Entangled Dynamics in Macroscopic Quantum Tunneling of Bose-Einstein Condensates

Tunneling of a quasibound state is a nonsmooth process in the entangled many-body case. Using time-evolving block decimation, we show that repulsive (attractive) interactions speed up (slow down) tunneling. While the escape time scales exponentially with small interactions, the maximization time of the von Neumann entanglement entropy between the remaining quasibound and escaped atoms scales quadratically. Stronger interactions require higher-order corrections. Entanglement entropy is maximized when about half the atoms have escaped.

Figure 1

Initial quasibound state. The many-body wave function for $N=20$, with $NU/J=+0.15$ (blue shaded region, points show actual TEBD results for the density average $⟨{\widehat{n}}_i⟩$), is first localized to the left behind the barrier (red line, red and pink shaded areas) via relaxation in imaginary time with a barrier of height $h$ and initial width $w_I$. At $t=0$, in real-time propagation, the barrier is reduced to width $w$ (solid red line, red shaded area) so the now-quasibound Bose gas can commence macroscopic quantum tunneling. The hard wall at the left and relatively small barrier area push the density tail to partially extend to the right.

Figure 2

Many-body tunneling and the calculation of decay time. Barrier widths (a),(d) $w=1$, (b),(e) $w=3$, and (c),(f) $w=5$. Top row: average atom number per site. Bottom row: number in well $⟨{\widehat{n}}_{\text{well}}⟩$ (magenta), number in barrier $⟨{\widehat{n}}_{\text{bar}}⟩$ (cyan), and escaped $⟨{\widehat{n}}_{\mathrm{esc}}⟩$ (tan) atoms; the $1/e$ decay time all $\pm 0.1$. All plots for $NU/J=+0.30$, with $N=20$.

Figure 3

Many-body (MB) vs mean-field (MF) escape time predictions. Solid lines: repulsive (REP). Dashed lines: attractive (ATT). (a)–(b) Dependence of $t_{\mathrm{esc}}^{\mathrm{MB}}$ on barrier area and atom number for (a) $NU/J=\pm 0.15$ and (b) $NU/J=\pm 0.30$. (c) $t_{\mathrm{esc}}^{\mathrm{MB}}$ plateaus towards $t_{\mathrm{esc}}^{\mathrm{MF}}$ for 10 to 80 atoms, as shown for $NU/J=\pm 0.15$ and $w=5$. (d) Depletion for $NU/J=\pm 0.30$ and $w=5$; attractive markers semitransparent for readability. Curves are a guide to the eye; points represent actual data with error bars smaller than data point in all panels. Panel (d) legend corresponds to (a),(b), and (d).

Figure 4

Many-body quantum measures. (a) Nearly universal curve for the entropy of entanglement vs the average number of trapped atoms. Black line: no interaction. Darker red/blue corresponds to higher $|NU/J|$. (b) Observables demonstrate very different scaling with interaction. Points show actual data (error bars smaller than points), while lines are best fit curves. All plots treat $N=6$, with 11 equally spaced values of $NU/J\in [-0.60,0.60]$.

Figure 5

Time dependence of density-density correlations. (a)–(c) $g^{(2)}$ shows correlations between trapped and escaped atoms. The barrier, indicated by white lines, breaks up negatively correlated regions (red); shown are time slices at (a) $t=0$, (b) $t=62\approx t_s$, and (c) $t=125\approx t_{\mathrm{esc}}^{\mathrm{MB}}$. (d) Quantum depletion grows rapidly for $N=2$, with $NU/J=\pm 0.15$. Solid lines: repulsive. Dashed semitransparent lines: attractive. Curves are a guide to the eye; points represent actual data (error bars smaller than points).