This paper investigates the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BECs) confined in a double-well magnetic trap. The dynamics are described by two Gross-Pitaevskii equations coupled by a transfer matrix element, which accounts for the inter-well phase difference and population imbalance. The study reveals a novel "macroscopic quantum self-trapping effect" due to the non-linear dynamics of the BEC tunneling, where a self-maintained population imbalance is sustained. This effect is a generalization of the familiar ac Josephson effect and plasma oscillations observed in superconductor junctions. The authors also explore different regimes of the dynamics, including the non-interacting limit, linear regime, and non-linear regime, and provide numerical solutions to illustrate the behavior of the system. The critical behavior and the onset of the self-trapping effect are analyzed, and the parameters governing the dynamics are derived. The findings highlight the unique quantum mechanical properties of BEC tunneling and its potential for observing new quantum phenomena on macroscopic scales.This paper investigates the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BECs) confined in a double-well magnetic trap. The dynamics are described by two Gross-Pitaevskii equations coupled by a transfer matrix element, which accounts for the inter-well phase difference and population imbalance. The study reveals a novel "macroscopic quantum self-trapping effect" due to the non-linear dynamics of the BEC tunneling, where a self-maintained population imbalance is sustained. This effect is a generalization of the familiar ac Josephson effect and plasma oscillations observed in superconductor junctions. The authors also explore different regimes of the dynamics, including the non-interacting limit, linear regime, and non-linear regime, and provide numerical solutions to illustrate the behavior of the system. The critical behavior and the onset of the self-trapping effect are analyzed, and the parameters governing the dynamics are derived. The findings highlight the unique quantum mechanical properties of BEC tunneling and its potential for observing new quantum phenomena on macroscopic scales.