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Extracting atoms one by one from a small matter-wave soliton

Excitations of small one-dimensional matter-wave solitons are considered within a framework of the attractive Bose-Hubbard model. The initial eigenstates of the system are found by exact diagonalization of the Bose-Hubbard Hamiltonian. We drive transitions between the eigenstates by inducing a weak modulation of the tunnelling rate and show that a single atom can be extracted while the remaining atoms stay localized despite the persistent external modulation. This scheme suggests the experimental realization of small matter-wave solitons with deterministic number of atoms. In addition, the knowledge of exact eigenstates allows identification of the selection rules for transitions between the different eigenstates of the Hamiltonian. One selection rule is related to the translation symmetry of the system. Another one is strictly applicable only on a subspace of the total Hilbert space and is related to the parity symmetry. We show that in the strongly interacting limit this selection rule has implications on the entire Hilbert space. We discuss its signatures on the system's dynamics and consider how it can be observed experimentally with ultracold atoms.