Bernard Caron
email: .caron@vjf.cnrs.fr

Zaar-Hausa-English Code mixing

According to [NEWMAN, 1990 : 202], Zaar (which he calls Saya) belongs to the the West-B3 group of Chadic languages, with some doubts arising from the properties it shares with Angas.

The names derived from the root 'Saya' (i.e. Bàsayèe (pl. Sayaawaa) for the speakers, and Sayancìi for the language) are the names used by the Hausas. The speakers call themselves  (pl. ) meaning human being, and call their language  (lit. the mouth of men). They consider the term "Saya" derogatory, and associate it with the Hausa verb sàyaa, to buy, taking it as a reference to the fact that Hausas used to take them as slaves. In this work, we will use the term Zaar to refer both to the people and the language.

Most Zaar people of the younger generation are bilingual Hausa-Zaar. They are schooled through the medium of Hausa in primary school, before learning English. The Zaars are Christians, and use a Hausa translation of the Bible. The older generation are not very confident in Hausa, whereas the younger educated elite who sometimes hold positions in the administration, police and education, switch between Zaar, Hausa and English.

The purpose of this paper is to examine how a minority language that is healthy is used in two different contexts : (A) an urban context, with educated speakers who are fluent in the two main vehicular languages of the area : English and Hausa ; (B) a rural context with speakers who hardly speak a few words of Pidgin English, but are quite fluent in Hausa.

Sources

This presentation was done using the transcription of two dialogues of around half an hour each. A a conversation between three teachers of English in Azare Teachers' College, in the North of Bauchi State. The conversation was recorded in a hotel in Azare in 1991, on the first day of my field work on Zaar. It was transcribed the following year in Ibadan with Sunday, my Zaar language assistant. Topic : one of the teachers narrates the attack of his taxi by a gang of highway robbers on the way from Ibadan to Jos. B a conversation between a mechanic (30) two unemployed young men (in their early 20s) and a young boy (10). It was recorded in 1996 and transcribed in 1999 in the village of Tudun Wada (B'ogoro LGA, South of Bauchi State). Tudun Wada is in the middle of the area where the Zaar language is spoken. The conversation bears on various topics : purchase of material to repair shoes ; radio equipment ; sexual behaviour of one neighbour, prostitution, teasing the young boy, etc. On the whole, the texts are samples of unmistakably « good » Zaar spoken by competent speakers, with various degrees of code mixing.

Historical stratification

It is possible to distinguish various strata of borrowing depending on the degree of integration into the matrix language structure, ranging from the more to the less integrated.

(Integrated) loan-words

They are part of the architecture of the sentence, both from the point of view of lexicon and grammar. They follow the phonological and grammatical rules of Zaar (noun determination ; integration into the verb class system).

Lexical

Nouns

English (sometimes via Hausa

) :         

Verbs

The integration of verbs borrowed from Hausa can be seen in the way they adopt the Zaar class system, with the two-syllable verb tone scheme Mid-Hi :

   2.1.1.3 Prepositions dàgà (from) ; zúwà (to).

Grammatical

Aspectual verbs

Zaar doesn't have any auxiliaries and makes a heavy use of Hausa aspectual verbs, which are integrated into the Zaar verb class system (cf. supra 2.1.1.2)



As in Hausa, the aspectual verb can be followed by a verbal noun (here :  , talk-ing) :

 if/ 3s

Exclamations

Emotional reactions are expressed through exclamations, insults, etc. borrowed from Hausa. It is not easy to assess the degree of integration of these borrowings.





Set phrases, quotations

They are not assimilated into Zaar, they do not follow the phonological and morphological rules of Zaar.

Hausa

Types of code mixing

Different types of code mixing can be associated to various contents.

Borrowing

Borrowing is omnipresent in Zaar in the complexification of sentences (2.1.2.3), and discourse continuity :

Tag switching

Tag switching, on the other hand, is associated with contents that escape the usual rural setting associated with the original Zaar lexicon stock. In the following instances, the wallet and the ID card are items associated with « modern » life, and are designated by loanwords.

Metaphorical code-switching

, he was courageous.

This type of code mixing is present mostly in the conversation of the educated speakers. In the case of the uneducated speakers, one observes the occasional quotation of a Hausa saying.

The most common function of tag-switching is pragmatic : highlighting an event, setting off a personal reaction to what has been said. (cf. 2.2)

In the second conversation, one notices a very stong rise in code-mixing when the topic switches to sexual jokes. The small boy is eating some fried groundnut cake, and the adults around him start teasing him, drawing a parallel between eating sweets and the sexual act. In this case, it corresponds to an expression of strong emotional reaction.

[somebody has just said that « totuwa », i. The interest of this study lies in the way it reveals the processes at work in borrowing and grammaticalisation. It gives a synchronic view of diachronic processes at work. Words are borrowed, beginning in the pragmatic domain (emotions, feelings, modality) then get integrated into the area of topicality verging on sentence complexification (with the so-called Hausa modal particles heavily borrowed). Then finally, aspectual verbs build a sub-area of Zaar grammar where auxiliarity is being developed.

  .Cond/ come/ again/ say/ trouble/ this/ again If he keeps talking about this problem...

	
	1pl.Perf/ already/ 1pl.Narr/ get up/ 1pl.Narr/ stand
	we had already	risen and gone
	2.1.2.2 Modality verbs, adverbs and particles
	Various quality judgements or modalities (epistemic, deontic) are expressed through borrowing
	from Hausa.
	
	
	
	if/ 2s.Fut/ buy/ even/ should/ 2sg.Subj/ buy-Dir/ small / this
	If you buy, you should	buy this small one.
	2.1.2.3 Discourse particles
	Topicalization and discourse continuity draw heavily on Hausa lexical ressources :
	
	

or a finite verb (here :  in the Narrative)

  These loanwords are nouns that get a Zaar determinant suffix : This is generally set off by an event or a thing associated with the world outside Zaar rural life.

	
	3pl.Narr/ go+Déf/ wallet+Def
	X -Did they go away with the wallet ?
	  
	
	3pl.Narr/ go+Déf/ completely/ indeed// ID card+1s.Gen/ even/ there is
	Y -They took everything. Even my ID card	there was.
	3.3 Interlarded speech
	
	
	We left there at night, well, (not) until they had finished... you know... « sanitation ». Well, we
	left there during the day. Because there were many cars. Do you understand ? So... we left...
	before we went away...
	
