Classical Chaos with Bose-Einstein Condensates in Tilted Optical Lattices

A widely accepted definition of “quantum chaos” is “the behavior of a quantum system whose classical limit is chaotic.” The dynamics of quantum-chaotic systems is nevertheless very different from that of their classical counterparts. A fundamental reason for that is the linearity of Schrödinger equation. In this paper, we study the quantum dynamics of an ultracold quantum degenerate gas in a tilted optical lattice and show that it displays features very close to classical chaos. We show that its phase space is organized according to the Kolmogorov-Arnold-Moser theorem.

Figure 1

Example of a chaotic evolution of a population. The BEC is initially prepared on three neighbor WS states around $n=0$, with $I_{-1}=0.1$, $I_0=0.6$, $I_1=0.3$, with phases ${\phi }_{-1}=0$, ${\phi }_0=0$, ${\phi }_1=\pi$. Plot (a) displays the evolution of the population $I_0=|⟨{\phi }_0|\psi ⟩{|}^2$ of the central (most populated) well. Plot (b) displays the spectrum of this evolution. The result, obtained by numerical integration of the GPE, shows a chaotic behavior. Parameters are $V_0=5$, $F=0.25$, and $g=0.2$.

Figure 2

Same as Fig. 1, but the calculation was made using the discrete model of Eq. (9). Plot (a) displays the same kind of chaotic dynamics as in Fig. 1, and plot (b) shows that the spectrum of the evolution is virtually the same.

Figure 3

Poincaré section of the generalized (quantum) phase space, corresponding to the plane $(I_0,{\theta }_1-{\theta }_0)$. The plane is defined by $I_{-1}=0.1$, ${\theta }_{-1}={\theta }_0$, all other angles and actions are zero, except $I_1=0.9-I_0$, imposed by the normalization condition. One observes different types of dynamics, ranging from regular to chaotic. The arrow indicates the $1:1$ resonance, and one can see around it various frequency-locking signatures evidenced by the presence stability islands. We verified that the calculations made either with the discrete model or by direct numerical integration of the GPE produce the same topology. Parameters are the same as in Fig. 1, except that $g=0.25$.