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
To begin with the DFR, this is closely related with the photodressed miniband formation [36]. Here, the cw laser with a constant amplitude
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
The equation of motion for the microscopic polarization is given by the semiconductor Bloch equation
with
where 〈λ|
where the rotating wave approximation is employed by replacing
and
The concerned function
with
where the temporally periodic boundary condition
where
Since the macroscopic polarization is given by
where
with
2.1.2. Multichannel scattering problem
The absorption coefficient of Eq. (9) is obtained by evaluating a set of the wave functions, {
where
where
The Floquet exciton in the laser-driven SL system pertains to the multichannel scattering problem, because
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
2.2. Theoretical model for TFR in the CP generation
2.2.1. Introduction of polaronic quasiparticle operators
The total Hamiltonian
where Ω
where
The equation of motion of a composite operator
where the total electronic Hamiltonian is defined as
Bearing in mind this situation, we tackle left and right eigenvalue problems of
where the operator
Eq. (13) is rewritten as
Here, the non-Hermitian matrix
where
In addition, the left vector
Given Eq. (18), Eq. (16) becomes adiabatic coupled equations for
where
2.2.2. Transient induced photoemission spectra
A weak external potential
based on the linear response theory [59, 60] with
Let
In terms of
Based on the PQ model developed in Section 2.2.1,
where
where
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
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
First of all, the calculated quasienergy bands {
To deepen the understanding of the DFR exciton, its characteristic quantities determining the spectral profiles are extracted from
in the vicinity of an excitonic resonance quasienergy
For the purpose of confirming such an effect of DL and the pronounced
where
The alteration pattern of
Finally, one mentions in brief the
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
Transient induced photoemission spectra
In Figures 7 and 8,
Figure 7(a) shows
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
Next, discussion is made on how such difference of
where a set of Shore’s spectral parameters of
and the natural spectral width is represented by Γ
An asymmetric spectral profile is exclusively determined by
Finally, the manifestation of TFR of Si is discussed from the viewpoint of the allocation of time constants
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
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
Acknowledgments
This work was supported by JSPS KAKENHI Grants No. JP21104504, JP23540360, and JP15K05121.
References
- 1.
Fano U. Physics Review. 1961; 124 . DOI: 10.1103/PhysRev.124.1866 - 2.
Bohn JL, Julienne PS. Physical Review A. 1997; 56 . DOI: 10.1103/PhysRevA.56.1486 - 3.
Ciurylo R, Tiesinga E, Julienne PS. Physical Review A. 2006; 74 . DOI: 10.1103/PhysRevA.74.022710 - 4.
Enomoto K, Kasa K, Kitagawa M, Takahashi Y. Physical Review Letters. 2008; 101 . DOI: 10.1103/PhysRevLett.101.203201 - 5.
Kouchi K. Superexcited states of molecules. In: Hiraoka K, editor. Fundamentals of Mass Spectrometry. Berlin: Springer-Verlag; 2013. DOI: 10.1007/978-1-4614-7233-9. Chap. 5. - 6.
Kobayashi K, Aikawa H, Katsumoto S, Iye Y. Physical Review Letters. 2002; 88 . DOI: 10.1103/PhysRevLett.88.256806 - 7.
Xu B, Dai YM, Zhao LX, Wang K, Yang R, Zhang W, Liu JY, Xiao H, Chen GF, Trugman SA, Zhu J-X, Taylor AJ, Yarotski DA, Prasankumar RP, Qiu XG. Nature Communications. 2017; 8 . DOI: 10.1038/ncomms14933 - 8.
Bar-Ad S, Kner P, Marquezini MV, Mukamel S, Chemla DS. Physical Review Letters. 1997; 78 . DOI: 10.1103/PhysRevLett.78.1363 - 9.
Hino K. Physical Review B. 2000; 62 . DOI: 10.1103/PhysRevB.62.R10626 - 10.
Hino K. Physical Review B. 2001; 64 . DOI: 10.1103/PhysRevB.64.075318 - 11.
Hino K, Toshima N. Physical Review B. 2005; 71 . DOI: 10.1103/PhysRevB.71.205326 - 12.
Chu S-I, Telnov DA. Physics Reports. 2004; 390 . DOI: 10.1016/j.physrep.2003.10.001 - 13.
Potvliege RM, Shakeshaft R. In: Gavrila M, editor. Atoms in Intense Laser Fields. New York: Academic Press; 1992. p. 373 - 14.
