Quantum Breaking Time near Classical Equilibrium Points

In the evolution of distributions localized around classical equilibrium points, the quantum-classical correspondence breaks down at a time, the so-called quantum breaking, or Ehrenfest time, which is related to the minimal separation of the quantum levels in proximity of the classical equilibrium energy. By studying one-dimensional systems with single- and double-well polynomial potentials, we find that the Ehrenfest time diverges logarithmically with the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. In all the other cases, we have a power law divergence with the exponent determined by the degree of the potential near the equilibrium point.