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arXiv:2304.00013 · Results in Physics

Active SU(2) operation on Poincaré sphere

Shinichi Saito

Published Mar 11, 2024• 7 citations
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Abstract

Polarisation is described by an SU(2) state due to the coherence of photons emitted from a ubiquitous laser source. We propose a polarisation modulator, called as a Poincaré rotator, which enables dynamic control over a polarisation state by executing an arbitrary rotation on the Poincaré sphere. The Poincaré rotator works as an arbitrary operator in the SU(2) Lie group, combining two U(1) operations to change the phase and the amplitude of the state, respectively. By modulating both the phase and amplitude, we have achieved distinguishable 4×4=16, 8×8=64, and 10×10=100 states on the Poincaré sphere. As a locus of the polarisation states on the sphere, we have successfully drawn the molecular structure of Buckminsterfullerene (C60) and the coastline of the Earth.

DOI
10.1016/j.rinp.2024.107567

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It is common to use the Poincaré sphere for the representation of the polarization1 (or, in other words, the spin components of the angular momentum) of a radiation beam. This approach has recently been generalized2–4 to the case of Gaussian-type modes which possess orbital angular momentum. In the present paper we propose to use this formalism for the description of an arbitrary scalar twodimensional signal, which might be deterministic or stochastic (coherent or partially coherent).\nA point on the Poincar´e sphere represents – as we may call it – a certain ‘direction’ of the orbital angular momentum, whose expectation value can be found from the ten second-order moments of the signal’s Wigner distribution,5, 6 arranged in a 4 × 4 matrix M. Transforming the signal into its canonical form, corresponding to a diagonal moment matrix,7 we associate this state with the intersection of the main meridian with the equator, where Cartesian coordinates are more appropriate for signal description. Applying orthosymplectic transformations8 (which, in particular, include the antisymmetric fractional Fourier transformation, the rotation transformation, and their cascades) to this state, the entire sphere can be populated, leading to a generalized canonical form. At the poles, we thus have the states with a possible z-component of the orbital angular momentum, where polar coordinates are the best choice for signal analysis. The way in which the signal is transformed into its canonical form defines principal axes (including relative coordinate scaling) for signal representation in phase space. The proposed approach is useful for different applications: beam characterization, adaptive filtering, signal analysis and synthesis, etc.

•Jan 1, 2008•1 citation
Polarisation is described by an SU(2) state due to the coherence of photons emitted from a ubiquitous laser source. We propose a polarisation modulator, called as a Poincaré rotator, which enables dynamic control over a polarisation state by executing an arbitrary rotation on the Poincaré sphere. The Poincaré rotator works as an arbitrary operator in the SU(2) Lie group, combining two U(1) operations to change the phase and the amplitude of the state, respectively. By modulating both the phase and amplitude, we have achieved distinguishable 4×4=16, 8×8=64, and 10×10=100 states on the Poincaré sphere. As a locus of the polarisation states on the sphere, we have successfully drawn the molecular structure of Buckminsterfullerene (C60) and the coastline of the Earth.
The coherent state from a laser source has spin and orbital degrees of freedom, which allow an arbitrary superposition state among orthogonal states with varying amplitudes and phases. Here, we theoretically show coherent photons with SU( N ) symmetry are characterised by expectation values of angular momentum shown on a hypersphere in SO( N 2 − 1 ) space. To demonstrate expected unitary transformations in experiments, we have constructed generators of transformations in the Lie group simply by combining widely available optical components such as waveplates and vortex lenses. We show a superposition state between twisted and Gaussian states is characterised by the dynamics of the topological charge upon the transformation in SU(3) states. We also realised photonic singlet and triplet states corresponding to SU(4) states, which were projected to SU(2)×SU(2) states upon passing through a rotated polariser.
In the framework of optical quantum computing and communications, a major objective consists in building receiving nodes that implement conditional operations on incoming photons, using the interaction with a single stationary qubit. In particular, the quest for scalable nodes motivated the development of cavity-enhanced spin-photon interfaces with solid-state emitters. An important challenge remains, however, to produce a stable, controllable, spin-dependant photon state, in a deterministic way. Here we use a pillar-based high-Q cavity, embedding a singly-charged semiconductor quantum dot, to demonstrate the control of giant polarisation rotations induced by a single electron spin. A complete tomography approach is used to deduce the output polarisation Stokes vector, conditioned by a single spin state. We experimentally demonstrate rotation amplitudes such as ±π2\pm \fracπ{2}±2π​ and πππ in the Poincaré sphere, as required for applications based on spin-polarisation mapping and spin-mediated photon-photon gates. In agreement with our modeling, we observe that the environmental noise does not limit the amplitude of the spin-induced rotation, yet slightly degrades the polarisation purity of the output states. We find that the polarisation state of the reflected photons can be manipulated in most of the Poincaré sphere, through controlled spin-induced rotations, thanks to moderate cavity birefringence and limited noise. This control allows the operation of spin-photon interfaces in various configurations, including at zero or low magnetic fields, which ensures compatibility with key protocols for photonic cluster state generation.
We propose a geometric hybrid Poincaré sphere (GHPS) as a unified geometrical framework for describing structured photon states with independently controllable spin angular momentum (SAM) and orbital angular momentum (OAM). Unlike the conventional higher-order Poincaré sphere, in which the SAM and OAM are intrinsically coupled through fixed basis states, the GHPS is constructed by defining its poles as direct products of arbitrary orthogonal bases on the Poincaré sphere (PS) and orbital Poincaré sphere (OPS) and by superposing these pole states. Using numerical simulations, we analyze representative GHPS states and show that the GHPS spherical coordinates govern the amplitude ratio and relative phase between the pole bases. This framework enables spatially inhomogeneous polarization distributions and intensity patterns, including nonseparable structures in which polarization and intensity are intrinsically intertwined, and provides a systematic state-space description for the coherent geometrical control of advanced structured light fields.
It is common to use the Poincaré sphere for the representation of the polarization1 (or, in other words, the spin components of the angular momentum) of a radiation beam. This approach has recently been generalized2–4 to the case of Gaussian-type modes which possess orbital angular momentum. In the present paper we propose to use this formalism for the description of an arbitrary scalar twodimensional signal, which might be deterministic or stochastic (coherent or partially coherent).\nA point on the Poincar´e sphere represents – as we may call it – a certain ‘direction’ of the orbital angular momentum, whose expectation value can be found from the ten second-order moments of the signal’s Wigner distribution,5, 6 arranged in a 4 × 4 matrix M. Transforming the signal into its canonical form, corresponding to a diagonal moment matrix,7 we associate this state with the intersection of the main meridian with the equator, where Cartesian coordinates are more appropriate for signal description. Applying orthosymplectic transformations8 (which, in particular, include the antisymmetric fractional Fourier transformation, the rotation transformation, and their cascades) to this state, the entire sphere can be populated, leading to a generalized canonical form. At the poles, we thus have the states with a possible z-component of the orbital angular momentum, where polar coordinates are the best choice for signal analysis. The way in which the signal is transformed into its canonical form defines principal axes (including relative coordinate scaling) for signal representation in phase space. The proposed approach is useful for different applications: beam characterization, adaptive filtering, signal analysis and synthesis, etc.