Z. Phys. B 99, 197—205 (1996)

Microscopic foundation of the phenomenological few-level approachto coherent semiconductor optics

K. Victor1, V.M. Axt1, G. Bartels1, A. Stahl1, K. Bott2, P. Thomas2

1Institut fur Theoretische Physik B, Rheinisch Westfalische Technische Hochschule Aachen, Sommerfeldstrasse, D-52056 Aachen, Germany2Fachbereich Physik und Wissenschaftliches Zentrum fur Materialwissenschaften, Philipps-Universitat Marburg, Renthof 5,D-35032 Marburg, Germany

Received: 28 June 1995

Abstract. It is shown how a phenomenological few-leveldescription of coherent semiconductor optics is related tomicroscopic density-matrix theory. It turns out that thefew-level dynamics is not obtained by simply projectingthe microscopic dynamics onto a suitable chosen set ofenergy eigenstates. Therefore, a new hierarchy of micro-scopic densities is introduced with the property that theirexpansion in terms of energy eigenstates yields the leveldynamics. The transformation rules between the few-levelvariables and the traditional density matrices are estab-lished. The problem of incorporating the coupling tocontinua into a few-level model is discussed. A refinedfew-level model approximating the influence of the con-tinua by modified couplings between the levels is present-ed. The modified couplings turn out to be similar instructure to phenomenologically introduced refinementslike local fields. The analysis makes clear that intuitivelysimilar approximations have a different meaning whenapplied in the context of a few-level model or a truncatedmicroscopic hierarchy.

PACS: 42.65.!k; 42.50.Md; 71.35.#z

I. Introduction

Coherent optical experiments in semiconductors havebeen described by means of phenomenological few-levelmodels [1—9] as well as microscopic density-matrix the-ories. [10—18, 21] Few-level models treat the opticallyexcited charge carriers using a phenomenologically se-lected set of many-particle eigenstates that diagonalize theparticle interaction. Sometimes, additional couplingsadded by hand, e.g. local fields, are necessary to fit experi-mental data. [3, 19, 20] In contrast, microscopic density-matrix theories are formulated in a basis of single particlemodes that are localized either in real space or in k-space.Excitations of these modes are coupled by the Coulombinteraction and the Pauli principle. Both models are ableto explain many of the experimental features observed andoften lead to similar results. Figure 1 gives an example.

The biexciton beating observed for negative time delays inFour-Wave-Mixing (FWM) experiments is well repro-duced by both theories. [3, 12] Considering the differencesin foundation and formulation, the similarity in the out-come of both models is surprising.

The exact relation between the two approaches is upto now not well understood. The two models make use ofdifferent sets of dynamic variables which obey equationsof motions that differ significantly in structure. In particu-lar the transformation connecting the two sets of coupleddifferential equations is not evident, as pointed out in[22].

In the present paper we show that the naive approachof projecting the microscopic equations of motion ontoa set of eigenfunctions does not yield the correspondingfew-level model. We therefore introduce a new class ofmicroscopic correlation functions with three crucial prop-erties: First, their equations of motion can be derived fromthe same Hamiltonian as used in the traditional microscop-ic approach. Second, the projection onto eigenfunctions ofthese new variables exactly reproduces the few-level dy-namics. Finally, the explicit relation between the new andthe usual microscopic densities can be established.

In principle, the reduction of the microscopic equa-tions of motion towards a few-level approach can beachieved starting from either a real space or a k-spaceformulation. Here, we will start from the microscopicequations of motion in a real space represention. Thisapproach has the advantage to be applicable to in-homogeneous, disordered or structured semiconductors.Furthermore, the microscopic treatment of biexciton be-ats, mentioned above, has been worked out in a real-spacerepresentation using a suitably truncated hierarchy ofequations of motion.

In the course of this discussion it will become evidentthat a simple few-level treatment finds its limits whenevercontinua are involved. We present an approximationscheme in which these continua can be eliminated intro-ducing modified couplings between the discrete levels. Theresult is a closed set of equations with a structure similarto refined few-level models including local-field effects[3, 19, 20].

