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PHYSICAL REVIEW B 15 AUGUST 1998-IIVOLUME 58, NUMBER 8
Effective Frenkel Hamiltonian for optical nonlinearities in semiconductors:Application to magnetoexcitons
V. Chernyak, S. Yokojima, T. Meier, and S. MukamelDepartment of Chemistry and Rochester Theory Center for Optical Science and Engineering,
University of Rochester, Rochester, New York 14627~Received 10 November 1997; revised manuscript received 7 April 1998!
Closed Green-function expressions for the third-order response of semiconductors are derived by mappingthe two-band model onto the much simpler molecular~Frenkel! Hamiltonian. The signatures of two-excitonresonances are incorporated through the exciton scattering matrix, totally avoiding the explicit calculation oftwo-exciton states. Exact expressions for the nonlinear optical response of two-dimensional semiconductornanostructures in a strong perpendicular magnetic field are derived by truncating at then51 Landau level, andusing the symmetry of the system and a group-theoretical analysis. We find that the nonlinear optical responsedepends crucially on asymmetries between particle-particle and particle-hole Coulomb interactions.@S0163-1829~98!03732-1#
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I. INTRODUCTION
The influence of geometric confinement on the optiproperties of semiconductor nanostructures has been invgated extensively over the past decade.1–5 It is well estab-lished that exciton confinement on length scales smaller tits Bohr radius leads to increased exciton binding energand larger oscillator strengths, and consequently to a stroresonant optical response. These effects have been stuin quasi-two-dimensional, one-dimensional, and zedimensional systems~quantum wells, wires, and dotsrespectively!.1–4 In addition to geometric confinement, thdimensionality of semiconductor systems may also beduced by applying external fields. A strong static electfield applied in the growth direction of semiconductor suplattices suppresses the coupling between the quantum wof the superlattice, resulting in a Stark localization of chacarriers6 and changing the exciton from anisotropic threemensional to basically two dimensional.7 The same reductionof the effective exciton dimensionality can be inducedapplying an appropriately chosen alternating field~dynamiclocalization!.8 Strong confinement effects on the relatielectron-hole motion have also been achieved by applystatic and homogeneous magnetic fields, which restrictfree motion of electrons perpendicular to the field, and leto the formation of Landau levels.9–14
In a strong magnetic field one can neglect proceswhich involve transitions of an electron or a hole betwedifferent Landau levels. In this approximation, when excitstates are formed, relative electron-hole motion is only psible in the direction of the field. This implies that, in thredimensional structures, strong magnetic fields result in odimensional relative motion,9–11 while relative motion intwo-dimensional semiconductors in strong perpendicumagnetic fields becomes zero dimensional.12–14 In semicon-ductor quantum wells in a strong magnetic field, we havsingle exciton for a given value of the center of mass mmentum once the Landau levels of the electron and theare fixed.
PRB 580163-1829/98/58~8!/4496~21!/$15.00
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Linear and nonlinear optical properties of magnetoextons in both bulk semiconductors9–11,15 and in quantumwells12–14 have been extensively studied experimentally.the theoretical side, investigations of the linear spectramagnetoexcitons in quantum wells16 have pointed out theimportance of valence-band mixing effects. SemiconducBloch equations~SBE’s! for magnetoexcitons in two17 andthree18 dimensions have been developed. The SBE’s, whfurther allow the analysis of nonlinear optical propertiehave been used to investigate many-particle effects, whictwo dimensions results in coupling between the individuLandau states, and in three dimensions also in a couplinhigher-energy Landau states to the continuum of lower Ldau states. This gives rise to Fano resonances in the lispectra that exhibit a unique and not yet fully understofour-wave-mixing ~FWM! response.9,10 Calculations ofFWM signal at Fano resonances neglecting many-parteffects19 indicate that those should be responsible forexperimental findings.
Similar to the field-free case, it is crucial to include thmany-particle Coulomb interaction for a proper descriptiof the linear and especially the nonlinear optical propertiesmagnetoexcitons. The SBE’s are based on the timdependent Hartree-Fock~TDHF! approximation, where two-exciton variables are not included as independvariables.2,20–22 They have been successfully applied to eplain nonlinear optical phenomena in semiconductorssemiconductor nanostructures.2 Recently, the importance oincluding correlations beyond the TDHF has been demstrated mainly by analyzing the dependence of the nonlinresponse on the polarization of the exciting pulses. Theperimental findings have been explained in termsexcitation-induced dephasing processes23–26 which were re-lated to excitonic screening. It has been shown that thterms are related to the two exciton continuum,27–29 i.e., re-sult from exciton-exciton scattering. The role of bound biecitons in four-wave mixing on semiconductors has been fther pointed out.28,30–36 Recently, using a polarizationconfiguration that eliminates the contribution of bound biecitons, the signatures of unbound two-exciton states on fo
4496 © 1998 The American Physical Society
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PRB 58 4497EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
wave mixing signals from GaAs layers have been invegated both experimentally and theoretically.11
Similar problems appear in optical excitations of moleclar crystals and nanostructures which are represented byFrenkel exciton Hamiltonian. The role of two-exciton statin the third-order optical response of Frenkel-exciton stems has been well understood.37 The analog of the SBE foFrenkel-exciton systems has been called the local-fieldproximation ~LFA!,37 and can be derived by taking the epectation value of the Heisenberg equation of motionexcitons operators and factorizing the expectation valuehigher-order products into products of expectation valuesexciton operators, which yields a closed system of equatfor one-exciton variables.37 The LFA accounts for excitonexciton scattering, and gives correct results far from the twphoton resonances on two-exciton states. These equahave been generalized by introducing closed equationmotion for one- and two-exciton variables,38 which yieldsexact results for the third-order optical response in thesence of dephasing, and shows the limitations of the LThis formalism is based on the fact that correlation functiocan be classified according to their order in the exciting fieThis allows a systematic truncation of the many-body prlem at a given order in the fields. The third-order respofunctions was later rederived using the concept of excitexciton scattering39 which finally resulted in Green-functionexpressions forx (3).40 x (3) is then represented as a produof four one-exciton Green functions and the exciton-exciscattering matrix. Radiative damping has been aincorporated.41 Coupled equations of motion for one-excitoand exciton population variables which take into accodephasing induced by exciton-phonon coupling, were deoped in Ref. 42. This theory provides an adequate desction of degenerate four-wave mixing, where populationlaxation is of extreme importance; however, treating excitexciton scattering on the LFA level does not reproduce twexciton resonances explicitly. The approach of Ref.which deals with coupled equations of motion for onexciton, two-exciton, and exciton-population variables intpolates between the theories of Refs. 38 and 42; howevedoes not describe the combined effects of exciton popularelaxation and resonant exciton-exciton scattering.
The LFA has been generalized to include populationlaxation and resonant exciton-exciton scattering. A thewhich accounts for these combined effects developed in R44 is based on Green-function techniques and perturbatreatment of exciton-phonon interaction with partial infinresummation of perturbative contributions. This theory hbeen applied to interpret pump-probe measurements intosynthetic antenna complexes.45 An alternative, more intui-tive, derivation is based on a closed system of coupled eqtions for one-exciton, two-exciton, exciton-population, acombined one-plus-two-exciton coherence variables elinating the phonons using projection operator technique46
and applying certain factorization to relaxation kernels ratthan to exciton variables.44
Recent developments in semiconductors follow precisthe same line of thought: thex (3) formalism developed toinclude correlations beyond the TDHF in semiconducsystems27,47–51manifests itself in a straightforward generazation of the theory of Refs. 2 and 38. This has be
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demonstrated52 by mapping the two-band semiconductHamiltonian onto an effective exciton model. By establising the connection between the two models, we can apmany of the results developed for Frenkel systems to seconductors. The Frenkel Hamiltonian restricts the dynamof the two-band model to a relevant subset of states.doing so the problem is simplified considerably, makingeasier to develop higher levels of theories which include crelation, scattering and phonon effects.
In this paper, starting with a two-band model, we useformal similarity between semiconductor and molecular stems to derive closed equations of motion for one- and twexciton variables which are exact for up to third-order nolinear optical response. By solving these equatioperturbatively in the external field we obtain compact Grefunction expressions~GFE’s! for the third-order susceptibil-ity. Using a formalism originally developed for Frenkeexciton systems,x (3) is expressed in terms of one-excitoGreen functions and the exciton-exciton scattering mawhich accounts for two-photon resonances on two-excistates. These expressions include the effects of both boand unbound two-exciton states. We apply GFE’s to callate x (3) of two-dimensional magnetoexcitons in the limit ostrong magnetic fields, where we can restrict the calculato the lowest (n51) Landau exciton. Using the symmetry othe system, a group-theoretical analysis enables us to deexact expressions for the optical response including bbound and unbound two-exciton states. Calculating the texciton states is a four-particle problem, which in two dmensions means that the wave functions depend on ecoordinates. Neglecting the small photon momentum, otwo-exciton states with zero total momentum can be excitwhich reduces the number of coordinates to six. In stromagnetic fields we can restrict the analysis to then51 Lan-dau level, and eliminate the relative motion of the electrohole pair. The two-exciton states with zero momentumthen constructed out of pairs of one-exciton states fromlowest Landau level with opposite center-of-mass momeand the calculation of these states involves only two coonates. We do not have to consider trial-variational wafunctions for the two-exciton states, and no further appromations are necessary beyond making use of the symmeof the system and truncating at then51 Landau level, whichin strong magnetic fields is energetically well separated frall other higher-lying levels. It was shown previously that tthermodynamical properties of spinless magnetoexcitonspend crucially on asymmetries between the particle-partand particle-hole Coulomb interactions.53 In particular, ifthese two interactions are identical, an ensemble of magtoexcitons at low concentrations behaves like an ensembnoninteracting bosons.54 We will show that there is still afinite nonlinear optical response in this case. We extethese theories by considering excitons with different angumomenta, and investigate the spectra of two-exciton stfor different ratios of the various Coulomb interactions. Bsolving the equations of motion derived here, we predictsignatures of two-exciton states in nonlinear optical expments conducted on GaAs quantum wells in strong magnfields.
