方差作为关联的灵敏探针承受无限粒子极限
Variance as a sensitive probe of correlations enduring the infinite particle limit
摘要：
由超冷玻色子原子构成的玻色-爱因斯坦凝聚体已成为研究相互作用多体量子系统的中心领域。已证明在无限粒子数和恒定相互作用参数的极限下，捕获的玻色-爱因斯坦凝聚体的基态是100%凝聚的[Lieb和Seiringer，Phys. Rev. Lett. { \bf 88}，170409（2002）]。这个结果的意义是凝聚体的性质，特别是它的能量和密度，收敛到通过最小化Gross-Pitaevskii能量泛函得到的那些。这自然提出了一个问题，在这个极限下，相关性是否重要。在这里，我们证明了分析和数值计算，即使在无限粒子限制多体相关性可以导致一个重大修改的\textit{variance}的任何运营商相比，预期从Gross-Pitaevskii的结果。方差的强偏差源于其对约化的两体密度矩阵的项的明确依赖，否则这些项对该极限中的能量和密度没有贡献。这使得方差成为多体相关性的敏感探针，即使系统的能量和密度已经收敛到Gross-Pitaevskii结果。我们使用质心位置算子来证明这种相关性的持久性。这种多体效应的影响进行了讨论。
英文摘要：
Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. {\bf 88}, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a substantial modification of the \textit{variance} of any operator compared to that expected from the Gross-Pitaevskii result. The strong deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the center-of-mass position operator to exemplify this persistence of correlations. Implications of this many-body effect are discussed.
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