We investigate finite-size quantum effects in the dynamics of $N$ bosonic particles which are tunneling between two sites adopting the two-site Bose-Hubbard model. By using time-dependent atomic coherent states (ACSs) we extend the standard mean-field equations of this bosonic Josephson junction, which are based on time-dependent Glauber coherent states. In this way we find $1/N$ corrections to familiar mean-field (MF) results: the frequency of macroscopic oscillation between the two sites, the critical parameter for the dynamical macroscopic quantum self-trapping (MQST), and the attractive critical interaction strength for the spontaneous symmetry breaking (SSB) of the ground state. To validate our analytical results we perform numerical simulations of the quantum dynamics. In the case of Josephson oscillations around a balanced configuration we also find that, for a few atoms, the numerical results are in good agreement with the predictions of time-dependent ACS variational approach, provided that the time evolution is not too long. Also, the numerical results of SSB are better reproduced by the ACS approach than with the MF approach. Instead, the onset of MQST is correctly reproduced by ACS theory only in the large $N$ regime and, for this phenomenon, the $1/N$ correction to the MF formula is not reliable.

中文翻译：

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玻色约瑟夫森结中的有限尺寸效应

我们研究了动力学中的有限尺寸量子效应 $ñ$采用两点Bose-Hubbard模型在两个点之间隧穿的Bosonic粒子。通过使用与时间有关的原子相干态（ACS），我们扩展了该玻色约瑟夫森结的标准均场方程，该方程基于与时间有关的格劳伯相干态。这样我们发现$\mathrm{1个}/ñ$对熟悉的平均场（MF）结果的修正：两个位点之间的宏观振荡频率，动态宏观量子自陷（MQST）的临界参数以及自发对称断裂（SSB）的有吸引力的临界相互作用强度基态。为了验证我们的分析结果，我们执行了量子动力学的数值模拟。在约瑟夫森（Josephson）振荡围绕平衡构型的情况下，我们还发现，对于几个原子，只要时间演化时间不太长，数值结果就与时变ACS变分方法的预测非常吻合。同样，与MF方法相比，ACS方法可以更好地再现SSB的数值结果。取而代之的是，仅在大范围内，ACS理论才能正确再现MQST的发作$ñ$ 政权，对于这种现象， $\mathrm{1个}/ñ$ MF公式的校正不可靠。