The Bloch sphere concept is a useful tool to represent the actions of various quantum-mechanical operators on a spinor. Most importantly, it provides a link between the rather abstract concept of a spinor and the more intuitive way (although not rigorous) of thinking of the spin of an electron as being associated with a magnetic moment along the axis of self rotation. The Bloch sphere is particularly useful when describing the action of a spatially uniform external (including time-dependent) magnetic field on the spin [1]. This will be illustrated in the next chapter through the derivation of the Rabi formula. Bloch sphere concepts are also frequently invoked in discussions of spin-based quantum computing, so that a sound understanding of this notion is imperative to comprehend the literature in that field.
The Bloch sphere concept is a useful tool to represent the actions of various quantum-mechanical operators on a spinor. Most importantly, it provides a link between the rather abstract concept of a spinor and the more intuitive way (although not rigorous) of thinking of the spin of an electron as being associated with a magnetic moment along the axis of self rotation. The Bloch sphere is particularly useful when describing the action of a spatially uniform external (including time-dependent) magnetic field on the spin [1]. This will be illustrated in the next chapter through the derivation of the Rabi formula. Bloch sphere concepts are also frequently invoked in discussions of spin-based quantum computing, so that a sound understanding of this notion is imperative to comprehend the literature in that field.
The Bloch sphere concept is a useful tool to represent the actions of various quantum-mechanical operators on a spinor. Most important, it provides a link between the rather abstract concept of a spinor and the more intuitive way (although not rigorous) of thinking of the spin of an electron as being associated with a magnetic moment along the axis of self-rotation. The Bloch sphere is particularly useful when describing the action of a spatially uniform external (including time-dependent) magnetic field on the spin [1]. This will be illustrated in the next chapter through the derivation of the Rabi formula. Bloch sphere concepts are also frequently invoked in discussions of spin-based quantum computing, so that a sound understanding of this notion is imperative to comprehend the literature in that field.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip,” i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip”, i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli Equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.
Electron spin, alone or in conjunction with charge, can be used to process information and build up quantum computers because it is a natural quantum bit.In this work, a Java simulator has been constructed and a 3D display has been created using Java 3D API to animate the motion of a qubit or spinor based on the Bloch sphere and Bloch ball concepts.The simulator is capable of illustrating the Larmor precession of a qubit, the Rabi formula which calculates the probability of spin flip in a time dependent magnetic field, and the density matrix concept where the time evolution of an ensemble of spinors can be examined.Control panels are constructed on the user interface to allow manipulation of qubits and adjustment of simulation parameters such as external time-independent and time-dependent magnetic fields.The simulator provides a user friendly software package useful for teaching and self-learning in the field of spintronics.