CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / A transmissÃo de impulsos nervosos à feita por meio das sinapses, envolvendo neurotransmissores e receptores. Os receptores ionotrÃpicos glutamatÃrgicos (GluRs) sÃo importantes canais iÃnicos do sistema nervoso central, encontrados em sinapses de excitaÃÃo rÃpida, e estÃo relacionados a funÃÃes cerebrais importantes como aprendizado e memÃria. AlÃm disso, os GluRs tambÃm estÃo associados com certas doenÃas neurolÃgicas e psiquiÃtricas, como por exemplo: a doenÃa de Alzheimer, o mal de Parkinson, a epilepsia, o acidente vascular cerebral, a esclerose lateral amiotrÃfica e a esquizofrenia. Neste trabalho, tiramos vantagem dos dados disponÃveis na literatura da co-cristalizaÃÃo dos seguintes agonistas glutamato (C5H9NO4) e AMPA (C7H10N2O4), do agonista parcial cainato (C10H15NO4) e do antagonista DNQX (C8H2N4O6) com o receptor GluR2 com resoluÃÃo de 1,9 Ã, 1,7 Ã, 1,9 à e 1,8 Ã, respectivamente, para estudar a interaÃÃo destes quatro ligantes com a GluR2 por meio de mÃtodos computacionais ab initio. Os hidrogÃnios ausentes dos dados de difraÃÃo de raios-X foram colocados atravÃs de um processo semi-clÃssico de minimizaÃÃo da energia total GluR2-ligante. A seguir, as simulaÃÃes foram feitas usando a Teoria do Funcional da Densidade (DFT), tanto ao nÃvel da aproximaÃÃo da densidade local (LDA), como da aproximaÃÃo do gradiente generalizado (GGA), para descriÃÃo dos efeitos de troca e correlaÃÃo. A utilizaÃÃo do mÃtodo de fragmentaÃÃo molecular com capas conjugadas (MFCC) tornou possÃvel analisar a interaÃÃo dos ligantes com cada um dos resÃduos prÃximos e pÃs-prÃximos do GluR2. Considerou-se tambÃm a relevÃncia da blindagem dos resÃduos pÃs-prÃximos que interagem com os ligantes, bem como se fez uma anÃlise da energia de interaÃÃo dos resÃduos (prÃximos e pÃs-prÃximos) considerados com os Ãtomos dos ligantes (resultados apresentados nos grÃficos BIRD), sem e com mediaÃÃo das molÃculas de Ãgua existentes no sÃtio de ligaÃÃo (o que permite se determinar ab initio a relevÃncia da Ãgua na energÃtica da interaÃÃo ligante-GluR2). Obteve-se a energia total de interaÃÃo GluR2-ligante em funÃÃo da distÃncia dos centroides dos ligantes aos resÃduos, o que permitiu correlacionÃ-la à intensidade de ativaÃÃo e antagonismo dos neurotransmissores em questÃo. Demonstrou-se que ela segue a ordem AMPA > glutamato > cainato > DNQX somente quando um raio do sÃtio de ligaÃÃo suficientemente grande à considerado, o que explica dados experimentais publicados sobre a ativaÃÃo e antagonismo do receptor glutamatÃrgico GluR2, sugerindo que os resÃduos pÃs-prÃximos podem ser importantes para determinar o funcionamento do receptor. Para o glutamato, os resultados obtidos indicam que os resÃduos atrativos mais relevantes sÃo: Arg485, Lys730 (mediado pela Ãgua W39), Ser654, Leu650 mediado por W69, e Lys656 mediado por W22; os resÃduos repulsivos mais relevantes para o glutamato sÃo Glu402 (pÃs-prÃximo) mediado por W36, Glu657 e Asp651 (pÃs-prÃximos). Para o AMPA os resÃduos atrativos mais relevantes sÃo: Arg485, Thr655 mediado por W134, Lys730 mediado por W137, Lys656 mediado por W138, Lys449 e Arg684 (pÃs-prÃximos); os resÃduos repulsivos mais relevantes para o AMPA sÃo Glu402 mediado por W3, Asp651 mediado por W96 e W139 (pÃs-prÃximo), e Glu657 (pÃs-prÃximo) mediado por W140. Para o cainato os resÃduos atrativos mais relevantes sÃo Arg485, Ser654, Thr655 e Arg684 (pÃs-prÃximo); os resÃduos repulsivos mais relevantes para o Cainato sÃo Glu402, Glu657 mediado por W78 (pÃs-prÃximo) e Asp651. Para o DNQX os resÃduos atrativos mais relevantes sÃo Arg485, Glu705 e Tyr450 mediado por W26 e W137; o resÃduo repulsivo mais relevante para o DNQX à Leu498. Uma plÃiade de perspectivas relacionadas aos resultados obtidos reluz e dentre elas podemos destacar a possibilidade do desenvolvimento de agonistas e antagonistas glutamatÃrgicos com especificidades voltadas à diminuiÃÃo de efeitos colaterais quando utilizados no tratamento