Abstract

We analyze high-order harmonic generation (HHG) in a disordered semiconductor within the context of the Anderson model of disorder. Employing the theoretical methods pioneered for the study of disordered metals, we show that disorder is a source of ultrafast dephasing of the HHG signal in semiconductors. Furthermore, it is shown that the dephasing effect induced by disorder on HHG spectra depends on both strength and correlation length of the disorder and very weakly on the frequency and intensity of the laser. Our results suggest that HHG has the potential to be a new spectroscopic tool for the analysis of disordered solids.

© 2020 Optical Society of America

1. INTRODUCTION

In a typical solid high-order harmonic generation (HHG) experiment, a wide bandgap semiconductor [15], or insulator [68], is made to interact with an intense laser field (approximately $1\;{\rm TW}/{{\rm cm}^2}$) with mid-infrared wavelength (2–5 µm). Driven by the external field, the crystal emits characteristic electromagnetic spectra containing many well-resolved peaks in correspondence to the harmonics of the laser frequency. It has already be demonstrated that solid HHG has many useful applications, such as the optical reconstruction of the band structure [9,10] or the investigation of ultrafast electronic processes in solids [11]. Furthermore, the potential of HHG for the study of ferromagnetic materials [12], Mott insulators [13], and other exotic materials [14] has been carefully investigated. It is therefore not surprising that solid HHG has attracted much attention in recent years. Nevertheless, as yet, a complete physical understanding of solid HHG has not been obtained.

Actually, it was shown in [4,15] that simulations based on the semiconductor Bloch equations (SBEs) [16] can reproduce the main features of HHG experiments. Nevertheless, to do so, it is required to introduce in the SBE a semi-phenomenological dephasing time of the order of 1 fs. These ultrashort dephasing times are problematic because a physical mechanism that can produce them has not clearly been identified [17]. It has been conclusively proven that Coulomb interactions cannot be responsible for the ultrashort dephasing times in HHG. In fact, simulations based either on the time-dependent density functional approach (TDDFT) [1820] or the Hartree–Fock approximation [21] show that, even in the presence of Coulomb interactions, the simulated HHG spectra do not show clean harmonic peaks resembling those observed in experiments. Furthermore, these simulations have confirmed the validity of the single-particle picture in the analysis of HHG from solids. For this reason, HHG has been almost universally modeled [2231] as the interaction of a set of independent electrons with a spatially uniform and time-dependent electric field and a time-independent and spatially periodic crystalline field.

A few papers have of late analyzed how the lack of perfect translation symmetry in the crystal [3241] may affect the HHG process in solids. These studies show beyond doubt that the HHG process is strongly sensitive to the presence of impurities or dopants in the crystal. In particular, in Refs. [32,33], we have pointed out that disordered crystals can emit HHG spectra that strongly resemble those observed in experiments. In fact, a model disordered crystal possesses HHG radiation composed of well-defined odd harmonics of the fundamental laser frequency. In the past work, we have used only a simple heuristic argument to explain how disorder may affect the HHG process in solids. In this paper, we extend the results of our previous work and, adopting the tools developed for the analysis of disordered metals [4246], show that disorder is a source of ultrafast dephasing for HHG in solids. The calculations reveal that the dephasing effect induced by disorder depends on both strength and correlation length of the disorder and, very weakly, on the intensity and frequency of the laser field. This result suggests that HHG could be used in the future as a source of detailed information about disordered solids.

2. TWO-BAND DISORDERED MODEL

We consider the interaction of a semiconductor with a strong short laser pulse. As reported in the introduction, in the study of solid HHG, it is acceptable to simulate the semiconductor as an ensemble of noninteracting electrons characterized by a one-particle Hamiltonian ${\hat H_0}$ and a one-particle density matrix $\rho$ [18]. To further simplify the problem, we only consider a one-dimensional two-band tight-binding Hamiltonian. The generalization to a more complex situation is straightforward.

The single-particle Hamiltonian of the electron in the solid is ${\hat H_0} = {\hat H_I} + {\hat H_D}$, where ${\hat H_I}$ is the Hamiltonian of an ideal crystal that possesses a perfect crystalline structure, and ${\hat H_D}$ is a small correction that models the effects of impurities and defects in the semiconductor. Specifically, we assume that the ideal crystal Hamiltonian ${\hat H_I}$ has an energy spectrum made of only one valence and one conduction band,

$${ \epsilon _c}(k) = {E_g} - 2\Delta \cos (ka),\quad{ \epsilon _v}(k) = - {E_g} + 2\Delta \cos (ka),$$
with $2{E_g} - 4\Delta$ the bandgap at $k = 0$ between the valence and conduction band, and $4\Delta$ the widths of the bands. The eigenfunctions of ${\hat H_I}$ for the conduction and valence bands are denoted as $|{k_c}\rangle ,|{k_v}\rangle$, respectively. The Hamiltonian ${\hat H_I}$ of a semiconductor with $2N + 1$ lattice sites in the two-band tight-binding approximation can be written as
$$\begin{split}{\hat H_I} & = \sum\limits_{l = - N}^N ({E_g}|{w_\textit{lc}}\rangle \langle {w_\textit{lc}}| - {E_g}|{w_\textit{lv}}\rangle \langle {w_\textit{lv}}| \\ &\quad -\Delta [|{w_{l,c}}\rangle \langle {w_{l + 1,c}}| + |{w_{l + 1,c}}\rangle \langle {w_\textit{lc}}|]) \\ &\quad +\Delta \sum\limits_{l = - N}^N (|{w_\textit{lv}}\rangle \langle {w_{l + 1,v}}| + |{w_{l + 1,v}}\rangle \langle {w_{l,v}}|),\end{split}$$
where $|{w_\textit{lc}}\rangle ,|{w_\textit{lv}}\rangle$ are the Wannier states for the conduction and valence bands,
$$|{w_\textit{lc}}\rangle = \sum\limits_k \frac{{{e^\textit{ikal}}|{k_c}\rangle }}{{\sqrt {2N + 1} }},\quad |{w_\textit{lv}}\rangle = \sum\limits_k \frac{{{e^\textit{ikal}}|{k_v}\rangle }}{{\sqrt {2N + 1} }}.$$
In Eq. (1), periodic boundary conditions are adopted, with the formal identification $|{w_{N + 1,v}}\rangle = |{w_{ - N,v}}\rangle$ and similarly for the conduction band. To facilitate our analysis, we assume that disorder does not affect the valence band and does not induce interband transitions. Furthermore, we assume that the matrix representing the operator ${\hat H_D}$ can have only diagonal matrix elements with respect to the Wannier functions (Anderson model) [47,48],
$$\begin{split}& \langle {w_\textit{nv}}|{\hat H_D}|{w_\textit{mv}}\rangle = \langle {w_\textit{nc}}|{\hat H_D}|{w_\textit{mv}}\rangle = 0, \\& \langle {w_\textit{nc}}|{\hat H_D}|{w_\textit{mc}}\rangle = {\alpha _n}{\delta _\textit{nm}}.\end{split}$$
Since the exact form of the disorder in the semiconductor is not known, we take the coefficients ${\alpha _n}$ as random numbers. The other parameters of the Hamiltonian Eq. (1) are chosen as $\frac{\Delta }{{{E_g}}} = 0.20$, and (in atomic units)${E_g} = 0.10\;{\rm a.u}$. The lattice spacing is set equal to $a = 5\;{\rm a.u}$. With this choice, the bands of the Hamiltonian resemble the valence and conduction bands of ZnO. In addition, we assume that (1) the system is in thermodynamic equilibrium at very low temperature $T$ and (2) its density matrix at $t = 0$ is $\rho (0) = {c_0}\sum\nolimits_{m = - N}^N |{w_\textit{mv}}\rangle \langle {w_\textit{mv}}|$, with ${c_0}$ a normalization constant.

In the presence of a linearly polarized spatially uniform laser field $F(t)$, the crystal Hamiltonian can be written as $\hat H(t) = {\hat H_0} + \textit{eF}(t)x$ or in Wannier representation as

$$\begin{split}\hat H(t) & = {\hat H_0} + \textit{eF}(t)\sum\limits_{l = - N}^N ({d_\textit{cv}}[|{w_\textit{lc}}\rangle \langle {w_\textit{lv}}| + |{w_\textit{lv}}\rangle \langle {w_\textit{lc}}|] \\&\quad +\textit{la}[|{w_\textit{lc}}\rangle \langle {w_\textit{lc}}| + |{w_\textit{lv}}\rangle \langle {w_\textit{lv}}|]).\end{split}$$
The matrix elements of the position operator are calculated adopting the Blount form of this operator [49] and assuming that the interband matrix elements ${d_\textit{cv}} = \langle {k_c}|x|{k_v}\rangle$ do not depend on the quasi-momentum $k$. The numerical value of ${d_\textit{cv}}$ is taken to be 4.0 a.u. Furthermore, in what follows, we formally consider $N$ as going to $\infty$.

The density matrix of the solid satisfies the Liouville equation, $i\hbar \frac{{d\rho (t)}}{{dt}} = [\hat H(t),\rho (t)]$, with the formal solution, $\rho (t) = \hat U\rho (0){\hat U^\dagger }$, and the evolution operator $\hat U$ solves the Schrödinger equation $i\hbar \frac{{d\hat U}}{{dt}} = \hat H(t)\hat U$. For $t \gt 0$, the density matrix of the solid has the form $\rho (t) = \sum\nolimits_{m = - N}^N {c_0}|m(t)\rangle \langle m(t)|$, where the vector $|m(t)\rangle$ is the solution of the one-particle Schrödinger equation,

$$i\hbar \frac{{d|m(t)\rangle }}{{dt}} = ({\hat H_0} + \textit{exF}(t))|m(t)\rangle ,$$
with initial condition, $|m(t = 0)\rangle = |{w_\textit{mv}}\rangle$. We can expand $|m(t)\rangle$ as $|m(t)\rangle = \sum\nolimits_{n = - N}^N [b_c^m(n,t)|{w_\textit{nc}}\rangle + b_v^m(n,t)|{w_\textit{nv}}\rangle ]$, where the time-dependent coefficients $b_c^m(n,t),b_v^m(n,t)$ are solutions of the coupled differential equations,
$$\begin{split} \dot b_c^m(n,t)& = \frac{{{\eta _\textit{nc}}(t)}}{{i\hbar }}b_c^m(n,t) - \frac{\Delta }{{i\hbar }}[b_c^m(n + 1,t) \\&\quad + b_v^m(n - 1,t)] + \frac{{e{d_\textit{cv}}F(t)}}{{i\hbar }}b_v^m(n,t),\end{split}$$
$$\begin{split} \dot b_v^m(n,t) &= \frac{{{\eta _\textit{nv}}(t)}}{{i\hbar }}b_v^m(n,t) + \frac{\Delta }{{i\hbar }}[b_v^m(n + 1,t) \\ &\quad +b_v^m(n - 1,t)] + \frac{{e{d_\textit{cv}}F(t)}}{{i\hbar }}b_c^m(n,t),\end{split}$$
and
$$\begin{split}{\eta _\textit{nc}}(t)& = {E_g} + {\alpha _n} + \textit{enaF}(t), \\ {\eta _\textit{nv}}(t)& = - {E_g} + \textit{enaF}(t),\end{split}$$
with the initial conditions $b_v^m(n,0) = {\delta _{n,m}}$ and $b_c^m(n,0) = 0$, for each $n = - N, \ldots ,N$.

Once the one-particle density matrix is known, it is possible to calculate any one-particle observable. In particular, we can write the current as

$$\begin{split} J(t) & = tr[\rho \hat p] = \sum\limits_{m = - N}^N {c_0}tr[|m(t)\rangle \langle m(t)|\hat p] \\ &= \sum\limits_{m = - N}^N {c_0}\langle m(t)|\hat p|m(t)\rangle ,\end{split}$$
where $\hat p$ is the ordinary momentum operator of the electron. The harmonic spectrum is proportional to the Fourier transform of the current $S(\omega ) = |J(\omega )|^2 = |\!\sum\nolimits_{m = - N}^N {c_0}{p_m}(\omega )|^2$, where ${p_m}(\omega ) = \int {e^{i\omega t}}\langle m(t)|\hat p|m(t)\rangle {\rm d}t$.

