Asymmetric Nonlinear System is Not sufficient for Non-Reciprocal Quantum Wave Diode

We demonstrate symmetric wave propagations in asymmetric nonlinear quantum systems. By solving the nonlinear Sch\"ordinger equation, we first analytically prove the existence of symmetric transmission in asymmetric systems with a single nonlinear delta-function interface. We then point out that a finite width of the nonlinear interface region is necessary to produce non-reciprocity in asymmetric systems. However, a geometrical resonant condition for breaking non-reciprocal propagation is then identified theoretically and verified numerically. With such a resonant condition, the nonlinear interface region of finite width behaves like a single nonlinear delta-barrier so that wave propagations in the forward and backward directions are identical under arbitrary incident wave intensity. As such, reciprocity re-emerges periodically in the asymmetric nonlinear system when changing the width of interface region. Finally, similar resonant conditions of discrete nonlinear Sch\"ordinger equation are discussed. Therefore, we have identified instances of Reciprocity Theorem that breaking spatial symmetry in nonlinear interface systems is not sufficient to produce non-reciprocal wave propagation.