Exact Quantum Dynamics of a Bosonic Josephson Junction

The quantum dynamics of a one-dimensional bosonic Josephson junction is studied by solving the time-dependent many-boson Schrödinger equation numerically exactly. Already for weak interparticle interactions and on short time scales, the commonly employed mean-field and many-body methods are found to deviate substantially from the exact dynamics. The system exhibits rich many-body dynamics such as enhanced tunneling and a novel equilibration phenomenon of the junction depending on the interaction, which is attributed to a quick loss of coherence.

Figure 1

Full quantum dynamics of a 1D bosonic Josephson junction below ($\Lambda <{\Lambda }_c$) and above ($\Lambda >{\Lambda }_c$) the transition to self-trapping as defined by the two-mode GP theory. Shown is the full many-body result [solid (blue) lines] for the probability of finding a boson in the left well, $p_L(t)$. For comparison, the respective GP [dotted (black) lines] and BH [dashed (magenta) lines] results are shown as well. The parameter values are (a) $N=20$, $\lambda =0.152$, (b) $N=100$, $\lambda =0.152$ ($\Lambda <{\Lambda }_c$), (c) $N=20$, $\lambda =0.245$, and (d) $N=100$, $\lambda =0.245$ ($\Lambda >{\Lambda }_c$). The GP and BH results are found to deviate from the full many-body results already after short times. The insets show the convergence of the full many-body results. In the color online: (a), (c) $M=2$ (solid purple line), $M=4$ (solid red line), $M=6$ (solid green line), and $M=8$ (solid blue line). The $M=2$ results are seen to deviate slightly from the converged results for $M\ge 4$. (b), (d) The results for $M=2$ (solid purple line) and $M=4$ (solid blue line) are shown. All quantities shown are dimensionless.

Figure 2

Emergence of equilibration of the density at interaction strength $\lambda =4.9$. Top: same as Fig. 1, but for $N=10$ [solid black (blue) line] and $N=100$ [solid gray (green) line]. The respective BH [dashed (magenta) line] results are on top of each other. In contrast to the BH dynamics, which is completely self-trapped, the full many-body dynamics is not. $p_L(t)$ tends towards its long-time average $p_L=0.5$. For $N=100$ particles, $M=4$ orbitals were used. The inset shows the convergence of the full many-body solution for $N=10$ bosons. In the color online: $M=4$ (solid black line), $M=10$ (solid blue line), and $M=12$ (solid red line). The $M=4$ result follows the trend of the converged $M=12$ result. Bottom: corresponding natural-orbital occupations for $N=10$ bosons. The system becomes fragmented and roughly four natural orbitals are macroscopically occupied. All quantities are dimensionless.

Figure 3

Dynamics of the first-order correlation function for $\lambda =4.9$ at which the equilibration phenomenon of Fig. 2 occurs. Shown is $|g^{(1)}(x^{\prime },x;t){|}^2$ of $N=10$ bosons at different times. Top left: full many-body result at $t=0$. The initial state exhibits coherence over the entire extent of the system. Top right: full many-body result at $t=10t_{\mathrm{Rabi}}$. The coherence is lost. The system is incoherent even on short length scales. Bottom left: BH result at $t=0$. Bottom right: BH result at $t=10t_{\mathrm{Rabi}}$. In contrast to the full many-body result, the BH wave function remains completely coherent.