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Acta Physica Sinica

THE COHERENCE OF RADIATION AND INCREASE IN ENTROPY

Guang-jiong Ni, Suqing Chen, ZHOU GU-SHENG, 复旦大学物理系

Published Jan 1, 1982• 0 citations
Publisher

Abstract

A model of radiation by multi-atoms is discussed. System A, composed of N atoms with two energy levels, is coupled with a single-mode radiation B. A and B are in turn coupled with two thermal reservoirs D and C respectively, and we also assume that all atoms which are located within a wave length of the radiation field have the same phase. Then under the approximation of long wave, rotating wave and mean fields, by introducing Markov approximation after using the method of density matrix, we have derived the of equations motion for systems A and B respectively. These equations are concerning with the following processes: the spontaneous emission of single atom, the absorption and induced emission stimulated by thermal radiation, the induced emission of N atoms stimulated by single mode coherent state radiation as well as the coherent spontaneous emission of this multi-atoms system. The latter two processes are actually the two kinds of superradiance originated from Dicke model and had been investigated previonsly. We have analysed these five processes by numerical calculations. Besides, the influence of these processes upon the evolution of atomic system attaining thermal equilibrium and the increase of entropy is also discussed. It is pointed out that the correlations inside a multi-atoms system would not be destroyed by coherent spontaneous emission, in contrast with the incoherent one which leads to the change of entropy in atomic system. However, only when the entropy of the two reservoirs has been taken into account, will it be possible to demonstrate the mono-tonic increase in total entropy.

