Emergent properties of small systems of trapped ultracold atoms

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Brandt, Benedikt Bruno
Landman, Uzi
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Systems of ultracold atoms are among the most versatile tools in condensed matter physics. Their applications range from the study of fundamental quantum mechanics to applications in quantum information. Once fundamental control and understanding of small (<10 sites and particles) systems of ultracold atoms is achieved, scaling up these systems to larger numbers of sites and particles will allow realization of Feynman's vision of a quantum simulator that behaves just like nature. Towards this aim, we study the properties of small systems of ultracold atoms using exact numerical diagonalization of the Hamiltonian (configuration interaction methodology) and use those exact microscopic results to assess and guide the development of appropriate spin models. The fundamental building block of ultracold atom systems consists of two atoms in a double well for which we have analyzed the wave function anatomy, revealing interesting phenomena like the formation of Wigner Molecules at strong repulsion. Furthermore, we studied the von Neuman entropy and the structure and occupation numbers of the natural orbitals for the full range of interaction strengths for select states. A natural progression is to assemble the double well building blocks to realize spin chains, which enable the study of quantum magnetism. To that aim we analyzed three and four particle systems in different configurations and interpreted their results in the context of the well-known Heisenberg and t-J model. Using our microscopic methodology, we assessed the accuracy and applicability of these spin models, provided insight into the wave function anatomy and demonstrated how the Wigner molecule formation enables the mapping to spin chains. We then moved to the study of plaquette systems and were able to observe the energetics of pairing in hole-doped plaquettes, which is believed to be at the heart of unconventional superconductivity. Using conditional probability densities and symmetry braking we furthermore directly observed the presence of hole pairing in the many body wavefunction. Our microscopic calculations provide essential guidance in the choice of parameters and demonstrate the shortcomings of the commonly used t-J and Hubbard model in certain parameter ranges. A third promising aspect of ultracold atom systems that was explored in this thesis is the study of second order correlation functions in position and momentum space, which have recently become experimentally accessible. Measurements of such correlation functions can be interpreted in the context of the famous Hong-Ou-Mandel effect, which was first studied in the context of quantum optics and allows for the probing of fundamental quantum statistics. We show detailed results for the full range of interaction strengths from the strongly attractive to the strongly repelling (Tonks–Girardeau) limit, for the ground and several excited states. We furthermore developed an analytical methodology for calculating the second order position and momentum space correlation functions from approximate models like the Hubbard model and confirmed their accuracy using our microscopic configuration interaction calculations.
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