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arXiv:quant-ph/0305083 · Physical Review A

Measuring Pancharatnam’s relative phase for SO(3) evolutions using spin polarimetry

Peter Larsson, Erik Sjöqvist

Published Oct 20, 2003• 2 citations
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Abstract

In polarimetry, a superposition of internal quantal states is exposed to a single Hamiltonian and information about the evolution of the quantal states is inferred from projection measurements on the final superposition. In this framework, we here extend the polarimetric test of Pancharatnam's relative phase for $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ proposed by Wagh and Rakhecha [Phys. Lett. A 197, 112 (1995)] to spin $j>~1$ undergoing noncyclic SO(3) evolution. We demonstrate that the output intensity for higher spin values is a polynomial function of the corresponding $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ intensity. We further propose a general method to extract the noncyclic SO(3) phase and visibility by rigid translation of two $\ensuremath{\pi}/2$ spin flippers. Polarimetry on higher spin states may in practice be done with spin polarized atomic beams.

DOI
10.1103/physreva.68.042109

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Physical Review A•Jan 5, 2016•5 citations
In polarimetry, a superposition of internal quantal states is exposed to a single Hamiltonian and information about the evolution of the quantal states is inferred from projection measurements on the final superposition. In this framework, we here extend the polarimetric test of Pancharatnam's relative phase for spin\ensuremath−12\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ proposed by Wagh and Rakhecha [Phys. Lett. A 197, 112 (1995)] to spin j> 1j>~1j> 1 undergoing noncyclic SO(3) evolution. We demonstrate that the output intensity for higher spin values is a polynomial function of the corresponding spin\ensuremath−12\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ intensity. We further propose a general method to extract the noncyclic SO(3) phase and visibility by rigid translation of two \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 spin flippers. Polarimetry on higher spin states may in practice be done with spin polarized atomic beams.
The polarimetric test of Pancharatnam's relative phase for spin−1/2-{1/2}−1/2 proposed by Wagh and Rakhecha [Phys. Lett. A {\\bf 197}, 112 (1995)] is extended to spin jgeq1j\\geq 1jgeq1 undergoing noncyclic SO(3) evolution. We demonstrate that the output intensity for higher spin values is a polynomial function of the corresponding spin−1/2-{1/2}−1/2 intensity. We further propose a general method to extract the noncyclic SO(3) phase and visibility by rigid translation of two pi/2\\pi /2pi/2 spin flippers. Polarimetry on higher spin states may in practice be done with spin polarized atomic beams.
We demonstrate that Pancharatnam's relative phase for mixed spin−1/2-{1/2}−1/2 states in noncyclic SU(2) evolution can be measured polarimetrically.
A critique of a recent experiment [Wagh et.al., Phys.Rev.Lett.81, 1992 (7 Sep 1998)] to measure the noncyclic phase associated with a precessing neutron spin in a neutron interferometer, as given by the Pancharatnam criterion, is presented. It is pointed out that since the experiment measures, not the noncyclic phase itself, but a quantity derived from it, it misses the most interesting feature of such a phase, namely the different sign associated with states lying in the upper and the lower hemispheres, a feature originating in the existence of a phase singularity. Such effects have earlier been predicted and seen in optical interference experiments using polarization of light as the spinor [Bhandari, Phys.Rep.281, 1 (Mar 1997)].
We report on theoretical calculations and experimental observations of Pancharatnam's phase originating from arbitrary SU(2) transformations applied to polarization states of light. We have implemented polarimetric and interferometric methods, which allow us to cover the full Poincar\'e sphere. As a distinctive feature, our interferometric array is robust against mechanical and thermal disturbances, showing that the polarimetric method is not inherently superior over the interferometric one, as previously assumed. Our strategy effectively amounts to feeding an interferometer with two copropagating beams that are orthogonally polarized with respect to each other. It can be applied to different types of standard arrays, such as a Michelson, a Sagnac, or a Mach-Zehnder interferometer. We exhibit the versatility of our arrangement by performing measurements of Pancharatnam's phases and fringe visibilities that closely fit the theoretical predictions. Our approach can be easily extended to deal with mixed states and to study decoherence effects.
We report robust polarimetric measurements of mixed-state geometric phases. An all-optical setup was used to generate geometric phases with great versatility, thereby extending the scope of analogous methods used in neutron polarimetry. Our method allows us to explore geometric phases generated by unitarily evolving mixed states. These are realized as partially polarized single-photon states whose degree of polarization and normalized Stokes vector can be fixed independently of one another. The geometric phases correspond to both open and closed trajectories in Poincar\'e space. By exploiting the gauge invariance of geometric phases we nullify their dynamical part, thereby making the geometric phase coincide with the total (Pancharatnam) phase, which can be directly addressed by employing standard techniques.