In absence of disorder, all the matrix elements $\langle m(t)|\hat p|m(t)\rangle$ are identical. When disorder is present, the $\langle m(t)|\hat p|m(t)\rangle$ are in general different, and a sum over all the possible initial conditions in Eq. (5) is needed to evaluate the current. For each different initial condition, the ket $|m(t)\rangle$ describes an electron that, at $t = 0$, is localized on the lattice site m. This particle evolves under the action of the laser field, the crystalline field, and a given disorder potential (characterized by a given realization of the ${\alpha _n}$, for $n = - N, \ldots ,N$). We note that summing over the $2 N+ {1}$ initial conditions for a given configuration of disorder (a single realization of the ${\alpha _n}$ for $n = - N, \ldots ,N$ and $2 N+ {1}$ different initial conditions $|m(t = 0)\rangle = |{w_\textit{mv}}\rangle$, $m = - N \ldots N$) is equivalent to summing over $2 N+ {1}$ different disorder configurations for a single initial condition (a fixed m, for example $m = 0$, and $2 N+ {1}$ disorder configurations characterized by $2 N+ {1}$ different sets of $\alpha _n^m$ that satisfy the condition $\alpha _n^m = {\alpha _{n - m}}$ for $m = - N\ldots N$). Since the number of sites $2 N+ {1}$ is a macroscopic number in a crystal, we make the assumption that the sum over $2 N+ {1}$ initial conditions is equivalent to that over all the possible configurations of disorder.

3. KELDYSH APPROXIMATION AND LONG-WAVELENGTH LIMIT

In Keldysh approximation [50] and letting the number of sites $2N + 1$ in the crystal go to infinity, we can write the equation of motion [Eq. (3)] of the electron in the two-band model in quasi-momentum space as

$$\begin{split} i\hbar {\dot b_c}(p,t) & = a\!\int\! \alpha (p - p^\prime ){b_c}(p^\prime ,t){\rm d}p^\prime -\left( {i\frac{{\partial {b_c}}}{{\partial p}} - {d_\textit{cv}}{b_v}} \right)\\&\quad \times \textit{eF}(t) - 2\Delta \cos (ap){b_c}(p,t),\end{split}$$
$$i\hbar {\dot b_v}(p,t)\def\LDeqtab{} = ( - 2{E_g} + 2\Delta \cos (ap)){b_v} - i\frac{{\partial {b_v}}}{{\partial\! p}}\textit{eF}(t),$$
with $|p| \le \frac{\pi }{a}$. In the above equations, ${b_c}(p,t) = \sum\nolimits_{n = - \infty }^\infty {e^\textit{ipna}}{b_c}(n,t)$ and similarly for ${b_v}(p,t)$. To simplify the notation, we have set ${b_v}(n,t) = b_v^{m = 0}(n,t)$ since from now on we only consider a particle that at the initial time $t = 0$ is located on the lattice position $m = 0$ ($b_v^m(n,t = 0) = \delta (n,0)$). The function $\alpha (p - p^\prime ) = \sum\nolimits_{n = - \infty }^\infty {e^{i(p - p^\prime )na}}{\alpha _n}$ is the discrete Fourier transform of the disorder coefficients ${\alpha _n}$. The Keldysh approximation implies that the depletion of the valence band is entirely negligible. This simplification has been found to be acceptable for the relatively weak fields employed in solid HHG experiments [15]. In general, we are unable to solve Eq. (6), even in Keldysh approximation and for a simple lattice, when disorder is present. Nevertheless, we can gain at least a qualitative understanding of the dynamics of the field-driven crystal electrons if we employ the long-wavelength approximation that has been successfully employed in the past to study the dynamics of a disordered semiconductor in photon-echo experiments [51].

The long-wavelength approximation is based on the following observation. Since the lasers employed in typical solid HHG experiments have only moderate intensities (${\simeq} 1\;{\rm TW/cm}^2$ [1]), only electrons with small quasi-momentum can tunnel from the valence to the conduction band, thus contributing to the HHG process [22]. For this reason, we can approximate the $\cos$ functions in Eq. (6) as quadratic polynomials ($\cos (ap) \simeq 1 - {a^2}{p^2}/2$) and formally consider the quasi-momentum $p$ as defined on the whole real axis ($p \in ( - \infty , + \infty )$). We then define two continuous functions ${\phi _c}(x,t),{\phi _v}(x,t)$ that satisfy the Scrhödinger-like equations of motion,

$$\begin{split} i\hbar {\dot \phi _c}(x,t) & = (\textit{exF}(t) + D(x)){\phi _c}(x,t)\\& \quad -\frac{{{\hbar ^2}}}{{2{m_c}}}\partial _x^2{\phi _c}(x,t) + {\textit{ed}_\textit{cv}}{\phi _v}(x,t)F(t),\end{split}$$
$$i\hbar {\dot \phi _v}(x,t) \def\LDeqtab{}= ( - { \epsilon _g} + exF(t)){\phi _v}(x,t) - \frac{{{\hbar ^2}}}{{2{m_v}}}\partial _x^2{\phi _v},$$
where ${ \epsilon _g} = 2({E_g} - 2\Delta )$, ${m_c} = - {m_v} = {\hbar ^2}/(2\Delta {a^2})$, and the function $D(x)$ satisfy the condition $D(x = ma) = {\alpha _m}$ for all integers m. The function $D(x)$ is otherwise arbitrary, but we always assume that it is a slowly varying function over a distance comparable to the lattice spacing a. This is equivalent to the assumption that two generic random variables ${\alpha _n},{\alpha _{n + m}}$ are correlated over distance largely with respect to the lattice spacing ($m \gg 1$). Using a simple discretization of the operator $\frac{{{d^2}}}{{d{x^2}}}$, it appears that Eq. (7) is the long-wavelength limit of Eq. (3). Since the disorder potential $D(x)$ is assumed to be slowly varying over distances comparable to the lattice spacing, it is a reasonable approximation to set ${b_\textit{nc}} = {\phi _c}(x = na),{b_\textit{nv}} = {\phi _v}(x = na)$.

As explained earlier, to simulate the effect of disorder in HHG experiments, the equations of motion Eq. (7) are solved for the initial same conditions ${\phi _c}(x,t = 0) = 0,{\phi _v}(x,t = 0) = \delta (x)$ and different configurations of disorder [different realization of the function D(x)]. Within the long-wavelength approximation, we then approximate the interband dipole moment for a single realization of disorder $d(t) = {d_\textit{cv}}\sum\nolimits_n (b_\textit{nc}^*(t){b_\textit{nv}}(t) + {\rm c.c})$ as $d(t) = ({d_\textit{cv}}/a)\int {\rm d}x(\phi _c^*(x,t){\phi _v}(x,t) + {\rm c.c})$. The total dipole moment of the system is the sum over all the configurations of disorder that we write formally as $\bar d(t) = ({d_\textit{cv}}/a)$$\int {\rm d}x(\bar \phi _c^*(x,t){\phi _v}(x,t) + {\rm c.c})$. ${\bar \phi _c}(x,t)$ is the average of the function ${\phi _c}(x,t)$ over the different configurations of disorder.

The equation for ${\phi _c}$ can be solved using the Green function approach [47,52],

$$\begin{split} & {\phi _c}(x,t) = e{d_\textit{cv}}\int_{ - \infty }^\infty \int_{ - \infty }^\infty \\ &{G_c}(x,x^\prime ,t,t^\prime ){\phi _v}(x^\prime ,t^\prime )F(t^\prime ){\rm d}x^\prime {\rm d}t^\prime ,\end{split}$$
where the Green function is the solution of the equation
$$\begin{split}&i\hbar {\dot G_c} - (exF(t) + D(x)){G_c} + \frac{{{\hbar ^2}}}{{2{m_c}}}\partial _x^2{G_c}\\&\quad = \delta (x - x^\prime )\delta (t - t^\prime ).\end{split}$$
According to Feynman, ${G_c}(x,x^\prime ,t,t^\prime )$ can be written as a path integral over all the continuous paths $x(\tau )$ [42,52] satisfying the boundary conditions $x(t^\prime ) = x^\prime ,x(t) = x$ (for short, we dub $[x]_{tt^\prime }^{xx^\prime }$ a generic trajectory satisfying these boundary conditions), ${G_c}(x,x^\prime ,t,t^\prime ) = \Theta (t - t^\prime )\int_{x(t^\prime ) = x^\prime }^{x(t) = x} [{\rm d}x]{e^{\frac{i}{\hbar }S( {[x]_{tt^\prime }^{xx^\prime }} )}}$, where $S( {[x]_{tt^\prime }^{xx^\prime }} ) = \int_{t^\prime }^t [\frac{{{{\dot x}^2}(\tau )}}{{2{m_c}}} - ex(\tau )F(\tau ) - D(x(\tau ))]{\rm d}\tau$ is the action for the trajectory $[x]_{tt^\prime }^{xx^\prime }$ and $\Theta$ the step function.

In order to describe the role of disorder in HHG, we need to find the average propagator ${\bar G_c}$, which is obtained by integrating over all possible configurations of disorder [4247] ${\bar G_c}(x,x^\prime ,t,t^\prime ) \;\;=\;\; \Theta (t - t^\prime )\int [dD]\int_{x(t^\prime ) = x^\prime }^{x(t) = x} [{\rm d}x]{e^{\frac{i}{\hbar }S\big( {[x]_{tt^\prime }^{xx^\prime }} \big)}}$. The functional integration of the disorder configuration is represented in the formula by the formal expression $\int [{\rm d}D]$. In particular, we consider a disorder potential of the form $D(x) = {W_0}\sum\nolimits_{i = 1}^{{N_D}} v(x - {r_i})$, where $v(x)$ is a dimensionless function, ${W_0}$ characterizes the strength of the disorder, and ${r_i}$ are random variables. This is the model of disorder introduced by Edwards and Gulyaev [45,47]. The function D describes the potential generated by ${N_D}$ impurities randomly positioned in space. The random variables ${r_i}$ are assumed to be described by a uniform probability distribution. It is convenient to choose $v(x) = {e^{ - {x^2}/{L^2}}}$. With the above choices, the functional integral over disorder configurations, equivalent to an integral over the variables ${r_i}$, can be exactly calculated, and the result is for weak and dense disorder (small ${W_0}$, large ${N_D}$) [4246] (and from now always for $t \gt t^\prime $),

$$\begin{split}& {\bar G_c}(x,x^\prime ,t,t^\prime ) = \int_{x(t^\prime ) = x^\prime }^{x(t) = x} [{\rm d}x]{e^{\frac{i}{\hbar }{S_c}\left( {[x]_{tt^\prime }^{xx^\prime }} \right)}} \\& {e^{ - \frac{{\rho W_0^2}}{{2{\hbar ^2}}}\int_{t^\prime }^t {\rm d}s^\prime \int_{t^\prime }^t {\rm d}s\gamma (x(s^\prime ) - x(s))}},\end{split}$$
where ${S_c} = \int_{t^\prime }^t [\frac{{{{\dot x}^2}(\tau )}}{{2{m_c}}} - ex(\tau )F(\tau )]{\rm d}\tau$ is the action for a particle in a uniform electric field, $\gamma (x,x^\prime ) = \int v(q - x)v(q - x^\prime ){\rm d}q$ is the spatial correlation of the disorder potential $v(x)$, and $\rho = \mathop {\lim }\nolimits_{V \to \infty } {N_D}/V$ (with V the volume of the crystal) is the density of the impurities present in the crystal.

