Cooperative Spontaneous Emission Research Articles

Spontaneous emission by a fully excited system of three identical atoms

The Lehmberg-Agarwal master equation is used to study cooperative spontaneous emission by a system of three identical two-level atoms. A choice of basis states is described which allows an exact solution to be obtained for any geometrical configuration, any sample size, and any initial conditions of the atomic system, fully including both near-field and super-radiance effects.

Vector coherence initiation of multi-level superfluorescence

An SU( n) coherent state representation is developed for the theory of multi-level cooperative spontaneous emission. A general Fokker-Planck equation is derived, whose characteristics are equivalent to the semiclassical equations for large numbers of atoms. The characteristics equations have non-classical initial values described by a vector coherence probability distribution. The theory can be used for the multilevel case with coherent pumping, to calculate the effects of pump interactions.

Cooperative Spontaneous Emission from an Initially Fully Excited System of Three Identical Two‐level Atoms

AbstractUsing Lehmberg's master equation and taking into account dipole‐dipole near‐field interaction the time‐dependent intensity pattern, intensity – and angular correlations of the cooperative spontaneous radiation from an initially fully excited system of three identical two‐level atoms are calculated. The intensity pattern developes lobes in different directions at different times in such a way that contrary to the two‐atom system the spatial distribution of the total number of the emitted photons differs from that of the single‐atom radiation pattern. In particular three atoms placed in proper distances on a straight line emit preferentially their energy along this line. At the beginning of the emission process the probability to find two photons simultaneously in the same (or geometrical equivalent) direction is always highest but in the course of time, contrary to the two‐atom case, it may well happen that the probability of finding two photons in quite different directions becomes more favoured.

Altered branching ratios and relaxation oscillations in superfluorescent cascades

A non-linear rate equation describing superfluorescence in many-level systems is presented. It is shown that when a large atomic population is excited into Rydberg levels the cascade back to lower levels by cooperative spontaneous emission leads to a number of new effects including 'trapping' of population in excited states, alteration of branching ratios, truncation of superfluorescent pulses and multiple pulsing of a single transition. When a collection of atoms is excited by a laser to a Ryberg state, it will cascade back down through a series of levels to the ground state. Due to the long wavelengths and large dipole moments of transitions between Rydberg states, the decaying atoms may be strongly coupled to each other by their common near field. Superfluorescence which can result from this strong coupling has been observed from Rydberg atoms by several groups. Gross et a1 (1979) in particular have emphasised the strength of the atom-atom coupling by observing superfluorescence in samples of only lo5 excited atoms. In this paper we show that superfluorescence can dramatically alter branching ratios and even the fundamental dynamics of these multi-level cascades. We find that transitions with the largest dipole moments are greatly enhanced so that substantially all the population goes down one channel without branching. We also find that the superfluorescence can be quenched by coherent trapping of the population in an excited level, but can subsequently be reinitiated, leading to a relaxation oscillation decay. The analysis of this cascade is inherently a multi-level problem with the possibility of successive cooperative decays. However, the extensive literature of superfluores- cence is almost completely devoted to the study of two-level systems. Agarwal and Trivedi (1976) and Ito (1976) have considered the three-level atom in the small-sample approximation. Ikeda (1977) discusses the swept excitation model for a three-level system. Crubellier et a1 (1980) have discussed both superradiance and subradiance in three-level atoms. Senitzky (1974) has derived the equations of small-sample multi-level superfluorescence from a boson second-quantisation formalism. However, none of these papers discuss the way in which superfluorescence changes branching ratios and the dynamics of cascades in a fundamental way. In this paper we present an intuitively appealing non-linear rate equation which we have derived from source-field quantum electrodynamics. Since the derivation is analogous to a number of two-level calculations including that in the textbook of Allen and Eberly (1975), details of the derivation are not given here. Only the necessary assumptions and approximations are pointed out. Our chief aim here will

Fluctuation of Polarized Light in Cooperative Spontaneous Emission

Macroscopic fluctuation of superradiance most sensitively appears as the fluctuation of the polarized photon number. This fluctuation is not much affected by unexpected variation of the exciting pulse area. In the superradiance limit the relative fiuctuation of the polarized photon number is equal to \(\sqrt{1/3}\), and it decreases rapidly as the atomic number density is reduced below a threshold density. Such a characteristic change is attributed to a transition from the superradiant emission to the amplified spontaneous emission.

