Classical and quantum chaos in a three-mode bosonic system

We study the dynamics of a three-mode bosonic system with mode-changing interactions. For large mode occupations the short-time dynamics is well described by classical mean-field equations allowing us to study chaotic dynamics in the classical system and its signatures in the corresponding quantum dynamics. By introducing a symmetry-breaking term we tune the classical dynamics from integrable to strongly chaotic, which we demonstrate by calculating Poincaré sections and Lyapunov exponents. The corresponding quantum system features level statistics that change from Poissonian in the integrable to Wigner-Dyson in the chaotic case. We investigate the behavior of out-of-time-ordered correlators (OTOCs), specifically the squared commutator, for initial states located in regular and chaotic regions of the classical mixed phase space and find marked differences between the two cases. The short-time behavior is well captured by semi-classical truncated Wigner simulations directly relating these features to properties of the underlying classical mean-field dynamics. We discuss a possible experimental realization of this model system in a Bose-Einstein condensate of rubidium atoms, which allows reversing the sign of the Hamiltonian required for measuring OTOCs.

Figure 1

Poincaré sections illustrating the phase-space structure for increasing values of $r$. The plane of intersection is defined by ${\mathrm{\Theta }}_m=0$. In (a) the points of intersection of each trajectory for a given energy are confined to a line. This changes in (b) and (c) for increasing values of $r$: In (b) we can see a mixed phase space where for some trajectories the points of intersection spread over a larger area of phase space while for others the points are still confined to lines; in (c) the phase space is strongly chaotic where all allowed regions compatible with energy and norm are reached from any initial point. To be able to show Poincaré sections for different initial states in one plot, the initial states in (a) correspond to different energies. Here the initial states are chosen by varying ${\rho }_0$ while keeping, ${\mathrm{\Theta }}_s=0$, $m=0.1$, and ${\mathrm{\Theta }}_m$ fixed. The separatrix between vibrational and rotational motion is shown as a solid black line. In (b) and (c) we only use initial states corresponding to an energy of $E=1.005$ chosen equal to the energy of the separatrix in the unperturbed case. To adjust the energy of the initial states for a given initial ${\rho }_0$ and ${\mathrm{\Theta }}_s$, we choose $m$ and ${\mathrm{\Theta }}_m$ accordingly. Different colors represent trajectories corresponding to different initial states.

Figure 2

Lyapunov exponents $\lambda$ (color encoded) for a raster of $80\times 80$ values of ${\rho }_0$ and ${\mathrm{\Theta }}_s$, ${\mathrm{\Theta }}_m=0$. At each point the value of $m$ has been adjusted to match a given energy, here $E=1.005$. The choice of energy mainly influences the allowed compatible regions in phase space but not the qualitative features of the phase-space portrait. The plots are shown for increasing values of $r$. As in Fig. 1, we see the transition from a (a) regular through a (b) mixed to a (c) strongly chaotic phase space.

Figure 3

Histograms of the level spacing distributions for different values of $r$, corresponding to regular (a) and largely chaotic (b) classical phase space. We show the combined distributions of the subspaces defined by the spin flip symmetry, see text. $N=100$ atoms, leading to 5091 energy eigenvalues, are used. The solid red lines are fits of a Brody distribution to the data.

Figure 4

(a) Husimi distributions of the two initial coherent states (color encoded) plotted on top of the outline of the classical Lyapunov exponent for $r=0.15$ (cf., Fig. 2). The state on the right-hand side corresponds to the coherent state centered at ${\rho }_0=0.2$, ${\mathrm{\Theta }}_s=4\pi /3$, ${\mathrm{\Theta }}_m=0$, and $m$ chosen such that its energy corresponds to $E=〈\widehat{H}〉=1.005$, which lies in the classically regular region of the phase space for the given parameters. The left one is centered in the classically chaotic region at ${\rho }_0=0.2$, ${\mathrm{\Theta }}_s=-2\pi /3$, ${\mathrm{\Theta }}_m=0$, and $m$ chosen again such that its energy corresponds to $E=1.005$. Panels (b) and (c) show the Husimi distributions of the time-evolved states at $t=10$ corresponding to both initial states in (a).

Figure 5

(a) Time evolution of the OTOC $C\left(t\right)=〈{\psi }_0\left|{\left|\left[{\widehat{\rho }}_0\left(t\right),{\widehat{\rho }}_0\right]\right|}^2\right|{\psi }_0〉$ for the two initial coherent states $\left|{\psi }_0\right.〉$ shown in Fig. 4 for $N=100$ particles (dotted blue line for classically regular state, solid red line for classically chaotic state) from the exact quantum mechanical calculations. The inset shows the short-time behavior (quantum calculation marked as crosses in the respective color) where the appropriately scaled OTOCs calculated using TWA are shown as dotted blue and solid red lines, respectively. Note that around $t\gtrsim 3.5$ the TWA starts to deviate from the exact quantum mechanical solution for the classically chaotic initial state. (b) OTOCs $C\left(t\right)$ calculated using TWA for the same parameters as in panel (a) and Fig. 4. Two OTOCs are shown in red for the classically chaotic initial state; for $N=100$ particles (solid) and $N={10}^4$ particles (dash-dotted). We also show an exponential fit for the OTOC with $N={10}^4$ as a black dashed line using the data in the shaded area. The OTOC for the classically regular initial state also for $N={10}^4$ is shown as a dotted blue line to illustrate that at sufficiently high particle numbers the OTOC can indeed be used for a proxy of chaotic behavior in the limiting classical dynamics.

Figure 6

Comparison of the time evolution of (a) the expectation value of ${\widehat{N}}_0$ and (b) its standard deviation calculated with TWA (solid blue line) to the exact quantum mechanical calculation (dashed black line) for $N=100$ particles. As initial state we used the classically chaotic state from Fig. 4 along with the same parameters. For the TWA we averaged over 1000 samples. We also compared the TWA to the exact quantum calculations for different observables and initial states yielding similar results, for classically regular dynamics the agreement is even better.