Wavepacket dynamics in a quantum double-well system with Razavy's potential coupled to a harmonic oscillator

We have studied wavepacket dynamics in the Razavy hyperbolic double-well (DW) potential which is coupled to a harmonic oscillator (HO) by linear and quadratic interactions. Taking into account the lowest two states of DW and $(N+1)$ states of HO ($N=1$ to 10), we evaluate eigenvalues and eigenfunctions of the composite system. An analytical calculation is made for $N=1$ and numerical calculations are performed for $1 < N \leq 10$. Quantum tunneling of wavepackets is realized between two bottoms of composite potential $U(x,y)$ where $x$ and $y$ denote coordinates in DW and HO potentials, respectively. It has been shown that with increasing $N$ and/or the coupling strength, the tunneling period is considerably increased. Phase space plots of $\langle x \rangle$ vs. $\langle p_x \rangle$ and $\langle y \rangle$ vs. $\langle p_y \rangle$ are elliptic, where $\langle \cdot \rangle$ denotes an expectation value for the two-term wavepacket. This result is quite different from the relevant one previously obtained for the quartic DW potential with the use of the quantum phase space representation [Babyuk, arXiv:0208070]. Similarity and difference between results calculated for linear and quadratic couplings, and the uncertainty relation in the model are discussed.