Kroner M, Govorov AO, Remi S, Biedermann B, Seidl S, Badolato A, Petroff PM, Zhang W, Barbour R, Gerardot BD, Warburton RJ, Karrai K. Nature. 2008; 451 . DOI: 10.1038/nature06506 - 15.
Maeshima N, Hino K. Physical Review B. 2012; 85 . DOI: 10.1103/PhysRevB.85.205305 - 16.
Maeshima N, Yamada K, Hino K. Journal of Physics: Condensed Matter. 2013; 25 . DOI: 10.1088/0953-8984/25/43/435801 - 17.
Russell JP. Applied Physics Letters. 1965; 6 :223. DOI: 10.1063/1.1754144 - 18.
Parker JH Jr, Feldman DW, Ashkin M. Physics Review. 1967; 155 . DOI: 10.1103/PhysRev.155.712 - 19.
Hart TR, Aggarwal RL, Lax B. Physical Review B. 1970; 1 . DOI: 10.1103/PhysRevB.1.638 - 20.
Cerdeira F, Cardona M. Physical Review B. 1972; 5 . DOI: 10.1103/PhysRevB.5.1440 - 21.
Belitsky VI, Cantarero A, Cardona M, Trallero-Giner C, Pavlov ST. Journal of Physics: Condensed Matter. 1997; 9 . DOI: 10.1088/0953-8984/9/27/022 - 22.
Jin K-J, Zhang J, Chen Z-H, Yang G-Z, Chen ZH, Shi XH, Shen SC. Physical Review B. 2001; 64 . DOI: 10.1103/PhysRevB.64.205203 - 23.
Fano U, Rau ARP. Atomic Collisions and Spectra. Orlando: Academic Press, Inc; 1986 Chaps. 7–11 - 24.
Knight PL, Lauder MA, Dalton BJ. Physics Reports. 1990; 190 . DOI: 10.1016/0370-1573(90)90089-K - 25.
Kukuu A, Amano T, Karasawa T, Maeshima N, Hino K. Physical Review B. 2010; 82 . DOI: 10.1103/PhysRevB.82.115315 - 26.
Nemoto Y, Hino K, Maeshima N. Physical Review B. 2013; 87 . DOI: 10.1103/PhysRevB.87.205305 - 27.
Meier T, Schulze A, Thomas P, Vaupel H. Physical Review B. 1995; 51 . DOI: 10.1103/PhysRevB.51.13977 - 28.
Siegner U, Mycek M-A, Glutsch S, Chemla DS. Physical Review Letters. 1995; 74 . DOI: 10.1103/PhysRevLett.74.470 - 29.
Siegner U, Mycek M-A, Glutsch S, Chemla DS. Physical Review B. 1995; 51 . DOI: 10.1103/PhysRevB.51.4953 - 30.
Hino K, Goto K, Toshima N. Physical Review B. 2004; 69 . DOI: 10.1103/PhysRevB.69.035322 - 31.
Watanabe Y, Hino K. M, Hase, and N. Maeshima. Physical Review B. 2017; 95 . DOI: 10.1103/PhysRevB.95.014301 - 32.
Bartal B, Kozma IZ, Stepanov AG, Almási G, Kuhl J, Riedle E, Hebling J. Applied Physics B: Lasers and Optics. 2007; 86 . DOI: 10.1007/s00340-006-2512-7 - 33.
Karpowicz N, Dai J, Lu X, Chen Y, Yamaguchi M, Zhang L, Zhang C, Price-Gallagher M, Fletcher C, Mamer O, Lesimple A, Johnson K. Applied Physics Letters. 2008; 92 . DOI: 10.1063/1.2828709 - 34.
Hirori H, Doi A, Blanchard F, Tanaka K. Applied Physics Letters. 2011; 98 . DOI: 10.1063/1.3560062 - 35.
Shirley J. Physics Review. 1965; 138 . DOI: 10.1103/PhysRev.138.B979 - 36.
Holthaus M. Physical Review Letters. 1992; 69 . DOI: 10.1103/PhysRevLett.69.351 - 37.
Grifoni M, Hänggi P. Physics Reports. 1998; 304 . DOI: 10.1016/S0370-1573(98)00022-2 - 38.
Dunlap DH, Kenkre VM. Physical Review B. 1986; 34 . DOI: 10.1103/PhysRevB.34.3625 - 39.
