The aim of this project is to extend a novel aspect of fermionic quantum gas that we have recently established, into two different, but complimenting directions. In our work, we have found that an ideal gas of fermions exhibits solitonlike structures that have roots in the phenomenon known in the literature as the quantum carpet pattern. Such a structure was shown to have a deep connection with many different branches of physics, from quantum fractals, through number theory, to quantum information processing. Two appropriate routes that we plan to explore are as follows: (i) Correlations in fermionic quantum carpets of two-component repulsive gas In our previous work, we did not accommodate for quantum correlations in the system. However, in a strongly interacting regime, they undoubtedly play a significant role, supposedly changing the character and behavior of coherent, solitonlike landscape we can observe in a weakly interacting gas. We plan to investigate our system with more realistic Ansa ̈tze for the wave functions, that inherently contain non-trivial quantum correlations and interspecies entanglement. (ii) Quantum carpets as a searching tool for exotic quantum states Quantum carpets appear in plethora of different systems, many of which was not studied from this per- spective. Emergence of such structures usually signifies a complicated evolution that can engineer exotic quantum states. We aim to use the quantum carpet pattern to search for such states, as fractional revivals of the initial configuration can suggest a highly entangled state present in the system. Our particular interest would be focused on the so-called coherent phase state that undergoes an evolution generated by Bose-Hubbard Hamiltonian without tunneling. Research project methodology In our project each of the tasks demands different theoretical approaches. The method we employ to deal with the dynamics of repulsive two-component Fermi gas is the atomic- orbital approach, also known as the Hartree-Fock method. This method is inherently mean field and as such, neglects some of the correlations present in the system. In order to accommodate for correlations, we would use the Jastrow-Slater variational wavefunction for a two-component Fermi gas. The Jastrow-Slater Ansatz includes two-body correlations between different fermions, that can be calculated within the lowest-order constrained variational approximation. The second task would involve quite a different machinery. We plan to analyze the time evolution of the coherent, superfluid state in the optical lattice that is generated by the Bose-Hubbard Hamiltonian. The structure of such a state has not been fully investigated yet and may yield new exotic states. We would like to use a quantum carpet pattern to search for such states. In order to do that, we have to use tools coming from the general quantum mechanics, like SU(k)-coherent states, and also from quantum information science, like Fisher information. The latter would be used to analyze found states in the context of quantum metrology. Expected impact of the research project on the development of science Recapitulating, each of the proposed tasks introduces a new quality by either connecting problems from dif- ferent branches of physics or investigating a novel tool that a broad audience of quantum information society might find useful. Taking correlations into account in the dynamical system of repulsive two-component Fermi gas would not only study behavior of intriguing patterns known as quantum carpets, but would also shed light on the long standing problem of the ferromagnetic instability in such a binary spin-mixture. Moreover, quan- tum carpets on their own offer a visual probe to scan for exotic quantum states in various physical setups. We propose to investigate this feature in order to test its effectiveness by studying a system widely used in the quantum information science.
