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1

Intersubband polaritonics revisited

O. Kyriienko1 and I. A. Shelykh1, 2

1Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland2International Institute of Physics, Av. Odilon Gomes de Lima, 1772, Capim Macio, 59078-400, Natal, Brazil

(Dated: December 2, 2011)

We revisit the intersubband polaritonics — the branch of mesoscopic physics having a hugepotential for optoelectronic applications in the infrared and terahertz domains. We show thatcontrary to the general opinion the Coulomb interactions play crucial role in the processes of light-matter coupling in the considered systems. We demonstrate that electron-electron and electron-holeinteractions radically change the nature of the elementary excitations in these systems. We arguethat intersubband polaritons represent the result of the coupling of a photonic mode with collectiveexcitations, and not non-interacting electron-hole pairs as it was supposed in the previous works onthe subject.

I. INTRODUCTION

Intersubband optical transitions in semiconductorquantum wells (QWs) are widely used in a variety ofmodern optoelectronic devices operating in a broad wave-length range spanning from mid-infrared to terahertz.1–5

Their practical realization needs high radiative quantumefficiency. In this context, the implementation of the con-cepts of strong light-matter coupling is a promising toolto improve the functionality of the devices as compare tothose operating in the weak-coupling regime.6,7

The achievement of the strong coupling is possible ifthe absorbing media is placed inside a photonic cavityand coherent light-matter coupling overcomes the dissi-pative processes in the system. For intersubband tran-sitions the experimental realization of strong couplingregime was for the first time reported in the pioneeringwork of D. Dini et al.8 The elementary excitations in thiscase have hybrid, half-light half-matter nature and arecalled intersubband polaritons. They have a number ofpeculiarities distinguishing them from conventional cav-ity polaritons formed by interband excitons. First, theyare formed only in TM polarization, as optical selectionrules prohibit the absorption of the TE mode in inter-subband transitions. Second, the strength of the couplingcan be an important fraction of the photon energy, whichmakes possible the transition to so-called ultrastrong cou-pling regime2,9. The light-matter coupling constant andresulting Rabi splitting depend on the geometry of theQW and photonic cavity and the electronic density inthe lowest energy subband10, which opens a possibility totune this parameter by application of the external gatevoltage.

The broad variety of the applications of intersubbandpolaritonics makes important the understanding of the

nature of intersubband polaritons. The question is: whatkind of the elementary excitation in a QW is coupled witha photonic mode and participate in the formation of thepolariton doublet? The former can be devided into twocategories: single-particle excitations (SPE) and collec-tive excitations, appearing from electron-electron inter-actions and absent in the non-interacting system. The

earlier works devoted to theoretical description of theintersubband polaritons neglected Coulomb interactionscompletely10–13 and the main qualitative conclusion wasthat the formation of the polaritons is a result of thecoupling between non-interacting electron-hole pairs anda cavity mode. Moreover, it was claimed that bosoniza-tion approach is valid for the description of unboundedfermion pairs.13

The opinion that Coulomb effects play no substantialrole in the intersubband polaritonics seems controversial,as their important role in photoabsorption of individualQWs (in the absence of a photonic cavity) is an estab-lished fact, studied extensively from both experimentaland theoretical points of view.14–17 Electron-electron in-teractions lead to the appearance of the collective exci-tation modes such as intersubband plasmon (ISP) whichunder certain conditions can give a dominant impact tothe optical response.14–22

The only work considering the role of many-electroninteractions for intersubband transition in the microcav-ity known to us is a paper by M. Pereira23. However, theinfluence of interactions on quasiparticle spectrum wasnot the main focus of the paper. The qualitative aspectsof strong light-matter interaction of collective intersub-band modes with microcavity photons remained unin-vestigated and the role of the many-body corrections ofvarious types thus remained unclear.

