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state of the system. Since the BdG spectrum refers to all excitations of the state, one can attribute in the linear limit to each mode an excited state of the linear problem. The energy gap between the ground state and the first excited state corresponds to the energy of the first mode. Since the frequency of the so-called bosonic Josephson junction , namely the oscillation frequency characterizing the transfer of atoms between the two wells, is equal to this frequency in the linear limit, this mode is connected to the oscillation of atoms between the two wells. We investigate this fact in more detail below (see Sec. V) by direct simulations in the framework of Eq. (1). The frequency value of the second nonzero BdG mode of the ground state spectrum is correspondingly equal to the energy gap between the ground state and the second excited state in the linear limit, and so forth. We define the difference in the way that these frequency differences are positive. 