Cooperative Spontaneous Emission from a Single‐quantum Excited Three‐atom System

AbstractThe total radiation rate of the cooperative spontaneous emission from an initially single‐quantum excited three‐atom system is calculated using Lehmberg's master equation and taking into account dipole‐dipole near field interaction. We find a strong influence of the precise configuration of the atomic dipoles and of the position of the initially excited atom on the temporal behaviour of the total radiation rate, especially if the atoms are separated by distances much smaller than the resonance wavelength.

Theory of cooperative spontaneous emission

An analysis is made of the deexcitation kinetics of strongly excited extended systems of initially uncorrelated two-level radiators. The possibility of partially cooperative states of the system is predicted when the radiation intensity acquires values intermediate between the spontaneous decay intensity and the Dicke superradiance intensity. The conditions for the onset of cooperative processes are obtained. The threshold for the oscillatory emission conditions is determined and the oscillation frequency is found. Modulation of the emission intensity is predicted with different modulation frequencies before and after the superradiant pulse. The modulation frequencies are determined. Formulas are derived for the superradiant pulse delay times and intensities. The possibility of weakly anisotropic cooperative emission is predicted.

Effect of short-range Coulomb interaction on cooperative spontaneous emission

We follow in detail the spontaneous decay of fully excited samples of two and three resonant two-level atoms separated by distances much smaller than a wavelength. We include both the static dipole-dipole interaction and an in rinsic detuning (inhomogeneous broadening) of the atoms. We find that the Coulomb interaction may have both a decohering and a recohering effect. The decohering effect is seen in three atoms not equivalently placed. Here the Coulomb interaction diminishes super-radiance by subjecting the different atoms to different frequency shifts. The recohering effect is seen in all cases and consists in a protection of super-radiance against an inhomogeneous broadening far in excess of the inverse radiation time. Both effects have analogs in small many-atom samples studied in previous literature.

Exact solution of the superradiance master equation. II. Arbitrary initial excitation

The cooperative spontaneous emission from a system of $N$ identical two-level atoms, excited arbitrarily and interacting with a single mode of radiation, is studied by using the Bonifacio-Schwendimann-Haake approach. The evolution of the probability distribution of the atomic system is considered as a Markov process described by a time-evolution matrix (TEM). The exact analytic expressions for the elements of this TEM are obtained by solving the superradiance master equation. It is proved that the TEM satisfies the normalization condition and the Chapman-Kolmogorov equation as required. The exact general formulas for the expectation values of various physical observables, expressed in terms of an arbitrary initial distribution, are obtained. Finally, these results are applied to a specific example, the superradiant initial state.

Exact solution of the superradiance master equation. I. Complete initial excitation

The cooperative spontaneous emission from a completely excited system of $N$ identical two-level atoms is studied under the condition that the cooperation number is conserved. The master equation for such a system has been solved analytically. The exact expressions for the probability distribution and the expectation values of various observable physical quantities are presented. Through a power-series expansion of the exact photon-number expectation value and by using properties of Stirling numbers of the second kind, it is shown that our result agrees with that of Haake and Glauber asymptotically. As by-products, some new identities involving binomial coefficients and Stirling numbers of the second kind have also been obtained. The exact solution of the same master equation for arbitrary initial conditions will be presented in a subsequent publication.

Quasi-Probability for Three-Level System and Superradiance

A quasi-probability distribution function based on the coherent state for U(3) is intro­ duced, 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 C 2 • In particular, the limit of validity of the semi­ classical 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.

Superfluorescence from magnetic insulator as a model crystal

We study incoherently-excited cooperative spontaneous emission from a linear lattice of interacting two-level atoms, to which a radiation field is very weakly coupled as compared with the atom-atom coupling. A realistic example of such a system is a group of magnetic insulators. It is found both numerically and analytically that the cooperative emission may come out if certain conditions are satisfied. Non-radiative relaxation after the incoherent excitation is shown to play a crucial role.

