Verification of exceptional points in the collapse dynamics of Bose-Einstein condensates

In Bose-Einstein condensates with an attractive contact interaction the stable ground state and an unstable excited state emerge in a tangent bifurcation at a critical value of the scattering length. At the bifurcation point both the energies and the wave functions of the two states coalesce, which is the characteristic of an exceptional point. In numerical simulations signatures of the exceptional point can be observed by encircling the bifurcation point in the complex extended space of the scattering length, however, this method cannot be applied in an experiment. Here we show in which way the exceptional point effects the collapse dynamics of the Bose-Einstein condensate. The harmonic inversion analysis of the time signal given as the spatial extension of the collapsing condensate wave function can provide clear evidence for the existence of an exceptional point. This method can be used for an experimental verification of exceptional points in Bose-Einstein condensates.

Figure 1

Mean-field energy of the ground and excited state of the BEC as a function of the scattering length computed with $N_g=1$ to 4 coupled Gaussian functions. (Results obtained with more than two Gaussian functions agree within the linewidths.) The two states emerge in a tangent bifurcation at a critical value of the scattering length.

Figure 2

Potential $V\left(q\right)$ in Eq. (9) for parameters $Na$ above, at, and below the critical value ${\left(Na\right)}_{\mathrm{cr}}$ of the tangent bifurcation.

Figure 3

(a) Trajectories $q\left(t\right)$ for the time evolution of the condensate extension computed with a single Gaussian function, $N_g=1$, and for various strengths $Na$ of the contact interaction in the range $-0.675\le Na\le -0.664$. A few trajectories are highlighted and labeled. The initial state of the condensate is prepared at $s=Na=-0.665$. The plus symbols indicate for some trajectories the points with vanishing third derivatives $\stackrel{⃛}{q}\left(t_0\right)=0$. (b) Time derivatives $\dot{q}\left(t\right)$ of the trajectories in (a).

Figure 4

(a) Frequencies and (b) and (c) amplitudes obtained by the local harmonic inversion analysis of trajectories $q\left(t\right)$ computed with a single Gaussian function, $N_g=1$, and an initial state prepared at $s=Na=-0.57$. Note that $\mathrm{Im}{\omega }_1=\mathrm{Im}{\omega }_2$ and $\mathrm{Re}{d}_1=\mathrm{Re}{d}_2=0$ and thus the corresponding lines cannot be distinguished. See text for discussion.

Figure 5

(a) Trajectories $q\left(t\right)$ and (b) time derivatives $\dot{q}\left(t\right)$ as in Fig. 3 but computed with $N_g=4$ coupled Gaussian functions and the initial state prepared at $s=Na=-0.57$.

Figure 6

(a) Frequencies and (b) and (c) amplitudes obtained by the local harmonic inversion analysis of trajectories $q\left(t\right)$ as in Fig. 4 but computed with $N_g=3$ coupled Gaussian functions and the initial state prepared at $s=Na=-0.57$.

Figure 7

Dependence of the critical parameter ${\left(Na\right)}_{\mathrm{cr}}$ on the strength $s=Na$ of the contact interaction of the initial state and the number of Gaussian functions $N_g$ used for the computations. The lines mark the values given in Table 1 for $N_g=2$ and $N_g=3$.