Macroscopic Quantum Self-Trapping in Dynamical Tunneling

It is well known that increasing the nonlinearity due to repulsive atomic interactions in a double-well Bose-Einstein condensate suppresses quantum tunneling between the two sites. Here we find analogous behavior in the dynamical tunneling of a Bose-Einstein condensate between period-one resonances in a single driven potential well. For small nonlinearities we find unhindered tunneling between the resonances, but with an increasing period as compared to the noninteracting system. For nonlinearities above a critical value we generally observe that the tunneling shuts down. However, for certain regimes of modulation parameters we find that dynamical tunneling reemerges for large enough nonlinearities, an effect not present in spatial double-well tunneling. We develop a two-mode model in good agreement with full numerical simulations over a wide range of parameters, which allows the suppression of tunneling to be attributed to macroscopic quantum self-trapping.

Figure 1

(a) Poincaré section for the classical Hamiltonian Eq. (1) with $\kappa =2.3$, $ϵ=0.3$, showing regular islands $I_{+}$, $I_{-}$ separated by a region of chaos. (b) Husimi function $Q(x,p)$ of the even tunneling Floquet state: $Q(x,p)[\Psi ]=|⟨\alpha |\Psi ⟩{|}^2/(2\pi {\hslash }_{\mathrm{eff}})$ for ${\hslash }_{\mathrm{eff}}=0.5$, where $|\alpha ⟩$ is a coherent state centered on momentum $p$ and position $x$.

Figure 2

(a)–(c) Stroboscopic momentum space density showing the transition from (a) DT to (b) MQST and (c) back to DT, with $\kappa =1.3$, $ϵ=0.2$ and $U=\{1.2,2.3,3.4\}\times {10}^{-2}$ respectively. (d)–(f) Comparison of GPE Floquet state populations $d_{\pm }(t)=\int dx{\varphi }_{\pm }^{*}(x,t)\Psi (x,t)$ (lines) with two-mode model results [crosses, $|c_{\pm }{|}^2$ see Eqs. (3, 5)]. Black solid line: $|d_{+}{|}^2$, red dashed line: $|d_{-}{|}^2$, black dotted line: $n_{\mathrm{tot}}\equiv |d_{+}{|}^2+|d_{-}{|}^2$, black crosses $|c_{+}{|}^2$, red crosses: $|c_{-}{|}^2$.

Figure 3

(a) Dependence of the tunneling rate $T^{-1}$ on the nonlinearity $U$, showing intermittent MQST for $U\in [1.4,2.2]\times {10}^{-2}$, with $\kappa =1.3$, $ϵ=0.2$ as in Fig. 2. Black solid line: GPE solution, crosses: two-mode-model [Eq. (5)] with overlap coefficients from Eq. (4). Red open circles: $T_{\mathrm{nl}}^{-1}=|E_e(U)-E_o(U)|/2\pi {\hslash }_{\mathrm{eff}}$. (b),(c) Critical nonlinearity and tunnel splitting for $\kappa =1.2$ (b) as a function of $ϵ$ for ${\hslash }_{\mathrm{eff}}=0.15$, and (c) and as a function of ${\hslash }_{\mathrm{eff}}$ for $ϵ=0.2$. Blue dots: estimates from linear Floquet states using $U_{\mathrm{crit}}=2|\Delta E|/|{\Lambda }_0|$. Red open circles: nonlinear two-mode model, using Eq. (7). Black crosses: extracted from full GPE simulations. Magenta squares: energy difference of Floquet states $|\Delta E|$.