Energy evolution in time-dependent harmonic oscillator

Abstract

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 time-dependent one-dimensional harmonic oscillator, whose Newton equation $\ddot{q} + ω^2(t) q=0$ cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of energy $E_1$ after time $t=T$, which is fully (all moments, including the variance $μ^2$) determined by the first moment $\bar{E_1}$. For example, $μ^2 = E_0^2 [(\bar{E_1}/E_0)^2 - (ω(T)/ω(0))^2]/2$, and all higher even moments are powers of $μ^2$, 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 $\bar{E_1} = ω(T) E_0/ω(0)$, and the variance $μ^2$ is zero, whilst for finite $T$ we calculate $\bar{E_1}$, and $μ^2$ for the general case using exact WKB-theory to all orders. We prove that if $ω(t)$ is of class ${\cal C}^{m}$ (all derivatives up to and including the order $m$ are continuous) $μ\propto T^{-(m+1)}$, whilst for class ${\cal C}^{\infty}$ it is known to be exponential $μ\propto \exp (-αT)$.