Abstract
Tunneling of two particles in synchronous and asynchronous regimes is studied in the framework of dissipative quantum tunneling. The critical temperature ${T}_{c}$ corresponding to a bifurcation of the underbarrier trajectory is determined. The effect of a heat bath local mode on the probability of two-dimensional tunneling transfer is also investigated. At certain values of the parameters, the degeneracy of antiparallel tunneling trajectories is important. Thus, 4, 6, 12, etc., pairs of the trajectories should be taken into account (a cascade of bifurcations). For parallel particle tunneling the bifurcation resembles a phase transition of the first kind, while for antiparallel transfer it behaves as a second-order phase transition. The proposed theory allows for an explanation of experimental data on quantum fluctuations in two-proton tunneling in porphyrins near the critical temperature.