de doenÃas relacionadas à neurotransmissÃo glutamatÃrgica. / The transmission of nerve impulses occurs through the synapses, involving neurotransmitters and receptors. The ionotropic glutamate receptors GluRs are important ionic channels of the central nervous system, founded in rapid excitation synapses, and related to important cerebral functions like learning and memory. Besides this, GluRs are also associated with important neurological and psychiatric diseases like Alzheimer, Parkinson, epilepsy, cerebral ischemia, amyotrophic lateral sclerosis, and schizophrenia. In this work, we take advantage of the available data in the literature of co-crystallization of the following full agonists glutamate (C5H9NO4) and AMPA (C7H10N2O4), the partial agonist kainate (C10H15NO4) and the antagonist DNQX (C8H2N4O6) with the GluR2 receptor with resolution of 1.9 Ã, 1.7 Ã, 1.9 à and 1.8 Ã, respectively to study the interaction of these four ligands with GluR2 by means of ab initio computational methods. The absent hydrogens in the GluR2-ligand X-ray diffraction data were inserted through a semi-classical total energy minimization process. Next, the simulations were performed within the scope of the Density Functional Theory (DFT), both in the local density approximation (LDA) as generalized gradient approximation (GGA) for the description of exchange-correlation effects. The use of the molecular fragmentation method with conjugated caps (MFCC) allowed to analyze the interaction between the ligands with each one close and next-closed GluR2 residues. It was also considered the relevance of the screening of the next-closed residues with interact with the ligands, and it was performed an analysis of the interaction energy between the focused residues (close and next-closed) with the atoms of the ligands (results depicted in the BIRD panels), without and with the mediation of water molecules existing in the binding pocket (which allows to determine ab initio the relevance of water in the GluR2-ligands energetic). It was obtained the GluR2-ligand total energy interaction as a function of the distance between the ligand centroid and the residues, which allowed to correlate it to the activation strength and antagonism of the ligands focused. It was demonstrated that it follows the order AMPA > glutamate > kainite > DNQX only when a large enough binding pocket radius is taken into account, explaining the experimental data published on the activation and antagonism of the glutamatergic receptor GluR2 and suggesting the next-closed residues can be important to determine the receptor functioning. For the glutamate, the obtained results point that the most important attractive residues are Arg485, Lys730 (water W39 mediated), Ser654, Leu650 (water W69 mediated), and Lys656 (water W22 mediated); the most important repulsive residues for the glutamate are Glu402 (next-closed water W36 mediated), Glu657 and Asp651 (nex-closed). For AMPA, the most important attractive residues are Arg485, Thr655 (water W134 mediated), Lys730 (water W137 mediated), Lys656 (water W138 mediated), Lys449 and Arg684 (next-closed); the most important repulsive residues for AMPA are Glu402 (water W3 mediated), Asp651 (next-closed, water W96 and W139 mediated), and Glu657 (next-closed, water W140 mediated). For kainate the most important attractive residues are Arg485, Ser654, Thr655 and Arg684 (next-closed); the most important repulsive residues for kainite are Glu402, Glu657 (next-closed, water W78 mediated) and Asp651. For DNQX, the most important attractive residues are Arg485, Glu705 and Tyr450 (water W26 and W137 mediated); the most important repulsive residue for DNQX is Leu498. A pleiade of perspectives related with the obtained results shines, among which one can highlight the possibility to develop glutamatergic agonists and antagonists with specificities related to decrease side effects when used in the treatment of maladies related with the glutamatergic neurotransmission.