In the current paper we bridge this evident gap.We propose a semi-analytical way of the description ofthe coupling between intersubband excitations (single-particle or collective) to a cavity mode in terms of Feyn-man diagrams corresponding to the different physicalprocesses in the system. This makes our calculationstransparent and allows a simple qualitative interpreta-tion of the role of many body interactions in the consid-ered system. Moreover, since we sum up infinite seriesof the diagrams, our treatment is non-perturbative andincludes all orders of the interaction. As well, it treatsthe resonant and anti-resonant terms in light-matter cou-pling Hamiltonian on equal footing and thus allows thedescription of the phenomenon of ultra-strong couplingas well. We show that Coulomb interactions play cru-

2

cial role in the intersubband polaritonics, especially forhigh electron concentrations necessary for the achieve-ment of strong coupling regime. We claim that in thiscase the intersubband polariton is formed due to the cou-pling of the collective excitation known as intersubbandplasmon (ISP) with the cavity mode. In the opposite caseof small electron concentrations we show that excitoniccorrections become crucial.

II. THE MODEL

We consider a system with GaAs/AlGaAs quantumwell (QW) embedded into microcavity in the configura-tion usually used for intersubband transitions with TMpolarized light (Fig. 1(a,b)). As our goal here is topresent qualitative results and not to perform a fit of anyexperimental data, we do not consider the case where thephotonic cavity contains several QWs which is often usedin order to obtain larger values of Rabi splitting and nec-essary for the achievement of the ultra-strong coupling.2,9

Our method, however, can be easily generalized for thisconfiguration as well. We choose a doping of a single QWin such a way that only the lowest subband is filled by theelectrons at T = 0 while upper subbands remain empty(Fig. 1(c)). Later on we concentrate on T = 0 case only.

When quantum well is embedded into a microcavity,the photons interact continuously with electrons mov-ing them from fundamental subband to the first excitedsubband, thus creating electrons and holes which can in-teract with each other. This electron-hole pairs can thenagain disappear, re-emitting a photon. In diagrammaticlanguage such process can be described by a polarizationbubble Π (Fig. 2). The photon in a cavity thus becomes”dressed” by such bubbles, and its Green function G canbe described as a sum of the terms containing one, two,three etc bubbles as it is shown in Fig. 2(a). The sum-mation up to the infinite order gives the Dyson equation,whose solution yields

G =G0

1− g2G0Π, (1)

where Π denotes to the full polarization operator de-scribed by the intersubband bubble containing all pos-sible Coulomb interactions. In this expression G0 is abare photon Green function,

G0(ω, q) =2~ω0(q)

~2ω2 − ~2ω20(q) + 2iΓω0(q)

, (2)

where ω0(q) describes the cavity mode dispersion and Γis the broadening of the photonic mode due to the finitelifetime (taken to be ≈ 10ps) and g is a matrix elementof electron-photon interaction which reads10

g(q) =

√∆ · d210

~2ǫǫ0LcavAω0(q)

q2

(π/Lcav)2 + q2, (3)

FIG. 1: (Color online) Geometry of the system. (a), GaAsquantum well (QW) placed into the microcavity created bydistributed Bragg reflectors or air. Using the total internalreflection of light, TM-polarized beam travels through thestructure exciting the excitations between upper and lowersubbands. The current figure shows the creation of inter-subband plasmon – charge-density excitation. (b), Sketch ofthe QW of width L = 12.8 nm with plotted wave functionsfor fundamental and upper subbands. The separation energybetween levels is (∆ = 100 meV). (c), Dispersions of elec-trons for two subbands with Fermi level energy for electronconcentration nel = 1.25 × 1012 cm−2. The single-particleexcitations are schematically described as blue arrows. ThekF label denotes to the Fermi wave vector.

where Lcav is cavity length, ∆ is separation energy be-tween levels, ǫ0 and ǫ are vacuum permittivity and rela-tive material dielectric constant, respectively, d10 standsfor the dipole matrix element of the transition and A isan area of the sample.The dispersion of elementary excitation in such system

is determined by the poles of G(ω, q) and can be foundby solving a transcendental equation

1− g2G0Π = 0. (4)

