Classification of FR into three categories.

## 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 [2–4], 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 [8–11], and the latter is a multiphoton process [12–16]. 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 [17–22], 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 [2–4, 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 [27–31].

Category | Characteristic | |||
---|---|---|---|---|

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) |

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 [32–34]. 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*) = *F*_{0}(*t*) cos(*ωt*) with an envelope function *F*_{0}(*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 *F*_{0}(*t*) ≡ *Fac* gives rise to a nonlinear optical interaction with electron to result in a photodressed miniband with effective width *Δeff* = *Δ*_{0}∣*J*_{0}(*x*)∣, where *Δ*_{0} and *J*_{0}(*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 *ℏ* represent the elementary charge, a lattice constant of the SLs, and Planck’s constant divided by 2*π*, 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 *J*_{0}(*x*) = 0 [36–38]. 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 [42–44].

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.

Acronyms | Meanings |
---|---|

CP | Coherent phonon |

cw | Continuous wave |

DFR | Dynamic FR |

DL | Dynamic localization |

FR | Fano resonance |

fs | Femtosecond |

LO | Longitudinal optical |

PQ | Polaronic-quasiparticle |

SL | Superlattice |

TFR | Transient FR |

THz | Terahertz |

## 2. Theoretical framework

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

#### 2.1.1. Optical absorption spectra

The total Hamiltonian *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 *F*_{ac} ≫ *fp* and *ω* ≪ *ωp*. The microscopic polarization defined as *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, **k**_{∥} in band *s*.

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

with *T*_{2} dephasing time. For the practical purpose of tackling the multichannel scattering problem of exciton, it is convenient to transform it into the equation for

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

where the rotating wave approximation is employed by replacing *f*(*t*) by **ρ**, *zv*, *zc*) for the in-plane motion is given by *m*_{∥} and *V*(*r*) = − 1/(*ε*_{0}*r*) represent an in-plane reduced mass and the Coulomb interaction, respectively, with *ε*_{0} the dielectricity of vacuum. The nearest-neighbor tight-binding Hamiltonian of the laser-driven SLs is given by

and *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 *b* ≠ *b*′ induces the ac-Zener tunneling, which plays a significant role of quantum control of DFR, as shown below.

The concerned function *ψEβ*(**ρ**, *zv*, *zc*, *t*)}, that is,

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

where the temporally periodic boundary condition *ψEβ*(**ρ**, *zv*, *zc*, *t*) = *ψEβ*(**ρ**, *zv*, *zc*, *t* + *T*) is imposed on it with *E* and *T* = 2*π*/*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 *ψEβ*(**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 *aEβ* is readily obtained by inserting Eq. (5) into Eq. (3) in view of Eq. (6) as

where *γ* = 1/*T*_{2}.

Since the macroscopic polarization is given by *χ*(*t*) with respect to the weak probe laser *f*(*t*) is cast into [54]

where *χ*(*t*) ≡ ∑*jeijωtχj*(*ωp*; *ω*), leads to the expression of the absorption coefficient to be calculated as

with *C* the speed of light; *Fac*→0.

#### 2.1.2. Multichannel scattering problem

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

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 *μ* is the *μ*th quasienergy. The index *μ* is considered as the approximate quantum number *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 *F*_{ac} 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,

where *Lμν* is an operator given by *Lμν* = *δμν*[−(2*m*_{∥})^{1}(*d*^{2}/*dρ*^{2} + *ρ*^{1}*d*/*dρ*) + *μ*] + *Vμν*(*ρ*) and *Vμν*(*ρ*) is a Coulomb matrix element defined as

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* > *μ* is an open channel, while the channel *μ* satisfying *E* < *μ* 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 *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 *F*_{ac} 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 *F*_{ac} 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].

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

#### 2.2.1. Introduction of polaronic quasiparticle operators

The total Hamiltonian *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 **q** is represented as

where Ω_{bb′}(*t*) = *d*_{bb′}*F*(*t*) with *d*_{bb′} an electric dipole moment between *b* and *b*^{′} bands, and

where *g*_{bq} 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, *F*_{0}(*t*) = _{0}*θ*(*t* + *τL*/2)*θ*(*τL*/2 − *t*) with _{0} constant, where temporal width *τL* is of the order of a couple of 10 fs at most, satisfying

The equation of motion of a composite operator **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

where the total electronic Hamiltonian is defined as *T*_{q}(**k***bb*^{′}) represents a phenomenological relaxation time constant relevant to *t* = ± *τL*/2.

