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Physical review. B, Condensed matter

Quantum statistical theory of fluorescence of low density Frenkel excitons in a crystal slab

Chang-qi Cao, Hui W. Cao, Yi-xi Liu

Published Dec 15, 2000• 4 citations
Publisher

Abstract

The fluorescence of Frenkel excitons in low density regime is studied without the aid of rotating wave approximation and Markov approximation. The evolution of the emitted field is derived in terms of its initial conditions. It is found that the usual interaction Hamiltonian of $(e/mc)\mathbf{P}\ensuremath{\cdot}\mathbf{A}$ type leads to unreasonable characteristic equation for decay rates. Only when the term ${(e}^{2}{/2\mathrm{mc}}^{2}){\mathbf{A}}^{2}$ is added to the $(e/mc)\mathbf{P}\ensuremath{\cdot}\mathbf{A},$ the result becomes reasonable. The case of single lattice layer is studied in detail. Different features of statistical properties of the superfluorescence are shown as compared with that of atom aggregate. Double and triple lattice-layer cases are also studied to show the effect of coupling between the excitons of different wave vectors.

DOI
10.1103/physrevb.62.16453

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Physical Review A•Mar 1, 1979•253 citations
The fluorescence of Frenkel excitons in low density regime is studied without the aid of rotating wave approximation and Markov approximation. The evolution of the emitted field is derived in terms of its initial conditions. It is found that the usual interaction Hamiltonian of (e/mc)P\ensuremath⋅A(e/mc)\mathbf{P}\ensuremath{\cdot}\mathbf{A}(e/mc)P\ensuremath⋅A type leads to unreasonable characteristic equation for decay rates. Only when the term (e2/2mc2)A2{(e}^{2}{/2\mathrm{mc}}^{2}){\mathbf{A}}^{2}(e2/2mc2)A2 is added to the (e/mc)P\ensuremath⋅A,(e/mc)\mathbf{P}\ensuremath{\cdot}\mathbf{A},(e/mc)P\ensuremath⋅A, the result becomes reasonable. The case of single lattice layer is studied in detail. Different features of statistical properties of the superfluorescence are shown as compared with that of atom aggregate. Double and triple lattice-layer cases are also studied to show the effect of coupling between the excitons of different wave vectors.
We systematically study the fluorescence of low-density Frenkel excitons in a crystal slab containing NT{N}_{T}NT​ V-type three-level atoms. Based on symmetric quasispin realization of SU(3) in large N limit, the two-mode exciton operators are invoked to depict various collective excitations of the collection of these V-type atoms starting from their ground state. By making use of the rotating-wave approximation, the light intensity of radiation for the single-lattice layer is investigated in detail. As a quantum coherence effect, the quantum beat phenomenon is discussed in detail for different initial excitonic states. We also test the above results analytically without the consideration of the rotating-wave approximation and the self-interaction of radiance field is also included.
A formal, regorous theory of Frenkel excitons interacting with an external electromagnetic field is doveloped within the framework of a two-level-atom model in which the dipole approximation is employed in describing the atom-field (taken as a c-number) interaction. The model exciton Hamiltonian is shown to be identical in form with the Heisenberg model Hamiltonian for S=1/2 spins with anisotropic long-range exchange interactions. Instead of treating excitons as a non-ideal Bose gas, atomic polarization operators and level population defference are taken up as two fundamental quantities to describe Frenkel excitons. A pair of equations satisfied by these two quantities are derived. It is shown that Frenkel excitons are generally expressed as quantum-mechanical nonlinear polarization waves, the nonlinearity being characterized by the level population. A discussion is given on the interrelationship between such polarization waves and those derivable from the classical Drude-Lorentz model. With the aid of several approximation procedures similar to those employed in nonlinear optics, a set of equations describing nonlinear exciton-photon coupled modes are derived. Brief discussion is given on the eigenfrequencies of nonlinear Frenkel excitons and nonlinear polaritons. The existence of nonlinear exciton-photon coupled modes whose properties are different from those of polariton modes is suggested.
Dynamical and statistical behaviors of Frenkel excitons in a matter system consisting of N two-level-atoms and interacting with an external field are studied with the molecular field and random phase approximations. Existence of a phase transition is suggested both in the exciton system and exciton-photon coupled system. This phase transition is the second order transition whose order parameter is the polarizability. A new type of condensation of the photon-exciton coupled wave, in which a macroscopic number of excitons and photons participate, has been found as a result of appreciable contribution of photons to the phase transition. When the incident light is coherent, the condition of phase transition is modified by photon induced exciton-exciton interaction. The energy dispersion of elementary excitations is derived.
Superradiance of high density Frenkel excitons in an RRR-phycoerythrin single crystal is observed at room temperature for the first time. No fluorescence is observed except the emission at the sharp exciton band when the superradiance of excitons occurs, and the higher the pump density, the sharper the emission bandwidth. A redshift and a blueshift are observed at the rise time and the fall time of the emission pulse, respectively. The experimental results also imply deformed-boson properties of high density Frenkel excitons.
For a pencil-shaped extended medium with Fresnel number equal to 1, we have quantum mechanically derived a description of the initiation of superfluorescence in terms of Maxwell-Bloch equations with a fluctuating source due to the zero-point fluctuations of the vacuum field. By the introduction of classical behavior, these equations are extended to include nonlinear behavior due to decreasing atomic inversion. The principal assumption in the derivation is that the main features of superfluorescence are governed by the interaction of atoms with field modes inside two small solid angles around the pencil axis. The delay ${\ensuremath{\tau}}_{D}$, defined as the time at which the mean-squared tipping angle of the collective Bloch vector attains the value 1, turns out to be given by ${\ensuremath{\tau}}_{D}=(\frac{{\ensuremath{\tau}}_{R}}{4}){[\mathrm{ln}{(2\ensuremath{\pi}N)}^{\frac{1}{2}}]}^{2}$, where ${\ensuremath{\tau}}_{R}$ is the radiation time for collective decay and NNN is the number of atoms. The corresponding effective initial tipping angle roughly equals 2(N)12\frac{2}{{(N)}^{\frac{1}{2}}}(N)21​2​. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}$ of the delay of the superfluorescence pulse satisfies approximately $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}=\frac{2.3}{\mathrm{ln}N}$. A brief comparison with previous treatments of superfluorescence is given.