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Semiclassical limit and separability of the Jaynes-Cummings model

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

Published Jan 1, 2004• 0 citations
Publisher

Abstract

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

DOI
10.58837/chula.the.2004.1018

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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
The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics. The relative simplicity of the JCM and the ease with which it can be extended through analytic expressions or numerical computation continues to motivate attention, as evidenced by the growing abundance of publications. We here present an overview of the theory of the JCM and some of the many extensions and generalizations that have appeared.
. The Jaynes-Cummings Model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probability for fields with sharply defined energy (or photon number). For fields having a statistical distribution of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive – only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further nonclassical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics.
In this paper, by using the notion of negativity, we study the degree of entanglement of a two-level atom interacting with a quantized radiation field, described by the Jaynes–Cummings model (JCM). We suppose that initially the field is in a pure state and the atom is in a general mixed state. In this case the negativity fully captures the entanglement of the JCM. We investigate the case that the initial state of the field is a coherent state. The influences of the detuning on the degree of entanglement are also examined.
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.
In this paper, we study the evolution of two two-level atoms interacting with a single-mode quantized radiation field, namely, the two-atom multiphoton Jaynes–Cummings model (JCM). We assume that the field and the atoms are initially prepared in the superposition of displaced number states and excited atomic states, respectively. For this system, we investigate the atomic inversion, Wigner function, phase distribution and entanglement. We show that for symmetric (asymmetric) atoms, the system can generate asymmetric (symmetric) cat states at a quarter of the revival time. Furthermore, the degrees of entanglement for the field–atoms and the one-atom–remainder tangles depend on the rate of energy flow between the parties. The interference in phase space decreases the degree of entanglement in the bipartite.
Summary form only given. We study the behavior of classical and quantum correlations between a two level atom and a quantised electromagnetic field mode in a thermal state interacting according to the Jaynes Cummings model (JCM). This model predicts collapses and revivals of Rabi oscillations which is also seen in experiments, providing direct evidence for the quantum mechanical nature of the electromagnetic field. It also has an analytical solution for arbitrary coupling constants and can easily be extended to include a variety of initial conditions,dissipation and damping, multiple atoms, multilevel atoms, multimode fields and more elaborated interactions such as multiphoton or intensity-dependent interaction.