PHYSICAL REVIEW B, VOLUME 65, 085306

Optically induced coherent intraband dynamics in disordered semiconductors

C. Schlichenmaier,1 I. Varga,1,2 T. Meier,1 P. Thomas,1 and S. W. Koch11Fachbereich Physik und Wissenschaftliches Zentrum fu¨r Materialwissenschaften, Philipps–Universitat,

Renthof 5, D-35032 Marburg, Germany2Elmeleti Fizika Tansze´k, Fizikai Intezet, Budapesti Mu¨szaki es Gazdasa´gtudoma´nyi Egyetem, H-1521 Budapest, Hungary

~Received 11 October 2001; published 30 January 2002!

On the basis of a tight-binding model for a strongly disordered semiconductor with correlated conduction-and valence-band disorder a coherent dynamical intraband effect is analyzed. For systems that are excited bytwo specially designed ultrashort light-pulse sequences delayed byt relatively to each other echolike phenom-ena are predicted to occur. In addition to the interband photon echo which shows up at exactlyt52t relativeto the first pulse, the system responds with two spontaneous intraband current pulses preceding and followingthe appearance of the photon echo. The temporal splitting depends on the electron-hole mass ratio. Calculatingthe population relaxation rate due to Coulomb scattering, it is concluded that the predicted new dynamicaleffect should be experimentally observable in an interacting and strongly disordered system, such as theQuantum Coulomb Glass.

DOI: 10.1103/PhysRevB.65.085306 PACS number~s!: 72.40.1w, 78.47.1p, 72.80.Ng

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I. INTRODUCTION

Echo phenomena, the prototype being the Hahn secho1 for spin-1/2 systems, rely on the generation of a cherent ensemble of excitations with a continuous distributof frequencies. After pulsed excitation the macroscopicsponse of the ensemble decays as a consequence of detive interference effects in the continuum of excited frequcies. A second delayed excitation pulse induces a rephaof the individual excited species such that at twice the detime the ensemble shows a spontaneous macroscopicsponse, the echo. A necessary requirement for the observity of this coherent dynamics are sufficiently weak dephasinteractions on the time scale of the pulse delay. The micscopic reason for the appearance of an echo is that theond pulse causes phase conjugation of the coherent extion generated by the first pulse. In close analogy to the secho also photon echoes have been observed in ensembtwo-level absorbers.2 There are also phonon echoes3 andtemperature echoes,4 even classical mechanic ensemblespendulums can show echo phenomena.5

Photon echoes have also been studied in more comcated systems, such as ordered6 and disordered7 semiconduc-tors. The optically excited interband transitions in a semicductor cannot be considered as an ensemble of indepentwo-level absorbers due to the strong interaction ofelectron-hole pairs. Photon echoes therefore may show acay as a function of the delay time due to the Coulointeractions, due to disorder, and due to combininteraction-disorder effects.6,8–10In Ref. 11 an echo phenomenon has been proposed for disordered conductors or Anson insulators. On the basis of a noninteracting odimensional tight-binding band with diagonal disorder fillwith a low density of carriers the current response to shexternally applied voltage pulses was calculated. Assumexcitation with two short voltage pulses att50 and t5t,wheret was chosen larger than the typical elastic scattertime tel , it was predicted that the system spontaneouslysponds with a current pulse exactly at timet52t. In contrast

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to spin echoes and photon echoes this current echo isrelated to phase conjugation.

The influence of Coulomb interactions on the current ecwas investigated in Ref. 12. The numerically exact calcution for a small tight-binding system showed that the currecho should remain visible in the presence of the many-binteraction, however, so far no experimental observationbeen reported.

In a series of papers van Driel, Sipe, and co-workers hshown that in semiconductors currents can be induced ocally on an ultrashort time scale using a coherent conscheme.13 In addition, a resulting current pulse could posbly be observed using THz-detection techniques.

Stimulated by these results, we studied optically induccurrent phenomena in disordered model systems.14 By solv-ing the equation of motion for the intraband current innoninteracting tight-binding model we concluded that onbasis of the coherent control scheme it should be indpossible to generate a current pulse also in a strongly didered semiconductor. The current pulse decays due to elscattering. In addition, the current traces show signatureAnderson localization~in a one-dimensional noninteractindisordered system all states are localized!. The application oftwo delayed optical pulses, both generating a current puwas found to result in a sizable echo in the intraband currthat appears att52t. At the same time the interband polaization shows the conventional photon echo.

