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Optical Manipulation and Structured Materials Conference (OMC)

Electron spin states on a higher-order Bloch sphere

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

Published Sep 20, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.3008914

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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.
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.
Leveraging analogies between precessing quantum spin systems and charge-monopole systems, we construct Bloch hyper-spheres with exact spherical symmetries in arbitrary dimensions. Such a Bloch hyper-sphere is realized as a collection of the orbits of precessing quantum spins, and its geometry mathematically aligns with the quantum Nambu geometry of a higher dimensional fuzzy sphere. Stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states and gives rise to Wilczek-Zee geometric phases of non-Abelian monopoles associated with the hyper-sphere holonomies. The degenerate spin-coherent states naturally induce matrix-valued quantum geometric tensors also. While the physical properties of Bloch hyper-spheres with minimal spin in even and odd dimensions are quite similar, their large spin counterparts differ qualitatively depending on the parity of dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties such as von Neumann entropies and Bures quantum metrics.
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.
Spin is a quantum property of electrons. For spin-based quantum information technology, preparation and read-out of the electron spin state should be spin coherent. We demonstrate that the polarization coher ence of light can be transferred to the spin coherence of electrons in a semiconductor quantum nanostructure [1], and the prepared coherence of the electron spin can also be read out with light by the developed tomographic Kerr rotation method [2]. We also demonstrate that a single photon is efficiently converted (~27%) into a single electron trapped in a gate-defined quantum dot, where the g-factor of electrons is tuned to zero, and the charge state is detected with an adjacent quantum point contact without destructing the spin state [3]. We further demonstrate that the spin coherence of a single electron trapped in one of double quantum dots is electrically manipulated with a microwave applied to the gate and read out via the Pauli spin blockade phenomenon [4]. These demonstrations were carried out in a condition where the up/down spin basis states of electrons remain ed degenerated under an in-plain magnetic field. As this condition ensures the energy conservation between photons and electrons, the entire Poincare sphere representing polarization states of photons can be mapped onto the Bloch sphere representing spin polar ization states of electrons. We theoretically showed that relative spin coherence of two electrons can be also measured with the help of spin-flip tunneling of electrons between the dots [5]. Full Bell state measurement is also possible by the single -spin manipulation and Pauli spin blockade [6]. All of these functions are needed to build all semiconductor quantum repeaters and distributed quantum computers.
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.