Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer

We study the quantum dynamics of the central-depleted-well (CDW) regime in a three-mode Bose Hubbard model subject to a confining parabolic potential. By introducing a suitable set of momentum-like modes we identify the microscopic variables involved in the quantization process and the dynamical algebra of the model. We describe the diagonalization procedure showing that the model reduces to a double oscillator. Interestingly, we find that the parameter-space domain where this scheme entails a discrete spectrum well reproduces the two regions where the classical trimer excludes unstable oscillations. Spectral properties are examined in different limiting cases together with various delocalization effects. These are shown to characterize quantum states of the CDW regime in the proximity of the borderline with classically unstable domains.

Figure 1

The $\tau v$ plane features two regions, ${\mathcal{D}}_{+}$ and ${\mathcal{D}}_{-}$, where the diagonalization process reduces the trimer Hamiltonian to a double-oscillator system with a discrete spectrum. ${\mathcal{D}}_{+}$ is the upper domain bounded from below by the red (upper) curve ${\Gamma }_{+}$ defined by $f\left(v\right)$ [see Eq. (19)]. ${\mathcal{D}}_{-}$ is delimited by the black straight line ${\Gamma }_0$ ($v=-0.5$) and bounded from below by the blue (lower) curve ${\Gamma }_{-}$ defined by $g\left(v\right)$ [see Eq. (27)].