	But, the driver was courageous, you hear ? I swear
	

  that they sell groundnut cakes and it is finished ? As for vagina, do they sell and there is none left ? [... only] if you die it is finished.

	
	
	Y -I swear, I thought that it was vagina, what ! Have you ever (heard) it finishes ? Here you
	are, have you already
	4. Conclusion

e. groundnut cakes, can never be finished]  X -Is it groundnut cakes that they call « totuwa » ? I thought they were saying vagina, what !  Z -Come on ! (Excl.) You bastard, X. ! heard


Simone De Liberato

Cristiano Ciuti

Quantum theory of intersubband polarons

We present a microscopic quantum theory of intersubband polarons, quasiparticles originated from the coupling between intersubband transitions and longitudinal optical phonons. To this aim we develop a second quantized theory taking into account the Fröhlich interaction between phonons and intersubband transitions and the Coulomb interaction between the intersubband transitions themselves. Our results show that the coupling between the phonons and the intersubband transitions is extremely intense, thanks both to the collective nature of the intersubband excitation and to the natural tight confinement of optical phonons. Not only the coupling is strong enough to spectroscopically resolve the resonant splitting between the modes (strong coupling regime), but it can become comparable to the bare frequency of the excitations (ultrastrong coupling regime). We thus predict the possibility to exploit intersubband polarons both for applied optoelectronic research, where a precise control of the phonon resonances is needed, and also to observe fundamental quantum vacuum physics, typical of the ultrastrong coupling regime.

I. INTRODUCTION

The theory of polarons, the quasiparticles describing electrons moving in a polarizable medium, dates back to the early days of quantum theory 1 , and it has been an active field of research ever since 2 . Polarons have been studied both for three 3 and two 4,5 dimensional electron gases, leading to a number of very precise estimates of the quasiparticle mass and energy.

In this paper we will talk of strongly and ultrastrongly coupled polarons. In polaron theory it is customary to distinguish between weak coupling 6 and strong coupling 7 polarons, accordingly to the value of the Fröhlich coupling constant, that quantifies the strength of the electron-phonon coupling, and that is used as development parameter in perturbative calculations [START_REF] Mahan | Many Particle Physics[END_REF] . In this paper anyway we will use the terms strong and ultrastrong coupling in a different, even if closely related, meaning.

The term strong coupling is commonly used in modern atomic and solid state physics with a rather precise meaning: two systems are strongly coupled if it is possible to resolve, at resonance, the anticrossing due to the coupling. Strong coupling in this latter sense has been predicted [START_REF] Varga | [END_REF]10 and observed 11,12 very early for polaritons coupled to three dimensional plasmons.

The figure of merit of the strong coupling regime is the ratio between the coupling strength, quantified by the so-called vacuum Rabi frequency Ω, and the width of the resonance Γ. If Ω Γ is bigger than one, the two resonances are spectroscopically resolved, and the system is in the strong coupling regime. An interesting point to be made is that the fact of whether a system is in the strong coupling regime or not, does not depend on the intrinsic strength of the coupling, but only on the ratio between the coupling and the linewidth; the first observations of the strong coupling regime in atomic systems were made with Rydberg atoms in superconducting cavities, where the strength of the coupling is negligible when compared to the bare frequency of the involved transitions ω, but where the astounding quality factor of the superconducting cavities allows to resolve the resonant splitting [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF] .

It is easy to prove that the value of the ratio Ω ω for atomic systems is always much smaller than one, its smallness being linked with the value of the fine structure constant [START_REF] Devoret | [END_REF] . Still this severe limitation is valid only for single particle effects, while collective excitations can beat it thanks to their coherent nature. In a phenomenon reminiscent of the Dicke superradiance 15 , the strength of the coupling scales as the square root of the number of particles involved, leading to the possibility to obtain values of Ω ω of the order of one. This observation has prompted the development of the theory of the so-called ultrastrong coupling regime, that is the regime in which the coupling between two systems is of the same order of their bare frequencies. Such theory was initially developed 16,17 in the case of intersubband polaritons 18 in semiconductor quantum wells, where it was also observed for the first time [19][20][21] . The ultrastrong coupling regime has since been studied in a lot of different systems, ranging from circuit quantum electrodynamics 22,23 , to organic semiconductors 24 , to high mobility electron gases in presence of a perpendicular magnetic field 25,26 .

The interest in such ultrastrong coupling regime is motivated by a whole range of new physics that can be observed: from spectral deformations 20 , to quantum vacuum emission phenomena [27][28][29] and even quantum phase transitions 30,31 .

This started a race to get the system with the highest value of the light-matter coupling ratio Ω ω , with actual record values of 0.12 in superconducting circuits 23 , 0.16 in organic molecules 24 , 0.24 in semiconductor quantum wells 21 and 0.36 for two dimensional electron gases in presence of an applied magnetic field 26 (all coupled to a confined electromagnetic field).

It is natural at this point to ask how strong the coupling of polarons really is. It turns out that, in the case of coupling to collective excitations, it is quite big. Already for the first observation of three dimensional plasmon-polaritons 11,12 we have Ω ω ≃ 0.2 and values only marginally smaller were observed for polarons in quantum dots 32,33 .

In this paper we will develop a microscopic theory of the ultrastrong coupling between intersubband transitions and longitudinal optical (LO) phonons in semiconductor quantum wells, that is the theory of ultrastrongly coupled intersubband polarons. The coupling between intersubband transitions and LO-phonons is relevant for a number of optoelectronic applications, as it determines the lifetime of carriers in excited subbands 34 . In particular a precise knowledge of LO-phonons intersubband scattering rates is important in the engineering of heterostructures for quantum cascade lasers 35 . Normally optoelectronic devices are designed to avoid being in resonance with optical phonon transitions, due to the high absorption between transverse and longitudinal optical phonon frequencies (Restrahlen band). A notable exception is provided by quantum cascade lasers operating near such optical resonances 36,37 in which instead the transitions between different subbands are almost resonant with LO-phonon modes.

