We theoretically study nonlinear optics of excitons under intense THz
irradiation. In particular, the linear near infrared absorption and resonantly
enhanced nonlinear sideband generation are described. We predict a rich
structure in the spectra which can be interpreted in terms of the quasienergy
spectrum of the exciton, via a remarkably transparent expression for the
susceptibility, and show that the effects of strongly avoided quasienergy
crossings manifest themselves directly, both in the absorption and transmitted
sidebands.Comment: 4 pages RevTex, 3 eps figs included, as publishe

A.-P. Jauho

B. W. Shore

C. A. Brau

G. Wannier

H. Haug

J. Cerne

J. H. Shirley

J. Kono

J. S. Bakos

K. Johnsen

K. Nordstrom

L. V. Keldysh

M. Grifoni

M. Holthaus

R. J. Elliot

S. H. Autler

T. Meier

V. I. Ritus

V. Nathan

W. Franz

W. Xu

X.-G. Zhao

Y. T. Rebane

Y. Yacoby

Ya. B. Zel'dovich

English

arXiv

Kristinn Johnsen

Antti-Pekka Jauho

Crossref

Physical Review Letters

Quasienergy Spectroscopy of Excitons

We theoretically study nonlinear optics of excitons under intense THz irradiation. In particular, the linear near-infrared absorption and resonantly enhanced nonlinear sideband generation are described. We predict a rich structure in the spectra which an be interpreted in terms of the quasienergy spectrum of the exciton, via a remarkably transparent expression for the susceptibility, and show that the effects of strongly avoided quasienergy crossings manifest themselves directly, both in the absorption and transmitted sidebands