A Quasi-Probability Approach to the Cooperative Spontaneous Emission

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

Differential equations of spin dynamics and asymptotic expansion of quantum expectation values near the classical limit. I

The method developed in Part I is used to obtain asymptotic expansions of the functions that represent spin operators and of the quantum expectation values near the classical limit J→∞. We calculate with accuracy to the first quantum correction the correlation functions responsible for the statistical properties of cooperative spontaneous emission (super-radiance) of a gas of twolevel molecules. It is shown that the light emitted by a finite number of molecules does not have coherence of higher orders. The statistics of the field of quantum fluctuations is discussed.

Adiabatic formulae for superflourescence

A two-level atomic system giving rise to single-pulse co-operative spontaneous emission (superfluorescence or superradiance) is considered. The most general adiabatic formula, reducing the expectation values of field observables and atom-field correlations to expectation values of atomic observables, is deduced. The effects of inhomogeneous broadening and finite temperature are analysed.

A new method to treat open systems

Let Q be a quantum system composed of two mutual ly interacting parts S and R. If only the dynamics S is of interest, we are natural ly led to the problem of deducing the evolution of the reduced statistical operator ~(t) of the open system S from the Von Neumann equation (or the master equation, if Q is not isolated) which gives the evolution of the full statistical operator W(t) of Q. Usually this goal is achieved by means of the projection technique (1) with the projection of ARGYRES and KELLEr (3). Such projection is characterized by the introduction of a (~ reference state ~) for the reservoir R, which a pr ior i can be chosen arbitrarily. Introducing such projection into the usual machinery one gets a generalized master equation (GME) for Q(t), i.e. an integro-differential equation with a kernel formally expressed as a power expansion in the coupling between S and R. However, the GME is meaningful only if one can truncate this expansion at a low order. This is possible when the state of R does never deviate appreciably from the reference state. Hence such state is chosen as the equil ibrium or stat ionary state of R (typically the initial state of R) which natural ly arises from each particular problem. However there are eases in which the state of the reservoir is changed by the interaction with S in a way relevant for the dynamics of S itself. Typical examples are laser ~)scillations and, under suitable conditions, co-operative spontaneous emission (superradiance and superfluorescence). In such cases the projection technique becomes unsuitable, because one must take into account several or rather infinite terms of the power expansion of the kernel of the GME. In this paper we very briefly sketch a different method which avoid the introduction of a projection operator and allows one to go far beyond the Born approximation of the GME, bypassing resummation difficulties. The details are given elsewhere (3). Let the statistical operator W(t) of the system Q = s % R obey the equation

Conditions of availability of the Markoff approximation for the description of cooperative spontaneous emission

We show that the conditions of validity of the Markoff approximation for the description of cooperative spontaneous emission depend on the Fresnel number associated with the radiating system. Let $\frac{S}{\ensuremath{\lambda}L}$ be the Fresnel number, where $S$ and $L$ are, respectively, the section and the length of the radiating system, and $\ensuremath{\lambda}$ is the radiation wavelength. In the case of large Fresnel numbers, we find that the characteristic time ${\ensuremath{\tau}}_{R}$ of the cooperative emission of radiation must be much greater than $\frac{L}{c}$; but for small Fresnel numbers, we show that ${\ensuremath{\tau}}_{R}$ must be much greater than $\frac{S}{\ensuremath{\lambda}c}$.

A Quasi-Probability Approach to the Co-operative Spontaneous Emission

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”.

Cooperative radiation processes in two-level systems: Superfluorescence

We give a general nonperturbative treatment of cooperative emission in systems of $N$ two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to ${N}^{2}$, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length $L$ of the active volume is much larger than a suitable threshold length ${L}_{T}$ (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length $L$ is smaller or of the same order of a suitable cooperation length ${L}_{c}$ (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to $N$). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to $N$, so that superfluorescence effects disappear.

Effects of Inhomogeneous Broadening on Cooperative Spontaneous Emission of Radiation

Effects of Inhomogeneous Broadening on Cooperative Spontaneous Emission of Radiation