Even if the coupling between intersubband transitions and LO-phonons in semiconductor quantum wells has indeed received some attention [38][39][40] and intersubband polaron resonances have been clearly and unambiguously observed 41 , the fact that such mixed excitations can be extremely easily in the strong (or even ultrastrong) coupling regime seems overlooked by the community working on intersubband optoelectronic devices. Moreover, to the best of our knowledge, there is no microscopic theory of such excitations, as the spectra of intersubband polarons are normally calculated in an indirect way.

In the Sec. II we explain how our microscopic theory differs from the dielectric function approach normally employed to calculate the spectra of intersubband polarons. In order to build such microscopic theory, we will need to develop an exactly solvable Hamiltonian theory of electron-phonon and electron-electron interactions, heavily relying on the concept of superradiant, or coherent, excitation. In Sec. III, we will explain the general concept of superradiant excitations, that will then be extensively used in Sec. IV to construct the actual theory. In Sec. V we will apply the theory to the case of GaAs quantum wells, showing how the ultrastrong coupling regime can be reached even with such relatively weakly polar material. Finally a few considerations on the impact of our results and on possible future developments will be drawn in Sec. VI.

II. LONGITUDINAL EXCITATIONS

Electromagnetism in condensed matter physics is usually studied in the Coulomb gauge. This gauge, while not adapted to high energy physics due to its non-manifestlycovariant form, is particularly well suited to study low energy phenomena, thanks to a natural separation of dy-namical and instantaneous degrees of freedom. In such gauge we have that, in absence of charges, div[E(ω)] = 0, (

i.e., all propagating waves are transverse. When we consider the propagation of an electromagnetic wave in a dispersive medium, Eq. (1) becomes

div[D(ω)] = div[ǫ(ω)E(ω)] = 0, (2)
that is, in a dispersive medium, it is possible to have propagating longitudinal waves at the frequencies for which

ℜ[ǫ(ω)] = 0, (3)
where ℜ indicates the real part. In order to find the frequencies of longitudinal modes it is thus customary to solve Eq. (3) for ω.

If we consider light propagating in a polar material, we can write the equation of the electromagnetic field coupled to the transverse optical (TO) phonons and from there we can calculate the dielectric function and finally use it in Eq. ( 3) to find the frequency of the LO-phonons. An interesting consequence of this approach is that we are calculating the frequency of the LO-phonon mode, considering the coupling of two fields that are not even coupled to it, as the propagating electromagnetic field couples to the TO-phonons, not to the LO-phonons. Microscopically this stems from the fact that LO-phonons couple to local charges through Coulomb interaction, that is physically the same quantity to which propagating electromagnetic waves couple. Only the particular choice of the Coulomb gauge makes them appear as distinct coupling mechanisms.

Polarons are also longitudinal excitations, being partially formed of LO-phonons, and as such their energies can be calculated with a simple dielectric function approach as the one sketched just above, if we take into account also the electronic component in the dielectric function. This approach is used, e.g., in Ref. [41]and it indeed gives the right results, also allowing to calculate all the optical observables of the system directly from the dielectric function.

Still physically such approach may appear rather unsatisfactory because, while allowing to calculate a number of physical observables, it completely neglects the microscopic nature of the coupling that creates the polarons. The polarons are due to the coupling of LOphonons and electrons (Fröhlich interaction) and in the dielectric function approach, neither the LO-phonons, nor the Fröhlich interaction, are taken explicitly into account.

Apart from this rather formal considerations, missing a microscopic approach makes it difficult, or altogether impossible, to study what happens in more complex situations, as for example in inhomogeneous samples or when, due to the strength of the coupling, it could become possible to observe exotic phenomena as quantum vacuum effects or quantum phase transitions.

In the following sections we will develop a detailed microscopic theory for the intersubband transitions coupled to LO-phonons, in the Appendix we will compare the developed theory with the results of the standard dielectric function approach.

III. SUPERRADIANT EXCITATIONS

In 1954 Dicke 15 noticed that a set of coherently excited identical dipoles relaxes radiatively much faster than a single, isolated dipole. This is due to the phenomenon of superradiance: N identical dipoles behave as a single collective dipole √ N times bigger. The reason of such collective enhancement can be understood considering that, being all the dipoles identical, all the transitions are degenerate and a simple basis change allows us to describe the system of N dipoles as a single superdipole coupled to the electromagnetic field and N -1 other modes that are not coupled to the field. The particular superposition that couples to the field (usually called bright mode) thus concentrates all the oscillator strength and has a dipole √ N times bigger.

The concept of superradiance has been thoroughly applied to the study of intersubband polaritons 18,[42][43][44] in microcavity embedded doped quantum wells. In quantum wells the confinement of electrons along the growth direction splits the electron bands, giving rise to multiple parallel subbands. Being the subbands parallel, and being the electromagnetically induced transitions between them almost vertical (the wavevectors of the photons are negligible compared to the ones of the electrons), the situation is very similar to the one considered initially by Dicke. The N electrons that populate the lower conduction subband can jump vertically in the excited one and indeed can be considered as N independent dipoles. As expected 16,17 , in these systems the strength of the coupling between light and matter is proportional to the square root of the number of electrons in the quantum well.

In this paper we will not consider the coupling of intersubband transitions with light, but we will study instead their coupling with longitudinal optical phonons, considering also the role of Coulomb electron-electron interaction. Such couplings are extremely rich and, in order to limit the complexity of our investigation, we will need to determine which scattering channels are dominant and which are negligible. Given that, in general, if N electrons undergo a certain transition in a coherent way the strength of the coupling is enhanced by a factor √ N , transitions involving a macroscopic number of electrons will be treated exactly, while the others (involving only few electrons ) will be treated perturbatively (or ignored altogether).