Well know therapies for the treatment of osteoporosis, syndrome characterized by increased bone fragility and fracture consist primarily of drugs which prevent bone losses, such as bisphosphonates. Among them, alendronate (PO3)-(OH)-C-((CH2)3 NH3)-(PO3) is one of the chosen treatments in the clinic, IC50 = 50 nM. In this work a study of the molecular alendronate and three of its crystals (sodium alendronate trihydrate and anhydrous and calcium alendronate ) is performed in the scope of Density Functional Theory (DFT). In the first case, the focus is the lowest energy conformations of the molecular alendronate in vacuum, in an aqueous medium in accordance with the model of polarization continuum (PCM), and interacting with three molecules of water of a sodium atom based on the characteristics structural relevadas diffraction X-ray crystal alendrontao sodium trihydrate; their vibrational infrared and Raman spectra are calculated to explain in detail the abundant signatures of the phosphate group in the frequency range 400-1400 cm −1 with the assignments of the most important vibrational modes. In the case of crystals, their structural, electronic and optical properties are obtained in the generalized gradient approximation taking into account the description of the term correlation and exchange Tchatschenko (GGA+TS ); Hirshfeld population analysis in which verified is presented alendronate is that the three crystals is in zwitterionic state. From the calculation of the band structure was obtained GAPs with very close values for the three crystals, however, with the density of states and characteristic for each crystal. The effective masses, dielectric function and theoretical optical absorptions for all crystals are presented. / Terapias estabelecidas para o tratamento da osteosporose, sÃndrome caracterizada por aumento na fragilidade Ãssea e fraturas, consistem primariamente de drogas que previnem a perda Ãssea, como os bisfosfonatos. Entre eles, o alendronato (PO3)-(OH)-C-((CH2)3 NH3)-(PO3) à um dos tratamentos escolhidos na clÃnica, pois possui o IC50=50 nM. Neste trabalho à realizado um estudo do alendronato molecular e de trÃs de seus cristais (alendronato de sÃdio trihidratado e anidro, e alendronato de cÃlcio) utilizando da Teoria do Funcional da Densidade (DFT). No primeiro caso, o foco sÃo os confÃrmeros de menor energia do alendronato molecular no vÃcuo, em meio aquoso de acordo com o modelo contÃnuo polarizÃvel (PCM), e interagindo com trÃs molÃculas de Ãgua e um Ãtomo de sÃdio de acordo com as caracterÃsticas estruturais reveladas por difraÃÃo de raios-X do cristal do alendrontao de sÃdio trihidratado; seus espectros vibracionais no infravermelho e Raman sÃo calculados para explicar com detalhes as abundantes assinaturas do grupo fosfato no intervalo de frequÃncia 400-1400 cm−1, com as atribuiÃÃes dos modos vibracionais mais importantes. No caso dos cristais, sÃo obtidas suas propriedades estruturais, eletrÃnicas e Ãpticas na aproximaÃÃo da gradiente generalizado levando-se em conta a descriÃÃo do termo de correlaÃÃo e troca de Tchatschenko (GGA+TS); à apresentada a anÃlise populacional de Hirshfeld na qual verificou-se que o alendronato nos trÃs cristais encontra-se no estado zwitterionico. A partir do cÃlculo de estrutura de banda, foi obtido GAPs com valores muito prÃximos para os trÃs cristais, porÃm, com densidade de estados bem caracterÃstica para cada cristal. SÃo apresentadas as massas efetivas, funÃÃo dielÃtrica e absorÃÃes Ãpticas teÃricas para todos os cristais.

Apresentamos aqui um formalismo, baseado no Principio de Máxima Entropia, que permite estudar sistemas celulares aleatórios bidimensionais compostos de muitas células que cobrem uma dada superfície plana sem poros ou superposições e são caracterizadas pela sua área, perímetro, número de lados e posição. Calculamos a função partição do sistema num espaço de fases generalizado e a partir desta obtemos quantidades como a área e o perímetro medi() das células de n lados, a entropia e a energia livre. Impomos ao sistema vínculos topol6gicos, geométricos e referentes â- energia do sistema. Considerando a energia de interface e de curvatura das paredes celulares, a evolução quase-estática e estudada a partir da variação de parâmetros como o comprimento médio dos lados das células, a energia média das células e o segundo momento da distribuição em número de lados. Os resultados estão em boa concordância com dados experimentais de sistemas naturais como espumas de sabão bidimensionais e agregados metalúrgicos e também com simulações numéricas. Além disso, são obtidas as condições para que uma estrutura ordenada seja estável e para a sua transição para um estado desordenado com o aumento da temperatura. / We present here a formalism, based on the Maximum Entropy Principie, which enables us to study bidimensional random cellular structures made of many cells that cover a given flat surface without pores or overlaps and are characterized by their arcas, perimeters, nuniber of sides and position. We calculate the partition function of the system defined in a generalized phase space and we obtai n variables as the average arca and perimeter of n-sided cells, the entropy and free energy. We impose upon the system some constraints refering to topology,geometry and to the energy of the system. Considering the interface energy and the one related to the curvature of cellular walls, we study the quasi-sia tic evolution from the variation of parameters as the average side length and energy of the cells and as the second moment of the distribution in number of sides. The results are in good agreement wi th experimental data of natural systems as bidimensional soap froths and metallurgical aggregates and also with numerical simulations. We also obtain the conditions for an ordered structure to be stable and for its transition to a disordered state as temperature increases.

