PACS numbers: 05.30.-d,67.85.-d,67.85.Pq
I. INTRODUCTION
The recent progresses on the experimental control of- fered by ultra-cold atoms setups [1–4] provides new plat- forms for the theoretical and experimental activity on one dimensional (1D) **strongly** **correlated** **quantum** **systems**. In the spirit of **quantum** simulations [5], they now allow to study many important phenomena like for instance superfluidity, integrable models [4], **quantum** phase tran- sitions and **quantum** magnetism [6]. In the latter case, multi-component **strongly** interacting **quantum** particles, living in continuous space, appear to be a promising al- ternative to lattice **systems** where magnetic interaction parameters are hardly tunable [7]. Indeed, in the low- energy regime, the κ internal degrees of freedom (in- teracting via highly-symmetric terms) can be mapped onto an effective SU (κ) spin chain subjected to a Suther- land Hamiltonian [8, 9], which reduces to the most tra- ditional SU (2) Heisenberg model in the case of two com- ponents [10–12]. Such **systems** are therefore currently at the focus of an intense experimental [7, 13, 14] and the- oretical [15–18] activity. Inevitably, a compelling ques- tion arises about probing the magnetic-like properties of these gases, via experimental techniques as elementary as possible, besides in-situ spin-resolved imaging of the cloud. Our purpose is here to endorse the use of the stan-

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4.1.1 Theoretical methods
On the theoretical side several methods are in principle available, but each has advantages and limitations. Among the most successful are techniques related to the density functional theory such as the local density approximation (LDA), which can make allowance for local Coulomb interaction in the so-called LDA+U method [ 92 ]. The latter has led to results showing good agreement with experiments [ 93 ]. Regarding model Hamiltonians such as the Hubbard model, dynamical mean eld theory (DMFT) can be applied to interfaces or heterostructures involving a large number of atomic layers [ 94 ], while its cluster extensions allow, to a certain extent, to take into account spatial correlations. These spatial correlations are described better by way of **Quantum** Monte Carlo techniques, but a major obstacle is the number of sites available to sim- ulations. While the single band Hubbard model in two-dimensions has been extensively studied by means of QMC methods, little has been done in the eld of heterostructures. Recent progress in both the numerical eciency of algorithms and the computational power of workstations has made it possible to deal with the case of multi-orbital 2D **systems**, such as the periodic An- derson model. In principle, such a plane involving two electronic orbitals can equivalently describe the coupling of two adjacent planes in which only one orbital is taken into account. It is then tantalizing to go further and try to understand the properties of thicker **systems** involving more than two atomic planes. Even though the accessible geometries are restricted, by numerical limitations, to a few layers, the perspectives oered by these models to under- stand better the electronic properties at interfaces or other heterostructures that are already accessible experimentally, make this eld very exciting. Be- cause theoretical models allow to explore systematically vast regions of the phase diagrams, such studies could potentially lead to the design of thin lms with novel properties.

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The field of electronic correlations is largely dominated by applications to **strongly** **correlated** materials such as high-T c superconductors or heavy fermions. As a re- sult, the large effort made by the community to build new numerical techniques to address correlations aims chiefly at reaching **strongly** **correlated** regimes for sys- tems whose one-body dynamics is rather simple (the archetype of these **systems** being the Hubbard model). There are, however, many situations where the correla- tions are either small or moderate, yet their interplay with one-body dynamics might be very interesting. Ex- amples include, for instance, the zero-bias anomaly in dis- ordered **systems** 1 , the Fermi- edge singularity in a quan- tum dot 2 , a Kondo impurity embedded in an electronic interferometer 3 and possibly the 0.7 anomaly in a quan- tum point contact 4 . While for a few situations, e.g. zero- dimensional (Kondo effects) and one-dimensional (Lut- tinger liquids) **systems** there exist exact analytical and numerical techniques 5,6 , the vast majority of these prob- lems remains elusive to theoretical approaches. The aim of this article is to design a technique that could ad- dress moderate interactions for a large variety of out-of- equilibrium situations.

