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Paralelo entre as teorias quântica e semi-clássica da interação átomo-campo no modelo Jaynes-Cummings

Luísa. A. P. Filpi

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

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.

Related papers

Efeitos dissipativos na inversão atômica segundo o modelo de Jaynes-Cummings

Clarissa do Valle da Silva Lima

We study the Jaynes-Cummings model (JCM) which describes the interaction between a two-level atom and a single-mode resonant or quasi-resonant electromagnetic field in the non-dissipative and issipative cases. In particular, we study the atomic population inversion whose behavior presents some aspects of purely quantum origin: collapse and revivals. This observables expectation value can be btained analytically without use of approximations for the non-dissipative case and using first-order perturbation in the dissipative case. We make use of the sum of Poisson and the steepest descents method to obtain analytical formulas for the collapse and for each revival. In this way, we can obtain relevant information like the occurrence time, the maximum amplitude, the approximate frequency, and width. The dissipative JCM models the action of the environment on the quantum system. We show that the characteristic time of the revival diminishes as the interaction with the environment enlarges. We consider the limit in which it becomes experimentally impossible to distinguish the obtained curve from the classical one.

•Apr 7, 2006•0 citations

The Jaynes–Cummings Model is Alive and Well

Pierre Meystre

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.

Cambridge University Press eBooks•Sep 2, 1993•0 citations

Semiclassical limit and separability of the Jaynes-Cummings model

Supitch Khemmani, Virulh Sa-Yakanit, Briggs, John S.

The Jaynes-Cummings model (JCM) of a two-level atom interacting with a quantized radiation field has been investigated theoretically. Of particular interest is the transition of a quantized electromagnetic field of the model to a classical one in the limit of very large average photon number. In this thesis, two aspects of the JCM are examined. Firstly, it is shown that, in contrast to the general case, for specic initial atom-field states, there is exact correspondence between the atomic inversion predicted by the JCM and that predicted by the semiclassical model in which the field is treated classically from the outset. Secondly, the degree of quantum entanglement of the JCM is investigated quantitatively and it is shown that, at any nite absolute temperature greater than zero, the atom and field are never separable and the degree of quantum entanglement approaches zero in the high and low temperature limits

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Atom–field interaction – quantum theory

Marlan O. Scully, Muhammad Suhail Zubairy

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.

Cambridge University Press eBooks•Sep 4, 1997•7 citations

Quantum states of radiation and their interaction with atomic matter

Stephen Mark Barnett

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

•Jan 1, 1985•0 citations

Influence of field dynamics on Rabi oscillations: Beyond the standard semiclassical Jaynes—Cummings model

A. Cives-esclop, Alfredo Luis, Luis Lorenzo Sánchez-Soto

The semiclassical interaction of a two-level atom with a singlemode field, taking into account the atomic reaction onto the field, is considered. Analytical expressions for the field and atomic dynamics are found. An expansion of this solution in terms of the inverse of the field intensity is performed. To zeroth order, the classical Rabi solution is recovered, while first-order corrections give rise to a slow modulation of the atomic dipole, in agreement with a fully quantum model. The evolution of the atomic dipole is studied in the framework of the reduced Q function, both in the semiclassical and quantum regimes. PACS 42.50.Dv — 03.65.-w.

Journal of Modern Optics•Mar 1, 1999•8 citations
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.
We study the Jaynes-Cummings model (JCM) which describes the interaction between a two-level atom and a single-mode resonant or quasi-resonant electromagnetic field in the non-dissipative and issipative cases. In particular, we study the atomic population inversion whose behavior presents some aspects of purely quantum origin: collapse and revivals. This observables expectation value can be btained analytically without use of approximations for the non-dissipative case and using first-order perturbation in the dissipative case. We make use of the sum of Poisson and the steepest descents method to obtain analytical formulas for the collapse and for each revival. In this way, we can obtain relevant information like the occurrence time, the maximum amplitude, the approximate frequency, and width. The dissipative JCM models the action of the environment on the quantum system. We show that the characteristic time of the revival diminishes as the interaction with the environment enlarges. We consider the limit in which it becomes experimentally impossible to distinguish the obtained curve from the classical one.
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.
The Jaynes-Cummings model (JCM) of a two-level atom interacting with a quantized radiation field has been investigated theoretically. Of particular interest is the transition of a quantized electromagnetic field of the model to a classical one in the limit of very large average photon number. In this thesis, two aspects of the JCM are examined. Firstly, it is shown that, in contrast to the general case, for specic initial atom-field states, there is exact correspondence between the atomic inversion predicted by the JCM and that predicted by the semiclassical model in which the field is treated classically from the outset. Secondly, the degree of quantum entanglement of the JCM is investigated quantitatively and it is shown that, at any nite absolute temperature greater than zero, the atom and field are never separable and the degree of quantum entanglement approaches zero in the high and low temperature limits
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.
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 semiclassical interaction of a two-level atom with a singlemode field, taking into account the atomic reaction onto the field, is considered. Analytical expressions for the field and atomic dynamics are found. An expansion of this solution in terms of the inverse of the field intensity is performed. To zeroth order, the classical Rabi solution is recovered, while first-order corrections give rise to a slow modulation of the atomic dipole, in agreement with a fully quantum model. The evolution of the atomic dipole is studied in the framework of the reduced Q function, both in the semiclassical and quantum regimes. PACS 42.50.Dv — 03.65.-w.