to calculate the electronic properties. In this section we show the relation of a SBE theory with a TD-CIS, in which a configuration interaction expansion with single excitations is evolved in time by integrating the time-dependent Schrödinger equation.

Several TD-CIS approaches have been developed in these recent years with the aim of describing complex systems interacting with ultrashort light pulses [33–37]. One of the main advantages of these approaches is the possibility to account for electron correlations in a time-dependent framework. Here we re-interpret the SBE equations (5) from a TD-CIS perspective.

In Hartree–Fock theory, the Bloch orbitals with periodic conditions satisfy the hartree–fock operator

$$\begin{eqnarray}&&{h}_{{\rm{HF}}}{{\rm{\Phi }}}_{i{\bf{k}}}({\bf{x}})={\epsilon }_{i}({\bf{k}}){{\rm{\Phi }}}_{i{\bf{k}}}({\bf{x}})\end{eqnarray} \tag{ 15 }$$

in which the wavefunction is given by

$$\begin{eqnarray*}&&{0}_{{\bf{k}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2},\ldots ,\,{{\bf{x}}}_{N})\equiv \hat{A}[{{\rm{\Phi }}}_{i{\bf{k}}}({{\bf{x}}}_{1}){{\rm{\Phi }}}_{j{\bf{k}}}({{\bf{x}}}_{2})...{{\rm{\Phi }}}_{n{\bf{k}}}({{\bf{x}}}_{N})],\end{eqnarray*}$$

where $\hat{A}[]$ stands for the antisymmetry operator. The total electron density of the periodic system in the ground state is then

$$\begin{eqnarray}&&n({\bf{x}})=\displaystyle \sum _{{\bf{k}}}\displaystyle \sum _{i}| {{\rm{\Phi }}}_{i{\bf{k}}}({\bf{x}}){| }^{2}.\end{eqnarray} \tag{ 16 }$$

In order to simplify the TD-CIS interpretation, we assume a physical system at temperature zero and the conduction and valence bands are well separated in energy at the Fermi energy level.

When the system is interacting with an external time-dependent electric field, it is convenient to cast the total Hamiltonian as

$$\begin{eqnarray}&&H(t)={H}_{{\rm{HF}}}+V+{V}_{I}(t)\end{eqnarray} \tag{ 17 }$$

in which we define the Hartree–Fock Hamiltonian as ${H}_{{\rm{HF}}}={\sum }_{j}{h}_{{\rm{HF}}}(j)$, where the sum j runs over all electrons in the occupied orbitals, while V is the difference between the Hamiltonian with no external field and the Hartree–Fock Hamiltonian. Following the recipe of TD-CI approaches, the solution of the evolving system is expanded as

$$\begin{eqnarray}\begin{array}{rcl}| {{\rm{\Psi }}}_{{\bf{k}}}(t)\rangle & = & {b}_{0}({\bf{k}},t)| {0}_{{\bf{k}}}\rangle +\displaystyle \sum _{\alpha }\displaystyle \sum _{i}{b}_{i}^{\alpha }({\bf{k}},t){\hat{c}}_{\alpha }^{\dagger }{\hat{c}}_{i}| {0}_{{\bf{k}}}\rangle \\ & & +\displaystyle \sum _{\alpha \beta }\displaystyle \sum _{{ij}}{b}_{i,j}^{\alpha ,\beta }({\bf{k}},t){\hat{c}}_{\alpha }^{\dagger }{\hat{c}}_{\beta }^{\dagger }{\hat{c}}_{i}{\hat{c}}_{j}| {0}_{{\bf{k}}}\rangle +\,...,\end{array}\end{eqnarray} \tag{ 18 }$$

where ${\hat{c}}^{\dagger }$ and $\hat{c}$ stand for the annihilation and creation operator of orbitals, eigenstates of the Hartree–Fock Hamiltonian. The order of the truncation depends on the problem under study. For example, first-order truncation will be sufficient for investigating laser-induced intraband dynamics of electron-hole pairs, while Pauli blocking effects, from different energy bands, will require a second-order truncation. Here we focus on the wavefunction created by single excitations, that is truncating to first excitations and assuming that second and higher excitations are zero. The ansatz is then reduced to

$$\begin{eqnarray}&&| {{\rm{\Psi }}}_{{\bf{k}}}(t)\rangle ={b}_{0}({\bf{k}},t)| {0}_{{\bf{k}}}\rangle +\displaystyle \sum _{\alpha }\displaystyle \sum _{i}{b}_{i}^{\alpha }({\bf{k}},t){\hat{c}}_{\alpha }^{\dagger }{\hat{c}}_{i}| {0}_{{\bf{k}}}\rangle .\end{eqnarray} \tag{ 19 }$$

