Quasiparticle tunneling in a periodically driven bosonic Josephson junction

A resonantly driven bosonic Josephson junction supports stable collective excitations, or quasiparticles, which constitute analogs of the Trojan wave packets previously explored with Rydberg atoms in strong microwave fields. We predict a quantum beating effect between such symmetry-related many-body Trojan states taking place on time scales which are long in comparison with the driving period. Within a mean-field approximation, this quantum beating can be regarded as a manifestation of dynamical tunneling. On the full $N$-particle level, the beating phenomenon leads to an experimentally feasible, robust strategy for probing highly entangled mesoscopic states.

Figure 1

Principle of Trojan beating for $\nu =2$: The unperturbed energy eigenstates $|n\rangle$ of a weakly anharmonic oscillator, with energy levels spaced by about $\hslash \omega /2$ in the vicinity of $n=r$ as indicated by the horizontal lines, are subjected to an external monochromatic perturbation with frequency $\omega$. Resonant coupling then occurs between next-to-nearest neighbors, as expressed by Eq. (6), giving rise to two almost uncoupled “ladders” of states indicated by the vertical braces. Each ladder corresponds to a set of $2\pi /\omega$-periodic Floquet states enumerated by a new quantum number $k$. Odd and even superpositions of the Floquet states with $k=0$ yield two $4\pi /\omega$-periodic Trojans, which beat among themselves on much longer time scales.

Figure 2

Quasienergy spectrum of the tilted, driven bosonic Josephson junction (12) for $N=200$ particles, $N\kappa /\Omega =0.95,\omega /\Omega =3.258$, and ${\mu }_0/\Omega =0.5$. Under these conditions $r=142$ is a resonant level with $\nu =2$, leading to two almost identical sets of quasienergies displaced against each other by $\hslash \omega /2$, as predicted by the Mathieu theory. The Trojan doublet with $k=0$ is indicated by the arrows.

Figure 3

Scaled population imbalance $2\langle J_z\rangle /N$ evolving from the Trojan initial states $|{\tilde{\Psi }}_0^{(+)}\rangle$ (solid lines) and $|{\tilde{\Psi }}_0^{(-)}\rangle$ (dashed lines). While the short-time evolution is monitored continuously, the long-time evolution is recorded stroboscopically at each multiple of $2T$. (a, c) $N=100,2{\mu }_1/\omega =0.025$; the other parameters are as in Fig. 2. (b, d) $N=50,2{\mu }_1/\omega =1.0,N\kappa /\Omega =0.95,\omega /\Omega =1.6$, and ${\mu }_0/\Omega =0.8$.

Figure 4

Poincaré surfaces of section generated by the classical driven pendulum (14). (a) For the parameters underlying Figs. 3 and 3, the 2:1 resonance leads to two banana-shaped zones surrounding the central elliptic fixed point. (b) For the parameters of Figs. 3 and 3, the two resonant islands are separated by a chaotic sea; an additional regular island is visible in the lower left corner.

Figure 5

Grey lines: Exact quasienergy spectrum of the tilted, driven bosonic Josephson junction for $N=200$ particles, as also shown in Fig. 2 $\left(N\kappa /\Omega =0.95,\omega /\Omega =3.258,{\mu }_0/\Omega =0.5\right)$. Black lines: Quasienergies provided for $\nu =2$ by the Mathieu approximations (A27) and (A28), with $r=142$. The Mathieu parameter (A33) is given by $q\omega /\left(2{\mu }_1\right)=-498.26$.