o trabalho de Krim e Windom [Phys. Rev B 38, 12184(1988)], que revelou a natureza viscosa do atrito em escala atômica, gerou uma intensa atividade tanto teórica quanto experimental. Contudo, questões fundamentais ainda se matém em aberto com opor exemplo a relação entre o coeficiente de atrito viscoso e a topologia do substrato assim como a dependência com a temperatura do substrato. Neste trabalho apresentamos os resultados, obtidos usando dinâmica molecular, para um modelo unidimensional de um sistema adsorvato/substrato. Pesquisamos diferentes relações de comensuração entre o adsorvato e o substrato assim como também a dependência do coeficiente de atrito fonônico, Nph, com a temperatura. Para todas as configurações estudadas obtivemos que o coeficiente de atrito fonônico depende quadráticamente da amplitude de corrugação do substrato, mas tem uma dependência não trivial com a razão de comensuração adsorvato/substrato. O resultado mais impressionante é que para relações de comensuração entre 0.65 e 0.9 o atrito fonônico é muito fraco. Nosso resultados podem explicar as diferencias encontradas na literatura em relação à magnitude do atrito de origem eletrônico e o de origem fonônico.

As Manganitas, como são conhecidos os óxidos de Manganês, tornaram-se objeto de incontáveis artigos na literatura científica devido ao seu potencial uso em ciência e tecnologia. Estes materiais apresentam, entre outros, o fenômeno da magneto-resistência colossal e um variado diagrama de fases em baixas temperaturas. Neste trabalho descrevemos este tipo de compostos magnéticos, mostrando as características físicas que os distinguem, os diagramas de fases que identificam as diferentes famílias e algumas propriedades termodinâmicas. O ponto central neste trabalho é demonstrar a importância da anisotropia dos hoppings, assim como a importância da degenerescência orbital dos níveis eletrônicos eg nas propriedades de baixas temperaturas nestes compostos, na fase ferromagnética. A partir do estudo do modelo de dois orbitais, em temperatura zero, e utilizando expressões exatas para os hoppings entre os átomos de Manganês e Oxigênio, mostramos a nível de campo médio que, para o caso do La1−xSrxMnO3 e com uma interação de Hund finita entre spins localizados e itinerantes, o sistema não é totalmente semi-metálico, como acreditava-se anteriormente na literatura, apresentando uma proporção finita de elétrons itinerantes alinhados anti-paralelamente aos spins localizados, o que concorda com experiências recentes. Em temperatura finita, calculamos algumas propriedades termodinâmicas do hamiltoniano de dois orbitais, incluindo os termos da interação coulombiana, utilizando o método de diagonalização exata em pequenos clusters. Mostramos que o calor específico e a magnetização variam fortemente com a mudança do hopping e também com a abertura dos níveis eg. Algumas aplicações destes resultados são discutidas e mostramos como eles podem ser relevantes para a identificação da relação entre transições magnéticas e estruturais em Manganitas dopadas. / The manganites, as they are known the manganese oxides, had become subject of countless articles in the scientific literature due to its potential use in science and technology. These materials present, among others, the phenomenon of colossal magneto-resistance and a rich phase diagram at low temperatures. In this work we describe this type of magnetic compounds, showing the physical characteristics that distinguish them, the phase diagrams that identify the different families and some thermodynamic properties. The central point in this work is to demonstrate the importance of the anisotropy of hoppings, as well as the importance of the orbital degeneracy of the electronic eg levels at the low temperature properties of these compounds, in the ferromagnetic phase. From the study of the two orbitals model, at zero temperature, and using accurate expressions for the hoppings among manganese and oxygen atoms, we show at the mean-field level that, for the case of La1−xSrxMnO3 and with a finite Hund interaction between local and itinerant spins, the system is not totally half-metallic, as it was obtained previously in the literature, presenting a finite ratio of anti-alligned itinerant electrons to the local spins, according with recent experiments. In finite temperature, we calculate some thermodynamic properties of the two orbitals hamiltonian, including the coulomb interaction terms, using the method of exact diagonalization in small clusters. We show that the specific heat and the magnetization vary strongly with the change in hopping and also with the opening of the eg levels. Some applications of these results are discussed and we show how they can be relevant for the identification of the relationship between magnetic and structural transitions in doped manganites.