According to equation (4), we define the time-dependent TD-CIS coherences as

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{\alpha i}^{(S)}({\bf{k}},t) & \equiv & \langle {{\rm{\Psi }}}_{{\bf{k}}}(t)| {\hat{c}}_{\alpha }^{\dagger }{\hat{c}}_{i}| {{\rm{\Psi }}}_{{\bf{k}}}(t)\rangle ={b}_{i}^{\alpha * }({\bf{k}},t){b}_{0}({\bf{k}},t)\\ {\rho }_{\alpha \alpha ^{\prime} }^{(S)}({\bf{k}},t) & \equiv & \langle {{\rm{\Psi }}}_{{\bf{k}}}(t)| {\hat{c}}_{\alpha }^{\dagger }{\hat{c}}_{\alpha ^{\prime} }| {{\rm{\Psi }}}_{{\bf{k}}}(t)\rangle =\displaystyle \sum _{i}{b}_{i}^{\alpha * }({\bf{k}},t){b}_{i}^{\alpha ^{\prime} }({\bf{k}},t)\\ {\rho }_{{ii}^{\prime} }^{(S)}({\bf{k}},t) & \equiv & \langle {{\rm{\Psi }}}_{{\bf{k}}}(t)| {\hat{c}}_{i}^{\dagger }{\hat{c}}_{i^{\prime} }| {{\rm{\Psi }}}_{{\bf{k}}}(t)\rangle ={\delta }_{{ii}^{\prime} }-\displaystyle \sum _{\alpha }{b}_{i^{\prime} }^{\alpha * }({\bf{k}},t){b}_{i}^{\alpha }({\bf{k}},t).\end{array}\end{eqnarray} \tag{ 20 }$$

Note while the coherence of an electron in the conduction and a hole in the valence is just the multiplication of a single excitation amplitude with the unperturbed amplitude, coherences between single excitations involve a sum among all the holes or particle states. Now we need to find the time evolution of the TD-CI coherences. We insert the ansatz given by equation (19) into the time-dependent Schrödinger equation, and projecting over the different eigenstates of the basis ($| {0}_{{\bf{k}}}\rangle$ and $| \alpha ,i{\rangle }_{{\bf{k}}}={\hat{c}}_{\alpha }^{\dagger }{\hat{c}}_{i}| {0}_{{\bf{k}}}\rangle$), we obtain the equations of motion (EOM) for the amplitudes

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\displaystyle \frac{\partial }{\partial t}{b}_{0}({\bf{k}},t) & = & \left[\displaystyle \sum _{i}{\epsilon }_{i}({\bf{k}})\right]\,{b}_{0}({\bf{k}},t)+{\rm{i}}{\boldsymbol{\varepsilon }}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{b}_{0}({\bf{k}},t)+\displaystyle \sum _{\alpha ^{\prime} i^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{i^{\prime} \alpha ^{\prime} }({\bf{k}})\,{b}_{i^{\prime} }^{\alpha ^{\prime} }({\bf{k}},t)\\ {\rm{i}}\displaystyle \frac{\partial }{\partial t}{b}_{i}^{\alpha }({\bf{k}},t) & = & \left[\displaystyle \sum _{i}{\epsilon }_{i}({\bf{k}})+{\epsilon }_{\alpha }({\bf{k}})-{\epsilon }_{i}({\bf{k}})\right]\,{b}_{i}^{\alpha }({\bf{k}},t)+\,{\rm{i}}{\boldsymbol{\varepsilon }}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{b}_{i}^{\alpha }({\bf{k}},t)+{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha i}({\bf{k}})\,{b}_{0}({\bf{k}},t)\\ & & +\displaystyle \sum _{\alpha ^{\prime} \ne \alpha ,i^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha \alpha ^{\prime} }({\bf{k}})\,{b}_{i^{\prime} }^{\alpha ^{\prime} }({\bf{k}},t){\delta }_{{ii}^{\prime} }-\displaystyle \sum _{\alpha ^{\prime} ,i^{\prime} \ne i}{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{i^{\prime} i}({\bf{k}})\,{b}_{i^{\prime} }^{\alpha ^{\prime} }({\bf{k}},t)\,{\delta }_{\alpha \alpha ^{\prime} },\end{array}\end{eqnarray} \tag{ 21 }$$

