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New Journal of Physics

Modelling electromagnetically induced transparency media using the finite-difference time-domain method

Curtis W. Neff, L. Mauritz Andersson, Min Qiu

Published Mar 2, 2007• 4 citations
Publisher

Abstract

Time-domain electromagnetic modelling of complex structures which include both non-dispersive media and media exhibiting electromagnetically induced transparency (EIT) require a general, robust calculation method such as finite-difference time-domain (FDTD). We propose a complex-valued, exact two-pole representation of the permittivity of a three-level system which is suitable for integration into the FDTD algorithm via the auxiliary differential equation method. Our calculation model confirmed reported results which were calculated with an approximate representation of the EIT permittivity. Additionally, propagation calculations which mimic slow light experiments were performed. A major advantage of our representation is the ease with which changes in the control field Rabi frequency can be implemented by using a time-dependent permittivity.

DOI
10.1088/1367-2630/9/3/048

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Time-domain electromagnetic modelling of complex structures which include both non-dispersive media and media exhibiting electromagnetically induced transparency (EIT) require a general, robust calculation method such as finite-difference time-domain (FDTD). We propose a complex-valued, exact two-pole representation of the permittivity of a three-level system which is suitable for integration into the FDTD algorithm via the auxiliary differential equation method. Our calculation model confirmed reported results which were calculated with an approximate representation of the EIT permittivity. Additionally, propagation calculations which mimic slow light experiments were performed. A major advantage of our representation is the ease with which changes in the control field Rabi frequency can be implemented by using a time-dependent permittivity.
The influence of a phase-modulated control field on the phenomenon of electromagnetically induced transparency (EIT) is investigated theoretically. We show that the phase modulation changes the dispersive properties of the medium considerably since it results in temporal oscillations of the transparency window in frequency space. This is in marked contrast to the standard EIT setup, where the transparency window is fixed and determined by the two-photon resonance condition. In particular, we find that the phase modulation enables the propagation of probe pulses with disjoint frequency spectra at different times and allows the shifting of the central frequency of a probe pulse almost without distortion of its shape. We employ the time-dependent susceptibility of the medium to explain and analyze our results and demonstrate that this concept yields qualitative as well as quantitative agreement with the numerical integration of Maxwell-Bloch equations. Our theoretical model can be applied to other media with time-dependent susceptibilities.
We employ another approach to quantize electromagnetic fields in the coordinate space, instead of the mode (or Fourier) space, such that local features of photons can be efficiently, physically, and more intuitively described. To do this, coordinate-ladder operators are defined from mode-ladder operators via the unitary transformation of systems involved in arbitrary inhomogeneous dielectric media. Then, one can expand electromagnetic field operators through the coordinate-ladder operators weighted by non-orthogonal and spatially-localized bases, which are propagators of initial quantum electromagnetic (complex-valued) field operators. Here, we call them QEM-CV-propagators. However, there are no general closed form solutions available for them. This inspires us to develop a quantum finite-difference time-domain (Q-FDTD) scheme to numerically time evolve QEM-CV-propagators. In order to check the validity of the proposed Q-FDTD scheme, we perform computer simulations to observe the Hong-Ou-Mandel effect resulting from the destructive interference of two photons in a 50/50 quantum beam splitter.
Abstract Slow light with electromagnetically induced transparency (EIT) in the core of optical fibre containing three-level atoms is investigated. The guided modes are treated in the weakly guiding approximation which renders the analysis into a manageable form. The transparency window and permittivity profile of the core due to the strong pump field in the EIT scheme is calculated. For a specific permittivity profile of the core due to EIT, the propagation constant of the weak signal field and spatial shape of fundamental guided mode are calculated by solving the vector wave equation using the finite difference method. It is found that the transparency window and slow light field can be controlled via the optical fibre parameters. The reduced group velocity of slow light in this configuration is useful for many technological applications such as optical memories, effective control of single photon fields, optical buffers and delay lines.
This technical note describes the physical model, numerical implementation, and validation of multilevel atomic media for lasers and saturable absorbers in Meep: a free/open-source finite-difference time-domain (FDTD) software package for electromagnetics simulation. Simulating multilevel media in the time domain involves coupling rate equations for the populations of electronic energy levels with Maxwell's equations via a generalization of the Maxwell--Bloch equations. We describe the underlying equations and their implementation using a second-order discretization scheme, and also demonstrate their equivalence to a quantum density-matrix model. The Meep implementation is validated using a separate FDTD density-matrix model as well as a frequency-domain solver based on steady-state ab-initio laser theory (SALT).
The Electromagnetically Induced Transparency (EIT) is associated with a Λ three-level system and the spectral position of EIT window can be changed by varying the frequency of the coupling field, however, at large detuning the EIT will evolve into a dispersion-like feature and transparency property of EIT become less obvious. In this paper, it is shown that we can perform EIT frequency tuning by a radio-frequency (rf) field. In the cascade quasic-Λ four-level system, the absorption profile of probe field is calculated by solving the equations of motion of the density matrix. It is shown that the Autler-Townes doublet originates from the rf-field induced dynamic Stark effect and the spectral position of EIT window is determined by the frequency detuning of the coupling field. When the frequency detuning of the coupling field is half of the rf Rabi frequency, the EIT feature remain its absorptive profile. The frequency tuning rang of EIT is determined by the rf Rabi frequency, and can be explained using a dressed-state analysis. Therefore, frequency tuning range of EIT can be controlled by the rf Rabi frequency.
We developed the Adomian's decomposition method to work for the electromagnetically induced transparency (EIT) problem. The method is general and capable to solve the coupled nonlinear partial differential equations for a light pulse passing through a three-level -type coherent medium. This EIT system is described by the coupled Maxwell-Schrödinger equations and optical Bloch equations. In the weak probe field case, the results agree with perturbation solutions and experimental data. In the stronger probe field case while perturbation may fail, our results reproduce experimental data well. With the techniques of spatial and time partitions, we extend the decomposition method that will be versatile for the investigation of the light pulse propagating through a coherent atomic medium.