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Mathematics » "Resonance", book edited by Jan Awrejcewicz, ISBN 978-953-51-3634-7, Print ISBN 978-953-51-3633-0, Published: November 29, 2017 under CC BY 3.0 license. © The Author(s).

# Laser-Induced Fano Resonance in Condensed Matter Physics

By Ken-ichi Hino, Yohei Watanabe, Nobuya Maeshima and Muneaki Hase
DOI: 10.5772/intechopen.70524

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## Overview

Figure 1. Schematic diagram of the DFR formation in the Floquet excitonic system. This shows the coupling mechanism that a bound state supported by the sideband μ¯ interacts with a continuum state belonging to the sideband μ¯′ by the ac-Zener tunneling to result in the Fano decay (from Ref. [15] with partial modification).

Figure 2. Schematic diagram of the TFR dynamics based on the PQ picture, where the LO-phonon state α2 is embedded in the quasiboson state β. The PQ FR state composed of α2 and β is deexcited by induced photoemission process. For more detail, consult the text (from Ref. [31] with partial modification).

Figure 3. The quasienergy ℰμ as a function of Fac. ω is set to 91 meV. The vertical double arrows represent the original SL miniband widths corresponding to the photon sidebands of μ1 and μ2 (from Ref. [16] with partial modification).

Figure 4. Absorption spectra α(ωp; ω) as a function of ωp for Fac=50–450 (kV/cm) with ω= 91 meV. A series of the arrowed spectral profiles are examined exclusively in the text. The quasienergies shown in Figure 3 are also plotted (dotted lines) (from Ref. [16] with partial modification).

Figure 5. The DFR-related quantities as a function of Fac with the fixed value of ω = 91 meV. The calculated results represented by the filled symbols are connected by the solid lines in order to aid the presentation. (a) ∣1/q(F)∣ and Γ and (b) α0 and αmax (from Ref. [16] with partial modification).

Figure 6. The DFR-related quantities as a function of ω with the fixed value of Fac = 180 meV. The calculated results represented by the filled symbols are connected by the solid lines in order to aid the presentation. (a) ∣1/q(F)∣ and Γ and (b) ∣rd∣ (from Ref. [16] with partial modification).

Figure 7. A¯qtpωp of undoped Si (solid line) as a function of ωp at tp equal to (a) 15 fs, (b) 65 fs, and (c) 100 fs. Separate contributions to the spectra from χ˜qtpωp and χ˜q'tpωp are also shown by chain and dashed lines, respectively. A¯qtpω is reckoned from structureless background due to electron-hole continuum states β that are almost constant in the ωp region concerned. The widths of the spectral peaks are determined by a phenomenological damping constant Tph of LO-phonon due to lattice anharmonicity: 2/Tph = 0.27 meV (from Ref. [31] with partial modification).

Figure 8. The same as Figure 7 but for undoped GaAs (from Ref. [31] with partial modification).

# Laser-Induced Fano Resonance in Condensed Matter Physics

Ken-ichi Hino1, 2, Yohei Watanabe3, Nobuya Maeshima2 and Muneaki Hase1
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## Abstract

Recent development of laser technology toward the realization of high-power laser has opened up a new research area exploring various fascinating phenomena governed by strongly photoexcited electronic states in diverse fields of science. In this chapter, we review the laser-induced Fano resonance (FR) in condensed matter systems, which is one of the representative resonance effects successfully exposed by strong laser field. The FR of concern sharply differs from FR effects commonly observed in conventional quantum systems where FR is caused by a weak external perturbation in a stationary system in the following two aspects. One is that the present FR is a transient phenomenon caused by nonequilibrium photoexcited states. The other is that this is induced by an optically nonlinear process. Here, we introduce two physical processes causing such transient and optically nonlinear FR in condensed matter, followed by highlighting anomalous effects inherent in it. The first is a Floquet exciton realized in semiconductor superlattices driven by a strong continuous-wave laser, and the second is the coherent phonon induced by an ultrashort pulse laser in bulk crystals.

Keywords: laser, Fano resonance, photodressed states, exciton, dynamic localization, Floquet theorem, coherent phonon, ultrafast phenomena, polaronic quasiparticle

## 1. Introduction

In quantum systems where discrete levels are embedded in energetically degenerate continuum states, resonance phenomenon is likely manifested, that is, characteristic of asymmetric spectral profiles consisting of both a peak and a dip. This is known as Fano resonance (FR) [1]; this is also termed as either Feshbach resonance or many-channel resonance. FR is one of the common and fundamental concepts in diverse fields of physics and chemistry; FR processes are observed, for instance, in strongly interacting Bose-Einstein condensates in an ultracold atomic system [24], superexcited states of molecules [5], a semiconductor quantum dot in an Aharonov-Bohm interferometer [6], an electronic transition near Weyl points strongly coupled with an infrared-active phonon in a Weyl semimetal [7].

In particular, within the restriction just to the FR processes triggered by laser irradiation, these may be classified in terms of the three categories as shown in Table 1. The first category is regarding whether a process is a linear one or a nonlinear one with respect to an order of a laser-matter interaction, as categorized as (a1) and (a2), respectively. For instance, the former is a photoabsorption process [811], and the latter is a multiphoton process [1216]. The second category is regarding whether the process results from a built-in interaction between the discrete level and continuum that is intrinsic to a material itself or from a coupling induced extrinsically by a laser, as categorized as (b1) and (b2), respectively. For instance, the former is the interaction of an electron with a longitudinal optical (LO)-phonon in incoherent Raman scattering [1722], and the electron-electron interaction brings about autoionization and the Auger process [23]. The latter FR process is known as a laser-induced continuum structure [24, 24]. The third category is regarding whether the process is a (quasi)stationary one or a transient one, as categorized as (c1) and (c2), respectively. In other words, this is whether (quasi)time-independent or time-dependent. For instance, the former is induced by a continuous-wave (cw) laser (monochromatic laser) [15, 16, 25, 26], and the latter is by a short pulsed laser [2731].

CategoryCharacteristic
Optical process(a1)Linear (perturbative)(a2)Nonlinear (nonperturbative)
Interaction causing FR(b1)Intrinsic (built-in)(b2)Extrinsic (external)
Light source(c1)Monochromatic, continuous wave (stationary/quasistationary)(c2)Pulsed (transient)

#### Table 1.

Classification of FR into three categories.