where we have defined

$$\begin{eqnarray*}\begin{array}{rcl}\langle \alpha ,i| {\bf{r}}| 0\rangle & \equiv & {{\bf{d}}}_{\alpha i}({\bf{k}})\\ \langle \alpha ,i| {\bf{r}}| \alpha ^{\prime} ,i^{\prime} \rangle & = & \langle \alpha ,i| {\bf{r}}| \alpha ^{\prime} ,i\rangle {\delta }_{{ii}^{\prime} }-\langle \alpha ,i| {\bf{r}}| \alpha ,i^{\prime} \rangle {\delta }_{\alpha \alpha ^{\prime} }\\ & \equiv & {{\bf{d}}}_{\alpha \alpha ^{\prime} }({\bf{k}}){\delta }_{{ii}^{\prime} }-{{\bf{d}}}_{i^{\prime} i}({\bf{k}}){\delta }_{\alpha \alpha ^{\prime} }.\end{array}\end{eqnarray*}$$

In the previous equations we have neglected electron correlations from the V potential of the Hamiltonian. Once we obtain the EOM for the amplitudes, after a cumbersome derivation, the time-dependent equations for the coherences (20) read as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}{\dot{\rho }}_{\alpha i} & = & [{\epsilon }_{i}-{\epsilon }_{\alpha }]\,{\rho }_{\alpha i}+{\rm{i}}{\boldsymbol{\varepsilon }}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\rho }_{\alpha i}+\displaystyle \sum _{\alpha ^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{i\alpha ^{\prime} }\,{\rho }_{\alpha \alpha ^{\prime} }-\displaystyle \sum _{\alpha ^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha \alpha ^{\prime} }^{* }\,{\rho }_{\alpha ^{\prime} i}\\ & & +\displaystyle \sum _{i^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{{ii}^{\prime} }\,{\rho }_{\alpha i^{\prime} }-\displaystyle \sum _{i^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha i^{\prime} }^{* }\,{\rho }_{i^{\prime} i}\\ {\rm{i}}{\dot{\rho }}_{\alpha \alpha ^{\prime} } & = & [{\epsilon }_{\alpha ^{\prime} }-{\epsilon }_{\alpha }]\,{\rho }_{\alpha \alpha ^{\prime} }+{\rm{i}}{\boldsymbol{\varepsilon }}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\rho }_{\alpha \alpha ^{\prime} }+\displaystyle \sum _{i}{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha ^{\prime} i}\,{\rho }_{\alpha i}-\displaystyle \sum _{i}{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha i}^{* }\,{\rho }_{i\alpha ^{\prime} }\\ & & +\displaystyle \sum _{\beta \ne \alpha ^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha ^{\prime} \beta }\,{\rho }_{\alpha \beta }-\displaystyle \sum _{\beta \ne \alpha }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{\alpha \beta }^{* }\,{\rho }_{\beta \alpha ^{\prime} }\\ {\rm{i}}{\dot{\rho }}_{{ii}^{\prime} } & = & [{\epsilon }_{i^{\prime} }-{\epsilon }_{i}]{\rho }_{{ii}^{\prime} }+{\rm{i}}{\boldsymbol{\varepsilon }}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{\rho }_{{ii}^{\prime} }+\displaystyle \sum _{\alpha }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{i^{\prime} \alpha }\,{\rho }_{i\alpha }-\displaystyle \sum _{\alpha }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{i\alpha }^{* }\,{\rho }_{\alpha i^{\prime} }\\ & & +\displaystyle \sum _{j\ne i^{\prime} }{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{i^{\prime} j}\,{\rho }_{{ij}}-\displaystyle \sum _{j\ne i}{\boldsymbol{\varepsilon }}(t)\cdot {{\bf{d}}}_{{ij}}^{* }\,{\rho }_{{ji}^{\prime} }.\end{array}\end{eqnarray} \tag{ 22 }$$

By separating the conduction and the valence bands in the SBE equations (5), we note that they are identical to the dynamical equations (22) found from the TD-CIS amplitudes. In conclusion, there is a correspondence between the SBE theory and the TD-CIS approach, in which the coherences (20) are built by tracing out the electron or hole of the single excitation. A similar correspondence is found by truncating the TD-CI expansion (18) at higher degree of excitations and finding the coherences via equations (20). This correspondence is particularly interesting to visualize the dynamical charge density of the system. In the following we further exploit this property to develop an analytical model to describe the intraband effects on ATA.

