Positive quantum Lyapunov exponents in experimental systems with a regular classical limit

Quantum chaos refers to signatures of classical chaos found in the quantum domain. Recently, it has become common to equate the exponential behavior of out-of-time order correlators (OTOCs) with quantum chaos. The quantum-classical correspondence between the OTOC exponential growth and chaos in the classical limit has indeed been corroborated theoretically for some systems and there are several projects to do the same experimentally. The Dicke model, in particular, which has a regular and a chaotic regime, is currently under intense investigation by experiments with trapped ions. We show, however, that for experimentally accessible parameters, OTOCs can grow exponentially also when the Dicke model is in the regular regime. The same holds for the Lipkin-Meshkov-Glick model, which is integrable and also experimentally realizable. The exponential behavior in these cases are due to unstable stationary points, not to chaos.

Figure 1

Top: Energy surface for the classical LMG model for two values of the parameter $\mathrm{\Omega }$ fixing $\xi =-1$. The stationary point ${\mathit{x}}_0=(Q=0,P=0)$ is marked with a red sphere. It is a saddle point for $\mathrm{\Omega }=1$ (a) and a minimum for $\mathrm{\Omega }=3$ (b). Panel (c): Each colored point corresponds to the maximal classical Lyapunov exponent for the Dicke model in a plane resulting from the intersection of an energy shell (with energy indicated by the vertical axis) and the hyperplane $p=0$. This is done for different values of ${\omega }_0$ as indicated by the horizontal axis. We fix $\gamma =0.66$ and $\omega =0.5$. The red square at ${\omega }_0=3$ is the unstable point studied in Fig. 3. The green circle at ${\omega }_{0c}=3.48$ is the critical point that marks the ground state quantum phase transition.

Figure 2

The classical LE $\lambda$ (solid line) and the quantum LE $\mathrm{\Lambda }$ (symbols) for the LMG (a) and the Dicke (b) model at the unstable point. The results for $\mathrm{\Lambda }$ for the LMG model are obtained with the exact quantum evolution and for the Dicke model, the TWA is used. For the LMG model, the FOTOC corresponds to ${\sigma }_Q^2\left(t\right)+{\sigma }_P^2\left(t\right)$, $\xi =-1$, and $j=500$. For the Dicke model, the FOTOC is ${\sigma }_Q^2\left(t\right)+{\sigma }_P^2\left(t\right)+{\sigma }_q^2\left(t\right)+{\sigma }_p^2\left(t\right)$, $\omega =0.5$, $\gamma =0.66$, and the $j$'s are indicated.

Figure 3

Energy surface of the LMG model (a) and of the Dicke model with $P=p=0$ (c). FOTOC ${\sigma }_Q^2\left(t\right)+{\sigma }_P^2\left(t\right)$ for the LMG model (b) and FOTOC ${\sigma }_Q^2\left(t\right)+{\sigma }_P^2\left(t\right)+{\sigma }_q^2\left(t\right)+{\sigma }_p^2\left(t\right)$ for the Dicke model (d). The FOTOC is computed for coherent states centered at the unstable point O and around it, at points A, B, and C. The (black) straight line in (b) and (d) corresponds to the exponential curve with rate given by twice the classical LE. The initial growth rate of the FOTOC for all points and for both models is $2\mathrm{\Lambda }\approx 2\lambda$. For the LMG model: $\xi =-1$, $\mathrm{\Omega }=1$, $j=500$. The points A, B, and C have constant $P=0$ and $Q=0.1,0.2,\text{and}0.3$, respectively. For the Dicke model: $\omega =0.5$, $\gamma =0.66$, ${\omega }_0=3$, and $j=500$. The points A, B, and C have $P=p=0$, $Q=0.1,0.2$, and 0.3, respectively, and $q$ is chosen so that $H_D=-{\omega }_0$ for all four points.