The degree of collective enhancement of a scattering process will be evaluated looking at the number of electrons that can participate to the process given fixed amounts of transferred impulsion and energy. In Fig. 1 we show a few illustrative examples. Panel (a) represents a transition from the first to the second subband, with a negligible amount of transferred momentum. All electrons can undergo this transition approximately at the same energy. This process will thus benefit from a large superradiant enhancement factor, proportional to the square root of the total number of the electrons in the gas. Panel (b) and (c) represent two possible transitions within the first subband. In panel (b) the transferred momentum is much smaller than the Fermi momentum k F and the majority of electrons, being Pauli blocked, do not participate to the process. In panel (c) the transferred momentum is of the order of 2 k F , the minimum to allow all the electrons to undergo a transition avoiding Pauli blocking. Anyway there is a large energy spread between the different single electron transitions. Electrons on the two opposite borders of the Fermi sea, with initial momenta ± k F parallel to the transferred momentum, have initially the same energy

2 k 2 F 2m * = E F ,
where m * is the electron effective mass and E F the Fermi energy. After the transition they will end up with final momenta of k F and 3 k F , corresponding to final energies

2 k 2 F 2m * = E F and 9 2 k 2 F 2m * = 9E F
respectively. This implies that, even if the transition is not blocked, only a small fraction of the single electron transitions can be resonant at the same time. In both cases, given that only few electrons can participate to the collective transitions, the superradiant enhancement factor will be small.

IV. THEORETICAL FRAMEWORK A. Free fields

We will consider a symmetric quantum well of length L QW in a bulk of height L BK , S will be the surface of the sample. For the moment, we will limit ourselves to the case of a single quantum well, the general case of multiple quantum wells will be addressed later in this Section.

The quantum well is supposed to be doped in such a way that its Fermi level is between the first and the second conduction subbands, separated between them by the intersubband gap energy ω 12 .

We will develop our theory using a zero temperature formalism (T = 0), anyway our results will remain quantitatively accurate while the thermal population of the second subband remains negligeable. Depending on material parameters and doping level, this could imply the necessity to perform experiments in different kinds of cryogenic environments.

Electron states will be indexed by the subband index j and by the value of the in-plane wavevector k. Their wavefunctions will be given by

ψ j,k (ρ, z) = χ j (z) e ikρ √ S , j = 1, 2, (4)
where, for simplicity, we will choose χ j (z) to be real and, due to the symmetry of the quantum well, the χ j (z) have well defined and opposite symmetry. Wavefunctions in Eq. ( 4) are chosen as basis for second quantization, the creation operator for an electron in the state described by Eq. ( 4) will be denoted as c † j,k . The free Hamiltonian of electron gas in the two considered subbands thus reads

H el = j={1,2},k ω j (k)c † j,k c j,k , (5)
where

ω 1 (k) = 2 k 2
2m * and ω 2 (k) = ω 1 (k)+ω 12 . In Eq. ( 5), as well as in the rest of this paper, we will omit the electron spin index. This is justified by the fact that all interactions we consider are spin conserving. Given that we will consider only in-plane wavevector exchanges q much smaller than the typical electron wavevector k, we can make the approximation ω j (k + q) ≃ ω j (k), and introduce the operators describing intersubband transitions with a well defined and dispersionless energy ω 12 16,17

b † q = 1 √ N k c † 2,k+q c 1,k , (6)
b q = 1 √ N k c † 1,k c 2,k+q ,
where N is the number of electrons in the quantum well. The b † q operators are bosonic in the dilute regime 45 , that is, if there are n of such excitations in the system

[b q , b † q ′ ] = δ q,q ′ + O( n N ) ≃ δ q,q ′ . (7)
Using Eq. ( 6) we can rewrite the Hamiltonian of the free electron gas in Eq. ( 5) in terms of bosonic intersubband excitations

H el = q ω 12 b † q b q . (8)
In this work we are interested in the resonant case in which ω 12 is equal, or close, to the LO-phonon frequency ω LO . We can thus neglect confinement effects on the phonons and consider bulk values for their frequencies 46,[START_REF] Stroscio | Phonons in Nanostructures[END_REF] . We will thus describe LO-phonons by means of the three dimensional boson operators d q,qz

[d q,qz , d † q ′ ,q ′ z ] = δ q,q ′ δ qz ,q ′ z , (9)
indexed by their in-plane and out-of-plane wavevectors.

While we know that LO-phonon modes are confined inside the quantum well, we do not need to impose this constraint in the mode definition because, as we will see, intersubband transitions end up coupling with linear superpositions of phonon modes that are anyway confined inside the quantum well. We will consider only the case of one single longitudinal optical branch, the expansion to the case of multiple branches not presenting any fundamental difficulty. Moreover, we are interested only in phonons with small in-plane wavevectors (in order to couple with coherent intersubband excitations), we can thus ignore phonon dispersion and write the free phonon Hamiltonian as

H ph = q,qz ω LO d † q,qz d q,qz . (10)

B. Electron phonon interaction

Interaction between electrons and LO-phonons can be described using the Fröhlich Hamiltonian 3

H F r = ω LO e 2 2ǫ 0 ǫ ρ SL BK q,qz e -i(qρ+qz z) q 2 + q 2 z d † q,qz + h.c., (11)
where

1 ǫ ρ = 1 ǫ ∞ - 1 ǫ s , (12)
and ǫ s and ǫ ∞ are respectively the static and high frequency dielectric constants [START_REF] Stroscio | Phonons in Nanostructures[END_REF] . The Hamiltonian in Eq. ( 11) can be written in second quantization (neglecting incoherent intrasubband scattering 34 ) as

H F r = ω LO e 2 2ǫ 0 ǫ ρ SL BK q,qz F (q z ) q 2 + q 2 z (13) (d † q,qz + d -q,-qz )(c † 1,k c 2,k+q + c † 2,k-q c 1,k ),
where we have defined

F (q) = dzχ 1 (z)χ 2 (z)e -iqz . (14)
From Eq. ( 13) we see that, due to the three dimensional character of the LO-phonons [START_REF] Stroscio | Phonons in Nanostructures[END_REF] , each electronic transition couples to multiple phonon modes, indexed by different values of the wavevector along the growth direction. It is thus convenient to introduce second quantized operators corresponding to the particular linear superposition of phonon modes that are coupled to electronic transitions

r † q = 1 √ A qz F (q z )d † q,qz q 2 + q 2 z , r q = 1 √ A qz F (q z )d q,qz q 2 + q 2 z , (15)
whose spatial wavefunctions along the z axis is