By exploiting the correspondence between the SBE and the TD-CIS approach introduced in a previous section, we develop a simple two-band model to describe the main features of the laser-driven intraband dynamics. We restrict our dynamics to the evolving state

$$\begin{eqnarray}&&| \psi (t)\rangle =\displaystyle \sum _{{\bf{k}}}[{a}_{g}({\bf{k}},t)| g,{\bf{k}}\rangle +{a}_{c}({\bf{k}},t)| c,{\bf{k}}\rangle ],\end{eqnarray} \tag{ 23 }$$

where $| g,{\bf{k}}\rangle$ stands for the ground state, while $| c,{\bf{k}}\rangle$ is the excited state formed by the core electron promoted into the conduction band. The amplitudes are found to obey the EOMs

$$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,{\mathop{a}\limits^{}}_{g}({\bf{k}},t) & = & {E}_{g}({\bf{k}}){a}_{g}({\bf{k}},t)+{\rm{i}}{{\boldsymbol{\varepsilon }}}_{0}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{a}_{g}({\bf{k}},t)+{{\boldsymbol{\varepsilon }}}_{x}(t)\cdot {\bf{d}}({\bf{k}})\,{a}_{c}({\bf{k}},t)\\ {\rm{i}}\,{\mathop{a}\limits^{}}_{c}({\bf{k}},t) & = & \left[{E}_{c}({\bf{k}})-{\rm{i}}\displaystyle \frac{{\rm{\Gamma }}}{2}\right]{a}_{c}({\bf{k}},t)+{\rm{i}}{{\boldsymbol{\varepsilon }}}_{0}(t)\cdot \displaystyle \frac{\partial }{\partial {\bf{k}}}{a}_{c}({\bf{k}},t)+{{\boldsymbol{\varepsilon }}}_{x}(t)\cdot {\bf{d}}({\bf{k}})\,{a}_{g}({\bf{k}},t),\end{array}\end{eqnarray} \tag{ 24 }$$

where E_g (k) and E_g (k) are the energies for the ground and excited state, ${{\boldsymbol{\varepsilon }}}_{0}(t)$ and ${{\boldsymbol{\varepsilon }}}_{x}(t)$ the electric field of the laser and x-ray pulse, d (k) the electric dipole transition, and Γ is the core-hole decay. We consider that the attosecond x-ray pulse excites a small fraction of the system in a very short time, i.e. the system is suddenly excited. Within the sudden approximation, the second line of equation (24) gives rise to the solution

$$\begin{eqnarray}&&{a}_{c}({\bf{K}},t)=-{{\rm{ie}}}^{-\displaystyle \frac{{\rm{\Gamma }}}{2}(t-{t}_{0})}{{\rm{e}}}^{-{\rm{i}}{\int }_{{t}_{0}}^{t}{\rm{d}}{t}^{\prime} [{E}_{c}({\bf{K}}+{\bf{A}}(t^{\prime} ))]}{a}_{c}({\bf{K}},{t}_{0}),\end{eqnarray} \tag{ 25 }$$

where we use the Volkov transformation, ${\bf{K}}={\bf{k}}-{\bf{A}}(t)$, as it is usual in strong-field theory [38], where A(t) is the vector potential of the laser pulse and a_c (K, t₀) is the small fraction of excited state produced by the attosecond pulse, considered as an impulse at t₀. Now that we have an analytical expression for the amplitudes as a function of time, we calculate the dipole response of the system (11) that is proportional to the absorption (14). The dipole evolution of the system, in the Volkov picture, is given by

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{\mu }}(t) & = & \displaystyle \sum _{{\bf{K}}}\,[{\bf{d}}({\bf{K}}+{\bf{A}}(t)){a}_{{c}}^{* }({\bf{K}},t){a}_{g}({\bf{K}},t)+{\rm{c}}{\rm{.}}{\rm{c}}.]\\ & \propto & \displaystyle \sum _{{\bf{K}}}\,[b({\bf{K}},t){{\rm{e}}}^{-{\rm{i}}({\epsilon }_{{ch}}({\bf{K}})-{\epsilon }_{{c}}({\bf{K}})-{\rm{i}}{\rm{\Gamma }}/2)(t-{t}_{0})+\phi (t,{t}_{0},{\bf{K}})}+{\rm{c}}{\rm{.}}{\rm{c}}.],\end{array}\end{eqnarray} \tag{ 26 }$$

where the energy difference of the transition is given by the energy difference of the core-hole and the conduction band ${E}_{c}({\bf{K}})-{E}_{g}({\bf{K}})={\epsilon }_{c}({\bf{K}})-{\epsilon }_{{ch}}({\bf{K}})$, b(K, t) is a slow-variant function in time, and the phase $\phi (t,{t}_{0},{\bf{K}})$ is, using that the core-hole energy band has a small dependence with the quasi-momentum k

$$\begin{eqnarray}&&\phi (t,{t}_{0},{\bf{K}})\approx {\rm{i}}{\int }_{{t}_{0}}^{t}{\rm{d}}{t}^{\prime} [({\epsilon }_{c}({\bf{K}}+{\bf{A}}(t^{\prime} ))-{\epsilon }_{c}({\bf{K}}))].\end{eqnarray} \tag{ 27 }$$

