Quantum state purity versus average phonon number for characterization of mechanical oscillators in cavity optomechanics

TL;DR

This paper advocates using quantum state purity instead of average phonon number to characterize mechanical oscillators in cavity optomechanics, providing exact expressions for maximum purity and analyzing differences in various coupling regimes.

Contribution

It introduces quantum state purity as a more versatile measure for oscillator states and derives analytical expressions for maximum achievable purity in optomechanical systems.

Findings

Purity relates directly to a thermal occupation number for 1D oscillators.

Exact formulas for maximum purity in the quantum backaction limit.

Differences between purity-based and phonon number-based measures in strong coupling regimes.

Abstract

Quantum oscillators in Gaussian states are often characterized by average occupation numbers that refer to a basis of eigenstates of the non-interacting oscillator Hamiltonian. We argue that quantum state purity is a more appropriate characteristic of such states, which can be applied to oscillators of any dimensionality. For a one-dimensional oscillator, the state purity is directly related to a thermal occupation number defined with respect to the number state basis in which the oscillator's quantum state is thermal. Thus, it naturally introduces a more versatile definition of an average occupation number. We study optomechanical sideband cooling of one- and two-dimensional mechanical oscillators in particular, and derive exact analytical expressions for the maximal mechanical state purity achievable in the quantum backaction limit. In the case of a one-dimensional oscillator, we show that the thermal occupation number related to purity can be well approximated by the average phonon number in the weak-coupling regime, but that the two differ in the regime of ultrastrong optomechanical coupling or in cases where the oscillator's resonance frequency is strongly renormalized.