Dynamics in multiple-well Bose-Einstein condensates

We study the dynamics of three-dimensional weakly linked Bose-Einstein condensates using a multimode model with an effective interaction parameter. The system is confined by a ring-shaped four-well trapping potential. By constructing a two-mode Hamiltonian in a reduced highly symmetric phase space, we examine the periodic orbits and calculate their time periods both in the self-trapping and Josephson regimes. The dynamics in the vicinity of the reduced phase space is investigated by means of a Floquet multiplier analysis, finding regions of different linear stability and analyzing their implications on the exact dynamics. The numerical exploration in an extended region of the phase space demonstrates that two-mode tools can also be useful for performing a partition of the space in different regimes. Comparisons with Gross-Pitaevskii simulations confirm these findings and emphasize the importance of properly determining the effective on-site interaction parameter governing the multimode dynamics.

Figure 1

Isosurfaces of the ground-state density and the trapping potential of the four-site system with $N={10}^4$ and $V_b/\hslash {\omega }_x=25$ (in arbitrary units).

Figure 2

Bare on-site interaction energy parameter $U$ as a function of the phase $\eta$. The inset shows the dispersion $\sigma$ of the WL functions in the $xy$ plane as a function of $\eta$, with $l_x=l_y=\sqrt{\hslash /\left(m{\omega }_x\right)}$.

Figure 3

WL wave-function densities $|w_0{|}^2$ (in arbitrary units) at the $z=0$ plane for $\eta =0$ (left panel) and $\eta =\pi /4$ (right panel).

Figure 4

Condensate dynamics arising from the GP equation, and from the EMM and the MM models for an initial condition $N_0=2530$ and $N_1=2470$ corresponding to the Josephson regime. Top panel: Populations in each well $N_k$. Bottom panel: Phase differences ${\phi }_k$.

Figure 5

Condensate dynamics for the initial values $N_0=2640$, and $N_1=2360$ in the ST regime. We depict the GP simulation result, together with those arising from the EMM and MM models. Top panel: Populations in each well $N_k$. Bottom panel: Phase differences ${\phi }_k$.

Figure 6

Characteristic Floquet multipliers for the orbits of the effective TM model as functions of the initial values of $N_0-N_1$. The dashed line marks the critical condition $N_0-N_1=116$ for the transition between Josephson and ST regimes, and the dotted ones indicate the initial values of $N_0-N_1$ for the evolutions shown in Fig. 7.

Figure 7

Time evolution of the populations $N_k$ for initial conditions close to the TM symmetric case for $Z_i=0.016$ (top), 0.024 (middle), and 0.04 (bottom), and $(n_0-n_2)/\left(2Z_i\right)=0.1$. The vertical dashed lines mark the orbit periods for the effective TM model at each value of $Z_i$.

Figure 8

Monodromy matrix elements ${\mathbb{M}}_{ij}$ as functions of the initial imbalance $N_0-N_1$. The empty circles mark the values corresponding to Fig. 7.

Figure 9

Phase-space diagram of the four-mode model with ${\phi }_i=0$ and $N_1=N_3$. The solid line marks the symmetric case with $N_0=N_2$ and $N_1=N_3$. The dashed lines indicate the local conditions $A_{ij}=B_{ij}$. The triangles correspond to the critical condition $N_0-N_1=N_2-N_3=\pm 116$ extracted from Eq. (14), the circles indicate the symmetric initial conditions of Figs. 4 and 5, and the crosses indicate the nonsymmetric ones in Figs. 10, 11, 12, 13.

Figure 10

Dynamics arising from the GP simulation and the EMM model, for initial values $N_0=2650,N_1=2360,N_2=2630$, and $N_3=2360$, with $V_b/\left(\hslash {\omega }_x\right)=25$. Top panel: Populations $N_k$. Bottom panel: Phase differences ${\phi }_k$.

Figure 11

Same as Fig. 10 for $N_0=2482,N_1=2482,N_2=2554$, and $N_3=2482$.

Figure 12

Same as Fig. 10 for $N_0=2490,N_1=2420,N_2=2670$, and $N_3=2420$.

Figure 13

Same as Fig. 10 for $N_0=2581,N_1=2440,N_2=2539$, and $N_3=2440$.