Reproducible mesoscopic superpositions of Bose-Einstein condensates and mean-field chaos

In a parameter regime for which the mean-field (Gross-Pitaevskii) dynamics becomes chaotic, mesoscopic quantum superpositions in phase space can occur in a double-well potential, which is shaken periodically. For experimentally realistic initial states, such as the ground state of some 100 atoms, the emergence of mesoscopic quantum superpositions in phase space is investigated numerically. It is shown to be reproducible, even if the initial conditions change slightly. Although the final state is not a perfect superposition of two distinct phase states, the superposition is reached an order of magnitude faster than in the case of the collapse-and-revival phenomenon. Furthermore, a generator of entanglement is identified.

Figure 1

Comparison of the Poincaré surface of section (a) and the QFI map (b) for parameters $2{\mu }_0/\Omega =3.0$, $N\kappa /\Omega =0.5$, $\omega /\Omega =3.0$, and $2{\mu }_1/\omega =0.1$. Plot (b) shows the time-averaged QFI for a system with $N=100$ particles and averaging time $10/\Omega$. The QFI is enhanced close to the hyperbolic fixed point in the Poincaré surface of section ($z\approx 0.21$, $\varphi \approx 0.54\pi$), which acts as a generator of mesoscopic entanglement (c).

Figure 2

Entanglement flag $F_{\mathrm{ent}}$ as a function of time $\tau =\Omega t$ and scaled driving amplitude $2{\mu }_1/\omega$ for (a) the ground and (b) the first excited state. (c) and (d): Contrast for the ground and the first excited state. $N=110$, $N\kappa /\Omega =1.045$, ${\mu }_0/\Omega =0.75$, and $\omega /\Omega =3.0.$

Figure 3

Entanglement flag (a) $F_{\mathrm{ent}}$ and (b) contrast for $2{\mu }_1/\omega =1.0$. Other parameters as in Fig. 2.

Figure 4

(a) Entanglement flag and (b) contrast at $\tau =7.1$ versus ${\mu }_0/\Omega$ and $\omega /\Omega$. $N=110$, $N\kappa /\Omega =1.045$, $2{\mu }_1/\omega =1.0$.

Figure 5

(a) $F_{\mathrm{ent}}$ and (b) contrast for $\tau =7.1$ (black crosses) and $\tau =9.0$ (blue stars) for different particle numbers such that $\kappa /\Omega =\mathrm{const}$. $N\kappa /\Omega (N=110)=1.045$, ${\mu }_0/\Omega =0.75$, $\omega /\Omega =3.0$, and $2{\mu }_1=1.0$.

Figure 6

Averaged phase distribution $p_{\varphi }$ (red solid line) with standard deviation (blue dashed lines). The initial particle number is modeled by a truncated renormalized Poisson distribution of the initial particle number, which ranges from $N=100$ to $N=120$, with mean value $N=110$. (a) Initial state, (b) entangled state at $\tau =6.5$, (c) time-developed state at $\tau =8.9$, and (d) assumed exemplary two-component statistical mixture at $\tau =8.9$. The interaction strength $\kappa /\Omega$ is kept constant such that $N\kappa /\Omega (N=110)=1.0$. ${\mu }_0/\Omega =1.3$, $\omega /\Omega =3.3$, and $2{\mu }_1/\omega =0.7$.

Figure 7

(a) $F_{\mathrm{ent}}$ for $\tau =7.1$ and (c) $\tau =9.0$. Black solid line: QFI $F_{\mathrm{ent},\mathrm{mixed}}$ for mixed states [cf. Eq. (18)], blue dashed line: $\langle F_{\mathrm{ent}}\rangle {}_{\mathrm{st}}$ statistically averaged over all 111 states, red dotted line: $\langle F_{\mathrm{ent}}\rangle {}_{\mathrm{st}}$ statistically averaged over the three lowest-lying experimentally important states [28]. (b) and (d): Contrast $\langle C\rangle {}_{\mathrm{st}}$ for $\tau =7.1$ and $\tau =9.0$ [cf. Eq. (16)]. Blue dashed-dotted line: statistical average over all states, red dotted line: statistical average over three lowest-lying eigenstates. Parameters same as in Fig. 3, $2{\mu }_1/\omega =1.0$.