The coefficient of the photoabsorption, α(q, ω), is pro-portional to the imaginary part of the full polarizationoperator of intersubband polariton accounting for multi-ple re-emissions and re-absorptions of the cavity photonΠpol(q, ω):

α(q, ω) ∼ ωℑΠpol(q, ω). (5)

The diagrammatic representation of the equation forΠpol(q, ω) is shown in Fig. 2(b) and yields

Πpol =Π

1− g2G0Π. (6)

3

FIG. 2: Diagrammatic representation of intersubband polari-tons. (a), The Dyson equation for microcavity photon inter-acting with intersubband quasiparticles Π leading to the for-mation of polariton. The label g corresponds to quasiparticle-photon interaction constant, indices 1 and 0 denote to excitedand fundamental subband, respectively. (b), Series for ab-sorption by intersubband quasiparticle coupled to the cavityin the general form with accounting all many-body effects.(c), The Dyson equation for self-energy corrections of theGreen function of the electron. (d), The Hartree-Fock self-energy which consists of direct (Hartree) and exchange (Fock)diagrams.

As it is seen from the Eqs. (4)-(6), all properties ofintersubband polaritons can be determined if the expres-sion of the polarization operator of an individual QWΠ(q, ω) accounting for Coulomb interactions is known. Ingeneral, the calculation of this quantity is a tricky taskwhich can be performed only in some particular cases,which we are now going to consider.

III. NON-INTERACTING CASE AND

HARTREE-FOCK APPROXIMATION

This case has a methodological interest and representsa test for our approach, allowing to compare the resultsit gives with those obtained earlier in the Refs. [10–13]by using the bosonisation scheme. For non-interactingparticles the calculation of the polarization operator Π0

represented by a single bubble without any Coulomb in-

teractions is straightforward and gives18

Π0(ω, q) = 2

∫dk

(2π)2n0k

~ω + E(0)k − E

(1)k+q + iγ

, (7)

where E(1)k+q = ∆+ ~

2(k + q)2/2m and E(0)k = ~

2k2/2mare dispersions of electrons in the excited and fundamen-tal subbands, respectively, and n0k is the Fermi distribu-tion in the fundamental subband which can be replacedby a step-like function at T = 0. The quantity γ rep-resents a non-radiative broadening of SPE. This integralcan be calculated analytically (see Appendix A). The useof Eqs. (4)-(6) allows the determination of the disper-sions of the intersubband polaritons and photoabsorptionof the system. It is instructive to consider the case γ → 0and assume that the transferred momentum of photon qis small as compare to the Fermi momentum of the elec-tron gas kF . In this case, the equation for the energiesof the polariton modes (4) reads

(~ω − ~ω0 + iΓ)(~ω + ~ω0 − iΓ)(~ω −∆) = 2ω0nAg2(q).(8)

If we are outside the ultrastrong coupling regime (ω0 ≫nAg2(q)) this reduces to

(~ω − ~ω0 + iΓ)(~ω −∆) = nAg2(q), (9)

which is nothing but the equation for two coupled har-monic oscillators corresponding to a cavity mode andsingle-particle excitations in the QW. The dispersions ofintersubband polaritons deduced from this equation co-incide with those obtained in the earlier works [10–13].One should note that in the mean field (Hartree-Fock)

approximation the electron-electron corrections can beeasily introduced into consideration without substantialmodification of the formalism. No new collective exci-tations appear in this approach and the results remainqualitatively the same as for the non-interacting case.We thus consider both situations in the same section.In diagrammatic representation the Hartree-Fock ap-

proximation results into renormalization of electronGreen functions which can be described by the Dysonequation shown in Fig. 2(c). Only the first order dia-grams corresponding to the direct Hartree term and ex-change Fock term (Fig. 2(d)) are retained in the expres-sion for the self-energy in this approach. The Hartreeterm diverges in the limit A → ∞ but is compen-sated by the interaction of the electrons with positivebackground20 and the Fock exchange self-energy correc-tion can be written in the form

