Nested SU(2) symmetry of photonic orbital angular momentum

Abstract

The polarization state is described by a quantum mechanical two-level system, which is known as special unitary group of degree 2 [SU(2)]. Polarization is attributed to an internal spin degree of freedom inherent to photons, while photons also possess an orbital degree of freedom. A fundamental understanding of the nature of spin and orbital angular momentum of photons is significant to utilize the degrees of freedom for various applications in optical communications, computations, sensing, and laser-patterning. Here, we show that the orbital angular momentum of coherent photons emitted from a laser diode can be incremented using a vortex lens, and the magnitude of orbital angular momentum increases with an increase in the topological charge inside the mode. The superposition state of the left and right vortices is described by the SU(2) state, similar to polarization; however, the radius of the corresponding Poincaré sphere depends on the topological charge. Consequently, we expect a nested SU(2) structure to describe various states with different magnitudes in orbital angular momentum. We have experimentally developed a simple system to realize an arbitrary SU(2) state of orbital angular momentum by controlling both amplitudes and phases of the left and right vortices using a spin degree of freedom, whose interplays were confirmed by expected far-field images of dipoles and quadruples.

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.…”

“…In order to discuss structured lights in our theoretical framework of a quantum field theory, we need to extend the SU(2) symmetry to have the SU(N ) symmetry with degree N > 2 [6,9,69,71,72,98,99,[102][103][104][105][106][107][108][109][110]. We are developing both theoretical [71] and experimental [72,98,99] platforms to discuss coherent photons with the higher order SU(N ) symmetry.…”

“…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].…”