DOI
10.7498/aps.31.585

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A model of radiation by multi-atoms is discussed. System A, composed of N atoms with two energy levels, is coupled with a single-mode radiation B. A and B are in turn coupled with two thermal reservoirs D and C respectively, and we also assume that all atoms which are located within a wave length of the radiation field have the same phase. Then under the approximation of long wave, rotating wave and mean fields, by introducing Markov approximation after using the method of density matrix, we have derived the of equations motion for systems A and B respectively. These equations are concerning with the following processes: the spontaneous emission of single atom, the absorption and induced emission stimulated by thermal radiation, the induced emission of N atoms stimulated by single mode coherent state radiation as well as the coherent spontaneous emission of this multi-atoms system. The latter two processes are actually the two kinds of superradiance originated from Dicke model and had been investigated previonsly. We have analysed these five processes by numerical calculations. Besides, the influence of these processes upon the evolution of atomic system attaining thermal equilibrium and the increase of entropy is also discussed. It is pointed out that the correlations inside a multi-atoms system would not be destroyed by coherent spontaneous emission, in contrast with the incoherent one which leads to the change of entropy in atomic system. However, only when the entropy of the two reservoirs has been taken into account, will it be possible to demonstrate the mono-tonic increase in total entropy.
Interaction with a multimode quantized radiation field of an assembly of NNN two-level atoms contained in a small volume is considered for the special cases of (i) highly excited, (ii) weakly excited, and (iii) superradiant atomic assemblies, and solutions of equations of motion are obtained in closed form. For (i) and (ii) we show that, if the atomic assembly is in the Dicke state ∣r,m>|r,m>∣r,m>, the spontaneously emitted radiation is Lorentzian, and its spectral width and the numerical value of the Lamb shift are r+∣m∣r+|m|r+∣m∣ times that for a single excited atom. For (i) the sign of Lamb shift is opposite to that for (ii) and for a single excited atom. Change in the frequency shift gives chirping in the pulse radiation emitted spontaneously by an assembly of atoms. We obtain the reduced density operator for a single mode and for the complete spontaneous emitted radiation, and discuss its characteristic function, weight function in diagonal representation, coherence functions, and normalized variance. We find that the emitted radiation is chaotic if all atoms are excited or if the assembly is weakly excited and is in thermal equilibrium. For a superradiant assembly, the radiation is coherent; it is only amplitude coherent if the initial atomic density operator is nondiagonal. We also study stimulated emission and scattering for the cases (i) and (ii). For case (iii) these phenomena do not take place.
In this paper we discuss the quantum features of cooperative radiation processes in a two-level system taking into account non-Markovian effects. If the initial condition is spatially uniform, the many-mode atom-field master equation which has been derived and semiclassically discussed in a previous paper, reduces to the well-known single-mode model laser master equation. Hence, using a suitable projection operator, we derive a generalized non-Markovian master equation which turns out to be correct already in the first Born approximation. This master equation incorporates the effects both of stimulated emission and absorption and of spontaneous emission. When stimulated processes can be neglected, it reduces to a previously known Markovian superradiance master equation, whereas in general it describes non-Markovian oscillations of the radiated intensity. Finally we derive a linear system of equations for the probability of occupation of the Dicke states which generalizes a previously discussed system, taking into account the contribution of stimulated processes.
Spontaneous and stimulated emission from atoms in a free one-dimensional space is considered from the quantum-mechanical point of view. The resolvent method is used to calculate the field density matrix, average electric field, and electromagnetic field energy density for a spontaneously emitting atom. For an atom emitting in the field of another atom, calculations are made of the electric field of the spontaneous and stimulated radiation. It is shown how the coherence length and coherence time of the spontaneous radiation increase. The application of these results in laser theory to explain the formation of a radiation mode structure and consequent appearance of coherent radiation is discussed.
In this paper we attempt to unify the theoretical treatment of a variety of cooperative interaction phenomena occurring in resonantly interacting radiation-matter systems. The system consists of a multimode optical field interacting with NNN resonant atoms in a region of large spatial extent. The field and atoms may or may not be coupled to reservoirs. We derive a set of coupled master equations for the reduced matter and radiation density operators of the composite system in the first Born approximation, using a method due to Bogoliubov. These equations describe within a single framework phenomena involving superradiant emission and coherent propagation. In the first place, our theory generalizes existing formulations of the superradiance problem by treating the radiation field as part of the dynamical system rather than as a reservoir. Hence, our equations describe stimulated interaction effects, which are entirely absent in the usual superradiance master equations. Moreover, our equations contain terms describing the self-consistent interaction of an average field with a set of polarizable atoms, as in the coupled Bloch and Maxwell equations of the theory of coherent pulse propagation, and enable us to describe fluctuations, arising from incoherent emission, about the average behavior described by the Bloch-Maxwell theory. We solve formally the dynamical equation for the radiation density operator and obtain a displaced multivariate Gaussian distribution centered on the solution of the Maxwell wave equation, which has as parameters the first and second moments of the matter density operator. Then we derive and discuss the field and atomic-moment equations, both in the wave-number and position representations. The moment equations enable us to show the relationship between descriptions of superradiant damping based on the Bloch equations and those based on the existence of correlated atomic states, including both as special cases. We introduce reduced matter moments, which allow us to separate incoherent- and coherent-emission effects and enable us to discuss the question of the closure of the moment equations. Then we consider cases where the behavior of the atomic system is dominated by a strong coherent pump wave or where the atoms are in a symmetric correlated state and show how the slowly-varying-envelope approximation simplifies the radiation-moment equations in these two cases. In the former case, we obtain an approximate theory which consists, in lowest order, of the Bloch-Maxwell equations with radiative-damping terms, plus an equation describing the time development of the correlation function for field fluctuations. Returning to our general formalism, we relate the usual parameters of the theory of radiative decay and of resonant interactions to the more general quantities appearing in our spatial analysis. Finally, we discuss the importance of the stimulated-emission terms appearing in our theory and describe several extensions, including the introduction of inhomogeneous broadening.
Coherence properties of the single-mode electromagnetic radiation emitted by a system of identical two-level atoms which is initially in the superradiant state are determined approximately. The approximation is valid only for short times when the state of the atomic system does not differ appreciably from the superradiant state. The electromagnetic field radiated spontaneously possesses full coherence in the sense of Glauber's definition, its density operator having the form of that for the ideal laser radiation. This coherence disappears when, because of external radiation, stimulated and spontaneous emission both take place.
The interaction between a coherently oscillating radiation field, such as is generally encountered in micro-wave experiments, and a number of similar atomic systems coupled to the field through their electric dipole moments is analyzed for the case of resonance between atomic system and field, with both the field and molecules treated quantum-mechanically. Expressions are derived for the expectation value of field strength and the energy, and comparison between the two makes it possible to distinguish between coherent and incoherent parts of the energy, the former having its counterpart in the expectation value of the field strength, while the latter has no such counterpart. Terms corresponding to induced and spontaneous emission are identified, and it is shown that the latter includes both coherent and incoherent components. Special situations related to masers and to the coherence of spontaneous radiation are discussed. The distinction between the behavior of correlated and uncorrelated states is examined, and it is shown that both the coherent emission from uncorrelated states and the incoherent emission from correlated states may be proportional to the square of the number of molecules. The phase of the field is predictable for the coherent emission but unpredictable for the incoherent emission.