In this thesis we first review the theory of the normal state of the unitary Fermi gas at T = 0 and the main properties of the normal-to-superfluid transition. Then we study the trapped gas under adiabatic rotation, i.e., avoiding the formation of vortices. We show that for polarized systems the rotation enhances the Chandrasekhar-Clogston limit due to pair breaking at the border between the superfluid and the normal phase, while it leaves the global critical polarization Pc of the trapped system unaffected. In the case of an unpolarized unitary superfluid the rotation causes a phase separation between a superfluid core and an unpolarized normal shell, in which the densities of the spin-up and spin-down atom numbers is equal. For both the polarized and the unpolarized systems we calculate experimental observables such as the density profiles and the angular momenta. From the study of Bose-Einstein condensates it is well known that an adiabatic rotation induces a quadrupole deformation of the trapped atomic cloud when the rotation exceeds a certain angular velocity. In Fermi gases the situation is different due to the phase separation discussed above, and the quadrupole instabilities are found to set on at smaller angular velocity than in the BEC case. This phenomenon together with a more general discussion concerning not only the energetic but also the dynamic instabilities of the phase separated system is presented. We use the present knowledge of the equation of state of Fermi mixtures with unequal masses to give quantitative predictions for the phase separation between the normal and superfluid components. The analysis is based on the study of the zero temperature Î1⁄4-h phase diagram of the uniform two component gas. The phase diagram at unitarity is determined thanks to the knowledge of the equation of state available from diagrammatic techniques applied to highly polarized configurations and from Monte Carlo simulations. The phase diagram is then used, in the local density approximation, to calculate the density profiles of the two Fermi components in the presence of harmonic trapping. Eventually we investigate the polarization produced by the relative displacement of the potentials trapping two spin species of a unitary Fermi gas with population imbalance. We investigate the dipole polarizability of a polarized system both in the two-fluid and the three-fluid model at zero temperature and point out the major differences between the two treatments.

Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.

In this thesis I present results of my fully general relativistic magnetohydrodynamic (GRMHD) binary neutron star merger (BNS) simulations, conducted by using the numerical code "Whisky" under various conditions, such as, different Equation of State (EOS) for neutron matter (APR4, Ideal fluid and H4 EOSs), masses (with equal/unequal masses for two neutron stars), different magnetic field configurations (both fields of two neutron stars aligned with the inspiral axis, one aligned and one anti-aligned, and both anti-aligned) to investigate the effect of these parameters on the dynamics of the simulations and possibility of forming relativistic jets, which is thought to be one of the necessary conditions for the central engine of short gamma-ray bursts (SGRBs).

