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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Higher-order Bloch sphere: geometric representation of Larmor precession of the higher-order spin states

Sota Sato, Toshiki Matsumoto, Satoshi Iba, Katsuhiko Miyamoto, Takashige Omatsu, Ken Morita

Published Jan 1, 2022• 0 citations
Publisher

Abstract

We propose an analogous sphere, herein referred a higher-order Bloch sphere (HOBS), to visualize the Larmor precession of higher-order spin. Our proposed HOBS is potentially applied to advanced quantum information processing.
DOI
10.1364/cleopr.2022.p_cth7_09

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Using the similarity between spin states on the Bloch sphere (BS) and polarization states on the Poincare sphere (PS), we construct higher-order spin states on the higher-order BS corresponding to higher-order polarization states of photons on the higher-order PS. We investigate the time evolution of higher-order spin states in a magnetic field and establish an extended form of the conventional Larmor precession. The results provide insights on coherent transfer from extended photons to extended spin qubit systems with spin and orbital angular momenta and the operation of extended spin qubits.
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Based on the correspondence between the conventional Poincaré and Bloch spheres, we proposed a higher-order Bloch spheres, which is an extension of the conventional Bloch spheres. By calculating the expectation value of the spin components using the Pauli spin matrices, we found that the new spin state is formed in a ring shape and the orientation of the spin changes depending on the azimuthal angle and topological charge of orbital angular momentum. We also realize the coherent transfer of the azimuth-dependent polarization state of photons to the electron spin state in a GaAs quantum well with a V-shaped three-level system.
A two-sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-1∕21∕21∕2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU(N)\mathrm{SU}(N)SU(N) in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation and entangled states in quantum-information sciences. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various subgroups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real, many-dimensional vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4). The latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint. This geometrical picture for two spins provides the extension and generalization of the Bloch sphere that has proved invaluable for the understanding of the dynamics of a single spin.