玻色粒子关联态：平均场以外的非微扰处理
Bosonic Particle-Correlated States: A Nonperturbative Treatment Beyond Mean Field
摘要：
在超低温下，所有粒子都处于同一量子态的（平均场）乘积态Ancor精确地预言了稀玻色气体的许多有用性质。然而，在粒子与粒子之间的相关性变得重要的情况下，产品Ancestion失败了。为了非微扰地包含关联，我们考虑了一组新的态：$N=l\times n$玻色子的粒子关联态是通过对称化$l$粒子量子态的$n$ -折叠积得到的。粒子关联态可以有效地模拟大的$N$，因为它们的参数空间，这取决于$l$，不增长与$n$。在这里，我们制定和发展非常详细的纯态的情况下，$l=2$，其中的多体状态是从一个两粒子的纯态。这些成对的波函数，我们称之为成对关联态（PCS），是由A。J. Leggett [Rev. Mod. Phys. ${\bf 73}$，307（2001）]作为Bogoliubov近似的粒子数守恒版本。我们提出了一个迭代算法，解决了减少（边际）密度矩阵（RDM），即，与PCS在时间上相关的关联函数为$O(N)$。RDM也可以从PCS的归一化因子导出，该因子在大$N$限值中解析导出。为了检验PCS的有效性，我们通过最小化PCS~态的能量来分析两格点Bose- Hubbard模型的基态，包括PCS ~态的精确形式和大- $N$近似形式，并与精确基态进行比较.对于$N=1\,000$，在从单个凝聚体到Mott绝缘体的整个参数区域上，两种情况下基态能量的相对误差都在$10^{-5}$之内。我们提出的数值结果表明，PCS可能是有用的描述在强相互作用制度的动态。
英文摘要：
Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\times n$ bosons is derived by symmetrizing the $n$-fold product of an $l$-particle quantum state. The particle-correlated states can be simulated efficiently for large $N$, because their parameter spaces, which depend on $l$, do not grow with $n$. Here we formulate and develop in great detail the pure-state case for $l=2$, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. ${\bf 73}$, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time $O(N)$. The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-$N$ limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS~state, both in its exact form and in its large-$N$ approximate form, and comparing with the exact ground state. For $N=1\,000$, the relative errors of the ground-state energy for both cases are within $10^{-5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.
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