Loschmidt Echo and Classicality of the Gamma Model

TL;DR

The paper investigates the classical limit of a solvable nonlinear gamma oscillator by analyzing Loschmidt Echo and roughness as diagnostic tools. It reveals that overlap dynamics can be periodic or non-periodic depending on the nonlinearity parameter γ, with long-time saturation tied to the effective Hilbert space rather than chaos. Through the overlap operator and its Wigner transform, the study shows that non-diagonal phase-space coherence largely governs the dynamics and that high non-classicality can persist even when the overlap suggests classical behavior. These results clarify that Loschmidt Echo alone is insufficient to diagnose classicality in nonlinear quantum systems and highlight the crucial role of phase-space coherence in shaping quantum dynamics.

Abstract

The classicality of the Gamma Model, an analytically solvable quantum oscillator with non-linear dynamics, is investigated using the overlap dynamics, also known as the Loschmidt Echo, and roughness, a classicality measure based on the Wigner representation of a state. Though the overlap dynamics would indicate a chaotic regime, here the model is integrable. The time mean of the overlap function decays inversely with the effective Hilbert space occupied by the initial state for the non-periodic case. Two different stationary regimes were found for the overlap mean and overlap variance. The state non-classicality was investigated using the roughness measure, and its mean also has two stationary regimes. For large effective Hilbert space, the roughness time mean depends more on the effective space than the initial state. While the Wigner function is dominated by the non-diagonal terms, the overlap operator has some contribution on each part, but it is more affected by the diagonal terms than the density matrix.

Figures (13)

• Figure 1: Overlap function $O_{\Delta H}$ as function of time for $\epsilon=1$, $\hbar=1$ and for a coherent state, ($\left|\,\alpha\,\right\rangle$) as initial state with $\alpha =2$. (A) $\gamma=1$, (B) $\gamma=1.1$, (C) $\gamma =2$ and (D) $\gamma=-1$.
• Figure 2: Overlap function $O_{\Delta H}$ as function of time for $\epsilon=1$, $\hbar=1$ and for a coherent state ($\left|\,\alpha\,\right\rangle$) as initial state with $\alpha =2$. (A) $\gamma=3$, (B) $\gamma=3.1$, (C) $\gamma =-0.5$ and (D) $\gamma=-2$.
• Figure 3: Time mean of the overlap function $O_{\Delta H}$ as a function of time for $\epsilon=1$, $\hbar=1$ and for a coherent state ($\left|\,\alpha\,\right\rangle$) as the initial state with $\alpha =2$. In red $\gamma=1$, blue $\gamma=2$, green $\gamma =3$, pink $\gamma=4$, yellow $\gamma=1.7$, brown $\gamma=3.5$, purple $\gamma=-0.5$ and gray $\gamma=-1$.
• Figure 4: (A) - Time mean of the overlap function $O_{\Delta H}$ as a function of time. (B) Time variance of the overlap function $O_{\Delta H}$ as a function of time. The initial state is a phase state $\left|\,\phi\,\right\rangle=\frac{1}{\sqrt{r+1}}\sum_{k=0}^r \left|\,k\,\right\rangle$, for $r=6$.In red $\gamma=1$, blue $\gamma=2$, green $\gamma =3$, pink $\gamma=4$, yellow $\gamma=1.7$, brown $\gamma=3.5$, purple $\gamma=-0.5$ and gray $\gamma=-1$.
• Figure 5: Time mean of the overlap function $O_{\Delta H}$ for $t \rightarrow \infty$ as a function of deviation of the operator $N$ in the state $\rho(0)$ time for $\epsilon=1$. Red circles $\gamma=3.1$ and green $\gamma=2.4$. The blue dotted line is the fitting function $Z$ equation \ref{['Zf']} for $\mu =0.923$. The initial state is a phase state for the circles and a coherent state for the cross.