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La plupart des objets quantiques individuels développés jusqu’à aujourd’hui ne permettent pas de satisfaire toutes les conditions nécessaires pour la construction d’un simulateur quantique. Une possibilité pour obtenir un système quantique robuste est de combiner plusieurs de ces approches. Dans cette thèse, nous décrivons les résultats obtenus sur deux systèmes expérimentaux développés dans ce but.La première partie de cette thèse décrit un système hybride d’atomes neutres couplés à des qubits supraconducteurs, en construction à l’Université du Maryland. La solution envisagée pour placer un ensemble d’atomes froids à proximité de la surface supraconductrice est de piéger les atomes dans le champ évanescent se propageant autour d’une fibre optique nanométrique. Nous avons développé un dispositif permettant la production de fibres optiques nanométriques de transmission optique supérieure à 99.95% dans le mode fondamental. Nous avons également optimisé la transmission de quelques modes d’ordres supérieurs, ce qui pourra s’avérer utile pour le piégeage d’atomes.La seconde partie de cette thèse décrit un système développé à l’Institut d’Optique et comprenant des atomes neutres piégés dans des matrices de pinces optiques. Dans ce cas, nous excitons les atomes dans des états de Rydberg afin de bénéficier de fortes interactions interatomiques. Nous avons caractérisé les interactions de van der Waals et les interactions résonantes entre deux atomes individuels, et démontré le caractère cohérent de l’interaction dipolaire. Nous avons enfin simulé la dynamique d’une chaine élémentaire de spins dans une matrice de trois atomes / Most platforms that are being developed to build quantum simulators do not satisfy simultaneously all the requirements necessary to implement useful quantum tasks. Robust systems can be constructed by combining the strengths of multiple approaches while hopefully compensating for their weaknesses. This thesis reports on the progress made on two different setups that are being developed toward this goal.The first part of this thesis focuses on a hybrid system of neutral atoms coupled to superconducting qubits that is under construction at the University of Maryland. Sub-wavelength diameter optical fibers allow confining an ensemble of cold atoms in the evanescent field surrounding the fiber, which makes them ideal for placing atoms near a superconducting surface. We have developed a tapered fiber fabrication apparatus, and measured an optical transmission in excess of 99.95% for the fundamental mode. We have also optimized tapered fibers that can support higher-order optical modes with high transmission, which may be useful for various optical potential geometries.The second part of this thesis focuses on a system of neutral atoms trapped in arrays of optical tweezers that has been developed at the Institut d’Optique. Placing the atoms in highly excited Rydberg states allows us to obtain strong interatomic interactions. Using two individual atoms, we have characterized the pairwise interactions in the van der Waals and resonant dipole-dipole interaction regimes, providing a direct observation of the coherent nature of the interaction. In a three-atom system, we have finally simulated the dynamics of an elementary spin chain

This Thesis focuses on different aspects of quantum computation theory: adiabatic quantum algorithms, decoherence during the adiabatic evolution and quantum simulators. After an overview on the area of quantum computation and setting up the formal ground for the rest of the Thesis we derive a general error estimate for adiabatic quantum computing. We demonstrate that the first-order correction, which has frequently been used as a condition for adiabatic quantum computation, does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections and shows that the computational error can be made exponentially small – which facilitates significantly shorter evolution times than the first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time of order of the inverse minimum energy gap is sufficient and necessary. Furthermore, exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm. It turns out that (in contrast to first-order transitions) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. Finally, we propose the use of electron systems to construct laboratory systems based on present-day technology which reproduce and thereby simulate the quantum dynamics of the Ising model and the O(3) nonlinear sigma model.

Ultracold atomic systems have provided an ideal platform to study the physics of strongly interacting many body systems in an unprecedentedly controlled and clean environment. And, since fermions are the building blocks of visible matter, being naturally motivated we focus on the physics of ultracold fermionic systems in this thesis. There have been many recent experimental developments in these systems such as the creation of synthetic gauge fields, realization of dimensional crossover and realization of systems with synthetic dimensions. These developments pose many open theoretical questions, some of which we address in this thesis. We start the discussion by studying the spectral function of an ideal spin-12 Fermi gas in a harmonic trap in any dimensions. We discuss the performance of the local density approximation (LDA) in calculating the spectral function of the system by comparing it to exact numerical results. We show that the LDA gives better results for larger number of particles and in higher dimensions. Fermionic systems with quasi two dimensional geometry are of great importance because of their connections to the high-Tc superconducting cuprate materials. Keeping this in mind, we consider a spin-12 fermionic system in three dimensions interacting with a contact interaction and confined by a one dimensional optical potential in one direction. Using the Bogoliubov-de Gennes formalism, we show that with increasing the depth of the optical potential the three dimensional superfluid evolves into a two dimensional one by looking at the shifts in the radio-frequency spectrum of the system and the change in the binding energy of the pairs that are formed. The next topic of interest is studying the effect of synthetic gauge fields on the ultracold fermionic systems. We show that a synthetic non-Abelian Rashba type gauge field has experimentally observable signatures on the size and shape of a cloud of a system of non-interacting spin-12 Fermi system in a harmonic trap. Also, the synthetic gauge field in conjunction with the harmonic potential gives rise to ample possibilities of generating novel quantum Hamiltonians like the spherical geometry quantum Hall, magnetic monopoles etc. We then address the physics of fermions in “synthetic dimensions”. The hyperfine states of atoms loaded in a one dimensional optical lattice can be used as an extra dimension, called the synthetic dimension (SD), by using Raman coupling. This way a finite strip Hofstadter model is realized with a tunable flux per plaquette. The experimental realization of the SD system is most naturally possible in systems which also have SU(M) symmetric interactions between the fermions. The SU(M) symmetric interactions manifest as long-ranged along the synthetic dimension and is the root cause of all the novel physics in these systems. This rich physics is revealed by a mapping of the Hamiltonian of the system to a system of particles interacting via an SU(M) symmetric interaction under the influence of an SU(M) Zeeman field and a non-Abelian SU(M) gauge field. For example, this equivalence brings out the possibility of generating a non-local interaction between the particles at different sites; while the gauge filed mitigates the baryon (SU(M) singlet M-body bound states) breaking effect of the Zeeman field. As a result, the site localized SU(M) singlet baryon gets deformed and forms a “squished baryon”. Also, finite momentum dimers and resonance like states are formed in the system. Many body physics in the SD system is then studied using both analytical and numerical (Density Matrix Renormalization Group) techniques. This study reveals fascinating possibilities such as the formation of Fulde-Ferrell-Larkin-Ovchinnikov states even without any “imbalance” and the possibility to evolve a “ferromagnet” to a “superfluid” by the application of a magnetic field. Other novel fermionic phases with quasi-condensates of squished baryons are also demonstrated. In summary, the topics addressed in this thesis demonstrate the possibilities and versatilities of the ultracold fermionic systems used in conjunction with synthetic gauge fields and dimensions