The average propagator can be rewritten as

$${\bar G_c}(x,x^\prime ,t,t^\prime ) = {G_c}(x,x^\prime ,t,t^\prime ) \langle {e^{\Phi (x,x^\prime ,t,t^\prime )}} \rangle ,$$
where ${G_c}(x,x^\prime ,t,t^\prime ) = \int_{x(t^\prime ) = x^\prime }^{x(t) = x} [{\rm d}x]{e^{i{S_c}([x]_{tt^\prime }^{xx^\prime })/\hbar }}$ is the Green function for the free particle in the laser field and ${ \langle }{e^{\Phi (x,x^\prime ,t,t^\prime )}} \rangle $ is defined as
$$\begin{split}& { \langle }{e^{\Phi (x,x^\prime ,t,t^\prime )}} \rangle = \int_{x(t^\prime ) = x^\prime }^{x(t) = x} [{\rm d}x]\frac{{{e^{\frac{i}{h}{S_c}([x]_{tt^\prime }^{xx^\prime })}}}}{{{G_c}(x,x^\prime ,t,t^\prime )}} \\& {e^{ - \frac{{\rho W_0^2}}{{2{\hbar ^2}}}\int_{t^\prime }^t {\rm d}s^\prime \int_{t^\prime }^t {\rm d}s\gamma (x(s^\prime ) - x(s))}}.\end{split}$$

The functional integral above cannot usually be performed in closed form, and it is usually estimated in the so-called first cumulant approximation [42,5254] ${ \langle }{e^{\Phi (x,x^\prime ,t,t^\prime )}} \rangle \simeq {e^{\bar \Phi (x,x^\prime ,t,t^\prime )}}$, where

$$\begin{split}\frac{{\bar \Phi (x,x^\prime ,t,t^\prime )}}{{{2^{ - 1}}{\hbar ^{ - 2}}\rho\! W_0^2}} & = - \int_{t^\prime }^t {\rm d}s^\prime \int_{t^\prime }^t {\rm d}s\int_{x^\prime }^x [{\rm d}x(\tau )] \\& \frac{{{e^{\frac{i}{\hbar }{S_c}\left( {[x]_{tt^\prime }^{xx^\prime }} \right)}}\gamma (x(s^\prime ) - x(s))}}{{{G_c}(x,x^\prime ,t,t^\prime )}}.\end{split}$$
After a series of algebraic manipulations [5254], detailed in Appendix A, we find the following integral expression for $\bar \Phi$:
$$\begin{split}\bar \Phi (x,x^\prime ,t,t^\prime ) &= - \sqrt {\frac{\pi } 2 } \frac{{\rho\! W_0^2{L^2}}}{{{\hbar ^2}}} \\& \int_{t^\prime }^t {\rm d}s\int_{t^\prime }^t {\rm d}s^\prime \frac{{{e^{ - A(x,x^\prime ,s,s^\prime ,t,t^\prime )}}}}{{{B^{\frac 1 2 }}(s,s^\prime ,t,t^\prime )}},\end{split}$$
with
$$\begin{split} & A(x,x^\prime ,s,s^\prime ,t,t^\prime ) = (s - s^\prime {)^2} \\& \frac{{{{[\frac{{x - x^\prime }}{{t - t^\prime }} + \frac{{\lambda (s) - \lambda (s^\prime )}}{{s - s^\prime }} - \frac{{\lambda (t) - \lambda (t^\prime )}}{{t - t^\prime }}]}^2}}}{{4B(s,s^\prime ,t,t^\prime )}}, \\& B(s,s^\prime ,t,t^\prime ) = \frac 1 2 \left[{L^2} + \frac{{i\hbar }}{{{m_c}}} \right.\\& \left.\left((s - s^\prime )\Theta (s - s^\prime ) + (s^\prime - s)\Theta (s^\prime - s) - \frac{{{{(s - s^\prime )}^2}}}{{t - t^\prime }}\right)\right],\end{split}$$
where $\ddot \lambda (t) = \textit{eF}(t)$ and $\Theta$ is the step function ($\frac{{d\Theta (x)}}{{dx}} = \delta (x)$). In general, it is not possible to find the exact value of Eq. (10). Nevertheless, the integral can be exactly evaluated for $x = x^\prime $ and in the weak field limit ($F \to 0$) [53,55,56], resulting in the closed form expression
$$\begin{split}\bar \Phi (x = x^\prime ,t,t^\prime ) & = - \frac{{\rho W_0^2{L^2}}}{{{{(\pi /2)}^{ - 1/2}}{\hbar ^2}}} \\& \frac{{{{(t - t^\prime )}^2}}}{{\sqrt { - i\frac{{\hbar (t - t^\prime )}}{{2{m_c}}}} }}\log \left( {\frac{{1 + \sqrt { - i\frac{{\hbar (t - t^\prime )}}{{4{m_c}{L^2}}}} }}{{1 - \sqrt { - i\frac{{\hbar (t - t^\prime )}}{{4{m_c}{L^2}}}} }}} \right).\end{split}$$
The above calculations can be extended to a three-dimensional system with the final result [53,55,56],
$$\begin{split}& \bar \Phi (x = x^\prime ,t,t^\prime ) = - \frac{{2\rho W_0^2{L^5}}}{{{{(\pi )}^{ - 3/2}}{\hbar ^2}}} \\& \frac{{4{L^2}{{(t - t^\prime )}^2} - i(\hbar /{m_c})(t - t^\prime {)^3}}}{{\left( {\frac{{{\hbar ^2}{{(t - t^\prime )}^2}}}{{m_c^2}} + 16{L^4}} \right)}}.\end{split}$$
Since we have now an explicit formula for the average propagator, albeit only approximate, we can calculate the expected value of any one-particle observable.

A. Dipole Moment

The interband part of the dipole moment $\bar d(t)$ averaged over the disorder configurations is written in term of the Green functions as

$$\begin{split}\frac{{\bar d(t)}}{{e{{\rm d}_\textit{cv}}}} & = \int dx\frac{{{{\bar \phi }_c}(x,t)\phi _v^*(x,t)}}{{{\textit{ed}_\textit{cv}}}} + {\rm c.c.} \\& = \int {\rm d}x\int {\rm d}{x_1}\int_0^t {\rm d}t^\prime F(t^\prime ) \\ & {\bar G_c}(x,{x_1},t,t^\prime ){G_v}({x_1},0,t^\prime ,0)G_v^*(x,0,t,0) + {\rm c.c.},\end{split}$$
using Eq. (8) for ${\bar \phi _c}$ and considering that ${G_v}$ is the solution of Eq. (7b) with initial condition ${\phi _v}(x,0) = \delta (x)$. As depicted in Fig. 1(a), we can formally read the above expression in terms of the following space-time events or Feynman trajectories: (1) at time $\tau = 0$, at the position $x = 0$, an electron ${e_v}$ is created in the valence band and then propagated up to the position ${x_1}$ at time $\tau = t^\prime $ (${G_v}({x_1},0,t^\prime ,0)$); (2) at time $\tau = t^\prime $, at ${x_1}$, the valence-band electron ${e_v}$ is transformed into a conduction-band electron ${e_c}$ and then propagated up to the position $x$ at time $\tau = t$ (${\bar G_c}(x,{x_1},t,t^\prime )$); (3) one valence-band hole ${\bar e_v}$ is created at $t = 0$ at $x = 0$ and propagated up to the position $x$ at time $\tau = t$ ($G_v^*(x,0,t,0)$), where it encounters the conduction-band electron and annihilates it. The valence-band electron ${e_v}$ has charge $e$ and mass ${m_v}$, the conduction-band electron ${e_c}$ has charge $e$ and mass ${m_c} = - {m_v}$, and the valence-band hole ${\bar e_v}$ has charge ${-}e$ and mass ${-}{m_v}$.
 figure: Fig. 1.

Fig. 1. (a) Schematic pictorial description of the generic Feynman trajectories contributing to the dipole moment. (b) Generic Feynman trajectories satisfying the stationary-phase conditions Eq. (15). The curves are not solutions of the Newton equations of motion but only a pictorial description of the Green functions ${\bar G_c}$ (blue asterisks), ${G_v}$ (red circles), $G_v^*$ (solid black line). The vertical dotted line represents the time instant $t^\prime $ when the conduction-band electron is created.

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The Fourier spectrum of the HHG radiation $|P(\omega )|^2$ is related to Fourier transform of $\bar d(t)$ via the relation $P(\omega ) = \int {e^{i\omega t}}\dot {\bar d}(t){\rm d}t$. Using the expression in Eq. (13) for $\bar d(t)$, the Pauli–Van Vleck form of the propagators [42] (${\bar G_c} = A{e^{\frac{i}{\hbar }(S_c^\textit{cl} - i\sigma )}},{G_v} = A{e^{\frac{i}{\hbar }(S_v^\textit{cl} + { \epsilon _G}t)}}$) and adopting the stationary-phase approximation in calculating the integrals in Eq. (13), we find that the Feynman trajectories that significantly contribute to the HHG process must satisfy the conditions [15,5762]

$$\frac{{\partial S_c^\textit{cl}}}{{\partial t}}([x]_{tt^\prime }^{xx^\prime }) + \frac{{\partial {\sigma _I}}}{{\partial t}} - { \epsilon _G} - \hbar \omega = \frac{{\partial S_v^\textit{cl}}}{{\partial t}}([x]_{t0}^{x0}),$$
$$\frac{{\partial S_c^\textit{cl}}}{{\partial t^\prime }}([x]_{tt^\prime }^{xx^\prime }) + \frac{{\partial {\sigma _I}}}{{\partial t^\prime }} + { \epsilon _G} = - \frac{{\partial S_v^\textit{cl}}}{{\partial t^\prime }}([x]_{t^\prime 0}^{x^\prime 0}),$$
$$\frac{{\partial S_c^\textit{cl}}}{{\partial x}}([x]_{tt^\prime }^{xx^\prime }) + \frac{{\partial {\sigma _I}}}{{\partial x}} = \frac{{\partial S_v^\textit{cl}}}{{\partial x}}([x]_{t0}^{x0}),$$
$$\frac{{\partial S_c^\textit{cl}}}{{\partial x^\prime }}([x]_{tt^\prime }^{xx^\prime }) + \frac{{\partial {\sigma _I}}}{{\partial x^\prime }} = - \frac{{\partial S_v^\textit{cl}}}{{\partial x^\prime }}([x]_{t^\prime 0}^{x^\prime 0}),$$
where we have set ${\hbar ^{ - 1}}{\sigma _R} = \text{Re}(\bar \Phi )$ the real part of the average phase and ${\hbar ^{ - 1}}{\sigma _I} = \text{Im}(\bar \Phi )$ the imaginary one and $\sigma = {\sigma _R} + i{\sigma _I}$. These stationary-phase conditions resemble those for a perfect crystal but with an explicit dependence on the parameters of disorder through the function ${\sigma _I}$. The solutions of Eq. (14) will therefore in general depend on $\rho ,L,{W_0}$.

Since $\bar \Phi$ depends only on the difference $x - x^\prime $, we can combine the last two conditions into a single relation,

$$\begin{split}& \frac{{\partial S_v^\textit{cl}}}{{\partial x}}([x]_{t0}^{x0}) - \frac{{\partial S_c^\textit{cl}}}{{\partial x}}([x]_{tt^\prime }^{xx^\prime }) - \frac{{\partial S_c^\textit{cl}}}{{\partial x^\prime }}([x]_{tt^\prime }^{xx^\prime }) \\ & = \frac{{\partial S_v^\textit{cl}}}{{\partial x^\prime }}([x]_{t^\prime 0}^{x^\prime 0}).\end{split}$$
It is useful to recall that the valence-band electron is a particle with mass ${m_v}$ and charge $e$, and the conduction-band electron is a particle with mass ${m_c} = - {m_v}$ and charge $e$. We note that the right-hand side of Eq. (15) is the momentum of a valence-band electron moving along the trajectory $[x]_{t^\prime 0}^{x^\prime 0}$ at $\tau = t^\prime $ [63], ${p^v}([x]_{t^\prime 0}^{x^\prime 0},\tau = t^\prime ) = \frac{{\partial S_v^\textit{cl}}}{{\partial x^\prime }}([x]_{t^\prime 0}^{x^\prime 0})$. Whereas, the left-hand side of Eq. (15) is identified as the momentum of a valence-band electron for the trajectory $[x]_{t0}^{x0}$ at time $\tau = t^\prime $. In fact, ${p^v}([x]_{t0}^{x0},\tau = t) = \frac{{\partial S_v^\textit{cl}}}{{\partial x}}([x]_{t0}^{x0})$ is the momentum of a valence-band electron, at time t, moving along the trajectory $[x]_{t0}^{x0}$, and $\Delta\! P = \frac{{\partial S_c^\textit{cl}}}{{\partial x}}([x]_{tt^\prime }^{xx^\prime }) + \frac{{\partial S_c^\textit{cl}}}{{\partial x^\prime }}([x]_{tt^\prime }^{xx^\prime })$ is the momentum difference accumulated by a conduction-band electron moving in the uniform electric field in the time interval $[t^\prime ,t]$. Noting that $\Delta P$ depends only on the time interval $(t - t^\prime )$ and is the same for all particles with the same electric charge, the difference ${p^v}([x]_{t0}^{x0},\tau = t) - \Delta P$ is therefore the momentum of the valence-band electron at time $t^\prime \;{p^v}([x]_{t0}^{x0},\tau = t^\prime )$. As a result, Eq. (15) implies that $[x]_{t0}^{x0}(\tau = t^\prime ) = x^\prime $. In fact, as shown in Fig. 1(b), the classical trajectories $[x]_{t0}^{x0}$, $[x]_{t^\prime 0}^{x^\prime 0}$ must be identical in the time interval $0 \le \tau \le t^\prime $ because they have the same position at time $\tau = 0$ and the same velocity at time $\tau = t^\prime $.

