Emergence of nonlinear behavior in the dynamics of ultracold bosons

We study the evolution of a system of interacting ultracold bosons which presents nonlinear, chaotic behaviors in the limit of a very large number of particles. Using the spectral entropy as an indicator of chaos and three different numerical approaches—exact diagonalization, the truncated Husimi method, and the mean-field (Gross-Pitaevskii) approximation—we put into evidence the destructive impact of quantum noise on the emergence of nonlinear dynamics.

Figure 1

Quasiclassical chaos in a tilted lattice, obtained by integrating the Gross-Pitaevskii equation (4). (a) Poincaré section corresponding to $I_{-1}=0.1,{\theta }_0-{\theta }_{-1}=0$. (b) Spectral entropy as a function of the initial conditions $I_0-I_1$ and ${\theta }_1$. Parameters are $V_0=5,F=0.25$, and $gN=0.2$.

Figure 2

Evolution of the average position of a wave packet calculated with different methods: Gross-Pitaevskii equation (cyan squares), Lanczos diagonalization (green), and the truncated Husimi method (red triangles), for (a) $N=30$ and (b) $N=500$. In (a), we also represented by blue circles the result obtained by direct diagonalization of the full many-body problem, which is in perfect agreement with the result of the Lanczos method, illustrating the accuracy of the later. The initial state is an SU(3) coherent state with $c_{-1}=\sqrt{0.5},c_0=\sqrt{0.25}$, and $c_1=i\sqrt{0.25}$. Other parameters are $gN=0.2,V_0=7$, and $F=0.1$.

Figure 3

Spectral entropy distribution in phase space obtained by LD (left column) and by THM (right column) with $N=30$ (top row) and $N=400$ (bottom row) and the same parameters as Fig.  1. When $N$ increases, the THM provides a quite good representation of the mean-field behavior for large $N$, and the phase-space structure of the many-body problem becomes similar to the Poincaré section obtained from the GPE, Fig. 1.

Figure 4

Variance of the wave-packet position as a function of the initial conditions $I_0-I_1$ and ${\theta }_1$, calculated between $t=0$ and $t=800$ from the GPE (left column), LD (center column), and THM (right column) with $N=30$ (top row) and $N=400$ (bottom row) and the same parameters as in Fig. 1.