SU(2) symmetry of coherent photons and application to Poincaré rotator

Abstract

Lie algebra is a hidden mathematical structure behind various quantum systems realised in nature. Here, we consider SU(2) wavefunctions for polarisation states of coherent photons emitted from a laser source, and discuss the relationship to spin expectation values with SO(3) symmetry based on isomorphism theorems. In particular, we found rotated half-wave-plates correspond to mirror reflections in the Poincaré sphere, which do not form a subgroup in the projected O(2) plane due to anti-hermitian property. This could be overcome experimentally by preparing another half-wave-plate to realise a pristine rotator in SU(2), which allows arbitrary rotation angles determined by the physical rotation. By combining another 2 quarter-wave-plates, we could also construct a genuine phase-shifter, thus, realising passive control over the full Poincaré sphere.

Quantum field theory for coherent photons: isomorphism between Stokes parameters and spin expectation values

“…As applications of our formalism, we consider several typical optical components to control the polarisation states [1,2,5,[20][21][22]28,[45][46][47][64][65][66][67][68][69][70][71][72]. Practically, this is nothing new compared with well-established Jones matrix formulation, but the purpose of this consideration is to establish a fundamental basis to justify the calculation of polarisation states using Jones matrices based on a many-body quantum physics and an SU(2) group theory.…”

“…So far we have discussed coherent photons of a single mode to discuss about the origin of polarisation in a frame work of a quantum field theory with an SU(2) symmetry [1][2][3][4][5][6][7][8][9][20][21][22][23][24][45][46][47][64][65][66][67][68][69][70][71][72]. It is beyond the scope of this work to include multiple modes with orbital angular momentum for discussing about the higher-order Poincaré sphere [12,[75][76][77][78][79][80][81][82][83][84][85].…”

Dirac equation for photons in a fibre: Origin of polarisation

“…We believe it is reasonable to use the SU(2) symmetry for describing the complex wavefunction of a 2-level system, and our formulation is suitable to explore the impact of a geometrical phase. In fact, we have recently demonstrated that the phase between orthogonal polarised states could be controlled by the proposed Poincar'e rotator, which relies on the phase difference of SU(2) upon propagation [94] [95] [96] .…”

Nested SU(2) symmetry of photonic orbital angular momentum

“…Here, we consider coherent photons emitted from a conventional laser diode (LD) [9,10,[40][41][42]. The rotational symmetry of photons emitted from LD is spontaneously broken down upon lasing, and the spin of a macroscopic number of photons is aligned to a direction with the minimum loss for propagation in a cavity to form a coherent state [26,[48][49][50][51]. We assumed a single-mode operation of LD [9,41,42] for simplicity.…”

“…If the beam propagates in vacuum, air, or a fiber with no polarization dependence, the polarization state can be maintained and controlled by various optical components, such as wave-plates, rotators, phaseshifters, and polarizers [9,10,[40][41][42]. We have previously demonstrated, both theoretically [48] and experimentally [50], that an arbitrary polarization state can be realized by a combination of half-wave plates (HWPs) and quarter-wave plates (QWPs) using the proposed Poincaré rotator. The idea was to realize a proper rotator using two HWPs, of which one of HWPs is physically rotated, while the other is fixed.…”