In terms of the trajectories for the valence-band electron ${e_v}$, valence-band hole ${\bar e_v}$, and conduction-band electron ${e_c}$ introduced in the discussion following Eq. (13), Eq. (15) stipulates that the Feynman trajectories contributing to HHG are those describing a valence-band electron ${e_v}$ and a valence-band hole ${\bar e_v}$ propagating along the same trajectory from the initial position $x = 0$ at time $t = 0$ to the position $x = x^\prime $ at $t = t^\prime $, where the valence-band electron is destroyed and a conduction-band electron ${e_c}$ is created. Later on, in the time interval $(t^\prime ,t)$, ${e_c}$ and ${\bar e_v}$ propagate along their respective trajectories until they meet each other again at the position $x$ at time $t$. Since in the time interval $(0,t^\prime )$ the valence-band electron and valence-band hole move along the same path, the Feynman trajectories relevant for HHG are those describing an electron–hole pair created at time $t^\prime $ at $x^\prime $ and recolliding at time $t$ at $x$.

The first two conditions in Eq. (14) enforce energy conservation at the moment of the creation $t^\prime $ and destruction $t$ of the particle–antiparticle system. In particular, the sum of Eqs. (14b) and (14a) tells us that to emit a photon of energy $\hbar \omega$, the electron–hole pair must acquire from the external field, on average, a kinetic energy equal to $\hbar \omega - (\frac{{\partial {\sigma _I}}}{{\partial t}} + \frac{{\partial {\sigma _I}}}{{\partial t^\prime }})$. The term $\frac{{\partial {\sigma _I}}}{{\partial t}} + \frac{{\partial {\sigma _I}}}{{\partial t^\prime }}$ accounts for the fact that the electron, on average, will dissipate energy in inelastic collisions while traveling in the disordered medium. Equation (14b) mandates that at $t^\prime $ energy of the valence-band electron [63], ${E_v}(t^\prime ) = - \frac{{\partial S_v^\textit{cl}}}{{\partial t^\prime }}([x]_{t^\prime 0}^{x^\prime 0})$ differs from the energy of the electron ${E_c}(t^\prime ) = \frac{{\partial S_c^\textit{cl}}}{{\partial t^\prime }}([x]_{tt^\prime }^{xx^\prime })$ by ${\tilde \epsilon _G} = { \epsilon _G} - \frac{{\partial {\sigma _I}}}{{\partial t^\prime }}$ or, more explicitly, $\Delta E(t^\prime ) = \frac{{\partial S_v^\textit{cl}}}{{\partial t^\prime }}([x]_{t^\prime 0}^{x^\prime 0}) + \frac{{\partial S_c^\textit{cl}}}{{\partial t^\prime }}([x]_{tt^\prime }^{xx^\prime }) = - {\tilde \epsilon _G}$. We observe that for the trajectories contributing to HHG, those satisfying Eq. (15), the energy of the valence-band hole at ${t_1}$ is the opposite of that of the valence-band electron ${E_v}(t^\prime )$. The quantity $\Delta E(t^\prime )$ can therefore be considered as the total energy of the electron–hole pair at time ${t_1}$. We can write the generic electron–hole trajectories ${x_{{e_c}}},{x_{{{\bar e}_v}}}$ for the pair created at $x^\prime $ at $t^\prime $ and recolliding at $x$ at $t$ as

$${x_{{e_c}}}(\tau ) = x^\prime + \frac{{\pi + P}}{{{m_c}}}(\tau - t^\prime ) + \frac{{\lambda (\tau ) - \lambda (t^\prime )}}{{{m_c}}},$$
$${x_{{{\bar e}_v}}}(\tau ) = x^\prime - \frac{{\pi - P}}{{{m_c}}}(\tau - t^\prime ) - \frac{{\lambda (\tau ) - \lambda (t^\prime )}}{{{m_c}}},$$
with $\ddot \lambda (\tau ) = eF(\tau )$, the initial total momentum $P = \frac{{{m_c}(x - x^\prime )}}{{(t - t^\prime )}}$, $\pi = - \frac{{\lambda (t) - \lambda (t^\prime )}}{{(t - t^\prime )}}$. The initial energy is $\Delta E(t^\prime ) = - \frac{{{{(\pi + \dot \lambda (t^\prime ))}^2} + {P^2}}}{{{m_c}}}$. In what follows, we assume that the quantity ${-}{\tilde \epsilon _G}$ is negative (and very close to ${-}{ \epsilon _G}$). We will show in the numerical calculations that, for reasonable values of disorder and laser field strength, this condition is always met. Under the given assumption, Eq. (14b) has no real solutions as in the cases of HHG in perfect crystal [15] or atoms [62]. The classical trajectories of the electron and hole with zero initial momenta are closest to satisfy the condition Eq. (14b), and therefore give the most relevant contributions to the HHG spectra. These trajectories describe electron–hole pairs that are created with zero initial momenta and that recollide in the vicinity of the point where they are created ($P \simeq 0,\pi + \dot \lambda (t^\prime ) \simeq 0$, and consequently $x \simeq x^\prime $).

Finally, Eq. (14d) requires that, at time $t^\prime $ when the electron–hole pair is created, the momentum of the conduction-band electron ${p^c}([x]_{tt^\prime }^{xx^\prime },\tau = t^\prime ) = - \frac{{\partial S_c^\textit{cl}}}{{\partial x^\prime }}([x]_{tt^\prime }^{xx^\prime })$ differs from the momentum of the valence-band electron ${p^v}([x]_{t^\prime 0}^{x^\prime 0},\tau = t^\prime ) = \frac{{\partial S_v^\textit{cl}}}{{\partial x^\prime }}([x]_{t^\prime 0}^{x^\prime 0})$ by the amount $\frac{{\partial {\sigma _I}}}{{\partial x^\prime }}([x]_{tt^\prime }^{xx^\prime })$,

$${p^c}([x]_{tt^\prime }^{xx^\prime },\tau = t^\prime ) - {p^v}([x]_{t^\prime 0}^{x^\prime 0},\tau = t^\prime ) = \frac{{\partial {\sigma _I}}}{{\partial x^\prime }}([x]_{tt^\prime }^{xx^\prime }),$$
which leads to the relation
$$\begin{split} & \frac{{4{m_c}\hbar (x - x^\prime )}}{{\sqrt {2\pi } \rho W_0^2{L^2}}} = - \int_{t^\prime }^t {\rm d}s\int_{t^\prime }^t {\rm d}s^\prime \\& {(s - s^\prime )^2}\left( {\frac{{x - x^\prime }}{{t - t^\prime }} + \Lambda } \right)\text{Im}\left( {\frac{{{e^{ - A}}}}{{{B^{3/2}}}}} \right),\end{split}$$
using the definition Eq. (16) for the electron–hole trajectories and Eq. (11) for the average phase, where we have set $\Lambda (s,s^\prime ,t,t^\prime ) = \frac{{\lambda (s) - \lambda (s^\prime )}}{{s - s^\prime }} - \frac{{\lambda (t) - \lambda (t^\prime )}}{{t - t^\prime }}$.

Equation (17) defines the distance $(x - x^\prime )$ covered by the electron–hole pair before annihilation as an implicit function of the recollision time $(t - t^\prime )$ and of the parameters of the laser field and disorder. It is not possible to write explicitly the solutions of Eq. (17), but it is possible to see that there are solutions with small travel distance $(x - x^\prime \lt L)$, when the correlation length is large ($L \gg {a_0}$) and laser strength and frequency have the values commonly used in HHG experiments. In fact, usually in HHG experiments, the ratio ${F_0}/\omega _L^2$ is of the order of ${a_0}$ (with ${F_0}$ the laser peak strength and ${\omega _L}$ its frequency); therefore, for $(x - x^\prime ) \ll L$, we can set ${e^{ - A}} \simeq 1$ [Eq. (10)] and write

$$\begin{split}(x - x^\prime ) \simeq \frac{{\sqrt {2\pi } \rho W_0^2{L^2}}}{{4{m_c}\hbar }}{I_1}{\left( {1 + \frac{{\sqrt {2\pi } \rho W_0^2{L^2}}}{{4{m_c}\hbar }}{I_2}} \right)^{ - 1}},\end{split}$$
where ${I_1} = - \int_{t^\prime }^t {\rm d}s\int_{t^\prime }^t {\rm d}s^\prime {(s - s^\prime )^2}\Lambda (s,s^\prime ,t,t^\prime ){\rm Im}( {{B^{ - 3/2}}} )$ and similarly ${I_2} = - \int_{t^\prime }^t {\rm d}s\int_{t^\prime }^t {\rm d}s^\prime \frac{{{{(s - s^\prime )}^2}}}{{t - t^\prime }}{\rm Im}( {{B^{ - 3/2}}} )$. Finally, we observe that, in HHG experiments with solids, total interaction time is limited to only a few optical cycles (usually less than 10) to avoid permanent damage to the crystal. Furthermore, the high harmonics in the spectrum are radiated only in the short period when the laser field envelope is close to its peak value. In other words, the time intervals $(t - t^\prime )$ that are relevant to HHG are of the order of only two or three optical cycles. Since in most experiments ${\omega _L} \simeq {10^{ - 2}}$ a.u., when $L$ is large with respect to ${a_0}$ ($L$ approximately 1 nm or more), we can approximate, for the time interval $(t - t^\prime )$ relevant for HHG, the function ${B^{ - 3/2}}$ as
$$\begin{split}& {B^{ - \frac 3 2 }} \simeq \frac 1 {{{2^{ - 3/2}}{L^3}}} + \frac{{3i\hbar }}{{2{m_c}{L^2}}} \cdot \\& \frac{{( - (s - s^\prime )\Theta (s - s^\prime ) + (s - s^\prime )\Theta (s^\prime - s) + \frac{{{{(s - s^\prime )}^2}}}{{t - t^\prime }})}}{{{2^{ - 3/2}}{L^3}}},\end{split}$$
If we substitute this approximation for the function B in Eq. (18), we note that ${I_2} \simeq \frac{{3\sqrt 2 \hbar }}{{10{m_c}{L^5}}}{(t - t^\prime )^4}$, and similarly ${I_1}$ can be estimated to be of the order ${I_1} \simeq \frac{{3\sqrt 2 \hbar }}{{10{m_c}{L^5}}}({F_0}/\omega _L^2)(t - t^\prime )^4$. For all the recollision times such that ${I_2} \ne 1$, Eq. (18) gives a recollision distance $x - x^\prime $ of the order or smaller than ${F_0}/\omega _L^2$ that is in itself smaller than $L$ in agreement with the initial assumptions. To solve Eq. (18) for value of $(t - t^\prime )$ such that ${I_2} \simeq 1$, it is required to consider a Taylor expansion of the function ${e^{ - A}}$ containing quadratic terms in $(x - x^\prime )$. We have therefore shown that trajectories with short recollision distance ($x - x^\prime \le L$) that satisfy the conditions Eq. (15) exist and that these trajectories contribute significantly to HHG. We do not expect to find a trajectory with long recollision distance ($x - x^\prime \gg L$) satisfying condition Eq. (17) for reasonable value of disorder because, for ($x - x^\prime \gg L$), the factor ${e^{ - A}}$ goes to zero exponentially fast.