As in the absence of a magnetic field, polarizatiodependent two- and three-pulse FWM experiments in wh
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4498 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
the signal is analyzed for different polarization configutions of the exciting pulses are most likely to pinpoint tinfluence of two-exciton states on the nonlinear opticalsponse. In such experiments two-exciton states with diffesymmetries may be investigated separately by varyingpolarizations of the excitation pulses. Here we focus onfrequency-domain two-color pump-probe signal.
This paper is organized as follows. In Sec. II, the twband Hamiltonian of semiconductors is mapped onto a Fkel exciton system which is exact up to the third order ofoptical response. The Hamiltonian is used to derive the eqtions of motion for one- and two-exciton variables in SeIII. Using these equations of motion, the Green-functionpressions for the third-order optical susceptibility are drived. In order to apply the Hamiltonian to a quantum wella strong magnetic field, in Sec. IV we calculate relevant oand two-exciton states. Here we choose the Coulomb potial in a form which accounts for the asymmetry in thparticle-particle and particle-hole interactions. Closedpressions for the third-order susceptibility of magnetoextons are given in Sec. V. Numerical results are presenteSec. VI. Energies of the two-exciton eigenstates show vous distributions depending on different asymmetries ofCoulomb interaction. Using the expression for the thiorder susceptibility of magnetoexcitons, two-color pumprobe signals are calculated. The signal strongly dependasymmetries of the Coulomb interaction.
II. MAPPING THE TWO-BAND HAMILTONIANONTO A FRENKEL FORM
We start with the two-band model Hamiltoniansemiconductors52
H5 (m1n1
tm1n1
~1! am1
† an11 (
m2n2
tm2n2
~2! bm2
† bn2
1 12 (
m1n1k1l 1Vm1n1k1l 1
~1! am1
† an1
† ak1al 1
1 12 (
m2n2k2l 2Vm2n2k2l 2
~2! bm2
† bn2
† bk2bl 2
1 12 (
m1n2k2l 1Wm1n2l 1k2
am1
† bn2
† bk2al 1
. ~2.1!
Herean1(an1
† ) and thebn2(bn2
† ) are the Fermi annihilation
~creation! operators of electrons and holes respectively, wthe commutation relations
an1am1
† 1am1
† an15dm1n1
,~2.2!
bn2bm2
† 1bm2
† bn25dm2n2
.
We adopt the following convention for indices: Latin indicwith a subscript 1~2!, e.g., m1 (m2) stand for electron~holes!, including spin variables. Electron-hole pairs are dnoted by Latin indices without subscriptsm5(m1m2).
The Hamiltonian of the systemHT(t) in the presence othe driving fieldE(r ,t) has the form
-
-ntee
-n-ea-.--
-n-
-i-ini-e--on
h
-
HT~t!5H2E dr m~r !E~r ,t!, ~2.3!
with the dipole operator
m~r 
--
-
n
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PRB 58 4499EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
tion. The operatorU acting in the two-exciton space is represented by a tetradic matrixUnm,pq , and describes excitonexciton Coulomb interactions
U5V13~1!1V24
~2!1W141W32. ~2.11!
In Eq. ~2.11! we used the following convention: the twoexciton space can be represented as (V1^ V2) ^ (V3^ V4),where V15V35V is the electron space, whereasV25V45V* is the hole space, andV1^ V2 and V3^ V4 representtwo identical replicas of the single-exciton space. For aoperatorQ acting inVi ^ Vj , we denote byQi j an operatorin the two-exciton space which acts asQ in Vi ^ Vj and as aunit operator in the product of the remaining two singparticle spaces.
The first two terms on the right-hand side of Eq.~2.11!represent electron-electron and hole-hole interactiowhereas the third term represents the electron-hole intetions. Note that electron-hole Coulomb energy for an eltron and a hole which belong to the same exciton is incluin h but does not appear inU. The tetradic matrixPnm,pq inEq. ~2.8! also represents a two-exciton operator.
Equations~2.8! are not closed, and a proper truncatiprocedure needs to be developed in order to convert tinto a practical computational method. In the simplest trucation scheme, known as the local-field approximation,factorize all products of normal ordered operators into sinoperator expectation values. This yields a closed equationthe expectation values.
id^Bn&
dt5(
mhnm^Bm&2En1(
mpqUnm,pq^Bm&* ^Bp&^Bq&
1(mpq
Pnm,pq^Bm&* ~^Bp&Eq1^Bq&Ep!. ~2.12!
Equation~2.12! constitutes the LFA for the exciton Hamitonian of Eq. ~2.6! and is equivalent to the SBE for thtwo-band model written in terms of exciton operators intduced by Eq.~2.5!. Note that since Eq.~2.12! was derivedusing Eqs.~2.6! and ~2.7!, it is equivalent to the SBE up tothe third-order response, and should be modified whigher-order response functions are calculated.
The same equation@Eq. ~2.12!# was derived in Ref. 55 forthe model of a Frenkel-exciton aggregate made of three-lmolecules. In that case Latin indices represent the molecuand Bn(Bn
†) are Frenkel exciton annihilation~creation! op-erators in the molecular representation. We assume thatmolecule has three electronic levels, whereVn
(1) andVn(2) are
the transition frequencies between consecutive levels~Fig.1!. We further denote the ratio between consecutive tration dipoles of themth molecules bykm . Expressions forh,U, andP were given in Ref. 55. We then have
Umn,kl5dmn12 @dmkdnlgm1~km
2 22!
3~hmkdnl1dmkhnl!#Pmn,kl , ~2.13!
with
y
-
s,c--d
m-eeor
-
n
els,
ch
i-
gn52Vn~2!kn
222Vn~1! ,
Pmnkl5dmndmkdnkS 12km
2
2 D .
Neglecting the momentum of the optical field, Eq.~2.12!is reduced to the equation of motion for the zero-momentexciton operatorsc[(mBm , which assumes the form of thGinzburg-Landau equation
idc
dt5Vc2mE1Vucu2c1mbucu2E. ~2.14!
Equation~2.14! applies to both the Frenkel exciton modand the semiconductor two-band model. Within theseproximations the third-order response is characterized byratio V/b, which determines the relative importance of ttwo optical nonlinearities. For the two-band model, the prametersV and b are determined by the shape of the onexciton wave function. Takingf(k) to be the real wavefunction of the 1s exciton ink space~herek is the momen-tum associated with the relative motion of electrons aholes!, and denoting the Coulomb-interaction byV(k2k8),the two nonlinear terms are given by56,57
b52(k
f~k!3, ~2.15!
V52(k,k8
V~k2k8!@f~k!3f~k8!2f~k!2f~k8!2#. ~2.16!
~Similar equations in real space were given in Ref. 58.! brepresents phase-space filling, andV denotes the Coulombinteraction.b is always finite, with a value which is determined by the exciton wave function. In contrast, the exprsion for V @Eq. ~2.16!# involves a difference between twterms, and the sign ofV can be different for different wavefunctions.
FIG. 1. The optical response of the two-band model of semicductors calculated using semiconductor Bloch equations~SBE’s!can be mapped onto an effective lattice of three-level systemsscribed by the Frenkel Hamiltonian. The parameters of this simmodel are given in this figure. Two sources of nonlinearity~anhar-monicity! are identified@k[mn
(2)/mn(1) andg ~Eq. ~2.13!#.
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4500 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
For the effective Frenkel exciton model we obtain, froEq. ~2.13!,
b522k2,~2.17!
V5~k222!h1k2g
2.
Now b represents non-Bose statistics, whereasV depends onstatistics as well as anharmonicity.
III. COUPLED ONE- AND TWO-EXCITONDYNAMICS; THIRD-ORDER SUSCEPTIBILITIES
BEYOND THE LFA
In this section we make use of Eqs.~2.6! and ~2.7! toderive equations of motion for one- and two-exciton vaables, whose solution results in closed expressions forthird-order susceptibility which goes beyond the local-fieapproximation. In the absence of pure dephasing, the pstate factorization can be applied, and Eq.~2.8! leads to thefollowing closed equations of motion for one- and twexciton variables:52
id^Bn&
dt5(
mhnm^Bm&2En1(
mpqUnm,pq^Bm
† &^BpBq&
1(mpq
Pnm,pq^Bm† &~^Bp&Eq1^Bq&Ep!, ~3.1!
id^BnBn8&
dt2 (
mm8~hnmdn8m81dnmhn8m8!^BmBm8&
2 (mm8
Unn8,mm8^BmBm8&
52~dnmdn8m82Pnn8,mm8!~Em^Bm8&1^Bm&Em8!.~3.2!
These equations are exact for the response up to third oand are equivalent to the equations derived in Ref. 47–4
Equations~3.1! and ~3.2! can be solved perturbatively ithe external field. Switching to the frequency domain yieGreen function expressions for optical susceptibilities whmay be conveniently represented by introducing operatortation. For the linear susceptibility we have
x~1!~2vr s ;vr !5^m~r s!uR~1!~2v;v!um~r !&, ~3.3!
wherem(r ) is treated as a vector in the one-exciton spawith componentsmn(r ), whereasR(1) is an operator in thesame space:
R~1!~2v ;v!5G~v!1G†~2v!. ~3.4!
G(v) is the one-exciton Green function
G~v!5@~v1 ih!I 2h#21, ~3.5!
whereh is phenomenological damping.The GFE for the third-order susceptibility has the form
-he
re
er,.
sho-
e
x~3!~2vsr s ;v1r1 ,v2r2 ,2v3r3!
