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arXiv:0705.2508

Theorie des longitudinalen Atomstrahl-Spinechos und paritaetsverletzende Berry-Phasen in Atomen

T. Bergmann

Published May 17, 2007• 0 citations
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Abstract

We present a nonrelativistic theory for the quantum mechanical description of longitudinal atomic beam spin echo experiments, where a beam of neutral atoms is subjected to static electric and magnetic fields. The atomic wave function is the solution of a matrix-valued Schroedinger equation and can be written as superposition of local (atomic) eigenstates of the potential matrix. The position- and time-dependent amplitude function of each eigenstate represents an atomic wave packet and can be calculated in a series expansion with a master formula that we derive. The zeroth order of this series expansion describes the adiabatic limit, whereas the higher order contributions contain the mixing of the eigenstates and the corresponding amplitude functions. We give a tutorial for the theoretical description of longitudinal atomic beam spin echo experiments and for the so-called Fahrplan model, which is a visualisation tool for the propagation of wave packets of different atomic eigenstates. As an example for the application of our theory, we study parity violating geometric (Berry-)phases. In this context, we define geometric flux densities, which for certain field configurations can be used to illustrate geometric phases in a vector diagram. Considering an example with a specific field configuration, we prove the existence of a parity violating geometric phase.

DOI
10.48550/arxiv.0705.2508

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We present a nonrelativistic theory for the quantum mechanical description of longitudinal atomic beam spin echo experiments, where a beam of neutral atoms is subjected to static electric and magnetic fields. The atomic wave function is the solution of a matrix-valued Schroedinger equation and can be written as superposition of local (atomic) eigenstates of the potential matrix. The position- and time-dependent amplitude function of each eigenstate represents an atomic wave packet and can be calculated in a series expansion with a master formula that we derive. The zeroth order of this series expansion describes the adiabatic limit, whereas the higher order contributions contain the mixing of the eigenstates and the corresponding amplitude functions. We give a tutorial for the theoretical description of longitudinal atomic beam spin echo experiments and for the so-called Fahrplan model, which is a visualisation tool for the propagation of wave packets of different atomic eigenstates. As an example for the application of our theory, we study parity violating geometric (Berry-)phases. In this context, we define geometric flux densities, which for certain field configurations can be used to illustrate geometric phases in a vector diagram. Considering an example with a specific field configuration, we prove the existence of a parity violating geometric phase.
In the present thesis Berry phases for metastable states of hydrogen and helium exposed \nto external electromagnetic fields are investigated. One main goal is the derivation of \na parity-violating observable incorporating the electroweak mixing angle. The study is \ndone in view of an available high-precision atomic beam spin echo interferometer and \nwith respect to the theoretically very well-known structure of hydrogen. We determine \nthe general properties of parity-violating and parity-conserving Berry phases and their \nflux-density vector fields distinguished by their behaviour under proper and improper \nrotations. This allows for extensive analytical investigations and cross-checks of our \nnumerical computations. Regarding hydrogen, we reveal geometry-dependent modifications \nof atomic lifetimes due to parity-conserving imaginary Berry phases. Deriving \nthe evolution of the atomic wave packets, we establish the theoretical description of an \natom interferometer. An analytical expression for the spin echo signal, easily accessible \nin experiment, is presented and numerically evaluated to extract parity-violating \nsignals. We discuss the implications of the computed parity-violating effects on interferometric \nprecision measurements of the proton’s weak charges. Moreover, we outline \nthe concept of Heisenberg-limited atom interferometry to improve the resolution for \nmeasurements of Berry phases. Finally, we present predictions for parity-conserving \nBerry phases of metastable helium states studied in the Heidelberg experiment.