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tems in presence of both driving and dissipation mechanisms represent natural platforms to understand and explore such dynamical phases. A well know example is provided by exciton-polariton condensates where superfluidity has an order parameter oscillating in time [ 30 , 31 , 70 , 122 ]. Yet the oscillating condensate is successfully described by semi- classical theories such as driven-dissipative Gross-Pitaevski equations that are valid in the regime of weak interactions. More recently the attention has shifted toward **strongly** **correlated** **quantum** lattice models with drive and dissipation, where several works have re- vealed the existence of limit cycles, i.e. non-stationary solutions of the **quantum** dynamics for a macroscopic order parameter, at least at the mean field level [ 16 , 17 , 20 , 93 , 187 – 189 ]. In this chapter we focus on a paradigmatic model of driven-dissipative interacting bosons on a lattice, which is directly relevant for the upcoming generation of circuit QED arrays experiments [ 61 , 62 , 86 ]. We argue that a dynamical susceptibility of such an open **quantum** many body system, which in thermal equilibrium is finite and small since non-zero frequency modes are typically damped by interactions, can display a genuine singularity at finite frequency, as a result of strong interactions and non-equilibrium effects. The critical frequency is non-trivial and set by a competition of interactions with drive and dissipation. Eventually, the system undergoes a dynamical phase transition where the order parameter emerges with a finite oscillation frequency and in the broken symmetry phase oscillates in time without damping, thus breaking the continuous time-translational symmetry. This stationary-state instability is controlled by both dissipative and coherent couplings, in particular by the ratio between hopping and local interaction, thus providing the **strongly** **correlated** analogue of weak coupling non-equilibrium bosons condensation.

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source. As a result, instead of a thermal equilibrium with a well defined number of particles, the system can only reach a steady-state, in which losses are exactly compensated by the external driving. As we shall see shortly, this also has important consequences.
The first studies on photon hydrodynamics were motivated by the close analogy between the Gross-Pitaevskii equation describing atoms in a condensate and the equation of the electromagnetic field in a dielectric nonlinear medium. [ 9 , 10 ]. Since then, “**quantum** fluids of light” in various **systems** and setups have attracted a great deal of interest [ 11 ]. Concerning the genesis of this PhD thesis, some of the most influential results were those obtained in semiconductor microcavities. In these structures (which will be described in more details in the next chapter), it is possible, due to the confinement of the electromagnetic field, to reach so strong a coupling between light and matter that the two can no longer be distinguished. They form instead hybrid quasiparticles called polaritons. In the last decade, tremendous experimental progress in creating and controlling polariton fluids in microcavities has been made, which lead to important achievements. In particular, observation of polariton Bose-Einstein condensation was reported in 2006 [ 12 ]. Superfluidity of polartions in planar microcavities was predicted in 2004 [ 13 ] and observed experimentally for the first time in 2009 [ 14 ]. Other aspects of **quantum** fluids, such as the presence of quantized vortices or other topological excitations has also been investigated [ 15 , 16 , 17 , 18 , 19 ]. In all these experiments, the observed hydrodynamic behavior resulted from the interactions of a very large number of particles, but interactions between single photons or polaritons remained relatively weak. An important step further in the investigation of **quantum** many-body effects in photon fluids would be to be able to increase the strength of the interactions and enter the so-called “**strongly** **correlated** regime”, where the effect of interactions becomes significant as soon as there is more than one particle in the system. To achieve this goal, arrays of nonlinear optical cavities, which are the main focus of this work, are a very promising candidate. As for the photon fluids presented above, there are exciting analogies between photons propagating in an array of cavities and cold atoms moving in an optical lattice. In the latter case, interesting **quantum** many-body effects, such as the celebrated Mott-insulator-to-superfuid phase transition have already been reported [ 20 ] and one of the objective of this PhD thesis was to determine whether a similar phenomenon could be observed with photons. Besides, apart from semiconductor microcavities, such arrays of cavities can also be implemented with superconducting circuits composed of microwave resonators and Josephson junctions [ 21 ].

