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arXiv:2009.01877 · Physical Review A

Spin-state estimation using the Stern-Gerlach experiment

Javier Martínez-Cifuentes, K. M. Fonseca-Romero

Published Apr 2, 2021• 2 citations
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Abstract

We present a state estimation scheme for spins using a modified setup of the Stern-Gerlach experiment in which a beam of neutral spin-1/2 point particles interacts with a quadrupolar magnetic field. The proposed estimation procedures, based either on a quadrant or a continuous intensity distribution detection, require a suitable initial spatial state of the beam. The statistical characterization of the estimators of the initial spin state allows us not only to associate an error to the estimated parameters, but also to define a measure for comparing estimation procedures corresponding to different Stern-Gerlach setups.

DOI
10.1103/physreva.103.042202

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We present a state estimation scheme for spins using a modified setup of the Stern-Gerlach experiment in which a beam of neutral spin-1/2 point particles interacts with a quadrupolar magnetic field. The proposed estimation procedures, based either on a quadrant or a continuous intensity distribution detection, require a suitable initial spatial state of the beam. The statistical characterization of the estimators of the initial spin state allows us not only to associate an error to the estimated parameters, but also to define a measure for comparing estimation procedures corresponding to different Stern-Gerlach setups.
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 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.
We use a stochastic theory approach to discuss different scenarios such as ``locked moment'' and ``superparamagnetic'' behavior of magnetic clusters seen in Stern-Gerlach (SG) experiments. A dimensionless parameter $\ensuremath{\lambda}{\ensuremath{\tau}}_{E}$ (where \ensuremath{\lambda} is an internal spin relaxation rate and ${\ensuremath{\tau}}_{E}$ is the time spent by the cluster in the field gradient region) and the initial polarization of the cluster moment are shown to determine sensitively the position of the cluster beam on the SG screen.
The Stern-Gerlach experiment has played an important role in our understanding of quantum behavior. We propose and analyze a modified version of this experiment where the magnetic field of the detector is in a quantum superposition, which may be experimentally realized using a superconducting flux qubit. We show that if incident spin-1/2 particles couple with the two-state magnetic field, a discrete target distribution results that resembles the distribution in the classical Stern-Gerlach experiment. As an application of the general result, we compute the distribution for a Gaussian waveform of the incident fermion. This analysis allows us to demonstrate theoretically: (1) the quantization of the intrinsic angular momentum of a spin-1/2 particle, and (2) a correlation between EPR pairs leading to nonlocality, without necessarily collapsing the particle’s spin wavefunction.
We study the quantum dynamics of a nonrelativistic neutral particle with spin in inhomogeneous external magnetic fields. We first consider fields with one-dimensional inhomogeneities, both unphysical and physical, and construct the corresponding analytic propagators. We then consider fields with two-dimensional inhomogeneities and develop an appropriate numerical propagation method. We propagate initial states exhibiting different degrees of space localization and various initial spin configurations, including both pure and mixed spin states. We study the evolution of their spin densities and identify characteristic features of spin density dynamics, such as the spatial separation of spin components, and spin localization or accumulation. We compare our approach and our results with the coverage of the Stern-Gerlach effect in the literature, and we focus on nonstandard Stern-Gerlach outcomes, such as radial separation, spin focusing, spin oscillation, and spin flipping.