Dynamical crossing of an infinitely degenerate critical point

TL;DR

This paper analyzes the evolution of a driven harmonic oscillator at an infinitely degenerate critical point, revealing that the resulting squeezed state is independent of the sweep rate and highlighting the failure of the adiabatic approximation.

Contribution

It provides an explicit analysis of the oscillator's state evolution at an infinite degeneracy point and explores the geometry of the squeezed state manifold.

Findings

Squeezing parameters are independent of the sweeping rate.

Adiabatic approximation fails at infinite degeneracy points.

The manifold of squeezed states is modeled by the Poincaré disk as a Kähler manifold.

Abstract

We study the evolution of a driven harmonic oscillator with a time-dependent frequency $\omega_t \propto |t|$. At time $t=0$ the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past then the asymptotic future state is a squeezed state whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points where infinitely many eigenvalues collide. We extend our analysis to the situation where the gap at $t=0$ remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincar\'e disk model viewed as a K\"ahler manifold.