Macroscopic quantum tunneling of Bose-Einstein condensates with long-range interaction

The ground state of Bose-Einstein condensates with attractive particle interactions is metastable. One of the decay mechanisms of the condensate is a collapse by macroscopic quantum tunneling, which can be described by the bounce trajectory as a solution of the time-dependent Gross-Pitaevskii equation in imaginary time. For condensates with an electromagnetically induced gravity-like interaction, the bounce trajectory is computed with an extended variational approach using coupled Gaussian functions and is simulated numerically exact within the mean-field approach on a spacetime lattice. It is shown that the variational computations converge very rapidly to the numerically exact result with increasing number of Gaussians. The tunneling rate of the condensate is obtained from the classical action and additional parameters of the bounce trajectory. The converged variational and numerically exact results drastically improve by several orders of magnitude the decay rates obtained previously with a simple variational approach using a single Gaussian-type orbital for the condensate wave function.

Figure 1

(a) Potential $V\left(q\right)$ of self-trapped BEC with attractive $1/r$ interaction at various values of scattering length $a$. (b) Scheme of the inversion of the potential and the periodic orbits approaching the bounce trajectory.

Figure 2

Variational parameters (a) $A_k\left(\tau \right)$, ${\overline{A}}_k\left(\tau \right)$ for $k=1,\cdots ,N_g$ and (b) ${\gamma }_k\left(\tau \right)-{\gamma }_1\left(\tau \right)$, ${\overline{\gamma }}_k\left(\tau \right)-{\overline{\gamma }}_1\left(\tau \right)$ for $k=2,\cdots ,N_g$ of a periodic trajectory with period $\beta =47.4$ at scaled scattering length $a=-0.9$ described by $N_g=3$ coupled Gaussian functions.

Figure 3

Density $\left[\overline{\psi }\psi \right]\left(r\right)$ of a condensate wave function for a periodic trajectory with period $\beta =42.45$ computed on the spacetime lattice at increasing values $\left|\tau \right|$ (from top to bottom line) of the imaginary time. The scaled scattering length is $a=-0.9$.

Figure 4

(a) Euclidean action of periodic trajectories at constant scaled scattering length $a=-0.9$. (b) Euclidean action of the bounce trajectory as function of the scaled scattering length $a$. Shown are the results obtained by the variational approach using up to $N_g=5$ coupled Gaussians and by numerically exact simulations on grids. The variational results with $N_g>3$ agree within the line width with the grid computations.

Figure 5

Decay rate by macroscopic quantum tunneling of a BEC with (a) $N=30$ and (b) $N=100$ particles obtained by Eq. (8) with the parameters $S_{\mathrm{b}}$, ${\omega }_0$, and $v_0$ of the bounce trajectory. The decay rate obtained by variational calculations with $N_g=1$ to 5 coupled Gaussians converges rapidly to the decay rate obtained by grid computations. For $N_g\ge 3$ deviations are less than the line width.

Figure 6

Sketch of potential with inverted harmonic saddle at the origin.