In the past century, two great discoveries revolutionized our understanding of the Universe. The first was the study of the NGC 3198 galaxy in the 80s. Looking at the rotation velocity of the galaxy objects with respect to its center, the so-called rotational curve of the galaxy, an anomaly was found. The rotational curve did not seem to obey the known laws of physics as the velocities of objects far away from the center of NCG 3198 were too big with respect to any theoretical prediction. The explanation for these anomalies involves the presence of an additional unknown matter in the galaxies, called dark matter. An interesting property of dark matter is its interaction nature. Since we have not yet observed this matter with telescopes but only looked at its effect on galaxies objects, we believe dark matter interacts with standard matter (baryons, leptons, ...), radiation and neutrinos only gravitationally and not through electromagnetic interactions. In fact, up to now, dark matter has not been directly observed with ground detection experiments. The second discovery came from the observation of type 1A supernovae emissions. Although we already knew that the Universe was expanding since the beginning of the 20th century, at the end of the millennium, two independent experiments discovered that this expansion was accelerating. Within the theory of General Relativity, the acceleration can not be explained with the known standard matter, radiation, neutrinos, or even with a different spatial curvature. This contradiction led to the resolutive hypothesis that a new kind of matter, which throughout the whole history of the Universe has a constant density achieved with negative pressure, should be considered. This matter is called dark energy. These two breakthroughs are the basis of the standard model of cosmology. However, even though this model has been proven astonishingly accurate in describing the history of the Universe, cosmologists still struggle with some fundamental questions about dark energy and dark matter. It is important to stress that both dark matter and dark energy are called "dark" to underline our ignorance about the fundamental nature of these additional matter types. We should think of them as a way to parametrize our ignorance about the actual nature of these two hypotheses rather than the solution of the two problems mentioned before. We know that there should be some matter with specific properties to explain the two aforementioned observational phenomena, and its inclusion in the theoretical model leads to satisfactory theory-observation accordance. However, we know nothing about the fundamental nature of dark matter and dark energy, nor any direct observation has proven their existence. Moreover, additional fundamental and mathematical questions arise when postulating dark matter and dark energy in the form proposed in the standard model. In the decades after these discoveries, we are witnessing two phenomena in theoretical and observational cosmology. On one hand, experiments are becoming more accurate and precise in detecting information from the Universe. One of the most recent examples is the Planck experiment, whose space telescope observed the photons coming from a moment in the Universe's history called recombination, which happened 13 billions of years ago. At that moment, photons and electrons decoupled, letting the firsts travel freely and unscattered. This radiation is called Cosmic Microwave Background (CMB). We can collect these photons with a detector to create a snapshot of the photon intensity distribution at that time. This distribution is tightly linked with dark matter and dark energy properties, helping physicists to shed some light on the two dark components. In general, the accuracy of new experiments puts very tight constraints on theoretical models. On the other hand, many theoretical models have been proposed as alternatives to the standard model to solve the problems mentioned above. These models modify the General Relativity equations of motion, the Einstein equations, either replacing the dark matter and dark energy matter contents with respect to the standard model or modifying the geometry of spacetime. They achieve this by including additional dynamical quantities or degrees of freedom, whose evolution can explain the accelerated expansion acceleration or dark matter effects (or both). For instance, a scalar field can be considered, but other models with more complex additional degrees of freedom have been proposed. All these models are usually called Modified Gravity theories. In the last few years, most of the modified gravity models have been under scrutiny due to increased observational data. For instance, the predictions of the CMB might change when we consider modified gravity models for dark energy or dark matter, putting constraints on the theory parameters or ruling the model out. The data are becoming accurate enough to put very tight constraints on the modified gravity models. Nevertheless, the analysis of the CMB power spectrum or similar observables is not an easy task. One of the main obstacles in checking the viability of the theoretical models against experimental data is the complexity of the theoretical study of these crucial observables. No analytical solution of the equations of motion valid at all times of the Universe's history can be found. Moreover, additional degrees of freedom can increase the complexity of the evaluations for modified gravity models. We should also consider that a single evaluation of the Universe's history is not enough to conclude anything about the viability of the model. When we compare a theoretical model with experiments, we should minimize the difference between the predictions and the data, varying the value of the theory parameters. This procedure usually needs an enormous number of evaluations that require significant computational power, even with the most efficient Monte Carlo algorithms. Moreover, especially in the modified gravity context, it is also essential to distinguish the theory predictivity given by the new proposals' real physical and mathematical power rather than the simple addition of new degrees of freedom. It is easier to fit three points with a parabola with respect to a straight line at the price of adding a new parameter to the theory. However, is it always necessary? In physics, like in other fields, Occam's razor principle tells us that the most straightforward theory should always be preferred. With the introduction of the Bayesian probability, we can perform comparisons between models to find the ones that fit the data with fewer parameters. But, again, this procedure is computationally expensive. Finally, we can ask ourselves if there is a way to parametrize the modified gravity models in a model-independent way. In other words, does it exist a way to write a general action or Lagrangian which can include all modified gravity models? The power of such a generalization would be undeniable: we would be able to compute the equations of motion from one single action and apply it for every modified gravity model. Such a theory, which we will call Effective Field Theory (EFT) of Gravity, has been developed, and it works for any theory with an additional degree of freedom with respect to General Relativity. The major drawback is that the general form of the EFT of Gravity does not provide an immediate physical interpretation of its Lagrangian terms, and therefore a mapping between a "standard" modified gravity theory and its EFT counterpart is always preferred.