It is stressed that for the FR categorized as (a2), its physical characters—such as asymmetry in spectral profile, spectral intensity, resonance position, and spectral width—are controllable in a quantum-mechanic manner by tuning various laser parameters. Thus, it is expected that underlying physics is enriched by intriguing effects inherent in this sort of FR. This differs from most of FR processes observed thus far because of being simply classified as (a1)-(b1)-(c1).

Currently, new research areas have been opened up owing to the progress of laser technology toward the realization of sophisticated high-power light sources. In particular, in the field of condensed matter physics, the development of high-intensity terahertz (THz) wave enables us to explore a photodressed quantum state in which a temporally periodic interaction of THz wave with matter is renormalized in the original quantum state in a nonperturbative manner [3234]. Such an anomalous state is termed as a Floquet state because of ensuring the Floquet theorem [35]. Further, the development of ultrashort pulse laser—with its temporal width being of an order of 10 femtosecond (fs)—enables us to explore ultrafast transitory phenomena governed by strongly photoexcited electronic states. Bearing in mind such current situations, here, we focus exclusively on the laser-induced FR effects realized in the following two physical systems. One is a Floquet exciton formed in semiconductor superlattices (SLs) driven by a strong THz wave, and the other is a coherent phonon (CP) generated by ultrashort pulse laser in bulk crystals. In the light of Table 1, the FR effects of concern sharply differ from those commonly observed in conventional quantum systems classified as (a1)-(b1)-(c1) in the following aspects. Both of the Floquet exciton and the CP are induced by optically nonlinear processes, and hence the significant quantum controls of FR are feasible by means of tuning the respective applied light sources. Further, the Floquet exciton forms manifolds of quasistationary states with quasienergy as a constant of motion, where the FR is mediated by the ac-Zener tunneling caused by the THz wave. Hence, this is classified as (a2)-(b2)-(c1) and is herein termed as dynamic FR (DFR). On the other hand, the CP is a transient phenomenon caused by the built-in interaction of an LO-phonon with nonequilibrium photoexcited carriers. Hence, this is classified as (a2)-(b1)-(c2) and is herein termed as transient FR (TFR).

Below, we survey the present research backgrounds of DFR and TFR in brief. In both cases, the applied electric field of pumping laser is represented as F(t) = F0(t) cos(ωt) with an envelope function F0(t) at time t and the center frequency ω.

To begin with the DFR, this is closely related with the photodressed miniband formation [36]. Here, the cw laser with a constant amplitude F0(t) ≡ Fac gives rise to a nonlinear optical interaction with electron to result in a photodressed miniband with effective width Δeff = Δ0J0(x)∣, where Δ0 and J0(x) represent the width of the original SL miniband and the zeroth-order Bessel function of the first kind with x = eFacd/ℏω, respectively, and e, d, and represent the elementary charge, a lattice constant of the SLs, and Planck’s constant divided by 2π, respectively. Each photodressed miniband forms a sequence of photon sidebands arrayed at equidistant energy intervals of ℏω following the Floquet theorem. The DFR is caused by the interaction due to the ac-Zener tunneling between photon sidebands pertaining to different sequences, and this is coherently controlled by tuning Fac and ω. In particular, it is expected that an anomalous effect attributed to dynamic localization (DL) on DFR is revealed on the occasion that all of the photodressed minibands collapse by tuning x to ensure J0(x) = 0 [3638]. The DL was first observed in electron-doped semiconductor SLs driven by a THz wave [39]. In addition, this has also been observed in diverse physical systems such as a cold atomic gas in one-dimensional optical lattices [40], a Bose-Einstein condensate [41], and light in curved waveguide arrays [4244].

As regards the TFR, this was observed in a lightly n-doped Si crystal immediately after carriers were excited by an ultrashort laser pulse [45], where the speculation was made that the observed FR would show the evidence of the birth of a polaronic-quasiparticle (PQ) likely formed in a strongly interacting carrier-LO-phonon system in a moment [46]. The TFR of concern has been observed exclusively in this system and semimetals/metals such as Bi and Zn [47, 48] till now, however, not observed in p-doped Si and GaAs crystals [49, 50]. Thus far, there are a number of theoretical studies regarding these experimental findings. The time-dependent Schrödinger equation in the system of GaAs was calculated to show the asymmetric shape featuring FR spectra, though apparently opposed to existing experimental results, as mentioned above [51]. Further, the classical Fano oscillator model was presented based on the Fano-Anderson Hamiltonian [52, 53], and the close comparison of the experimental results of the CP signals of Bi was made with the time signal obtained by taking the Fourier transform of Fano’s spectral formula into a temporal region [48]. Recently, the authors have constructed a fully quantum-mechanical model based on the PQ picture in a unified manner on an equal footing between both of polar and nonpolar semiconductors such as undoped GaAs and undoped Si [31]. Here, it has been shown that the TFR is manifested in a flash only before the carrier relaxation time (∼100 fs) in undoped Si, whereas this is absent from GaAs.

Acronyms used in the text and the corresponding meanings are summarized in Table 2. The remainder of this chapter is organized as follows. In Section 2, the theoretical framework is described, where the models of the DFR and TFR are presented separately in Sections 2.1 and 2.2, respectively. The results and discussion are given in Section 3, and the conclusion with summary is given in Section 4. Atomic units (a.u.) are used throughout unless otherwise stated.

AcronymsMeanings
CPCoherent phonon
cwContinuous wave
DFRDynamic FR
DLDynamic localization
FRFano resonance
fsFemtosecond
LOLongitudinal optical
PQPolaronic-quasiparticle
SLSuperlattice
TFRTransient FR
THzTerahertz

#### Table 2.

Summary of acronyms used in text in alphabetical order and corresponding meanings.

## 2. Theoretical framework

### 2.1. Theoretical model for DFR in the photodressed exciton

#### 2.1.1. Optical absorption spectra

The total Hamiltonian ĤDFRt concerned comprises a SL Hamiltonian consisting of field-free Hamiltonians of the conduction (c) and valence (v) bands, a Coulomb interaction between electrons, an intersubband interaction caused by the driving laser F(t) polarized in the direction of crystal growth (the z-axis), and an interband interaction caused by the probe laser f(t) = fp cos(ωpt) with the center frequency ωp and the constant amplitude fp; it is assumed that Fac ≫ fp and ω ≪ ωp. The microscopic polarization defined as pλλ'ktaλkvaλ'kc is examined to shed light on the detail of DFR of the Floquet exciton; 〈O〉 represents the expectation value of the operator O. Here, λ() = (b(), l()), which represents the lump of the SL miniband index b() and the SL site l(). In addition, k represents the in-plane momentum associated with the relative motion of the pair of c band and v band electrons, where the in-plane is defined as the plane normal to the z-axis; hereafter, the relative position conjugate to k is represented as ρ. Further, aλksaλks represents the creation (annihilation) operator of the electron with λ and k in band s.