Johnsen, Kristinn

Jauho, Antti-Pekka

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Quasienergy Spectroscopy of ExcitonsJohnsen, Kristinn; Jauho, Antti-PekkaPublished in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.83.1207Publication date:1999Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Johnsen, K., & Jauho, A-P. (1999). Quasienergy Spectroscopy of Excitons. Physical Review Letters, 83(6),1207-1210. DOI: 10.1103/PhysRevLett.83.1207VOLUME 83, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 AUGUST 1999Quasienergy Spectroscopy of ExcitonsKristinn Johnsen and Antti-Pekka JauhoMikroelektronik Centret, Technical University of Denmark, Bldg. 345east, DK-2800 Lyngby, Denmark(Received 3 February 1999)We theoretically study nonlinear optics of excitons under intense THz irradiation. In particular, thelinear near-infrared absorption and resonantly enhanced nonlinear sideband generation are described.We predict a rich structure in the spectra which can be interpreted in terms of the quasienergy spectrumof the exciton, via a remarkably transparent expression for the susceptibility, and show that the effectsof strongly avoided quasienergy crossings manifest themselves directly, both in the absorption andtransmitted sidebands.PACS numbers: 71.35.Cc, 42.65.Ky, 78.20.BhExcitons, strongly correlated pairs of electrons andholes, dominate the band-edge optics of semiconductors:their bound states are observed optically as resonances inthe absorption spectrum in the gap of the semiconductorand in the continuum as enhanced absorption [1]. Theinterband susceptibility describes the linear optical prop-erties of the semiconductor and can be expressed in termsof the well-known equation [2],xv  2Xnjcnr  0j2h¯v 2 En 1 i01, (1)where cnr are the Wannier wave functions of the exci-ton with spectrum En. This compact formula is extremelyuseful since it relates the spectrum of the exciton directlyto the resonances occurring in the absorption spectrum,which is proportional to the imaginary part of (1).The development of coherent sources of intense elec-tromagnetic radiation in the THz regime has opened upa new exciting area of semiconductor physics [3]: e.g.,two color spectroscopy of excitons in the dynamical Franz-Keldysh (DFK) regime [4]. One color has weak intensityin the near infrared (NIR) regime inducing interband tran-sitions, creating the excitons. The other color, which hashigh intensity, is in the far infrared regime (FIR), whichcorresponds to transitions between the internal degrees offreedom of the created excitons. In this regime the fieldstrength, EFIR, and the frequency, V, of the FIR are suchthat both strong field effects [dc Franz-Keldysh (FK) ef-fect [5]] and multiphoton processes (MP) [6] are impor-tant. This regime is quantified by [7]g e2E2FIR4mrh¯V3, (2)which is the ratio between the ponderomotive energy andthe energy of a FIR photon. Here mr is the reducedmass of the exciton and e is the electronic charge. Theregime of FK corresponds to g ¿ 1, MP correspondsto g ø 1, while g  1 corresponds to the regime weconsider presently, the DFK regime.Study of the response of the weak NIR probe as afunction of the two frequencies and the intensity of theFIR beam yields considerable insight into the nature ofexcitons and the fundamental electro-optical processesoccurring in semiconductors. Experiments have beenreported on the effects of the FIR on luminescence[8], absorption [4], and resonantly enhanced nonlinearmixing of the fundamental NIR with the FIR [9], evenin the presence of strong magnetic fields [10]. Thefree particle properties of such experiments have beenstudied intensively [6,11–14], while excitonic effects onabsorption have only been reported recently [4,15,16].The goal of this Letter is to report a generalization ofthe textbook result (1) to the new dynamical regime g  1.We find that the macroscopic polarization can be expressedasPt  d2 ENIRXnxnveiv1nVt , (3)wherexnv  2Xan0fan01nfan0h¯v 2 e˜a 2 n0h¯V 1 i01. (4)Here v is the frequency of the NIR, d is the inter-band dipole matrix element, assumed to be constant, andfan are the temporal Fourier components of the Floquetstates far  0, t of the exciton, with quasienergy e˜a[17]. A Floquet state is the temporal analog to a Blochstate in a spatially periodic potential. The spectral de-composition of the polarization [Eq. (3)] shows that thelinear absorption of the NIR is proportional to Imx0v,while Inv  jxnvj2 is proportional to the intensity ofphotonic