Classicality of a quantum oscillator

TL;DR

This paper investigates the classicality of quantum oscillators using Hilbert space and phase-space formalisms, revealing fundamental inconsistencies in hybrid classical-quantum dynamics and limitations of classical simulations of quantum effects.

Contribution

It demonstrates that hybrid classical-quantum systems exhibit dynamical inconsistencies and quantum violations, challenging the classical simulation of quantum phenomena.

Findings

Failures of consistency in hybrid dynamics from both formalisms

Non-conservation of energy in the quantum sector

Violations of the Heisenberg Uncertainty Principle in short-time evolution

Abstract

Gaussian quantum systems exhibit many explicitly quantum effects but can be simulated classically. Using both the Hilbert space (Koopman) and the phase-space (Moyal) formalisms we investigate how robust this classicality is. We find failures of consistency of the dynamics of a hybrid classical-quantum systems from both perspectives. By demanding that no unobservable operators couple to the quantum sector in the Koopmanian formalism, we show that the classical equations of motion act on their quantum counterparts without experiencing any back-reaction, resulting in non-conservation of energy in the quantum system. Using the phase-space formalism we study the short time evolution of the moment equations of a hybrid classical-Gaussian quantum system and observe violations of the Heisenberg Uncertainty Relation in the quantum sector for a broad range of initial conditions. We estimate the time scale for these violations, which is generically rather short. This inconsistency indicates that while many explicitly quantum effects can be represented classically, quantum aspects of the system cannot be fully masked. We comment on the implications of our results for quantum gravity.