Σ(i)HF = −

∑

k1

Vi00i(|k− k1|)nk1, (10)

where i = 0, 1 and the index 0 corresponds to the funda-mental subband and index 1 to the first excited subband,nk1

denotes the Fermi distribution in the fundamentalband and sign ”−” shows that the exchange interaction

4

decreases the energy. Vijkl(q) denotes a matrix elementof the Coulomb interaction which reads

Vijkl(q) =e2

2ǫǫ0Aq

∫dzdz′φi(z)φj(z

′)φk(z′)φl(z)e

−q|z−z′|

(11)with indices i, j, k and l corresponding to the initial andfinal subbands from which particles interact and φi(z)being the envelope wave functions in the direction of thestructure growth axis.18,24

The calculation of the polarization operator accountingfor the Hartree-Fock corrections can be done by substi-tuting the energies of bare electrons in the fundamentaland first subbands by their renormalized values, calcu-lated as

E(i)(k) = E(i)(k) + Σ(i)HF (k) (12)

This renormalization has the following consequences.First, the correction for populated fundamental subbandis greater then for the empty first subband leading to therenormalization of the transition energy ∆. Accountingfor the negative sign of exchange correction, one con-

cludes that the effective gap ∆ is increased. Second, ingeneral the self-energy is a function of momentum whichleads to the non-parabolicity of the renormalized disper-sions and contributes to the broadening of the absorptionline. For intersubband polaritons the first effect shifts theanticrossing point in the region of larger momenta (sev-eral meV in the geometry we consider) and second leadsto the slight decrease of the observed Rabi splitting.

IV. INTERSUBBAND PLASMON-POLARITON

In general, electron-electron interactions can not beneglected and Π 6= Π0. Their account is a compli-cated task. However, for high enough electron con-centrations (n ∼ 1012cm−2) the polarization operatorcan be estimated using Random Phase Approximation(RPA) whose diagrammatic representation is shown inFig. 3(c).15,22 In this regime the system demonstratesthe appearance of intersubband plasmon (ISP) — charge-density collective excitation arising from intersubbandtransitions.The summation of the infinite series of the diagrams

represented in Fig. 3(c) allows us to obtain a compactexpression for the polarization operator corresponding toISP:

ΠISP =Π0

1− V1010Π0. (13)

The dispersion of intersubband plasmon is plotted inFig. 3(a). The absorption by intersubband plasmon isgiven by imaginary part of polarization Π (Fig. 3(b)).One sees that the system still has an absorption peakcorresponding to the single-particle excitations. How-ever, another peak corresponding to the absorption of

0 0.2 0.4 0.6 0.8 1.0

40

60

80

100

120

140

Wave vector 108m

1

En

erg

ym

eV

single-particle excitations

intersubband plasmon

0

1

=

0

1

0

1

0

1+ + …Π0 Π0

Π0

Π ISP =

(a)

90 95 100 105 110 115 120 1250.0

0.2

0.4

0.6

0.8

1.0

Energy meV

Ab

so

rptio

na

.u.

SPE

δ = 0.77 meV

ISP

δ = 0.04 meV

(b)

(c)

=

0

1

0

1

+Π0 Π0

0

1

Π ISP

FIG. 3: (Color online) Intersubband plasmon. (a), The dis-persion of intersubband plasmon (ISP, red line) and single-particle excitations spectrum (SPE, violet parabolas) plottedfor the long range of momentum. For the considered concen-tration nel = 1.25 × 1012 cm−2 ISP dominates over SPE inthe small wave vectors range. (b), Absorption spectrum ofintersubband quasiparticles plotted for momentum q = 106

m−1. While the intersubband plasmon peak is sharp (δ = 0.05meV ) and high, the SPE continuum is much broader (δ = 0.77meV ) and has smaller oscillator strength. (c), Random PhaseApproximation (RPA) series for intersubband electron-holepairs interacting by direct Coulomb matrix element V1010.These diagrams correspond to the formation of intersubbandplasmon.

