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Chinese Physics Letters

Phase-Dependent Effects in Stern–Gerlach Experiments

Xu Xu, Xiaoji Zhou

Published Jan 1, 2010• 5 citations
Publisher

Abstract

In the frame of quantum mechanics, we consider an ensemble of spin-½ neutral particles passing through a Stern–Gerlach apparatus and explore how their motions depend on the initial phase difference between two internal spin states. Assuming the particles moving along y-axis, due to the initial phase difference between spin states, they not only split along the longitudinal direction (z-axis) but also separate along the lateral direction (x-axis). The dependence of the lateral displacement on the initial phase difference reminds one of the picture of a quantum interference. This generalized interference provides an alternative approach to measuring the initial phase difference. The experimental realization with ultracold atoms or Bose–Einstein condensates is also discussed.

DOI
10.1088/0256-307x/27/1/010309

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Phase-Dependent Effects in Stern-Gerlach Experiments

许顼, 周小计

在量力学的框架，我们认为 spin-1/2 的一个整体是通过一台 Stern-Gerlach 仪器的中立粒子并且探索他们的运动怎么取决于二个内部旋转状态之间的起始的阶段差别。假定沿着 Y 轴移动的粒子，由于纺纱之间的起始的阶段差别说，他们不仅沿着纵的方向(Z 轴) 切开而且分开侧面的方向(X 轴) 。起始的阶段差别上的侧面的排水量的依赖提醒一量干扰的图画。这概括干扰提供一条其他的途径给测量起始的阶段差别。有 ultracold 原子或 Bose 爱因斯坦冷凝物的试验性的实现也被讨论。[从作者抽象]

Chinese Physics Letters•Jan 1, 2010•0 citations
In the frame of quantum mechanics, we consider an ensemble of spin-½ neutral particles passing through a Stern–Gerlach apparatus and explore how their motions depend on the initial phase difference between two internal spin states. Assuming the particles moving along y-axis, due to the initial phase difference between spin states, they not only split along the longitudinal direction (z-axis) but also separate along the lateral direction (x-axis). The dependence of the lateral displacement on the initial phase difference reminds one of the picture of a quantum interference. This generalized interference provides an alternative approach to measuring the initial phase difference. The experimental realization with ultracold atoms or Bose–Einstein condensates is also discussed.
Wave mechanics and interference have always been at the heart of quantum mechanics. Many ofthe most striking quantum effects, at their heart, arise from the wave nature that quantum particles exhibit. In this thesis, I describe two experiments which showcase two different interference effects using a rubidium Bose-Einstein condensate. The first experiment examines the ways geometric phases enter into two-dimensional spin systems by studying their effect on an atomic Ramsey interferometer. The addition of an off-resonantly coupled level causes the location of interference minima to shift on frequency scales comparable to relevant Rabi frequencies of the system, as well as potentially creating locations with increased or decreased sensitivity to magnetic noise. The second experiment is a weak Stern-Gerlach experiment, showing how the components of a superposition state, with different magnetic moments, can interfere to show a greatly increased shift due to a magnetic gradient. The effect is observed using the second-order magnetic sensitivity of the clock states of 87Rb, with a greater than fivefold amplification of the momentum shift created by the magnetic gradient.
The Stern-Gerlach experiment consists of a beam composed by neutral spin 1/2 particles that enters on a non-uniform magnetic field, which splits the beam into two, one of which consists of spin +1/2 particles and the other one consists of spin -1/2 particles. We initially consider that the particles are independent and do not interact. The objective of this work is to study and compare two cases. The first one is the case in which the particle with spin and spatial degrees of freedom initially independents is subjected to two consecutive Stern-Gerlach. The second one is similar to the first, differing only by it's initial state, which is an entangled state of spin with a second spin 1/2 particle.
The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.
A successful transverse Stern–Gerlach experiment has been performed, using a beam of neutral potassium atoms and an inhomogeneous time-dependent magnetic field of the form[Formula: see text]A classical analysis of the Stern–Gerlach experiment is given for a rotating inhomogeneous magnetic field. In general, when space quantization is achieved, the spins are quantized along the effective magnetic field in the reference frame rotating with angular velocity ω about the z axis. For ω = 0, the direction of quantization is the z axis (conventional Stern–Gerlach experiment), while at resonance (ω = −γH 0 ) the direction of quantization is the x axis in the rotating reference frame (transverse Stern–Gerlach experiment). The experiment, which was performed at 7.2 Mc, is described in detail.
The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semiclassical framework. The concept of coherent internal states is used in the formulation of the semiclassical approximation from the full quantum-mechanical expression. The classical trajectories are defined only for certain spin states, that satisfy the conditions for being coherent internal states. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
在量力学的框架，我们认为 spin-1/2 的一个整体是通过一台 Stern-Gerlach 仪器的中立粒子并且探索他们的运动怎么取决于二个内部旋转状态之间的起始的阶段差别。假定沿着 Y 轴移动的粒子，由于纺纱之间的起始的阶段差别说，他们不仅沿着纵的方向(Z 轴) 切开而且分开侧面的方向(X 轴) 。起始的阶段差别上的侧面的排水量的依赖提醒一量干扰的图画。这概括干扰提供一条其他的途径给测量起始的阶段差别。有 ultracold 原子或 Bose 爱因斯坦冷凝物的试验性的实现也被讨论。[从作者抽象]