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).

The analysis and the results contained in this work are rooted in a first contact between the quantum theory and the general theory of relativity. By first contact it is meant that we are not considering candidates for “unified theories", but rather we focus on aspects of the full quantum theory in changing geometric backgrounds: the analysis of such an interaction already had important applications in cosmology, e.g. in the description of the evolution of fields in inflationary scenarios. Another compelling – and still growing – area of application is the study of thermodynamical properties of gravitional systems, which covers the main bulk of this thesis.

In this Thesis is presented a study on dynamical properties of mixtures of ultraold Bose gases. The behaviour of this system in different regimes is analysed: with and without coherent coupling between the two components, in homogeneous and harmonic shaped trapping potentials and in different dimensions and geometries. Most of the results presented here have been obtained by means of numerical solutions of coupled Gross-Pitaevskii equations and have been compared with theoretical predictions (and sometimes experiments), describing the same phenomena. In particualr the stability of persistent currents in a two-component Bose-Einstein condensate in a toroidal trap is studied in both the miscible and the immiscible regime. In the miscible regime we show that superflow decay is related to linear instabilities of the spin-density Bogoliubov mode. We find a region of partial stability, where the flow is stable in the majority component while it decays in the minority component. We also characterize the dynamical instability appearing for a large relative velocity between the two components. In the immiscible regime the stability criterion is modified and depends on the specific density distribution of the two components. The effect of a coherent coupling between the two components is also discussed. A study on the collective modes of the minority component of a highly unbalanced Bose-Bose mixture is also presented. In the immiscible case we find that the ground state can be a two-domain walls soliton. Although the mode frequencies are continuous at the transition, their behaviour is very different with respect to the miscible case. The dynamical behaviour of the solitonic structure and the frequency dependence on the inter- and intra-species interaction is numerically studied using coupled Gross-Pitaevskii equations. The results of the study on the static and the dynamic response of coherently coupled two component Bose-Einstein condensates due to a spin-dipole perturbation is also sown. The static dipole susceptibility is determined and is shown to be a key quantity to identify the second order ferromagnetic transition occurring at large inter-species interactions. The dynamics, which is obtained by quenching the spin-dipole perturbation, is very much affected by the system being paramagnetic or ferromagnetic and by the correlation between the motional and the internal degrees of freedom. In the paramagnetic phase the gas exhibits well defined out-of-phase dipole oscillations, whose frequency can be related to the susceptibility of the system using a sum rule approach. In particular in the interaction SU (2) symmetric case, when all the two-body interactions are the same, the external dipole oscillation coincides with the internal Rabi flipping frequency. In the ferromagnetic case, where linear response theory is not applicable, the system shows highly non-linear dynamics. In particular we observe phenomena related to ground state selection: the gas, initially trapped in a domain wall configuration, reaches a final state corresponding to the magnetic ground state plus small density ripples. Interestingly, the time during which the gas is unable to escape from its initial configuration is found to be proportional to the square root of the wall surface tension.

This thesis is devoted to the study of the conformation of monosacchrides in six-membered ring form. The main goal is to develop and apply new computational tools to investigate conformational properties and to improve the description of carbohydrates in the framework of molecular dynamics simulations. In the field of monosaccharides, modeling the system within the molecular dynamics framework presents troublesome aspects. The most important issue is that some force fields (e.g., the chosen gromos 45a4 parameter set) fail in reproducing the conformational preferences of the sugar constituents, with the appearance of unphysical conformations. This lack stems from the fact that the conformational behavior, dominated by few structures, generates a severe bottleneck: the non-ergodicity of the system by any practical means. This aspect explains the interest in free energy calculations, and methods exist, such as umbrella sampling or metadynamics, that allow to accelerate the sampling of different conformations by adding bias forces. In general, accelerated sampling methods are based on the choice of collective variables (CVs), which is of particular importance for the proper reconstruction of free energy landscapes. In the field of conformational analysis, suitable CVs have to be considered to describe non-planar, puckered conformations of cyclic structures. One of the main goals of this work is the enhancement of the gromos 45a4 force field for carbohydrates, with respect to the ability to describe ring conformation (that is, puckering) of six-membered rings. To this end, the development of efficient computational tools for the investigation of the general puckering problem are presented. In particular, we indicate how to exploit the capabilities of the metadynamics algorithm applied to the investigation of puckered ring conformers, exploring also different parametrizations of puckered structures to assess their respective advantages as collective variables for metadynamics.

Quantum Monte Carlo Methods applied to strongly correlated and highly inhomogeneous many-Fermion systems

We use mean-field theory to investigate the dynamics of superfluid fermions. This thesis includes our two works. The first one is to study Josephson oscillations and self-trapping of superfluid fermions in a double-well potential with time-dependent Bogoliubov-de Gennes equations. We investigate the behaviour of a two-component Fermi superfluid. We numerically solve the time-dependent Bogoliubov-de Gennes equations and characterize the regimes of Josephson oscillations and self-trapping for different potential barriers and initial conditions. In the weak link limit the results agree with a two-mode model where the relative population and the phase difference between the two wells obey coupled nonlinear Josephson equations. A more complex dynamics is predicted for large amplitude oscillations and large tunneling. The second one is to calculate the dynamic structure factor of unitary fermions. We have studied the dynamic structure factor of unitary fermions both at zero and finite temperature using the Bogoliubov-de Gennes theory and also Superfluid Local Density Approximation. We have derived the expression of the linear response function and the dynamic structure factor in the random phase approximation. At zero temperature, the SLDA+RPA formalism indeed provides a better accuracy at low momentum transfer and also its static structure factor is closer to quantum Monte Carlo value than that in BdG+RPA; however SLDA seems to give worse results for the molecular excitations at large momentum transfer. We have discussed the role of temperature and the comparison between SLDA and BdG, as well as with experimental data. The analysis is still at a preliminary level, but it suggests that mean-field theories can indeed be used to extract quantitative information about the order parameter and the excitations of the system by two-photon Bragg scattering experiments.