Study Of The Hybridand Time-domain Algorithms For The Maxwell-Schrodinger System

With rapid development of the modern electronic science and technology,the length of the electronic devices has been shrunk down to tiny-scale structures.One of the heated areas in the engineering science and computational electromagnetic is the highly integrated circuit.When dealing with nano-scale or sub-nano-scale devices,the researchers have found that the Maxwell equations had no capacity for the simulations alone,due to the quantum effects.Hence,there has been a revival of interest in coupling electromagnetic theories with quantum formulas.For electromagnetic theories,the Maxwell equations can systematically describe the change and influence of the electromagnetic fields.Most typical electromagnetic algorithms take the Maxwell equations as the springboard.And as the analytical methods of these algorithms cannot be applied to practical project design easily,the scientists cooperate the computational science with the Maxwell equations to deal with the complex simulations.This so-called computational electromagnetic includes the finite difference time domain method,the moment method,the finite element method and so on.For the quantum part,the famous Schrodinger equation is the key to the quantum mechanical system and can be discretized for computational methods.Through the Schrodinger equation,the wave function and the eigenvalues of the particle can be figured up.And for the study of nano-scale devices,the important and efficient way is to get the wave function and the eigenvalues.Then,the charge density and the current can be calculated.As a result,it is important to find a way to couple the electromagnetic system with the quantum part.The excited electron by the electromagnetic waves is a superposition of several states and becomes a local current source that can radiate new electromagnetic waves.Thus,the fields near the electron should be the sum of the incident waves and the new radiated ones and will influence the wave function in return.We note that many researchers have done precursory works and these work can be divided into two main kinds.For the Schrodinger equation under electromagnetic waves gets the scalar and vector potentials involved,the steps of the first kind are:1.Calculate the electromagnetic fields through the Maxwell equations;2.Update the scalar and vector potentials from the electromagnetic fields;3.Calculate the wave function by the Schrodinger equation under electromagnetic waves;4.Get the new current source through the wave function and the potentials,and this term will be added into the next iteration of the fields.Two extra steps should be introduced into the hybrid system to transform the electromagnetic fields of each step into the vector and scalar potentials.If the symplectic method is introduced into this system,the computational cost will be much more and the speed will be slow down.The other kind aims at simplifying the conventional Maxwell-Schrodinger system into a form without the electromagnetic potentials involved.Since the radiation gauge is introduced in this scheme,the feedback term is neglected based on the assumption that the electron motions may have much less influence than the control pulse.The program of this kind can be simple:1.Update the electromagnetic waves;2.Calculate the Schrodinger system without the electromagnetic potentials involved;3.Start the iteration of the next time step.These methods require less computer memory and have a high speed of the simulations.But with the radiation gauge and the assumption,the errors from the theories cannot be neglected in the long time simulation of the nano-scale devices.To address the problems above,the novel approaches are proposed in this paper to solve the hybrid Maxwell-Schrodinger simulations.The main research work of this paper includes:(1)A brief introduction of the computational electromagnetic algorithms is given.Taking the Maxwell equations as an example,the finite difference time domain method is studied as well as the numerical stability and the numerical dispersion.Then we analyze the influence of the different orders of the finite difference time domain method.(2)The novel potential equations are proposed since the quantization of theelectromagnetic effects can be done more expediently with the vector and scalarpotential than with electronic and magnetic field.The vector and scalar potential formulation is suitable for both quantum theory and classical electromagnetic simulation.The discretized forms of the new equations are given through the finite difference time domain method.(3)In many cases,most real-world applications are set in a seemingly infinite space.So the perfectly matched layers for the potential equations are studied with the complex axes theory and the reflect theory.The experiments of the perfectly layers show that the layers for the potential equations can absorb the incident potentials without any reflection.(4)The computational form of the Schrodinger equation is given in the paper.Then the symplectic method is introduced into the Schrodinger part,and the symplectic parameters are figured up.The accuracy of the second-order difference method,the fourth-order difference order and the symplectic method is tested through the experiments.The results show that the symplectic method is suitable for the long-time simulation of the quantum devices.(5)A high-order symplectic FDTD scheme for the Maxwell-Schrodinger system is proposed with the vector and scalar potentials involved.This simulation incorporates two parts:Maxwell equations and the Schrodinger equation.For the Maxwell part,the magnetic vector potential and electric scalar potential are employed instead of magnetic and electric fields.This approach can reduce the computational cost.For the Schrodinger part,the time-domain Schrodinger equation is used and modified to take radiation into account.The particle current term couples the two systems.The symplectic part of the scheme makes the algorithm more accurate and energy-saved.(6)The given controller in the paper focuses on the system of a single particle and studies the quantum states of the particle affected by electromagnetic incident pulses.The results shows the pulses can lead to the change of the states accurately.In the Maxwell part,the potential equations are updated to a new and simple form than our previous work.To summarize,the paper focuses on the numerical research of the hybrid Maxwell-Schrodinger simulations.We establish the new approaches for the increasing need of the multi-physics simulations.The main highlights of this paper are:(1)The novel electromagnetic potential equations are proposed in the paper as well as the perfectly matched layers theory for the equations.With these improvements,we can study the electromagnetic phenomena in the infinite regions without the conventional electromagnetic fields and the Maxwell equations involved.(2)With the symplectic method introduced,a new high order scheme for the Maxwell-Schrodinger system is studied.The novel method requires less computer memory and can give more accurate results in long time simulations.(3)A new quantum state controller based on the electromagnetic potentials is given in the paper.This new algorithm is without any assumption or gauge that may cause theoretical errors.The control current can be calculated through the algorithm and then excites the accurate change of the quantum states.