sidebands which radiate at frequency v 1 nV.Equation (4) thus relates the nonequilibrium optical prop-erties directly to the excitonic quasienergy spectrum.We first outline the derivation of (3) and then present anumerical study of a quantum well (QW) exciton, whichillustrates the applicability of the theory. We show thatavoided crossings in the quasienergy spectrum lead totwo clear experimental consequences: (i) the sidebandintensities In have a resonant behavior and (ii) the 1sresonance in optical absorption displays a strong Autler-Townes splitting [18].0031-90079983(6)1207(4)$15.00 © 1999 The American Physical Society 1207VOLUME 83, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 AUGUST 1999In the following we describe the semiconductor viaa simple spinless effective mass two-band model. TheHamiltonian is given byH Xk,m[c,yemh¯ k 1 e Atcymkcm k 1 Hi , (5)where m labels conduction or valence bands, and Hidescribes the Coulomb interaction via the potential yj kj.Using nonequilibrium Green function methods, one canshow that the retarded susceptibility obeys the integralequation [19],xr k; t, t0  x¯r  k; t, t01Z dn k02pnZdt00 x¯r k; t, t003 yj k 2 k0jxr k0; t00, t0 , (6)where x¯r k; t, t0 is the free-particle susceptibility [20].Equation (6) can be transformed into an infinite-dimensional matrix equation and solved numerically byintroducing a cutoff frequency [4,16]. This method hassuccessfully described some effects in the linear absorp-tion [4], but has failed to describe the nonlinear mixingadequately. The representation introduced here does notintroduce cutoffs, is numerically much faster, and clarifiesthe physics involved. In the rotating wave approximationwith respect to the NIR probe, the irreducible interbandsusceptibility for an undoped semiconductor obeys [13]Ωih¯≠≠t2 eh¯ k 1 e Atæx¯r k; t, t0  dt 2 t0 , (7)where eh¯ k  ech¯ k 2 eyh¯ k and e  jej. The uni-form FIR field is described by the vector potential At 2 EFIR sinVtV. With (7) we transform (6) toΩih¯≠≠t2 e∑h¯i=r 1 e At∏1e24pkræxrr; t, t0 drdt 2 t0 , (8)where k is the effective dielectric constant in the semicon-ductor. To proceed, we first consider the solutions to thehomogeneous part of (8),Ω h¯i =r 1 e At22mr1eg 2e24pkræCr , t ih¯≠Cr , t≠t.(9)This is a Schrödinger equation for a hydrogenlike parti-cle, the exciton, in the presence of the intense FIR field.When the FIR field is absent this equation reduces to theWannier equation [1]. We have thus arrived at a gen-eralized Wannier equation with a time periodic Hamil-tonian, Ht  Ht 1 T , where T  2pV. Such adiscrete time symmetry gives rise to the temporal ana-log of the Bloch wave functions, the Floquet states [17]:car, t  e2ie˜a t h¯far , t, where e˜a are the quasiener-gies and far , t  far , t 1 T . The Floquet statescan be viewed as the stationary states of the periodi-cally driven system. They form complete sets of solu-tions to the Schrödinger equation, which is to say thatany wave function obeying (9) is of the form Cr , t Pa cae2ie˜at h¯far , t, where ca are c numbers. Further-more, at equal times the states fulfill the closure relationPa far , tfar 0, t  dr 2 r 0. Using these proper-ties, we expand the susceptibility in terms of the Floquetstates and their quasienergies, and findxrr; t, t0 ut 2 t0ih¯Xae2ie˜a t2t0 h¯fa0, t0far, t .(10)We next set r  0, define fat  fa0, t, and expandthe states as fat Pn fane2inVt. The macroscopicpolarization is then readily expressed as (3) and (4). Thisconcludes the outline of our derivation.We next apply the present theory to the nonlinearoptical properties of a quantum well exciton. Specificallywe choose ea0EFIR  1ERy , where ERy  h¯22mra20is the effective Rydberg energy of the medium, anda0  h¯2ke2mr  is the effective Bohr radius of theexciton. As an example, for InGaAs, ERy  2 3 meVand a0  200 Å which leads to EFIR  105 Vm, wellwithin the range of free electron lasers [3]. The FIR islinearly polarized and the field oscillates in the plane ofthe QW. We sweep the THz frequency such that it probesthe various internal resonances of the exciton and studythe absorption and the nonlinear sideband generation.A two-dimensional exciton in equilibrium has thebound state spectrum En  Egap 2 ERyn 2 0.52,n . 