The intensity of radiation, stimulated by laser pulses of arbitrary shape, is derived up to fourth-order terms. It is shown that there is a linear fourth-order mechanism of emission that depends on linear susceptibilities of molecules. This linear mechanism reveals collective, super-radiant features. The intensity of laser-driven spontaneous emission (LDSE) is proportional to for samples smaller than , where is a wavelength of emitted radiation, and N is the number of molecules. For extended volumes , the intensity of LDSE is proportional to . Equations of motion describing collective radiation damping are derived.
The intensity of radiation stimulated by laser pulses of arbitrary shape is derived up to the fourth order terms. It is shown that there is a linear fourth order mechanism of emission that depends on linear susceptibilities of molecules. This linear mechanism reveals collective, superradiant features. The intensity of laser-driven spontaneous emission (LDSE) is proportional to N2 for samples smaller than (lambda) 3, where (lambda) is a wavelength of emitted radiation, and N is the number of molecules. For extended volumes >> (lambda) 3, the intensity of LDSE is proportional to N4/3. Equations of motion describing collective radiation damping are derived.
By considering a radiating gas as a single quantum-mechanical system, energy levels corresponding to certain correlations between individual molecules are described. Spontaneous emission of radiation in a transition between two such levels leads to the emission of coherent radiation. The discussion is limited first to a gas of dimension small compared with a wavelength. Spontaneous radiation rates and natural line breadths are calculated. For a gas of large extent the effect of photon recoil momentum on coherence is calculated. The effect of a radiation pulse in exciting "super-radiant" states is discussed. The angular correlation between successive photons spontaneously emitted by a gas initially in thermal equilibrium is calculated.
In order to study the cooperative emission of radiation from an extended system of
N two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled
j and
k via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the
z axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the
z dependence predominates, in the opposite case the (
x,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the
z dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the
z axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).
We present a numerical study of the spatial coherence of light that is radiated from a dilute ensemble of atoms. The spatial coherence is established as a result of the collective (cooperative) coupling of the atoms to the light and is qualitatively different from the coherence of a laser. Specifically, the coherence in collective spontaneous emission does not rely on population inversion and stimulated emission, is governed by antiphasing of the dipoles (subradiance), and the key figure of merit for the observed coherence is $\ensuremath{\approx}N/(L/\ensuremath{\lambda})$. Here,
N is the number of atoms in the ensemble,
L is the size of the sample, and $\ensuremath{\lambda}$ is the wavelength of the emitted light.
Spontaneous decay is a phenomenon that has been described for the first time by Dirac, Wigner
and Weisskopf, at the early ages of the development of quantum electrodynamics. The theory is
applicable to a multi-body system only when atoms don’t interact during their decay. Dicke introduces
in his seminal paper the case where this assumption doesn’t hold.
The results he obtains not only tell that the system emits pulses, but the pulse duration now depends
on 1/N, where N is the number of participating atoms to the process. The pulses temporal profile
is described by a squared hyperbolic secant function. The resulting emission intensity scales as N^2.
The atomic system radiates more intensively than the ordinary fluorescent emission, this is the reason
why the process is called superradiance, or by a generalization to extended systems, superfluorescence.
The superradiant emission has preferential escape directions, depending on the atomic sample shape.
Superradiance is a collective process and it takes place when the group of excited atoms is left unper-
turbed during the formation of of the macroscopic dipole. A fundamental difference compared to the
spontaneous and stimulated emission processes is that all atoms act together as a single entity and it
isn’t possible to identify single atoms involved.
This remarkable behaviour is manifest when the atomic coherence of the system, while influenced by
environmental interactions that cause dephasing in atoms, persists for a sufficient interval of time.
Typical dephasing mechanisms, such as collisions in gases or lattice phonon interactions in crystals,
make its experimental observation difficult. A possible solution to this problem is the use of optical
crystals doped with rare-earths that are characterized by long and narrow levels lifetimes.
We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix
R to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where
N is the number of two-level systems per unit volume,
r0 is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.