ISP appears. This peak is blueshifted by a value of de-polarization shift nV1010(q). It is much more intensiveand narrow that the peak corresponding to SPE. There-fore, it is natural to suppose that after placing of the QWin a photonic cavity this peak will give main contributionto the formation of intersubband polariton.

This conclusion is supported by the results of calcula-tions shown in Fig. 4. The dispersion of the elementary

5

FIG. 4: (Color online) Intersubband plasmon polariton. (a),Density plot of the intersubband plasmon polariton spectrumshowing the dispersion of excitations and absorption (colorintensity) in the system. The detuning of cavity mode is 10meV. (b), Absorption spectrum of the system plotted for wavevector q = 1.25 × 106 m−1 where the anticrossing point forISP-photon exists. The two peaks corresponding to upper andlower plasmon polaritons are clearly observed, while single-particle excitations are suppressed. (c), Dispersions of inter-subband plasmon polariton modes (red and violet lines) instrong coupling regime with corresponding anti-crossing andRabi frequency VR = 2.4 meV . The SPE dispersion (pinkline) is weakly coupled to the cavity mode.

excitations of the hybrid QW-cavity system was calcu-lated using Eq. (4), where the Hartree-Fock correctionswere accounted for in Π0. The corresponding spectrumof intersubband excitations in the semiconductor micro-cavity contains three branches (Fig. 4(c)). Two of themcorresponding to the cavity photons and ISP reveal an-ticrossing and give birth to the intersubband polaritonmodes, which in this case can be called more correctly in-tersubband plasmon-polaritons. The third mode denotesto the single-particle excitations. The corresponding dis-persion line crosses the dispersion of the cavity photon,which means that for SPE the weak coupling regime isrealized. In the absorption spectra shown in Fig. 4(b)three peaks appear. Two of them corresponding to inter-subband plasmon-polaritons are very pronounced and thethird one corresponding to coupling with single-particleexcitations is very weak but still observable (Fig. 4(a)).

V. EXCITONIC EFFECTS

In the previous section we investigated the case of highelectron concentration in QW where plasmonic effectsdominate and RPA can be successfully used. However,the opposite limit of low concentrations is also interest-

FIG. 5: Diagrammatic representation of intersubband exci-ton. (a), The ladder series for the intersubband excitationleading to the creation of intersubband exciton. The wavy linecorresponds to V1001 Coulomb interaction and double wavyline denotes to effective interaction between particles. (b),Two-particle integral equation for effective interaction. q isthe transferred momentum. (c), The general Dyson equationwhich describes the situation of mixed excitonic and plas-monic effects leading to the formation of mixed collectivemodes.

ing. In this case plasmonic corrections can be neglectedand excitonic effects corresponding to the interaction be-tween the photoexcited electron in the first subband witha hole in fundamental subband become dominant. In di-agrammatic language they can be described by ladder di-agrams shown in Fig. 5(a).25,26 Summation of the ladderdiagrams up to an infinite order gives birth to the forma-tion of the attraction between the electron and hole, andcan give a peak i photoabsorption lying below the con-tinuum of SPE (contrary to the the peak correspondingto ISP).Mathematically, the dispersion of the excitonic mode

can be deduced from the integral Bethe-Salpeter equa-tion for the effective electron-hole interaction, whose dia-grammatic representation is given in Fig. 5(b) and whichreads25

W (k,k1, ω,q) = −V1001(k− k1)− (14)

−

∫dk2V1001(k− k2)Π0(ω,k2,q)W (k2,k1, ω,q).