Continuing our earlier work14 and making our modemore realistic we were surprised to discover that the ocally induced intraband dynamics in a disordered semicductor with correlated conduction- and valence-band disder shows features that differ profoundly from both tphoton and the current echo. By allowing the effectimasses of the electrons and holes to differ from each otwe find that the related spontaneous signals appear at dent times. While the interband photon echo always showsat t52t, the intraband signals split into an advanced anretarded signal current pulse. This splitting depends onmass ratio. For the semiconductor model we find that thtwo current pulses appear only if the disorder is correla

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SCHLICHENMAIER, VARGA, MEIER, THOMAS, AND KOCH PHYSICAL REVIEW B65 085306

for electrons and holes, which is often at least approximavalid in real low-dimensional semiconductor nanosystewhere disorder may be due to the fluctuations of the confiment potential.

We furthermore find that only the first excitation pulset50 has to be designed according to the coherent conscheme, which is necessary to generate an intraband cuFor the second pulse att5t a normal resonant optical exctation is sufficient. This simple excitation pulse does not gerate a current pulse att5t, however, it is still able to ini-tiate the pair of spontaneous signals. These signals areprominent for high excitation density. Since the manparticle Coulomb interaction has been ignored in theseculations, one might suspect that Coulomb scattering leadsufficiently rapid dephasing such that the new dynamics islonger observable. The consideration of the Coulomb inaction for the situation at hand requires one to calculateresponse at least up to fifth order in the external light fiand to treat the interaction consistently with all the relevcorrelations~for a third-order treatment in a disordered semconductor model see, e.g., Refs. 10,15,16!. Even for a one-dimensional system this task is beyond present computepacity. In order to get a feeling for the relevant time scawe consider a much simpler case. For a strongly disordeone-dimensional single-band model we calculate the poption relaxation time due to Coulomb scattering by treatthe many-particle interaction in second Born approximaton the basis of Hartree-Fock states. The situation envisais the so-called Quantum Coulomb Glass model, one ofmost challenging problems in modern many-particle physThe results we obtain can be taken as an indication thafact strong disorder leads to a slowing down of the dynacal Coulomb effects. For the time being this suggests thatcurrent echo should be observable in experiment.

This paper is organized as follows. In Sec. II the modeintroduced. The relevant observables, the interband andtraband currents and the polarization, are defined in SecThe equations of motion are given in Sec. IV and the cohent control excitation scheme is described in Sec. V. Finain Sec. VI the numerical results are presented and discuin Sec. VII. In the concluding Sec. VIII we consider thpossible influence of the Coulomb many-particle interactand give evidence for the observability of the predicted snals in experiments.

II. THE MODEL

We consider a one-dimensional two-band tight-bindmodel with nearest-neighbor couplingJl defined to model adirect gap semiconductor. TheN sites i have nearest-neighbor separationuRW u and periodic boundary conditions aapplied. Every sitei at positionRW i carries two energiese i

c

and e iv . They are distributed randomly according to a bo

shaped distribution of widthWl. In the electron-hole pictureall electron energies and hole energiese i

l as well as the cou-plings Jl are positive quantities if the energy is taken tozero in the gap between the valence and the conduction bThe Hamiltonian matrix for bandl is then

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nd.

Ti jl 5d i j e i

l2Jl~d i j 211d i j 11!. ~1!

For an ordered situation (e il5e0

l ,Wl50 eV) we have a di-rect semiconductor with cosine bands and a gap in the ce(kW50) of the Brillouin zone. Close tokW50 the bands arecharacterized by effective massesml related to the couplingsJl

ml5\2

2JluRW u2. ~2!

In the disordered case the relevant disorder parameter is

hl5Wl

Jl. ~3!

In our one-dimensional system all single-particle stateslocalized for nonzeroW. The disorder is called uncorrelateif the site energies are distributed independently from eother in the two bands. It is called correlated if

e ic2^e i

c&

Jc5

e iv2^e i

v&

Jv, ~4!

where^e ic,v& are the expectation values of the site energi

In this work we exclusively treat correlated disorder. It moels, e.g., a disorder potential in a semiconductor heterostture with effective dimensionality less than three, whichproduced by local fluctuations in the confining potential.

The total Hamiltonian then reads

H5H01HI , ~5!

H05 (i , j

l5c,v

Ti jl al i

1al j , ~6!