The equation of motion for the microscopic polarization is given by the semiconductor Bloch equation

 iddt+1T2pλλ′k∥t=[aλk∥v†aλk∥cĤDFRt] (1)

with T2 dephasing time. For the practical purpose of tackling the multichannel scattering problem of exciton, it is convenient to transform it into the equation for p¯ρzvzct defined in the real-space representation as

 p¯ρzvzct=eiωpt∑λ,λ′∫dk∥eik∥⋅ρzvλpλλ′k∥tλ′zc, (2)

where 〈λ| z〉 represents the Wannier function at position z − ld in SL miniband b. The resulting equation becomes

 iddt+1T2−iωpp¯ρzvzct+2π2eiωptf0+td0vcδρδzv−zc=∫dzp¯ρzvztHTBczzct−HTBvzvztp¯ρzzct+Hρzvzcp¯ρzvzct, (3)

where the rotating wave approximation is employed by replacing f(t) by f0+tfp/2eiωpt and d0vc represents the interband dipole moment of a bulk material. Here, the Hamiltonian H(ρ, zv, zc) for the in-plane motion is given by Hρzvzc=ρ2/2m+Vr, where m and V(r) =  − 1/(ε0r) represent an in-plane reduced mass and the Coulomb interaction, respectively, with r=ρ2+zvzc2 and ε0 the dielectricity of vacuum. The nearest-neighbor tight-binding Hamiltonian of the laser-driven SLs is given by HTBszztzĤTBstz, where

 ĤTBst=∑λ=blε0bsλ〉〈λ+−1b+σs4Δbslb〉l+1b+l+1b〈lb−Ft∑λλ′∣λ〉Zλλ′s〈λ′∣, (4)

and ε0bs and Δbs represent the band center and the band width of b, respectively, with σs = 0 (for s = c) and 1 (for s = v). The last term of Eq. (4) represents the dipole interaction induced by the driving laser F(t) with Zλ,λs as a dipole matrix element. It should be noted that the off-diagonal contribution of Zλ,λs with b ≠ b′ induces the ac-Zener tunneling, which plays a significant role of quantum control of DFR, as shown below.

The concerned function p¯ρzvzct can be expressed in terms of the complete set of the Floquet wave functions {ψ(ρ, zv, zc, t)}, that is,

 p¯ρzvzct=∫dE∑βaEβψEβρzvzct (5)

with a as an expansion coefficient. Here, the Floquet wave function ensures the following homogeneous equation associated with the inhomogeneous equation of Eq. (3) as

 iddt+EψEβρzvzct=∫dzψEβρzvztHTBczzct−HTBvzvztψEβρzzct]+HρzvzcψEβρzvzct, (6)

where the temporally periodic boundary condition ψ(ρ, zv, zc, t) = ψ(ρ, zv, zc, t + T) is imposed on it with E and T = 2π/ω as quasienergy and the time period of the driving laser field, respectively. Equation (6) is the Wannier equation of the Floquet exciton of concern. It should be noted that this is cast into the multichannel scattering equations and the Floquet state of ψ(ρ, zv, zc, t) forms a continuum spectrum designated by both E and β with β representing the label of an open channel. Such a multichannel feature is introduced by the strong driving laser F(t) that closely couples an excitonic-bound state with continua; more detail of the multichannel scattering problem is described in Section 2.1.2. The expansion coefficient a is readily obtained by inserting Eq. (5) into Eq. (3) in view of Eq. (6) as

 aEβ=2π2d0vcfp/2E−ωp−iγT∫0Tdt′ψ¯Eβt′, (7)

where ψ¯t=dzψ0zzt and γ = 1/T2.

Since the macroscopic polarization is given by Pt=λ,λdkd0vcpλλkt, the linear optical susceptibility χ(t) with respect to the weak probe laser f(t) is cast into [54]

 χt=d0vc2ε0∫dE∑βOEβtE−ωp−iγ, (8)

where Ot=ψ¯t/T0Tdtψ¯t. Taking the Fourier transform of χ(t) ≡ ∑jeijωtχj(ωp; ω), leads to the expression of the absorption coefficient to be calculated as

 αωpω=ωpC∑jImχjωpω (9)

with C the speed of light; χj=0ωpω vanishes in the limit of Fac→0.

#### 2.1.2. Multichannel scattering problem

The absorption coefficient of Eq. (9) is obtained by evaluating a set of the wave functions, {ψ(ρ, zv, zc, t)}. To do this, first, the wave function is expanded as

 ψEβρzvzct=∑μΦμzvzctFμβρ, (10)

where ρ = ∣ρ∣, and just the contribution of the s-angular-momentum component is incorporated because of little effects from higher-order components. Here, Φμ(zv, zc, t) is the real-space representation of the Floquet state ∣Φμ〉, that is, Φμ(zv, zc, t) = 〈zv, zc| Φμ〉, satisfying ĤTBi/tΦμ=EμΦβ, where ĤTBĤTBc+ĤTBv and Eμ is the μth quasienergy. The index μ is considered as the approximate quantum number μμ¯k with μ¯bcbvnp as a photon sideband index, where bc and bv are SL miniband indexes belonging to the c- and v-bands, respectively, and k and np represent the Bloch momentum of the joint miniband of (bc, bv) and the number of photons relevant to absorption and emission, respectively. The quantum number μ¯ becomes a set of the good quantum numbers with Fac decreasing, while k always remains conserved because of the spatial periodicity in the laser-driven SLs of concern. In view of Eq. (10), Eq. (6) is recast into the coupled equations for the radial wave function Fνβ(ρ), that is,

 ∑μLμνFνβρ=EFμβρ, (11)

where Lμν is an operator given by Lμν = δμν[−(2m)1(d2/2 + ρ1d/) + Eμ] + Vμν(ρ) and Vμν(ρ) is a Coulomb matrix element defined as Vμνρ=T10TdtdzvdzcΦμzvzctVρzvzcΦνzvzct.