Keay BJ, Zeuner S, Allen SJ Jr.,Maranowski KD,Gossard AC, Bhattacharya U, and Rodwell MJW, Physical Review Letters 75 , (1995). DOI: 10.1103/PhysRevLett.75.4102 - 40.
Madison KW, Fischer MC, Diener RB, Niu Q, Raizen MG. Physical Review Letters. 1998; 81 . DOI: 10.1103/PhysRevLett.81.5093 - 41.
Eckardt A, Holthaus M, Lignier H, Al Z, Ciampini D, Morsch O, Arimondo E. Physical Review A. 2009; 79 . DOI: 10.1103/PhysRevA.79.013611 - 42.
Longhi S, Marangomi M, Lobino M, Ramponi R, Laporta P, Cianci E, Foglietti V. Physical Review Letters. 2006; 96 . DOI: 10.1103/PhysRevLett.96.243901 - 43.
Della Valle G, Ornigotti M, Cianci E, Foglietti V, Laporta P, Longhi S. Physical Review Letters. 2007; 98 . DOI: 10.1103/PhysRevLett.98.263601 - 44.
Szameit A, Garanovich IL, Heinrich M, Sukhorukov AA, Dreisow F, Pertsch T, Nolte S, Tünnermann A, Longhi S, Kivshar YS. Physical Review Letters. 2010; 104 . DOI: 10.1103/PhysRevLett.104.223903 - 45.
Hase M, Kitajima M, Constantinescu AM, Petek H. Nature (London). 2003; 426 :51. DOI: 10.1038/nature02044 - 46.
Gaal P, Kuehn W, Reimann K, Woerner M, Elsaesser T, Hey R. Nature. 2007; 450 . DOI: 10.1038/nature06399 - 47.
Hase M, Ishioka K, Demsar J, Ushida K, Kitajima M. Physical Review B. 2005; 71 . DOI: 10.1103/PhysRevB.71.184301 - 48.
Misochko OV, Lebedeva MV. JETP 120 ; 2015. DOI: 10.1134/S1063776115020168 - 49.
Misochko OV. JETP. 2001; 92 . DOI: 10.1134/1.1354682 - 50.
Kato K, Ishizawa A, Oguri K, Tateno K, Tawara T, Gotoh H, Kitajima M, Nakano H. Japanese Journal of Applied Physics. 2009; 48 . DOI: 10.1143/JJAP.48.100205 - 51.
Lee JD, Inoue J, Hase M. Physical Review Letters. 2006; 97 . DOI: 10.1103/PhysRevLett.97.157405 - 52.
Mahan GD. Many-Particle Physics. New York: Plenum; 1981. DOI: 10.1007/978-1-4757-5714-9. Chaps. 4 and 5. - 53.
Riffe DM. Physical Review B. 2011; 84 . DOI: 10.1103/PhysRevB.84.064308 - 54.
Hino K, Tong XM, Toshima N. Physical Review B. 2008; 77 . DOI: 10.1103/PhysRevB.77.045322 - 55.
Maeshima N, Hino K. Computer Physics Communications. 2012; 183 . DOI: 10.1016/j.cpc.2011.07.022 - 56.
Glutsch S, Bechstedt F. Physical Review B. 1999; 60 . DOI: 10.1103/PhysRevB.60.16584 - 57.
Meystre P, Sargent M III. Elements of Quantum Optics. 3rd ed. Berlin: Springer-Verlarg; 1999. DOI: 10.1007/978-3-540-74211-1. Chaps. 3 and 15 - 58.
Morse PM, Feshbach H. Methods of Theoretical Physics. New York: McGraw-Hill; 1953 Chap. 7 - 59.
Schäfer W, Wegener M. Semiconductor Optics and Transport Phenomena. Berlin: Springer-Verlag; 2002. DOI: 10.1007/978-3-662-04663-0. Chaps. 2, 10, and 11. - 60.
Fetter AL, Walecka JD. Quantum Theory of Many-Particle Systems. New York: McGraw-Hill, Inc.; 1971 Chaps. 3–5 - 61.
Yu PY, Cardona M. Fundamentals of Semiconductors. 4th ed. Berlin: Springer-Verlarg; 2010. DOI: 10.1007/978-3-642-00710-1. Chaps. 3 and 7 - 62.
Shore BW. Reviews of Modern Physics. 1967; 39 . DOI: 10.1103/RevModPhys.39.439