II. Microscopic density-matrix theory

The microscopic description of optically excited semicon-ductors is usually [10—20, 21, 23—28] based on a Hamil-tonian that in site representation reads

H"+i, j

¹ #ijcL sicLj#+

i,j

¹ 7ijdK sidKj

#+ij

E(ij)

(t) (MijdK sicL sj#M*

ijcLjdKi)

#

1

2+i, j

(cL sicLi!dK s

idK si)»

ij(cL s

jcLj!dK s

jdK sj) .

This Hamiltonian is illustrated in Fig. 2a. c'i(dK

i) are annihi-

lators for electrons (holes) in Wannier states i. The in-traband matrix-element ¹ #@7

ijincludes the band structure,

the energy gap and possible electrostatic potentials. Theinterband dipole-matrix-element M

ijcouples to the op-

tical field E(ij)

that is evaluated at a point (ij) somewherebetween the sites of i and j. The two-particle interactionsof monopole-monopole type are taken into accountby »

ij.

The equations of motion of many-point density ma-trices derived from the above Hamiltonian form an openhierarchy of coupled differential equations. This hierarchyhas been analyzed in detail in [28] using the excitationstrength as a classification parameter. In the following, wecollect some aspects that are of special interest for thecomparison with few-level models. The dynamical objectsof the density-matrix approach are expectation values ofthe following form:

X[m, ng, n

u]:"ScL s[n

u]j½K s[q]l{l ½K [p]k{k cL [n

u]T. (2.2)

Here, c'[nu]i for positive (negative) n

uis an annihilator for

DnuD electrons (holes) at the sites summarized by the multi

index i. ½] [p]k{k is the annihilator for p electron-hole pairswith electrons at sites k and holes at sites k@. X hasresonances near the gap-order n

g:"p!q times the gap-

frequency. The majority number m :"p#q#2DnuD is the

lowest order in the optical field to which X is excited,provided that the excitation starts from the ground state.This allows a truncation of the hierarchy if one is onlyinterested in the optical response up to a given order.[28, 29, 38] The hierarchy of density matrices forms apyramid centred along the m-axis in the (m, n

g, n

u) -space.

[28]The equation of motion of a many-point density

matrix of type (2.2) has the form

!i+LtScs[n

u]j½ s[q]l{l ½ [p]k{k c[n

u]T

"(+)[q, nu]l{l,j!+)[p, n

u]k{k,i)

]Sc' s[nu]j½] s[q]l{l ½] [p]k{k c' [n

u]iT

#SE#S

V. (2.3)

The operator +) forms the homogeneous parts of theequation and thus determines the resonance frequencies.+)[p, n

u] is identical to the Schrodinger Hamiltonian of

an unperturbed (not illuminated) system with p pairs,including the interaction between all 2p#Dn

uD particles

(see Appendix 43A).

The source terms SV

and SE

(cf. Appendix 43A) couplethe equations of motion of different density matrices viathe two-particle interaction » and the optical excitationE, respectively. Fig. 3(a) illustrates the coupling projectedonto a plane spanned by the parameters m and n

g.

If no carriers exist before the optical excitation, thedynamics is completely described by the two dimensionalsubset of density-matrices with n

u"0. For more informa-

tion on the hierarchy we refer to [28] and Appendix 43A,where the source terms S

Eand S

Vare expressed within this

subset.The electromagnetic observable of most interest, the

optical polarization P, can be derived from a single two-point density matrix:

P (r1, t)" :

rÇ~r*

M*i1SdK

1c'iT#c.c. (2.4)

The structure of (5) suggests to expand the density ma-trices in terms of eigenfunctions of the operators +)[p][10—12]. This procedure, however, does not yield thefew-level equations.

III. Few-level model

The Hamiltonian underlying few-level models is for-mulated in a basis of eigenstates of the not illuminatedcrystal (see Fig. 2b))

H"+p

+a

+up,aDp, aTSp, aD

#+p

+a,b

E (t) (kp`1,ap,b Dp#1, aTSp, bD#H.c.) (3.1)

Each eigenstate Dp, aT with energy +up,a contains a fixed

number p of electron-hole pairs and is further character-ized by a quantum number a. The optical excitationcauses transitions with dipole moment k between stateswith pair numbers differing by one. The phenomenologi-cal nature of the few-level approach manifests itself in theselection of a set of relevant states Dp, aT already in theformulation of the model [1]. This selection is usuallybased on resonance arguments. The summations over theindices p, q, a, b are limited to the relevant set.