Dans cette thèse nous abordons des aspects topologiques de la matière condensée. Les états topologiques sont insensibles à un large spectre des perturbations externes et au désordre – une propriété indispensable dans le domaine d'information quantique. L’effet des interactions dans des systèmes topologiques est pourtant loin d’être bien maîtrisé à ce jour. Dans ce travail, nous étudions la corrélation entre la description topologique et l'effet des interactions. Afin d'accomplir notre but, nous utilisons des méthodes analytiques et numériques. Nous nous intéressons aussi à des sondes expérimentales qui peuvent être utilisées pour vérifier nos prédictions théoriques. Tout d’abord, nous étudions la version bosonique en interactions du modèle de Haldane – le modèle célèbre qui décrit l’effet Hall anomal. Nous proposons son implémentation expérimentale dans des circuits quantiques, basée sur l’application de perturbation périodique dépendantes du temps – méthodologie qui s’appelle l’ingénierie de Floquet. En poursuivant ces idées, nous étudions la version bosonique du modèle de Kane-Mele d’un isolant topologique. Ce modèle possède un diagramme de phase très riche. En particulier, lorsque les interactions sont fortes, nous observons l’apparition d’un modèle de magnétisme frustrée présentant une variété d'états exotiques. La mise en œuvre de ces modèles dans des réseaux d'atomes ultra-froids ou des circuits quantiques permettra de sonder expérimentalement les propriétés exotiques que nous avons observées. Ensuite, nous abordons d’une manière plus détaillée la réalisation expérimentale des modèles topologiques dans des circuits quantiques, en considérant le cas particulier du modèle de Su-Schrieffer-Heeger en couplage fort. Nous testons aussi des nouvelles sondes qui peuvent être utilisées afin de mesurer la phase de Zak et en déduire la topologie du système. Finalement, nous nous intéressons aux sondes hors d’équilibre et des méthodes pour tester les propriétés spectrales de systèmes quantiques, en utilisant l’approche de purification, pertinent pour le numérique et les expériences. / In this thesis we study the topological aspects of condensed matter physics, that received a revolutionary development in the last decades. Topological states of matter are protected against perturbations and disorder, making them very promising in the context of quantum information. The interplay between topology and interactions in such systems is however far from being well understood, while the experimental realization is challenging. Thus, in this work we investigate analytically such strongly correlated states of matter and explore new protocols to probe experimentally their properties. In order to do this, we use various analytical and numerical techniques. First, we analyze the properties of an interacting bosonic version of the celebrated Haldane model – the model for the quantum anomalous Hall effect. We propose its quantum circuit implementation based on the application of periodic time-dependent perturbations – Floquet engineering. Continuing these ideas, we study the interacting bosonic version of the Kane-Mele model – the first model of a topological insulator. This model has a very rich phase diagram with an emergence of an effective frustrated magnetic model and a variety of symmetry broken spin states in the strongly interacting regime. Ultra-cold atoms or quantum circuits implementation of both Haldane and Kane-Mele bosonic models would allow for experimental probes of the exotic states we observed. Second, in order to deepen the perspectives of quantum circuit simulations of topological phases we analyze the strong coupling limit of the Su-Schrieffer-Heeger model and we test new experimental probes of its topology associated with the Zak phase. We also work on the out-of-equilibrium protocols to study bulk spectral properties of quantum systems and quantum phase transitions using a purification scheme which could be implemented both numerically and experimentally.

This Thesis focuses on different aspects of quantum computation theory: adiabatic quantum algorithms, decoherence during the adiabatic evolution and quantum simulators. After an overview on the area of quantum computation and setting up the formal ground for the rest of the Thesis we derive a general error estimate for adiabatic quantum computing. We demonstrate that the first-order correction, which has frequently been used as a condition for adiabatic quantum computation, does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections and shows that the computational error can be made exponentially small – which facilitates significantly shorter evolution times than the first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time of order of the inverse minimum energy gap is sufficient and necessary. Furthermore, exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm. It turns out that (in contrast to first-order transitions) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. Finally, we propose the use of electron systems to construct laboratory systems based on present-day technology which reproduce and thereby simulate the quantum dynamics of the Ising model and the O(3) nonlinear sigma model.