Resonance solutions of the nonlinear Schrödinger equation: Tunneling lifetime and fragmentation of trapped condensates

It is shown here how the lifetimes and energies of resonance states can be calculated by applying the complex scaling transformation to the nonlinear Schr\"odinger equation. It is essential to first apply the complex scaling transformation to the full Hamiltonian and to subsequently derive from the result the correct complex scaled nonlinear Schr\"odinger equation. The latter equation is physically relevant and amenable to numerical calculations. To analyze the results obtained by solving this equation, it is necessary to realize the close association of resonance phenomena with fragmentation of the system. As an illustrative example, we apply this theory to the Gross-Pitaevskii nonlinear equation to calculate the tunneling lifetime of a condensate inside an external (either optical or magnetic) trap. We show that by varying the scattering length, the external potential acts like a ``selective membrane'' which controls the direction of the flux of the cold atoms through the barriers and, thereby, controls the size of the stable condensate inside the trap.