Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems

The exponential growth of the out-of-time-ordered correlator (OTOC) has been proposed as a quantum signature of classical chaos. The growth rate is expected to coincide with the classical Lyapunov exponent. This quantum-classical correspondence has been corroborated for the kicked rotor and the stadium billiard, which are one-body chaotic systems. The conjecture has not yet been validated for realistic systems with interactions. We make progress in this direction by studying the OTOC in the Dicke model, where two-level atoms cooperatively interact with a quantized radiation field. For parameters where the model is chaotic in the classical limit, the OTOC increases exponentially in time with a rate that closely follows the classical Lyapunov exponent.

Figure 1

Percentage of chaos over energy shells as a function of energy and coupling strength. The thick (green) solid line follows the ground-state energy and the diamond marks the critical point. The (blue) vertical dotted line indicates the coupling $\gamma =2{\gamma }_c$ and the circle marks the energy chosen for the studies below.

Figure 2

Panel (a): Exponential growth of the OTOC for an eigenstate with $E_n/(j{\omega }_0)\approx -1.1$; numerical results (solid line), fit for $C_n^{qp}(t)$ (stars) and for $C_{n}qq(t)$ (circles); saturation times (square and triangle). Panel (b): Log-log plot for the evolution of the OTOC and saturation value (dotted lines). Inset: short time behavior compared with ${\sin }^2(t)$ and ${\cos }^2(t)$ (dotted lines). We use $j=100$, $n=1625$.

Figure 3

Panel (a): Poincaré surfaces of section projected on the plane $(q,p)$ for $\varphi =0$ and energy $-1.1{\omega }_0$ for various initial conditions. Panel (b): map of chaos over the same Poincaré surface in terms of the finite-time classical Lyapunov exponents. These exponents are evaluated for each trajectory up to 10000 units of time, which is enough to have stable results.

Figure 4

Comparison between the classical Lyapunov exponent ${\tilde{\lambda }}_{⟨\ln ⟩}$ (lower green horizontal line), the maximum classical Lyapunov exponent ${\lambda }_{\max }$ (black dotted horizontal line), and the quantum Lyapunov exponents ${\mathrm{\Lambda }}_Q$ (red circles) for $C_n^{qp}(t)$ for fifty-one states of different energies around $\epsilon /{\omega }_0\approx -1.1$. The solid orange line depicts the average value of ${\mathrm{\Lambda }}_Q$ and the shaded region represents the standard deviation around it.