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In conclusion, to study periodically driven **strongly** **correlated** electrons, we have considered the Fermi- Hubbard model with time-periodic interaction. Within nonequilibrium DMFT we have calculated the evolution of local observables and of the local Green function, which provide evidence for thermalization or prethermal- ization. We have showed the existence of three dynam- ical regimes: (i) Thermalization to infinite temperature at moderate interaction, as expected for generic isolated **quantum** many-body **systems**; (ii) Floquet prethermal- ization at large interaction, characterized by oscillations of local observables around a non-thermal plateau and a stationary non-thermal distribution function; (iii) Reso- nant thermalization at large interaction for an isolated critical frequency Ω ∗ , where local observables relax ex- ponentially to the infinite-temperature thermal value, together with a damping of oscillations and a flat dis- tribution function. We have then developed a periodic Schrieffer-Wolff transformation which captures the quali- tative features of the Floquet prethermal state and whose

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Introduction
Contex and Motivations
Heavy fermions are a family of **strongly** **correlated** electron **systems** behaving like Fermi liquids with renormalized parameters, notably the electronic effective mass. This renormalization is due to the presence of magnetic moments coming from the partially filled f orbitals, close to the Fermi energy, of lanthanide or actinide atoms. Indeed, the f shells can hybridize with broader bands such as s, p and d bands leading to competing magnetic orders on the system. This competing order can be explained, in general, through the Doniach phase diagram and a special interest is focused on the study of **quantum** critical points. In addition, heavy fermion compounds have a Fermi energy that is quite small, contrary to other **strongly** **correlated** electron **systems** such as High-Temperature superconductors (like cuprates or pnictides sys- tems) that also present competing orders. The low Fermi energy of heavy fermion **systems** gives the opportunity to explore this competition quite easily by applying pressure, doping and magnetic field as external parameters. This easy access made that these **systems** became so popular.

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system. The low energy excitations and the thermodynamics are therefore universal functions of T and T K .
The Kondo s-d exchange model
The exact origin of the anomalous resistance minimum was famously discovered by J. Kondo [ 83 ] about 30 years after its experimental discovery. His explanation revealed the many-body nature of this phenomenon. Non-magnetic impurities — with zero total spin — affect conduction electrons only through a scattering potential which adds up to the lattice potential. The induced resistance is independent from the temperature. However, a magnetic impurity has an internal degree of freedom (its spin) whose fluctuations affect how electrons are scattered on it. The coupling of an electron to the impurity indeed depends on the current state of the impurity spin, hence on the previous scatterings. The conduction electrons cannot be considered independent, they are **correlated**.

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combination with square fit techniques will provide the exit.
7. Conclusion
The fascinating properties observed in the new materials synthesized in the last couple of decades are commanded by their electronic structure and more specifically by strong electron-electron cor- relations. The understanding of the observed col- lective effects necessitates a clear picture of the interplay between the dominant microscopic local interactions. **Quantum** chemists with their exten- sive culture of correlation effects in finite **systems**, of valence correlation but also dynamical correla- tion and its retroactive effects on the valence shell, in the ground state but also in excited, multi- reference states, have the intellectual tools to an- swer this question. We have seen in this paper how up-to-date **quantum** chemical spectroscopic meth- ods associated to a controlled description of the crystalline embedding, a clear understanding of the model hamiltonians aims, an appropriate ex- tension and usage of effective hamiltonian methods that allow a controlled reduction of the ab initio information into a few dominant interactions, are the clues to a controlled and accurate modelisation of these materials.

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the quasiparticles and hIi the average value of the dc current flowing across the conductor. Correlations between subsequent transfers are encoded in the Fano factor F, defined as the ratio between the shot noise and its Poissonian value. Fermionic statistics tend to impose some order on charge transport [2,3] , characterized by reduced fluctuations (F < 1), or, in the case of perfectly ballistic conductors, fully noiseless transport (F ¼ 0). While Coulomb interactions tend to do the same [4] , they can, in some remarkable cases, give rise to positively **correlated** transport processes with super-Poissonian (F > 1) fluctua- tions, where charges flow in bursts through the conductor. Coulomb-blockaded **quantum** dots are very rich **systems**, as they can not only present Poissonian and sub-Poissonian transport regimes [1,4] , but also, depending on their internal structure, **strongly** **correlated** transport with super-Poissonian current fluctuations [5 –17] . The latter regime corresponds to non-Markovian transport processes where the transfer of a charge across the dot changes its state, thereby influencing the next transfer event [5] . As a result, the **quantum** dot randomly switches between highly and poorly conducting channels while other parameters

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The problems of stability and stabilizability are of great importance in the the- ory of delayed **systems** [1–3]. In this context note that the majority of works deals with so-called exponential stability or stabilizability. In this case the conditions of stability (stabilizability) are well explored for the both **systems** with ordinary delay and **systems** of neutral type [1–3,18,4]. Note also that the mentioned type of stability is similar to the stability for finite-dimensional lin- ear **systems**. However, for **systems** of neutral type there appears an essentially different kind of stability – the so-called strong stability.