The solution of the Bethe-Salpeter equation is a compli-cated task which goes beyond the scope of the presentpaper. Here we restrict ourselves by a simplified treat-

6

FIG. 6: (Color online) Intersubband exciton polariton. (a),The dispersion of intersubband exciton (green line) whichlies below the single-particle excitations continuum (violetparabolas). The concentration of electrons in QW is nel =1011 cm−2. (b), Density plot of intersubband exciton coupledto the photonic mode. The intensity describes the absorp-tion in the system. (c), Dispersions of intersubband excitonpolariton modes (green and violet lines) in strong couplingregime (VR = 0.5 meV ). The SPE dispersion (pink line) isweakly coupled to a cavity mode.

ment of the screened 2D Coulomb potential24,27

V1001(q) ≈e2

2ǫǫ0A(|q|+ κ), (15)

where κ = me2

2πǫǫ0~2 is the screening wave vector. In 2Dsystems it does not depend on electron density and forthe structure we consider is equal to κ = 2.4 × 108m−1.Then, since we are working with zero temperatures andsmall concentrations (kF < κ), the interaction betweenparticles is slowly varying function of q and can be ap-proximated by a constant value V0 ≈ e2/2ǫǫ0Aκ. In thislimit the equation for the polarization operator corre-sponding to the intersubband exciton can be found ana-lytically as15

Πexc =Π0

1 + V0Π0, (16)

where the negative sign of Coulomb interaction corre-sponding to the electron-hole attraction is accounted forin the denominator. The calculated dispersion of inter-subband exciton is shown in Fig. 6(a) for the QW withdoping nel = 1011 cm−2, where assumption kF < κ isappropriate.The dispersion of the elementary excitations of the QW

coupled to a cavity mode accounting for the excitonic ef-fects are shown in Fig. 6(c). Similarly to the case of the

high concentrations with the strong ISP-photon coupling,one sees three dispersion branches. Two of them corre-sponding to the excitonic and photonic modes reveal an-ticrossing and form the intersubband exciton polaritons.The third one corresponding to single-particle excitationsremains in a weak coupling regime. In the photoabsorp-tion spectrum shown in Fig. 6(b) three peaks of differentintensities are observed. The excitonic effects lead to theredistribution of the oscillator strength, which becomessmall for SPE transitions and corresponding peak is con-sequently very weak. In this case the Rabi splitting isreduced due to the concentration dependence of interac-tion constant and is about 0.5 meV.

One should also comment on the case of the interme-diate densities, where strictly speaking neither RPA norladder approximation can be applied. In some works20

RPA and ladder corrections were accounted simultane-ously to describe the Coulomb correlations in this regime.In the diagrammatic representation this results into theequation for the polarization operator shown in Fig. 5(c)which leads to the appearance in the system of somemixed collective exciton-plasmon modes. As depolariza-tion and excitonic shifts have opposite signs, it may bepossible that there exists a regime when they almost com-pensate each other and the energy of the collective modeis close to the energy of the single-particle transition.The consideration of this case, however, goes beyond thescope of the present paper.

VI. CONCLUSIONS

In conclusion, we analyzed the elementary excitationsarising from the strong coupling of a photonic cavitymode with an intersubband transition of a single QW.We have shown that contrary to the current opinionCoulomb interactions can play crucial role in the sys-tem and lead to the qualitative changes of the nature ofintersubband polaritons. We predict theoretically thatstrong coupling of the cavity mode occurs with collec-tive excitations, while single-particle excitations remainin the weak coupling regime.

This work was supported by Rannis ”Center of Ex-cellence in Polaritonics” and FP7 IRSES project ”PO-LAPHEN”. I.A.S. acknowledges the support from COSTPOLATOM program. O.K. acknowledges the help ofEimskip Foundation.

Appendix A: Calculation of a polarization operator

for non-interacting particles

The generic form of the electron-hole polarization op-erator for non- interacting particles can be written as

iΠ0(ω,q) = 2

∫dkdν

(2π)4G(k + q, ν + ω)G(k, ν), (A1)

7

where G(k, ν) denotes the Green function of particle withmomentum k and energy ν. q and ω correspond to thetransferred momentum and energy, respectively. For theintersubband transition case the polarization bubble de-scribes the excitation process where electron is trans-ferred to the upper subband while the hole is createdin the lower subband. Thus, the Eq. (A1) can be repre-sented as

