A solvable model of the breakdown of the adiabatic approximation

Abstract

Let L ≥ 0 and 0 < ɛ ≪ 1. Consider the following time-dependent family of 1D Schrödinger equations with scaled harmonic oscillator potentials iε∂tuε=−12∂x2uε+V(t,x)uε, uɛ(−L − 1, x) = π−1/4 exp(−x2/2), where V(t, x) = (t + L)2x2/2, t < − L, V(t, x) = 0, − L ≤ t ≤ L, and V(t, x) = (t − L)2x2/2, t > L. The initial value problem is explicitly solvable in terms of Bessel functions. Using the explicit solutions, we show that the adiabatic theorem breaks down as ɛ → 0. For the case L = 0, complete results are obtained. The survival probability of the ground state π−1/4 exp(−x2/2) at microscopic time t = 1/ɛ is 1/2+O(ε). For L > 0, the framework for further computations and preliminary results are given.

Quantum adiabatic cycles and their breakdown

“…zν → ∞. The same investigation strategy discussed in the present letter may be employed to analytically study and generalise other interesting problems of quantum dynamics, such as the case of a confined quantum system dynamically projected onto a de-confined one [33].…”