0. The n  1 state is a nondegenerate s state,while n ﬁ 1 are degenerate containing also p states,etc. [21]. With linear polarization of the FIR the doublydegenerate p states may be decomposed into p andpk, where the pk has the same spatial symmetry as thefield. Only pk contributes to the dynamics. We includeboth s and pk states in our calculations since both arephysically important as they couple strongly in the pres-ence of the THz field [22]. The Floquet states and theirquasienergies are determined numerically. In the figures,we have introduced a phenomenological damping of theresonances in (4). In Fig. 1 we show the results of ourcalculation for fixed h¯V  2.55ERy as a function of NIRfrequency, illustrating the basic effects due to the THzfield. Considering the absorption, the THz frequency isbelow the 1s ! 2pk equilibrium transition frequency,h¯v012 	 3.56ERy . The redshift of the 1s resonance, awayfrom its equilibrium position, is due to the ac-Stark effect[23], which is pronounced even though the frequencyis considerably detuned from the 1s ! 2pk transition.The effect is maximal in the g  1 regime. The 2s1208VOLUME 83, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 AUGUST 1999FIG. 1. The main effects due to intense FIR irradiation,colinear with weak NIR field. Linear optical absorption Imx0:(i) ac-Stark redshift of the 1s resonance, (ii) DFK blueshiftof the 2s resonance and the band edge, (iii) suppression ofoscillator strength, and (iv) emergence of a photon replica of thepk state. Nonlinear mixing I62: sideband emission at v 6 nV.The inset illustrates the experimental geometry.resonance is blueshifted with respect to the equilibriumposition. The band edge is blueshifted as well. This isdue to the DFK effect which shifts all main features by theponderomotive energy [6,11,12]. The oscillator strengthof the 1s resonance is suppressed. These effects havebeen observed in quantum well excitons [4]. Furthermore,a new resonance appears in the absorption spectrum: asingle photon replica of the dark pk state which underirradiation becomes optically active. The tail to the bluefrom the resonance is due to replicas of the pk symmetricstates in the continuum. The sideband intensity for I62is shown as well. In reflection symmetric systems onlysidebands with n even appear: the optical properties areinvariant under the transformation EFIR ! 2 EFIR, andhence the “effective” frequency is 2V. Breaking thereflection symmetry introduces odd sidebands as well.The sideband generation is at a maximum if either v orv 2 nV, n . 0, is tuned to the 1s resonance. We findthat Inv  I2nv 1 nV. However, in an experimentusing a multiple quantum well sample the radiation dueto I2n will tend to be reabsorbed, its maximum beingtuned to the main absorption resonance while In hasits maximum 2nV away from it. This behavior of thesideband generation has been observed [9].In Fig. 2 we show the spectra for a constant fieldstrength with a THz frequency sweep h¯V  0.5 5ERy .In Imx0, one observes how photon replicas traverse fromh¯v  E2 in a fan as the THz frequency is increased.The fan blades have been tagged with the order in theTHz frequency involved, which is roughly proportionalto the inverse slope of the blade. In view of (4), whenthe replicas reach the main resonance a strongly avoidedFIG. 2. The effect of the THz frequency at fixed fieldstrength. From below; the absorption Imx0, two-, and four-photon sideband generation. The THz frequency is indicatedon the right vertical axis. Here several avoided crossings inthe quasienergy spectra can be distinguished, as splittings inthe absorption and as resonant enhancement in the sidebandgeneration. The order of the processes involved is indicated.I4 is multiplied by 3 compared to I2.crossing in the quasienergy spectrum results, which isdirectly visible in the spectra. For the first order processan Autler-Townes splitting results. The sidebands, in theupper two panels, show clear evidence of the fan shape.In this case, however, the underlying avoided crossing inthe quasienergy spectrum results in resonantly enhancedsideband generation. Generation of sideband In calls foroptical processes of order jnj 1 1 or higher and thusthe predominant order determines if a strongly avoidedcrossing results in a resonantly enhanced sideband.The oscillator strength reflects the resonant conditions.In Fig. 3 we compare the oscillator strength to the maxi-mum of the generated sidebands at each frequency. Weremark in passing that the n  2 sideband was recentlydetected for h¯V  4ERy [9], which in view of Fig. 31209VOLUME 83, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 AUGUST 1999FIG. 3. The maximum absorption and sideband generation asa function of the FIR frequency illustrating the relation be-tween the absorption suppression and the resonantly enhancedsideband generation.suggests that the predicted results should be detectable withpresent technology. Resonant suppression of the oscillatorstrength occurs when the sideband generation is enhanced.The resonance conditions are met by the quasienergies ac-cording to (4). The range in which resonant enhancementoccurs is determined by the coupling of the internal excitonlevels. The enhancement mainly involves the 1s ! 