The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k^−4 laws. The weights of these tails, denoted as Tan’s contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan’s contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattistheorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan’s contact as a function of the interaction, temperature and transverse confinement. These laws. Display displadisplay display interesting effects related to strong correlations and dimensionality.
The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k^−4 laws. The weights of these tails, denoted as Tan’s contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan’s contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattistheorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan’s contact as a function of the interaction, temperature and transverse confinement. These laws. Display displadisplay display interesting effects related to strong correlations and dimensionality.
The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical k−4 laws. The weights of these tails, denoted as Tan's contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan's contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan's contact as a function of the interaction, temperature and transverse confinement. These laws display interesting effects related to strong correlations and dimensionality.
We consider multi-component quantum mixtures (bosonic, fermionic, or mixed) with strongly repulsive contact interactions in a one-dimensional harmonic trap. In the limit of infinitely strong repulsion and zero temperature, using the class-sum method, we study the symmetries of the spatial wave function of the mixture. We find that the ground state of the system has the most symmetric spatial wave function allowed by the type of mixture. This provides an example of the generalized Lieb–Mattis theorem. Furthermore, we show that the symmetry properties of the mixture are embedded in the large-momentum tails of the momentum distribution, which we evaluate both at infinite repulsion by an exact solution and at finite interactions using a numerical DMRG approach. This implies that an experimental measurement of the Tan’s contact would allow to unambiguously determine the symmetry of any kind of multi-component mixture.
A universal ${k}^{\ensuremath{-}4}$ decay of the large-momentum tails of the momentum distribution, fixed by Tan's contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. Here we consider a repulsive multicomponent Fermi gas under harmonic confinement, as in the experiment of G. Pagano et al. [Nat. Phys. 10, 198 (2014)], realizing a gas with tunable $\text{SU}(\ensuremath{\kappa})$ symmetry. We exploit an exact solution at infinite repulsion to show a direct correspondence between the value of the Tan's contact for each of the $\ensuremath{\kappa}$ components of the gas and the Young tableaux for the SN permutation symmetry group identifying the magnetic structure of the ground state. This opens a route for the experimental determination of magnetic configurations in cold atomic gases, employing only standard (spin-resolved) time-of-flight techniques. Combining the exact result with matrix-product-state simulations, we obtain the Tan's contact at all values of repulsive interactions. We show that a local-density approximation (LDA) on the Bethe-ansatz equation of state for the homogeneous mixture is in excellent agreement with the results for the harmonically confined gas. At strong interactions, the LDA predicts a scaling behavior of the Tan's contact. This provides a useful analytical expression for the dependence on the number of fermions, number of components, and interaction strength. Moreover, using a virial approach, we study the Tan's contact behavior at high temperature and in the limit of infinite interactions, and we show that it increases with the temperature and the number of components. At zero temperature, we predict that the weight of the momentum distribution tails increases with interaction strength and the number of components if the population per component is kept constant. This latter property was experimentally observed in G. Pagano et al. [Nat. Phys. 10, 198 (2014)].
Ultracold quantum gases are an ideal toolbox for simulating complex condensed or nuclear matter systems and to investigate fundamental quantum properties of matter. In this thesis, we will investigate universal properties connecting the high-momentum tail of the momentum distribution, the short-range correlations and the thermodynamics, encapsulated in Tan's relations. These relations are especially useful in the strongly interacting case, where perturbative approaches usually fail. With the aid of Tan's universal relations, we can still come to general conclusions about strongly interacting quantum gases. In particular, the momentum distribution exhibits a characteristic algebraic decay, unlike the exponential decay of the non-interacting case. The main focus in this thesis is on the one-dimensional, fermionic case, where we study the highly polarized case (the one-dimensional Fermi polaron), verifying Tan's relations using a variety of theoretical tools. In addition, we show that localized systems exhibit a universal, dynamical instability to delocalization when a short-range interaction between particles is switched off rapidly. This delocalization process relies on the algebraic decay of the momentum distribution, which guarantees that at least some of the delocalized single-particle states are occupied with a finite probability. Finally, we investigate the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover for the three-dimensional Fermi gas and develop a novel method to describe the breakdown of the Fermi liquid description in the vicinity of the critical temperature for superfluidity, in good agreement with a recent experiment.