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energy dispersive spectroscopy. The low frequency suscepti- bility has been determined with an ac field modulation up to 1.7 mT at a frequency of 1 kHz superposed to a constant field B using an ac magnetometer of a **Quantum** Design physical properties measurement system. A stacking of few single crystals, all oriented with c axis parallel to the mag- netic field, has been used to reach a mass of 20 mg for these measurements. Figure 2 displays the temperature depen- dences of the susceptibility for selected values of magnetic field, which basically span three regimes with specific behav- iors as highlighted by the areas. In particular, a magnetic field induced Fermi liquid regime characterized by a tem- perature independent susceptibility is revealed at the lowest temperatures 共blue or gray area兲. This behavior is followed by a power law T-dependence at higher temperatures 共yellow or light gray area 兲. At very low fields, an anomaly can be seen around 6 K indicating a possible enhancement of the magnetic correlations. These anomalies disappear above nearly 0.2 T as exemplified in Fig. 3共a兲 by delimitating the third regime in Fig. 2 共red or dark gray area兲. Contrasting

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The arrows connecting different compo- nents of HQS in Fig. 1 are labeled by ap- proximate values of geff that can be realistically achieved with present-day technology. Some of the larger coupling strengths in Fig. 1 seem to contradict our observation concerning the weak coupling between very different phys- ical **systems**. This contradiction is resolved by noting that the coupling of a mesoscopic system via light or microwave fields to ensembles rather than to single atoms or spin dopants (see below). The red and blue arrows in the figure indicate the single-sys- tem and ensemble coupling strengths, re- spectively. Fig. 1 also shows various examples for the coupling of **systems** with **strongly** dissimilar excitation energies (dashed lines). In such cases, the coupling mechanism in- volves an external source or sink, such as a (classical) laser or microwave field, which bridges the energy mismatch to make the processes described by Eq. 1 resonant when they are accompanied by absorption or stimulated emission of photons. This cou- pling mechanism applies to the well-known processes of laser-assisted optical Raman transitions in atoms and molecules. In opto- mechanics (20), parametric coupling via an applied control field is used to bridge the energy difference between mechanical vibra- tional modes and optical photons and to enhance the interaction strength geff .

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9 73, 118 In spite of this, it is highly desirable, at least for applications to communication and other systems, to have a common framework for treating quantum[r]

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Our results further support the hope that, for realistic errors in physical systems, clever concatenation between active and passive error control codes carries the [r]

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as clean model **systems**, for which the classical dynamics is well understood [36, 30]. The popularity of **quantum** maps mostly stems from the much simplified numerical study they offer, both at the **quantum** and classical levels, compared with the case of Hamiltonian flows or the corresponding Schr¨odinger operators. For instance, the distribution of resonances and resonant modes has proven to be much easier to study numerically for open **quantum** maps, than for realistic flows [7, 37, 29, 25, 27]. Precise mathematical definitions of **quantum** maps on the torus phase space are given in [29, §4.3-4.5].

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DOI: 10.1103/PhysRevB.68.155305 PACS number~s!: 78.67.De, 42.65.Sf
I. INTRODUCTION
There is currently an interest in exploiting intersubband transitions in semiconductor **quantum** wells for detectors, emitters, **quantum** cascade lasers, and lasers without inver- sion in the far-infrared and terahertz region. 1 Many of the physical concepts considered in this context originate from atomic physics, where one manipulates well-defined quan- tum levels. 2 In **quantum** wells, however, a macroscopic num- ber of carriers participates in intersubband transitions 3 and analogies with atomic physics are not obvious. We are par- ticularly motivated by recent experiments on **quantum** foun- tain laser in coupled **quantum** wells by Liu et al. 4 In these experiments, two subbands of a modulation-doped **quantum** well are optically driven by a CO 2 laser. In related experi- ments, intersubband transitions in intense laser fields have been studied experimentally 5– 8 and theoretically. 9–14 In the- oretical works, the effects of electron-electron interactions have been included mainly in the Hartree approximation by, e.g., Zaluzny 10 and Birnir and Galdrikian. 13 The effect of the exchange and correlations were studied using semiconductor Bloch equations ~SBE’s! ~exchange only! 11,12,14,15 and the time-dependent density functional approach. 16 The effects of electron-electron correlations in **quantum** dots in intense la- ser field have also been studied using exact diagonalization techniques. 17,18 The SBE approach typically relies on the equation of motion approach which is not transparent to ex- perimentalists, as it resorts early on to numerical analysis, and often neglects exchange and correlations.