The dynamical objects of interest are the density-matrix elements between the eigenstates

oq,ap,b"SDq, aTSp, bDT. (3.2)

The equations of motion are

[!i+Lt#(+u

p,a!+uq,b)]oq,a

p,b

"E A+ckq`1,cq,a oq`1,c

p,b #+c

kq,a*q~1,coq~1,c

p,b

!+c

kp,bp~1,coq,a

p~1,c!+c

kp`1,cp,b *oq,a

p`1,cB . (3.3)

The coupling between the dynamical variables asgiven by (10) is illustrated in Fig. 3(b). Similar to the

198

Fig. 1. Two theoretical results describing the four-wave mixing sig-nal of a GaAs/AlGaAs quantum well in the vicinity of the biexci-tonic resonance. The dotted curve is calculated using a few-levelmodel including local-field effects [3]. The solid line is calculatedwithin a microscopic density-matrix theory, along the lines de-scribed in 12. Both curves reproduce the rising and decay as well asexciton-biexciton beating in the rising part of the signal as experi-mentally observed in [3]. The curves have been shifted vertically

truncation of the hierarchy of microscopic density ma-trices (2.2), only p-pair states with p4[n`1

2] have to be

incorporated in a s(n) calculation.The optical polarization in general is a sum over

contributions from all transitions between states with pairnumbers differing by one

P"+k

+a,b

(kk`1,ak,b ok`1,a

k,b #c.c.). (3.4)

An often used modification of the few-level model is toreplace the optical field by a local field [3, 15, 19, 20]E-0#

"E#¸P, where ¸ is a phenomenological para-meter. In the context of exciton dynamics the parameter¸ is commonly interpreted to represent couplings due toCoulomb effects like exciton-exciton interactions. Some-times it also has been interpreted in terms of a classicalLorentzian local field. This replacement introduces addi-tional couplings and has been successful in describingexperimental data. The curve in Fig. 1 is an example.Within the few-level approach both, the beating and therising of the signal can only be reproduced when local-field corrections are taken into account [3].

IV. Connection between few-level modelsand microscopic density-matrix theories

The two models described in Sect. II and III have severalsimilarities. In particular both models can be formulatedas an open two dimensional hierarchy of dynamical ob-jects. Nevertheless, there is no one-to-one correspondencebetween the variables. This becomes obvious consideringthe expressions for the optical polarization in the twoapproaches as given by (2.4) and (3.4). Whereas in themicroscopic density-matrix theory this important observ-able is derived from a single variable, in the few-levelapproach it becomes a sum of contributions from differenttransition densities.

In the following we therefore construct a modifiedmicroscopic hierarchy that has the desired one-to-onecorrespondence to the few-level system. The transforma-tion rules between the new and the conventional micro-scopic hierarchy are derived in the second subsection.

A. Microscopic representation of level systems

The few-level objects o in (3.2) are defined using eigenstatesDp, aT of p-pair problems. Thus the site representations

$&&&&&&&&&&&&&&&&&&&&&&&&

Fig. 2. Comparison of model Hamiltonians: a A microscopic den-sity-matrix theory is formulated in single-particle modes. These mayeither be Wannier-states coupled by intra-band hopping matrix-elements in real space (as illustrated in the figure) or uncoupledBloch states with dispersion in pseudo-momentum space. Themany-particle interaction is explicitely included into the Hamil-tonian. The optical polarization is given by a single interbandtransition density. b A few-level model is formulated in many-par-ticle eigenstates. According to their definition these states diagonal-ize the many-particle interaction. The optical polarization consistsof several contributions, each describing transitions between stateswith a given number of pairs

199

fp,a of the states Dp, aT are eigenfunctions of the Schrodin-ger operators +)[p, ],+)[p, 0], that also occurred inthe equations of motion (2.3) of the microscopic theory.Denoting the ground state by D0T and choosing a conve-nient normalization we can write

Dp, aT"p!~2 +k{,k

fp,a(k{k )½] s[p]k{k D0T (4.1)

with

+)[p] fp,a(k{k )"+u

p,a fp,a (k{k ), (4.2)

+k,k{

fp,a(k{k )*f

p,a{ (k{k )"p!2daa{, (4.3)

