Introduction In 1963, E. T. Jaynes and F. W. Cummings published a paper entitled “Comparison of Quantum and Semiclassical Radiation Theories, with Application to the Beam Maser.” This paper was written at a time when the quantum theory of the laser had not yet been worked out, and represented an attempt at discussing the new behaviour that could be expected from treating the field quantum mechanically instead of classically. Jaynes and Cummings began by considering the interaction between a single two-level molecule and a single field mode, a simple situation which has become known as the Jaynes-Cummings model. Treating then the maser medium as a beam of two-level systems initially in a mixture of their upper and lower states characterized by a temperature T , they found that the field mode reached a thermal equilibrium with temperature T f = Ω T /ω, where ω is the transition frequency of the two-level systems and Ω is the field mode frequency, provided that T > 0. For T < 0, the steady state field was found to contain an infinite amount of energy, an unphysical result resulting from the neglect of losses in the model. Jaynes and Cummings then proceeded to develop an alternative semi-classical theory of the beam maser, introducing their famous neo-classical theory of spontaneous emission. An important follow-up paper was published by F. W. Cummings, who studied the long-time behavior of a two-level system interacting with a quantized single-mode field initially in a coherent state.
Introduction In 1963, E. T. Jaynes and F. W. Cummings published a paper entitled “Comparison of Quantum and Semiclassical Radiation Theories, with Application to the Beam Maser.” This paper was written at a time when the quantum theory of the laser had not yet been worked out, and represented an attempt at discussing the new behaviour that could be expected from treating the field quantum mechanically instead of classically. Jaynes and Cummings began by considering the interaction between a single two-level molecule and a single field mode, a simple situation which has become known as the Jaynes-Cummings model. Treating then the maser medium as a beam of two-level systems initially in a mixture of their upper and lower states characterized by a temperature T , they found that the field mode reached a thermal equilibrium with temperature T f = Ω T /ω, where ω is the transition frequency of the two-level systems and Ω is the field mode frequency, provided that T > 0. For T < 0, the steady state field was found to contain an infinite amount of energy, an unphysical result resulting from the neglect of losses in the model. Jaynes and Cummings then proceeded to develop an alternative semi-classical theory of the beam maser, introducing their famous neo-classical theory of spontaneous emission. An important follow-up paper was published by F. W. Cummings, who studied the long-time behavior of a two-level system interacting with a quantized single-mode field initially in a coherent state.
We studied a very simple model of the interaction between radiation and matter, named Jaynes-Cummings model. Matter is represented by an atom, treated as a two level system, which interacts with one single mode of the electromagnetic field. We looked at the problem in two different ways: first, treating both systems as quantum and later, considering the field as classical and the atom as quantum. In particular,we concentrate in obtaining the population inversion in both cases. In the context of semiclassical theory, we still used two distinct approachs: one, taking into account the field reaction of the atom and, the other, ignoring any effect of the atom onto the field. We verified that the first gives a more realistic description than the second because it accounts for spontaneous emission by the atom. We still included the environment on the description of both quantum and classical dynamics.
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.
In this thesis we present a theoretical analysis of some of the implications of the quantum nature of the electromagnetic field.In particular, we are interested in how atomic dynamics are modified by coupling to a quantized radiation field.In chapter two we review the Jaynes-Cummings model of atom-field interactions.This model consists of a single two-level atom coupled to a single mode of the radiation field.The evolution of the atom depends upon the initial quantum state of the field mode.Chapter three is a review of the quantum theory of dissipation.In chapter four we examine the properties of the thermofield states of the two-mode, quantized radiation fields.We investigate the use of these states as a representation of single mode thermal states.The thermo field representation is used to describe interactions involving thermal averages and in an analysis of two-mode squeezing in parametric amplifiers.Chapters five to seven are concerned with the Rydberg atom maser.In chapter five we review some of the properties of Rydberg atoms in cavities and consider the validity of the Jaynes-Cummings model as a description of the maser.In chapter six we discuss the modification of the atomic dynamics when more than one atom is present in the cavity.We show how the collective atomic inversion is influenced by many-atom effects, as well as the photon statistical effects present in the single atom Jaynes-CHAPTER
The Jaynes-Cummings paper treats the problem of one-atom interacting with one cavity mode with two alternative theories: quantum electrodynamics and semiclassical theory, and compares the predictions. The semiclassical theory was later extended to treat a single atom in free space interacting with a classical field including radiation reaction. The resulting theory was called Neoclassical theory, and is described by nonlinear optical Bloch equations. These nonlinear equations have shown up in a number of physical systems and were the subject of the famous Jaynes-Franken bet that was a prominent feature of the early Rochester Coherence and Quantum Optics Conferences. We will touch on all of these topics in a historical review.
