It is shown that the dynamics of an N-level quantum system driven by a classical radiation field can be derived rigorously from an associated fully quantum-mechanical model with the help of an asymptotic limit. By formally expanding the full quantum dynamics around the semiclassical limit, quantum corrections to the semiclassical description are systematically constructed. The observables describing these corrections are shown to exist in any order, and explicit expressions for their time dependence are given.
It is shown that the dynamics of an N-level quantum system driven by a classical radiation field can be derived rigorously from an associated fully quantum-mechanical model with the help of an asymptotic limit. By formally expanding the full quantum dynamics around the semiclassical limit, quantum corrections to the semiclassical description are systematically constructed. The observables describing these corrections are shown to exist in any order, and explicit expressions for their time dependence are given.
Semiclassical descriptions of a few-level system coupled to an electromagnetic field mode reduce the field to a time-dependent driving term. Although such methods are widely used, the underlying quantum character of the field generates corrections to the dynamics that can become significant. Here, we develop a general approach for systematically calculating quantum corrections to the time-dependent Floquet dynamics that emerges in the semiclassical description. Taking the Rabi model of a spin-field system as an example, we show how approximate analytical expressions for short-time quantum corrections to the semiclassical dynamics can be obtained. This framework helps explain the emergence of semiclassical behavior, sheds light on collapse and revival dynamics in the Rabi model, and provides a tool for assessing corrections to semiclassical control techniques.
A classical model for the interaction of molecular systems with infrared radiation is presented. It differs from the usual ’’semiclassical theory of radiation and matter’’ in that the molecular system is treated by classical rather than quantum mechanics, and the radiation field is described, as in Jaynes’ neoclassical theory, as classical (mechanical) harmonic oscillators rather than as a classical field (i.e., via Maxwell’s equations). The classical Hamiltonian for the composite system—molecules, radiation, and their interaction—is thus that of a completely mechanical system, and its classical dynamics is determined by computing the classical trajectories of the system. Quantum mechanical interference and tunneling effects can be built into the description within the framework of the classical S-matrix theory. Even within the strictly classical limit of the model, it is shown that all dynamical effects in the interaction of radiation and matter are obtained; in the perturbative limit, for example, it is shown that absorption and induced emission, and even spontaneous emission, which is often difficult to obtain in other classical or semiclassical models, appear in a completely straightforward manner.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2/3c3)(d〈v〉/dt)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
We investigate the dynamics of an arbitrary atomic system (n-level atom or many n-level atoms) interacting with a resonant quantized mode of an electromagnetic field in a lossless cavity. Initially, the field is taken in a coherent state with a large average photon number. If the initial atomic state is chosen as an eigenstate of the atomic Hamiltonian in the classical field limit (semiclassical eigenstates), then the evolution of the system wave function is approximately factorized into the field and atomic parts. Explicit analytical expressions are given for the factorized wave functions. The time evolution of an arbitrary initial atomic state can be fully described in terms of factorized states. As examples, we discuss trapping states, and collapses and revivals of the atomic energy.