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arXiv:1905.10105 · Optics Express

Simulation technique of quantum optical emission process from multiple two-level atoms based on classical numerical method

Hideaki Taniyama, Hisashi Sumikura, Masaya Notomi

Published Apr 16, 2019• 2 citations
arXivHTMLPDF

Abstract

In this paper, we report a numerical method for analyzing optical radiation from a two-level atom. The proposed method can consistently consider the optical emission and absorption process of an atom and also the interaction between atoms through their interaction with a radiation field. The numerical model is based on a damping oscillator description of a dipole current, which is a classical model of atomic transition and is implemented with a finite-difference time-domain method. Using the method, we successfully simulate the spontaneous emission phenomena in a vacuum, where the interaction between an atom and a radiated field plays an important role. We also simulate the radiation from an atom embedded in a photonic crystal (PhC) cavity. As a result, an atom-cavity field interaction is sucessfuly incorporated in the simulation, and the enhancement of the optical emission rate of an excited atom is explained. The method considers the effect of the interaction between atoms through the radiated field. We simulate the optical emission process of the multiple atoms and show that an enhancement of the emission rate can occur owing to an atom-atom interaction (superradiance) (R. H. Dicke, Phys. Rev. 93, 99 [1954]). We also show that the emission rate is suppressed by the effect of the destructive dipole-dipole interaction under an out-of-phase excitation condition (subradiance).

DOI
10.1364/oe.27.012070

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In this paper, we report a numerical method for analyzing optical radiation from a two-level atom. The proposed method can consistently consider the optical emission and absorption process of an atom and also the interaction between atoms through their interaction with a radiation field. The numerical model is based on a damping oscillator description of a dipole current, which is a classical model of atomic transition and is implemented with a finite-difference time-domain method. Using the method, we successfully simulate the spontaneous emission phenomena in a vacuum, where the interaction between an atom and a radiated field plays an important role. We also simulate the radiation from an atom embedded in a photonic crystal (PhC) cavity. As a result, an atom-cavity field interaction is sucessfuly incorporated in the simulation, and the enhancement of the optical emission rate of an excited atom is explained. The method considers the effect of the interaction between atoms through the radiated field. We simulate the optical emission process of the multiple atoms and show that an enhancement of the emission rate can occur owing to an atom-atom interaction (superradiance) (R. H. Dicke, Phys. Rev. 93, 99 [1954]). We also show that the emission rate is suppressed by the effect of the destructive dipole-dipole interaction under an out-of-phase excitation condition (subradiance).
In this paper, we report the numerical simulation of an atom-cavity interaction within photonic crystal nano-cavities. The numerical model is based on a damping oscillator description of a dipole current and it is implemented with a finite-difference time-domain method. Using the method, we successfully simulate the atom-cavity mode field interactions of a two-level system embedded in a photonic crystal cavity under several coupling strength conditions. We show that enhancement and suppression of optical emission rate from a two-level system are also shown by this model.
Spontaneous emission from two two-level atoms within an anisotropic photonic crystal is investigated when direct dipole-dipole interaction is taken into account. The radiation properties of the present two atoms system are similar to that of a V-type three-level atom embedded in an anisotropic photonic crystal. The direct dipole-dipole interaction can enhance or depress the spontaneous emission depending on the relative position of the upper levels from the band edge of the photonic crystals. Owing to quantum interference and the dipole-dipole interaction, the spontaneous emission spectra are complicated non-Lorentzian shapes and depend strongly on the relative position of the upper levels from the bandgap and the initial state of the system.
We investigate spontaneous photon emission processes of two-level atoms in parabolic and ellipsoidal cavities thereby taking into account the full multimode scenario. In particular, we calculate the excitation probabilities of the atoms and the energy density of the resulting few-photon electromagnetic radiation field by using semiclassical methods for the description of the multimode scenario. Based on this approach photon path representations are developed for relevant transition probability amplitudes which are valid in the optical frequency regime where the dipole and the rotating-wave approximations apply. Comparisons with numerical results demonstrate the quality of these semiclassical results even in cases in which the wave length of a spontaneously emitted photon becomes comparable or even larger than characteristic length scales of the cavity. This is the dynamical regime in which diffraction effects become important so that geometric optical considerations are typically not applicable.
The dynamic evolution of atomic spontaneous emission in one-dimensional photonic crystals (1DPC) is investigated by using mode functions for the 1DPC. Our attention is focused on the non-Markovian processes in the first several optical cycles of the spontaneous emission decay. The integral-differential dynamic equation is directly solved by numerical calculation without using the Markovian approximation. The interference of the multireflected fields plays an important role in the non-Markovian processes. Due to the interference of the fields successive arrived at the atom due to the reflection at the interfaces, the decay rate shows a series of pulselike peaks. If the interference is constructive, the spontaneous decay is enhanced, while if the interference is destructive, the spontaneous decay is suppressed. The steady decay rate is approximately equal to the vacuum decay rate, when the atomic transition frequency corresponds to an allowed normal mode of the 1DPC. The spontaneous emission decay is suppressed if the atomic transition frequency is within the forbidden gap of the 1DPC.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.