We study the equilibrium properties of a one-dimensional mixture of two Tonks-Girardeau gases on a ring geometry in the limit of strongly repulsive interspecies interactions. We derive the exact many-body wave function and compare it to the SU(2) solution where intra- and interspecies interactions are also diverging but equal. We focus on the role of the SU(2) symmetry breaking on the behavior of the large- and short-distance correlations by studying the zero-momentum occupation number and the Tan's contact from the asymptotic behavior of the momentum distribution. Although the symmetry is only weakly broken, it has important consequences on spin correlations in the system as the reduction by a factor of two of the zero-momentum occupation number with respect to the SU(2) case in the thermodynamic limit and the decrease of the Tan's contact.
We study the equilibrium properties of a one-dimensional mixture of two Tonks-Girardeau gases on a ring geometry in the limit of strongly repulsive interspecies interactions. We derive the exact many-body wave function and compare it to the SU(2) solution where intra- and interspecies interactions are also diverging but equal. We focus on the role of the SU(2) symmetry breaking on the behavior of the large- and short-distance correlations by studying the zero-momentum occupation number and the Tan's contact from the asymptotic behavior of the momentum distribution. Although the symmetry is only weakly broken, it has important consequences on spin correlations in the system as the reduction by a factor of two of the zero-momentum occupation number with respect to the SU(2) case in the thermodynamic limit and the decrease of the Tan's contact.
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.
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.
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.
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.
In one-dimensional bosonic quantum mixtures with SU(2)-symmetry breaking Hamiltonian, the dynamical evolution explores different particle exchange symmetry sectors. For the case of infinitely strong intra-species repulsion, the hallmark of such symmetry oscillations are time modulations of the momentum distribution [Phys. Rev. Lett. 133, 183402 (2024)], an observable routinely accessed in experiments with ultracold atoms. In this work we show that this phenomenon is robust in strongly interacting quantum mixtures with arbitrary inter-species to intra-species interaction strength ratio. Taking as initial state the ground state of the SU(2) symmetric Hamiltonian and time-evolving with the symmetry breaking Hamiltonian, we analyze how the amplitude and frequency of symmetry oscillations, and thus of the momentum distribution oscillations, depend on the strength of the symmetry-breaking perturbation. We find that the set of symmetry sectors which are coupled during the time evolution is dictated by the spin-flip symmetry of the initial state and show that the population of the initial state may vanish periodically, even in the thermodynamic limit, thus revealing the robustness and universality of the symmetry oscillations.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures, and in particular our new exact solution for the case of symmetric-breaking mixtures.The solutions allow one to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach, as well as to describe finite-temperature conditions, spin, and magnetization effects. I will present some uses of these solutions to obtain the exact quantum dynamics, both for the small oscillation regime and for the strongly out-of-equilibrium spin dynamics, where I will propose a protocol to observe symmetry oscillations in binary bosonic mixtures.