Josephson plasma oscillations and the Gross-Pitaevskii equation: Bogoliubov approach versus two-mode model

We show that the Josephson plasma frequency for a condensate in a double-well potential, whose dynamics is described by the Gross-Pitaevskii (GP) equation, can be obtained with great precision by means of the usual Bogoliubov approach, whereas the two-mode model—commonly constructed by means of a linear combination of the low-lying states of the GP equation—generally provides accurate results only for weak interactions. A proper two-mode model in terms of the Bogoliubov functions is also discussed, revealing that in general a two-mode approach is formally justified only for not too large interactions, even in the limit of very small amplitude oscillations. Here we consider specifically the case of a one-dimensional system, but the results are expected to be valid in arbitrary dimensions.

Figure 1

Oscillations of the population imbalance $z\left(t\right)$, for $u_0=16\left(\mu /V_0=0.1\right)$.

Figure 2

Josephson frequency as obtained from the different approaches discussed in the text [Eqs. (14), (15), and the Bogoliubov frequency ${\omega }_B]$, compared with the results of the GP equation (points), as a function of $u_0$, for $V_0=50$. The upper $x$ axis shows the corresponding value of of $\mu /V_0$ (that is a monotonic increasing function of $u_0$).

Figure 3

Josephson frequency as obtained from the different approaches discussed in the text, compared with the results of the GP equation (points), as a function of $u_0$ for fixed $\mu /V_0=0.25$.

Figure 4

(Top) Plot of the ground-state wave function ${\psi }_0\left(x\right)$, and of the Bogoliubov functions $\tilde{u}\left(x\right)$ and $\tilde{v}\left(x\right)$, for (a) $\mu /V_0=0.1\left(u_0=16\right)$ and (b) $\mu /V_0=0.5\left(u_0=220\right)$. (Bottom) Left and right states obtained from symmetric and antisymmetric combination of ${\psi }_0\left(x\right)$ and $\tilde{u}\left(x\right)$; see Eq. (30). In both panels, the double-well potential is also shown.