Bearing in mind this situation, we tackle left and right eigenvalue problems of _{q} and the associated biorthogonal set of adiabatic eigenvectors *t* fixed as a parameter. The orthogonality relation and the completeness are read as

where the operator _{qα}(*t*) is complex adiabatic energy at time *t* associated with the operator *α*} is composed of continuum states represented as *β* with eigenenergy _{qβ} and a single discrete energy state represented as *α*_{1} with eigenenergy _{qα1}, that is, {*α*} = ({*β*}, *α*_{1}); the state *β* corresponds to electron-hole continuum arising from interband transitions, and the state *α*_{1} corresponds to a plasmon-like mode. It is noted that the relation of *B*_{qα}(*t*) and

Eq. (13) is rewritten as

Here, the non-Hermitian matrix *h*_{q} ≡ {*h*_{qγγ′}} given by *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 *α*_{1} and *α*_{2}, respectively: {*γ*} = ({*β*}, *α*_{1}, *α*_{2}). In the system of the TFR of concern, the case is exclusively examined that both of the discrete levels of *α*_{1} and *α*_{2} are embedded into the continua {*β*}. Thus, the following coupled equations for the multichannel scattering problem are taken account of

where _{qβ}; 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,

In addition, the left vector *F*_{qβ}(*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 _{qβ}(*t*) is read as the single-PQ adiabatic energy at time *t* with mode **q***β*.

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

where *F*_{q} and

#### 2.2.2. Transient induced photoemission spectra

A weak external potential *f*_{q}(*t*) additionally introduced in the transient and nonequilibrium system of concern induces an electron density

based on the linear response theory [59, 60] with *V* the volume of crystal. It is noted that *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 *χ*_{q}(*t*, *t*^{′}) and

Let *f*_{q}(*t*^{′}) be assumed to be *f*_{q}(*t*^{′}) = *f*_{q0} *δ*(*t*^{′} − *tp*) in the present system; *f*_{q0} is independent of *t*^{′}, and *tp* represents the time at which *f*_{q}(*t*^{′}) probes transient dynamics of concern. Thus, it is seen that *tp*, since Eq. (21) becomes

In terms of *ε*_{q}(*tp* + *τ*, *tp*) is readily obtained, and by taking the Fourier transform of it as *α*_{q}(*tp*; *ωp*) at time *tp*. This is given by *α*_{q}(*tp*; *ωp*) = *ωA*_{q}(*tp*; *ωp*)/*n*(*tp*; *ωp*)*C*, where *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 *A*_{q}(*tp*; *ωp*) < 0, peak at positive *ωp*, while transient photoabsorption spectra, where *A*_{q}(*tp*; *ωp*) > 0, peak at negative *ωp*. For the sake of the later convenience, the transient induced photoemission spectra are defined as

Based on the PQ model developed in Section 2.2.1, *χ*_{q}(*t*, *t*^{′}) and

where

where *g*_{cq} + *g*_{vq})/2 and

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 *α*_{2} 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 *α*_{2}, 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 *α*_{1} is omitted because of a negligibly smaller effect on the TFR.

## 3. Results and discussion

### 3.1. DFR in the photodressed exciton

For the calculations of DFR spectra, the semiconductor SLs of GaAs/Ga_{0.75}Al_{0.25}As are employed with 35/11 monolayers (ML) for the well and barrier thickness, where 1 ML = 2.83 Å. Here, 14 photon sidebands of *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 {*μ*} as a function of *Fac* are shown in Figure 3 to illustrate the effect of ac-Zener coupling. The two photon sidebands labeled by *μ*_{1} = [1, 1, 0, *k*] and *μ*_{2} = [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 *μ*_{1} and *μ*_{2} 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* = *F*_{DL1} ≡ 170 kV/cm and *F*_{DL2} ≡ 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*_{+}.

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]:

in the vicinity of an excitonic resonance quasienergy *ex*, where *ε* = 2(*ωp* − *ex*)/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) = *α*_{0}[*q*^{(F)}]^{2} ≡ *αmax* and background spectra *α*(±∞) = *α*_{0} as a function of *Fac*. It is seen that these functions are affected pronouncedly by *Fac*; in particular, extrema are formed around *Fac* = *F*_{DL1}. 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* = *F*_{DL2} in the concerned quantities except Γ. Therefore, the DL is considered to fulfill a special role of the quantum control of photodressed excitonic states.

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 *μ*_{1} and *μ*_{2} 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 *μ*_{1} and *μ*_{2} 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 *μ*_{1} and *μ*_{2}, and the matrix element of the ac-Zener coupling between them, respectively, *M*(*Fac*) is provided as

where *z* ≡ tan *φ* = 2∣*v*∣/∣*Δε*∣. With *x* = *Facd*/*ω*, *v* and *Δε* are evaluated as *v* ∝ *x* and *Δε* ∝ cos(*kd*)*J*_{0}(*x*), respectively; see Section 1. Thus, one has *z* = 2*x*/*ηJ*_{0}(*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 *J*_{0}(*x*), which just correspond to DL concerned here; that is, *M*(*Fac*) shows extrema at *Fac* = *F*_{DL1} and *F*_{DL2}. In fact, *M*(*Fac*) shows a clear extremum at *Fac* = *F*_{DL1}, while the second extremum at *Fac* = *F*_{DL2} is not obviously discernible. This is understood by the behavior that the oscillating component incorporated in *J*_{0}(*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, *α*_{max}, and *α*_{0} 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* = *F*_{DL1}. 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 *μ*_{1} and *μ*_{2} 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*.

