旋转框架下的超冷玻色气体——极尽深寒的量子世界里的奇妙涡旋

Ultracold Bose Gases in Frame Rotating—The Beautiful Vortex in the Extremely Deep and Cold Quantum World

摘要: 玻色–爱因斯坦凝聚与量子霍尔效应是近现代理论与凝聚态物理研究的两大热点，旋转框架的构建为玻色系统研究两者之间提供了很好的平台。本文以超冷玻色气体为研究主体，通过谐振势与旋转构造特殊的旋转框架，获得与磁场中二维电子气体相似的哈密顿模型。同时，选取朗道规范和最低朗道能级近似，通过理论和数值上研究相互作用系数对系统基态能量和基态波函数的影响，其结果无量纲化表示。结果表明，在较弱的相互作用下，系统的基态为经典的玻色–爱因斯坦凝聚态，并随着相互作用增加，出现准粒子激发，借助Bogoliubov近似可以观测到与超流氦4类似的旋子激发谱结构。随后，继续增加相互作用则会出现涡旋阵列，其涡旋阵列的数目与相互作用系数密切相关，而涡旋的大小与旋转频率有关。最后，在较强的相互作用下，可以想象系统会进入一个涡旋高度密集的状态，当涡旋的数目与粒子数目出现数量级的特殊关系时，预计会形成量子霍尔结晶相，并可能出现分数量子霍尔现象，这为进一步研究量子化涡旋物理和量子霍尔效应提供了一个新的途径。

Abstract: Bose-Einstein condensation and the quantum Hall effect represent two prominent areas of interest in modern theoretical and condensed matter physics research. The construction of rotating frames provides an excellent platform for the study of Bose system bridging the two. This paper focuses on ultracold Bose gases and establishes a specialized rotational framework using harmonic potentials and rotation, yielding a Hamiltonian model akin to that of two-dimensional electron gases in a magnetic field. Utilizing the Landau gauge and specifically approximating the lowest Landau level, the study investigates the influence of interaction coefficients on the ground state energy and wave function of the system through both theoretical and numerical approaches, with results presented in dimensionless units. The findings indicate that under weak interactions, the system’s ground state resembles a classical Bose-Einstein condensate. As the interaction strength increases, quasi-particle excitations emerge, with an observable roton-like excitation spectrum using the Bogoliubov approximation akin to superfluid helium-4. Subsequently, further escalation of interactions results in the formation of vortex arrays, with the number of vortex array closely tied to the interaction coefficient and their size dependent on the rotation frequency. Finally, under strong interactions, it is conceivable that the system will enter a highly dense state of vortices, when the number of vortices correlates with the particle count in a specific magnitude relationship, a quantum Hall crystal phase is anticipated to form, potentially exhibiting fractional quantum Hall phenomena. This provides a new approach for further research on quantum vortex physics and quantum Hall effects.