Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems

📝 Abstract

A numerical model based on the finite-difference time-domain method is developed to simulate fluctuations which accompany the dephasing of atomic polarization and the decay of excited state’s population. This model is based on the Maxwell-Bloch equations with c-number stochastic noise terms. We successfully apply our method to a numerical simulation of the atomic superfluorescence process. This method opens the door to further studies of the effects of stochastic noise on light-matter interaction and transient processes in complex systems without prior knowledge of modes.

💡 Analysis

A numerical model based on the finite-difference time-domain method is developed to simulate fluctuations which accompany the dephasing of atomic polarization and the decay of excited state’s population. This model is based on the Maxwell-Bloch equations with c-number stochastic noise terms. We successfully apply our method to a numerical simulation of the atomic superfluorescence process. This method opens the door to further studies of the effects of stochastic noise on light-matter interaction and transient processes in complex systems without prior knowledge of modes.

📄 Content

arXiv:0901.0430v2 [physics.optics] 25 May 2009 1 Finite-difference time-domain formulation of stochastic noise in macroscopic atomic systems Jonathan Andreasen, Student Member, IEEE, Hui Cao, Fellow, OSA Abstract—A numerical model based on the finite-difference time-domain method is developed to simulate fluctuations which accompany the dephasing of atomic polarization and the decay of excited state’s population. This model is based on the Maxwell- Bloch equations with c-number stochastic noise terms. We successfully apply our method to a numerical simulation of the atomic superfluorescence process. This method opens the door to further studies of the effects of stochastic noise on light-matter interaction and transient processes in complex systems without prior knowledge of modes. Index Terms—Noise, spontaneous emission, stochastic pro- cesses, FDTD methods, Maxwell equations I. INTRODUCTION T HE finite-difference time-domain (FDTD) method [1] has been extensively used in solving Maxwell’s equations for dynamic electromagnetic (EM) fields. The incorporation of auxiliary differential equations, such as the rate equations for atomic populations [2] and the Bloch equations for the density of states of atoms [3], has lead to comprehensive studies of light-matter interaction. Although the FDTD method has become a powerful tool in computational electrodynamics, it has been applied mostly to classical or semiclassical problems without noise. Noise plays an important role in light-matter interaction. Marcuse solved the rate equations for light intensity and elec- tron population including noise terms [4] to illustrate the effect of noise on lasing mode dynamics [5]. Gray and Roy extended the formulation by adding noise to the field equation in order to study the laser line shape [6]. Starting from a microscopic Hamiltonian, Kira et al. developed a semiconductor theory including spontaneous emission to describe semiconductor lasers [7]. While considerable progress has been made, these models remain in the modal picture. Knowledge of mode prop- erties is required to characterize the noise, making it difficult to study complex systems in which the mode information is unknown a priori. Without invoking the modal picture, Hofmann and Hess obtained the quantum Maxwell-Bloch equations including spatiotemporal fluctuations [8]. Although it was useful to study spatial and temporal coherence in diode lasers, this formalism was based on the assumption that the temporal fluctuations of carrier density and photon density were statistically independent, which often broke down above the lasing threshold. A FDTD simulation of microcavity lasers including quantum fluctuations was also done recently [9]. This simplified model added white Gaussian noise as a source The authors are with the Department of Applied Physics, Yale University, New Haven, CT 06520 USA and J. Andreasen is also with the the Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208 USA (email: hui.cao@yale.edu; j-andreasen@northwestern.edu). to the electric field. The noise amplitude depended only on the excited state’s lifetime. The dephasing process, which was much faster than the excited state’s population decay, should have induced more noise but was neglected. Our goal is to develop a FDTD-based numerical method to simulate fluctuations in macroscopic systems caused by interactions of atoms and photons with reservoirs (heatbaths). Such interactions induce temporal decay of photon number, atomic polarization and excited state’s population, which can be described phenomenologically by decay constants. The fluctuation-dissipation theorem demands temporal fluctuations or noise to accompany these decays. We intend to incorporate such noise in a way compatible with the FDTD method, that allows one to study the light-matter interaction in complex systems without prior knowledge of modes. In a previous work [10], we included noise caused by the interaction of light field with external reservoir in an open system. In this paper, we develop a numerical model to simulate noise caused by the interaction of atoms with reservoirs such as lattice vibrations and atomic collisions. As an example, we apply the method to a numerical simulation of superfluorescence in a macroscopic system where the dominant noise is from the atoms rather than the light field. We start with the Bloch equations for two-level atoms in one dimension (1D) where the direction of light propagation is along the x-axis.   ˙ρ1 ˙ρ2 ˙ρ3  =   0 ω0 0 −ω0 0 2ΩR 0 −2ΩR 0     ρ1 ρ2 ρ3   −   1/T2 0 0 0 1/T2 0 0 0 1/T1 + Pr     ρ1 ρ2 ρ3 −ρ(s) 3  , (1) where ΩR ≡γEz/¯h is the Rabi frequency, ω0 the atomic tran- sition frequency, Ez the electric field which is parallel to the z-axis, γ the dipole coupling term. Phenomenological decay times due to decoherence T2 and the excited state’s lifetime T1 (which includes spontaneous emission and non-radiative recombination) are appended. In the

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