In summary, the Feynman trajectories that contribute to the HHG process in our model disordered semiconductor are almost identical to those of a defect-free solid [15,58]. Nevertheless, in the HHG process in disordered solids, the contribution of each trajectory to the HHG signal is weighted by a factor ${e^{{\sigma _R}(x \simeq x^\prime ,t,t^\prime )}}$. This factor, calculated in Eq. (11), strongly suppresses the contribution of trajectories with long travel times (large $t - t^\prime $) [32]. It has been observed in Refs. [15,32,58] that to obtain a clean HHG spectrum it is required to eliminate the contribution of all the trajectories with travel time longer than a quarter of the laser period ${T_L}/4 = \pi /2{\omega _L}$. In our model, this condition is met if the parameters that characterize the disorder, $\rho ,{W_0},L$, are such that ${-}{\sigma _R}(x = x^\prime ,t - t^\prime = {T_L}/4) \ge 1$. When only short trajectories contribute to HHG, the current $J(t)$ satisfies the relation [64] $J(t) \simeq - J(t - {T_L}/2)$ for $t$ in the generic ${m}$th half-cycle, $t \in [m{T_L},(m + 1/2){T_L}]$. In fact, the current is generated by electron–hole quantum trajectories with short recombination times. These short trajectories in adjacent half-cycles are almost identical because the laser envelope does not change significantly over such a short time interval. As a result, the Fourier transform of the current vanishes for $\omega = 2n{\omega _L}$ (with $n$ any integer), and the spectrum $|J(\omega )|^2$ is made up of odd harmonics of the laser frequency.

4. NUMERICAL RESULTS

We pause at this point to stress that the main finding of this paper is that the HHG spectrum generated by a disordered crystal is made up only of peaks in correspondence of, or very close to, the harmonics of the laser frequency ${\omega _L}$, when a sufficient amount of disorder is present. This result is important because all the reported experimental HHG spectra are indeed built of a series of sharp peaks at frequencies that are integer multiples (or very close to integer multiples), i.e., harmonics, of the laser frequency. This fact is clearly shown in the experimental papers Refs. [110]

We further stress that ab initio numerical calculations based on model Hamiltonian describing a perfect crystal, i.e., a crystal without impurities or defects, are not able to reproduce the experimental HHG spectra. As a support to this statement, we mention, just as a few examples among many others, the results in Ref. [21], where the spectra are calculated using the Hartree–Fock approach, the numerical computations in Refs. [17,20] with spectra obtained within the Kohn–Sham density functional scheme, or the findings in Refs. [15,29] based on the use of the SBE with or without dephasing time. All these simulations show HHG spectra with both harmonic peaks and a large number of nonharmonic peaks at variance with the experiments. The only simulations, other than those based on the disordered Hamiltonian used in our paper and in our previous works [32,33], that produce HHG spectra that resemble the experimental ones are those based on the SBE with ultrashort dephasing time (dephasing time of the order of 1 fs). Nevertheless, simulations based on the SBE with dephasing time cannot considered as ab initio simulations. In fact, the SBE with a dephasing time cannot be derived from a microscopic many-body Schrödinger equation (see, for example, [16,65,66]). The SBEs with a dephasing time are semi-phenomenological equations where the effects such as the lattice vibrations (phonons), the Coulomb interaction beyond the Hartree–Fock approximation, or disorder are approximately treated with the introduction of an arbitrary parameter (the dephasing time) whose value is fixed by trying to match the simulated value of some observable quantity to its experimental counterpart as closely as possible. In the case of HHG, the value of dephasing time is chosen to reproduce the emitted spectra. The SBEs with dephasing time are certainly very useful; nevertheless, they cannot reveal which physical phenomenon is responsible for the fact that the experimental HHG spectra are composed of harmonic peaks only.

We now compare the predictions of the long-wavelength model against the numerical solutions of the equation of motions [Eq. (3)]. In the numerical calculations, the external field has the form

$$F(t) = {F_0} \sin ^2 (t/\!{T_{\rm tot}})(\Theta (t) - \Theta (t - {T_{\rm tot}}))\sin ({\omega _L}t),$$
where $\Theta$ is the step function, and ${T_{\rm tot}} = 16\pi /{\omega _L}$ is the total interaction time. Equation (3) is solved using a Runge–Kutta fourth-order method. For an electron initially localized at the site ${\nu _0} = 0$, we solve Eq. (3) for $n \in [ - \nu /2, + \nu /2]$. To avoid spurious reflections at the boundaries, we need to include approximately $\nu = 300$ lattice sites in the calculations. To simulate the entire crystal, we perform the calculations for approximately 800 different disorder configurations. For the disorder coefficients we set, following the Edwards model of disorder, ${\alpha _n} = {\Gamma _1}\sum\nolimits_{i = 1}^{{N_D}} {e^{ - \frac{{{{(2n - \nu {x_i})}^2}}}{{4{l^2}}}}}$, with ${x_i}$ independent random variables with uniform distribution in $(- 1, 1)$. We fix $l = 8$ and ${N_D} = \nu /8$ in all simulations corresponding to density $\rho = 1/(8a)$ and correlation length $L = la$. We have chosen these values for the laser field ${F_0}$ and the disorder ${W_0}$ and $L$ in such a way that the conditions for the validity of the long-wavelength approximation are satisfied. We remember that these conditions are as follows: (1) the laser intensity is not very strong so that predominantly electrons with small quasi-momentum tunnel into the conduction band (2) the correlation length of the disorder is large compared to the lattice spacing $a$.

In Fig. 2, we plot the spectrum emitted by a disordered semiconductor interacting with a field of intensity $I = 2.0\;{\rm TW}/{{\rm cm}^2}$ and wavelength $\lambda = 2.5\;\unicode{x00B5}{\rm m}$. The strength of the disorder potential is ${\Gamma _1} = 0.25\hbar {\omega _L}$. We observe that, in this case, the HHG spectrum presents both harmonic peaks and nonharmonic peaks. In detail, we note that in the HHG spectrum between the bandgap frequency (approximately $8{\omega _L}$) and the cutoff (approximately at $15{\omega _L}$), there are three odd harmonic peaks at $11{\omega _L}$, $13{\omega _L}$, $15{\omega _L}$; one even harmonic peak at $12\hbar {\omega _L}$; and 5 nonharmonic peaks. This result is in agreement with the long-wavelength model since ${-}{\sigma _I}(x = x^\prime ,T/4) \ll 1$. In fact, such a small value of ${\sigma _I}$ means that the disorder is too weak to effectively suppress the contributions of trajectories with long recollision times to the dipole moment. As a result, the HHG spectrum above the bandgap frequency contains a large number of nonharmonic peaks.

 figure: Fig. 2.

Fig. 2. HHG spectrum generated by a disordered semiconductor with ${\Gamma _1} = 0.25\hbar {\omega _L}$, interacting with a laser field with $I = 2.0\;{\rm TW}/{{\rm cm}^2}$ and $\lambda = 2.5 \;{\unicode{x00B5}{\rm m}}$. Blue line is the total spectrum; red line is the intraband contribution.

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In Fig. 3, we present the spectrum radiated by a disordered semiconductor with ${\Gamma _1} = 0.70\hbar {\omega _L}$, with all other parameters of the semiconductor and of the laser field being the same as in the previous calculation. We note that, for the given value of disorder, the long-wavelength model predicts that ${-}{\sigma _I}(x = x^\prime ,T/4) \simeq 1$. According to the discussion in the previous sections, this means that the disordered solid should radiate an HHG spectrum composed only of harmonic peaks. In fact, as we have shown already in Refs. [32,33], the HHG spectrum of the disordered solid is for all frequencies above the bandgap (${\simeq} 8{\omega _L}$) much weaker than that generated by the perfect solid, and it is made almost entirely of odd harmonics of the laser frequency. In particular, we note that, in the region [$8{\omega _L}$-$15{\omega _L}$], there are now four harmonic peaks ($9{\omega _L}$, $11{\omega _L}$, $13{\omega _L}$, $15{\omega _L}$), and only one nonharmonic peak close to $8{\omega _L}$.

 figure: Fig. 3.

Fig. 3. As in Fig. 2, but with ${\Gamma _1} = 0.7\hbar {\omega _L}$.

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To further illustrate the role of disorder in the ultrafast optical response of a crystal, we plot in Fig. 4 HHG spectra, in the range between the bandgap frequency and the cutoff, as a function of the amount disorder in the crystal. In particular, the results presented in this figure support our claim that the role of disorder in HHG is well described through the long-wavelength model. In fact, it is clearly shown in this figure that, only when the condition ${\sigma _R}(x = x^\prime ,t - t^\prime = \pi /2{\omega _L}) \ge 1$ is met (for disorder strength ${\Gamma _1} = 0.7\hbar {\omega _L}$), the HHG spectra are made up, almost entirely, of odd harmonics of the laser frequency. It is useful to stress that, even in experimental HHG spectra (for example [1,3]), some nonharmonic peaks are visible close to the bandgap frequency.

 figure: Fig. 4.

Fig. 4. Detailed view of the HHG spectrum as a function of the disorder level in the solids. The topmost curve (black solid line with crosses) is the HHG spectrum generated by a crystal without disorder ${\Gamma _1} = 0.0$, the middle curve (dashed red) is the spectrum emitted by a solid with disorder strength ${\Gamma _1} = 0.25\hbar {\omega _L}$, and the bottom curve (blue line) is the spectrum radiated by the solid when ${\Gamma _1} = 0.7\hbar {\omega _L}$. Intensity and frequency of the laser are as in Fig. 2. The three curves have been shifted by arbitrary factors to improve visibility.

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In conclusion, we have investigated HHG from an imperfect semiconductor, and we have observed that the HHG spectra generated resemble those observed in experiments when sufficiently strong disorder is present in the solid. Further, a simple theory has been presented that explains how the impurities in the crystal can generate the ultrafast dephasing effect that is responsible for the clean and well-resolved HHG spectra in the numerical simulations. This dephasing effect depends critically on the strength and correlation length of the disorder. Our results therefore show that (1) disorder is a source of ultrafast dephasing of the HHG signals, and (2) it may be responsible for the well-resolved experimental HHG spectra. Furthermore, our results suggest that it may be possible to utilize HHG as a novel experimental tool for the analysis of disordered solids.

