We consider simple models of tunneling of an object with intrinsic degrees of freedom. This important problem was not extensively studied until now, in spite of numerous applications in various areas of physics and astrophysics. We show possibilities of enhancement for the probability of tunneling due to the presence of intrinsic degrees of freedom split by weak external fields or by polarizability of the slow composite object.Quantum tunneling is a subject of constantly renewed interest, both experimentally and theoretically. The standard textbook approach describes the tunneling process for a point-like particle in an external static potential. Chemical and nuclear subbarrier reactions [1], especially in astrophysical conditions, as a rule, involve complex objects with their intrinsic degrees of freedom. As stated in Ref. [2], "Although a number of theoretical works have studied tunneling phenomena in various situations, quantum tunneling of a composite particle, in which the particle itself has an internal structure, has yet to be clarified." There are experimental data [3,4] indicating that at low energies the penetration probability for loosely bound systems, such as the deuteron, can noticeably exceed the conventional estimates.The problems of tunneling and reflection of a composite particle were discussed recently with the help of various models [5,6,7,8,9,10,11]. It was stressed that new, usually ignored, effects are important for nuclear fusion and fission, nucleosynthesis in stars, molecular processes, transport phenomena in semiconductors and superconductors, both in quasi-one-dimensional and three-dimensional systems. The resonant tunneling associated with the intrinsic excitation, finite size effects, polarizability of tunneling objects, evanescent modes near the barrier, real [12,13] and virtual [14] radiation processes are the examples of interesting new physics. Below we consider simple models which illustrate how "hidden" degrees of freedom can show up in the process of tunneling leading to a considerable enhancement of the probability of this process.Let the tunneling particle possess two degenerate intrinsic states and the incident wave comes to the barrier in a pure state "up" (it is convenient to use the spin-1/2 language with respect to the z-representation). We assume one-dimensional motion with the simplest rectangular potential barrier of height U 0 located at 0 < x < a. At low energy E ≪ U 0 , when the imaginary action κ 0 a = [2m(U 0 − E)] 1/2 a is very large, the transmission coefficient T 0 ∝ exp(−2κ 0 a) is exponentially small. This probability can be exponentially enhanced by a weak "magnetic" field applied in the area of the barrier. We assume that the interaction of this field with the particle is −hσ x , where h is proportional to the transverse magnetic field.Indeed, this field creates the "down" spin component and splits the states inside the barrier according to the value of σ x . In the z-representation the regions x ≤ 0 and x ≥ a acquire the down component in the reflected a...