We propose a variation of the Jaynes-Cummings Model (JCM), which consists of a Two-Level Atom (TLA) within a cross cavity configuration. This new geometry is a suitable tool to understand the origin of new QED features that the original JCM does not reveal, showing revivals in cases where they are not expected to appear. Full Text: PDF References E.T. Jaynes, F.W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser", Proc. IEEE 51, 1 (1963). CrossRef M. Wilkens, P. Meystre, "Nonlinear atomic homodyne detection: A technique to detect macroscopic superpositions in a micromaser", Phys. Rev. A 43, 7 (1991). CrossRef I.V. Jyotsna, G.S. Agarwal, "The Jaynes-Cummings model with continuous external pumping", Opt. Commun. 99, 344 (1993). CrossRef M.S. Abdalla, M. Abdel-Aty, A.-S.F. Obada, "Quantum entropy of isotropic coupled oscillators interacting with a single atom", Opt. Comm. 211, 225 (2001). CrossRef M.A. Marchiolli, R.J. Missori, "Qualitative aspects of entanglement in the Jaynes?Cummings model with an external quantum field", J. Phys. A: Math. Gen. 36, 49 (2003). CrossRef J.C. García-Melgarejo, Ph.D. Thesis. INAOE (2015). J.R. Ackerhalt, K. Rzazewski, "Heisenberg-picture operator perturbation theory", Phys. Rev. A 12, 6 (1975) CrossRef J.H. Eberly, N.B. Narozhny, J.J. Sanchez-Mondragon, "Periodic Spontaneous Collapse and Revival in a Simple Quantum Model", Phys. Rev. Lett. 44, 20 (1980). CrossRef N.B. Narozhny, J.J. Sanchez-Mondragon, J.H. Eberly, "Coherence versus incoherence: Collapse and revival in a simple quantum model", Phys. Rev. A 23, 1 (1981). CrossRef L. Allen, J.H. Eberly, Optical resonance and two-level atoms (New York, Wiley 1975). A. Joshi, R.R. Puri, "Effects of the Binomial Field Distribution on Collapse and Revival Phenomena in the Jaynes-Cummings Model", J. Modern Optics 34, 11 (1987). CrossRef
This paper has two purposes: 1) to clarify the relationship between the quantum theory of radiation, where the electromagnetic field-expansion coefficients satisfy commutation relations, and the semiclassical theory, where the electromagnetic field is considered as a definite function of time rather than as an operator; and 2) to apply some of the results in a study of amplitude and frequency stability in a molecular beam maser. In 1), it is shown that the semiclassical theory, when extended te take into account both the effect of the field on the molecules and the effect of the molecules on the field, reproduces almost quantitatively the same laws of energy exchange and coherence properties as the quantized field theory, even in the limit of one or a few quanta in the field mode. In particular, the semiclassical theory is shown to lead to a prediction of spontaneous emission, with the same decay rate as given by quantum electrodynamics, described by the Einstein A coefficients. In 2), the semiclassical theory is applied to the molecular beam maser. Equilibrium amplitude and frequency of oscillation are obtained for an arbitrary velocity distribution of focused molecules, generalizing the results obtained previously by Gordon, Zeiger, and Townes for a singel-velocity beam, and by Lamb and Helmer for a Maxwellian beam. A somewhat surprising result is obtained; which is that the measurable properties of the maser, such as starting current, effective molecular Q, etc., depend mostly on the slowest 5 to 10 per cent of the molecules. Next we calculate the effect of amplitude and frequency of oscillation, of small systematic perturbations. We obtain a prediction that stability can be improved by adjusting the system so that the molecules emit all their energy h Ω to the field, then reabsorb part of it, before leaving the cavity. In general, the most stable operation is obtained when the molecules are in the process of absorbing energy from the radiation as they leave the cavity, most unstable when they are still emitting energy at that time. Finally, we consider the response of an oscillating maser to randomly time-varying perturbations. Graphs are given showing predicted response to a small superimposed signal of a frequency near the oscillation frequency. The existence of "noise enhancing" and "noise quieting" modes of operation found here is a general property of any oscillating system in which amplitude is limited by nonlinearity.