5 16 (
perm^m~r s! ^ m~r3!uR~2vs ;v1 ,v2 ,2v3!u
3m~r1! ^ m~r2!&1c.c., ~3.6!
where (perm denotes a sum over the six permutationsthree pairsv1r1 ,v2r2, and2v3r3. The operatorR acting inthe two-exciton space is given by
R~2vs ;v1 ,v2 ,2v3!
5G~vs! ^ G†~v3!G~v11v2!G~v1! ^ G~v2!, ~3.7!
where the two-exciton scattering matrix, treated as an optor in two-exciton space, has the form
G~v!522P@ F~v!#2112U$@ F~v!#212U%21
3~ I 2P!@ F~v!#21, ~3.8!
and F(v) is the two-exciton Green function for excitontreated as noninteracting bosons:
F~v
en
c
s fm
inliI
se-
nc
ri-elee-
neasid
ru
ere
umil-
o
b
-s
ym-all
a
sec-
the
on,ingf
tumThectlyinnalunt
unc-
e-
n,
in-the
PRB 58 4501EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
Pab,mn are matrix elements of the electron angular momtum permutation, whereasI ab,mn are matrix elements of theunit operator in the angular-momentum two-exciton spaand
H[h121h341U. ~3.13!
Equations~3.10!–~3.12! expressx (3) in terms of operatorsacting only in real space, and provide general expressionthe frequency-domain third-order optical response of seconductors. The two-exciton scattering matrix@Eq. ~3.12!#has two terms. The first includes exciton-exciton scatterdue to statistical properties, and the second reflects nonearities induced by the many-body Coulomb interaction.the coming sections we will apply Eqs.~3.10!–~3.12! to cal-culate the frequency-domain third-order optical responsequantum wells in a strong magnetic field. We will also uEqs. ~3.1! and ~3.2! to analyze the time-domain optical response of the same system.
IV. ONE- AND TWO-EXCITON STATES OF QUANTUMWELLS IN A STRONG MAGNETIC FIELD
Equations~3.10!–~3.13!, which give x (3) for a generaltwo-band model, require an extensive numerical effort si
Eq. ~3.12! for G(v) involves inverting of the operator (v2H12ih)in the two-exciton space. It is very difficult tosolve a four-particle problem either analytically or numecally. However, in the case of semiconductor quantum win a strong magnetic field, the large separations betwelectron~or hole! Landau levels, together with the high symmetry of the problem, makes it possible to calculate oexciton states analytically and to construct convenient bsets in the space of relevant two-exciton states. This provan efficient algorithm for computingx (3) as well as approxi-mations for limiting cases.
We consider a two-dimensional semiconductor nanostture, i.e., a thin quantum well, in thexy plane with a mag-netic fieldH applied in thez direction. WhenH is strong, wecan neglect the Coulomb coupling between excitons formby electrons or holes from different Landau levels, andstrict the electron and hole to the lowest Landau levels~otherLandau levels can be considered in a similar manner!. In thissection we start with a two-band Hamiltonian for a quantwell in strong magnetic field. We derive a two-band Hamtonian which only incorporates those electrons~holes! be-longing to the lowest electron~hole! Landau levels, We thencalculate the one-exciton Hamiltonianh, the permutation op-eratorsP, and the exciton-exciton interaction operatorU.These will allow to apply the GFE’s@Eqs.~3.9!–~3.12!# forcalculatingx (3) of magnetoexcitons.
We assume that the kinetic-energy operators of electrand holes possess two-dimensional Poincare´ symmetry~i.e.,with respect to translations and rotations!, and the charge-density operator responsible for the Coulomb interactionstween electrons and holes has a form
r~r !5e@Ce†~r !Ce~r !2Ch
†~r !Ch~r !#, ~4.1!
whereCe(r ) @Ch(r )# is the annihilation operator for an electron ~hole! at point r . We further assume that interaction
-
e,
ori-
gn-n
of
e
lsn
-ises
c-
d-
ns
e-
with the external transverse field conserve translational smetry ~since optical wave vectors are small compared toother length scales in the problem!.
The HamiltonianH describing electrons and holes inmagnetic field is given by
H5(i
Hi1(i j
Hi j , ~4.2!
with i , j 5e,h whereHe andHh contain the kinetic energieincluding the terms induced by the magnetic fields for eltrons and holes with effective massesme and mh , respec-tively:
H j5E Cj†~r !
1
2mjS pj1sj
e
cA~r ! D 2
C j~r !dr , ~4.3!
where se[1,sh[21, and p[2 i\]/]r . Hi j describe theCoulomb interaction, and are given by
Hi j 5si j
2 E C i†~r !C j
†~r 8!e2
« i j ur2r 8uC j~r 8!C i~r !drdr 8,
~4.4!
wheresee5shh[1, andseh5she[21.The vector potentialA~r ! is chosen in the holomorphic
gauge
A~r 
igowthleth
fo
or
ebeeymer
ia-b
oisaic
c-
in
est
b
t
on
be
a
.t
orn
4502 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
particle-hole part of the interaction, which has important snatures in two-exciton spectroscopy, as will be shown belAnother possible source of asymmetry may come fromsingle-particle dispersions, which especially for the homay not be isotropic. However, this effect, and alsovalence-band mixing effect,16 are neglected in the followingcalculations, where we employ the effective mass modelboth electrons and holes.
In our calculations we assumed the following forms fthe Coulomb potentials;
Veh~r ,r 8!52a
ur2r 8u$11keh exp@2ur2r 8u2/r 0
2#%,
Vee~r ,r 8!5Vhh~r ,r 8!
5a
ur2r 8u$11keeexp@2ur2r 8u2/r 0
2#%. ~4.8!
This form may be rationalized as follows. At large distancthe particle-particle and particle-hole potentials shouldhave like6aur2r 8u21, and electron-hole asymmetry in thCoulomb interactions should vanish. This short-range asmetry is modeled in Eqs.~4.8! by the Gaussian term. Thasymmetry length-parameterr 0 is chosen to be the same foall potentials, whereas the parameterskeh andkee representits magnitude.
It is important to note that although the vector potentgiven by Eq.~4.5! is not translationally invariant, translational symmetry is not destroyed: it should be redefinedadding a phase vector~see Appendix A!. The momentumoperators no longer commute, and the Poincare´ symmetry istransformed to its central extension, hereafter referred toPoincare´-Weyl-Heisenberg~PWH! symmetry~see AppendixA for details!. Classification of irreducible representationsthe PWH algebra is given in Appendix B. Adopting thlanguage, the electron and hole Landau levels can be treas irreducible representations of the PWH algebra, whshows an algebraic origin of the energetic degeneracyLandau levels.
Hereafter we switch to dimensionless coordinatesr / l H
scaled by the characteristic magnetic lengthl H[A\/eH. Us-ing holomorphic coordinatesze , and zh for electrons andholes (z5x1 iy) we obtain the electron and hole wave funtions on the lowest Landau levels:
c~re!5w~ ze!expF2uzeu2
4 G , ~4.9!
c~rh!5w~zh!expF2uzhu2
4 G , ~4.10!
wherew is an arbitrary function.Using Eqs.~4.9! and~4.10! we can introduce basis sets
the one-electron and one-hole spaces corresponding tolowest Landau level.
wm1~re!5
1
Ap2m111Am1!ze
m1expF2uzeu2
4 G , ~4.11!
-.
ese
r
s-
-
l
y
as
f
tedhof
the
wm2~rh!5
1
Ap2m211Am2!zh
m2expF2uzhu2
4 G . ~4.12!
Considering electrons and holes which belong to the lowLandau level and making use of Eqs.~4.11! and ~4.12!, we
represent the field operatorsC j (r ) in forms
Ce~r !5 (m150
`1
Ap2m111Am1!zm1expF2
uzu2
4 Gam1,
~4.13!
Ch~r !5 (m250
`1
Ap2m211Am2!zm2expF2
uzu2
4 Gbm2,
~4.14!
where am1(bm2
) are electron~hole! annihilation operatorsintroduced in Sec. II for the general two-band model.
Substituting Eqs.~4.13! and ~4.14! into Eqs.~4.2!–~4.4!,we recast the two-band Hamiltonian in the form of Eq.~2.1!,where indicesm1 (m2) represent electron~hole! states on thelowest Landau level. The parameters of the HamiltonianV( j )
andW are expressed in terms of integrals involving Coulompotentials and wave functions@Eqs. ~4.11! and ~4.12!#. Fort ( j ) we have
tm1n1
~e! 5E0~e!dm1n1
, tm2n2
~h! 5E0~h!dm2n2
~4.15!
whereE0(e)@E0
(h)# is the electron~hole! energy on the lowesLandau level.
