Quantum fluctuations stabilize an inverted pendulum

TL;DR

This paper analytically investigates the quantum dynamics of a pendulum, revealing that quantum fluctuations can stabilize the inverted position through oscillatory motion, akin to the Kapitza pendulum's stabilization.

Contribution

It introduces a dynamical system based on quantum variables that demonstrates stabilization of the inverted pendulum due to quantum fluctuations, a novel quantum analogue of classical stabilization.

Findings

Quantum fluctuations induce oscillations around the inverted position.

Stabilization occurs when the initial wave packet's variance exceeds that of a coherent state.

The behavior parallels the classical Kapitza pendulum's vibrational stabilization mechanism.

Abstract

We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.