The quantum effect of nonlinear co-tunnelling process,which is dependent on atom-pair tunneling and asymmetry of an double-well trap,is studied by using an asymmetrical extended Bose-Hubbard model.Due to the existence of atompair tunneling that describes quantum phenomena of ultracold atom-gas clouds in an asymmetrical double-well trap,the asymmetrical extended Bose-Hubbard model is better than the previous Bose-Hubbard model model by comparing with the experimental data cited from the literature.The dependence of dynamics and quantum phase transition on atom-pair tunneling and asymmetry are investigated.Importantly,it shows that the asymmetry of the extended Bose-Hubbard model,corresponding to the bias between double wells,leads to a number of resonance tunneling processes,which tunneling is renamed conditional resonance tunneling,and corrects the atom-number parity effect by controlling the bias between double wells.