Classical chaos in atom-field systems

The relation between the onset of chaos and critical phenomena, like quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs), is analyzed for atom-field systems. While it has been speculated that the onset of hard chaos is associated with ESQPTs based in the resonant case, the off-resonant cases, and a close look at the vicinity of the QPTs in resonance, show clearly that both phenomena, ESQPTs and chaos, respond to different mechanisms. The results are supported in a detailed numerical study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The appearance of chaos is quantified calculating the largest Lyapunov exponent for a wide sample of initial conditions in the whole available phase space for a given energy. The percentage of the available phase space with chaotic trajectories is evaluated as a function of energy and coupling between the qubit and bosonic part, allowing us to obtain maps in the space of coupling and energy, where ergodic properties are observed in the model. Different sets of Hamiltonian parameters are considered, including resonant and off-resonant cases.

Figure 1

Energy surface shell for $\gamma =2.0{\gamma }_c$, in coordinates $(u=\theta cos\varphi ,v=\theta sin\varphi ,q)$. Three energies are shown: (a) $\epsilon =-1.4{\omega }_0$, (b) $\epsilon =-1.1{\omega }_0$, and (c) $\epsilon =-0.5{\omega }_0$.

Figure 2

Solutions of the canonical variables versus time for two close trajectories each, for $\epsilon =-1.4{\omega }_0,\gamma =2{\gamma }_c$, with initial condition $(p_0,q_0,P_0,Q_0)=(0,q_{+}\left(\epsilon \right),0.648,1.371$).

Figure 3

Points of analysis (a) for $\gamma =2{\gamma }_c,\omega ={\omega }_0$, and $\epsilon =-1.4{\omega }_0$ (orange), $\epsilon =-1.1{\omega }_0$ (red), $\epsilon =-0.5{\omega }_0$ (green). The blue curve represents the ground state energy as a function of the coupling parameter $\gamma$. (b) Contours of the surface energy $\epsilon$ in the canonical variables ($Q,P$), for coupling value $\gamma =2{\gamma }_c$ and $\epsilon =-1.4{\omega }_0$ (orange), $\epsilon =-1.1{\omega }_0$ (red), and $\epsilon =-0.5{\omega }_0$ (green).

Figure 4

Poincaré sections (top) and Lyapunov exponents (bottom) in the projection of variables $(Q,P)$, for $\gamma =2{\gamma }_c$ and $\epsilon =-1.4{\omega }_0$ (a), $\epsilon =-1.1{\omega }_0$ (b), $\epsilon =-0.5{\omega }_0$ (c). In the Poincaré sections the colors are associated with different classical trajectories. For the Lyapunov exponents the color code is given on the bar at the bottom. Blue represents the regular regions.

Figure 5

Percentage of chaos and average Lyapunov exponent in phase space as a function of energy, with coupling $\gamma =2{\gamma }_c$ in resonance.

Figure 6

Probability distribution (top) for the Lyapunov exponent of three different surface energies in the respective phase space, (bottom) distribution of probability for three trajectories with initial condition ${\overline{x}}_1\left(0\right)=(p_o,q_o,P_o,Q_o)=(0,q\left(\epsilon \right),0,0.948),{\overline{x}}_2\left(0\right)=(0,q\left(\epsilon \right),0,0.707),{\overline{x}}_3\left(0\right)=(0,q\left(\epsilon \right),0,0.547)$ for $\epsilon =-1.4{\omega }_0$ (a), $\epsilon =-1.1{\omega }_0$ (b), and $\epsilon =-0.5{\omega }_0$ (c).

Figure 7

Mapping for average Lyapunov exponent on energy surface (left), percentage of chaoticity (right) for different values of $\omega$: $\omega ={\omega }_0/4$ (top), $\omega ={\omega }_0$ (center), $\omega =4{\omega }_0$ (bottom). The dotted lines represent the energies of the ESQPT.

Figure 8

The average Lyapunov exponent (left) and the percentage of chaoticity (right), plotted as functions of the photon energy $\omega$ for $\gamma ={\gamma }_c$ and different values of the excitation energy $\epsilon$ close to the ground state energy ${\epsilon }_0$.

Figure 9

Zoom around $\gamma ={\gamma }_c,\epsilon ={\epsilon }_0$ displaying the average Lyapunov exponent over the energy surface (left), and the percentage of chaoticity (right) for $\omega ={\omega }_0$.

Figure 10

Trajectories with initial condition ${\mathbf{x}}_r$ in the canonical variables ($q,p$) and ($Q,P$).

Figure 11

Poincaré section for the trajectory with initial condition ${\mathbf{x}}_r$ in the canonical variables ($Q,P$).

Figure 12

Trajectories with initial condition $x_c$ in the canonical variables ($q,p$) and ($Q,P$).

Figure 13

Poincaré section for the trajectory with initial condition ${\mathbf{x}}_c$ in the canonical variables ($Q,P$).

Figure 14

Geometric distance $dx\left(t\right)$ (black) and separation $\delta x\left(t\right)$ (red) in logarithmic scales between two close trajectories, in a regular regime (left) and a chaotic one (right).

Figure 15

Convergence of the Lyapunov exponents as functions of time, for regular (left) and chaotic (right) trajectories.

Figure 16

Histograms of values obtained in the sampling of Lyapunov exponents, for regular (left) and chaotic (right) trajectories.

Figure 17

Probability distribution of the Lyapunov exponents, evaluated for three different energies $\epsilon =-1.4{\omega }_0$ (a), $\epsilon =-1.1{\omega }_0$ (b), and $\epsilon =-0.5{\omega }_0$ (c), for three time scales $t_f$ for each trajectory, with initial conditions $\overline{x}\left(0\right)=(0,q\left(\epsilon \right),0,0.948)$.