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Journal of the Physical Society of Japan

Coherent States and Envelope Kinklike Exciton-Photon Coupled Modes in Nonlinear Frenkel Excitons Interacting with Radiation Fields

Shôzô Takeno

Published Aug 15, 1993• 3 citations
Publisher

Abstract

Interactions of nonlinear Frenkel excitons with an intense external radiation field (rf) are studied by taking the atomic polarization, the atomic population difference and the electric field as a set of variables to study the coherence and nonlinearity properties of the exciton-rf system. To equations for these quantities, terms are added phenomenologicaly to take care of pumping and dissipative effects. Two kinds of solutions are obtained under certain conditions, one representing the existence of the stationary coherent polarization and the macroscopic number of photons with wave number equal to that of the excitons and another envelope-kink-like, steady exciton-rf coupled modes around the stationary coherent states

DOI
10.1143/jpsj.62.2894

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Interactions of nonlinear Frenkel excitons with an intense external radiation field (rf) are studied by taking the atomic polarization, the atomic population difference and the electric field as a set of variables to study the coherence and nonlinearity properties of the exciton-rf system. To equations for these quantities, terms are added phenomenologicaly to take care of pumping and dissipative effects. Two kinds of solutions are obtained under certain conditions, one representing the existence of the stationary coherent polarization and the macroscopic number of photons with wave number equal to that of the excitons and another envelope-kink-like, steady exciton-rf coupled modes around the stationary coherent states
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.
A quantum theory of nonlinear excitations in a matter-photon system is developed by generalizing the conventional two-level-atom model of a matter in several directions, on the assumption that the overlapping of atomic wave functions on different atomic sites is negligible.The model matter Hamiltonian is shown to be expressible in a generalized form' of the Heisenberg model Hamiltonian with anisotropic long-range exchange interactions with spin operators replaced by quasi-Pauli operators.From approxirpate formal solutions of a set of nonlinear equations satisfied by the quasi-Pauli operators ~nd photon operators, brief discussion is given on nonlinear Frenkel excitons and nonlinear exciton-photon coupled modes.An attempt is made to formulate a theory of higher-order nonlinear physical properties of the matter-photon system by including interactions of magnetic as •well as electric dipoles of atoms with photons.A brief discussion is also given on the coherence properties. of the matter-photon system which are characterized by the simultaneo~s existence of •a photon coherent state and an atomic-polarization coherent state.*> Throughout this paper, we will often use a simplified .notation l, rather than l 1, whenever appropriate, to avoid the use of complicated symbols.Also, the complex conjugate of a quantity• A is denoted by A*.
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.
Nonlinear excitations in the Ising model in a transverse field or an equivalent Frenkel-exciton model for a system of two-level atoms are studied by using a method of diagonal coherent-state representation. It is shown that if the ratio of the Ising interaction to a transverse-field strength exceeds a certain critical value, stationary coherent states giving the ground state and excited states of the model Hamiltonian are characterized by two degenerate structures resulting from spatially uniform coherent states and static domain-wall type excitations, respectively. Particular solutions for the time evolution of spin operators corresponding to atomic polarization are then studied. It is shown that moving domain-wall solutions or moving solitary-wave solutions can exist under certain conditions.
A model is presented for a system of N two-level excitons interacting with each other via optical near fields represented as localized photons. In a low exciton density limit, quantum dynamics of the dipole moments or quantum coherence between any two energy levels is linear. As the exciton density becomes higher, the dynamics becomes nonlinear, and the system has several kinds of quasi-steady states of the dipole distribution depending on the system parameters. These quasi-steady states are classified with the help of the effective Hamiltonian that is derived from the renormalization of degrees of freedom of localized photons with a unitary transformation. Among them there exist a “ferromagnetic” state (dipole-ordered state), in which all electric dipoles are aligned in the same direction, and an “anti-ferromagnetic” state, where all dipoles alternatingly change the direction. In addition, we show that an arbitrary state can be transformed into a dipole-ordered state by manipulating initial values of the population differences appropriately. For example, if we initially prepare a dipole-forbidden state, which is similar to the “anti-ferromagnetic” state and cannot be coupled with propagating far fields, and if we manipulate the distribution of the population differences properly, the initial state evolves into a dipole-ordered state. The radiation property of such dipole-ordered states is examined in detail. Neglecting energy dissipation by radiation, we find that some of the ordered states show strong radiation equivalent to Dicke's superradiance. Then by introducing a radiation reservoir, the dissipative master equation is derived. Solving the equation with and without quantum correlations, we numerically show that multiple peaks in the radiation profile can survive in both cases. The mechanism of this phenomenon is discussed, and a brief comment on an application to photonic devices on a nanometer scale is given.