Coherence in magnetic quantum tunneling

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Currently, spin tunneling at very low temperatures is assumed to proceed as an incoherent sequence of events that take place whenever a bias field h(t) that varies randomly with time t becomes sufficiently small, as in Landau-Zener transitions. We study the behavior of a suitably defined coherence time tₚ. Coherence effects become significant when tₚ >= tₕ, where tₕ is the correlation time for h(t). The theory of tunneling of Prokof'ev and Stamp (PS), which rests on the assumption that tₚ <= tₕ, is extended beyond this constraint. It is shown, both analytically and numerically, that tₚ >= tₕ when tₕ*dh <= h, where dh is the rms deviation of h. Equations that give tₚ and the tunneling rate as a function of tₕ*dh both for tₕ*dh>=h, where the theory of PS hold, and for tₕ*dh <= h, where it does not, are derived.