... physics:1.1
The charge of the electron $ e$, the vacuum permittivity  $ \epsilon_0$, the reduced Planck constant $ \hbar$ and the speed of light in vacuum $ c$. 
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... nature.1.2
In quantum dots. I use here a terminology that I shall define more precisely later. The main field where the terminology of polaritons apply, is currently that of quantum wells, where their quantum character is more disputable. However the term is gaining wide acceptance to describe superposition of light and matter, specifically in a quantum context. It is now even used in the atomic literature. 
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... SE.1.3
Even the famous exception to this rule, the ``collapse'' of the wavefunction, does not help to resolve the issue. 
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... divergences1.4
They would resurface much later when his work would be scrutinized by early quantum field theorists such as Low (1952) or quantum opticians such as Louisell (1973). 
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... famous.1.5
The historical background of this important theory can be found in the Interview of Weisskopf by T.S. Kuhn and J.L. Heilbron on July 10, 1965, Niels Bohr Library & Archives, American Institute of Physics. Weisskopf's humility brought him to conclude that he was the first author of the paper with Wigner only for reasons of alphabetical order. Sadly, the theory is now more frequently referred to as ``Wigner-Weisskopf'' theory. 
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... photons.1.6
Here we must quote again Low (1952) who, in his three complaints against the Weisskopf-Wigner theory, starts with the problem of the excitation scheme that is approximated as a mere initial condition. The divergence I mentioned before was his second complaint. 
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... elegantly.1.7
Stroud remembers the whole sequence of events in ``The Jaynes-Franken Bet'' §30 of ``A Jewel In The Crown'', Meliora Press, (2004). Jaynes' efforts have naturally been pursued long time after him, see for instance the attempts by Barut & (1996), whose claims have been, naturally, further disputed. I will leave aside further questions on to which extent is the full-field quantization necessary, holding to the mainstream view that it is and that both the Lamb shift and SE are two fundamentally quantum phenomena. 
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... enhancement.1.8
Of sodium atoms, with an increase to  $ 8\times10^4$s$ ^{-1}$ from the free-space value of 150s$ ^{-1}$. 
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... achievements.1.9
This much quoted paper is nowadays of interest mainly for its historical content. Better reviews for the modern reader are given by, among others, Raimond et al. (2001), Mabuchi & (2002), Vahala (2003), etc... There are also many excellent textbooks now available. 
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... cQED.1.10
The brevity and clarity of the full abstract of their text is exemplar and will fit comfortably in this footnote: ``The spontaneous-emission spectrum of an atom in an ideal cavity is calculated.'' This work was part of the Ph. D. thesis of José Javier Sánchez Mondragón. 
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... element,1.11
``Active'' in this context means the part that provides the electronic excitations, the excitons. A cavity without QWs or QDs between the two Bragg mirrors is an empty cavity, a passive element described by classical linear optics. 
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... field.1.12
Excellent reviews have been written on that topic, for instance by Skolnick et al. (1998), Khitrova et al. (1999) or, for the most recent developments, by Kavokin (2007); see also the textbook by Kavokin & (2003) for a dedicated coverage of QWs polaritons and the collection of texts edited by Deveaud (2007) for the views of some of the leading experts of these questions. Kavokin et al. (2007)'s textbook is addressing these questions in a larger context and will be a useful companion to this introduction for bridging between dimensions. 
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... heterostructure,1.13
There has been, prior to Weisbuch's line-splitting, reports of Purcell effect in planar cavities (e.g., fromYokoyama et al. (1990) and Björk et al. (1991)), but I will not discuss these because they are of interests in the context of 2D polaritons only. We shall focus on the QD case later. 
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... wavevectors.1.14
Often, the $ k=0$ mode is considered separately from the higher modes and some arguments of 0D cQED reappear in this particular context. Here neither, however, this polariton ground-state can be put on a par with the single mode of a real 0D system. 
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...ciuti04a1.15
Ciuti et al. (2000) had already pioneered the theory of OPO in 2D microcavities, that has been so far the system of choice for tracking polariton superfluidity, in a theoretical context that he also put to the front with Carusotto (Carusotto & (2004)). 
