The Coupling of a Spin System to a Cavity Mode

The problem of a non-interacting spin system coupled to a damped resonant cavity is formulated wave-mechanically. An exact solution of the Schrödinger equation is found for this system, and the expectation value Mz of the z component of total spin angular momentum Sz is determined in series form from an expansion of the solution. The expectation value is obtained exactly and in closed form only for very small values of the total spin quantum number S. Approximate solutions in closed form are obtained for the initial conditions M ≃ S and M ≃ -S. For the former condition, ⟨ t | Sz | t⟩ exhibits exponential decay for small values of t. The approximations break down for larger values of t, but damped oscillatory behaviour is suggested. The results are discussed in terms of the total number of spins, and a value of at least 1018 spins seems necessary for an appreciable exponential change in ⟨ t | Sz | t ⟩ to occur, under typical experimental conditions. Reasons are given why a simple, closed form for ⟨ t | Sz | t ⟩ seems unlikely for the general case of arbitrary S, and some mathematical difficulties are considered in more detail.