To calculate the matrix elements of operatorsh, U, andP,we start withh, which operates in the space of one-excit~one electron and one hole! states. According to Eqs.~4.9!and ~4.10!, a state with one electron and one hole canrepresented as
C~re ,rh!5w~ ze ,zh!expF2uzeu2
42
uzhu2
4 G . ~4.16!
Since the total charge of an excitonic state is zero@ p1 ,p2#50 wave functions with given center-of-mass momentapform a basis set of eigenstates of the Hamiltonianh in thespace of functions given by Eq.~4.16!, a normalized wavefunction with momentump can be easily found and hasform
Cp~1!~re ,rh!5
1
~2p!3/2expF2
p2
4 GexpH 2r 2
41
i
2n•@r ,R#
1 ip•R11
2n•@p,r #J , ~4.17!
whereR5 12 (re1rh), r5re2rh , andn is a unit vector in the
z direction.Hereafter we use the basis setCp represented by Eq
~4.17! in the one-exciton space instead of the basis sem5(m1m2). Due to translational symmetry, the operatorh isdiagonal in theCp basis set. Its eigenvaluesep may be ob-tained by evaluating the expectation values of the operathgiven by Eq.~2.10! using one-exciton wave functions giveby Eq. ~4.17!. A direct calculation yields
PRB 58 4503EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
ep5E02aAp
2 H S I 0S p2
4 D expF2p2
4 G1keh
1
A11 2/r 02
I 0S p2
4~112/r 02!D expF2
p2
4 S 11 4/r 02
112/r 02 D G J . ~4.18!
in-n
enonlaft
a
as
tro
ce-
o
n
th
ito
h
r-
-
llb
c
iton
itonfor
theg-
m
Herea is the constant of the Coulomb interaction writtendimensionless units of length,I 0 is the modified Bessel function, andE0[E0
(e)1E0(h) is the sum of Landau energies of a
electron and a hole.We next turn to the operatorsP andU acting in the two-
exciton space. It is not necessary to calculate matrix elemof these operators between all two-exciton states sincetwo-exciton states with zero total momentum and angumomentum are involved in the optical response. Hereawe refer to them as the relevant two-exciton states.~Usingmore formal language the relevant states are those whichsymmetric with respect to the PWH algebra.! This allows usto use the theory of representations to find convenient bsets in the space of relevant two-exciton states. IfV0 is thePHW algebra representation related to the lowest elecLandau level~see Appendixes A and B!, thenV0* ~the dualrepresentation! is related to the lowest hole level. The spaof one-exciton states isV0^ V0* , and the space of twoexciton states isW[(V0^ V0* ) ^ (V0^ V0* ).
Since we are interested only in zero-momentum twexciton states, we need to decomposeW into a sum of irre-ducible representations and keep only unit components~i.e.,one-dimensional representations of the vectors which areaffected by the action of the algebra!. This can be done intwo ways leading to two convenient basis sets~see AppendixC!. A discrete basis set is obtained by representingW5(V0
^ V0) ^ (V0* ^ V0* ), decomposingV0^ V0 andV0* ^ V0* into adirect sum of irreducible representations and by findingsymmetric components in the product (V0^ V0) ^ (V0*^ V0* ). This leads to a basis set represented by two-excwave functionsCn given by Eq.~C7!. The matrix elementsUmn and Hmn of the operatorsU @Eq. ~2.11!# and H @Eq.~3.13!# are evaluated and given in Appendixes C and D. Tmatrix elements ofP have the form~see Appendix C!
Pmn5~21!mdmn . ~4.19!
An alternative, continuous, basis set is obtained by repsentingW5(V0^ V0* ) ^ (V0^ V0* ). As shown before, oneexciton wave functionsCp
(1) @Eq. ~4.17!# form a basis set inV0^ V0* . Translationally invariant two-exciton wavefunctions are represented bycp defined by
cp[cp~1!
^ c2p~1! . ~4.20!
The relevant two-exciton states which are translationainvariant as well as have zero angular momentum canobtained by
cq~2![E dp d~ upu2q!cp ; ~4.21!
however, it is more convenient to work in a larger sparepresented bycp .
tslyr
er
re
is
n
-
ot
e
n
e
e-
ye
e
The matrix elementsUpp8 , Hqq8 , andPqq8 are evaluatedin Appendix E, yielding
Ppp851
4p~e2 in•[p,p8]1ein•[p,p8] ! ~4.22!
which implies that in the continuous basis setP acts as aFourier transform. The matrix elementsUpp8 and Hqq8 aregiven in Appendix E.
In summary, we found the eigenstates of the one-excHamiltonianh, and matrix elements of the operatorsU andHusing two basis sets in the space of relevant two-excstates. In Sec. V we combine these results with the GFEx (3) obtained in Sec. III to derive a closed expression forthird-order susceptibility of quantum wells in a strong manetic field.
V. THIRD-ORDER SUSCEPTIBILITYOF MAGNETOEXCITONS
We start our derivation by switching to the momentu(k) space. Introducing
ma~k
ito
y
tede
.
n
-set
n-
-on
ons
on-the
r ofe
sticpli-
tonsys-t
emlec-ita-e-ive
he
4504 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
ma~k! ^ mb~q!5mambf~k,q! ~5.4!
which follows from translational symmetry. Herema arenumbers andf(k1 ,k2)[ck1
(1)^ ck2
(1) denotes a two-exciton
state represented by a direct product of two one-excstates with momentak1 andk2, ma is the dipole of the ex-citon with polarizationa, ek is the exciton energy given bEq. ~4.18!, and
Gasa3 ,a1a2~v;ks ,k3 ;k1 ,k2!
[^f~ks ,k3!uGasa3 ,a1a2~v!uf~k1 ,k2!&. ~5.5!
To calculateG we note thatf(p,2p)5Cp , the latter isdefined by Eq.~E1!. Since in theCp basis set the operatorPacts as a Fourier transform, we have
^f~p,2p!uPuf~p8,2p8!&
51
4p~e2 in•[p,p8]1ein•[p,p8] !. ~5.6!
The wave functionf(0,0) represents a two-exciton stawith zero total and angular momentum. It can be expanthrough the statesCn given by Eq.~C7! @see Eqs.~C9!–~C11!#,
f~0,0!51
Ap(
nCn , ~5.7!
whereas
PCn5~21!nCn . ~5.8!
Setting ks5k35k25k150 in the right-hand side of Eq~5.5! and making use of Eqs.~5.5!–~5.8! we finally obtain
x~3!~2vs2ks ;v1k1 ,v2k2 ,2v32k3!
5 16 (
perm(
asa1a2a3
masma1
ma2ma3
31
v12ek11 ih
1
v22ek21 ih
1
v32ek32 ih
31
vs2eks1 ih
Gasa3 ,a1a2~v11v2!, ~5.9!
where
Gab,mn~v!5Gab,mn~1! ~v!1Gab,mn
~2! ~v!, ~5.10!
with
Gab,mn~1! ~v!52
1
p~v22e012ih!Pab,mn ,
n
d
Gab,mn~2! ~v!5
2
p~v22e012ih!
3(mn
@U~v2H12ih!21#mn
3@ I ab,mn2~21!nPab,mn#. ~5.11!
Equations~5.9!–~5.11! constitute our closed final expressiofor x (3).
Equations~5.11! expressesG (2)(v) using the discrete basis set. A similar expression using the continuous basiscan be derived in a similar way, yielding
Gab,mn~2! ~v!52~v22e012ih!
3E dqdpU0q@~v2H12ih!21#qp
3S I ab,mndp011
2pPab,mnD . ~5.12!
VI. TWO-EXCITON RESONANCES IN PUMP-PROBESPECTROSCOPY
The expressions forx (3) derived in Sec. V@Eqs. ~5.9!–~5.11!# represent the susceptibility in terms of the excito
exciton scattering matrixG(v), which is given by a sum of
two termsG (1)(v) and G (2)(v) @Eq. ~5.10!#. G (1) describeseffects of phase-space filling~or, using a different terminol-ogy, nonboson exciton statistics!. Exciton statistics is determined by exciton wave functions which do not depend
the exciton-exciton interaction operatorU; G (1) has therefore
a universalU-independent form given by Eq.~5.11!. G (2)
describes scattering induced by exciton-exciton interacti
and vanishes forU50. In this sectionG (2) will be calculatednumerically using Eq.~5.11!. Analytical expressions forG (2)
for some limiting cases are presented in Appendix F.It has been known since the early 1980s that if electr
electron, hole-hole, and electron-hole interactions havesame [email protected]., kee5keh in Eq. ~4.8!# and the bands areisotropic ~this is known as the symmetric model!, excitonsdo not interact at large distances, resembling the behaviofree bosons,54 and the exciton-exciton scattering amplitudvanishes for small exciton momenta. Weak asymmetry (keeÞkeh) leads to the appearance of exciton-exciton elascattering with a universal behavior of the scattering amtude S(k)5(Aklnk)21 ~Ref. 53! at small momenta, whichstrongly affects the thermodynamic properties of the excigas. Since the nonlinear optical response of harmonictems is known to vanish,37 S(k) should have some direcsignatures inx (3).
In order to calculate the optical response of this systwe need to take into account the angular momenta of etrons and holes. As discussed in Ref. 30, the optical exction of the energetically lowest exciton in bulk GaAs corrsponds to the excitation of an electron from an effectangular momentaJ5 3
2 valence band to aJ5 12 conduction
band. Since excitons are made of band states close to tG
a
thrm
gth
ci
of
ns
te
triarclolesninlsthfoe,
e-
o-
notthelytederd
he
eldf ad
ons
tric
PRB 58 4505EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
point p50, the conduction band can be described by a pof Jz56 1
2 states and the valence band by pairs ofJz56 32
~heavy-hole! and Jz56 12 ~light-hole! states. In a quantum
well, the confinement lifts the degeneracy betweenheavy-hole and light-hole excitons, and provides a natuquantization axis so that for fields polarized in the quantuwell plane only transitions withDJz561 are dipole al-lowed. As displayed in Fig. 2, we neglect in the followinthe light holes and consider only the transitions betweenheavy holes and the conduction-band, which gives risetwo degenerate single exciton resonances that can be exby circularly polarized lights1 ands2, respectively.30
Since the eigenvalues ofHmn determine the resonances
G (2)(v), we first examine the two-exciton states usingHmn@Eq. ~C12!#. We call the eigenvalues ofHmn as e2 in thefollowing. The variation of the two-exciton energiese2 withincreasing dimensionalityN0 (N0 is taken to be even! of thetwo-exciton matrixHi j for keh50 andkee>0 is displayedin Figs. 3 and 4. We denote the one-exciton energy imagnetic fieldH asV(H). The energy of two one-excitonHamiltonians in the absence of a magnetic field 2v0[2V(0) is taken to be zero. We use the GaAs parameme50.0665,mh50.457, ande0513.74.2 Shown are calcu-lations for three different asymmetry lengthsr 0 / l H51, 4,and 16, where the magnetic lengthl H58.1 nm forH510 T.
