Quantifying the nonlinearity of a quantum oscillator

We address the quantification of nonlinearity for quantum oscillators and introduce two measures based on the properties of the ground state rather than on the form of the potential itself. The first measure is a fidelity-based one and corresponds to the renormalized Bures distance between the ground state of the considered oscillator and the ground state of a reference harmonic oscillator. Then, in order to avoid the introduction of this auxiliary oscillator, we introduce a different measure based on the non-Gaussianity (nG) of the ground state. The two measures are evaluated for a sample of significant nonlinear potentials and their properties are discussed in some detail. We show that the two measures are monotone functions with respect to each other in most cases, and this suggests that the nG-based measure is a suitable choice to capture the anharmonic nature of a quantum oscillator, and to quantify its nonlinearity independently of the specific features of the potential. We also provide examples of potentials where the Bures measure cannot be defined, due to the lack of a proper reference harmonic potential, while the nG-based measure properly quantifies their nonlinear features. Our results may have implications in experimental applications where access to the effective potential is limited, e.g., in quantum control, and protocols rely on information about the ground or thermal state.