Effect of a laser dip in the semiclassical dynamics of bosonic Josephson junctions

We consider the standard double-well setup extended with a laser beam in the center to create a “triple-well” potential. The beam in the center is much narrower than the barrier, and it creates a tunable depth well which can support a localized state in the middle. We show that the presence of the localized state in the central well changes the sign of tunneling between the left and the right wells and therefore controls the fixed-point dynamics of the bosonic Josephson junction.

Figure 1

Top panel: the potential landscape with the energy eigenvalues for various laser intensities $I_0$. Bottom panel: the wave functions corresponding to the lowest three energies. The blue (solid) line corresponds to $v_1\left(x\right)$, i.e., the ground-state wave function, the red (dashed) line corresponds to $v_2\left(x\right)$, the wave function of the first excited state, and the green (dotted) line stands for $v_3\left(x\right)$, which is the second excited state. From left to right parameter $I_0$ varies as 0.0, 2.0, 4.0, and 8.0.

Figure 2

The three lowest-energy eigenvalues plotted as a function of $I_0$. One can observe an avoided crossing. The second energy level is unaffected by the perturbing potential, whereas the lowest-energy and the third energy eigenvalue tilt down with increasing $I_0$.

Figure 3

The tunneling ratio as a function of the depth of the central well $I_0$.

Figure 4

Time evolution of the solution of the BJJ equations when a laser dip is abruptly turned on at $t=10$ (time is measured in units of ${\left|J\right|}^{-1})$. For $t<10$ the dip potential is switched off, and the parameters are $UN/J=3.5$. The initial conditions are $\left[z\right(0)=0.5,\theta (0)=0]$. The system performs Josephson oscillations. At $t=10$ a dip potential is suddenly turned on and kept constant. For $t>10$ the parameters change to $UN/J=-3.5$, and the dynamics exhibits self-trapping. (b) shows the population imbalance as a function of time. (a) shows, for $t<10$, the phase space, fixed points $({\mathbf{X}}_1$ and ${\mathbf{X}}_4)$, and the oscillation (thick line) corresponding to the initial conditions. Panel (c) shows the phase space for the new system parameters valid from $t=10$ and the fixed points $({\mathbf{X}}_2$ and ${\mathbf{X}}_3)$. The thick line corresponds to the new trajectory of the system continuing its oscillation in the new energy landscape.

Figure 5

The oscillation frequencies as a function of the strength of the potential dip. The given fixed point becomes unstable when the curve goes below zero.