In 1950, Ramsey and Hahn separately described a method using separated oscillatory magnetic fields and electronic levels to measure interference in atomic states. Today, similar physics is exploited to make atom interferometer sensors utilizing what is called the spin-echo sequence. In 1956, Pancharatnam described the geometric phase difference between two polarized states of light. In 1984 Berry described the phase accrued in cyclic and adiabatic quantum systems. A Pancharatnam-Berry phase experiment involving a quarter-wave – half-wave – quarter-wave plate stack of optics and an experiment using the spin-echo sequence are based on seemingly different phenomena in optics versus atomic optics, respectively. However, the mathematical representations used to describe the Pancharatnam-Berry phase in optics are identical to those used to describe the spin-echo sequence. Therefore, the relationship between the spin-echo sequence and the Pancharatnam-Berry phase was explored. Experiments to test the comparisons using a pre-existing cold atom experiment, ~150 μK, were performed. The mathematical connection between the two types of experiments was explained more deeply.
We have constructed the geometric phases emerging from the non-trivial topology of a space-dependent magnetic field, interacting with the spin magnetic moment of a neutral particle. Our basic tool is the local unitary transformation which recasts the magnetic spin interaction under a diagonal form. Rewriting the kinetic term in the "rotated" frame requires the introduction of non-Abelian covariant derivatives, involving the gradients of the Euler angles which define the orientation of the local field. Within the rotated frame, we have built a perturbation scheme,assuming that the longitudinal non-Abelian field component dominates the transverse ones, to be evaluated to second-order. The geometry embedded in the longitudinal gauge vector field and its curl, the geometric magnetic field, is described by the associated Aharonov-Bohm phase. As an illustration, we study the physics of cold ^{171}Yb atoms dressed by two sets of circularly polarized beams, forming square or triangular 2D optical lattices. The geometric field is computed explicitly from the Euler angles. The magnitude of 2nd-order corrections due to transverse fields can be reduced to the percent level by a choice of light intensity which keeps the dressed atom loss rate below 5 s^{-1}. An auxiliary optical lattice confines the atoms within 2D domains where the geometric field is pointing upward.
We have constructed the geometric phases emerging from the non-trivial topology of a space-dependent magnetic field, interacting with the spin magnetic moment of a neutral particle. Our basic tool is the local unitary transformation which recasts the magnetic spin interaction under a diagonal form. Rewriting the kinetic term in the "rotated" frame requires the introduction of non-Abelian covariant derivatives, involving the gradients of the Euler angles which define the orientation of the local field. Within the rotated frame, we have built a perturbation scheme,assuming that the longitudinal non-Abelian field component dominates the transverse ones, to be evaluated to second-order. The geometry embedded in the longitudinal gauge vector field and its curl, the geometric magnetic field, is described by the associated Aharonov-Bohm phase. As an illustration, we study the physics of cold 171Yb atoms dressed by two sets of circularly polarized beams, forming square or triangular 2D optical lattices. The geometric field is computed explicitly from the Euler angles. The magnitude of 2nd-order corrections due to transverse fields can be reduced to the percent level by a choice of light intensity which keeps the dressed atom loss rate below 5 s^{-1}. An auxiliary optical lattice confines the atoms within 2D domains where the geometric field is pointing upward.
I derive the effective phase of the spin precession for a neutral particle with spin 1∕2 moving in a superposition of constant and radio frequency fields. The fields are perpendicular to each other at all times, and the radio frequency field is slowly rotating with angular speed ω. The derivation is accomplished with the help of the exact solution of the Schrödinger equation. I show that the geometrical (or Berry) phase is the linear term in the power series expansion of the effective phase in terms of ω. I also discuss the role of the Berry phase in the experimental search for an electric dipole moment of the neutron with ultracold neutrons.
Diese Arbeit beschreibt die erste systematische experimentelle Untersuchung geometrischer Berry-Phasen in Heliumatomen im metastabilen 2S-Triplettzustand. \nZu diesem Zweck wurde ein neuartiges Atom-Interferometer konstruiert, aufgebaut und sorgfältig charakterisiert. \nBasierend auf der etablierten Atomstrahl-Spinecho-Technik, ermöglicht das Interferometer die kontrollierte Manipulation und den hochpräzisen Nachweis dynamischer und geometrischer Phasen. \nDabei stellt das metastabile Heliumatom ein isoliertes, reines Spin-1-Quantensystem dar. \nIn unterschiedlichen Experimenten werden Berry-Phasen sowohl gemischt mit dynamischen Phasen, als auch als pure geometrische Phasen präpariert. In der ersten rein-geometrischen Spinrotation ohne dynamischen Phasenanteil ist es gelungen die Berry-Phase kontinuierlich zwischen 0 und 4π zu variieren.