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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Measurement and analysis of a coupled three-particle quantum system

S. Gómez

Published Mar 31, 2010• 0 citations
Publisher

Abstract

The intrinsic quantum properties shown in electrons and atoms allow us to explode a non-classic field for the information management. Through the Stern-Gerlach device (SG), the measurement of spin via a momentum is obtained with associated probabilities due to quantum principles. This paper studies the behavior of a three-particle coupled quantum system in which a spin particle is measured by the SG apparatus while the others are not. The response of the spin-1/2 uncharged particle because of the inhomogeneous magnetic field is also analyzed introducing as well non-linear variations of the magnetic field. The system solution will be determined by specials functions according to the variation field defined and three cases with Airy, Whittaker and Heun functions will be treated and supported by different simulations using symbolic computation with Maple.

DOI
10.1117/12.850515

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The intrinsic quantum properties shown in electrons and atoms allow us to explode a non-classic field for the information management. Through the Stern-Gerlach device (SG), the measurement of spin via a momentum is obtained with associated probabilities due to quantum principles. This paper studies the behavior of a three-particle coupled quantum system in which a spin particle is measured by the SG apparatus while the others are not. The response of the spin-1/2 uncharged particle because of the inhomogeneous magnetic field is also analyzed introducing as well non-linear variations of the magnetic field. The system solution will be determined by specials functions according to the variation field defined and three cases with Airy, Whittaker and Heun functions will be treated and supported by different simulations using symbolic computation with Maple.
The Stern-Gerlach (SG) apparatus for measuring the spin of an uncharged spin-1/2 particle is the archetypal quantum sensing device. We study this device for the new problem of measuring the spin of a particle that is coupled externally to another particle. Specifically, we treat two coupled particles in which a single particle is measured by the SG device while the other is not. We show simulations of how the binding energy associated with the external coupling is completely converted to potential energy and kinetic energy as the single particle separates spatially within the magnetic field of the SG device. Additionally we show simulations of how the initial particle acceleration within the SG devices relates to the coupling, the quantum state of the two-particle system, and the initial spatial dispersion of the particle within the SG device. The results of our analysis, though obtained specifically for the SG apparatus, may be generic to other quantum measurement devices with similar external coupling.
A dynamical model for the collapse of the wave function in a quantum measurement process is proposed by considering the interaction of a quantum system (spin-1/2) with a macroscopic quantum apparatus interacting with an environment in a dissipative manner. The dissipative interaction leads to decoherence in the superposition states of the apparatus, making its behaviour classical in the sense that the density matrix becomes diagonal with time. Since the apparatus is also interacting with the system, the probabilities of the diagonal density matrix are determined by the state vector of the system. We consider a Stern-Gerlach type model, where a spin- 1/2 particle is in an inhomogeneous magnetic field, the whole set up being in contact with a large environment. Here we find that the density matrix of the combined system and apparatus becomes diagonal and the momentum of the particle becomes correlated with a spin operator, selected by the choice of the system-apparatus interaction. This allows for a measurement of spin via a momentum measurement on the particle with associated probabilities in accordance with quantum principles.
Spin is an important quantum degree of freedom in relativistic quantum information theory. This paper provides a first-principles derivation of the observable corresponding to a Stern-Gerlach measurement with relativistic particle velocity. The specific mathematical form of the Stern-Gerlach operator is established using the transformation properties of the electromagnetic field. To confirm that this is indeed the correct operator we provide a detailed analysis of the Stern-Gerlach measurement process. We do this by applying a WKB approximation to the minimally coupled Dirac equation describing an interaction between a massive fermion and an electromagnetic field. Making use of the superposition principle we show that the +1 and -1 spin eigenstates of the proposed spin operator are split into separate packets due to the inhomogeneity of the Stern-Gerlach magnetic field. The operator we obtain is dependent on the momentum between particle and Stern-Gerlach apparatus, and is mathematically distinct from two other commonly used operators. The consequences for quantum tomography are considered.
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.
It is demonstrated that the collapse of the wavefunction in the quantum measurement process is reflected in the two-particle correlation function for the electron spin, which can in principle be studied by a triple Stern-Gerlach experiment. The change in momentum of a Stern-Gerlach magnet is itself sufficient to allow the determination of the electron spin, subject to the uncertainty principle, if the coherence of the two components of the spin is destroyed by the magnetic field. Similar considerations apply to the two-particle correlation function for the photon polarisation, and two-photon atomic cascades can be used in practice to investigate the measurement process and determine the extension of a photon wavepacket. The irreversibility of quantum mechanical measurements and its relation to the uncertainty principle is briefly considered.
The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.