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Noncyclic phase in SO(3) polarimetry

Peter Larsson, Erik Sjöqvist

Published May 15, 2003• 0 citations
OpenAlex

Abstract

The polarimetric test of Pancharatnam's relative phase for spin$-{1/2}$ proposed by Wagh and Rakhecha [Phys. Lett. A {\\bf 197}, 112 (1995)] is extended to spin $j\\geq 1$ undergoing noncyclic SO(3) evolution. We demonstrate that the output intensity for higher spin values is a polynomial function of the corresponding spin$-{1/2}$ intensity. We further propose a general method to extract the noncyclic SO(3) phase and visibility by rigid translation of two $\\pi /2$ spin flippers. Polarimetry on higher spin states may in practice be done with spin polarized atomic beams.

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

Peter Larsson, Erik Sjöqvist

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.

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Rajendra Bhandari

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)].

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In the mid-l950s, Pancharamam [l] propounded an exact prescription for the phase acquired in a completely general evolution of a quanta1 system. Without any loss of generality, we highlight this phase prescription here for spin-polarized neutrons, a two-state system, and describe an interferometric experiment observing phases and associated interference amplitudes in noncyclic evolutions.

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All-optical polarimetric generation of mixed-state single-photon geometric phases

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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.

Physical Review A•Jan 5, 2016•5 citations
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.
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> 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 $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ 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.
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 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.