Bifurcations in resonance widths of an open Bose-Hubbard dimer

We investigate the structure of resonance widths of a Bose-Hubbard dimer with intersite hopping amplitude $k$, which is coupled to a continuum at one of the sites with strength $\gamma$. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term $\chi$ the resonances undergo consequent bifurcations. For $\Lambda =k∕\gamma ⩾0.5$, the bifurcation points follow a scaling law ${\tilde{\chi }}_m\equiv {\chi }_{m}N∕k=f_{\Lambda }(m-0.5∕\Lambda )$, where $N$ is the number of bosons. For the function $f_{\Lambda }$ two different $\Lambda$ dependences are found around the minimum and maximum bifurcation points.

Figure 1

(Color online) Bifurcation diagram for the rescaled resonance widths $\tilde{\Gamma }$ vs the interaction strength $\tilde{\chi }$ for three representative coupling ratios: (a) $\Lambda =1$, (b) $\Lambda =0.5$, and (c) $\Lambda =0.4$. Circles (엯) and diamonds (◇) correspond to $k=1$ and 0.1, respectively. In this particular example we consider $N=10$. The solid lines are drawn for the eye and show the flow of resonance widths as $\tilde{\chi }$ is changing. The stars (⋆) indicate the predicted asymptotic values of the resonance widths [see Eqs. (6, 7)]. The dashed “blue” line is the “classical” result ${\Gamma }_{\mathrm{cl}}^{\pm }$.

Figure 2

(Color online) The bifurcation points ${\tilde{\chi }}_m$ plotted against the scaled index $\mu$ for representative values of the dimensionless ratio $\Lambda =0.65,1,{10}^6$. The symbols correspond to different boson numbers $N$ and different hopping amplitudes $k$: $N=10$, $k=1$ (black), $N=30$, $k=10$ (red), and $N=50$ $k=20$ (blue). The symbol $A$ corresponds to $N=20$, $k=0.1$. The solid line indicates the large $N$ limit of the case $\Lambda ={10}^6$, while the dashed line corresponds to the value $\Lambda =0.5$ and a system size of $N=50$. In the inset we show a magnification of the main figure.

Figure 3

(Color online) The rescaled bifurcation points ${\tilde{\chi }}_m∕{\tilde{\chi }}^{*}$ for three values of $\Lambda =0.65,1,{10}^6$. The symbols correspond to different boson numbers $N$ and different hopping amplitudes $k$ (see Fig. 2). The solid line indicates the large $N$ limit of the case $\Lambda ={10}^6$. Inset: the extracted scaling parameter ${\tilde{\chi }}^{*}$ as a function of the dimensionless ratio ${\Lambda }^{-1}$.