+a

fp,a(k{k )* f

p,a (l{l )"p!2dkk{dll{. (4.4)

It is useful to introduce a new set of dynamical variables,which by their definition have some similarity to theconventional density matrices (2.2)

o[q, p]k{l{kl :"S½] s[q]k{k D0TS0 D½] [p]l{l T. (4.5)

The essential difference between (4.5) and conventionaldensity matrices as defined by (2.2) is the additional pro-jector onto the ground state D0TS0 D. Therefore wesubsequently denote objects like those in (4.5) groundeddensities. The equations of motion for the grounded dens-ities again form a hierarchy which in principle exhaust-ively describes the dynamics of the system. Although theyare normally not discussed, we are interested in thegrounded densities because they are closely connected tothe variables of the few-level model. The following rela-tions hold:

oq,ap,b"(q!p!)~2 +

k,k{,l,l{fq,a(k{k ) o[q, p]k{l{kl f

p,b(l{l )* (4.6)

o[q, p]k{l{kl "+a,b

fq,a(k{k ) oq,a

p,b fp,b(l{l ) . (4.7)

The equations of motion for the grounded densities areobtained using the Heisenberg equations derived from themicroscopic Hamiltonian (2.1). We obtain

!i+Lto[q, p]k{l{kl "

(+)[q]k{k !+)[p]l{l ) o[q, p]k{l{kl

!

q+

s, t/1

M*k{tks

E(k{

tks)(!)s`t

]o[q!1, p]k@12k@

t~1k@t`12k@

ql@k12ks~1ks`12kql!+

i,j

MjiE

(ji)o[q#1, p]jk{l{

ikl

#

p+

s, t/1

Ml{tls

E(l{tls)(!)s`t

]o[q, p!1]k@l@12l@

t~1l @t`12l @

qkl12ls~1ls`12lq#+

i,j

M*jiE

(ji)o[q, p#1]k{l{jkli . (4.8)

Defining dipole matrix-elements as

Ekp,ap~1,b :"!p!~2+

k{kfp,a(k{k )*

p+

s, t/1

(!)s`tMk{tks

E(k{tks)

]fp~1,b(k

@12k@

t~1k @t`12k@

pk12ks~1ks`12kp

), (4.9)

the expansion of o[q, p] with respect to the fp,a is straight

forward. Using (4.6), (4.7), (4.3) and the antisymmetry ofo[q, p] resulting from the anti-commutation properties ofthe defining creators and annihilators we recover exactlythe equation of motion (3.3) that has been derived inSect. III using the few-level Hamiltonian. Thus we con-clude that the grounded densities are the single-particle-mode representation of the level variables.

Up to this point, things work quite well and one mightguess that the only approximation necessary to obtaina few-level model would be the truncation of the completeset of eigenfunctions. But unfortunately this simple pictureis only true as long as all eigenstates are discrete and thecrystal volume » is kept finite. As will be shown inSect. IV C the limit »PO will upset this simple picturebecause of the formation of continua. This makes it neces-sary to develop strategies to handle these continua. Onepossible strategy is presented later.

B. Relation between the fundamental variables

As shown in the last subsection, a level-like model can bededuced from the same Hamiltonian as the microscopictheory. Still, the models deal with different objects. To bespecific, it is the projector onto the ground state in (4.5)that makes the grounded densities level-like variables. Inorder to establish a connection between the two types ofdensities we introduce the following operator, alreadyconsidered in [28]:

P]l"

l+k/0

pk

+i,j

½] s[k]ij½] [k]ij (4.10)

with

p0"1, p

k"!

k~1+

k{/0

pk{

(k!k@)!2. (4.11)

This operator is a linear combination of pair annihilatorsand creators and behaves like D0TS0 D in the subspace ofpair states with at most l pairs. With the help of theexpansion theorem of [28] it is easy to show that

S½] s[q]k{k D0TS0 D½] [p]l{l T"S½] s[q]k{k P]l½] [p]l{l T

#O (Eq`p`2l`2). (4.12)

Inserting (4.10) into (4.12) we obtain the relation betweenthe fundamental variables of both hierarchies.

o[q, p]k{l{kl "

l+k/0

pk

+j,j{

S½] s[q#k]k{j{kj ½] [p#k]l{j{lj T

#O (Eq`p`2l`2). (4.13)

