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Progress of Theoretical Physics

Quasi-Probability for Three-Level System and Superradiance

Hiroshi T. ITO

Published Apr 1, 1976• 5 citations
Publisher

Abstract

A quasi-probability distribution function based on the coherent state for U(3) is introduced, and its properties are then investigated. With the aid of this function it is shown that the co-operative spontaneous emission from a system of the three-level atoms can be described as a diffusion-like process on C2. In particular, the limit of validity of the semiclassical approach is examined. The case in which the semiclassical approach is not applicable can also be treated approximately by means of the Wiener-Hermite expansion method.

DOI
10.1143/ptp.55.1038

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This chapter introduces a three-level atomic model and discusses two other coherent atomic phenomena; namely, coherent Raman scattering and lasing without inversion. Lasing without inversion may take place in three-level resonant atomic systems and, like super-radiance, results from atomic coherence. Super-radiation is initiated by spontaneous fluctuations of the polarization in the medium. The semiclassical method is a powerful tool that is useful for the treatment of propagation effects in super-radiance as well as in such coherent emission processes as self-induced transparency, photon echo and others. The use of the Maxwell—Bloch equations allows one to consider these phenomena in elongated systems. Fortunately, the semiclassical approach is readily applicable to low-density systems that are to gases. However, one has to exercise greater care when applying it to coherent emission in solids. In high-density systems there is a strong mixing of the field and atomic states in the vicinity of resonance.

CRC Press eBooks•Mar 29, 2018•0 citations
A quasi-probability distribution function based on the coherent state for U(3) is introduced, and its properties are then investigated. With the aid of this function it is shown that the co-operative spontaneous emission from a system of the three-level atoms can be described as a diffusion-like process on C2. In particular, the limit of validity of the semiclassical approach is examined. The case in which the semiclassical approach is not applicable can also be treated approximately by means of the Wiener-Hermite expansion method.
The dynamics of the co-operative. spontaneous emission· from many· two-level atoms is investigated by introducing a specific form of quasi-probability distribution which apparently excludes the vacuum fluctuation of the radiation field but includes the ·fluctuation of the atomic dipole moment. It is shown that the equation of motion for the random variables associated with the atom-field coupled system can be written in a quite classical form. As a result, the formula is derived for computing the multi-time correlation functions of the radi­ ation field-essential to its coherence properties-which generalizes Bonifacio, Schwendimann and Haake's adiabatic corresponding theorem. § 1. Introduction. The co-operative spontaneous emission (so-called superradiance) from an as­ sembly of many two-level atoms first discussed by Dicke in 1954n has been studied by many authors with different methods, roughly speaking, classicall/> and quantum mechanically. 3>-5> The classical approach, though simple, has a special significance for the dynamics of a large number of atoms participating in phase, and has succeeded except when the population of the atoms is completely inverted. The failure of the simple classical treatment in the case of complete inversion is due to the fact that the effect of spontaneous emission is not considered at all, so that emission is impossible from such a state as with no dipole moment.2>.s> Therefore, if one takes into account adequately the quantum fluctuation of the tdtal dipole moment which exists even when the atoms are fully excited, one should be able to describe the spontaneous emission from a classical point of view. This idea was suggested by Senitzky6> in a rather intuitive argument. In the present paper an attempt is made to formulate the above problem from a quasi-probability point of view. To this end we make here a particular choice of the representation of the quantum density operators (or quasi-probability. distri­ bution functions abbreviated by QPDF hereafter) among others7l-Io> by which the quantum mechanical equation of motion may be transformed into a form in an . associated c-number phase space, i.e., the yvell-known P-representation 9> of Glauber IS taken for the field variables and the so-~alled Q~representation based on the
The dynamics of the co-operative spontaneous emission from many two-level atoms is investigated by introducing a specific form of quasi-probability distribution which apparently excludes the vacuum fluctuation of the radiation field but includes the fluctuation of the atomic dipole moment. It is shown that the equation of motion for the random variables associated with the atom-field coupled system can be written in a quite classical form. As a result, the formula is derived for computing the multi-time correlation functions of the radiation field–essential to its coherence properties–which generalizes Bonifacio, Schwendimann and Haake's adiabatic “corresponding theorem”.
A semiclassical radiation theory is developed in which the spontaneous emission field of a two-level atom is a quantum entity represented by atomic operators acting in the Hilbert space of the atomic states. The theory is applied to an investigation of the coherence properties of spontaneous emission. It is found that incoherent and coherent components can be identified in the emission field, the former dominating in the early stages of the decay from the excited state and the latter dominating in the later stages or in the emission from a weakly excited atom. The single-atom coherence properties are shown to lead to well-known superradiance phenomena in the spontaneous emission from an assembly of similar atoms. The paper does not use the theory of quantum electrodynamics; it builds on knowledge at the undergraduate level of classical radiation theory and the quantum mechanics of atoms.
Coherent spontaneous emission from a system of NNN two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number" rrr is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\ensuremath{\sim}r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ensuremath{\ll}r$), the emitted radiation may be represented approximately by a coherent state.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
This chapter introduces a three-level atomic model and discusses two other coherent atomic phenomena; namely, coherent Raman scattering and lasing without inversion. Lasing without inversion may take place in three-level resonant atomic systems and, like super-radiance, results from atomic coherence. Super-radiation is initiated by spontaneous fluctuations of the polarization in the medium. The semiclassical method is a powerful tool that is useful for the treatment of propagation effects in super-radiance as well as in such coherent emission processes as self-induced transparency, photon echo and others. The use of the Maxwell—Bloch equations allows one to consider these phenomena in elongated systems. Fortunately, the semiclassical approach is readily applicable to low-density systems that are to gases. However, one has to exercise greater care when applying it to coherent emission in solids. In high-density systems there is a strong mixing of the field and atomic states in the vicinity of resonance.