The Floquet exciton in the laser-driven SL system pertains to the multichannel scattering problem, because Vμν(ρ) vanishes at ρ ≫ 1. Actually, for a given E, the channel μ satisfying E > Eμ is an open channel, while the channel μ satisfying E < Eμ is a closed channel. Thus, the label μ of Fμβ plays the role of the scattering channel. On the other hand, the label β means the number of independent solutions satisfying Eq. (11). Here, there are same number of independent solutions as open channels, since as many scattering boundary conditions are imposed on Fμβ at ρ ≫ 1; while evanescent boundary conditions that Fμβ vanishes at ρ ≫ 1 are imposed on closed channels. Eq. (11) can be numerically evaluated by virtue of the R-matrix propagation method, which is a sophisticated formalism providing a stable numerical algorithm with extremely high accuracy [55].

It is expected that the DFR of concern is caused by a coupling between photon sidebands mediated by ac-Zener tunneling, as mentioned in Section 1. To see this situation in more detail, Figure 1 shows the interacting two photon sidebands μ¯ and μ¯bcbvnp, where the discrete Floquet excitonic state is supported by the photon sideband μ¯, and this is also embedded in the continuum of the alternative photon sideband μ¯. It is likely that the DFR occurs due to a close coupling between these photon sidebands, and, eventually, the exciton state decays into the continuum state pertaining to μ¯. In fact, it is noted that the Coulomb interaction incorporated in Eq. (6) also gives rise to FR. Defining the difference between the photon numbers of both photon sidebands, namely, Δnp=npnp, the ac-Zener tunneling is featured by Δnp ≠ 0, while the Coulomb coupling is by Δnp = 0. The spectral profile and intensity of FR in the former can be even more effectively controlled than in the latter by modulating the laser parameters Fac and ω, since the degree of magnitude of ac-Zener tunneling depends exclusively on both of the external parameters, differing from the Coulomb interaction. In the region of Fac weak enough to suppress the ac-Zener tunneling, the FR is dominated by the Coulomb coupling, similarly to the conventional FR observed in the original SLs without laser irradiation [56].

### Figure 1.

Schematic diagram of the DFR formation in the Floquet excitonic system. This shows the coupling mechanism that a bound state supported by the sideband μ¯ interacts with a continuum state belonging to the sideband μ¯ by the ac-Zener tunneling to result in the Fano decay (from Ref. [15] with partial modification).

### 2.2. Theoretical model for TFR in the CP generation

#### 2.2.1. Introduction of polaronic quasiparticle operators

The total Hamiltonian ĤTFR of concern is given by ĤTFR=Ĥe+Ĥt+Ĥp+Ĥep. Here, Ĥe represents an electron Hamiltonian including an interelectronic Coulomb potential, where a two-band model is employed that consists of the energetically lowest c band and the energetically highest valence v band, and a creation (annihilation) operator of electron with band index b and Bloch momentum k is represented as abkabk. Ĥp represents an LO-phonon Hamiltonian, where a creation (annihilation) operator of LO-phonon with an energy dispersion ωqLO at momentum q is represented as cqcq. Further, Ĥt and Ĥep represent interaction Hamiltonians of electron with the pump pulse and the LO-phonon, respectively. These are given as follows:

 Ĥ′t=−12∑b,b′=b,kΩbb′tabk†ab′k+Ωbb′∗tab′k†abk, (12)

where Ωbb(t) = dbbF(t) with dbb an electric dipole moment between b and b bands, and

 Ĥe−p=∑b,q,kgbqcqabk+q†abk+gbq∗cq†abk†abk+q, (13)

where gbq is a coupling constant of b band electron with the LO-phonon. Here, let the envelope of F(t) be of squared shape just for the sake of simplicity, that is, F0(t) = F0θ(t + τL/2)θ(τL/2 − t) with F0 constant, where temporal width τL is of the order of a couple of 10 fs at most, satisfying τL2π/ωqLO.

The equation of motion of a composite operator Aqkbbab,k+qabk is considered below, where this represents an induced carrier density with spatial anisotropy determined by q; ∣q∣ is finite, though quite small, that is, q ≠ 0. It is convenient to remove from this equation high-frequency contributions by means of the rotating-wave approximation [57] by replacing Aqkbb by eiω¯bbtA¯qkbb, where ω¯cv=ω, ω¯vc=ω, and ω¯bb=0. Thus, the resulting equation of motion is as follows:

 −iddt+1Tqkbb′A¯q†kbb′=ĤetA¯q†kbb′−A¯q†kbb′ω¯bb′+Ĥe−pA¯q†kbb′≈∑k˜b˜b˜′A¯q†k˜b˜b˜′Z¯qk˜b˜b˜′kbb′+Ĥe−pA¯q†kbb′, (14)

where the total electronic Hamiltonian is defined as Ĥet=Ĥe+Ĥt, the first commutator in the right-hand side of the first equality is evaluated by making a factorization approximation, and Tq(kbb) represents a phenomenological relaxation time constant relevant to Aqkbb. Further, Z¯q represents a non-Hermitian matrix, which is a slowly varying function in time, since rapidly time-varying contributions are removed owing to the above rotating-wave approximation, aside from the discontinuity at t =  ± τL/2.

Bearing in mind this situation, we tackle left and right eigenvalue problems of Z¯q [58], described by UqLZ¯q=EqUqL and Z¯qUqR=UqREq, respectively, in terms of an adiabatic-eigenvalue diagonal matrix Eq and the associated biorthogonal set of adiabatic eigenvectors UqLUqR with time t fixed as a parameter. The orthogonality relation and the completeness are read as UqLUqR=1 and UqRUqL=1, respectively [58]. Given the relation Z¯q=UqREqUqL, Eq. (14) is recast into the form of adiabatic coupled equations:

 −idBqα†dt=Bqα†Eqα+i∑α′Bqα′†Wqα′α+Ĥe−pBqα†, (15)

where the operator Bqα is defined as BqαA¯qUqαR, WqααdUqαL/dtUqαR,+UqαLTq1UqαR, and Eqα(t) is complex adiabatic energy at time t associated with the operator Bqαt thus introduced. Hereafter, this operator is termed as a creation operator of quasiboson, and the corresponding annihilation operator is defined as BqαUqαRA¯q. The set of eigenstates {α} is composed of continuum states represented as β with eigenenergy Eqβ and a single discrete energy state represented as α1 with eigenenergy Eqα1, that is, {α} = ({β}, α1); the state β corresponds to electron-hole continuum arising from interband transitions, and the state α1 corresponds to a plasmon-like mode. It is noted that the relation of BqαtBqαt=δqqδαα is assumed, though Bqα(t) and Bqαt do not satisfy the equal-time commutation relations for a real boson, where X̂ means an expectation value of operator X̂ with respect to the ground state; the validity of the criterion of this relation is discussed in detail in Ref. [31].