SinceVmn(pp) is diagonal@see Eq.~C13!# andVmn
(eh)50 form1n5odd @see Eq.~C15!# in the discrete basis set@Eq.~C7!#, the two-exciton HamiltonianHmn is partitioned intotwo parts corresponding to bothm andn being either even orodd. The two-exciton wave functions ofHmn for m,n even~odd! are symmetric~antisymmetric! with respect to the ex-change of the two electrons and the two holes~without spinexchange!. Figures 3 and 4 show the energies of symmeand antisymmetric wave functions, respectively. Thereno bound biexciton states in this case, since the partiparticle Coulomb interaction is stronger than particle-hCoulomb interaction in exciton-exciton interaction. Figur3~a! and 4~a! show that for symmetric Coulomb interactiothe two-exciton energies change smoothly with increasmatrix dimensionalityN0, and form almost continuous levebetween twice the one-exciton energy and the sum oflowest Landau energy of two electrons and two holeslarge N0. Since the size of the two-exciton matrix is finitwe do not obtain a continuum.N0 is related to the size of thequantum well in thexy direction: It gives the area of thquantum well, in units ofl H
2 . The discrete energy levels appearing in the continuum reflect a finite-size effect.
FIG. 2. Exciton level scheme for a GaAs quantum well. Ttwo-exciton states are shown with indicesi (J52, Jz52), j (J52, Jz50), k (J52, Jz522), andl (J51, Jz50).
ir
eal-
etoted
a
rs
cee-e
g
er
The two-exciton energies are slightly modified by intrducing a small Coulomb asymmetry. Forkee51 @Fig. 3~d!#,we can see a symmetric eigenstate whose energy doeschange withN0. This means that there is an eigenstate ofsymmetric wave function in the continuum for an infinitelarge quantum well. The energy of this eigenstate is shifto the blue for larger 0. This is because we have a largasymmetry for largerr 0. This single level is also blueshiftewith increasingkee @Fig. 3~g!#, and finally splits out from the
FIG. 3. Energies of the symmetric two-exciton eigenstatese2 forkeh50 are displayed vs the dimensionalityN0 of the two-excitonmatrix Hmn . @We denote the one-exciton energy in a magnetic fiH as V(H).# The energy of two one-excitons in the absence omagnetic field 2v0[2V(0) is taken to be zero. Left, middle, anright columns representr 051, 4, and 16, respectively.~a! kee50.~b! and ~c! kee58. ~d!–~f! kee51. ~g!–~i! kee52. ~j!–~l! kee53.Twice the one-exciton energy~7.5 meV! is shown by the longdashed line. The sum of the lowest Landau energy of two electrand two holes~39.9 meV! is given by the short dashed line.
FIG. 4. Same as Fig. 3, but for the energies of antisymmetwo-exciton eigenstates.
de
aes
iton
-tly.
nn a
h to
-
c-lon-
onned
ate
.
am
tes
4506 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
continuum @Figs. 3~j!, 3~h!, and 3~i!#. For kee53 and r 054, we have also an antisymmetric eigenstate embeddethe continuum@Fig. 4~k!#. The larger the asymmetry, thlarger is the number of discrete eigenvalues. Figures 3~b! and4~c! show such an extreme case (kee58). The number ofdiscrete eigenvalues is 5~58! for r 054 (r 0516).
Figures 5 and 6 show the energies of symmetric andtisymmetric wave functions of the two-exciton eigenstatrespectively. Forr 054 and varyingkeh , bound biexcitonstates are observed forkeh.kee. Figures 5~a! and 6~a!,
FIG. 5. Energies of the symmetric two exciton eigenstates sas Fig. 3 but for nonzerokeh . r 054. Left column:kee50; rightcolumn:kee51. ~a! and~b! keh50.2. ~c! and~d! keh51. ~e! and~f!keh52. Twice the one-exciton energy is 1.4 meV@~a! and ~b!#,223.0 meV@~c! and ~d!#, 253.6 meV@~e! and ~f!#.
FIG. 6. Same as Fig. 5, but for the antisymmetric eigensta
in
n-,
which represent weak Coulomb asymmetry (keh50.2),show two bound biexciton states. For larger values ofkeh thediscrete part of the spectrum contains three bound biexcstates@Figs. 5~c! and 6~c! (keh51) and Figs. 5~e! and 6~e!(keh52)#. However, whenkeh is increased further, the number of bound biexciton states does not increase significanThis is because this number depends not only onkeh but alsoon r 0. For smallr 0, the exciton-exciton attractive interactiorapidly becomes shallow at small distances, resulting ismall number of bound biexciton states. For largerr 0, theexciton-exciton attractive interaction can be strong enougsupport many bound biexciton states.
For kee.keh , we expect a behavior similar to that displayed in Figs. 3 and 4. The spectrum displayed in Figs. 5~b!and 6~b! closely resembles that of Figs. 3~e! and 4~e!. Fig-ures 5~d! and 6~d! represent the symmetric Coulomb interation @like Figs. 3~a! and 4~a!#. However, the nonlinear opticaspectra are very different in this case, as will be demstrated later in Fig. 8. Figures 5~f! and 6~f! are similar toFigs. 5 and 6, except for the number of bound biexcitstates which is 2 and 3, respectively. This can be explaiby the larger arguments in the former (keh51, kee50) com-pared with the latter (keh52, kee51).
We next turn to calculatingx (3) using Eq. ~5.9!. G isevaluated using Eq.~3.12! utilizing the discrete basis setCnrather than Eqs.~5.10! and ~5.11!. The reason is the follow-ing: in deriving Eqs.~5.10! and~5.11! we invertF(v) in thefull ~infinite-dimensional! two-exciton space which yields
@ F(v)#215v22e012ih. However performing calcula-tions in theN0-dimensional subspace,F(v) should be in-verted in the subspace in order not to miss some deliccancellations.
The two-color pump-probe signal is given by
FIG. 7. The two-color pump-probe signalI pp is displayed as afunction of Dv2[v22v0, with Dv1[v12v050.4 eV fixed.h50.8 meV,N05100. Solid line: bothE1 andE2 haves1 polariza-tion. Dashed line:E1 is s1 polarized andE2 is s2 polarized. Left,middle, and right columns representr 051, 4, and 16, respectively~a! kee50. ~b! and ~c! kee58. ~d!–~f! kee51. ~g!–~i! kee52. ~j!–~l! kee53.
e
.
.
id
btei
nhela
t
bymthvisctint.
ueegsk,na
ubeev
hethti-
gatolineaaetritn
the
andr-
hethe
in-
mbthe
-
ell.ci-
om
. Inth,
cture
as
see
s
PRB 58 4507EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
I pp~v2 ,v1!5Im x~3!~2v2 ;v1 ,v2 ,2v1!, ~6.1!
where v1 (v2) represents the pump~probe! frequency. InFig. 7 we displayI pp for different values of the asymmetryIn all cases we usev12v050.4 eV, h50.8 meV, andH510 T, and tunev2 across the two-exciton band. The solline represents the case when both fields haves1 polariza-tion, whereas the dashed line corresponds to as1 polarizedpump and as2 polarized probe.
We have chosenv1 to be well abovev0 to avoid inter-ference between one- and two-exciton resonances and othe two-exciton resonances, in an unambigous way. Sincour model we consider only the lowest Landau level,makes no difference whetherv12v0 is positive or negative.In a realistic system one needs to tunev1 carefully to avoidresonances with higher Landau levels and higher subbawhich can lead to the creation of real excitations on higlevels, and affect the pump-probe signal because of reation. Another option is to choosev1 well below v0, wherethe structure of two-exciton resonances is the same as inprevious case.
Figure 7~a! shows that the signal for symmetric Coulominteraction is considerably smaller compared to the asmetric Coulomb interaction cases. This is consistent withdiscussion given in Appendix F. The reason why we hasmall but finite signal for symmetric Coulomb interactionthat we used a finite-size two-exciton matrix which reflethe finite quantum-well size. We will return to this issueFig. 9. Since the signal is very weak, we can see the effecone-exciton resonances in the blue part of the spectrumFigs. 7~d!–7~j! we clearly see a sharp peak which is blshifted with increasingkee. This peak corresponds to theigenstate with symmetric wave function appearing in Fi3~d!, 3~g!, and 3~j!. The solid line does not show a peasince a symmetric wave function contributes to the sigonly in the case of different polarizations ofE1 andE2. Wenote a large broad peak which originates in the continupart of the two-exciton levels. However this broad peakcomes almost negligible compared to the sharp discrete-lresonances for large asymmetrykee @see Figs. 7~b! and 7~c!#.
In Fig. 7~k! both signals show peaks at the top of ttwo-exciton band corresponding to the eigenstate insideband shown in Fig. 4~k!. Since this eigenstate has an ansymmetric wave function, it couples both cases whereE1and E2 have the same or different polarizations. For larkee @Figs. 7~b! and 7~c!#, we have sharp resonances origining from the discrete eigenstates located above the ctinuum. Scanning from blue to red, we first see a dashedpeak. Next we find a solid line peak and a dashed line pat the same energy. After that, the dashed line peakdegenerate solid and dashed line peaks alternate. This mthat the eigenstate with symmetric and with antisymmewave function appears alternately.~In particular the highesenergy eigenvalue corresponds to the symmetric wave fution.! This is consistent with Figs. 3 and 4.
