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Future Semiconductor Devices
Published in Lambrechts Wynand, Sinha Saurabh, Abdallah Jassem, Prinsloo Jaco, Extending Moore’s Law through Advanced Semiconductor Design and Processing Techniques, 2018
Wyn Lambrechts, Saurabh Sinha, Jassem Abdallah, Jaco Prinsloo
Recall that the quantum NOT gate is represented by one of a series of three Pauli matrices, as introduced in (6.23). These Pauli matrices are known as the Pauli-X, Pauli-Y and Pauli-Z matrices, representing quantum gates with the same names as these matrices. Each Pauli matrix (gate) also denotes one-qubit rotation operations along each of the three axes in the Bloch sphere in Figure 5.12 (Akama 2015). These quantum gates will now be briefly described.
Jones Matrix Data Reduction with Pauli Matrices
Published in Russell A. Chipman, Wai-Sze Tiffany Lam, Garam Young, Polarized Light and Optical Systems, 2018
Russell A. Chipman, Wai-Sze Tiffany Lam, Garam Young
The Pauli matrices were introduced into quantum mechanics by Wolfgang Pauli to describe the interaction of the angular momentum of electrons and nuclei with external magnetic fields.2–4 Light is a quantum phenomenon, and even though a quantum formulation is not used here, it is natural that the underlying mathematics of quantum mechanics should appear in the polarization calculus.5
Additional Aspects and Applications
Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019
Marlos A. G. Viana, Vasudevan Lakshminarayanan
The Pauli matrices are extensively used in physics and are a set of three 2×2 Hermitian and unitary matrices. Since Hermitian operators are quantum mechanical observables, the spin of a particle is in the space of observation in a complex Hilbert space. More specifically, σi represents the spin of a particle along the ith coordinate axis in the ℝ3. The Pauli matrices are related to the single 2-dim irreducible representation of the dihedral group D4 of (2.11) and (2.12) by XrkhℓX−1=(iσ1)kσ2ℓ for k=0,1,2,3 and ℓ=0,1, with X given by (2.13). The relationship Sj=tr Jσj/2 suggests that an indexing of the form x(τ)=tr β(τ−1)M, with τ∈D4 defined in the group algebra of D4, should be of interest to study the symmetry properties of G that are present in the optical effects introduced (or represented) by the linear element M. This indexing is discussed in Chapter 3.
Electron spin and donor impurity effects on the absorption spectra of pseudo-elliptic quantum rings under magnetic field
Published in Philosophical Magazine, 2021
Including the spin through the Zeeman effect and spin–orbit interaction, the total Hamiltonian becomes: with I2 is the 2 × 2 identity matrix, is the Bohr magneton, g is the Landé factor for the bound electron, σi (i = x, y, z) are the i component of the Pauli matrices vector and HR, HD introduce the Rashba and Dresselhaus spin–orbit interactions (SOIs), respectively. For x and y axes oriented parallel to [100] and [010] crystal directions, respectively, the spin–orbit coupling terms along z axis are: The Rashba SOI [41] is determined by structure inversion asymmetry along the growth direction and its intensity (the Rashba constant ) depends on the slope of the potential along the growth direction. The Dresselhaus SOI [42] is due to bulk inversion asymmetry of the lattice and its intensity (the Dresselhaus coupling constant ) depends only on the layer thickness in the growth direction.
Influence of spin–orbit interaction, Zeeman effect and light polarisation on the electronic and optical properties of pseudo-elliptic quantum rings under magnetic field
Published in Philosophical Magazine, 2020
In Equation (1), g represents the Landé factor for the bound electron, is the Bohr magneton, σz is the z component of the Pauli matrices vector σi (i = x, y, z) and HR, HD introduce the Rashba and Dresselhaus spin–orbit interactions (SOIs). For x and y axes oriented parallel to [100] and [010] crystal directions, respectively, the spin–orbit coupling terms along z axis are [7,8,15]:Taking into account that for confined electrons but , the Dresselhaus term simplifies to the well-known formula:The Dresselhaus coupling constant depends on the bulk Dresselhaus constant and on the layer thickness in the growth direction . The value of the Rashba constant () depends on the slope of the potential along the growth direction, which is partially defined by the growth conditions and can be tuned by electrical gating [1].
Efficient eigenvalue determination for arbitrary Pauli products based on generalized spin-spin interactions
Published in Journal of Modern Optics, 2018
We consider N two-level systems with logical basis of the l-th qubit () defined by the eigenstates of the z-component of a spin-1/2 angular momentum operator with . We write for the identity matrix in the state space of the l-th qubit and for the Pauli-matrices for . When setting for simplicity, the eigenvalues in the measurement basis are