In order to probe the transitions of a nuclear spin
s from one of its substate quantum numbers
m to another substate
m′, the experimenter applies a magnetic field
B0 in some particular direction, such along
z^, and then applies an weaker field
B1(t) that is oscillatory in time with the angular frequency
ω, and is normally perpendicular to
B0, such as
B1(t)=B1x^cos(ωt). In the rotating wave approximation,
B1(t)=B1[x^cos(ωt)+y^sin(ωt)]. Although this problem is solved for spin
21 in every quantum mechanics textbook, for the general spin
s case, its general solution has been published only for the overall probability of a transition between the states, but the time dependence of the probability of finding the nucleus in each of the substates has not previously been published. Here we present an elementary method to solve this problem exactly, and present figures for the time dependencies of the various substates states for a variety of initial substate probabilities for a variety of
s values. We found a new result: unlike the
s=21 case, for which if the initial probability of finding the particle in one of the substates was 1, and the time dependence of the probabilities of each of the substates oscillates between 0 and 1, for higher spin values, the time dependencies of the probabilities finding the particle in each of its substates, which periodic, is considerably more complicated.
This is the second in a series of three papers on nonlinear resonance effects in multilevel quantum systems in which quantum-mechanical systems are considered that are under the influence of any number of classically described driving fields (well defined in amplitude and phase) and are in weak contact with a thermal reservoir, such that relaxation may be described by means of phenomenological decay constants. In the first paper, referred to as I, a projection-operator formalism was developed for separating the equation of motion for the density operator into a part which may be treated by conventional time-dependent perturbation techniques and another part which, because it would give rise to secular terms using a time-dependent perturbation theory, must be treated by more exact means. In this paper three applications of the formalism developed in I are considered. The first application is to a system in which a multiple-quantum transition is taking place between two levels of the system which contains many levels between the two levels under consideration. In particular, the transition probability is calculated for these two levels, which are not directly connected by matrix elements of the interaction but are connected indirectly through intermediate states. The next application considered is the Raman effect in a four-level system. As a final application of the formalism developed in I an expression is derived for the magnetization
M+(t)=Mx(t)+iMy(t) of a system of
N noninteracting spins under the influence of a static field in the
z direction and an arbitrary number of rf fields in the
x direction. As in I, a phenomenological form is assumed for the relaxation in all of these applications. It is shown in this problem that fields which are not involved in directly causing transitions in this system give rise to frequency shifts in the resonance conditions, which are accounted for in this treatment in a very natural and straightforward manner. For these fields, the rotating as well as the counter-rotating fields give shifts in the resonance condition.
The time-dependent Schr\"odinger equation is solved formally for an atomic or molecular system which is subjected simultaneously to a rotating magnetic field of constant amplitude and angular velocity and to a constant magnetic field along the axis of rotation. The method yields the transition probabilities in terms of the solutions to an eigenvalue problem. This eigenvalue problem is solved both for (a) a normal Zeeman effect and for (b) the case where a transition from a given level is isolated in frequency from other transitions from the same level. Case (a) is exactly soluble and yields a solution which is shown to be the same as that of Bloch and Rabi, but is in a form which is more convenient for integration over the velocity distribution. Case (b) must be solved by an approximate method which results in a prediction of multiple quantum transitions as observed by Kusch.
Exact expressions for the transition probabilities of a three-level atom interacting with a rotating magnetic field are obtained, and used to obtain the double-quantum resonance shift of sixth order and the linewidths to fourth order of perturbation theory. The treatment is extended to deal with overlap effects, oscillating fields and atomic systems with more energy levels. Numerical calculations are performed to check the validity of the analytic expressions and to obtain results applicable at higher field strengths. Although the majority of the detailed work is concerned with the double-quantum resonance, the formalism is general, and the methods may be applied to single-quantum or higher multiple-quantum transitions.
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
The resonance behavior of magnetic and optical two-level systems under the influence of strong modulated fields is studied in detail by solving the Bloch equations in the limit of vanishing damping, namely, in terms of matrix continued fractions. Thus both analytical and numerical results are obtained for the positions of the various multiple quantum resonances in the time-averaged as well as modulated interaction components. Specifically, the role of the orientation of the optical pump beam in magnetic-resonance configurations is investigated, and it is shown that there are several classes of resonances for which the behavior of the system is independent of the orientation of the pump beam. These are the geometries in which the static and oscillating magnetic fields are either orthogonal or parallel, the geometries leading to Haroche-like resonances (which, however, play a special role in modulated interaction components also), and a previously unknown class, labeled saddle-point resonances. However, other than for these special classes, the behavior of the system depends generally on the orientation of the optical pump beam, unlike as implied in some previous work on this problem. As in the recently studied case of a fully amplitude-modulated optical interaction [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], it is found that the resonances in modulated interaction components are generally doubly branched, with one branch duplicating the resonances found in the time-averaged component. A new feature found for not-fully-amplitude-modulated and phase-modulated, off-resonance optical excitation (or magnetic resonance with nonparallel or nonorthogonal static and oscillating magnetic fields) is the occurrence of an extra type of resonance in modulated components of the interaction that cannot, like all other resonances of the system, be classified as power-shifted multiple quantum transitions.