ϕ q (z) = 1 √ AL BK qz F (q z )e iqz z q 2 + q 2 z . ( 16
)
From Eqs. ( 16) and ( 14) we see that the intersubband transitions naturally couple to phonon modes localized inside the quantum well (it is easy to verify that ϕ q (z) vanishes to the first order in q if z is outside the common support of χ 1 and χ 2 ). The normalization factor A can be fixed imposing bosonic commutation relations to the r † q operators [r q , r †

q ′ ] = 1 A qz ,q ′ z F (q z )F (-q ′ z ) (q 2 + q 2 z )(q ′2 + q ′ z 2 ) [d q,qz , d † q ′ ,q ′ z ] = δ q,q ′ L BK I(q) 2Aq , (17)
and thus

A = L BK 2 I(q) q , (18)
where we have defined

I(q) = dzdz ′ χ 1 (z)χ 2 (z)χ 2 (z ′ )χ 1 (z ′ )e -q|z-z ′ | . ( 19
)
We can thus write Hamiltonians in Eq. ( 10) and Eq. ( 13) in term of coherent r † q and b † q operators as

H ph = q ω LO r † q r q , (20)
and . Two electrons in subbands j and n, with momenta k and k ′ are scattered into subbands i and m, with momenta k + q and k ′ -q respectively.

H F r = q N 2DEG ω LO e 2 4ǫ 0 ǫ ρ I(q) q (b † q + b -q )(r q + r † -q ),
where

N 2DEG = N S , (21)
is the density of the two dimensional electron gas inside the quantum well. In order to pass from Eq. ( 10) to Eq. ( 20), we are neglecting higher order phonon modes confined inside the quantum well. This is justified by the fact that we limit ourselves to long-wavelength modes.

C. Resonant Electron-Electron interaction

In order to treat the Coulomb electron-electron interaction we start by the second quantized form of the Hamiltonian describing the Coulomb interaction 48 (see Fig. 2 (a) for a graphical representation of the interaction coefficients)

H c = 1 2 i,j,m,n=1,2 q,k,k ′ V imnj q c † i,k+q c † m,k ′ -q c n,k ′ c j,k , (22)
where

V imnj q = e 2 2ǫ 0 ǫ ∞ q dzdz ′ χ i (z)χ j (z)χ m (z ′ )χ n (z ′ )e -q|z-z ′ | , (23)
is the two dimensional Coulomb matrix element. It is important to notice that in Eq. ( 23) we used the high frequency dielectric constant ǫ ∞ instead of the static one. This is due to the fact that ǫ s includes the effect of the coupling to LO-phonons, that is already treated exactly in the Hamiltonian. Due to the symmetry of the wavefunctions, a certain number of matrix elements in Eq. ( 23) can be seen to be zero, in particular all the matrix elements with an odd number of 1 and 2 indices

V 1112 q = V 1121 q = V 1211 q = V 2111 q = 0, ( 24
)
V 2111 q = V 2212 q = V 2122 q = V 1222 q = 0.
The other elements can be evaluated as

V 1122 q = V 1212 q = V 2121 q = V 2211 q = e 2 I(q) 2ǫ 0 ǫ ∞ q , ( 25
)
V 1221 q = V 2112 q = e 2 2ǫ 0 ǫ ∞ q dzdz ′ χ 2 1 (z)χ 2 2 (z ′ )e -q|z-z ′ | , V 1111 q = e 2 2ǫ 0 ǫ ∞ q dzdz ′ χ 2 1 (z)χ 2 1 (z ′ )e -q|z-z ′ | , V 2222 q = e 2 2ǫ 0 ǫ ∞ q dzdz ′ χ 2 2 (z)χ 2 2 (z ′ )e -q|z-z ′ | ,
where I(q), defined in Eq. ( 19), is same integral we encountered studying the electron-phonon Fröhlich interaction.

The four distinct nonzero possible values of the matrix elements correspond to different kinds of scattering processes. In Fig. 2 (b) a graphical representation for each of these processes is shown.

It is important at this point to notice a major difference between the elements in the first line of Eq. ( 25) and the others. The elements in the first line (upper left subpanel in Fig. 2 (b)) represent intersubband excitations: each electron is scattered from one subband to the other. Such processes, responsible for the depolarization shift [START_REF] Nikonov | [END_REF] , describe a superradiant process in the sense defined in Sec. III, that is, at least for small values of q, a great number of electrons can coherently undergo the same transition, approximately at the same energy. This is not the case for the interactions described in the other lines of Eq. ( 25), that instead describe intrasubband excitations that, either due to Pauli blocking or to the non-flat energy dispersion, involve only few electrons (see Fig. 1).

The discussion in Sec. III thus implies that the strength of the terms in the first line of Eq. ( 25) strongly dominates over the others due to the superradiant enhancement. For this reason we have to treat them exactly in an Hamiltonian formalism, while we can limit ourselves to treat the others within a perturbative approach.

Here we will thus construct an exact, Hamiltonian approach, to treat the effect of the depolarization shift terms, neglecting the others. We will analyze later the effect of the intrasubband terms.

Le us start to rewrite the depolarization shift part of Eq. ( 22) in a more useful form

H c = q,k,k ′ e 2 I(q) 4ǫ 0 ǫ ∞ q c † 1,k+q c † 1,k ′ -q c 2,k ′ c 2,k + c † 1,k+q c † 2,k ′ -q c 1,k ′ c 2,k + c † 2,k+q c † 1,k ′ -q c 2,k ′ c 1,k + c † 2,k+q c † 2,k ′ -q c 1,k ′ c 1,k = q,k,k ′ e 2 I(q) 4ǫ 0 ǫ ∞ q c † 1,k+q c 2,k c † 1,k ′ -q c 2,k ′ + c † 1,k+q c 2,k c † 2,k ′ -q c 1,k ′ + c † 2,k+q c 1,k c † 1,k ′ -q c 2,k ′ + c † 2,k+q c 1,k c † 2,k ′ -q c 1,k ′ + q N e 2 I(q) 4ǫ 0 ǫ ∞ q . ( 26
)
We see from Eq. ( 26) that, thanks to its collective, superradiant nature, the depolarization shift can be naturally written in terms of the bosonic intersubband excitations defined in Eq. ( 6) as

H c = q N 2DEG e 2 4ǫ 0 ǫ ∞ I(q) q (b † q + b -q )(b † -q + b q ), (27)
where we have neglected the last constant term, that simply shifts the ground state energy.

