Spontaneous decay in arbitrary cavity size

We consider a complete study of the influence of the cavity size on the spontaneous decay of an atom excited state, roughly approximated by a harmonic oscillator. We confine the oscillator-field system in a spherical cavity of radius $R$, perfectly reflective, and work in the formalism of dressed coordinates and states, which allows to perform non-perturbative calculations for the probability of the atom to decay spontaneously from the first excited state to the ground state. In free space, $R\to\infty$, we obtain known exact results an for sufficiently small $R$ we have developed a power expansion calculation in this parameter. Furthermore, for arbitrary cavity size radius, we developed numerical computations and showed complete agreement with the exact one for $R\to\infty$ and the power expansion results for small cavities, in this way showing the robustness of our results. We have found that in general the spontaneous decay of an excited state of the atom increases with the cavity size radius and vice versa. For sufficiently small cavities the atom practically does not suffers spontaneous decay, whereas for large cavities the spontaneous decay approaches the free-space $R\to\infty$ value. On the other hand, for some particular values of the cavity radius, in which the cavity is in resonance with the natural frequency of the atom, the spontaneous decay transition probability is increased compared to the free-space case. Finally, we showed how the probability spontaneous decay go from an oscillatory time behaviour, for finite cavity radius, to an almost exponential decay, for free space.