Eq. (13) is rewritten as Ĥep=q,αcqBqαMqα+MqαBqαcq with an effective coupling between quasiboson and LO-phonon as Mqα=kbgbqUqαLkbb. Thus, the commutator in Eq. (15) is approximately evaluated as ĤepBqαMqα'cq, though Mqα'Mqα. On the other hand, the equation of motion of the LO-phonon is described by idcq/dt=αBqαMqα+cqωqLO. Both of the equations of motion for Bq and cq are integrated into a single equation in terms of matrix notations as follows:

 −iddtBq†cq†≈Bq†cq†hq+iBq†Wq0. (16)

Here, the non-Hermitian matrix hq ≡ {hqγγ} given by hq=EqMqMqωqLO is introduced with γ , γ = 1 ∼ N + 2, where N represents the number of electron-hole (discretized) continua, namely, β = 1 ~ N, aside from two discrete states attributed to a plasmon-like mode and an LO-phonon mode designated by α1 and α2, respectively: {γ} = ({β}, α1, α2). In the system of the TFR of concern, the case is exclusively examined that both of the discrete levels of α1 and α2 are embedded into the continua {β}. Thus, the following coupled equations for the multichannel scattering problem are taken account of

 ∑γ′hqγγ′Vqγ′βR=VqγβREqβ, (17)

where VqβR=VqγβR is the right vector representing the solution for given energy Eqβ; similarly to Eq. (11) for the DFR, the indices of γ and β play the roles of a scattering channel and the number of independent solutions, respectively. Eq. (17) provides the theoretical basis on which both of LO-phonon and plasmon-like modes are brought into connection with the CP dynamics on an equal footing. In terms of this vector, a set of N-independent operators, Fqββ=1N, is defined as

 Fqβ†=∑β′Bqβ′†Vqβ′βR+Bqα1†Vqα1βR+cq†Vqα2βR. (18)

In addition, the left vector VqβL=VqβγL associated with VqβR is introduced to ensure the inverse relations Bqα=FqVqαL and cq=FqVqα2L, where VqLVqR=1 and VqRVqL=1. Hereafter, the operator Fqβt thus introduced is termed as a creation operator of PQ, and then the corresponding annihilation operator is Fqβ(t); these are not bosonic operators. The bosonization scheme for the PQ operators is similar to that for the quasiboson operators, where the PQ ground state is given by the direct product of the ground states of quasiboson and LO-phonon and Eqβ(t) is read as the single-PQ adiabatic energy at time t with mode qβ.

Given Eq. (18), Eq. (16) becomes adiabatic coupled equations for Fq:

 −iddtFqβ†≈Fqβ†Eqβ+i∑β′Fqβ′†Iqβ′β, (19)

where Iq=dVqL/dtVqR+VqLWqVqR. In terms of Fq and Fq, the associated retarded Green function is given by [59]

 Gqββ′Rtt′=−iθt−t′〈FqβtFqβ′†t′〉. (20)

#### 2.2.2. Transient induced photoemission spectra

A weak external potential fq(t) additionally introduced in the transient and nonequilibrium system of concern induces an electron density nqindt given by

 nqindt=14πV∫−∞tdt′χqttt′fqt′, (21)

based on the linear response theory [59, 60] with V the volume of crystal. It is noted that nqindt is nonlinear with respect to the pump field. Here, χqttt represents the retarded longitudinal susceptibility that depends on passage of t and the relative time τ = t − t, differing from equilibrium systems depending solely on τ. Introducing a retarded longitudinal susceptibility due to the electron-induced interaction and that of an LO-phonon-induced interaction represented as χq(t, t) and χqtt, respectively, χqttt is given by [59]

 χqttt′=χqtt′+χq′tt′. (22)

Let fq(t) be assumed to be fq(t) = fq0 δ(t − tp) in the present system; fq0 is independent of t, and tp represents the time at which fq(t) probes transient dynamics of concern. Thus, it is seen that χqtttp reveals the way of alteration of nqindt after tp, since Eq. (21) becomes nqindt=fq0χqtttpθttp/4πV.

In terms of χqttp+τtp, the dielectric function εq(tp + τ, tp) is readily obtained, and by taking the Fourier transform of it as ε˜qtpωp=0eiωpτεqtp+τtp, this leads to a transient absorption coefficient αq(tp; ωp) at time tp. This is given by αq(tp; ωp) = ωAq(tp; ωp)/n(tp; ωp)C, where Aqtpωp=Imε˜qtpωp and n(tp; ωp) represents the index of refraction. It is remarked that according to the definition of the sign of ωp made above, transient photoemission spectra, where Aq(tp; ωp) < 0, peak at positive ωp, while transient photoabsorption spectra, where Aq(tp; ωp) > 0, peak at negative ωp. For the sake of the later convenience, the transient induced photoemission spectra are defined as A¯qtpωp=Aqtpωp.

Based on the PQ model developed in Section 2.2.1, χq(t, t) and χqtt can be explicitly expressed in terms of the retarded Green function given by Eq. (20). Here, the obtained results are shown below; for more detail, consult Ref. [31]:

 χq∗tt′=4πV∑αα′ββ′NqαL∗tVqαβLtGqββ′Rtt′Vqβ′α′L†t′Nqα′Lt′, (23)

where NqαL=kbUqαLkbb, and this is equivalent to a normalization constant of the left vector UqαL:

 χq′tt′=4πVgq′2D¯′qRtt′+D¯′−qRtt′∗, (24)

where gq is a constant in proportion to (gcq + gvq)/2 and

 D¯′qRtt′=∑ββ′Vqα2βLtGqββ′Rtt′Vqβ′α2L†t′. (25)

Finally, the TFR dynamics caused by the CP generation is mentioned based on the PQ picture. As shown in Figure 2, the LO-phonon state α2 is embedded in the quasiboson state β, and the effective coupling between both states induces the formation of transient PQ FR state. This composite state is deexcited into the PQ ground state via two paths: one is the transient photoemission from α2, and the other is from β. It is likely that these two paths interfere to give rise to asymmetry in spectra. It is remarked that the contribution from the plasmon-like mode α1 is omitted because of a negligibly smaller effect on the TFR.