HI52EW ~ t !•dW . ~7!

al i1 , al j are electron creation and annihilation operators,

spectively. The couplingsT in the electron picture are unchanged for the conduction band, i.e.,Ti j

c 5Ti jc , whereas a

change of sign is required for the valence band, i.e.,Ti jv

52Ti jv .

dW 5 (i

l,l85c,v

dW i ill8al i

1al8 i ~8!

and

dW i ill852e~RW idll81rWll8! ~9!

rWll8 is the interband optical dipole matrix element~see TableI!.

III. THE OBSERVABLES

The total polarization is given by

PW 5^dW &V

~10!

and the total current by

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OPTICALLY INDUCED COHERENT INTRABAND . . . PHYSICAL REVIEW B65 085306

jW5^dW &51

i\^@dW ,H0#&

5ie

\ (l i j

~RW i2RW j !Ti jl ^al i

1al j&

1ie

\ (ll8 i j

rWll8~ Ti jl82Ti j

l !^al i1al8 j&

5ie

\RW (

l iJl~^al i

1al i 11&2^al i1al i 21&!

2ie

\ (ll8 i

rWll8~ e il2 e i

l8!^al i1al8 i&

1ie

\ (ll8 i

rWll8~ Jl2 Jl8!~^al i1al8 i 11&

1^al i1al8 i 21&!,

whereRW 5RW i 112RW i . These observables have both intrabaand interband contributions due to the first and second tin Eq. ~9!, respectively.

Denoting the expectation values of the operators by

pi j 5^av i1ac j&, ~11a!

ni jc 5^aci

1ac j&, ~11b!

ni jh 5d i j 2^av j

1 av i&, ~11c!

wherepi j is the interband coherence related to the interbpolarization~and the interband current! and ni j

c and ni jh are

the intraband coherences (iÞ j ) and densities (i 5 j ) relatedto the intraband current.

The intraband currentJW intra, the interband currentJW inter,and the total polarizationPW are given by

JW intra52eRW

\ FJc(i

Im@ni 11,ic #2Jv(

iIm@ni 11,i

h #G , ~12!

TABLE I. Parameters of the semiconductor model and the lipulse.

^ec&1^ev& 1.316 eVJc 34 meVr cv 3.0 Å

uRW u 20 Å

Disorderh 2Number of sites 71

Light field described in Sec. Vf12 p/2Central wavelength of full-gap pulse 1.316 eV

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JW inter522erWcv

\ (i

~e ic1e i

v!Im@pii #

12erWcv

\~Jc1Jv!(

iIm@pi 21,i1pi 11,i #, ~13!

PW 52e

V (i

@RW i~niic 2nii

h 11!12rWcvRe@pii ##, ~14!

respectively. In Eq.~14! the first and second terms referthe intraband and interband polarization, respectively.

IV. EQUATIONS OF MOTION

Using the Heisenberg equation of motion and takingexpectation values everywhere we obtain

d

dtpi j 52

i

\~e i

v1e jc!pi j 1

i

\Jv~pi 21 j1pi 11 j !

1i

\Jc~pi j 211pi j 11!

1i

\eEW ~ t !@~RW i2RW j !pi j 1rWcv~ni j

c 1njih 2d i j !#, ~15!

d

dtni j

c 5i

\~e i

c2e jc!ni j

c 1i

\Jc~ni j 21

c 1ni j 11c 2ni 21 j

c 2ni 11 jc !

1i

\eEW ~ t !@~RW i2RW j !ni j

c 1rWcv~pi j 2pji* !#, ~16!

d

dtni j

h 5i

\~e i

v2e jv!ni j

h 1i

\Jv~ni j 21

h 1ni j 11h 2ni 21 j

h 2ni 11 jh !

2i

\eEW ~ t !@~RW i2RW j !ni j

h 1rWcv~pi j* 2pji !#, ~17!

which differ from the conventional optical Bloch equationfor a noninteracting tight-binding model10,15,16by the termscontaining the positionRW i of the sites. We numerically solvethese equations using the standard fourth-order-Runge-Kalgorithm.

