AUTHOR=Saito Shinichi 

TITLE=Topological polarisation states

JOURNAL=Frontiers in Physics

VOLUME=11

YEAR=2023

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2023.1225462

DOI=10.3389/fphy.2023.1225462

ISSN=2296-424X

ABSTRACT=<p>Polarisation states are described by spin expectation values, known as Stokes parameters, the trajectories of which in a rotationally symmetric system form a sphere named after Poincaré. Here, we show that the trajectories of broken rotational symmetric systems can exhibit distinct topological structures in polarisation states. We use a phase-shifter to form a polarisation circle <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, which interferes with the original input due to the phase change of the output state upon rotation. By rotating the circle using a rotator, the trajectories become a polarisation torus <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, which was experimentally confirmed in a simple setup using passive optical components together with the Mach–Zehnder interferometer. We also discuss the realisations of other topological features, such as a Möbius strip, a trefoil knot, Hopf links, and topological Dirac bosons, with a bulk-edge correspondence.</p>