Indeed, the equation of motion for o[q, p] can be repro-duced in a straight forward but tedious calculation using

200

Fig. 3. a Coupling scheme between density matrices projected ontothe (m, n

g)-plane. The coupling occurs by the optical field (solid lines)

and by the Coulomb interaction (dashed lines), the latter one con-necting only variables with equal gap order n

g. An arrow APB

means that A is present in a source for B. Quantities contributing toelectrodynamic observables (i.e. polarizations and charges) havebeen emphasized. b The dynamical variables of the few-level model

are coupled only by the optical field. Each coupling is bidirectional.All variables in the shaded area directly enter the calculation of theelectromagnetic observables. c Relation between grounded densitiesand standard density matrices: A given variable of one hierarchycorresponds to a stripe of the other hierarchy with equal gap-orderngand higher or equal majority-number m. Note that the correspond-

ence depends on the applicability of the contraction theorem [28]

(4.13) and (2.3). As shown in Appendix B the transforma-tion (4.13) can be inverted. Using in addition the contrac-tion theorem of [28] it is also possible to express anarbitrary conventional density matrix by grounded dens-ities. For the case n

u50 (unpaired electrons) one obtains

Sc' s[nu]g½] s[q]l{l ½] [p]k{k c' [n

u]g{T

"

l+k/0

1

(k#nu)!k!

] +j,j{,j{{

o[q#nu#k, p#n

u#k]j{{j{k{l{j{j{{

gjkljg{#O (Eq#p#2n

u#2l#2). (4.14)

The case of unpaired holes (nu40) is analogous. The

variables of one hierarchy thus can be expressed as a lin-ear combination of variables of the other hierarchy withthe same gap order n

g"p!q and higher or equal major-

ity number m"p#q as visualized in Fig. 3(c). As animportant special case of (4.14), the correct few-levelexpression (3.4) for the optical polarization is straightforwardly recovered using the corresponding microscopicexpression (2.4) together with (4.7).

C. Refined few-level models

As mentioned in Sect. IV A one is facing a problem whensolving the equations of motion for the grounded densities(21) in the limit of an infinite crystal. The source termscoupling these differential equations to higher densitiesare asymptotically proportional to the volume». This canbe seen from the example of the transition between the

ground state and a one-pair state o[0, 1]

(!i+Lt#+)[1]1

2) o[0, 1]1

2

"!+i,j

E(ji)

(Mjio[1, 1] j1

i2!M *

jio[0, 2]1j

2i)

#M12

E(12)

o[0, 0]. (4.15)

The critical term in this equation is the i, j-sum over thecrystal volume. For sites i, j sufficiently separated fromsites 1 and 2 the Coulomb interaction will become negli-gible and a Hartree-Fock factorization of the four-pointdensities will become asymptotically correct

+ij

E(ji)

(Mjio[1, 1] j1

i2!M*

jio[0, 2]1j

2i)

HFB o[0, 1]12+ij

E(ji)

(Mjio[0, 1] j

i*!M*

jio[0, 1] j

i)

#Fockterms. (4.16)

The sum is the generation rate of the total population inone-pair states and is thus proportional to the crystalvolume. Similar terms arise in the equations of motion ofother densities.

Similar volume proportionalities have previously beenreported [30, 31]. In a diagrammatic representation theycorrespond to unlinked diagrams [30, 32]. Although thedivergences cancel out in the calculation of observablequantities, a practical (numerical) evaluation of a set ofequations like (4.8) becomes impossible. Such compensa-tion effects have also been observed in the description ofoptical experiments in polymers [33, 34]. Whereas a levelmodel works fine for the case of small molecules, large

201

compensation effects occur when many atoms are in-volved. For the latter cases a density-matrix treatmentbased on one-particle sites seems to be more adequate.

The volume proportionality in our model has an intu-itive interpretation. The polarization is decomposed intocontributions connecting levels with a given pair number.The contribution describing transitions from one-pair totwo-pair states, for example, contains among others allthose processes where the already existing pair is arbitrar-ily far separated from the transition. The number of suchtwo-pair states with uncorrelated pairs is proportional tothe volume. A well established way to deal with suchdivergencies is to introduce suitably defined cumulants[35]. This, however, is out of the focus of our interest,because the resulting equations have no similarities witha few-level model [38].

