Abstract The behavior of a magnetic dipole in magnetic fields is of interest in many systems including magnetic resonance imaging. The explanation presented in this chapter combines quantum physics and electricity and magnetism. The chapter begins with an explanation of magnetic dipole Larmor precession in static magnetic fields. The discussion then shifts to time-dependent magnetic fields and how they lead to transitions between the dipole’s energy states. The calculations are carried out both by using the time dependence of energy eigenstates and by solving the differential equations describing the quantum state vector amplitudes. The results are expressed in terms of the Rabi frequency and the Rabi formula. For the special cases of π and π/2 pulses of the oscillating magnetic field, the results show how those pulses implement quantum gates for the dipole system.
Abstract The behavior of a magnetic dipole in magnetic fields is of interest in many systems including magnetic resonance imaging. The explanation presented in this chapter combines quantum physics and electricity and magnetism. The chapter begins with an explanation of magnetic dipole Larmor precession in static magnetic fields. The discussion then shifts to time-dependent magnetic fields and how they lead to transitions between the dipole’s energy states. The calculations are carried out both by using the time dependence of energy eigenstates and by solving the differential equations describing the quantum state vector amplitudes. The results are expressed in terms of the Rabi frequency and the Rabi formula. For the special cases of π and π/2 pulses of the oscillating magnetic field, the results show how those pulses implement quantum gates for the dipole system.
Abstract This chapter begins the discussion of the time evolution of an active quantum system. From the earlier chapters, time dependent physics has been observed through the presence of the time evolution of the phase of each eigenstate. The Hamiltonian itself is time independent. This represents the same kind of evolution of a classical system like the vibration of a tuning fork when it has been struck or the oscillation of an LC circuit if the capacitor is charged to some voltage and then the switch is closed. In the quantum case, the Hamiltonian has also been time independent. The time evolution evolves according the full-time dependent Schrödinger equation, depending only on a single initial condition of the state vector or wave function and the corresponding time evolution of the phase factor for each eigenstate. However in this chapter, we consider the case of when there is a time dependent Hamiltonian such as a sinewave generator or laser. As in the case of resonant tunneling, we see the importance in dynamics of resonant coupling. With an oscillating potential energy term, we see the presence of Rabi oscillations in the probability amplitude of a two-state system on resonance, which can be viewed as a quantum flip-flop between two states of a quantum bit (qubit).
In 1937, Rabi treated the problem of a magnetic moment in an applied time-dependent magnetic field. This became the well-known magnetic resonance situation. The Hamiltonian is often taken to be [Formula: see text] = – µ · [[Formula: see text]]. In this paper, the Rabi oscillations formula, describing the spin flipping, is derived in an unusual way. The method uses a modification of a method due to Heitler. In the Heitler method, one uses the Interaction Picture of quantum mechanics. Due to the time-dependence in the problem, the usual Heitler method fails. However, the solution is found after quantizing the electromagnetic field. To better understand the origin of the spin flipping, the analogous time-independent problem is also solved. It is made clear that the origin of the Rabi oscillations is not due to the time-dependent magnetic field. The spin flipping is essentially due to the fact that the spin system, when initially prepared, is not in an eigenstate of the Hamiltonian. Thus, as times progresses, the system naturally evolves through the noneigenstate basis states.PACS Nos.: 03.65.–w, 76.20.+q
We present a quantum mechanical treatment of particle beam magnetic resonance and related experiments, in terms of static plane waves, applicable to experiments with oscillating fields. The results yield an alternate view of Larmor precession, showing it as a beating between different waves, and help to shed new light on several aspects of the problems discussed. We discuss magnetic resonance experiments with both the Rabi and Ramsey methods, neutron resonance spin echo and a neutron interferometer experiment.
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 solution of the time-dependent wave equation describing the interaction of two coupled spins with a rotating magnetic field in the presence of a steady field, as in the usual magnetic resonance situation, has been reexamined. By extending the method previously developed by Salwen, an exact expression is derived for all components of the wave function describing a three-state system under magnetic-resonance conditions. This solution describes in analytical form all the single and multiple resonances of which the system is capable for fields of any intensity. It is shown that when z is the axis of quantization, the projection of the wave function on an eigenstate of Fx oscillates at twice the resonance frequency at exact double quantum resonance. An approximate form is derived for the transition probability near double quantum resonance, showing that our solution is consistent with Salwen's in the limit of well-separated single and double quantum resonances. The theory is applied to the uppermost three substates of the upper hyperfine multiplet of an alkali metal of arbitrary nuclear spin, a case of interest to the analysis of magnetic resonance in optically pumped vapors.