APPENDIX A: DERIVATION OF $\bar \Phi$

The starting point for the calculation of $\bar \Phi$ is Eq. (9) in the main text. Since we have chosen $v(x)$ a Gaussian function, the correlation function is a Gaussian $\gamma (x(\tau ^\prime ) - x(\tau )) = \sqrt \pi L{e^{ - {{(x(\tau ^\prime ) - x(\tau ))}^2}/2{L^2}}}$, where $L$ is the parameter that determines the length-scale of the correlation of the disorder. The first cumulant can now be written as

$$\begin{split}& \frac{{\bar \Phi (x,x^\prime ,t,t^\prime )}}{{\rho {L^2}W_0^2\sqrt {{2^{ - 3}}} {\hbar ^{ - 2}}}} = - \int_{t^\prime }^t {\rm d}s^\prime \int_{t^\prime }^t {\rm d}s \\& \int dk{\gamma _F}(k)\int_{x(t^\prime ) = x^\prime }^{x(t) = x} [dx]\frac{{{e^{\frac{i}{\hbar }{S_k}\left( {[x]_{tt^\prime }^{xx^\prime }} \right)}}}}{{{G_c}(x,x^\prime ,t,t^\prime )}},\end{split}$$
where ${\gamma _F}(k) = {e^{ - {L^2}{k^2}/2}}$ is the Fourier transform of the correlation function and
$$\begin{split}& {S_k}([x]_{tt^\prime }^{xx^\prime }) = {S_c} + \hbar k(x(s^\prime ) - x(s)) = \\& \int_{t^\prime }^t \left[\frac{{{{\dot x}^2}}}{{2{m_c}}} - x(\tau )(eF(\tau ) + {F_k}(\tau ,s,s^\prime ))\right]{\rm d}\tau .\end{split}$$
${S_k}$ is the action for a particle moving under the influence of the laser field $F(\tau )$ and of an additional force ${F_k}(\tau ,s,s^\prime ) = - \hbar k\delta (\tau - s^\prime ) + \hbar k\delta (\tau - s)$ that is made up of a couple of instantaneous delta interactions centered at $s$ and $s^\prime $. We can then write
$$\begin{split}& \bar \Phi (x,x^\prime ,t,t^\prime ) = - \frac{{\rho {L^2}W_0^2}}{{\sqrt {{2^3}} {\hbar ^2}}}\int {\rm d}s^\prime \int {\rm d}s \\& \int {\rm d}k{\gamma _F}(k)\frac{{{G_k}(x,x^\prime ,t,t^\prime )}}{{{G_c}(x,x^\prime ,t,t^\prime )}},\end{split}$$
where ${G_k} = \int_{x(t^\prime ) = x^\prime }^{x(t) = x} [{\rm d}x]{e^{i{S_k}}}$ and ${G_c}$ are Green functions corresponding, respectively, to the quadratic actions ${S_k},{S_c}$. These two Green functions can be written in the Pauli–Van Vleck form [42]
$$\begin{split}&{G_k}(x,x^\prime ,t,t^\prime ) = A(t,t^\prime ){e^{\frac{i}{\hbar }S_k^\textit{cl}([x]_{tt^\prime }^{xx^\prime })}}, \\& {G_c}(x,x^\prime ,t,t^\prime ) = A(t,t^\prime ){e^{\frac{i}{\hbar }S_c^\textit{cl}([x]_{tt^\prime }^{xx^\prime })}},\end{split}$$
where the superscript cl is used to remind that the actions $S_k^\textit{cl},S_c^\textit{cl}$ are calculated for the trajectory satisfying the classical Newton equation of motions with the boundary conditions $x(\tau = t^\prime ) = x^\prime ,x(\tau = t) = x$. The first cumulant approximation can therefore be written as
$$\begin{split}& \bar \Phi (x,x^\prime ,t,t^\prime ) = - \frac{{\rho {L^2}W_0^2}}{{\sqrt {{2^3}} {\hbar ^2}}}\int_{t^\prime }^t {\rm d}s^\prime \int_{t^\prime }^t {\rm d}s \\& \int_{ - \infty }^\infty {\rm d}k{\gamma _F}(k){e^{i(S_k^\textit{cl} - S_c^\textit{cl})}}.\end{split}$$
The classical trajectory ${x_k}(\tau )$ for a particle driven by the laser field $F(t)$ and the force ${F_k}(t)$ (${m_e}{\ddot x_k}(\tau ) = eF(\tau ) + {F_k}(\tau )$), with boundary conditions ${x_k}(t^\prime ) = x^\prime ,{x_k}(t) = x$, can be written as the sum ${x_k}(\tau ) = {x_c}(\tau ) + {x_{k,0}}(\tau )$, where ${x_c}(\tau )$ is defined as the classical trajectory for a particle in the laser field (${m_c}{\ddot x_c}(\tau ) = eF(\tau )$) with boundary conditions ${x_c}(t^\prime ) = x^\prime ,{x_c}(t) = x$, and ${x_{k,0}}(\tau )$ is the classical trajectory for a particle driven by the force ${F_k}$ (${m_c}{\ddot x_{k,0}}(\tau ) = {F_k}(\tau )$) with boundary conditions ${x_{k,0}}(t^\prime ) = {x_{k,0}}(t) = 0$. Correspondingly, the action $S_k^\textit{cl}([x]_{tt^\prime }^{xx^\prime })$ becomes $S_k^\textit{cl}([x]_{tt^\prime }^{xx^\prime })\; =\; S_c^\textit{cl}([x]_{tt^\prime }^{xx^\prime }) \;+\; S_{k0}^\textit{cl}([x]_{tt^\prime }^ 00 ) \,- \int {\rm d}\tau {F_k}(\tau ){x_c}(\tau )$, with $S_{k0}^\textit{cl}([x]_{tt^\prime }^ 00 ) = \int_{t^\prime }^t [\frac{{\dot x_{k,0}^2}}{{2{m_c}}} - {x_{k,0}}(\tau ){F_k}(\tau ,s,s^\prime )]{\rm d}\tau$. The $k$ integral in the last equation can be done exactly, and, as a result, the expression for the average phase factor becomes Eq. (10) of the main text.

Funding

Ministry of Science and Technology, Taiwan; National Taiwan University (108L893201 and 108L104048).

Disclosures

The authors declare no conflict of interests.

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References

  • View by:

  1. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
    [Crossref]
  2. G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522, 462–464 (2015).
    [Crossref]
  3. S. Gholam-Mirzaei, J. Beetar, and M. Chini, “High harmonic generation in ZnO with a high-power mid-IR OPA,” Appl. Phys. Lett. 110, 061101 (2017).
    [Crossref]
  4. O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8, 119–123 (2014).
    [Crossref]
  5. P. Xia, C. Kim, F. Lu, T. Kanai, H. Akiyama, J. Itatani, and N. Ishii, “Nonlinear propagation effects in high-harmonic generation in reflection and transmission from gallium arsenide,” Opt. Express 26, 29393–29400 (2018).
    [Crossref]
  6. T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521, 498–502 (2015).
    [Crossref]
  7. M. Garg, H. Y. Kim, and E. Goulielmakis, “Ultimate waveform reproducibility of extreme ultraviolet pulse by high-harmonic generation in quartz,” Nat. Photonics 12, 291–296 (2018).
    [Crossref]
  8. G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534, 520–524 (2016).
    [Crossref]
  9. G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115, 193603 (2015).
    [Crossref]
  10. A. A. Lanin, E. A. Stepanov, A. B. Fedotov, and A. M. Zheltykov, “Mapping the electron band structure by intraband high-harmonic generation in solids,” Optica 4, 516–519 (2017).
    [Crossref]
  11. M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. W. Koch, and R. Huber, “Real-time observation of interfering crystal electrons in high-harmonic generation,” Nature 523, 572–575 (2015).
    [Crossref]
  12. G. P. Zhang, M. S. Si, M. Murakami, Y. H. Baiand, and T. F. George, “Generating high-order optical and spin harmonics from ferromagnetic monolayers,” Nat. Commun. 9, 3031 (2018).
    [Crossref]
  13. Y. Murakami, M. Eckstein, and P. Werner, “High-harmonic generation in Mott insulators,” Phys. Rev. Lett. 121, 057405 (2018).
    [Crossref]
  14. D. Bauer and K. K. Hansen, “High-harmonic generation in solid with and without topological edge states,” Phys. Rev. Lett. 120, 177401 (2018).
    [Crossref]
  15. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generations in solids,” Phys. Rev. Lett. 113, 073901 (2014).
    [Crossref]
  16. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, 2004).
  17. I. Floss, C. Lemell, G. Wachter, M. Pickem, J. Burgdörfer, X.-M. Tong, K. Yabana, and S. A. Sato, “Ab initio simulation of high-order harmonic generation in solids,” Phys. Rev. A 97, 011401 (2018).
    [Crossref]
  18. N. Tancogne-Dejean, O. D. Mücke, F. X. Kärtner, and A. Rubio, “Impact of electronic band structure in high-harmonic generation spectra of solids,” Phys. Rev. Lett. 118, 087403 (2017).
    [Crossref]
  19. T. Otobe, “High-harmonic generation in α-quartz by electron-hole recombination,” Phys. Rev. B 94, 235512 (2016).
    [Crossref]
  20. K. K. Hansen, T. Deffge, and D. Bauer, “High-order harmonic generation in solid slabs beyond the single-active-electron approximation,” Phys. Rev. A 96, 053418 (2017).
    [Crossref]
  21. T. Ikemachi, Y. Shinohara, T. Sato, J. Yumoto, M. Kuwata-Gonokami, and K. L. Ishikawa, “Time-dependent Hartree-Fock study of electron-hole interactions effects on high-order harmonic generation from periodic crystals,” Phys. Rev. A 98, 023415 (2018).
    [Crossref]
  22. M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91, 043839 (2015).
    [Crossref]
  23. T. Higuchi, M. I. Stockman, and P. Hommelhoff, “Strong-field perspective on high-harmonic radiation from bulk solids,” Phys. Rev. Lett. 113, 213901 (2014).
    [Crossref]
  24. N. Moiseyev, “Selection rules for harmonic generation in solids,” Phys. Rev. A 91, 053811 (2015).
    [Crossref]
  25. P. G. Hawkins, M. Y. Ivanov, and V. S. Jakovlev, “ Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91, 013405 (2015).
    [Crossref]
  26. T.-Y. Du and X.-B. Bian, “Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids,” Opt. Express 25, 151–158 (2017).
    [Crossref]
  27. J.-B. Li, X. Zhang, S.-J. Yue, H.-M. Wu, B.-T. Hu, and H.-C. Du, “Enhancement of the second plateau in solid high-order harmonic spectra by the two-color fields,” Opt. Express 25, 18603–18613 (2017).
    [Crossref]
  28. X. Liu, X. Zhu, X. Zhang, D. Wang, P. Lan, and P. Lu, “Wavelength scaling of the cutoff energy in the solid high harmonic generation,” Opt. Express 25, 29216–29224 (2017).
    [Crossref]
  29. S. Jiang, J. Chen, H. Wei, C. Yu, R. Lu, and C. D. Lin, “Role of the transition dipole amplitude and phase on the generation of odd and even high-order harmonics in crystals,” Phys. Rev. Lett. 120, 253201 (2018).
    [Crossref]
  30. Y. W. Kim, T.-J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X.-B. Bian, and S.-W. Kim, “Spectral interference in high harmonic generation from solids,” ACS Photon. 6, 851–857 (2019).
    [Crossref]
  31. N. Tsatrafyllis, S. Kühn, M. Dumergue, P. Foldi, S. Kahaly, E. Cormier, I. A. Gonoskov, B. Kiss, K. Varju, S. Varro, and P. Tzallas, “Quantum optical signatures in a strong laser pulse after interaction with semiconductors,” Phys. Rev. Lett. 122, 193602 (2019).
    [Crossref]
  32. G. Orlando, C.-M. Wang, T.-S. Ho, and S.-I. Chu, “High-order harmonic generation in disordered semiconductors,” J. Opt. Soc. Am. B 35, 680–688 (2018).
    [Crossref]
  33. G. Orlando, T.-S. Ho, and S.-I. Chu, “Macroscopic effects in high-order harmonic generation in disordered semiconductors,” J. Opt. Soc. Am. B 36, 1873–1880 (2019).
    [Crossref]
  34. S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
    [Crossref]
  35. T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order harmonic generation of a doped semiconductor,” Phys. Rev. A 96, 043425 (2017).
    [Crossref]
  36. C. Yu, K. K. Hansen, and L. B. Madsen, “Enhanced high-order harmonic generation in donor-doped band-gap materials,” Phys. Rev. A 99, 013435 (2019).
    [Crossref]
  37. C. Yu, K. K. Hansen, and L. B. Madsen, “High-order harmonic generation in imperfect crystals,” Phys. Rev. A 99, 063408 (2019).
    [Crossref]
  38. J. Ma, C. Zhang, H. Cui, Z. Ma, and X. Miao, “Theoretical investigation of the electron dynamics in high-order harmonic generation process from the doped periodic potential,” Chem. Phys. Lett. 744, 137207 (2020).
    [Crossref]
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2020 (4)

