Adiabatic splitting, transport, and self-trapping of a Bose-Einstein condensate in a double-well potential

We show that the adiabatic dynamics of a Bose-Einstein condensate (BEC) in a double-well potential can be described in terms of a dark variable resulting from the combination of the population imbalance and the spatial atomic coherence between the two wells. By means of this dark variable, we extend, to the nonlinear matter-wave case, the recent proposal by Vitanov and Shore [Phys. Rev. A 73, 053402 (2006)] on adiabatic passage techniques to coherently control the population of two internal levels of an atom or molecule. We investigate the conditions to adiabatically split or transport a BEC as well as to prepare an adiabatic self-trapping state by the optimal delayed temporal variation of the tunneling rate via either the energy bias between the two wells or the BEC nonlinearity. The emergence of nonlinear eigenstates and unstable stationary solutions of the system as well as their role in the breaking down of the adiabatic dynamics is investigated in detail.

Figure 1

(a) Two-level model for the BEC in a double-well potential with $\Omega$ the tunneling rate and $\epsilon$ the energy bias. (b) Three-level correspondence of (a) for the density matrix variables and coupling strengths. $w$ is the population difference, $u/2$ ($v/2$) is the real (imaginary) part of the spatial coherence, and $U$ is the nonlinear self-interaction energy ($\hslash =1$).

Figure 2

Adiabatic splitting of a BEC. (a) Population difference $w(t_{\mathrm{out}})$ at the end of the splitting process in the parameter plane modulation amplitude of the energy bias ${\epsilon }_0$ versus the nonlinear interaction parameter $U_g$. Initial conditions: $w(t_{\mathrm{in}})=-1$ and $u(t_{\mathrm{in}})=v(t_{\mathrm{in}})=0$. Parameter setting: ${\sigma }_{\epsilon }={\sigma }_{\Omega }=212.8{\Omega }_0^{-1}$, $\Delta t_{\epsilon }=500{\Omega }_0^{-1}$, $t_{\Omega }=1500{\Omega }_0^{-1},$ and ${\Omega }_g={\epsilon }_g=n_{\epsilon }=U_0=n_U=0$. (b) Same as in (a) for the fixed value ${\epsilon }_0=1.5{\Omega }_0$. The temporal variation of the energy bias and the tunneling rate are shown in the inset. From a nonlinear dynamics perspective, we have classified the results into five different regions, from $a$ to $e$, depending on the number and type of bifurcations that the system suffers during its dynamics (see text). The detailed temporal dynamics corresponding to cases (i) and (ii) are plotted in Figs. 3 and 4, respectively. The adiabatic splitting succeeds in region $c$, corresponding to $U_g/{\Omega }_0=(-0.12,1.5)$.

Figure 3

Adiabatic splitting of a BEC. $U_g=0.5{\Omega }_0$ and the rest of the parameters are as in Fig. 2. Plotted as a function of time are (a) the nonlinear interaction, energy bias, and tunneling rate profiles; (b) energy eigenvalues; (c) stationary solutions for the population difference $w^{\mathrm{ss}}$; and (d) the population difference $w(t)$ after numerical integration of Eqs. (7). In (b) and (c), the dashed curve accounts for the unstable solution, while the short arrows indicate the adiabatic solution (in red) to be followed by the system. PB42 and PB24 denote the two pitchfork bifurcations that the system suffers at ${\Omega }_{0}t=777$ and ${\Omega }_{0}t=1750$, respectively. Note that the pitchfork bifurcation PB24 yields one unstable solution (dashed curve) plus two energetically degenerated stable solutions (solid curve), corresponding, in the limit of vanishing tunneling, to $w=\pm 1$.

Figure 4

Same as in Fig. 2 for $U_g=1.7{\Omega }_0$. SNB42 and PB24 account for a saddle-node and a pitchfork bifurcation at ${\Omega }_{0}t=1374$ and ${\Omega }_{0}t=1588$, respectively.

Figure 5

Adiabatic splitting of a BEC. Plotted is the population difference $w(t_{\mathrm{out}})$ at the end of the BEC splitting process as a function of the temporal delay $\Delta t_U$. The temporal variation of the nonlinear interaction and the tunneling rate is shown in the two upper figures for $\Delta t_U=-125{\Omega }_0^{-1}$ (left) and $\Delta t_U=425{\Omega }_0^{-1}$ (right). Initial conditions: $w(t_{\mathrm{in}})=-1$ and $u(t_{\mathrm{in}})=v(t_{\mathrm{in}})=0$. Parameter setting: $U_g=-0.4{\Omega }_0$, $U_0=1.5{\Omega }_0$, ${\sigma }_U={\sigma }_{\Omega }=212.8{\Omega }_0^{-1}$, $t_{\Omega }=1500{\Omega }_0^{-1},$ and ${\epsilon }_g={\epsilon }_0=n_{\epsilon }=n_U=0$. The adiabatic splitting succeeds in the parameter region ${\Omega }_0\Delta t_U\simeq (250,600)$.

Figure 6

Adiabatic transport of a BEC. Plotted is the population difference $w(t_{\mathrm{out}})$ at the end of the BEC transport process as a function of the nonlinear interaction parameter $U_g$. The temporal variation of the energy bias and the tunneling rate are shown in the inset. Initial conditions: $w(t_{\mathrm{in}})=-1$ and $u(t_{\mathrm{in}})=v(t_{\mathrm{in}})=0$. Parameter setting: ${\epsilon }_0=1.5{\Omega }_0$, ${\sigma }_{\epsilon }={\sigma }_{\Omega }=212.8{\Omega }_0^{-1}$, $\Delta t_{\epsilon }=500{\Omega }_0^{-1}$, $t_{\Omega }=1500{\Omega }_0^{-1}$, $n_{\epsilon }=-1,$ and $U_0={\epsilon }_g=n_U=0$. The adiabatic transport succeeds in the parameter region $U_g/{\Omega }_0=(-2,1.4)$.

Figure 7

Adiabatic self-trapping of a BEC. Plotted is the population difference at (a) the intermediate time $t=t_{\Omega }$ and (b) the end of the BEC transport process, as a function of the energy bias ${\epsilon }_g$. The temporal variation of the nonlinear interaction and the tunneling rate are shown in the inset for $U_0=1.5{\Omega }_0$. Initial conditions: $w(t_{\mathrm{in}})=-1$ and $u(t_{\mathrm{in}})=v(t_{\mathrm{in}})=0$. Parameter setting: $U_g=-0.4{\Omega }_0$, ${\sigma }_U={\sigma }_{\Omega }=212.8{\Omega }_0^{-1}$, $\Delta t_U=325{\Omega }_0^{-1}$, $t_{\Omega }=1500{\Omega }_0^{-1}$, $n_U=1,$ and ${\epsilon }_0=U_0=n_{\epsilon }=0$. The adiabatic self-trapping state does not succeed in the parameter region ${\epsilon }_g/{\Omega }_0=(0,0.8)$.