### 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*=0.12*π* and 0.20*π*, respectively, and the maximum excited electron density ^{17} and 5.30 × 10^{17} cm^{3}, respectively; by *Δω* > 0, it is meant that opaque interband transitions with real excited carriers are examined. Further, two time constants of *T*_{12} and *T*_{q12} 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* < *T*_{12} 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; *T*_{q12} is approximately equal to *T*_{q}(**k***bb*^{′}) introduced in Eq. (14). On the other hand, the temporal region *t* ≳ *T*_{12} is termed as the classical region. For the present calculations, *T*_{q12} and *T*_{12} are set equal to 20 and 90 fs, respectively. As regards experimental estimates of these time constants for Si, *T*_{q12} and *T*_{12} extracted from the CP measurements in Ref. [45] are 16 and 100 fs, respectively, at

Transient induced photoemission spectra *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, *τ* into the *ωp* domain. In the small transferred momentum **q** limit, **q**|^{2}, while **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, *ωp* is shown, respectively, by solid lines at *tp* equal to *t*_{1} ≡ 15 , *t*_{2} ≡ 65 and *t*_{3} ≡ 100 fs, where the separate contributions from *α*_{1} and the LO-phonon mode *α*_{2}, respectively. In both figures, it is seen that just

Figure 7(a) shows *tp* = *t*_{1} < *T*_{q12}, where the obtained continuum spectra are governed by the contribution from **q**|^{4}. In Figure 7(b) at *tp* = *t*_{2} with *T*_{q12} < *tp* < *T*_{12}, the contributions from *tp* = *t*_{3} > *T*_{12}. As regards *tp* = *t*_{1}, a pronounced peak due to the *α*_{2} mode, is superimposed with a continuum background composed of **q**|^{2}. The spectra at *tp* = *t*_{2} shown in Figure 8(b) are dominated by *tp* = *t*_{3} in Figure 8(c) are similar to those in Figure 7(c).

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 *M*_{qβ} between quasiboson and LO-phonon aside from less significant difference in other material parameters; this appears in the matrix *h*_{q} introduced in Eq. (16), and the approximation of *g*_{bq} incorporated in *M*_{qβ} consists of **q**, while *M*_{qβ} is almost determined by that of *g*_{bq}, since a residual factor defining *M*_{qβ} is almost considered real. Thus, *M*_{qβ} is a complex number given by *M*_{qβ} = ∣*M*_{qβ}∣*e*^{iφqβ} in general; *φ*_{qβ} = 0 , *π* for Si, while *φ*_{qβ} = ± *π*/2 for GaAs.

Next, discussion is made on how such difference of *M*_{qβ} affects the spectral profile of *α*_{2} to the PQ ground state, and the other is a two-step resonant path mediated by *M*_{qβ} from *α*_{2} to quasiboson state *β*, followed by a deexcited process mediated by an optical transition matrix *β* to the PQ ground state. Accordingly, owing to Shore’s model [62], the induced photoemission spectra in the proximity of

where a set of Shore’s spectral parameters of _{qα2}, and

and the natural spectral width is represented by Γ_{qα2} = 2*πρ*_{qα2}|*M*_{qα2}|^{2}; *ρ*_{qα2} and *M*_{qα2} are the density of state of quasiboson and the coupling matrix at *q* parameter is determined in terms of Shore’s parameters as

An asymmetric spectral profile is exclusively determined by *φ*_{qβ} = ± *π*/2 and *q*_{qα2}(*tp*)∣ infinite. The spectral profile of GaAs shown in Figure 8(b) corresponds to this case. For *φ*_{qβ} ≠ ± *π*/2, both _{qα2}(*tp*) are finite, and *q*_{qα2}(*tp*)∣ finite. The spectral profile of Si shown in Figure 7(b) corresponds to this case, where *φ*_{qβ} ≈ 0 , *π*. For Figures 7(c) and 8(c), since *M*_{qβ}∣ become negligibly small, _{qα2}(*tp*), and this becomes symmetric with Γ_{qα2} ≈ 0. To conclude, the effective coupling *M*_{qβ} around *φ*_{qβ} as long as ∣*M*_{qβ}∣ is still large.

Finally, the manifestation of TFR of Si is discussed from the viewpoint of the allocation of time constants *T*_{q12} and *T*_{12}, where one sets *T*_{q12} < *T*_{12}. This is an important issue for deepening the understanding of TFR. As shown in Figure 7(b), in the region of *T*_{q12} ≲ *tp* < *T*_{12}, the asymmetric spectral profile bursts into view from the structureless continuum *tp* < *T*_{12}, the excited carrier density is still populated enough around the energy region of *T*_{q12} ≲ *tp*. As regards a different allocation of these time constants, for instance, *T*_{q12} ∼ *T*_{12}, the TFR profile is no longer observed in the region of *tp* < *T*_{12}, since this is covered with still dominant contributions from *M*_{qβ} becomes too small to cause TFR in the region of *tp* ≈ *T*_{12}. Therefore, the allocation of time constants such as *T*_{q12} < *T*_{12} is a necessary condition for realizing the TFR of Si in

## 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 *μ*_{1} and *μ*_{2}, 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 *M*_{qβ}(*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 *T*_{q12} < *tp* < *T*_{12} with *T*_{q12} ≪ *T*_{12}.

## Acknowledgments

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