D. Residual Electron-Electron interaction

We showed how it is possible to treat exactly the depolarization shift effect in a bosonic excitation formalism. It remains to study the effect of the residual Coulomb contributions due to the intrasubband terms in the last three lines of Eq. ( 25) (schematized in the last three panels of Fig. 2 (b)).

An important result due to Lee and Galbraith 49,50 , is that such intrasubband terms do not contribute to the screening of the intersubband ones at the level of the random phase approximation (RPA). This can be seen writing the Dyson equation for the dynamically screened Coulomb potential 51 V q (ω)

V imnj q (ω) = V imnj q + rs V irsj q Π sr q (ω)V smnr q (ω), (28)
where Π sr (q, ω) is the RPA polarization function. In the case of an intersubband contribution (e.g. V 1122 q ), Eqs. ( 24) and ( 25) imply that

V 1122 q (ω) = V 1122 q + rs V 1rs2 q Π sr q (ω)V s12r q (ω) (29)
= V 1122 q + r =s V 1rs2 q Π sr q (ω)V s12r q (ω) = V 1122 q + V 1122 q (Π 12 q (ω) + Π 21 q (ω))V 1122 q (ω).
We have thus

V 1122 q (ω) = V 1122 q 1 -V 1122 q (Π 12 q (ω) + Π 21 q (ω)) , (30)
from which we see that the intrasubband Coulomb terms (V 1111 q ,V 2222 q , V 1221 q and V 2112 q ) do not intervene in the renormalization of the intersubband terms.

An analogous reasoning can be done for the phononelectron interaction. Calling M q,qz and M q,qz (ω) the bare and screened version of the potential defined in Eq. ( 13), we have the Dyson equation

M q,qz (ω) = M q,qz + rs V 1mn2 q Π sr q (ω)M q,qz (ω) (31)
= M q,qz + V 1122 q (Π 12 q (ω) + Π 21 q (ω))M q,qz (ω), and we have thus the formula for the screened potential M q,qz (ω) = M q,qz 1 -V 1122 q (Π 12 q (ω) + Π 21 q (ω))

.

Being the RPA screening only due to terms that are exactly treated in the Hamiltonian, we can thus neglect the screening due to the two dimensional electron gas.

E. Hopfield-Bogoliubov Hamiltonian

Putting together Eqs. ( 8), ( 21) and ( 27) we arrive to the full Hamiltonian for the intersubband transitions-LOphonons system

H = q ω 12 b † q b q + ω LO r † q r q (33) + N 2DEG ω LO e 2 4ǫ 0 ǫ ρ I(q) q (b † q + b -q )(r † -q + r q ) +N 2DEG e 2 4ǫ 0 ǫ ∞ I(q) q (b † q + b -q )(b † -q + b q ).
The Hamiltonian in Eq. ( 33) can be rewritten in a more compact form by introducing the intersubband transitions-LO-phonons coupling constant Ω and the depolarization shift constant D

Ω = N 2DEG ω LO e 2 4ǫ 0 ǫ ρ I(q) q , (34)
D = N 2DEG e 2 4ǫ 0 ǫ ∞ I(q) q ,
where we have dropped the dependences over the wavevector as we are interested in the long wavelength limit (from Eq. ( 19) we can verify that lim q→0 I(q) q tends to a constant value).

Using Eq. ( 34), Eq. ( 33) can be written as

H = q ω 12 b † q b q + ω LO r † q r q + Ω(b † q + b -q )(r † -q + r q ) +D(b † q + b -q )(b † -q + b q ), (35)
that can be cast in matrix form as

H = 2 q v † q η M q vq , (36)
where the column vector of operators vq is defined as

vq = [b q , r q , b † -q , r † -q ] T , (37)
η is the diagonal metric

η = diag[1, 1, -1, -1], (38)
and the Hopfield-Bogoliubov 52 matrix M q is defined as

M q =    ω 12 + 2D Ω 2D Ω Ω ω LO Ω 0 -2D -Ω -ω 12 -2D -Ω -Ω 0 -Ω -ω LO    . ( 39
)
Diagonalizing the matrix in Eq. (39) will yield the frequencies of the normal modes of the system ω ± , that are usually called polarons 32,33 . In our case we will name them more properly intersubband polarons, because the electronic part of the mixed excitation is given by an intersubband transition.

It is interesting to notice that, from Eq. ( 34), we can write the Coulomb coefficient D as

D = Ω 2 ω LO ǫ ρ ǫ ∞ ≥ Ω 2 ω LO . ( 40
)
As it has recently been shown in Ref. [53], Eq. ( 40) implies that it is impossible to observe a Dicke phase transition in the system we are considering, regardless of the strength of the coupling.

F. Multiple Quantum Wells

Until now we considered the case of a single quantum well. This choice has been motivated by the fact that, as we will show, the presence of multiple quantum wells does not modify our results.

Given that we are considering rather large quantum wells (in order for the transition to be resonant with the LO-phonon mode), the optical phonon spectrum is not modified [START_REF] Stroscio | Phonons in Nanostructures[END_REF] and the optical phonon modes we consider are confined in each quantum well. This is a rather important difference between the intersubband polaron case we consider in this paper and the physics of intersubband polaritons. For intersubband polaritons, the electromagnetic mode coupled to the intersubband transitions extends over all the quantum wells in the structure. It thus couples to all the electrons, regardless of the quantum well they are in. This means that the only meaningful parameter for intersubband polaritons is the total density of electrons, and the light-matter coupling thus scales as n QW N 2DEG , where n QW is the number of quantum wells inside the microcavity.

In the present case instead, being the phonon modes confined inside each quantum well, electrons in different quantum wells are completely decoupled. This can also be inferred from the coupling integral in Eq. ( 23). This integral does vanish, at least in the long wavelength limit (first order in q), if the wavefunctions for the two integration variables z and z ′ do not have a common support, i.e., if the two interacting electrons are in different quantum wells.

This means that, contrary to the intersubband polariton case, the intersubband polaron interaction scales only as √ N 2DEG and growing multiple quantum wells in the same sample will not increase the coupling.

V. RESULTS

In order to obtain some numerical predictions from Hamiltonian in Eq. ( 35), we need to fix a few parameters concerning the material and the quantum well.

