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Coherent Interactions

Stephen Mark Barnett, PAUL M. RADMORE

Published Nov 14, 2002• 0 citations
Publisher

Abstract

Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.

DOI
10.1093/acprof:oso/9780198563617.003.0002

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Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
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.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.
We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse-revival behavior.
Abstract In this chapter we study with the tools developed in Chapter 3 the basic models that are the foundations of light–matter interaction. We start with Rabi dynamics, then consider the optical Bloch equations that add phenomenologically the lifetime of the populations. As decay and pumping are often important, we cover the Lindblad form, a correct, simple and powerful way to describe various dissipation mechanisms. Then we go to a full quantum picture, quantizing also the optical field. We first investigate the simpler coupling of bosons and then culminate with the Jaynes–Cummings model and its solution to the quantum interaction of a two-level system with a cavity mode. Finally, we investigate a broader family of models where the material excitation operators differ from the ideal limits of a Bose and a Fermi field.
Introduction So far, we have discussed closed systems involving a single quantized mode of the field interacting with atoms, as for example in the Jaynes–Cummings model in Chapter 4. As we saw in this model, the transition dynamics are coherent and reversible: the atom and field mode exchange excitation to and fro without loss of energy. As we add more modes for the atom to interact with, the coherent dynamics become more complicated as the relevant atom–field states come in and out of phase and beat together to determine the total state occupation probabilities. As time goes on, these beats get out of phase, leading to an apparent decay of the initial state occupation probability. But at later times, the beating eigenfrequencies get back in phase in a manner rather reminiscent of the Jaynes–Cummings revival discussed earlier in this book, and this leads to a partial recurrence or revival of the initial state probability. The time scale for this partial revival depends on the number of participating electromagnetic field modes and as these increase to the level appropriate for an open system in free space the recurrence disappears off to the remote future, and the exponential decay law appropriate for decay is recovered as an excellent approximation. We have already discussed the origin of spontaneous emission and the Einstein A coefficient using perturbation theory in Chapter 4.
The interaction of a two-level atom with a single-mode quantized field is one of the simplest models in quantum optics. Under the rotating wave approximation, it is known as the Jaynes-Cummings model and without it as the Rabi model. Real-world realizations of the Jaynes-Cummings model include cavity, ion trap and circuit quantum electrodynamics. The Rabi model can be realized in circuit quantum electrodynamics. As soon as nonlinear couplings are introduced, feasible experimental realizations in quantum systems are drastically reduced. We propose a set of two photonic lattices that classically simulates the interaction of a single two-level system with a quantized field under field nonlinearities and nonlinear couplings as long as the quantum optics model conserves parity. We describe how to reconstruct the mean value of quantum optics measurements, such as photon number and atomic energy excitation, from the intensity and from the field, such as von Neumann entropy and fidelity, at the output of the photonic lattices. We discuss how typical initial states involving coherent or displaced Fock fields can be engineered from recently discussed Glauber-Fock lattices. As an example, the Buck-Sukumar model, where the coupling depends on the intensity of the field, is classically simulated for separable and entangled initial states.