### Figure 2.

Schematic diagram of the TFR dynamics based on the PQ picture, where the LO-phonon state α2 is embedded in the quasiboson state β. The PQ FR state composed of α2 and β is deexcited by induced photoemission process. For more detail, consult the text (from Ref. [31] with partial modification).

## 3. Results and discussion

### 3.1. DFR in the photodressed exciton

For the calculations of DFR spectra, the semiconductor SLs of GaAs/Ga0.75Al0.25As are employed with 35/11 monolayers (ML) for the well and barrier thickness, where 1 ML = 2.83 Å. Here, 14 photon sidebands of μ¯=1133 and [2, 1, −3 ∼ 3] are incorporated by setting ω to 91 meV; this equals to the difference between the centers of the joint minibands of (1,1) and (2,1). Other photon sidebands are neglected for the sake of simplicity.

First of all, the calculated quasienergy bands {Eμ} as a function of Fac are shown in Figure 3 to illustrate the effect of ac-Zener coupling. The two photon sidebands labeled by μ1 = [1, 1, 0, k] and μ2 = [2, 1, −1, k] are mixed by the coupling induced by the driving laser F(t). With the increase of Fac, the quasienergy bands are branched into two distinct photon sidebands, termed as the upper sideband μ+ and the lower sideband μ, where both labels of μ1 and μ2 are no longer good quantum numbers, aside from k. It is noted that both of μ+ and μ form dynamic localization showing band collapse around two points Fac = FDL1 ≡ 170 kV/cm and FDL2 ≡ 395 kV/cm. Figure 4 shows the absorption spectra α(ωp; ω) obtained by solving Eq. (9) in the range of Fac from 10 to 450 kV/cm. Asymmetric spectral profiles characteristic of DFR are discerned at the arrowed positions of ωp when Fac≥ 150 kV/cm, where all peaks are followed by dips. These peaks are located just below the upper sideband μ+, thereby being blue shifted. Consulting Figure 1, the DFR is dominantly formed by the interaction between one open channel μ and one closed channel μ+.

### Figure 3.

The quasienergy ℰμ as a function of Fac. ω is set to 91 meV. The vertical double arrows represent the original SL miniband widths corresponding to the photon sidebands of μ1 and μ2 (from Ref. [16] with partial modification).

### Figure 4.

Absorption spectra α(ωp; ω) as a function of ωp for Fac=50–450 (kV/cm) with ω= 91 meV. A series of the arrowed spectral profiles are examined exclusively in the text. The quasienergies shown in Figure 3 are also plotted (dotted lines) (from Ref. [16] with partial modification).

To deepen the understanding of the DFR exciton, its characteristic quantities determining the spectral profiles are extracted from α(ε) ≡ α(ωp; ω) arrowed in Figure 4 by being fitted to Fano’s Formula [1]:

 αε=α0ε+qF2ε2+1, (26)

in the vicinity of an excitonic resonance quasienergy Eex, where ε = 2(ωp − Eex)/Γ with the spectral width Γ and the asymmetry parameter (Fano’s q-parameter) q(F) < 0. Figure 5(a) shows the evaluated values of ∣1/q(F)∣ and Γ as a function of Fac, while Figure 5(b) shows the peak intensity α(0) = α0[q(F)]2 ≡ αmax and background spectra α(±∞) = α0 as a function of Fac. It is seen that these functions are affected pronouncedly by Fac; in particular, extrema are formed around Fac = FDL1. It is remarked that with the decrease in ∣1/q(F)∣ and Γ, the DFR state becomes a pure bound state. In addition, there still exist faint extrema around Fac = FDL2 in the concerned quantities except Γ. Therefore, the DL is considered to fulfill a special role of the quantum control of photodressed excitonic states.

### Figure 5.

The DFR-related quantities as a function of Fac with the fixed value of ω = 91 meV. The calculated results represented by the filled symbols are connected by the solid lines in order to aid the presentation. (a) ∣1/q(F)∣ and Γ and (b) α0 and αmax (from Ref. [16] with partial modification).

For the purpose of confirming such an effect of DL and the pronounced Fac dependence of related quantities on the excitonic DFR, one evaluates the transition probability between the photon sidebands of μ1 and μ2 due to the ac-Zener coupling; this value is represented as M(Fac) as a function of Fac. This corresponds to the degree of mixing between these two photon sidebands. M(Fac) is readily obtained by solving the associated coupled equations between μ1 and μ2 in an approximate manner of neglecting contributions from all other photon sidebands [16]. Given Δε and v as the difference of ac-Zener-free quasienergies between μ1 and μ2, and the matrix element of the ac-Zener coupling between them, respectively, M(Fac) is provided as

 MFac=sinφ/22=121−11+z2, (27)

where z ≡ tan φ = 2∣v∣/∣Δε∣. With x = Facd/ω, v and Δε are evaluated as v ∝ x and Δε ∝ cos(kd)J0(x), respectively; see Section 1. Thus, one has z = 2x/ηJ0(x) where η is a proportional constant between Δε and v. According to Eq. (27), for finite values of η, with the increase of Fac, M(Fac) increases from 0 to 1/2 in an oscillating manner; for more detail of the shape of M(Fac) for several values of η, consult Ref. [16].

The alteration pattern of M(Fac) with respect to Fac looks somewhat similar to the shapes of the DFR-related functions shown in Figure 5. In particular, it is noted that M(Fac) has extrema at zeros of J0(x), which just correspond to DL concerned here; that is, M(Fac) shows extrema at Fac = FDL1 and FDL2. In fact, M(Fac) shows a clear extremum at Fac = FDL1, while the second extremum at Fac = FDL2 is not obviously discernible. This is understood by the behavior that the oscillating component incorporated in J0(x) is overwhelmed by the ac-Zener coupling v for large x. Therefore, it is concluded that the characteristic Fac dependence of the functions of ∣1/q(F)∣, Γ, αmax, and α0 is attributed to the competition between the ac-Zener effect and the band width of the free electron-hole pair states in the vicinity of the DL positions.

