Master Thesis by Hjalte Frellesvig – Niels Bohr Institute - University of Copenhagen

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Master Thesis by Hjalte Frellesvig

In this thesis a theory for steady-state squeezing of an abstract collective spin-variable is developed for particles in a Bose-Einstein condensate. Such a theory can be put to direct use in the field of quantum-metrology, since squeezed states can help improve the precision of atomic clocks. In the two-mode approximation it is easy to see that a steady-state Bose-Einstein condensate will be spin-squeezed, since the Hamiltonian describing such a system has the maximally squeezed state as the lowest energy eigenstate. The calculation central to this thesis is done using a Bogoliubov transformation of perturbations of the quantum-field operator describing the particles in the condensate. This transformation will diagonalize the Hamiltonian, which enables us to find an (almost) complete set of eigenvalues and eigenstates. These states can be used to calculate the expectation values of the angular momentum operators, from which the spin-squeezing can be calculated. The atoms in the condensate are described using two constants U and Uab proportional to the scattering lengths for atoms of equal and different spin respectively. The result of a numerical implementation of the theory is that for zero temperature the optimal atoms have U ≈ Uab ≈ 0. This is contrary to the case of higher, physical temperatures, where atoms with U >> Uab and U + Uab >> 1 will maximize the squeezing.Further results are that the most spin-squeezing is produced for as large a particle number and as low a temperature as possible.