Collective quantum phenomena are fascinating, as they repeatedly challenge our comprehension of nature and its underlying mechanisms. The qualification ``quantum'' can be attributed to a generic many-body system whenever the interference effects related to the underlying wave nature of its elementary constituents can not be neglected anymore, and a naive classical description in terms of interacting billiard balls fails to catch its most essential features. This interference phenomenon called ``quantum degeneracy'' which occurs at weak temperatures, leads to spectacular collective behaviours such as the celebrated Bose-Einstein Condensation (BEC) phase transition, where a macroscopic fraction of a bosonic system of particles collapses below a critical temperature T_c on a single-particle state. Quantum coherence, when combined with inter-particle interactions, gives rise to highly non-classical frictionless hydrodynamic behaviours such as superfluidity (SF) and superconductivity (SC). Even more exotic quantum phases emerge in presence of important interactions as matter reaches a ``strongly correlated regime'' dominated by quantum fluctuations, where each particle is able to affect significantly the surrounding fluid: characteristic examples are the so-called Mott-Insulator (MI) quantum phase where particles are localized on a lattice due to a strong interaction-induced blockade, along with the Tonks-Girardeau (TG) gas where impenetrable bosons in one-dimension acquire effective fermionic statistics up to a unitary transformation, and the Fractional Quantum Hall (FQH) effect which occurs in presence of a gauge field, and features a special type of elementary excitation possessing a fractional charge and obeying to fractional statistics called `anyon'. These quantum many-body effects were explored in a first place in systems well isolated from the external environment such as ultra-cold atomic gases or electrons in solid-state systems, within a physical context well described by ``equilibrium statistical mechanics''. Yet, over the last two decades a broad community has started investigating the possibility of stabilizing interacting quantum phases in novel nonlinear quantum optics architectures, where interacting photons have replaced their atomic and electronic counterpart. Thanks to their high level of controllability and flexibility, and the possibility of reaching the quantum degeneracy regime at exceptionally high temperatures, these platforms appear as extremely promising candidates for the ``quantum simulation'' of the most exotic many-body quantum problems: while the precursors experiments in semiconductor exciton-polariton already allow to reach the Bose-Einstein Condensation and superfluid regimes, novel platforms such as superconducting circuits, coupled cavity arrays or photons coupled to Rydberg EIT (Electromagnetically induced Transparency) atoms have entered the so-called `photon blockade' where photons behave as impenetrable particles, and open a encouraging pathway toward the future generation of strongly correlated phases with light. A specificity of quantum optics devices is their intrinsic ``non-equilibrium'' nature: the interplay between the practically unavoidable radiative and non-radiative losses and the external drive needed to replenish the photon gas leads the many-body system toward a steady-state presenting important non-thermal features. One one hand, an overwhelmingly large quantity of novel quantum phenomena is expected in the non-equilibrium framework, as breaking the thermal equilibrium condition releases severe constraints on the state of a quantum system and on the nature of its surrounding environment. On the other hand, we do not benefit yet of an understanding of non-equilibrium statistical mechanics comparable with its well-established equilibrium counterpart, which relies on strong historical foundations. Understanding how to tame (and possibly exploit) non-equilibrium effects in order to stabilize interesting quantum phases in a controlled manner often reveals a hard challenge. In that prospect, an important conceptual issue in the non-equilibrium physics of strongly interacting photons regards the possibility of stabilizing ``incompressible quantum phases'' such as the Mott-Insulator or Fractional Quantum Hall states, and more generally to stabilize the ground-state of a given particle-number conserving Hamiltonian, in a physical context where dissipative losses can not be neglected. While being able to quantum simulate those emblematic strongly correlated quantum phases in this novel experimental context would strongly benefit to the quantum optics community, gaining such a kind of flexibility would also contribute to fill an important bridge between the equilibrium and the non-equilibrium statistical physics of open quantum systems, allowing to access in a controlled manner a whole new phenomenology at the interface between the two theories. In this thesis I address those questions, which I reformulate in the following manner: -What are the conditions for the emergence of analogue equilibrium properties in open quantum systems in contact with a non-thermal environment ? -In particular, is it possible to stabilize strongly correlated quantum phases with a perfectly defined particle number in driven-dissipative photonic platforms, in spite of environment-induced losses and heating effects ? The structure of the thesis is the following. [Chapter 1.] We give an overview of the physics of many-body photonic systems. As a first step we