Adiabatic versus instantaneous transitions from a harmonic oscillator to an inverted oscillator

TL;DR

This paper derives explicit formulas for the energy behavior of a quantum harmonic oscillator during adiabatic and instantaneous transitions through zero frequency, highlighting differences when the frequency remains real or becomes imaginary.

Contribution

It provides analytical expressions for energy and fluctuations during zero-crossing transitions, emphasizing the distinct behaviors in real versus imaginary frequency cases.

Findings

Energy increases when frequency remains real after crossing zero.

Energy grows exponentially when frequency becomes imaginary.

Small corrections are crucial for accurate energy estimates in unstable regimes.

Abstract

We have obtained explicit analytical formulas for the mean energy and its variance (characterizing the energy fluctuations) of a quantum harmonic oscillator with time-dependent frequency in the adiabatic regimes after the frequency passes through zero. The behavior of energy turns out to be quite different in two cases: when the frequency remains real and when it becomes imaginary. In the first case, the mean energy always increases when the frequency returns to its initial value, and the increment coefficient is determined by the exponent in the power law of the frequency crossing zero. On the other hand, if the frequency becomes imaginary, the absolute value of mean energy increases exponentially, even in the adiabatic regime, unless the Hamiltonian becomes time independent. Small corrections to the leading terms of simple adiabatic approximate formulas are crucial in this case, due to the unstable nature of the motion.