No volume proportionalities result from contributionsinvolving only bound many-pair states. Nevertheless, thecontinuum of two scattering excitons, for example, isalways in exact resonance with twice the transition energyto the single exciton. In nonlinear optical experimentswith a single central frequency, this continuum thus isexcited whenever the exciton is excited. Therefore, onlyvery specialized experimental arrangements [36] justifythe restriction of the dynamics to bound states only. Ascontinua in general cannot be neglected, we subsequentlydevelop a method that allows to describe the dynamics ofbound states as a few-level system and to approximatelytake into account the coupling to continua.

To this end we consider a new class of dynamicalobjects, denoted by oJ , from which in particular two prop-erties are desired. Similar to the case of the groundeddensities, the optical polarization expressed by these newvariables shall be decomposed into contributions corres-ponding to transitions between states of definite pair num-ber. The new variables shall differ from the old ones in sofar as only transitions between bound multi-pair statesshall be singled out from the two-point density S½] T. Thisis achieved defining spectrally truncated counterparts ofthe grounded densities o in analogy to (4.13) by

oJ [q, p]k{l{kl

:"l+k/0

pk

+j,j{

S(P] Bq`k

½] s[q#k]k{j{kj ) (P] Bp`k

½] [p#k]l{j{lj )T

#O (Ep`q`2l`2), (4.17)

where for p'1, P] Bp

is the projector onto the relevantsubspace of the few-level description.

P] Bpgij :" +

"06/$ 45!5%4 ap!~2 +

i{,j{fp,a(ij)* f

p,a (i{j{) gi{j{. (4.18)

P] B1is defined as the identity. Although usually the relevant

subspace will be spanned by bound states only, a moregeneral kind of model might be constructed, e.g. by replac-ing the edge of a continuum by a single representativestate.

As we shall see, the projected variables oJ q,ap,b associated

with (4.17) satisfy equations of motion identical to those ofa few-level model except for additional source terms de-scribing the interface between the discrete levels and the

continua. These corrections can approximatively be re-placed by modified couplings between the levels. In thisway we obtain a closed model.

Most calculations using few-level models for the de-scription of nonlinear optical experiments were restrictedto the important case of the s(3) response. Furthermore,bound states are only known for one and two pairs inmost materials. For this reason we subsequently alsoconcentrate on this regime. Inserting (2.3) into the defini-tion (4.17) we obtain the following equations of motionfor oJ .

!i+LtoJ 00#O(E4)"E +

b(k1,b

0oJ 1,b0

!k1,b*0

oJ 01,b), (4.19)

[!i+Lt#+ (u

1,a!u1,a{)]oJ 1,a{1,a #O(E4)

"E(!k1,a0

oJ 1,a{0

#k1,a{*0

oJ 01,a), (4.20)

[!i+Lt#+u

2,a]oJ 02,a#O(E4)"!E+

bk2,a1,boJ 01,b, (4.21)

[!i+Lt#+u

1,a]oJ 01,a#O(E5)

"EA!k1,a0

oJ 00!+

bk2,b*1,a oJ 0

2,b#+b

k1,b*0

oJ 1,b1,a

! +b,b{

kJ b,b{a oJ 1,b{1,b !¸

1,aB, (4.22)

[!i+Lt#+ (u

2,a!u1,a)]oJ 1,a

2,a{#O(E5)

"EAk1,a*0

oJ 02,a{!+

b(k2,a{

1,b!kJJJ 2,a{1,b )oJ 1,a

1,bB. (4.23)

These are the usual few-level equations (3.3) except forthree additional couplings defined by

EkJ b,b{a :" +i, j, i{,j{

f1,a( j{i{

)* f1,b{( ji

)*(1!P] B2)

][MjiE

(ji)f1,a ( j{i{)#M

j{i{E(j{i{)

f1,a(ji)!M

ji{E(ji{)

f1,a(j{i )

!Mj{i

E(j{i)

f1,a (ji{)],

EkJJ 2,a{1,b :" +

i,j, i{,j{

f2,a{ ( jj{ii{

)*(1!P] B2)

