This paper studies the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BECs) in a double-well magnetic trap. The dynamics are described by two coupled Gross-Pitaevskii equations, which account for the inter-well phase difference and population imbalance. The system exhibits a novel "macroscopic quantum self-trapping effect," where a self-maintained population imbalance arises due to non-linear interactions. This effect is analogous to the Josephson effects in superconductors, with the ability to tailor traps and atomic interactions compensating for electrical neutrality. The system is modeled using a two-state model, where the dynamics are governed by equations that describe the time evolution of the population imbalance $ z $ and the phase difference $ \phi $. These equations are derived from the Gross-Pitaevskii equation and describe the tunneling current between the two wells. The parameters $ \Lambda $, $ \Delta E $, and $ K $ are determined from the overlap integrals of the time-independent GPE eigenfunctions. The paper analyzes three regimes: the non-interacting limit, the linear regime, and the non-linear regime. In the non-interacting limit, the system exhibits Rabi-like oscillations. In the linear regime, the system behaves like a pendulum, with oscillations determined by the Josephson frequency. In the non-linear regime, the system exhibits anharmonic oscillations and a self-trapping effect, where the population imbalance becomes non-zero and time-averaged. The self-trapping effect is a result of the non-linear self-interaction in the BEC, which is a quantum phenomenon involving the coherence of a macroscopic number of atoms. This effect is distinct from single-polaron trapping and gravitational effects on He II baths. The paper also compares the BEC system to superconducting Josephson junctions, showing that the ac Josephson frequency and the Josephson "plasma" frequency are analogous to those in superconductors. The study concludes that the coherent atomic tunneling in a double-well trap induces nonlinear population oscillations that generalize the sinusoidal Josephson effects in superconductors. The self-trapping effect occurs for parameters beyond critical values, demonstrating a novel "macroscopic quantum self-trapping" effect.This paper studies the coherent atomic tunneling between two zero-temperature Bose-Einstein condensates (BECs) in a double-well magnetic trap. The dynamics are described by two coupled Gross-Pitaevskii equations, which account for the inter-well phase difference and population imbalance. The system exhibits a novel "macroscopic quantum self-trapping effect," where a self-maintained population imbalance arises due to non-linear interactions. This effect is analogous to the Josephson effects in superconductors, with the ability to tailor traps and atomic interactions compensating for electrical neutrality. The system is modeled using a two-state model, where the dynamics are governed by equations that describe the time evolution of the population imbalance $ z $ and the phase difference $ \phi $. These equations are derived from the Gross-Pitaevskii equation and describe the tunneling current between the two wells. The parameters $ \Lambda $, $ \Delta E $, and $ K $ are determined from the overlap integrals of the time-independent GPE eigenfunctions. The paper analyzes three regimes: the non-interacting limit, the linear regime, and the non-linear regime. In the non-interacting limit, the system exhibits Rabi-like oscillations. In the linear regime, the system behaves like a pendulum, with oscillations determined by the Josephson frequency. In the non-linear regime, the system exhibits anharmonic oscillations and a self-trapping effect, where the population imbalance becomes non-zero and time-averaged. The self-trapping effect is a result of the non-linear self-interaction in the BEC, which is a quantum phenomenon involving the coherence of a macroscopic number of atoms. This effect is distinct from single-polaron trapping and gravitational effects on He II baths. The paper also compares the BEC system to superconducting Josephson junctions, showing that the ac Josephson frequency and the Josephson "plasma" frequency are analogous to those in superconductors. The study concludes that the coherent atomic tunneling in a double-well trap induces nonlinear population oscillations that generalize the sinusoidal Josephson effects in superconductors. The self-trapping effect occurs for parameters beyond critical values, demonstrating a novel "macroscopic quantum self-trapping" effect.