V. CURRENT EXCITATION BY COHERENT CONTROL

We apply the coherent control scheme developedthe Toronto Group13 in order to optically generate a shointraband current pulse in both bands. The first excitationtime t50 is chosen to be due to a light field

EW ~ t !5EW 1e2(t/tL1)2cosFv2 t G1EW 2e2(t/tL2)2

cos@vt1f1,2#,

~18!

whereEW n are the amplitudes,tLn the temporal widths, andf1,2 is the relative phase of the two contributions havifrequencyv ~called full-gap pulse! andv/2 ~called half-gappulse!. We takev to be larger than the band gap, where

t

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SCHLICHENMAIER, VARGA, MEIER, THOMAS, AND KOCH PHYSICAL REVIEW B65 085306

v/2 is smaller than the band gap. In a previous paper14 wehave shown, using the above model, that this excitationsults in a current depending on sin@f1,2# that decays due toscattering at the disorder. This decay is modulated by oslations which are a fingerprint of Anderson localization.was also shown there~see Fig. 1! that a second identicapulse, arriving at timet5t, leads to a spontaneous curreecho at timet52t having opposite direction. In that woridentical electron and hole masses have been taken, i.eJc

5Jv. The process of generating a current is of at least thorder in the field amplitudes. The current echo requiresditional excitation with delayed pulses and is thereforehigher order. The analysis presented in the following secshows that the appearance of an echo is at least of fifth oin the external fields. Although the current pulses have bgenerated optically, the dynamics seems to strongly resemthe current echo as it was originally suggested11 for a singleband model with current generation by voltage pulses.will be shown in the next section, however, the intrabadynamics initiated in a semiconductor is profoundly differefrom the current echo in a single band situation.

VI. NUMERICAL RESULTS

A. Dependence on disorder and on electron-hole mass ratio

There are three main findings which point out that tdynamics initiated by the coherent control excitation charterizes a new coherent phenomenon.~i! A spontaneous signapulse is completely absent if instead of correlated disoraccording to Eq.~4! we consider a model with uncorrelatedisorder. It should be mentioned, that in this case for a sindisorder realization current fluctuations are excited evennormal band-band excitation. These can be suppressedby extensive configurational averaging over a large numof disorder realizations. After sufficient averaging no deteable signal can be seen in the resulting current traces.~ii ! Thespontaneous response results even for a simplified excitasequence. The first pulse att50 is taken to generate a curent, i.e., it is given by Eq.~18!. However, for the second

FIG. 1. The negative peaks are spontaneous responses folloa succession of two positive current pulses generated by cohcontrol. The effective masses of electrons and holes are idenNote the correct shift of the spontaneous signal with 2t. The excit-ing light pulses have the pulse areaAn50.64p ~Sec. VI B! and aduration of tLn520 fs. Each curve has been averaged over 1disordered realizations. From Ref. 14.

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pulse att5t we take a simple full-gap pulse with onlysingle central frequencyv. In order to have an influence othe dynamics of intraband quantities, this pulse has to ethe equations of motion at least in second order. As a rewe still see a spontaneous response at the timet52t, seeFig. 2~a!. We conclude, therefore, that the spontaneous sigdoes not require excitation by two current pulses. Insteonly one current pulse is necessary and the secondpulse can be a single full-gap pulse. The current pulse haprecede the full-gap pulse, since the reverse order of pudoes not yield a spontaneous response. Because of thising we avoid to call the spontaneous response an echo infollowing. ~iii ! For different electron and hole masses E~2!, Jv/Jc,1, we find that, surprisingly, there are two seprate intraband responses, one precedingt52t and one fol-lowing t52t. In particular, the delayed contribution is duto the valence band and appears at

tv5S 11Jc

JvD t ~19!

while the preceding contribution is due to the conductiband and appears at

tc5S 11Jv

JcD t. ~20!

ingental.

0

FIG. 2. ~a! Spontaneous response in the intraband current aexcitation att50 with coherent control pulses and att5800 fs witha single full-gap pulse. For different electron and hole massesspontaneous response splits into a pair of signals according to~19!,~20!. ~b! The interband current~and also the polarization! forthe same excitation scenario continue to rephase att52t. Pulseareas areA150.5p, A250.5p, A45p, the pulse durationtLn

510 fs. The curves are averages over 64 disordered model reations.

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OPTICALLY INDUCED COHERENT INTRABAND . . . PHYSICAL REVIEW B65 085306

Note that the interband photon echo in all these cases alwappears att52t, as shown in Fig. 2~b!.

B. Dependence on excitation density

The amplitude of the spontaneous signal dependsexcitation intensities of the various pulses in different waFigure 3 shows the amplitude as a function of tpulse area An of the pulse No. n, defined by An

5ueEW •rWcv /\u*2`` dt exp@2(t/tLn)

2#. A pulse withAn5p cor-responds to complete inversion of a two-level absorbercited resonantly by pulse No.n.

