论文标题
射线空间的出生和射线几何形状
Born-Oppenheimer and the Geometry of Ray Space
论文作者
论文摘要
众所周知，在出生的烟囱近似中，核运动的慢模式通过整合电子运动快速模式而产生的三种效应改变了核运动模式。第一个是来自电子状态特征值的有效标量潜在$ v _ {\ mathrm dyn} $，第二个是来自浆果相位矢量电位$ a $的有效磁场。第三学期是一个额外的潜在$ v _ {\ mathrm geom} $，起源于射线空间和fubini-study指标。在本文中，我们在简单的玩具模型的背景下说明了这些效果及其几何起源，该模型的中性子与强，空间变化的磁场相互作用。关于中子旋转是快速的自由度，我们弄清了中子运动的缓慢动力学。我们的处理是几何的，并带出了源自射线空间和福比尼 - 研究度量的Kähler几何形状的影响。然后，我们提供磁场配置的示例，这些磁场配置分离了这三个单独的效果。最后，我们将这些想法应用于捕获冷原子。我们的主要结果是几何电势$ v _ {\ mathrm geom} $以较小的陷阱为主，可用于将冷原子限制在静态陷阱中。这种观察可能会导致更好和较小的原子钟。本文在迈克尔·贝里（Michael Berry）80岁生日时致力于迈克尔·贝里（Michael Berry）。
It is known that, within the Born-Oppenheimer approximation, the slow modes of the nuclear motion are altered by three effects that emerge from integrating out the fast modes of the electronic motion. The first is an effective scalar potential $V_{\mathrm dyn}$ coming from the eigenvalue of the electronic state, the second is an effective magnetic field coming from the Berry phase vector potential $A$. The third term is an additional potential $V_{\mathrm geom}$ originating in the geometry of ray space and the Fubini-Study metric. In this article, we illustrate these effects and their geometric origin in the context of a simple toy model of a slow neutron interacting with a strong, spatially varying magnetic field. Regarding the neutron spin as a fast degree of freedom, we work out the slow dynamics of the motion of the neutron. Our treatment is geometrical and brings out the effects originating in the Kähler geometry of ray space and the Fubini-Study metric. We then give examples of magnetic field configurations which isolate these three separate effects. Finally we apply these ideas to the trapping of cold atoms. Our main result is that the geometric electric potential $V_{\mathrm geom}$ dominates for smaller traps and can be used to confine cold atoms in static traps. This observation could result in better and smaller atomic clocks. This paper is dedicated to Michael Berry on his 80th Birthday.
