Energy Evolution in Time-Dependent Harmonic Oscillator

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general timedependent one-dimensional harmonic oscillator, whose Newton equation q̈+ω(t)q = 0 cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy E0 at time t = 0 and calculate rigorously the distribution of energy E1 after time t = T , which is fully (all moments, including the variance μ) determined by the first moment Ē1. For example, μ 2 = E 0 [(Ē1/E0) 2 − (ω(T )/ω(0))]/2, and all higher even moments are powers of μ, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function ω(t) and is in this sense universal. In ideal adiabaticity Ē1 = ω(T )E0/ω(0), and the variance μ 2 is zero, whilst for finite T we calculate Ē1, and μ for the general case using exact WKB-theory to all orders. We prove that if ω(t) is of class Cm (all derivatives up to and including the order m are continuous) μ ∝ T, whilst for class C∞ it is known to be exponential μ ∝ exp(−αT ). PACS numbers: 05.45.-a, 45.20.-d, 45.30.+s, 47.52.+j Published in Open Systems & Information Dynamics 13 No.2 p.197-222 (2006)