J. Ma, C. Zhang, H. Cui, Z. Ma, and X. Miao, “Theoretical investigation of the electron dynamics in high-order harmonic generation process from the doped periodic potential,” Chem. Phys. Lett. 744, 137207 (2020).
[Crossref]

K. Chinzei and T. N. Ikeda, “Disorder effects on the origin of high-order harmonic generation in solids,” Phys. Rev. Res. 2, 013033 (2020).
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A. Pattanayak, M. S. Mrudul, and G. Dixit, “Influence of vacancy defects in solid high-order harmonic generation,” Phys. Rev. A 101, 013404 (2020).
[Crossref]

M. S. Mrudul, N. Tancogne-Dejean, A. Rubio, and G. Dixit, “High-harmonic generation by spin-polarized defects in solids,” Comput. Mater. 6, 10 (2020).
[Crossref]

2019 (6)

Y. W. Kim, T.-J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X.-B. Bian, and S.-W. Kim, “Spectral interference in high harmonic generation from solids,” ACS Photon. 6, 851–857 (2019).
[Crossref]

N. Tsatrafyllis, S. Kühn, M. Dumergue, P. Foldi, S. Kahaly, E. Cormier, I. A. Gonoskov, B. Kiss, K. Varju, S. Varro, and P. Tzallas, “Quantum optical signatures in a strong laser pulse after interaction with semiconductors,” Phys. Rev. Lett. 122, 193602 (2019).
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G. Orlando, T.-S. Ho, and S.-I. Chu, “Macroscopic effects in high-order harmonic generation in disordered semiconductors,” J. Opt. Soc. Am. B 36, 1873–1880 (2019).
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C. Yu, K. K. Hansen, and L. B. Madsen, “Enhanced high-order harmonic generation in donor-doped band-gap materials,” Phys. Rev. A 99, 013435 (2019).
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C. Yu, K. K. Hansen, and L. B. Madsen, “High-order harmonic generation in imperfect crystals,” Phys. Rev. A 99, 063408 (2019).
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S. Y. Kruchinin, “Non-Markovian pure dephasing in a dielectric excited by a few-cycle laser pulse,” Phys. Rev. A 100, 043839 (2019).
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2018 (10)

S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
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G. Orlando, C.-M. Wang, T.-S. Ho, and S.-I. Chu, “High-order harmonic generation in disordered semiconductors,” J. Opt. Soc. Am. B 35, 680–688 (2018).
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T. Ikemachi, Y. Shinohara, T. Sato, J. Yumoto, M. Kuwata-Gonokami, and K. L. Ishikawa, “Time-dependent Hartree-Fock study of electron-hole interactions effects on high-order harmonic generation from periodic crystals,” Phys. Rev. A 98, 023415 (2018).
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S. Jiang, J. Chen, H. Wei, C. Yu, R. Lu, and C. D. Lin, “Role of the transition dipole amplitude and phase on the generation of odd and even high-order harmonics in crystals,” Phys. Rev. Lett. 120, 253201 (2018).
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P. Xia, C. Kim, F. Lu, T. Kanai, H. Akiyama, J. Itatani, and N. Ishii, “Nonlinear propagation effects in high-harmonic generation in reflection and transmission from gallium arsenide,” Opt. Express 26, 29393–29400 (2018).
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M. Garg, H. Y. Kim, and E. Goulielmakis, “Ultimate waveform reproducibility of extreme ultraviolet pulse by high-harmonic generation in quartz,” Nat. Photonics 12, 291–296 (2018).
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G. P. Zhang, M. S. Si, M. Murakami, Y. H. Baiand, and T. F. George, “Generating high-order optical and spin harmonics from ferromagnetic monolayers,” Nat. Commun. 9, 3031 (2018).
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Y. Murakami, M. Eckstein, and P. Werner, “High-harmonic generation in Mott insulators,” Phys. Rev. Lett. 121, 057405 (2018).
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D. Bauer and K. K. Hansen, “High-harmonic generation in solid with and without topological edge states,” Phys. Rev. Lett. 120, 177401 (2018).
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I. Floss, C. Lemell, G. Wachter, M. Pickem, J. Burgdörfer, X.-M. Tong, K. Yabana, and S. A. Sato, “Ab initio simulation of high-order harmonic generation in solids,” Phys. Rev. A 97, 011401 (2018).
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2017 (9)

N. Tancogne-Dejean, O. D. Mücke, F. X. Kärtner, and A. Rubio, “Impact of electronic band structure in high-harmonic generation spectra of solids,” Phys. Rev. Lett. 118, 087403 (2017).
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A. A. Lanin, E. A. Stepanov, A. B. Fedotov, and A. M. Zheltykov, “Mapping the electron band structure by intraband high-harmonic generation in solids,” Optica 4, 516–519 (2017).
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S. Gholam-Mirzaei, J. Beetar, and M. Chini, “High harmonic generation in ZnO with a high-power mid-IR OPA,” Appl. Phys. Lett. 110, 061101 (2017).
[Crossref]

K. K. Hansen, T. Deffge, and D. Bauer, “High-order harmonic generation in solid slabs beyond the single-active-electron approximation,” Phys. Rev. A 96, 053418 (2017).
[Crossref]

T.-Y. Du and X.-B. Bian, “Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids,” Opt. Express 25, 151–158 (2017).
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J.-B. Li, X. Zhang, S.-J. Yue, H.-M. Wu, B.-T. Hu, and H.-C. Du, “Enhancement of the second plateau in solid high-order harmonic spectra by the two-color fields,” Opt. Express 25, 18603–18613 (2017).
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X. Liu, X. Zhu, X. Zhang, D. Wang, P. Lan, and P. Lu, “Wavelength scaling of the cutoff energy in the solid high harmonic generation,” Opt. Express 25, 29216–29224 (2017).
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T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order harmonic generation of a doped semiconductor,” Phys. Rev. A 96, 043425 (2017).
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G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B 50, 083001 (2017).
[Crossref]

2016 (2)

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534, 520–524 (2016).
[Crossref]

T. Otobe, “High-harmonic generation in α-quartz by electron-hole recombination,” Phys. Rev. B 94, 235512 (2016).
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2015 (8)

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. W. Koch, and R. Huber, “Real-time observation of interfering crystal electrons in high-harmonic generation,” Nature 523, 572–575 (2015).
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G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115, 193603 (2015).
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T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521, 498–502 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522, 462–464 (2015).
[Crossref]

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91, 043839 (2015).
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N. Moiseyev, “Selection rules for harmonic generation in solids,” Phys. Rev. A 91, 053811 (2015).
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P. G. Hawkins, M. Y. Ivanov, and V. S. Jakovlev, “ Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91, 013405 (2015).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91, 064302 (2015).
[Crossref]

2014 (3)

T. Higuchi, M. I. Stockman, and P. Hommelhoff, “Strong-field perspective on high-harmonic radiation from bulk solids,” Phys. Rev. Lett. 113, 213901 (2014).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8, 119–123 (2014).
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G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generations in solids,” Phys. Rev. Lett. 113, 073901 (2014).
[Crossref]

2011 (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

2008 (2)

V. Turkowski and C. A. Ullrich, “Time-dependent density-functional theory for ultrafast interband excitations,” Phys. Rev. B 77, 075204 (2008).
[Crossref]

J. Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78, 075204 (2008).
[Crossref]

1995 (1)

M. Protopapas, D. G. Lappas, C. H. Keitel, and P. Knight, “Recollisions, bremsstrahlung, and attosecond pulses from intense laser fields,” Phys. Rev. A 53, R2933–R2936 (1995).
[Crossref]

1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref]

1991 (2)

N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8, 1207–1211 (1991).
[Crossref]

D. Bennhardt, P. Thomas, A. Weller, M. Lindberg, and S. W. Koch, “Influence of Coulomb interactions on the photon echo in disordered semiconductors,” Phys. Rev. B 43, 8934–8945 (1991).
[Crossref]

1986 (2)

D. C. Khandekar, V. A. Singh, K. V. Bhagwat, and S. V. Lawande, “Functional integral approach to positionally disordered systems,” Phys. Rev. B 33, 5482–5486 (1986).
[Crossref]

D. C. Khandekar and S. V. Lawande, “Feynman path integrals: some exact results and applications,” Phys. Rep. 137, 115–229 (1986).
[Crossref]

1977 (1)

V. Sayakanit, “Mobility of an electron in a Gaussian random potential,” Phys. Lett. 59, 461–463 (1977).
[Crossref]

1973 (1)

V. Samathiakanit, “An average propagator of a disordered system,” J. Phys. A 6, 632–639 (1973).
[Crossref]

1970 (1)

V. Bezak, “Path integral theory of an electron gas in a random potential,” Proc. R. Soc. London A 315, 339–354 (1970).
[Crossref]

1968 (1)

A. V. Chaplik, “The Green’s function and mobility of an electron in a random potential,” Sov. Phys. JETP 26, 797–800 (1968).

1965 (2)

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

T. Lukes, “On the electronic structure of disordered systems,” Philos. Mag. 12(118), 719–724 (1965).
[Crossref]

1964 (1)

S. F. Edwards and V. B. Gulyaev, “The density of states of an highly impure semiconductor,” Proc. Phys. Soc. 83, 495–496 (1964).
[Crossref]

1962 (1)

E. I. Blount, “Formalisms of band theory,” Solid State Phys. 13, 305–373 (1962).
[Crossref]

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

Akiyama, H.

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Phonons (Cambridge University, 2007).

Almalki, S.

S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
[Crossref]

Baiand, Y. H.

G. P. Zhang, M. S. Si, M. Murakami, Y. H. Baiand, and T. F. George, “Generating high-order optical and spin harmonics from ferromagnetic monolayers,” Nat. Commun. 9, 3031 (2018).
[Crossref]

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref]

Bart, G.

S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
[Crossref]

Bauer, D.

D. Bauer and K. K. Hansen, “High-harmonic generation in solid with and without topological edge states,” Phys. Rev. Lett. 120, 177401 (2018).
[Crossref]

K. K. Hansen, T. Deffge, and D. Bauer, “High-order harmonic generation in solid slabs beyond the single-active-electron approximation,” Phys. Rev. A 96, 053418 (2017).
[Crossref]

Beetar, J.

S. Gholam-Mirzaei, J. Beetar, and M. Chini, “High harmonic generation in ZnO with a high-power mid-IR OPA,” Appl. Phys. Lett. 110, 061101 (2017).
[Crossref]

Bennhardt, D.

D. Bennhardt, P. Thomas, A. Weller, M. Lindberg, and S. W. Koch, “Influence of Coulomb interactions on the photon echo in disordered semiconductors,” Phys. Rev. B 43, 8934–8945 (1991).
[Crossref]

Bezak, V.

V. Bezak, “Path integral theory of an electron gas in a random potential,” Proc. R. Soc. London A 315, 339–354 (1970).
[Crossref]

Bhagwat, K. V.

D. C. Khandekar, V. A. Singh, K. V. Bhagwat, and S. V. Lawande, “Functional integral approach to positionally disordered systems,” Phys. Rev. B 33, 5482–5486 (1986).
[Crossref]

Bian, X.-B.

Y. W. Kim, T.-J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X.-B. Bian, and S.-W. Kim, “Spectral interference in high harmonic generation from solids,” ACS Photon. 6, 851–857 (2019).
[Crossref]

T.-Y. Du and X.-B. Bian, “Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids,” Opt. Express 25, 151–158 (2017).
[Crossref]

Blount, E. I.

E. I. Blount, “Formalisms of band theory,” Solid State Phys. 13, 305–373 (1962).
[Crossref]

Brabec, T.

S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
[Crossref]

G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B 50, 083001 (2017).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91, 064302 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522, 462–464 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115, 193603 (2015).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generations in solids,” Phys. Rev. Lett. 113, 073901 (2014).
[Crossref]

Browne, D. A.

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534, 520–524 (2016).
[Crossref]

Burgdörfer, J.

I. Floss, C. Lemell, G. Wachter, M. Pickem, J. Burgdörfer, X.-M. Tong, K. Yabana, and S. A. Sato, “Ab initio simulation of high-order harmonic generation in solids,” Phys. Rev. A 97, 011401 (2018).
[Crossref]

Chaplik, A. V.