Finally, one mentions in brief the ω dependence of the physical quantities ∣1/q(F)∣ and Γ at Fac=180 kV/cm in the vicinity of Fac = FDL1. As shown in Figure 6(a), ∣1/q(F)∣ decreases sharply with the increase in ω, while Γ is maximized around ω = 91 meV at which the centers of two photon sidebands μ1 and μ2 coincide. The tendency of ∣1/q(F)∣ is in harmony with the ω dependence of the ratio of dc to do, namely, rd = dc/do, as shown in Figure 6(b), where dc and do represent a dipole-transition matrix from the ground state to the closed channel μ+ and that to the open channel μ, respectively. Actually, rd is in proportion to q(F) [16]. Such alteration of rd is interpreted on the basis of the anticrossing formation of photon sidebands of μ+ and μ due to the Autler-Townes effect, though not discussed here; for more detail, consult Ref. [16]. Thus, it seems that comparing Figure 6(a) with Figure 5(a), the q parameter is even more controllable by changing ω than by Fac.

### Figure 6.

The DFR-related quantities as a function of ω with the fixed value of Fac = 180 meV. The calculated results represented by the filled symbols are connected by the solid lines in order to aid the presentation. (a) ∣1/q(F)∣ and Γ and (b) ∣rd∣ (from Ref. [16] with partial modification).

### 3.2. TFR in the CP generation

For the calculations of TFR spectra of undoped Si and undoped GaAs, the associated materials parameters employed in the present study are shown in Ref. [31], while parameters of a square-shaped pulse laser employed are as follows. For undoped Si and undoped GaAs, detuning with reference to energy band gap Δω=82 and 73 meV, respectively, temporal width τL=15 fs, pulse area AL=0.12π and 0.20π, respectively, and the maximum excited electron density Nex0=6.31 × 1017 and 5.30 × 1017 cm3, respectively; by Δω > 0, it is meant that opaque interband transitions with real excited carriers are examined. Further, two time constants of T12 and Tq12 are introduced, which represent phenomenological damping time constants of induced carrier density with isotropic momentum distribution and anisotropic momentum distribution with q, respectively. The temporal region t < T12 is termed as the early-time region during which a great number of carriers still stay in excited states, and the quantum processes govern the CP dynamics; Tq12 is approximately equal to Tq(kbb) introduced in Eq. (14). On the other hand, the temporal region t ≳ T12 is termed as the classical region. For the present calculations, Tq12 and T12 are set equal to 20 and 90 fs, respectively. As regards experimental estimates of these time constants for Si, Tq12 and T12 extracted from the CP measurements in Ref. [45] are 16 and 100 fs, respectively, at Nex0=4×1019cm3.

Transient induced photoemission spectra A¯qtpωp defined in Section 2.2.2 show the change of excited electronic structure due to the pump field at probe time tp, and this is crucial to understand the TFR accompanied by CP generation. The total retarded longitudinal susceptibility consists of the dynamically screened Coulomb interaction induced by electron and the LO-phonon-induced interaction. That is, χ˜qttpωp=χ˜qtpωp+χ˜q'tpωp, where this is a Fourier transform of Eq. (22) with respect to τ into the ωp domain. In the small transferred momentum q limit, χ˜qtpωp is proportional to |q|2, while χ˜q'tpωp is proportional to |q|2 for the Fröhlich interaction exclusively for a polar crystal such as GaAs and to |q|4 for the deformation potential interaction. This difference is attributed to the fact that the Fröhlich interaction is of long range, and the deformation potential interaction is of short range. It is noted that in a nonpolar crystal such as Si, a dipole transition for lattice absorption vanishes in the limit of q = 0 because of the presence of spatial inversion symmetry [61].

In Figures 7 and 8, A¯qtpωp of Si and GaAs as a function of ωp is shown, respectively, by solid lines at tp equal to t1 ≡ 15 ,  t2 ≡ 65 and t3 ≡ 100 fs, where the separate contributions from χ˜qtpωp and χ˜q'tpωp are also shown by chain and dashed lines, respectively. χ˜qtpωp and χ˜q'tpωp are mostly governed by the plasmon-like mode α1 and the LO-phonon mode α2, respectively. In both figures, it is seen that just χ˜q'tpωp contributes to the formation of spectral peaks and becomes dominant over χ˜qtpωp in the classical region.

Figure 7(a) shows A¯qtpωp of Si at tp = t1 < Tq12, where the obtained continuum spectra are governed by the contribution from χ˜qtpωp, whereas the contribution from χ˜q'tpωp is negligibly small due to the proportion of it to |q|4. In Figure 7(b) at tp = t2 with Tq12 < tp < T12, the contributions from χ˜qtpωp are damped to be comparable to those from χ˜q'tpωp. It is noted that asymmetric spectra characteristic of FR are manifested with a dip followed by a peak. This is in sharp contrast with a symmetric Lorentzian profile shown in Figure 7(c) at tp = t3 > T12. As regards A¯qtpωp of GaAs, it is shown in Figure 8(a) that at tp = t1, a pronounced peak due to the α2 mode, is superimposed with a continuum background composed of χ˜qtpωp and χ˜q'tpωp with comparable order, since both are in proportion to |q|2. The spectra at tp = t2 shown in Figure 8(b) are dominated by χ˜q'tpωp, differing a lot from those shown in Figure 7(b) of Si. The spectra at tp = t3 in Figure 8(c) are similar to those in Figure 7(c).

### Figure 7.

A¯qtpωp of undoped Si (solid line) as a function of ωp at tp equal to (a) 15 fs, (b) 65 fs, and (c) 100 fs. Separate contributions to the spectra from χ˜qtpωp and χ˜q'tpωp are also shown by chain and dashed lines, respectively. A¯qtpω is reckoned from structureless background due to electron-hole continuum states β that are almost constant in the ωp region concerned. The widths of the spectral peaks are determined by a phenomenological damping constant Tph of LO-phonon due to lattice anharmonicity: 2/Tph = 0.27 meV (from Ref. [31] with partial modification).

### Figure 8.

The same as Figure 7 but for undoped GaAs (from Ref. [31] with partial modification).