][MjiE

(ji)f1,a ( j{i{)#M

j{i{E(j{i{)

f1,a(ji)!M

ji{E(ji{)

f1,a ( j{i )

!Mj{i

E(j{i)

f1,a ( ji{)],

¸1,a :" +

i, j, i{,j{

f1,a(j{i{)*(»ji{

!»jj{!»

ii{#»

ij{)

S½] s[1]ji(1!P] B

2)½] [2]jj{

ii{T. (4.26)

The corrections due to kJ bb{a together with the third sourceterm in equation (4.22) describe the phase space fillingwithin this formulation of the dynamics. Their form indi-cates that our model is in a certain sense halfway betweenthe semiconductor Bloch equations and a strict few-levelmodel. While in the case of a strict few-level model onlydensities including the state D1, aT contribute to the block-ing of transitions towards this state, in a microscopictheory the blocking is local in the corresponding one-particle basis, which implies that arbitrary many-particleeigenstates are coupled. kJJ 2,a{

1,b is simply a renormalizationof the dipole matrix elements in (4.23).

202

The term ¸1,a is of particular importance. As it stands,

it leads to an open set of differential equations as long asthe six-point density S½] s[1]j

i(1!P] B

2)½] [2]jj{

ii{T is not

known. This term is similar to the Coulomb source (A5) inthe microscopic model. The only difference is the projec-tor (1!P] B

2) onto the continuum. In the derivation of the

semiconductor Bloch equations these terms are usuallytreated by a Hartree-Fock factorization leading to selfconsistent potentials, local fields and bandgap renormaliz-ations [19, 25, 27, 37]. As an approximation scheme wewill adopt this strategy. In our case, the factorization ofthe density including the projector is even better moti-vated, since the factorization means the neglect of correla-tions between pairs. The projector onto the continuumexcludes an important source of such correlations, thebound two-pair states. Therefore, we write

S½] s[1]ji(1!P] B

2)½] [2] jj{

ii{T

BoJ [1, 1] jjiioJ [0, 1] j{

i{!oJ [1, 1]jj

ii{oJ [0, 1] j{

i.

Therewith, the correction ¸1,a can be written as

¸1,a" +

b,c,d(Habcd#Fabcd)oJ 1,b1,coJ 01,d, (4.28)

with

Habcd" +i, j, i{,j{

f1,a( j{i{

)* f1,b(ji

)* f1,c(ji

) f1,d(j{i{

)

](»ji{#»

ij{!»

jj{!»

ii{), (4.29)

Fabcd" +i,j, i{,j{

f1,a(j{i{

)* f1,b (ji

)* f1,c( ji{

) f1,d( j{i

)

](»ji@#»ij@!»jj @!»ii @ ). (4.30)

Thus we have found a closed set of dynamical equations.The additional couplings of Hartree type Habcd vanish inhomogeneous systems. The couplings of Fock typeFabcd and non-vanishing Hartree type terms are similar instructure to those arising from phenomenologically intro-duced local fields. As the local fields they lead to sourceterms involving products of densities and transitions.

Figure 4 schematically illustrates the result of ourcalculation: The coupling to continuum states of scatter-

Fig. 4. Incorporation of scattering continua into a few-level ap-proach is only possible in an approximate sense. It amounts toa modification of the coupling between the levels

ing pairs has been eliminated in favour of modified coup-lings between the discrete levels.

V. Conclusion

We have shown that the few-level model results from thedynamics of a new hierarchy of dynamical objects. Thefew-level description is the energy-eigenstate representa-tion of this hierarchy. The microscopic representation,which we called grounded densities in this paper, allows tocalculate the dynamics starting from a microscopicHamiltonian. The new hierarchy has to be contrasted tothat of conventionally used density matrices. While con-ventional density matrices are formulated in terms ofcreators and annihilators, the family of densities that leadsto few-level systems is formulated in states with definiteparticle number. Thus, irrespective of the representationin many-particle eigenstates or in single-particle modeslike Wannier or Bloch functions, the equations of motionin either hierarchy have a different structure.