A. V. Chaplik, “The Green’s function and mobility of an electron in a random potential,” Sov. Phys. JETP 26, 797–800 (1968).

Chen, J.

S. Jiang, J. Chen, H. Wei, C. Yu, R. Lu, and C. D. Lin, “Role of the transition dipole amplitude and phase on the generation of odd and even high-order harmonics in crystals,” Phys. Rev. Lett. 120, 253201 (2018).
[Crossref]

Chini, M.

S. Gholam-Mirzaei, J. Beetar, and M. Chini, “High harmonic generation in ZnO with a high-power mid-IR OPA,” Appl. Phys. Lett. 110, 061101 (2017).
[Crossref]

Chinzei, K.

K. Chinzei and T. N. Ikeda, “Disorder effects on the origin of high-order harmonic generation in solids,” Phys. Rev. Res. 2, 013033 (2020).
[Crossref]

Chu, S.-I.

Ciappina, M.

Y. W. Kim, T.-J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X.-B. Bian, and S.-W. Kim, “Spectral interference in high harmonic generation from solids,” ACS Photon. 6, 851–857 (2019).
[Crossref]

Corkum, P. B.

S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91, 064302 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522, 462–464 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115, 193603 (2015).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generations in solids,” Phys. Rev. Lett. 113, 073901 (2014).
[Crossref]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref]

Cormier, E.

N. Tsatrafyllis, S. Kühn, M. Dumergue, P. Foldi, S. Kahaly, E. Cormier, I. A. Gonoskov, B. Kiss, K. Varju, S. Varro, and P. Tzallas, “Quantum optical signatures in a strong laser pulse after interaction with semiconductors,” Phys. Rev. Lett. 122, 193602 (2019).
[Crossref]

Cui, H.

J. Ma, C. Zhang, H. Cui, Z. Ma, and X. Miao, “Theoretical investigation of the electron dynamics in high-order harmonic generation process from the doped periodic potential,” Chem. Phys. Lett. 744, 137207 (2020).
[Crossref]

Deffge, T.

K. K. Hansen, T. Deffge, and D. Bauer, “High-order harmonic generation in solid slabs beyond the single-active-electron approximation,” Phys. Rev. A 96, 053418 (2017).
[Crossref]

Delone, N. B.

DiChiara, A. D.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

DiMauro, L. F.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

Dixit, G.

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic pictorial description of the generic Feynman trajectories contributing to the dipole moment. (b) Generic Feynman trajectories satisfying the stationary-phase conditions Eq. (15). The curves are not solutions of the Newton equations of motion but only a pictorial description of the Green functions ${\bar G_c}$ (blue asterisks), ${G_v}$ (red circles), $G_v^*$ (solid black line). The vertical dotted line represents the time instant $t^\prime $ when the conduction-band electron is created.
Fig. 2.
Fig. 2. HHG spectrum generated by a disordered semiconductor with ${\Gamma _1} = 0.25\hbar {\omega _L}$, interacting with a laser field with $I = 2.0\;{\rm TW}/{{\rm cm}^2}$ and $\lambda = 2.5 \;{\unicode{x00B5}{\rm m}}$. Blue line is the total spectrum; red line is the intraband contribution.
Fig. 3.
Fig. 3. As in Fig. 2, but with ${\Gamma _1} = 0.7\hbar {\omega _L}$.
Fig. 4.
Fig. 4. Detailed view of the HHG spectrum as a function of the disorder level in the solids. The topmost curve (black solid line with crosses) is the HHG spectrum generated by a crystal without disorder ${\Gamma _1} = 0.0$, the middle curve (dashed red) is the spectrum emitted by a solid with disorder strength ${\Gamma _1} = 0.25\hbar {\omega _L}$, and the bottom curve (blue line) is the spectrum radiated by the solid when ${\Gamma _1} = 0.7\hbar {\omega _L}$. Intensity and frequency of the laser are as in Fig. 2. The three curves have been shifted by arbitrary factors to improve visibility.

Equations (42)

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ϵ c ( k ) = E g 2 Δ cos ( k a ) , ϵ v ( k ) = E g + 2 Δ cos ( k a ) ,
H ^ I = l = N N ( E g | w lc w lc | E g | w lv w lv | Δ [ | w l , c w l + 1 , c | + | w l + 1 , c w lc | ] ) + Δ l = N N ( | w lv w l + 1 , v | + | w l + 1 , v w l , v | ) ,
| w lc = k e ikal | k c 2 N + 1 , | w lv = k e ikal | k v 2 N + 1 .
w nv | H ^ D | w mv = w nc | H ^ D | w mv = 0 , w nc | H ^ D | w mc = α n δ nm .
H ^ ( t ) = H ^ 0 + eF ( t ) l = N N ( d cv [ | w lc w lv | + | w lv w lc | ] + la [ | w lc w lc | + | w lv w lv | ] ) .
i d | m ( t ) d t = ( H ^ 0 + exF ( t ) ) | m ( t ) ,
b ˙ c m ( n , t ) = η nc ( t ) i b c m ( n , t ) Δ i [ b c m ( n + 1 , t ) + b v m ( n 1 , t ) ] + e d cv F ( t ) i b v m ( n , t ) ,
b ˙ v m ( n , t ) = η nv ( t ) i b v m ( n , t ) + Δ i [ b v m ( n + 1 , t ) + b v m ( n 1 , t ) ] + e d cv F ( t ) i b c m ( n , t ) ,
η nc ( t ) = E g + α n + enaF ( t ) , η nv ( t ) = E g + enaF ( t ) ,
J ( t ) = t r [ ρ p ^ ] = m = N N c 0 t r [ | m ( t ) m ( t ) | p ^ ] = m = N N c 0 m ( t ) | p ^ | m ( t ) ,
i b ˙ c ( p , t ) = a α ( p p ) b c ( p , t ) d p ( i b c p d cv b v ) × eF ( t ) 2 Δ cos ( a p ) b c ( p , t ) ,
i b ˙ v ( p , t ) = ( 2 E g + 2 Δ cos ( a p ) ) b v i b v p eF ( t ) ,
i ϕ ˙ c ( x , t ) = ( exF ( t ) + D ( x ) ) ϕ c ( x , t ) 2 2 m c x 2 ϕ c ( x , t ) + ed cv ϕ v ( x , t ) F ( t ) ,
i ϕ ˙ v ( x , t ) = ( ϵ g + e x F ( t ) ) ϕ v ( x , t ) 2 2 m v x 2 ϕ v ,
ϕ c ( x , t ) = e d cv G c ( x , x , t , t ) ϕ v ( x , t ) F ( t ) d x d t ,
i G ˙ c ( e x F ( t ) + D ( x ) ) G c + 2 2 m c x 2 G c = δ ( x x ) δ ( t t ) .
G ¯ c ( x , x , t , t ) = x ( t ) = x x ( t ) = x [ d x ] e i S c ( [ x ] t t x x ) e ρ W 0 2 2 2 t t d s t t d s γ ( x ( s ) x ( s ) ) ,
G ¯ c ( x , x , t , t ) = G c ( x , x , t , t ) e Φ ( x , x , t , t ) ,
e Φ ( x , x , t , t ) = x ( t ) = x x ( t ) = x [ d x ] e i h S c ( [ x ] t t x x ) G c ( x , x , t , t ) e ρ W 0 2 2 2 t t d s t t d s γ ( x ( s ) x ( s ) ) .
Φ ¯ ( x , x , t , t ) 2 1 2 ρ W 0 2 = t t d s t t d s x x [ d x ( τ ) ] e i S c ( [ x ] t t x x ) γ ( x ( s ) x ( s ) ) G c ( x , x , t , t ) .
Φ ¯ ( x , x , t , t ) = π 2 ρ W 0 2 L 2 2 t t d s t t d s e A ( x , x , s , s , t , t ) B 1 2 ( s , s , t , t ) ,
A ( x , x , s , s , t , t ) = ( s s ) 2 [ x x t t + λ ( s ) λ ( s ) s s λ ( t ) λ ( t ) t t ] 2 4 B ( s , s , t , t ) , B ( s , s , t , t ) = 1 2 [ L 2 + i m c ( ( s s ) Θ ( s s ) + ( s s ) Θ ( s s ) ( s s ) 2 t t ) ] ,
Φ ¯ ( x = x , t , t ) = ρ W 0 2 L 2 ( π / 2 ) 1 / 2 2 ( t t ) 2 i ( t t ) 2 m c log ( 1 + i ( t t ) 4 m c L 2 1 i ( t t ) 4 m c L 2 ) .
Φ ¯ ( x = x , t , t ) = 2 ρ W 0 2 L 5 ( π ) 3 / 2 2 4 L 2 ( t t ) 2 i ( / m c ) ( t t ) 3 ( 2 ( t t ) 2 m c 2 + 16 L 4 ) .
d ¯ ( t ) e d cv = d x ϕ ¯ c ( x , t ) ϕ v ( x , t ) ed cv + c . c . = d x d x 1 0 t d t F ( t ) G ¯ c ( x , x 1 , t , t ) G v ( x 1 , 0 , t , 0 ) G v ( x , 0 , t , 0 ) + c . c . ,
S c cl t ( [ x ] t t x x ) + σ I t ϵ G ω = S v cl t ( [ x ] t 0 x 0 ) ,
S c cl t ( [ x ] t t x x ) + σ I t + ϵ G = S v cl t ( [ x ] t 0 x 0 ) ,
S c cl x ( [ x ] t t x x ) + σ I x = S v cl x ( [ x ] t 0 x 0 ) ,
S c cl x ( [ x ] t t x x ) + σ I x = S v cl x ( [ x ] t 0 x 0 ) ,
S v cl x ( [ x ] t 0 x 0 ) S c cl x ( [ x ] t t x x ) S c cl x ( [ x ] t t x x ) = S v cl x ( [ x ] t 0 x 0 ) .
x e c ( τ ) = x + π + P m c ( τ t ) + λ ( τ ) λ ( t ) m c ,
x e ¯ v ( τ ) = x π P m c ( τ t ) λ ( τ ) λ ( t ) m c ,
p c ( [ x ] t t x x , τ = t ) p v ( [ x ] t 0 x 0 , τ = t ) = σ I x ( [ x ] t t x x ) ,
4 m c ( x x ) 2 π ρ W 0 2 L 2 = t t d s t t d s ( s s ) 2 ( x x t t + Λ ) Im ( e A B 3 / 2 ) ,
( x x ) 2 π ρ W 0 2 L 2 4 m c I 1 ( 1 + 2 π ρ W 0 2 L 2 4 m c I 2 ) 1 ,
B 3 2 1 2 3 / 2 L 3 + 3 i 2 m c L 2 ( ( s s ) Θ ( s s ) + ( s s ) Θ ( s s ) + ( s s ) 2 t t ) 2 3 / 2 L 3 ,
F ( t ) = F 0 sin 2 ( t / T t o t ) ( Θ ( t ) Θ ( t T t o t ) ) sin ( ω L t ) ,
Φ ¯ ( x , x , t , t ) ρ L 2 W 0 2 2 3 2 = t t d s t t d s d k γ F ( k ) x ( t ) = x x ( t ) = x [ d x ] e i S k ( [ x ] t t x x ) G c ( x , x , t , t ) ,
S k ( [ x ] t t x x ) = S c + k ( x ( s ) x ( s ) ) = t t [ x ˙ 2 2 m c x ( τ ) ( e F ( τ ) + F k ( τ , s , s ) ) ] d τ .
Φ ¯ ( x , x , t , t ) = ρ L 2 W 0 2 2 3 2 d s d s d k γ F ( k ) G k ( x , x , t , t ) G c ( x , x , t , t ) ,
G k ( x , x , t , t ) = A ( t , t ) e i S k cl ( [ x ] t t x x ) , G c ( x , x , t , t ) = A ( t , t ) e i S c cl ( [ x ] t t x x ) ,
Φ ¯ ( x , x , t , t ) = ρ L 2 W 0 2 2 3 2 t t d s t t d s d k γ F ( k ) e i ( S k cl S c cl ) .

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