If there is not laser pulse, the dipole response (26) has an oscillating term, given by the energy difference of the transition, that is damped by the Γ factor. By the presence of the laser pulse, the oscillating term acquires a phase (27) that depends in a trajectory in the reciprocal space driven along the polarization of the electric field. Depending on the position in the reciprocal space, the acquired phase is different, this gives rise to the complex ATA interference structure shown in figure 3(b). Note that in the limit of a core-hole lifetime much larger than the IR laser pulse, and when attosecond excitations only involve transitions around the gamma point, we will recover the absorption formula found in a two-band parabolic model [39].

We use our model to calculate the absorption of the system by using the following approach: (i) at different positions in the reciprocal space, we consider a trajectory driven by the vector potential of the laser field, (ii) for each trajectory, we calculate the phase (27) that depends on the energy band wandered by the carrier, (iii) we calculate the dipole response (26) by summing all trajectories, taking into account the energy of the transition and the phase at each position of the reciprocal space. By using this semiclassical model, we obtain the ATA spectrum shown in figure 4(a), in very good agreement with figure 3(b). This model also provides a simple interpretation of the main features. At the gamma point k = 0, the electron driven by the laser pulse can only move to higher energies, and the accumulated phase (27) for any trajectory is positive, see figure 4(b). This is quite different for a position at the middle of the band, such as at k = ±π/2a, where the laser-driven electron could go to positive and negative energies, and the accumulated phase could be positive or negative and more sensitive to the initial excitation, i.e. time delay between the laser and the attosecond pulse. Also, a 2ω₀ structure, being ω₀ the frequency of the laser pulse, is clearly observed at the middle of the band, as it is expected by the accumulated phase at symmetric points such as k = ±π/2a. At further points k = ±π/a, the accumulated phase only is negative, opposite to the behavior found at gamma point. In conclusion, the coherent phase accumulated by the carriers driven along the reciprocal space determines the spectral features in the attosecond absorption spectrum. Interestingly, the ATA features arisen from intraband dynamics contain the information of the energy bands. Note that this model assumes that the excitation is shorter than the period of the laser pulse and the decoherence or decay times. If the excitation is comparable to the period of the electric field, the coherent superimposed signal would be blurred in the absorption spectrum. Also, one expects that electron-hole interactions would play a role and modify the spectrum. This shows the potential of this unique approach to access ultrafast electron dynamics and obtain electronic properties of condensed-matter systems.

Zoom In Zoom Out Reset image size

In early experiments of attosecond transient spectroscopy, by studying Fano resonances in atomic systems, it was shown that the ultrashort IR laser has an important effect on the lineshape of the absorption spectrum [28]. In those experiments, an autoionizing state was excited with an XUV attosecond pulse. After excitation, and during the autoionization decay lifetime, a few-cycle IR pulse was sent to the atomic system to induce a dynamical Stark shift. This energy shift results in a phase change in the time-dependent dipole response of the system, giving rise to a change in the absorption lineshape.

Here we show that the intraband dynamics results in Fano lineshapes in the ATA spectrum similar to the effects found in atomic systems.

In the reciprocal space, the absorption at a particular quasi momentum is linked to the transition from the ground state to the core-hole state, i.e. the created pair electron and core hole at a particular k. During the subsequent decay of the electron-core-hole pair, the energy of the exciton is modified by the quiver induced by the laser, introducing then a complex dynamical Stark shift. The excursion of the electron introduces a quantum phase in the probability amplitudes, that is translated to a dephase in the dipole response, see equation (26) and the discussion in the previous section. Depending on the initial position in the reciprocal space, the induced phase shift is different because of the local energy dispersion.

Figure 5 shows, for the same positions shown in figure 4(b), the absorption lineshape changes induced by the laser pulse. When the laser is not present, blue filled lines in 5, we observe the expected Lorentzian lineshape for the three different positions in the reciprocal space. However, the presence of the laser puse, in this case synchronized at time delay zero with the attosecond x-ray pulse, modifies the lineshape and it depends on the reciprocal-space position. At k₁, the main change is due to an energy shift, while for k₂ and k₃ we observe substantial changes in the lineshape profile, similar to the observed laser-induced Fano lineshapes in atomic systems.

Zoom In Zoom Out Reset image size