For nonzerokeh , I pp is markedly different from thekeh50 case. Hereafter we set the asymmetry sizer 054. Theweak Coulomb asymmetry case is presented in Fig. 8~a!(keh50.2, kee50). The signal is slightly different from thesymmetric Coulomb interaction case given in Fig. 7~a!, andwe can still see the finite-size effect from the continuum. W
ainint
dsrx-
he
-ee
s
ofIn
.
l
m-el
e
e-n-ek
ndansc
c-
e
also see two peaks below the continuum corresponding tobound biexciton states@Figs. 5~a! and 6~a!#. Note that wehave an almost degenerate lowest state of the symmetricantisymmetric wave functions, for symmetric Coulomb inteaction@Figs. 3~a!, 4~a!, 5~d!, and 6~d!#. These two levels splitbecause of the asymmetry of the Coulomb interaction. Tsplitting becomes larger for increasing asymmetry ofCoulomb interaction. In Fig. 8~c! (keh51, kee50), we canclearly see the splitting of these states. The splitting iscreased in Fig. 8~e! (keh52, kee50). The magnitude of thesignal increases as well for larger asymmetry of Coulointeraction, as already seen in Fig. 7. Another feature oftwo-color pump-probe signal for nonzerokeh is a wide two-exciton band compared with thekeh50 case. Since the oneexciton bandwidth for fixedr 0 depend onkeh only, thisholds for the continuum part of the two-exciton band as wThis explains why the signal coming from the bound biexton in Fig. 8~f! (keh52, kee51) resembles that of Fig. 8~c!,whereas the signal of the continuum part in Fig. 8~f! is simi-lar to that of Fig. 8~e!.
Figure 8~b! (keh50.2,kee51) looks similar to Fig. 7~e!.However, because of the reduced asymmetry coming frkeh50.2, the magnitude of the signal in Fig. 8~b! is smaller.
Although both Figs. 7~a! and 8~d! represent symmetricCoulomb interaction, the signals show some differencesFig. 7~a! the continuum part of the spectrum is smoowhereas the signal shown in Fig. 8~d! has some structure.~Ifwe increase the phenomenological dephasing, these strudisappear, and the signal becomes smooth.! The reason forthe observed structure in Fig. 8~d!, in spite of the fact thatwe have used the same damping in both cases, isfollows: sincekeh51 in Fig. 8~d!, the energy range of thecontinuum part is much wider than Fig. 7~a! (keh50). Con-sequently, if we do not add a large dephasing, we can
FIG. 8. Same as Fig. 7 but for nonzerokeh . r 054. Left column:kee50; right column:kee51. ~a! and ~b! keh50.2. ~c! and ~d!keh51. ~e! and ~f! keh52. Twice the one-exciton energy minu2v0 is 1.4 meV@~a! and~b!#, 223.0 meV@~c! and~d!#, 253.6 meV@~e! and ~f!#.
he
e
ev
r--
he
n-
o
a
co
a
r
lun
es
eole
Fig.
erfor
e-and
o-
two-itonasing
4508 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
each peak which comes from the finite size effect. Anotdifference is that the height of the peak in Fig. 8~d! is abouttwice that of Fig. 7~a!. This is explained by considering thfact thatU in the case ofkeh5kee51 is almost twice that ofkeh5kee50 for ur2r 8u!r 0 in Eq. ~4.8!.
Because of the slow convergence of the two-exciton lels of the continuum part with increasingN0, the continuumpart of the signal strongly depends onN0. Figure 9 shows thetwo-color pump-probe signal for symmetric Coulomb inteaction (kee5keh50) for various dimensions of the two exciton matrixN0. The signal atDv250 becomes smaller withincreasingN0. The peak height approximately shows t}N0
21/2 scaling. Therefore the value of the signal atDv2
50 vanishes for an infinitely large quantum well. In Appedix F, we show howx (3) for infinite systems is given by itsLFA part, and, therefore, vanishes atDv250. Our numeri-cal results are consistent with Appendix F.
The symmetric wave functions of the three lowest twexciton eigenstates in the discrete basisCn in Eq. ~C7! aredisplayed as a function ofn of Cn ~x-axis! in Fig. 10. Sincethe two-exciton Hamiltonian~3.13! is decoupled into twoparts, we have onlyN0/2 nonzero components~x axis! foreither symmetric or antisymmetric wave function~for evenN0). In order to obtain the wave functions, we usedN05100 for the top panel andN05120 for the middle andbottom panels. For each panel the number of nodes increwith increasing two-exciton energy.
The eigenstates in Figs. 10~b! and 10~c! belong to thecontinuum part. From these panels we can see the slowvergence of the wave function with increasingN0. For ex-ample, the lowest wave function does not have a nodedoes not go to zero for largen. Upon increasing the matrixsize, the wave function becomes flatter and extends to lan. On the other hand, the wave functions in Fig. 10~a!, whichcorrespond to the bound biexcitons, converge. A small vaof n dominates the wave functions of the lowest two eigestates, and these wave functions rapidly vanish for largn.Even if we increase the size of the two-exciton matrix, thewave functions do not change appreciably. In Fig. 10~b! wehave zero amplitude for smalln. This is explained as fol-
FIG. 9. Same as Fig. 8 but for various dimensionalitiesN0 of thetwo-exciton matrix and symmetric Coulomb interaction (kee5keh
50). Upper six lines@E1 (s1) andE2 (s2) N0] from bottom totop: 50, 60, 70, 80, 90, and 100. Lower six lines@E1 s(2) andE2
(s2) N0] from bottom to top: 50, 60, 70, 80, 90, and 100.
r
-
-
ses
n-
nd
ge
e-
e
lows: for keh,kee the particle-particle Coulomb repulsivinteraction is significantly stronger than the electron-hCoulomb attractive interaction forur2r 8u!r 0 in Eq. ~4.8!.Here ur2r 8u!r 0 is dominant for smalln. As a result, weneed to have a smaller amplitude for smalln for the lowesteigenstate. The wave function of the lowest eigenstate in10~a! is peaked at smalln, since forkeh.kee the electron-hole Coulomb attractive interaction is significantly largthan the particle-particle Coulomb repulsive interactionur2r 8u!r 0.
VII. SUMMARY
We have derived the GFE for the third order optical rsponse of semiconductor systems by mapping the two-b
FIG. 10. The wave function of the three lowest symmetric twexciton eigenstates in the discrete basis~x axis!. ~a! keh52, kee
50. ~b! keh50, kee58. ~c! kee5keh50. Solid line: the lowesttwo-exciton eigenstates. Long dashed line: the second lowestexciton eigenstates. Short dashed line: the third lowest two-exceigenstates. Note that the number of nodes increases with incretwo-exciton energy.
n
tte
ch
-on
cu
tue
toe
acw
ndeouwogmthple
reitholarheya
etce--
thinr
thnto-eehe-illeline
affi-
grac-re
-our
-fully
ym-
eeous
d,
ta-ec-
f ald,
etic
at,the
PRB 58 4509EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
Hamiltonian onto an effective Frenkel exciton HamiltoniaThe GFE’s expressx (3) in the form of a product of fourone-exciton Green functions and the exciton-exciton sca
ing matrix G. The latter consists of two components whi
have a simple physical meaning:G (1) represents the excitonexciton scattering due to statistical properties of electr
hole pairs~phase filling!, whereasG (2) originates from theCoulomb interaction. We have applied the GFE’s to callate the pump-probe signals from GaAs quantum well instrong magnetic field. Here we make use of the fact thathe limit of strong magnetic fields we can restrict the calclation to the lowest Landau-level exciton. The high symmtry of the system allows us to treat the relevant two-excistates with a little computational effort. The role of asymmtry between the particle-particle and particle-hole intertions in optical nonlinearities was examined. We have shothat in the symmetric case~i.e., when the particle-particleand particle-hole interactions have the same magnitude! op-tical nonlinearities originate from effects of phase filling, athe exact analytical expression forx (3) has been derived. Thasymmetry between particle-particle and particle-hole Clomb interactions, which appears due to the fact thatstudy a pseudo-two-dimensional problem in a twdimensional scheme, is determined by the asymmetry lenr 0 , which is related to the length scale on which the asymetry occurs and the magnitudes of the deviation ofparticle-particle and particle-hole interaction from the simCoulomb law at distances shorter thanr 0, denoted bykeeand keh , respectively. The two-exciton wave functions aclassified into two types, symmetric and antisymmetric, wrespect to the exchange of the two electrons and two h~without spin exchange!. These correspond to total angulmomentumJ50 and 2 states, respectively. The form of tpump-probe signals depends crucially on the asymmetrthe Coulomb interaction. We found a small but finite signfor the symmetric case (keh5kee), which is attributed to thefinite size of the quantum well in thexy direction. Forkeh,kee and small values of asymmetry, a well-defined discrresonance appears inside the continuum. This resonanblueshifted with increasingkee, and splits out from the continuum at a certain value ofkee. The number of the discretelevel resonances for large asymmetry (keh!kee and 1!kee) depends crucially onr 0. For keh.kee the signal isdominated by bound biexcitons. The results presented inpaper are general for two-dimensional semiconductorsstrong perpendicular magnetic field. We expect that the pdicted results should be found experimentally.
It should be emphasized that in the symmetric caseoptical nonlinearity does not vanish: in this case the respois represented by that of an effective harmonic oscillawhich is nonlinearly driven by the optical field. In this language, effects of phase space filling show up as nonlinterms in the expansion of the polarization operator in powof the effective oscillator coordinate. This implies that, in tsymmetric case,x (3) vanishes atDv250. Under these conditions the main contribution to the optical nonlinearity woriginate due to transitions between different Landau levThese effects are not considered in the present paper sour goal is to study the structure of two-exciton resonancand for resonant measurements in the asymmetric case
.
r-
-
-ain--n--n
-e-th-e
es
ofl
eis
isa
e-
eser
arrs
s.ces,the
principal contribution comes from processes involvingsingle Landau level, provided the magnetic field is suciently strong.
Finally we note that a four-particle problem in a stronmagnetic field has been considered in the context of the ftional quantum Hall problem, where the four particles arepresented by a single hole with the chargee and threeparticles with the charge2e/3.60 For that problem the constraints due to the symmetry are even stronger than incase since there is only one hole in the complex, andx (3) canbe found analytically.
ACKNOWLEDGMENTS
The support of the Air Force Office of Scientific Research, and the National Science Foundation, are grateacknowledged.
