Quantum dynamics in the bosonic Josephson junction

We employ a semiclassical picture to study dynamics in a bosonic Josephson junction with various initial conditions. Phase diffusion of coherent preparations in the Josephson regime is shown to depend on the initial relative phase between the two condensates. For initially incoherent condensates, we find a universal value for the buildup of coherence in the Josephson regime. In addition, we contrast two seemingly similar on-separatrix coherent preparations, finding striking differences in their convergence to classicality as the number of particles increases.

Figure 1

Contour lines for $u>2$. Sea levels are colored blue, island levels are colored green, and the separatrix is colored red (left panel). Energy spectrum for $N=20$ and $u=10$. WKB energies (red x) are compared with exact eigenvalues (blue $+$). Dashed lines indicate slopes ${\omega }_J$ for low energies, ${\omega }_x$ for near-separatrix energies, and ${\omega }_{+}$ for high energies (right panel).

Figure 2

An illustration of twin-Fock ($n=0$) preparation (left), and of $\pi$ (“$\pi$”), zero (“$0$”), and edge (“e”) preparations (right) using Wigner plots on a sphere. The left panel is a three-dimensional plot, while the right panel is a Mercator projection of the sphere using $(\mathit{\phi },\mathit{n})$ coordinates.

Figure 3

The evolving quantum state of $N=40$ bosons with $u=5$ for twin-Fock ($n=0$) preparation. Here and below the units are such that $K=1$. The time is $t=4$. (a) Wigner function of the evolved quantum state. (b) Semiclassical evolved state. (c) Occupation statistics, with the semiclassical result shown as dashed-dotted line.

Figure 4

The variation of $S_x(t)$ with time for $N=40$ particles with $u=5$, for zero (a), $\pi$ (b), edge (c), and twin-Fock (d) preparations. Note the different vertical scale. The dashed-dotted lines are based on semiclassical simulation. Note that the fluctuations of a semiclassical preparation always die after a transient, which should be contrasted with both the classical (single trajectory) and quantum (superposition of $M>1$ eigenstates) behavior.

Figure 5

The LDOS (left) and the spectral content of the fluctuations (right) of $N=500$ bosons with $u=4$, for zero, $\pi$, edge, and twin-Fock preparations (top to bottom). The horizontal axes are $E-E_x$ and $\omega /{\omega }_J$. The lines in the LDOS figures are based on a semiclassical analysis (see text), while the circles are from the exact quantum calculation. Note that due to the mirror symmetry of the zero preparation the expected frequency should approach $2{\omega }_J$, while for the $\pi$ preparation it is bounded from below by $2{\omega }_x$ (both frequencies are indicated by vertical dashed lines). Note also the outstanding difference between the spectral support of zero and $\pi$ preparations compared with continuous-like support in the case of edge and twin-Fock preparations.

Figure 6

The participation number $M$ as determined from the LDOS for $N=100$ (circles), 500 (squares), and 1000 (diamonds) particles. The left panel contains the zero (lower set in blue) and $\pi$ (upper set in red) preparations, while the edge preparation is presented in the right panel. Note the different vertical scale. In the crudest approximation we expect in the edge case $M~N^{1/2}$, while in the $\pi$ case $M\ll N^{1/2}$ as long as $(u/N)\ll 1$.

Figure 7

The mean frequency of the $S_x(t)$ oscillations versus $u/N$ for $N=100$ (circles), 500 (squares), and 1000 (diamonds) particles. The preparations are (upper to lower sets of data points) zero (blue), edge (magenta), and $\pi$ (red). The doubled Josephson frequency $2{\omega }_J$ is marked by a dashed blue line. The theoretical predictions of Eq. (42), doubled due to mirror symmetry, are represented by red and magenta dashed lines, while Eq. (43) for $u/N\gg 1$ is represented by the black dash-double-dotted line (there is one fitting parameter as explained there).

Figure 8

(Left) The long-time average of $S_x(t)$ versus $u/N$ for $N=100$ (circles), 500 (squares), and 1000 (diamonds) particles. The preparations are (upper to lower sets of data points) zero (blue), twin-Fock (black), edge (magenta), and $\pi$ (red). The symbols are used for the quantum results and the dashed lines are the semiclassical prediction for 40 particles. Note that the scaling holds only in the Josephson regime $1\ll u\ll N^2$, and therefore, for a given $u/N$ range, becomes better for large $N$. (Right) The long-time $v_{\mathrm{RMS}}$ of $S_x(t)$ for the three coherent preparations (lower to upper sets) are zero (blue), edge (magenta), and $\pi$ (red). The implied $N^{1/4}$ scaling based on Eq. (69) is confirmed. In the inset, the $v_{\mathrm{RMS}}$ of $S_x(t)$ for edge (up triangle) and $\pi$ (down triangle) preparations is plotted versus $N$ while $u=4$ is fixed. The dashed lines are power-law fits that nicely agree with the predictions of Eq. (70).