The transformation rules connecting the two familiesof microscopic densities have been given. Although thedefinitions of the variables in both families suggest similarinterpretations of particular objects, a single variable ofone hierarchy can only be expressed by a linear combina-tion of the variables of the other hierarchy. This impliesthat approximations with similar motivations might havedifferent results in the two models. Examples of particularimportance are the restriction to a selected set of eigen-states and the introduction of phenomenological relax-ation times. The relaxation constants introduced forS½] [1]T and for o0

1, for instance, have both been denoted

as excitonic dephasing time in previous works [1, 10]. Inview of our results, it becomes clear that the meaning ofthis constant depends on whether it is defined for thefew-level system or within the microscopic theory.

Although it is possible to derive the equations ofmotion of the few-level model from a microscopic Hamil-tonian we realize that without further approximations, thelevel approach reaches its limits whenever continua areinvolved. An approximate way to handle the coupling tothe continua is to introduce modified couplings betweenthe discrete levels. Phenomenologically introduced addi-tional couplings like local fields thus can be interpreted asrepresenting scattering continua.

Appendix A: Equation of motionof microscopic density matrices

For nu"0, the operator +)[p],+)[p, 0] in (2.3) reads

+)[p]lkglk"p+t/1A+

i

¹cktiglkÇ2k

t~Çikt`Ç2k

p

#+j

¹ vltjglÇ2l

t~Çjlt`Ç2l

pk B#

1

2

p+

s, t/1

(»ktks

#»ltls

!2»ltks

)glk. (A1)

203

The source terms have the form:

SE"ScL s[n

u]j½K s[q]l{l SK

E(½K [p]k{k cL [n

u]i)T

!SSKE(½K [q]l{l cL [n

u]j)s (½K [p]k{k cL [n

u]iT (A2)

SV"ScL s[n

u]j½K s[q]l{l SK

V(½K [p]k{k cL [n

u]i)T

!SSKV(½K [q]l{l cL [n

u]j)s½K [p]k{k cL [n

u]iT (A3)

When the intrinsic semiconductor is excited from theground state, all expectation values containing unpairedoperators can be eliminated. [28] The source terms in theequations of motion for densities with completely pairedoperators are given by

S]E(½] [p]lk)"

p+

s, t/1

(!1)s`tMltks

E(ktls)½] [p!1]lÇ2l

t~Çlt`Ç2lpkÇ2k

s~Çk4`Ç2kp

!

l+k/0

g1,k

p+t/1

+i, j,j,j{

½] s[k#1] jj{ij

](MltiE(lti)½] [p#k]j{lÇ2l

t~Çjlt`Ç2lpjk

#Mjkt

E(jk

t)½] [p#k]j{lj{kÇ2k

t~Çikt`Ç2kp

)

#O (Ep`2l`4) (A4)

and

S]V(½] [p]lk)"

l+k/0

g1,k

p+t/1

+i,j,j,j{

(»jkt

!»jlt

!»ikt

#»ilt

)

]½] s[k#1]jj{ij ½] [p#k#1] jj{l

ijk

#O (Ep`2l`4). (A5)

The coefficient g1,k

stems from the application of thecontraction theorem [28] and is recursively defined by

g1,k

"

1

(k#1)!k!!

k+

k{/0

g1,k{

(k!k@) !2; g

1,0"1. (A6)

Appendix B: Inversion of the relation between microscopicdensity matrices and grounded densities

The inverse transformation of (4.13) is the special casenu"0 of (4.14).

S½] s[q]l{l ½] [p]k{k T"l+k/0

1

k!2+j,j{

o[q#k, p#k]j{k{l{j{jklj

#O (Eq`p`2l`2). (B1)

This identity can be proved inserting (4.13) into the sumon the right hand side.

l+k/0

1

k!2+j,j{

o[q#k, p#k]j{k{l{j{jklj #O(Eq`p`2l`2)

"

l+k/0

1

k!2+j,j{

l~k+

k{/0

pk{

+g,g{

]S½] s[q#k#k@]g{j{k{gjk ½] [p#k#k@]l{j{g{ljg T

s>/k§k{"

l+s/0

s+

k{/0

pk{

(s!k@)!2+

(gj), (g{j{)]S½] s[q#s] (g{j{)k{

(gj)k ½] [p#s]l{(j{g{)l(jg) T.

All members of the sum over s are zero except for s"0,what can be seen comparing the k@-sum with the definitionof the coefficients p

k(4.11). s"0 gives the left hand side of

(B1).

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