Understanding quantum many-body systems with interactions is an important problem common to various fields of physics, such as atomic physics, condensed matter physics, nuclear physics, and particle physics.Among others, ultracold atoms have provided ideal grounds to study strongly interacting systems and to explore exotic quantum states because of their high controllability.We can widely control the strengths of the interactions from weak to strong as well as the geometry of the systems.In order to deepen our understanding of quantum manybody systems with interactions, we in this thesis investigate two types of one-dimensional (1D) systems realizable with ultracold atoms; bosons and spinless fermions near two-body resonances and bosons near a three-body resonance.One-dimensional systems near two-body resonances have been intensively studied because, in the homogeneous cases, they are exactly solvable systems.The energy spectra and thermodynamic quantities exactly obtained by the Bethe ansatz have greatly contributed to deepening our comprehension of many-body systems in 1D.On the other hand, the exact computations of correlation functions, which encode information about excitations of the systems, are in general much more complicated even in the integrable systems.Motivated by this background, we derive a series of exact relations for correlation functions in bosons and fermions near two-body resonances.These relations called universal relations originate from the universal properties of the resonant systems: Their properties depend on the interactions only through the scattering lengths characterizing the low-energy scattering.The universal relations are strong constraints on the systems because the relations hold for any particle number, scattering length, temperature, and with or without a trapping potential.The relations include the asymptotic behaviors of correlation functions in high-energy regime as well as the energy relations, in which the energies are expressed in term of the momentum distributions.The universal relations involve two-and three-body contacts, which are the integrals of local pair and triad correlations, respectively.Firstly we derive, in both bosons and fermions, universal relations for static correlation functions such as static structure factors and momentum distributions within the first-quantized formalisms.The power laws of the static correlation functions for large momentum and the energy relations are obtained.The coefficients in these power-law behaviors are proportional to the two-body contact.We clarify that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one.Secondly, we derive universal relations for dynamic correlation functions for bosons.Using the operator product expansion (OPE) in the field theoretical formalism, we obtain the largeenergy-momentum behaviors of the dynamic structure factor, the dynamic current correlation, and the single-particle spectral density as well as the quasiparticle energy and the scattering rate in high-energy regime.While the behaviors of the dynamic structure factor and the current correlation are proportional to the two-body contact, that of the single-particle spectral density is proportional to the number density.Because of the interrelation between 1D bosons and fermions near two-body resonances, the results of the dynamic structure factor and the current i
We have conducted an analytical and numerical investigation of observables for one-dimensional ultracold atomic spin mixtures with contact interaction between atoms. These particles can be either bosons, fermions, or a combination of both. The focus was on studying the strong repulsion regimes for their high correlation features. Furthermore, we have extensively explored the symmetry properties of the corresponding systems. We first analyzed the correlation properties of ultracold atomic gases trapped in a box-shaped potential. We examined both the case of identical particles and the more complex spin mixtures. Our findings demonstrate that the hardwalls of the trap have a significant impact on the tail of the momentum distribution. Specifically, we observed that the trap's discontinuity causes a breakdown of Tan's relation. As a result, the momentum distribution showcases an oscillatory tail, which we utilized to propose an experimental method for determining the state of a bosonic spin mixture. Then, we considered a spin mixtures model, which breaks the system's symmetry. Throughout joint correlations and symmetry analysis, we have exposed that even a slight symmetry breaking has a tremendous effect on the long-distance coherence of the system. Alongside this, we have provided an experimental protocol for realizing such a system. By this means, we suggest that new physical states can be engineered to encode information on the system. Finally, we moved to a more involving system of ring-shaped trapped bosonic mixtures with an artificial gauge field generated via the application of flux. We were able to obtain the exact solution for balanced mixtures as well as the corresponding spectrum. Thanks to the symmetry analysis, we have assigned a well-defined symmetry to each part of the spectrum, enabling a deeper understanding of the system. This investigation constitutes an engaging step toward experimental setups for selecting a chosen spin mixture symmetry state.