APPENDIX A:PARTICLES IN A HOMOGENEOUS MAGNETIC FIELD
AND THE POINCARE -HEISENBERG-WEYLSYMMETRY
In the absence of a magnetic field the system has a smetry with respect to the two-dimensional Poincare´ algebra,which consists of translations
p[ i ]m , ~A1!
where]m[]/]r m , and rotations
M[2 i emnr m]n , ~A2!
whereemn is the two-dimensional Levi-Cevita tensor and wassume summations over repeating indices. A homogenmagnetic field in thez direction with the strengthH can bedescribed by a vector potential
Am[2H
2emnr n , ~A3!
with H5emn]mAn . To take into account the magnetic fielwe introduce the ‘‘long’’ derivative:
¹m5]m2 ieAm5]m1 ieH
2emnr n , ~A4!
and substitute¹m instead of]m everywhere in the Hamil-tonian. Now we need to redefine the symmetry. Since rotions do not change, not only the magnetic field but the vtor potential of Eq.~A3! as well, the operatorM from Eq.~A2! commutes with the Hamiltonian in the presence omagnetic field. Translations, conserving the magnetic fiechange the vector potential. However, since the magnfield is conserved, the shifted value ofA differs from itsoriginal value by a gauge transformation. This implies thto obtain operators of translations which commute withHamiltonian, a usual shift~translation! of a wave functionshould be accompanied by multiplication by exp@iw(x)# tocompensate for the change ofA; i.e.,
TaC~x!5exp@ iewa~x!#C~x1a!, ~A5!
with
l
f tth
,erroli
-r.
t
elo
ens-e
ans
a-o
ta-gfirst
m-by
l
-
nta-
ith
ism
4510 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
Am~x1a!2Am~x!5]mwa~x!. ~A6!
A straightforward calculation making use of Eqs.~A5!, ~A6!,and ~A3! yields the following expressions for infinitesimatranslationsDm :
Dm5]m2 ieH
2emnr n . ~A7!
Naturally, we have
@Dm ,¹n#50. ~A8!
We also obtain that the operators of translationsDm do notcommute:
@Dm ,Dn#5 ieHemn . ~A9!
This is a consequence of the fact that, in the presence omagnetic field, we have a projective representation ofPoincare´ group:
TaTbC5exp@2 iwab#Ta1bC, ~A10!
which originates from the fact that the functionwa(x) in Eq.~A5! is defined up to a constant@see Eq.~A6!#. According toa well-known theorem of the theory of representations59
each projective representation of a group can be considas a usual representation of a central extension of the gwhich leads to the appearance of nonzero commutatorsin Eq. ~A9!. Introducing the operators
pm[2 iD m , Q[eH, ~A11!
we obtain the Lie algebra generated bypm , M , andQ withthe following nonzero commutations
@ pm ,pn#5 i emnQ, @M ,pm#5 i emnpn , ~A12!
which forms a central extension of the Poincare´ algebra~generated bypm and M with the only nonzero commutations @M ,pm#5 i emnpn , Q being the central charge operatoWe will refer to this as the PHW~Poincare´-Heisenberg-Weyl! algebra. The PHW algebra can be also treated asalgebra of a harmonic oscillator withp1, p2, and M beingthe momentum, coordinate, and Hamiltonian, respectivandeH being the Planck constant. Expanding the actionthe PHW algebra to multiparticle states, we obtain the ctral charge to beeH, wheree is the total charge. This impliethat, in the zero charge sector~i.e., the same number of electrons and holes!, Q50, @ p1 ,p2#50, and one can define thmomentum of a state.
Splitting of the space of single-particle states into Landlevels in the language of theory of representations meadecomposition of the representation of the PHW algebrafunctionsC(r ) into a direct sum of irreducible representtions Vn , whereVn is formed by all states which belong tthe nth Landau level. The relationHC5EnC for any CPVn follows from the fact thatVn’s are irreducible, andVmÞVn for mÞn .
hee
edupke
he
y,f-
ua
in
APPENDIX B: CLASSIFICATION OF UNITARYREPRESENTATIONS OF THE PHW ALGEBRA
Here we present the classification of unitary representions of the PHW algebra; we will use it in the followinsections to determine one- and two-exciton states. Theparameter of a representation is the central chargeq; i.e., Q
5qI on the representation~where I is the unit operator!.~i! q50. We then have irreducible representations para
etrized byp0>0. The space of the representation is givenfunctions with the absolute value of momentum beingp0;i.e.,
C~r !5E d2pd~ upu2p0!g~p!exp@ ip•r #: ~B1!
for p0.0 a representation is infinite-dimensional; forp050 it is a one-dimensional unit representation.
~ii ! q.0. Each representation is determined byq and in-tegerm, which is defined as follows: LetuV& be the statewhich satisfies (p12 i p2)uV&50, thenM uV&5muV&.
~iii ! q,0. m is defined by (p11 i p2)uV&50 andM uV&5muV&. Using this language, thenth electron Landau leveis given by the representation withq52ueHu, m5n, and thenth hole Landau level byq5ueHu, m52n. These representations are dual to each other.
APPENDIX C: BASIS SET FOR TWO-EXCITON STATES
We first represent
W[~V0^ V0! ^ ~V0* ^ V0* !, ~C1!
and decompose
V0^ V05 % n50` V~n!, V0* ^ V0* 5 % n50
` V~n!* , ~C2!
where the orthonormal basis sets of irreducible represetions V(n) andV(n)* are given by
~Cn~e!!m[
22~m1n!
2pAn!m!~ z12 z2!n~ z11 z2!m
3expF2uz1u2
42
uz2u2
4 G , ~C3!
~Cn~h!!m[
22~m1n!
2pAn!m!~w12w2!n~w11w2!m
3expF2uw1u2
42
uw2u2
4 G , ~C4!
where (Cn(e))m is the mth component ofCn
(e) , and m50,1,2, . . . . Now wehave
W5 % m,n50` V~m!
^ V~n!* . ~C5!
The unit representations is contained only in the terms wm5n, i.e., V(n)
^ V(n)* , and eachV(n)^ V(n)* contains the
unit representation only once;59 this unit representation isgenerated by the functionCnPV(n)
^ V(n)* , which corre-sponds to the unit operator by the canonical isomorphV(n)
^ V(n)* >Hom(V(n),V(n)), and therefore has the form
w
f
ion
PRB 58 4511EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
Cn5 (m50
`
~Cn~e!!m^ ~Cn
~h!!m . ~C6!
Substituting Eqs.~C3! and ~C4! into Eq. ~C6!, we obtain adiscrete basis set in the space of zero-momentum texciton statesW0:
Cn5Ap
~2p!222nn!@~ z12 z2!~w12w2!#n
3expF2uz1u21uz2u21uw1u21uw2u2
4
1~ z11 z2!~w11w2!
4G . ~C7!
We easily find an alternative a continuous basis set inW0which is represented by the functions
Cp~2!~r1 ,R1 ,r2 ,R2!5E
0
2p
dwpCp~r1 ,R1!C2p~r2 ,R2!,
~C8!
whereCp is given by Eq.~4.17!, andwp denotes the angle op. To find the matrix elementsCn(p)[^CnuCp
(2)& of thetransformation between the basis sets, we setr15r250 andR152R25R, which yields
Cp~2!~0,R,0,2R!5
1
~2p!2expF2
p2
2 GJ0~2pR!
5 (n50
`
Cn~p!Ap
~2p!2n!R2nexp@2R2#
o-
5 (n50
`
uCn~0,R,0,2R!&^CnuCp~2!&. ~C9!
Expanding Eq.~C9! in R, we obtain
Cn~p!51
ApexpF2
p2
2 GLn~p2!, ~C10!
whereLn is the Laguerre polynomial:
Ln~x!5(l 50
n
~21! lxln!
~ l ! !2~n2 l !!. ~C11!
We can now evaluate the matrix elements
Hmn52E0dmn1Vmn~pp!1Vmn
~eh! . ~C12!
Here we only consider the symmetric Coulomb interactcasekeh5kee50 for simplicity. We will give the asymmet-ric part of the Coulomb interaction in Appendix D.V(pp) isproportional to the unit operator on eachV(n), Vmn
(pp)
5dmnVn(pp) , and
2^Cmua
uz12z2uuCn&5Vm
~pp!dmn
52^~Cn~e!!0u
a
r 1u~Cn
~e!!0&dmn , ~C13!
whereCm0(e) is given by Eq.~C3!. The operatorV(eh) is diag-
onal in theCp(2) basis set, and we therefore have
matrix
Vmn~eh!52^Cmu
2a
uz12w1uuCn&12^Cmu
2a
uz12w2uuCn& ~C14!
5H 8pE0
`
p dpU~p!Cn~p!Cm~p! ~m1n5even!
0 ~m1n5odd!,
~C15!
with U(p) given by the second term of the right-hand side of Eq.~4.18!. Calculating the right-hand side of Eqs.~C13! and~C15! yields
Vmn~pp!5dmn2Ap
~2n21!!!
2n11n!a, ~C16!
Vmn~eh!524aAp/2 (
l 150
n
(l 250
m
~21! l 11 l 2~ l 11 l 2!!m!n!
~ l 1! l 2! !2~n2 l 1!! ~m2 l 2!!S 4
5D l 11 l 211
FS l 11 l 211
2,l 11 l 212
2,1,
1
25D , ~C17!
whereF is the hypergeometric function.Evaluation of two-exciton energy by the two-exciton basis with variational principle is given in Appendix E.
APPENDIX D: EFFECT OF ASYMMETRY OF THE COULOMB INTERACTION
In Appendix C, we obtained the matrix element of the symmetric part of the Coulomb interaction. Here we give theelement of the asymmetric part of the Coulomb interaction which is given by Eq.~4.8!:
d
Sec. IV.