2pktransition for odd processes, while it is mainly the 1s ! 2stransition which is responsible for even resonances. Thefirst seven resonances are indicated in Fig. 3.In summary, we have studied the nonlinear optics ofexcitons subject to intense THz radiation. We have shownanalytically that observed resonances in the sidebandgeneration may be viewed as manifestations of stronglyavoided crossings in the quasienergy spectrum of theexciton. Our theory is consistent with experiment, whereavailable, and it leads to a number of predictions for theoutcome of new experiments. An example is the strongcorrelation between the oscillator strength suppression inabsorption and the associated resonant enhancement insideband generation, respectively.We would like to acknowledge our colleagues, S. J.Allen, B. Birnir, B. Y.-K. Hu, A. Ignatov, J. Kono,W. Langbein, K. Nordstrom, M. Sherwin, and M. Wagnerfor sharing details of their experiments and for manyenlightening discussions.[1] G. Wannier, Phys. Rev. 52, 191 (1937).[2] R. J. Elliot, in Polarons and Excitons, edited by C. G.Kuper and G. D. Whitefield (Oliver and Boyd, Edinburgh,1963), p. 269.[3] See, e.g., C. A. Brau, Free-Electron Lasers (AcademicPress, San Diego, 1990).[4] K. Nordstrom et al., Phys. Rev. Lett. 81, 457 (1998).[5] W. Franz, Z. Naturforsch. 13, 484 (1958); L. V. Keldysh,Sov. Phys. JETP 34, 788 (1958).[6] See, e.g., V. Nathan, A. H. Guenther, and S. S. Mitra,J. Opt. Soc. Am. B 2, 294 (1985), and references therein.[7] Y. Yacoby, Phys. Rev. 169, 610 (1968).[8] J. Cerne et al., Phys. Rev. B 51, 5253 (1995).[9] K. Nordstrom et al., Phys. Status Solidi (b) 204, 52(1997).[10] J. Kono et al., Phys. Rev. Lett. 79, 1758 (1997).[11] Y. T. Rebane, Sov. Phys. Solid State 27, 824 (1985).[12] A.-P. Jauho and K. Johnsen, Phys. Rev. Lett. 76, 4576(1996).[13] K. Johnsen and A.-P. Jauho, Phys. Rev. B 57, 8860(1998).[14] W. Xu, J. Phys. Condens. Matter 10, 10 787 (1998).[15] T. Meier, F. Rossi, P. Thomas, and S. W. Koch, Phys.Rev. Lett. 75, 2558 (1995).[16] K. Johnsen and A.-P. Jauho, Phys. Status Solidi (a) 164,553 (1997).[17] For general Floquet theory, see J. H. Shirley, Phys. Rev.138, B979 (1965); Ya. B. Zel’dovich, Zh. Eksp. Teor.Fiz. 51, 1492 (1966) [Sov. Phys. JETP 24, 1006 (1967)];V. I. Ritus, Zh. Eksp. Teor. Fiz. 51, 1544 (1966) [Sov.Phys. JETP 24, 1041 (1967)]. For recent applicationsof the Floquet formalism to transport in THz irradiatedsuperlattices and photon assisted tunneling, see, e.g.,M. Holthaus, Phys. Rev. Lett. 69, 351 (1992); M. Holthausand D. W. Hone, Phys. Rev. B 49, 16 605 (1994);X.-G. Zhao, J. Phys. Condens. Matter 6, 2751 (1994);M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998). Fora textbook review on the theory applied to atoms underintense irradiation, see, e.g., B. W. Shore, The Theory ofCoherent Atomic Excitation (Wiley, New York, 1990),Chap. 4, and references therein.[18] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703(1955).[19] The equilibrium case is discussed in H. Haug andS. Schmitt-Rink, Prog. Quantum Electron. 9, 1 (1984),while the nonequilibrium case is treated in [16].[20] This Letter is concerned with undoped semiconductors,where the low density limit is appropriate. It is possible toformulate the theory for screened interaction; this wouldlead to a renormalized form of the results presented here.[21] H. Haug and S. W. Koch, Quantum Theory of the Opticaland Electronic Properties of Semiconductors (WorldScientific, Singapore, 1993).[22] Note that the exciton state created due to a single photoninterband transition is always a s state. Coupling to d, fstates, and so on, are thus of third order or higher and arealways dominated by equivalent, energetically degenerates- to p-state transitions. Therefore only s and p states areincluded in the calculations.[23] See, e.g., J. S. Bakos, Phys. Rep. 31C, 209 (1977), andreferences therein.1210

Quasienergy Spectroscopy of Excitons

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