Π0(ω, q) = 2

∫dk

(2π)2n0k

~ω + E(0)k − E

(1)k+q + iγ

, (A2)

where E(1)k+q = ∆+ ~

2(k + q)2/2m and E(0)k = ~

2k2/2mare dispersions of electrons in the excited and fundamen-tal subbands, respectively, ∆ is an energy distance be-tween subbands and n0k is the Fermi distribution in thefundamental subband which can be replaced by a step-like function at T = 0. Parameter γ describes the lifetimeof the excitation.First, let us rewrite Eq. (A2) in the following form

Π0(ω, q) = 2

∫ kF

0

kdk

(2π)2

∫ 2π

0

dφ

~ω −∆− ~2q2/2m− ~2kq cosφ/m+ iγ,

(A3)where φ is an angle between two vectors k and q. Theassumption that transferred momentum of photon q issmall yields simple integration on φ and k, and the realpart of polarization operator is

ℜΠ0(ω)q→0 =nA

~ω −∆, (A4)

where n =k2

F

2π is 2D density of electron gas, A is QWarea and we assumed γ → 0.Now we return to the case of non-negligible photon mo-

mentum and calculate the imaginary part of polarizationoperator (A3). The analysis of the denominator showsthat integral on k in Eq. (A3) has poles only in certainangular range [φmin, φmax]:

arccos

[(~ω −∆− Eq)

~2kF q/m

]< φ < arccos

[−(~ω −∆− Eq)

~2kF q/m

].

(A5)Thereby, in this region one can find the imaginary part ofpolarization operator using residue theorem in the limitγ → 0:

ℑΠ0(ω, q)γ→0 =Am

4~21

Eq

(~ω−∆−Eq) tanφ|φmax

φmin≡ Iδ(ω, q),

(A6)which describes a peak in the region where single-particleexcitations exist, bounded by parabolas ∆ + ~

2q2/2m−~2kF q/m and ∆ + ~

2q2/2m+ ~2kF q/m.

However, to describe the realistic absorption spectralfunction one needs to account for the finite non-radiativelifetime of intersubband excitations. It can be done usingthe Sokhatsky-Weierstrass theorem with non-zero γ∫

f(x)

x+ iγdx = −iπ

∫γf(x)

π(x2 + γ2)dx+

∫xf(x)

(x2 + γ2)dx.

(A7)

The integral (A3) can be rewritten in the similar form

Π0(ω, q) = −2

∫ 2π

0

dφ

(2π)2

∫ kF

0

k

Bk + C + iγdk = (A8)

= −1

2π2

∫ 2π

0

dφ

B

∫ kF

0

k

k − k0 + iγdk =

= −1

2π2

∫ 2π

0

dφ

B

∫ kF−k0

−k0

k + k0

k + iγdk,

where B = ~2q cosφ/m, C = ~ω − ∆ − ~

2q2/2m,k0 = C/B and the φ-dependence of B and k0 shouldbe taken into consideration. In the latest expression the

new integration variable k = k − k0 was used. Thus,comparing the Eqs. (A8) and (A7), one can calculatenumerically the imaginary part of polarization operator(A3) with finite lifetime of excitations. One sees thatintegral (A8) can be separated into two parts. Detailedanalysis shows that the first integral gives obtained pre-viously delta-function like absorption which does not de-pend on γ, while the second integral ILor is proportionalto γ and has Lorentzian form. Therefore, for the sake ofsimplicity, in the purely analytical calculations it is pos-sible to use absorption spectrum as the sum of equation(A6) and phenomenological Lorentzian broadening dueto finite lifetime of excitation (τ ≈ µs) written in theform24

ILor(ω) = ℑ

[nA

~ω −∆+ iγ

]=

−γnA

(~ω −∆)2 + γ2. (A9)

Finally, the imaginary part of polarization operatorwhich describes optical absorption by intersubband tran-sition yields

ℑΠ0(ω, q) = Iδ(ω, q) + ILor(ω) (A10)

The real part of polarization operator can be found bydirect integration in Eq. (A3)18,24

ℜΠ0(ω, q) =Am

π~21

2Eq

((~ω −∆− Eq)∓

∓√(~ω −∆− Eq)2 − 4EFEq), (A11)

where ” − ” sign corresponds to the case ~ω > ∆ and” + ” for ~ω < ∆.

