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.

Figure 1

Nonlinearity measures for a weakly perturbed harmonic potential. In the left panel we show ${\eta }_{NG}\left[V\right]$ as a function of ${\eta }_B\left[V\right]$ for random values of ${\epsilon }_3\in [-0.1,0.1]$ and ${\epsilon }_4\in [-0.25,0.25]$. The right panel shows the corresponding values for ${\epsilon }_3\in [-0.2,0.2]$, ${\epsilon }_4\in [-0.25,0.25]$. The solid black line in both plots is the function ${\eta }_{NG}\left({\eta }_B\right)$ of Eq. (13).

Figure 2

The Morse potential. In the upper-left panel we show $V_M\left(x\right)$ for $\alpha =1$ (solid red), $\alpha =2$ (dashed blue), $\alpha =3$ (dotted black) and for a fixed depth parameter $D=1$. In the upper-right panel we show the corresponding g.s. probability densities, $|{\varphi }_M{|}^2\left(x\right)$. In the lower panel we show the nonlinearity measures ${\eta }_{NG}\left[V\right]$ and ${\eta }_B\left[V\right]$ as a functions of $\alpha$ for different values of $D$. We have $D=0.25$ (solid red curves), $D=0.5$ (dashed blue curves), $D=1$ (dotted black curves). The vertical black lines are placed in correspondence of the limiting values of $\alpha$.

Figure 3

The modified Pöschl–Teller potential. The upper panels show the MPT potential $V_P\left(x\right)$ and the corresponding ground-state probability density $|{\varphi }_P{\left(x\right)|}^2$ for $\alpha =1/2$ (solid red curves), $\alpha =1$ (blue dashed curves), and $\alpha =3$ (black dotted curves) and a fixed potential depth $D=1$. In the lower panel we show the nonlinearity measures ${\eta }_B\left[V_P\right]$ and ${\eta }_{NG}\left[V_P\right]$ as a function of $\alpha$ for $D=1$ (red solid curves), $D=2$ (blue dashed curves), and $D=3$ (black dotted curves). The inset is a parametric plot of ${\eta }_B$ as a function of ${\eta }_{NG}$, showing that the two measures are monotone functions with respect to each other, independently of the value of $D$.

Figure 4

The modified isotonic potential. The upper-left panel shows the MIO potential $V_T\left(x\right)$ for different values of the parameter: $a=5$ (solid red curve), $a=1$ (blue dashed curve), and $a=\frac{1}{2}$ (black dotted curve). The upper-right panel shows the probability density $|{\varphi }_T{\left(x\right)|}^2$ of the corresponding g.s. wave functions. In the lower panel we show the nonlinearity measures ${\eta }_B\left[V\right]$ (solid red curve) and ${\eta }_{NG}\left[V\right]$ (dashed blue curve) as a function of $a$.

Figure 5

The Fellows–Smith potential. The upper-left panel shows $V_F\left(x\right)$ for different values of the parameter: $p=-\frac{1}{10}$ (solid red curve), $p=-\frac{2}{5}$ (blue dashed curve), and $p=-\frac{9}{10}$ (black dotted curve). The upper-right panel shows the probability density $|{\varphi }_F{\left(x\right)|}^2$ of the corresponding g.s. wave functions. In the lower panel we show the nonlinearity measure ${\eta }_{NG}\left[V\right]$ (solid red curve) as a function of $p$, together with the measure ${\eta }_B\left[V\right]$ (dashed blue curve) in the regions where it can be evaluated. The vertical black lines denote the values $p=p_{\pm }$.