address the weakly interacting regime in the physical context of exciton-polaritons: after describing the microscopic aspects of typical experiments, we move to the discussion of non-equilibrium Bose-Einstein Condensation and the various mechanisms related to the emergence of thermal signatures at steady-state. The second part of this Chapter is dedicated to strongly interacting fluids. After drawing a quick overview of several experimental platforms presenting a good potential for the study of such physics in a near future, we discuss the relative performance of several schemes proposed in order to replenish the photonic population [Chapter 2.] We investigate the potential of a non-Markovian pump scheme with a narrow bandpass (Lorentzian shaped) emission spectrum for the generation of strongly correlated states of light in a Bose-Hubbard lattice. Our proposal can be implemented by mean of embedded inverted two-level emitters with a strong incoherent pumping toward the excited state. Our study confirms in a single cavity the possibility of stabilizing photonic Fock states in a single configuration, and strongly localized n=1 Mott-Insulator states in a lattice with n=1 density. We show that a relatively moderate hopping is responsible for a depletion of the Mott-state, which then moves toward a delocalized state reminiscent of the superfluid regime. Finally, we proceed to a mean-field analysis of the phase diagram, and unveil a Mott-to-Superfluid transition characterized by a spontaneous breaking of the U(1) symmetry and incommensurate density. The results of this Chapter are based on the following publications: - J. Lebreuilly, M. Wouters and I. Carusotto, ``Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping'', C. R. Phys., 17(8), 836, 2016. - A. Biella, F. Storme, J. Lebreuilly, D. Rossini, R. Fazio, I. Carusotto and C. Ciuti, ``Phase diagram of incoherently driven strongly correlated photonic lattice'', Phys. Rev. A, 96, 023839, 2017. [Chapter 3.] In view of improving the performance of the scheme introduced in last chapter, and reproducing in particular the equilibrium zero temperature phenomenology in driven-dissipative photonic lattices, we develop a fully novel scheme based on the use of non-Markovian reservoirs with tailored broadband spectra which allows to mimick the effect of tunable chemical potential. Our proposal can be implemented by mean of a small number of emitters and absorbers and is accessible to current technologies. We first analyse the case of a frequency-dependent emission with a square spectrum and confirm the possibility of stabilizing Mott insulator states with arbitrary integer density. Unlike the previous proposal the Mott state is robust against both losses and tunneling. A sharp transition toward a delocalized superfluid-like state can be induced by strong values of the tunneling or a change in the effective chemical potential. While an overall good agreement is found with the T=0 predictions, our analysis highlights small deviations from the equilibrium case in some parts of the parameters space, which are characterized by a non-vanishing entropy and the kinetic generation of doublon excitations. We finally consider an improved scheme involving additional frequency-dependent losses, and show in that case that the Hamiltonian ground-state is fully recovered for any choice of parameters. Our proposal, whose functionality relies on generic energy relaxation mechanisms and is not restricted to the Bose-Hubbard model, appears as a promising quantum simulator of zero temperature physics in photonic devices. The results of this Chapter are based on the following publication: - J. Lebreuilly, A. Biella, F. Storme, D. Rossini, R. Fazio, C. Ciuti and I. Carusotto, ``Stabilizing strongly correlated photon fluids with non-Markovian reservoirs'', Phys. Rev. A 96, 033828 (2017). [Chapter 4.] We adopt a broader perspective, and analyse the conditions for the emergence of analogous thermal properties in driven-dissipative quantum systems. We show that the impact of an equilibrated environment can be mimicked by several non-Markovian and non-equilibrated reservoirs. Chapter 2 already features a preliminary result in that direction, showing that in presence of a broad reservoir spectral density a given quantum system will evolve toward a Gibbs ensemble with an artificial chemical potential and temperature. In this chapter we develop a broader analysis focusing as a counterpart part on the exactly solvable model of a weakly interacting Bose Gas in the \acs{BEC} regime. Our formalism based on a quantum Langevin model, allows in particular to access both static and dynamical properties: remarkably, we demonstrate not only the presence of an equilibrium static signature, but also the validity of the fluctuation-dissipation theorem. While our results apply only for low-energy excitations for an arbitrary choice of reservoir spectral densities, we predict that a fine tuned choices of reservoirs mimicking the so-called Kennard Stepanov condition will lead to a full apparent equilibration. Such effect that we call ``pseudo-thermalization'' implies that under very specific conditions, an open quantum system can present all the properties of an equilibrated one in spite of the presence of an highly non equilibrated environment. The results of this Chapter are based on the following paper: - J. Lebreuilly, A. Chiocchetta and I. Carusotto, ``Pseudo-thermalization in driven-dissipative non-Markovian open quantum systems'', arXiv:1710.09602 (submitted for publication).