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Microoptics Conference

SU(4) States of Coherent Photons and Symmetry-Breaking: A Quantum-Classical Crossover

Shinichi Saito

Published Sep 24, 2023• 1 citation
Publisher

Abstract

Coherent photons possess both spin and orbital angular momentum as internal degrees of freedom. We show these states have a symmetry of Special Unitary group of degree 4, SU(4), which could be broken down to SU(2)$\times$ SU(2) states upon projection by a polariser to loose the classical entanglement.

DOI
10.23919/moc58607.2023.10302851

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Coherent photons possess both spin and orbital angular momentum as internal degrees of freedom. We show these states have a symmetry of Special Unitary group of degree 4, SU(4), which could be broken down to SU(2)×\times× SU(2) states upon projection by a polariser to loose the classical entanglement.
We show coherent photons with spin and orbital angular momentum have symmetry of special unitary group of degree four, SU(4), both theoretically and experimentally. We show the degree of polarisation could be a measure to characterise the quantum coherency of a spin state, which was significantly affected by the partial trace out of orbital degree of freedom. By changing the amplitudes of the classically entangled state, we observed a crossover from quantum Heisenberg spin to classical Ising spin. We also demonstrated projections of coherent states, which recover the coherency of the spin state, while the SU(4) state is projected to be a simple product state of SU(2)×SU(2). Based on the developed platform, we have explored spin texture of coherent photons, and found a unique spin profile reminiscent of the Bengal cat upon wrapping to a three-dimensional sphere. We have controlled the phase between singlet and triplet states and found the spin texture rotated upon increasing the phase.
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.
It is well established now that light carries both spin and orbital angular momentum which are associated with circular polarisation and helical phase fronts. Orbital angular momentum degrees of freedom recently have been used frequently in quantum information processing as their states are described by vectors in a higher-dimensional Hilbert space which enhances the possibility of realising superior quantum information protocols. On the other hand, quantum coherence, which arises from the superposition principle, is a distinct feature of quantum mechanics that cannot be satisfactorily described by classical physics. Coherence is also identified as essential ingredient for applications of quantum information, computation, and quantum thermodynamics. Three research projects, with their related background information, are presented in this thesis. In the first one, we design a linear optical system to transform the maximally entangled state of a down-converted photon pair into a genuine entangled χ-type state, as this class of genuine entangled states has been showed to have many interesting entanglement properties and can be employed in several quantum information protocols. In the second project, we study the mechanism of angular momentum transfer from light to a dielectric medium when it undergoes total internal reflection. The result shows that the torque associated with angular momentum transfer appears shortly, when the light pulse hits the interface. Finally, we study quantum coherence transfer from a coherence resource initialised in a coherence state to an atomic state by the Jaynes-Cummings model, and we compare it to the coherent operation that uses a resource prepared in a ladder state described by Åberg’s model. We found that a resource in a coherent state is more robust against failures.
We investigate two-photon entangled states using two important degrees of freedom of the electromagnetic field, namely orbital angular momentum (OAM) and spin angular momentum. For photons propagating in the same direction we apply the idea of entanglement duality and develop schemes to do entanglement sorting based on either OAM or polarization. In each case the entanglement is tested using appropriate witnesses. We finally present generalizations of these ideas to three- and four-photon entangled states.
Coherent photons have both spin and orbital angular momentum, and the superposition state among orthogonal states can describe a qubit. Here, we show the internal structure of coherent photons with spin and orbital angular momentum can be described by a representation theory of Lie algebra and Lie group. As an application of this theory, we have experimentally demonstrated the controlled-NOT (CNOT) operation, using a standard green laser diode. We have used beam splitters and waveplates to generate the macroscopic entangled light source with amplitudes and phases, controlled by rotating the waveplates. By applying the Bell projection to the entangled light, the far-field image of the light shows an expected dipole shape in the superposition state of left and right vortices, allowing to identify the state from the image. Then, we have used a polarisation dependent beam splitter to split the spin state, and we applied a NOT operation, made of a pair of cylindrical lenses, only for a vortex with a chosen polarisation, and recombined the beams after the operation. The proposed architecture could be used for a platform to manipulate qubits, made of macroscopically coherent photons.
Using the single SU(2) qutrit (spin-1 like system), we show that entanglement may take place for a single particle with respect to its internal degrees of freedom, in other words, beyond the conventional requirements of nonlocality and nonseparability. We show that the SU(2) or spin coherent states do not manifest entanglement, while the so-called squeezed spin states are entangled. We reveal the principle difference between the spin coherent and spin squeezed states in terms of quantum correlations. A number of physical realizations of the SU(2) qutrit is discussed.