For sake of simplicity we will consider the quantum well to be correctly approximated by a rectangular, infinite potential well of length L QW . We thus have

ω 12 = 3 2 π 2 2m * L 2 QW , (41)
and the electronic and phononic modes profiles are given by

χ 1 (z) = 2 L QW sin( πz L QW ), (42)
χ 2 (z) = 2 L QW sin( 2πz L QW ), ϕ 0 (z) = 16 5L QW sin 3 ( πz L QW ), (43)
inside the quantum well and zero outside. As explained in Sec. IV, we see here explicitly that the intersubband transitions couple to a linear superposition of phonon modes that is localized inside the quantum well (the cubic sinus in the third line of Eq. ( 42) comes from the integral of the two first two, as can be verified performing the integral in Eq. ( 16)). Inserting Eq. ( 42) into Eq. ( 19) and performing the integral we have lim q→0 I(q) → 10 9π 2 qL QW .

In Fig. 3 we plot the normalized coupling Ω ωLO as a function of the two dimensional electron gas, for a GaAs quantum well. In the inset of Fig. 3 we instead present a comparison of the values of Ω ωLO , at room temperature, for different semiconductors of the III-V and II-VI groups [START_REF] Adachi | Parameter for the III-V group were taken from: www.ioffe.ru/SVA/NSM/Semicond. For ZnSe we used data[END_REF] , as a function of the respective LO-phonon energies, for a reference doping N 2DEG = 10 12 cm -2 .

In Fig. 4 there is a plot of the intersubband polaron frequencies ω ± as a function of the intersubband frequency ω 12 , in GaAs, for N 2DEG = 10 12 cm -2 . Notice that, due to the effect of Coulomb interaction, the resonant anticrossing is not at ω 12 = ω LO but at a lower frequency. In the inset of the same figure we plot the same quantity as a function of the electron density. The quantum well length L QW has been chosen in this case to have the two uncoupled modes at resonance (ω 12 = ω LO , that is

L QW ≃ 23nm).
It is clear from the figure that intersubband polarons are not only strongly coupled, having coupling constants much larger than their linewidth (usual linewidths being not bigger than a few meV), but they are indeed in the ultrastrong coupling regime, with values of the renormalized coupling Ω ωLO comparable or larger than the best ones reported in the literature 21 . For physically realizable levels of doping, coupling values of a few tenths of the bare frequency of the excitation ω LO are predicted in GaAs, and it seems that values much larger can be obtained using more polar materials. The reason of such large coupling can be found in the superradiant nature of intersubband excitations and in the natural confinement of the phonons inside the quantum well, that gives an extremely small mode volume, when compared with what can be obtained with photonic microcavities.

The consequences of our results can be multiple, both for fundamental and applied research. On the fundamental side, intersubband polarons could become a new laboratory to test quantum vacuum physics, typical of the ultrastrong coupling regime 28 . On the applied side our theory can be naturally exploited in the study of quantum cascade lasers working in or near the Restrahlen band. It can, for example, help explaining the anticrossing observed in Ref. 36, near the LO-phonon frequency. Moreover the capability to strongly modify the LO-phonon spectrum could have an impact on the performances of optoelectronic devices, as the electron-LO-phonon scattering rate determines the lifetime of carriers in excited subbands 34 .

VI. CONCLUSIONS

In this paper we have developed a microscopic theory of intersubband polarons, mixed excitations resulting from the coupling between intersubband transitions in doped quantum wells and LO-phonons. We took into account the electron-electron Coulomb interaction and we were able to treat exactly the resulting depolarization shift. We proved that intersubband polarons can be in the ultrastrong coupling regime, reaching extremely high values of the coupling constant. We critically discussed the relevance of our result both for fundamental and applied research.

FIG. 1

 1 FIG. 1: a) An example of collective transition: an intersubband transition with small transferred momentum. b) An example of non-collective transition: an intrasubband transition with small transferred momentum in which most of the single particle transitions are Pauli blocked. c) An example of noncollective transition: an intrasubband transition with large transferred momentum. The extrema of the energy spread of single particle transitions are shown explicitly.

FIG. 2

 2 FIG. 2: a) Graphical representation of the scattering channel V imnj q

FIG. 3 :FIG. 4 :

 34 FIG. 3: Normalized coupling Ωω LO in GaAs as a function of the doping density N2DEG. Inset: the same quantity as a function of the LO-phonon frequency ωLO for different materials, for N2DEG = 10 12 cm -2 .

VII. ACKNOWLEDGMENTS

We would like to thanks D. Hagenmüller, M. Zaluzny, P. Nataf, L. Nguyen, J. Restrepo, C. Sirtori and Y. Todorov for useful discussions and comments.

Appendix: Comparison with dielectric function theory

As explained in Sec. II, the dispersion of intersubband polarons can be calculated by solving Eq. ( 3) for the dielectric function of the coupled quantum well-lattice system. In order to test our theory we will compare this approach, following what done in Ref. [41], with the microscopic one we developed in this manuscript.

The total dielectric function of the system is given by 21,41

where

is the plasma frequency of the two dimensional electron gas and d 12 is the intersubband dipole

The equation

thus reads

Note that Eq. ( 45) refers to the dipolar function along the z direction. In order to recover the same result from our microscopic approach, we will thus have to consider only q = 0 phonons. Moreover, the use of a simple dielectric function theory, requires to have an homogeneous medium. This is equivalent to consider only the mode with q z → 0, that is, from Eqs. ( 14) and ( 47)

Following exactly the same procedure done in Sec. IV, but with the F (q) defined in Eq. ( 50) and considering only the q z → 0 mode, we get

and thus, from Eq. ( 40)

In order to obtain the polaronic eigenfrequencies we have to diagonalize the matrix in Eq. ( 39) using the coupling coefficients for the homogeneous limit defined in Eqs. ( 51) and (52). We thus obtain the secular equation

that, using Eqs. ( 52) and ( 46) can be put into the form

Equating the coefficients of Eqs. ( 49) and ( 54), we obtain

that is the well known Lyddane-Sachs-Teller relation [START_REF] Stroscio | Phonons in Nanostructures[END_REF] .

We have thus proved that the homogeneous version of our theory gives the same results as the homogeneous dielectric function approach.