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... structures,1.16
Of the order of $ \sim 3\times10^4$. Same figures are also reported by Muller et al. (2006), and more recently, of  $ 1.65\times10^5$ for radius size of $ 4\mu$m by Reitzenstein et al. (2007). 
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... semiconductors:1.17
Yablonovitch (2001) himself dub them ``semiconductors of light'' in a personal recount of the early experimental efforts to the Scientific American, and is generally keen to relate the two systems in his academic discussions. Yablanovitch is one of the surest future Nobel laureate one can envision for the near future. 
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...nosich07a.1.18
Oddly, the SC issue is addressed but no literature is quoted in this review. 
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... crystal.1.19
``L3'' means that three holes in the PC patterns have been skipped to produce the defect region that serves as the microcavity. The PC in Fig. 1.10 is therefore L1. 
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... detail.1.20
In their case, they suspect the effect of a charged carrier; see their manuscript for more precisions. 
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... itself.1.21
Which is ultimately quantum anyway; if the photons are coming from the sun, for instance, they all originate independently from the spontaneous emission of an atom, or to much lower probability, from stimulated emission. If this seems like a moot statement, let us remember from the atomic QED case the controversy that arise anytime that field quantization is deduced. Did not Lamb Jr. (1995) himself support the view that ``there is no such thing as a photon''? Such controversies can be settled completely only with a direct, explicit demonstration of field quantization, rather than one of its many possible logical consequence. 
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... apply.1.22
Laussy et al. (2006) have considered QDs that would exhibit an intermediate case between bosons and fermions, but their approach does not lend itself to an exact computation of the PL lines. They have resorted to a Lorentzian approximation and followed a manifold method that I explain later and find also useful, although limited. 
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... regime2.1
$ a_\mathrm{B}$ is the exciton Bohr radius, $ D$ is the dimension of the system and $ d$ the density of excitons. 
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... particles,2.2
$ \vert\bra{f}a\ket{i}\vert^2=\bra{i}\ud{a}\ket{f}\bra{f}a\ket{i}$ and summing over $ f$ gives the result. 
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... distribution.2.3
Discovered by S. D. Poisson and published in 1838 in his work ``Research on the Probability of Judgments in Criminal and Civil Matters''. 
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... projectors,2.4
A projector is an operator such as $ \ket{\Psi}\bra{\Psi}$ that, when applied to $ \ket{\xi}$, returns the state  $ \ket{\Psi}$ with its weight in $ \ket{\xi}$. This is zero if $ \ket{\xi}$ and  $ \ket{\Psi}$ are orthogonal. 
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... mode2.5
The fact that a single mode of thermal light is first-order coherent was not clearly understood before Glauber. 
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...representation,2.6
The $ P$ representation is the counterpart of the Wigner representation for the density matrix, in normal ordering of the characteristic function. This is therefore an object of great importance in quantum optics, given the role of normal order with respect to detection, as I have discussed previously. It can also be understood as the weight of the state in the basis of coherent states, $ \rho=\int P\ket{\alpha}\bra{\alpha}d\alpha$. This representation has been advocated by Glauber in his foundations of quantum optics and by Sudarshan (1963), although with an antagonist interpretation of its meaning. 
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... operators:2.7
See footnote 4. 
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... level:2.8
The operator $ \ud{a}$ had implicit this possibility in its expression as $ \ud{a}=\sum_{n}\sqrt{n+1}\ket{n+1}\bra{n}$. 
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... rules2.9
I use the notation $ [A,B]_+\equiv AB+BA$. Another common notation is with curly brackets. 
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... Approximation2.10
The Rotating Wave Approximation in this context allows to write the coupling as Eq. (2.54), i.e., neglecting the energy non-conserving terms $ ab$ and $ \ud{a}\ud{b}$. 
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... reads:2.11
Note that this formula is, strictly speaking, an arcane mathematical result in the theory of stochastic processes. There, $ S(\omega)$ is a measurement of the strength of the fluctuations of the Fourier component at frequency $ \omega$. It has no strict connection with a physical signal, as both infinite negative and positive times are required for its demonstration, which violates causality among other things. A rigorous derivation of a physical optical spectrum based on the photon detection arguments introduced in Section 2.1, has been given by Eberly & Wodkiewicz (1977) and used in the seminal investigations of luminescence lines of strongly-coupled systems. We already explained why we still prefer the mathematical, ideal limit, over the physical spectrum. 