4512 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
a
e~ ur2r 8u!ur2r 8u5
a
ur2r 8ue2~ ur2r8u/r 0!2
. ~D1!
The asymmetric part of Coulomb interactionsVmnA(pp) , UA(p), andVmn
A(eh) are given by
VmnA~pp!5dmn2Ap
~2n21!!!
2n11n!aS 1
A11~2/r 0!2D 2n11
. ~D2!
In this case,UA(p) is given by
UA~p!52a1
A112/r 02Ap/2 I 0S p2
4~112/r 02!D expF2
p2
4S 114/r 0
2
112/r 02D G . ~D3!
Equation~C15! even holds for the asymmetric part of the Coulomb interaction just by replacingU(p) by UA(p), and we have
VmnA~eh!524aAp/2
1
A112/r 02
(l 150
n
(l 250
m
~21! l 11 l 2~ l 11 l 2!!m!n!
~ l 1! l 2! !2~n2 l 1!! ~m2 l 2!!S 4
5D l 11 l 211S 11
2
5r 02
1
112/r 02D 2 l 12 l 221
3FF l 11 l 211
2,l 11 l 212
2,1,
1
25
1
~112/r 02!2S 11
2
5r 02
1
112/r 02D 22G . ~D4!
The limit r 0→` for Eqs.~D2! and ~D4! reproduces Eqs.~C16! and ~C17!, respectively. The combination of symmetric anasymmetric terms gives the matrix element of the Coulomb interaction~4.8!.
APPENDIX E: CALCULATION OF TWO-EXCITON ENERGIES USING THE TWO-EXCITON BASIS SET
We have derived the matrix element of the Coulomb energy of a two-exciton Hamiltonian by a discrete basis set inHere we will utilize only a continuous basis set inW0 ,
Cp~2!~r1 ,R1 ,r2 ,R2!5Cp~r1 ,R1!C2p~r2 ,R2!
51
~2p!3expF2
p2
2 GexpF2r 1
2
42
r 22
41
i
2n•@r1 ,R1#1
i
2n•@r2 ,R2#1 ip•~R12R2!1
1
2n•@p,~r12r2!#G ,
~E1!
onu-oo
th,
nalrss
s.
to evaluate the Coulomb energy of the two-exciton forkeh5kee50. We have to calculate three terms: the electrelectron~hole-hole! Coulomb energy, the electron-hole Colomb energy in one of the excitons, and the electron-hCoulomb energy between two different excitons. We denthose asV(pp), V(eh1), and V(eh2) in that order. We havealready obtained the expression forV(eh1) in Eq. ~4.18!,
Vp8p~eh1!
521
N ^Cp8~2!u
a
ur1uuCp
~2!&
5221
NaAp/2I 0S p2
4 DexpF2p2
4 G3d~p2p8!
1
~2p!2E dR2 , ~E2!
where N is a normalization factor and cancels wi@1/(2p)2# *dR2. Integration overR2 appears here explicitly
-
lete
since we have the same total momentum for initial and fistates~the total momentum is 0!. The same thing happens foV(pp) and V(eh2). In those cases we find the center-of-macoordinateR1 instead ofR2. For calculatingV(pp), it is con-venient to transform the coordinate from (R1, R2, r1, r2) to(R1 , re, r1, r2), wherere is a vector between two electronThen we can carry out all the integration exceptR1 , and wehave
Vp8p~pp!
5^Cp8~2!u
a
ureuuCp
~2!&1^Cp8~2!u
a
urhuuCp
~2!&
51
Na
2p
1
up2p8uexpS 2
1
2up2p8u2D
3$exp~ in•@p,p8# !1exp~2 in•@p,p8# !%
31
~2p!2E dR1 . ~E3!
We can perform the similar calculation forV(eh2):
nthsi
ss
trixiththehis
54
e a
sonthe-
s
PRB 58 4513EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
Vp8p~eh2!
52^Cp8~2!u
a
ur12uuCp
~2!&
5221
Na
2p
1
up2p8uexpS 2
1
2up2p8u2D
31
~2p!2 E dR1 , ~E4!
wherer12 is a vector between an electron in one exciton aa hole in the other exciton. So the matrix element ofCoulomb energy for the two-exciton in the continuous baset is given by
Vp8p~pp!
1Vp8p~eh1!
1Vp8p~eh2!
5a
2p
1
up2p8uexpS 2
1
2up2p8u2D
3$exp~ in•@p,p8# !1exp~2 in•@p,p8# !%
22aAp
2I 0S p2
4 DexpF2p2
4 Gd~p2p8!
22a
2p
1
up2p8uexpS 2
1
2up2p8u2D . ~E5!
Let us evaluate the two-exciton Coulomb energy byvariational principle. As a trial function, we use first a Gauian type function.
Cp0~p!5A2
p
1
p0e2p2/p0
2. ~E6!
Then the expectation value becomes
^Cp0uV~pp!1V~eh1!1V~eh2!uCp0
&
52aA2pp0
p0212
22aA2p1
A41p02
2aA2p1
A11p02
.
~E7!
des
a-
This equation has two minima atp050 and`. The mini-mum energy is2aA2p which is nothing but twice the oneexciton energy.
APPENDIX F: EXCITON SCATTERING MATRIX FORWEAK AND STRONG COULOMB ASYMMETRY
In this appendix we evaluate the exciton scattering maG (2)(v) in some limiting cases. We start our analysis wthe symmetric case. Straightforward calculation usingcontinuous representation of Appendix E shows that in tcase Uuf0&50 ~where we have used the notationf0[f(0,0)). This yields
Gab,mn~2! ~v!52~v22e012ih!^f0uU~v2H12ih!21
3~ I I ab,mn2PPab,mn!uf0&50, ~F1!
which implies
Gab,mn~v!5Gab,mn~1! ~v!. ~F2!
The fact thatGÞ0 does not contradict the results of Refs.
and 53. since on the energy shell (v52e0) G (1) vanishes.This implies that magnetoexcitons in this case behave likharmonic system, with nonzerox (3) induced by nonlineari-ties in the dipole operators expanded in powers of bovariables.x (3) is then related to a three-level system wiequidistant levels but with the ratio of transition dipole btween consecutive levels different fromA2.
We next turn to the weak-asymmetry casekeh'kee. TocalculateG (2) we use the continuous basis setuq& @Eq. ~E1!#.The operatorsU andF(v) are given by the matrix elementUqq8 represented by Eq.~E5! and
@ F~v!#qq851
v22e022eq12ihdqq8 . ~F3!
Expanding the middle term in (v2H12ih)21 in Eq. ~F1!in powers ofU, and making use of@P,H#50, we obtain
n,er than
Gab,mn~2! ~v!52v0(
n50
` E )j 50
n
dqj S I ab,mn
1
v0U0q0
2 Pab,mnE dq1
2eq1v0Uqq0D
3)j 51
n1
2eqj1v0
Uqj 21qjd~qn! ~F4!
wheredq[(2p)22d2q, and
v0[2e02v22ih. ~F5!
We now note that at large distancesr , all Coulomb potentials scale liker 21, whereas in the sum of electron-electrohole-hole, and electron-hole potentials ther 21 terms are canceled. Assuming that the sum of these potentials decays fastr 22, Eq. ~E5! then implies thatUpq is finite atp,q→0. On the other hand, due to resonant terms (eq1v0)21 in Eq. ~F4!, themain contribution to the integral atv0→0 comes from the regioneqj
&v0. This allows us to setqj50 in Uqj 21qj, U0q0
, and
Uqq0which yields
e
4514 PRB 58V. CHERNYAK, S. YOKOJIMA, T. MEIER, AND S. MUKAMEL
Gab,mn~2! 52g(
n50
`
~ I ab,mn2 Pab,mnv0I ~v0!!@gI~v0!#n, ~F6!
where we have used the notation
g[1
2pU00, ~F7!
I ~v0
ber-
of
PRB 58 4515EFFECTIVE FRENKEL HAMILTONIAN FOR OPTICAL . . .
The angular momentum part of two-exciton statesx aregiven by the combination of thesef1,61,
x2,625f1,61~1!f1,61~2!,
x2,051
A2$f1,1~1!f1,21~2!1f1,21~1!f1,1~2!%, ~G2!
ro
d
et
ys
ys
.
x1,051
A2$f1,1~1!f1,21~2!2f1,21~1!f1,1~2!%,
where the argument in the parentheses indicates the numing of each electron~or hole!. The permutation operatorP iscomposed of two parts,P5( Pe1 Ph)/2, wherePe representsthe exchange of electrons andPh represents the exchangeholes.
Pex2,625 Phx2,625x2,62 , ~G3!
Pex2,051
A2@e1/2 , 1/2~1!h3/2 , 3/2~1!e1/2 ,2 1/2~2!h3/2 ,2 3/2~2!1e1/2 ,2 1/2~1!h3/2 ,2 3/2~1!e1/2 , 1/2~2!h3/2 , 3/2~2!#, ~G4!
Phx2,051
A2@e1/2 ,2 1/2~1!h3/2 ,2 3/2~1!e1/2 , 1/2~2!h3/2 , 3/2~2!1e1/2 , 1/2~1!h3/2 , 3/2~1!e1/2 ,2 1/2~2!h3/2 ,2 3/2~2!#. ~G5!
We can obtainPex1,0 and Phx1,0 by changing the sign of the second term of the right-hand side of Eqs.~G4! and ~G5!,respectively. These wave functions are mapped on the basis formed byx2,2, x2,22, x2,0, andx1,0 and we have
P5S 1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
D . ~G6!
The matrixPab,mn is written by a basis set:f1,1(1)f1,1(2), f1,21(1)f1,21(2), f1,1(1)f1,21(2), andf1,21(1)f1,1(2). Afterthe transformation in this basis set we have a expression forPab,mn , and it has the same form as Eq.~G6!.
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