Consequently, the polarization operator Π0 whichstands for the simple intersubband bubble is described bythe sum of real (Eq. (A11)) and imaginary (Eq. (A10))parts. Finally, using it in the Dyson equations, the opti-cal response and elementary excitation spectrum can befound.

8

1 E. Dupont, H. C. Liu, A. J. SpringThorpe, W. Lai, andM. Extavour, Phys. Rev. B 68, 245320 (2003).

2 G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol,L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leit-enstorfer, R. Huber, Nature Lett. 458, 07838 (2009).

3 C. Walther, M. Fischer, G. Scalari, R. Terazzi, N. Hoyler,and J. Faist, Appl. Phys. Lett. 91, 131122 (2007).

4 O. Cathabard, R. Teissier, J. Devenson, J. C. Moreno, andA. N. Baranov, Appl. Phys. Lett. 96, 141110 (2010).

5 D. Oustinov, N. Jukam, R. Rungsawang, J. Madeo, S. Bar-bieri, P. Filloux, C. Sirtori, X. Marcadet, J. Tignon, andS. Dhillon, Nat. Commun. 1:69 doi: 10.1038/ncomms1068(2010).

6 R. Colombelli, C. Ciuti, Y. Chassagneux, C. Sirtori, Semi-cond. Sci. Technol. 20, 985 (2005).

7 P. Jouy, A. Vasanelli, Y. Todorov, L. Sapienza, R.Colombelli, U. Gennser, and C. Sirtori, Phys. Rev. B 82,

045322 (2010).8 D. Dini, R. Kohler, A. Tredicucci, G. Biasiol, and L. Sorba,Phys. Rev. Lett. 90, 116401 (2003).

9 Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato,C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Phys. Rev.Lett. 105, 196402 (2009).

10 S. De Liberato and C. Ciuti, Phys. Rev. B 77, 155321(2008).

11 C. Ciuti, G. Bastard, I. Carusotto, Phys. Rev. B 72,

115303 (2005).12 C. Ciuti and I. Carusotto, Phys. Rev. A 74, 033811 (2006).13 S. De Liberato and C. Ciuti, Phys. Rev. Lett. 102, 136403

(2009).14 A. Pinczuk, S. Schmitt-Rink, G. Danan, J. P. Valladares,

L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 63, 1633(1989).

15 I. K. Marmorkos and S. Das Sarma, Phys. Rev. B 48, 1544(1993).

16 O. E. Raichev and F. T. Vasko, Phys. Rev. B 60, 777(1999).

17 M. F. Pereira and H. Wenzel, Phys. Rev. B 70, 205331(2004).

18 J. Dai, M. E. Raikh, and T. V. Shahbazyan, Phys. Rev.Lett. 96, 066803 (2006).

19 S. Das Sarma and I. K. Marmorkos, Phys. Rev. B 47,

16343 (1993).20 D. E. Nikonov, A. Imamoglu, L. V. Butov, and H. Schmidt,

Phys. Rev. Lett. 79, 4633 (1997).21 J. Li and C. Z. Ning, Phys. Rev. Lett. 91, 097401 (2003).22 J. C. Ryan, Phys. Rev. B 43, 12406 (1991).23 M. F. Pereira, Phys. Rev. B 75, 195301 (2007).24 H. Haug and S. W. Koch, Quantum Theory of the Op-

tical and Electronic Properties of Semiconductors (WorldScientific, Singapore, 1990).

25 O. Betbeder-Matiber and M. Combescot, Eur. Phys. J. B22, 17-29 (2001).

26 G. D. Mahan, Many Particles Physics (Plenum Press, NewYork, 1993).

27 M. E. Portnoi and I. Galbraith, Solid State Commun. 103,325-329 (1997).