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...sec:publications.3.1
The corresponding entries in the Bibliography appear as Laussy et al. (2008b,2009,2008a). 
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... character.3.2
An example of a fundamental complication is that the emission is enhanced in the cavity mode and suppressed otherwise, and the exciton lifetime is typically much longer, so the exciton direct emission is much weaker. An example of a technical complication is that the exciton detection should be made at some angle as compared to the direction of the cavity emission. Practically, a lot of samples are grown on the same substrate and, therefore, both the substrates and other samples hinder the lateral access to one given sample, whereas they are all equally accessible from above. 
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... functions3.3
This was first shown by C. Hermite in 1858 in his publication "Sulla risoluzione delle equazioni del quinto grado." Annali di math. pura ed appl. 1, 256-259. 
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... experiment3.4
The authors place it at $ 180\mu$eV but from a Lorentzian fit of the 5K curve in the assumption that the system is not strongly-coupled here, where our model shows this to be a poor approximation. 
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... correlators.4.1
In some form, such symmetry requirements are responsible for the exchange interaction. 
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...).4.2
That is, pictorially, when the structure of levels and transitions in Fig. 6.1 (which also represents our system if we ignore the cavity mode) are rotated by 180º or, mathematically, when the rising and lowering operators are inverted. We already pointed out this symmetry in Sec. 2.4 when we introduced the 2LS master equation. 
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...sec:publications.5.1
The corresponding entries in the Bibliography appear as del Valle et al. (2009,2008) and Laussy &Valle (2009). 
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... times.5.2
Cf. Eq. (2.101) for the HO alone and Eq. (3.4b) when in a cavity. 
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...#tex2html_wrap_inline38773#5.3
In this Section, the spectra is normalized to the mean number  $ \langle n_a\rangle $ instead of one. The reason is to better appreciate the spectral asymmetries with detuning that manifest in the lineshapes but also in the intensity. 
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...:5.4
The function $ \mathrm{sgn}(x)$ is defined as 0 for $ x=0$ and $ x/\vert x\vert$ otherwise. 
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...sec:publications,6.1
The corresponding entries in the Bibliography appear as del Valle, Laussy, Troiani & Tejedor (2007a); del Valle, Troiani & (2007); del Valle, Laussy & (2007); del Valle, Laussy, Troiani & Tejedor (2007b). 
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... Poissonian-like6.2
Numerically, $ g^{(2)}=1.42$ for 2PR and $ 1$ for 1PR. 
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...#tex2html_wrap_inline41657#6.3
We analyze again the not normalized spectra in order to better appreciate the combined effect in lineshape and intensity that the 2PR and 1PR produce. 
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... especificidades7.1
Ver en la Fig. 1.15 el esquema de los Hamiltonianos usados en cada Capítulo. 
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... emisión7.2
El espectro de un sistema es la probabilidad de emitir fotones dada la frecuencia. 
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... partículas7.3
La reflexión en las paredes de la cavidad no es perfecta, permitiendo la pérdida de fotones de vez en cuando. Esto provoca una ``interupción'' del acoplo con los excitones aunque también nos da la posibilidad de estudiar lo que sucede dentro de la cavidad. 
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...decoherencia7.4
Como decoherencia se conocen los efectos de la ``interrupción'' del acoplo luz-materia por el contacto incontrolado con el exterior, como el escape de fotones o la inyección de un flujo de partículas. 
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... buenos7.5
Se consideran sistemas buenos para cQDE aquellos que dejan escapar pocos fotones, producen interacción fuerte y son limpios en elementos ajenos. 
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... Rabi7.6
Frecuencia a la que oscilan los polaritones. 
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... sistema7.7
Un sistema es de naturaleza cuántica o se encuentra en un régimen cuántico cuando se ve afectado por la presencia o ausencia de una sola partícula. 
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...entanglement7.8
Entanglement es un tipo de correlación que se establece entre elementos de un sistema cuántico, provocando una dependencia mutua sin análogo en el mundo clásico. 
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... specificities,7.9
See Fig. 1.15 for the schema of Hamiltonian models used in each Chapter. 
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