It is anyway important to notice that the homogeneous limit in not exact, as a quantum well is, by definition, non-homogeneous. Ignoring the higher q z modes leads to underestimate the intersubband dipole of a factor roughly equal to √ 2.


BERNARD CARON

Review of : Newman, Paul. The Hausa Language: An Encyclopedic Reference Grammar.

New Haven, London : Yale University Press, 2000, xxxix + 760 p. Coll. Yale Language Series.

La parution de la grammaire haoussa de Paul Newman (760 pages !) est un événement considérable à l'instar de la publication en 1934 du dictionnaire haoussa de Bargery (Bargery, 1934) (56 000 entrés).

Si les études haoussa ont un histoire d'un siècle et demi, et ont produit un important corpus de publications (la bibliographie donnée en fin d'ouvrage comporte 305 titres), seuls deux grammaires méritaient jusqu'alors d'être mentionnées : (Abraham, 1959) pour la période prélinguistique et [START_REF] Wolff | Referenzgrammatik des Hausa[END_REF] pour un premier travail scientifique, [START_REF] Caron | Le haoussa de l'Ader[END_REF] ne concernant qu'un dialecte marginal.

L'ouvrage de P.N. se présente comme une grammaire encyclopédique : une suite de 80 articles classés par ordre alphabétique, de "Abstract Noun (Derived)" à "Writing System (Orthography)", précédés d'une introduction présentant un rapide historique des études haoussa. Une table des matières détaillé et un index fourni sont deux outils essentiels pour la consultation de cette encyclopédie.

Son ambition affichée est d'être un ouvrage de référence qui fait la somme et présente l'ensemble des connaissances actuelles sur le haoussa. Le corps de chaque article fait le tour d'un phénomène grammatical d'un point de vue factuel (morphologie, propriétés distributionnelles, caractérisation sémantique). Des commentaires de points particuliers intéressant les spécialistes du haoussa et des langues tchadiques sont relégués dans des notes : (i) notes analytiques qui mettent les données en perspective par rapport aux débats en cours en linguistique haoussa ; (ii) notes dialectales ; (iii) notes historiques sur le haoussa et comparaison avec les autres langues tchadiques.

Paul Newman fait appel à la tradition haoussaïste : Parsons en premier, dont les travaux sur la morphologie verbale et nominale (les degrés verbaux (verb grades), les Noms Abstraits de Qualité Sensible et leurs dérivés) ont fondé les études modernes du haoussa, mais aussi les travaux de grands haoussaïstes tels que Lukas, Gouffé, Zima, Wolff, Jaggar. On remarque également l'importance des travaux des linguistes haoussaphones (Bagari et Galadanci bien entendu, mais également la jeune génération des linguistes nigérians et nigériens). Et, bien entendu, on retrouve les nombreux travaux de P.N. lui-même sur la phonologie, la morphologie, la formation des pluriels, la dérivation, le système verbal et l'histoire du haoussa et des langues tchadiques.

La structure choisie pour cette grammaire reflète la position empiriste de P.N. dans ses travaux linguistiques, en faisant l'économie d'une introduction générale théorique sur le langage et la langue haoussa. La crainte de ne pas pouvoir y trouver l'information cherchée est levée par la possibilité d'utiliser un index généralement très bien fait et une table des matières détaillée. Par exemple, si le terme "déictique" n'y figure pas, on trouvera les informations sous celui de "démonstratifs". L'absence d'entrée "phrases complexes" obligera cependant à retrouver les articles éparpillés : "concessives, complementation, temporal clauses, conditionnal, purpose clauses, conjoined sentences, conjunctions, coordination, relative clauses", etc.

Pour la notation du haoussa, P.N. reprend le système mis au point pour le dictionnaire de Roxana Ma Newman [START_REF] Newman | An English-Hausa Dictionary[END_REF], la longueur vocalique y étant notée par un macron suscrit, et abandonne la notation de [START_REF] Newman | Modern Hausa-English dictionary (Sabon amus na Hausa zuwa Turanci)[END_REF] qui notait la voyelle brève par une cédille. Le ton modulé (tombant) n'apparaissant que sur une syllabe lourde, P.N. choisit de ne pas noter la longueur de la voyelle en syllabe ouverte, considérant que cette notation est redondante une fois le ton noté par un accent circonflexe.

On remarquera quelques innovations terminologiques dans la présentation du système de conjugaison (chapitre 70, "Temps, Aspect, Mode"), concernant en particulier le complétif, le prétérit, le potentiel et l'allatif. P.N. traite, dans le système des conjugaisons, du rhétorique (rhetorical), généralement passé sous silence, et intègre dans ces conjugaison des constructions habituellement laissées hors-système : l'allatif (construction en  + forme brève du pronom, indiquant un mouvement en direction de) et les constructions progressives (continuous) où le prédicat est un adverbe ou un syntagme prépositionnel. De même, P.N. introduit les termes d'infinitif et de gérondif pour caractériser respectivement les emplois de formes verbales finies en contexte nominal et « les noms verbaux fonctionnant comme des constituants dynamiques dans des contextes non-finis ».

Enfin, en ce qui concerne la présentation, bien que le corps très petit choisi pour imprimer cette grammaire impose un effort au lecteur, on apprécie le soin apporté à la mise en page extrêmement claire et lisible. Après avoir travaillé régulièrement avec cette grammaire pendant un mois, on signalera tout au plus une erreur p. 565, à la fin de la l. 25, où figure le début d'un exemple haoussa qui aurait dû être rejeté à la ligne suivante. C'est dire le soin qui a été apporté à la relecture du manuscrit.

Pour conclure, nous dirons que cette grammaire du haoussa est un formidable outil de travail qui non seulement fait la somme de connaissances jusqu'alors éparpillées dans des centaines de publications, mais apporte également des informations nouvelles dans certains domaines (ex. le chapitre 42 sur les jeux linguistiques : calembours et langages codés). C'est une grammaire qui fera date non seulement dans les études haoussa, mais dans la linguistique africaine, et dont l'exhaustivité, la profondeur et la qualité seront difficile à égaler.

  Références Abraham, R.C. The Language of the Hausa People. London :, 1959. Bargery, G.P. A Hausa-English Dictionary and English-Hausa Vocabulary. London : Oxford University Press, 1934, liv+1226 p.