It is seen that in particular the half-gap contribution hasbe strong enough, while the first full-gap pulse does not nto have such high intensity. While in the limit of low excitation intensity the amplitude of the signal depends linearlythe area of the first full-gap pulse, the dependence is qdratic for both the half-gap pulse and the second full-gpulse, reflecting the lowest relevant order of the variopulses. Therefore the spontaneous intraband responseleast of fifth order in the external light field.

VII. DISCUSSION

The appearance of the spontaneous intraband signasystems with correlated disorder can be easily understoothe basis of a simplified model. Let us assume that we donalize the conduction and valence band Hamiltonians,sulting in eigenstatesuvn& anducn& having energiesevn andecn for the valence and conduction band, respectively. THamiltonian is then given in the Appendix, Eq.~A1!. Theoptical Bloch equations describing the dynamics of the inband polarizationpnn85^avn

1 acn8& and the intraband variablesnn8n

h5dnn82^avn

1 avn8& andnnn8c

5^acn1 acn8& after exci-

tation with fieldE(t) are given in Eq.~A4! in the Appendix.Here we are interested in the response ofnn8n

h andnnn8c to the

excitation sequence. We assume thenn8nh andnnn8

c have beencreated by the coherent-control pulse and are now subje

FIG. 3. Dependence of the amplitude of the spontaneous sion the pulse areas. Solid line: variation of first half-gap pulse,A2

50.1p, A450.2p; dotted line: variation of first full-gap pulseA150.1p, A450.2p; dashed line: variation of second full-gapulse,A150.1p, A250.1p. tLn510 fs in all cases. The data habeen extracted from averages over 64 disorder realizations.effective masses of holes and electrons are equal.

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the excitation with the second full-gap pulse. This excitatiis resonant in the terms proportional to the interband dipmatrix elementmW and off-resonant in all terms proportionato the intraband dipole matrix elementDW nn8 . Thus we omitall terms proportional toDW nn8 in Eqs. ~21! and ~22!. Notethat because of the correlated disorder only interband dipmatrix elementsmW are nonzero between pairs of corresponing states, i.e., there is a strict selection rule. For tutoreasons we consider in the following only two states in eaband, i.e.,uv1&, uv2& and uc1&, uc2&, Fig. 4. The equationsof motion for nn8n

h andnnn8c for n51, n852 read

dn21h.

dt1 idvn21

h.5i

\mW •EW ~ t !@p21* 2p12#,

dn12c.

dt2 idcn12

c.5i

\mW •EW ~ t !@p21* 2p12#, ~21!

where dv5(ev12ev2)/\ and dc5(ec12ec2)/\ with dv,c.0. These equations of motion describe the intrabandnamics after the second full-gap pulse. Just before the arrof this second pulse then21

h, andn12c, have acquired phase

according todv anddc , respectively, due to their free motioin the interval between the first pulse and the second pusee Fig. 5. Just after the first excitation pulse their phawere such that~for the total ensemble ofM levels in eachband! a macroscopic intraband current was present. Wedenote these phases as initial phases. Now our aim is tothe time when the phases ofn21

h. andn12c. again equal these

initial phases. We assume for simplicity that the second puhas the formd(t2t), i.e., it arrives after a delay timetfollowing the first pulse~at t50) and is extremely shortFrom the equation of motion forp we find the values of thisvariable at timet5t which enter the driving term on theright-hand side of Eq.~21!:

al

he

FIG. 4. The four-level model we are using in Sec. VII to explahow the second excitation pulse att5t induces spontaneous signain the intraband current at later times. Note that there is a dipmoment between the two levelsv1, c1 and the levelsv2, c2 only.

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SCHLICHENMAIER, VARGA, MEIER, THOMAS, AND KOCH PHYSICAL REVIEW B65 085306

FIG. 5. ~a! The evolution ofphases for the case of microscoppolarizations which cause the photon echo.~b! Phase evolution ofelectron and hole intraband coheences that give rise to the spontneous signals of the intraband curent. Note, that the phases oelectron and hole coherences ainterchanged by the second excittion pulse.