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desired properties of an ideal entangled photon source, opens new opportunities in **quantum** optics, integrated **quantum** photonic circuits 37,38 and **quantum** information processing.
To realize an ideal entangled photon source in future work there are several properties of our source to consider. First, **quantum** dot-entangled photon sources have not yet reached the ﬁdelity or concurrence values of parametric down-conversion sources 2,39 . However, with recently available post-growth tuning methods to bring the ﬁne-structure splitting of almost any **quantum** dot near zero 25 and two-photon resonant excitation 3 , the ﬁdelity of these **quantum** dot sources are approaching that of parametric down- conversion sources. Second, the single-photon coherence of the emitted photon pairs is not yet Fourier-transform limited, which is needed for advanced **quantum** information-processing schemes. Such Fourier-transform-limited photons may be reached by combining two-photon resonant excitation techniques 3,40 , cooling of the **quantum** dot sample to 300 mK 28 and by accelerating the **quantum** dot emission via the Purcell effect 20 . Finally, the major advantage of tapered nanowire waveguides over other approaches is the light extraction efﬁciency, which promises entangled photon-pair extraction efﬁciencies exceeding 90% due to the broadband frequency of operation 29 . Such efﬁciencies would surpass the state-of-the-art entangled photon-pair efﬁciency of 12% 20 , without the stringent requirements needed to engineer both the exciton and biexciton into resonance with a cavity mode by using post-growth manipulation of pre-selected **quantum** dots.

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i−j i
, as defined by Robbins and Rumsey [RR86], for the particular value λ = −q. The latter is known to be related to the classical T -system with coefficients, a higher-dimensional version of the Q- system, itself having a cluster algebra formulation [DF13]. In particular, it was shown that the lambda-determinant of any matrix is the partition function for suitable families of non- intersecting lattice paths whose local weights involve the matrix entries, or alternatively of weighted domino tilings of the so-called Aztec diamond. The expression (2.25) should therefore allow to interpret the **quantum** determinant of the matrix W α,n as the partition

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in our design, where the heterostructure is only »20 nm thick. CONCLUSION
To conclude, we could spatially isolate single nanowires to perform several spectroscopic (bias-dependent photoluminescence, photocurrent) and structural (STEM) measurements on the same GaN nanowire containing a » 1-nm-thick Al(Ga)N/GaN **quantum** dot. From these **correlated** measurements, we could model the evolution of the photoluminescence with bias, taking into account the measured structural parameters of each dot. A systematic blue (red) shift is observed in µPL measurements for the application of an external electric field compensating(enhancing) the polarization-related internal electric field within the **quantum** dot structure. Spectral shifts of 12 and 20 meV/V are observed for **quantum** dots emitting at 307.5 and 327.5 nm, respectively. Three-dimensional modeling of the electronic structure taking the STEM-measured morphology into account allow estimating the internal electric field in the dots (around 3 and 4.8 MV/cm in the two nanowires under study, respectively) and predicting its variation with bias. A deviation from the theoretical trend is observed when the bias voltage is high enough to favor tunneling of the electron in the **quantum** dot towards the stem or the cap. In such a situation, the spectral shifts saturate and additional transitions associated to charged excitons can be observed. We demonstrate that small differences in opto- electrical properties between nominally identical nano-objects can be well understood taking into account their precise heterostructure dimensions and composition. Improved growth techniques may be developed to reduce these differences, while modeling shows its value in predicting the detailed opto-electrical properties of such nanowire heterostructures.

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