Fig. 4: Eigenvalues and eigenmodes of the system.
a, b Real (a) and imaginary (b) parts of the eigenvalues for the effective Hamiltonian [Eq. (2), curves] and the original Hamiltonian [Eq. (1), dots] as functions of \({\Delta}\phi\). c, d Expansion coefficients of the eigenmodes for the effective Hamiltonian (2) (c) and the original Hamiltonian (1) (d) as functions of \({\Delta}\phi\). The red solid and magenta dotted curves illustrate the distribution in mode am for each eigenmode (\(|\alpha _{1,2}|^2\)), and the blue solid and cyan dotted curves show the distribution in mode am+1 for each eigenmode (\(|\beta _{1,2}|^2\)). The parameters are \(\kappa ^{(1)} = \kappa ^{(2)} \equiv \kappa\), \(\gamma _m/\kappa = \gamma _{m + 1}/\kappa = 10^{ - 3}\), \({\Delta}^{(1)}/\kappa = 80\), \({\mathrm{{\Delta}}}^{(2)}/\kappa = - \kappa /[4{\mathrm{{\Delta}}}^{(1)}]\), \(g_L^{(1)}/\kappa = g_R^{(1)}/\kappa = 0.1\sqrt {|{\Delta}^{(1)}/\kappa + i/2|} ,\,g_L^{(2)}/\kappa = 0.1\sqrt {|{\Delta}^{(2)}/\kappa + i/2|}\), and \(g_R^{(2)} = g_L^{(2)}e^{ - i{\mathrm{{\Delta}}}\phi }\)