The origin of the manifestation of TFR in Si shown in Figure 7(b) is examined below. The principal difference between Si and GaAs observed here is attributed just to the effective coupling Mqβ between quasiboson and LO-phonon aside from less significant difference in other material parameters; this appears in the matrix hq introduced in Eq. (16), and the approximation of MqMq' is employed here. The primitive coupling constant gbq incorporated in Mqβ consists of gbqD and gbqF representing the coupling constants due to a phenomenological LO-phonon deformation potential interaction and the Fröhlich interaction, respectively, that is, gbq=gbqD+gbqF. Here, gbqD is real and approximately independent of q, while gbqF is pure imaginary and gbqFq1 [61]. In a nonpolar crystal such as Si, gbq=gbqD, whereas in a polar or partially ionic crystal such as GaAs, gbqF is much dominant to gbqD, namely, gbqgbqF. Actually, the phase of Mqβ is almost determined by that of gbq, since a residual factor defining Mqβ is almost considered real. Thus, Mqβ is a complex number given by Mqβ = ∣Mqβeiφqβ in general; φqβ = 0 , π for Si, while φqβ =  ± π/2 for GaAs.

Next, discussion is made on how such difference of Mqβ affects the spectral profile of A¯qtpωp based on the PQ picture depicted in Figure 2. It is seen that there are two transition paths for the process: one is a direct path mediated by an optical transition matrix Dqα2r from LO-phonon state α2 to the PQ ground state, and the other is a two-step resonant path mediated by Mqβ from α2 to quasiboson state β, followed by a deexcited process mediated by an optical transition matrix Dqα2c from β to the PQ ground state. Accordingly, owing to Shore’s model [62], the induced photoemission spectra in the proximity of ωpωqLO is read as

 A¯qtpωp≈Cqβ+Aqα2ωp−ωqLO+Bqα2Γqα2/2ωp−ωqLO2+Γqα2/22, (28)

where a set of Shore’s spectral parameters of Aqα2, Bqα2, and Cqβ are provided by

 Aqα2=2∣Dqβc∣∣Dqα2r∣∣Mqβ∣cosφqβBqα2=−2∣Dqβc∣∣Dqα2r∣∣Mqβ∣sinφqβ+Dqα2r2Mqβ2/Γqα2/2Cqβ=Dqβc2 (29)

and the natural spectral width is represented by Γqα2 = 2πρqα2|Mqα2|2; ρqα2 and Mqα2 are the density of state of quasiboson and the coupling matrix at Eqβ=ωqLO, respectively. The associated Fano’s q parameter is determined in terms of Shore’s parameters as qqα2tp=rqα2tp+σqα2tprqα2tp2+1 with rqα2tp=Bqα2/Aqα2 and σqα2tp=Aqα2/Aqα2.

An asymmetric spectral profile is exclusively determined by Aqα2. It is seen that Aqα2tp vanishes for φqβ =  ± π/2 and A¯qtpωp becomes of symmetric shape with ∣qqα2(tp)∣ infinite. The spectral profile of GaAs shown in Figure 8(b) corresponds to this case. For φqβ ≠  ± π/2, both Aqα2tp and Bqα2(tp) are finite, and A¯qtpω becomes of asymmetric shape with ∣qqα2(tp)∣ finite. The spectral profile of Si shown in Figure 7(b) corresponds to this case, where φqβ ≈ 0 , π. For Figures 7(c) and 8(c), since Dqα¯c and ∣Mqβ∣ become negligibly small, A¯qtpω is governed by the second term of the expression of Bqα2(tp), and this becomes symmetric with Γqα2 ≈ 0. To conclude, the effective coupling Mqβ around EqβωqLO plays the crucial role of the manifestation of TFR, and the asymmetry of profile is mostly determined by φqβ as long as ∣Mqβ∣ is still large.

Finally, the manifestation of TFR of Si is discussed from the viewpoint of the allocation of time constants Tq12 and T12, where one sets Tq12 < T12. This is an important issue for deepening the understanding of TFR. As shown in Figure 7(b), in the region of Tq12 ≲ tp < T12, the asymmetric spectral profile bursts into view from the structureless continuum χ˜qtpω. Actually, in the early-time region of tp < T12, the excited carrier density is still populated enough around the energy region of EqβωqLO to couple strongly with LO-phonon, while the effect of χ˜qtpω is much suppressed in the region of Tq12 ≲ tp. As regards a different allocation of these time constants, for instance, Tq12 ∼ T12, the TFR profile is no longer observed in the region of tp < T12, since this is covered with still dominant contributions from χ˜qtpω, and the effect of Mqβ becomes too small to cause TFR in the region of tp ≈ T12. Therefore, the allocation of time constants such as Tq12 < T12 is a necessary condition for realizing the TFR of Si in A¯qtpω; otherwise this never appears.

## 4. Conclusion

Transient and optically nonlinear FR in condensed matter is examined here, which differs from conventional FR processes caused by a weak external perturbation in a stationary system. In particular, the following two FR processes are discussed: one is the DFR of Floquet exciton realized in semiconductor superlattices driven by a strong cw laser, and the other is the TFR accompanied by the CP generated by an ultrashort pulse laser in bulk crystals of undoped Si and undoped GaAs.

It is shown that the physical quantities relevant to the DFR spectra can be controlled by modulating Fac and ω. In particular, the quantities as a function of Fac take the extrema due to the ac-Zener coupling between the photon sidebands of μ1 and μ2, when Fac is suitably adjusted to satisfy the DL condition. Further, the strong ω dependence is explained on the basis of the Autler-Townes effect forming the anticrossing between these two photon sidebands. It is remarked that the spectral width shown in Figures 5 and 6 seems too small to be confirmed by experiments. Actually, in the present calculations, the Coulomb many-body effect is neglected. At least at the Hartree-Fock level, the vertex correction to the Rabi energy would make the net ac-Zener coupling stronger to result in such a great DFR width that experimental measurement would be accessible.

As regards the TFR spectra, the PQ model succeeds in demonstrating the appearance of asymmetric spectral profile in Si in a flash, whereas the profile observed in GaAs remains symmetric; the obtained results are in harmony with the existing experimental ones [45]. The difference between Si and GaAs is attributed to the phase factor of the effective coupling Mqβ(tp). To conclude, it is found that in order to realize the TFR in the CP dynamics, the following conditions are to be fulfilled simultaneously. First, the coupling of an LO-phonon with an electron-hole continuum is conducted by the LO-phonon deformation potential interaction rather than by the Fröhlich interaction. Second, photoexcited carriers are populated enough around the energy region EqβωqLO in the early-time region Tq12 < tp < T12 with Tq12 ≪ T12.

## Acknowledgements

This work was supported by JSPS KAKENHI Grants No. JP21104504, JP23540360, and JP15K05121.

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