Differences between Mean-Field Dynamics and $N$-Particle Quantum Dynamics as a Signature of Entanglement

A Bose-Einstein condensate in a tilted double-well potential under the influence of time-periodic potential differences is investigated in the regime where the mean-field (Gross-Pitaevskii) dynamics become chaotic. For some parameters near stable regions, even averaging over several condensate oscillations does not remove the differences between mean-field and $N$-particle results. While introducing decoherence via piecewise deterministic processes reduces those differences, they are due to the emergence of mesoscopic entangled states in the chaotic regime.

Figure 1

Poincaré surface of section for the forced nonrigid pendulum [the mean-field dynamics (3) are plotted for various starting points at integer multiples of the oscillation period $2\pi /\omega$]. Closed loops are characteristic for stable orbits whereas irregular dots represent chaotic regions. For a BEC in a double well, the parameters correspond to: (a) a tilt of $2{\mu }_0/\Omega =3.0$, a driving frequency of $\omega =3\Omega$, an interaction of $N\kappa /\Omega =0.8$, and a driving amplitude of $2{\mu }_1/\Omega =0.9$ (i.e., a one-photon-resonance [26]); (b) the $3/2$-photon-resonance with $N\kappa /\Omega =0.1$, $2{\mu }_0/\Omega =3.0$, $\omega /\Omega =2.08$, and $2{\mu }_1/\Omega =1.8$; (c) all parameters as in (a) except for $N\kappa /\Omega =0.3$.

Figure 2

Quantum dynamics ($N=100$) versus mean-field dynamics using the parameters of Fig. 1a. (a) The difference of the time-averaged population imbalances $⟨J_z{⟩}_T/N$ and $⟨z/2{⟩}_T$ as a function of ${101}^2$ initial conditions ($z_0$, ${\varphi }_0$) in a two-dimensional projection of the resulting three-dimensional plot ($T=100/\Omega$). (b) The time-averaged root-mean-square fluctuations $⟨\Delta J_z{⟩}_T/N$ of the population imbalance as a function of the initial atomic coherent state (2).

Figure 3

Time-averaged population imbalance $⟨J_z{⟩}_T/N$ for various driving amplitudes ${\mu }_1$ in a tilted driven double well ($2{\mu }_0/\Omega =3.0$, $\omega =3\Omega$, $T=100/\Omega$). The BEC initially is in the lower well ($z_0=1$). Solid line: $⟨J_z{⟩}_T/N$ for $N=1000$ is a smooth curve as opposed to the mean-field results depicted in the inset, which display chaotic jumps for small changes of the driving amplitude. Dots in the main plot: if decoherence is included via the PDP process described around Eq. (6) with on average $\simeq 5$ jumps ($\alpha =1/20$) [33], the behavior is closer to the mean-field dynamics. Many dots lie in the area defined by the curves $(⟨J_z{⟩}_T\pm ⟨\Delta J_z{⟩}_T)/N$ (dashed lines).

Figure 4

Time-averaged population imbalances of quantum dynamics ($N=100$) versus mean-field dynamics at the $3/2$ photon resonance of Fig. 1b. (a) The difference is plotted as a function of the initial condition ($z_0$, ${\varphi }_0$) in a two-dimensional projection ($T=100/\Omega$). (b) As in (a) but the mean-field dynamics are replaced by an average over the distribution of initial conditions (7).

Figure 5

Entanglement (10) for parameters as in Fig. 1a (left column) and as in Fig. 1c (right column). (a),(b) Mesoscopic quantum superpositions were identified at times $\tau =5,10,15,\dots$; the maximum value of ${\sigma }_{\mathrm{ent}}$ is displayed for ${101}^2$ initial conditions ($z_0$, ${\varphi }_0$) and for (a) short times ($\tau =10$) and (b) longer times ($\tau =100$). (c) Projection of two characteristic entangled states (with maxima $m_1$, $m_2$) on the atomic coherent states (2). Black (gray or red) regions: $|⟨\theta ,\varphi |\psi ⟩{|}^2>0.16$ ($>0.05$). Left: $z_0=-0.6$, ${\varphi }_0=-2.764601535$, $\tau =80$, ${\sigma }_{\mathrm{ent}}\simeq 72.3\%$. Right: $z_0=-0.98$, ${\varphi }_0=-2.701769682$, $\tau =75$, ${\sigma }_{\mathrm{ent}}\simeq 33.5\%$. In the left plot, the large blue or gray circle is a typical set $R$ around $|\tilde{\theta },\tilde{\varphi }⟩$ with $|⟨\theta ,\varphi |\tilde{\theta },\tilde{\varphi }⟩{|}^2>0.001$ [cf. Eq. (9)].