FIG. 2. (Color online) BdG Spectrum of the states shown in Fig. 1. First panel shows the spectrum of the first branch, second panel the spectrum of the second branch and so forth. The (blue) ⋇ symbol denotes an eigenmode of positive Krein sign (or zero Krein sign for a vanishing eigenfrequency), the (green) × symbol an eigenmode of negative Krein sign, and the (red) + symbol an eigenmode associated to a vanishing Krein sign with complex/imaginary eigenfrequency. 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. 12  0  50  100  150  200  250 r  (2 Hz)  0  5  10  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150  200  250 r  (2 Hz)  0  20  40  60  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150 r  (2 Hz)  0  10  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150  200  250 r  (2 Hz)  0  5  10  3  3.3 i  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150 r  (2 Hz)  0  20  40  60  3  3.3 i  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150 r  (2 Hz)  0  20  40  60  2.4  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)  0  0.5  2.142 2.144 r  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150  200  250 r  (2 Hz)  0  5  10  2.4  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150 r  (2 Hz)  0  5  10  2.4  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)