-

r-s

f

s

ndx-omherrent

ase

o a

ies,torsthe

e-asesand

dp12

dt1 idc2 v1p1252

i

\mW •EW ~ t !~n21

h,1n12c,!,

dp21*

dt2 idc1 v2p21* 5

i

\mW •EW ~ t !~n21

h,1n12c,!, ~22!

where dc2 v15(ec21ev1)/\ and dc1 v25(ec11ev2)/\. Ifthese functions, taken at timet5t, are inserted into Eq.~21!,we find that the equation forn21

h. has a driving term proportional to 2n12

c,(t) while that for n12c. has a driving term

proportional to2n21h,(t). Note that due to correlated diso

der the free dynamics in the upper and lower pair of stateidentical up to a global scaling factor given bydv /dc . Con-sequently, n12

c.(t) has acquired the initial phase o2n21

h,(t) at time tc5(11dv /dc)t andn21h.(t) has acquired

the initial phase of2n12c,(t) at time tv5(11dc /dv)t. This

is schematically shown in Fig. 5~b!. Turning back to theensemble of more than two eigenstates in the bands wethat all these particular terms add up at timestv andtc to an

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intraband dynamics showing the initially generated intrabacurrent, while all other terms interfere destructively. This eplains the new intraband phenomenon. We also learn frthis consideration, i.e., the sign of the driving terms, why tspontaneous signals have the opposite sign to the first cupulse.

In contrast, the interband photon echo relies on the phconjugation of the interband polarizationspnn8 , as shown inFig. 5~a!. This interband-phase conjugation leads always trestoration of the initial phases at timet52t. If, however,one is interested in the dynamics of the intraband quantitwe have to consider that not only the interband phase facare conjugated by the second pulse, but in addition alsointraband phases.

VIII. CONCLUSION

The experimental verification or observation of the phnomena discussed in this paper should be possible in cwhere dephasing interactions do not destroy the electron

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OPTICALLY INDUCED COHERENT INTRABAND . . . PHYSICAL REVIEW B65 085306

hole coherences generated by the laser pulses. Interacwith the environment via phonons and for elevated excitatintensity, in particular, Coulomb scattering provides the mjor causes of fast loss of phase information in these systeTherefore it is necessary to discuss the circumstances uwhich these dephasing processes may still give enoughfor the coherent spontaneous signals to occur.

Coulomb scattering in ordered or weakly disordered stems is a rapid dephasing process.17,18 The question here ishow is this fact modified if disorder gets stronger. As whave seen, strong disorder proves to be more effective forealization of a current echo. It is expected that in the locized regime the Coulomb scattering between single partstates will be strongly suppressed due to the very small olap of these states.

In order to have a feeling of what the interplay betwestrong disorder and Coulomb scattering might be we hperformed numerical simulations of a one-dimensional oband quantum Coulomb-glass model19 with half-band filling.After obtaining the mean-field approximation by treating tdisorder exactly but the Coulomb interaction on a HartrFock level, we study the relaxation process of an initianonequilibrium occupation probability distribution over thHartree-Fock basis towards equilibrium caused by Couloscattering. This was treated within the second order Bapproximation. The initial nonequilibrium distribution wameant to be ‘‘generated’’ by a laser pulse.

Increasing the strength of disorder we see a very fastcrease in the relaxation rate as can be seen in Fig. 6.effect is attributed to the fact that with increasing disordthe single particle states become more localized and thfore reduce the probability of Coulomb scattering. At the eof the relaxation process the equilibrium distributionachieved to be a Fermi-Dirac distribution with an effectitemperature that increases with the excitation energy towthe band center.

The fact that the efficiency of Coulomb scattering canreduced due to the presence of disorder, may make itsible to have the intraband dephasing times sufficiently loin order to observe the spontaneous current responsedicted here. Ideally suited experiments should be performat low temperatures to reduce scattering with phononssamples that are characterized by not to weak disorder wneeds to be correlated in the valence and conduction baSemiconductor quantum wells with a significant amountwell width fluctuations seem to be good candidates for

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APPENDIX

The Hamiltonian for a system ofM states with energiesenv

being occupied in the ground state andM upper unoccupiedstates with energiesen

c reads

H05 (l5c,v

(n51

M

elnaln1 aln . ~A1!

In the electron-hole pictureenv anden

c are positive quantities

In the electron picture we haveenv52en

v and enc5en

c . Thelight-matter interaction is given by

HL52EW ~ t !•dW ~A2!

with the electric fieldEW (t) and the dipole operator

dW 5mW (n51

M

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FIG. 6. Population relaxation rateG/D as a function of disorderW/J for a single-band tight-binding model withN520 sites. Theinitial population is a Gaussian centered around the lower bedge.D is the inverse Heisenberg time~mean level spacing dividedby \! andJ is the nearest-neighbor coupling. The curves are paretrized according toU/J, the Coulomb potential at the nearesneighbor distance.

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SCHLICHENMAIER, VARGA, MEIER, THOMAS, AND KOCH PHYSICAL REVIEW B65 085306

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