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Acta Scientiarum Naturalium Universitatis Sunyatseni

Numerical Methods for Optical Bloch Equations

Xiangyang Yu

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The classical numerical treatments of ordinary differential equations, including Eider, Heun, fourth-order Runge-Kutta and a new fourth-order predictor-corrector methods, are used to solve the optical Bloch equations. By comparing the numerical results and with analytical results, the four numerical methods, especially for the new method's errors and reliability are analyzed. The numerical results prove that, if the steps are suitable, the four numerical methods for the Bloch equations are possible convengence and can keep the numerical method's corresponding errors. The new method is reliable, can be used to optical transient coherent process, and is significant for solving the optical Bloch equations.

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The classical numerical treatments of ordinary differential equations, including Eider, Heun, fourth-order Runge-Kutta and a new fourth-order predictor-corrector methods, are used to solve the optical Bloch equations. By comparing the numerical results and with analytical results, the four numerical methods, especially for the new method's errors and reliability are analyzed. The numerical results prove that, if the steps are suitable, the four numerical methods for the Bloch equations are possible convengence and can keep the numerical method's corresponding errors. The new method is reliable, can be used to optical transient coherent process, and is significant for solving the optical Bloch equations.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations that are used to describe the optical pulse propagation and interaction with inhomogeneously broadened two-level medium.The convergence and stability of this numerical method are proved by comparing with the analytic solutions and area theorem derived under special condition.The simulations of arbitrary conditions is discussed by employing this method and analyze the pulse evolution for different detunings,relaxation times and inhomogeneously broadened linewidthes.The established numerical method provides a good way to have further study about the coherent interaction of short optical pulses with resonant mediums,and can be used to solve the corrections of Maxwell-Bloch equations.
We discuss a new way of solving the optical Maxwell–Bloch equations governing the interaction of two-level inhomogeneously broadened atoms with a laser by expanding the components of the Bloch vector in power series of the detuning. This method leads to a set of infinitely many coupled ordinary differential equations in which the detuning no longer appears explicitly. These equations are solved numerically, reproducing some analytic solutions reported previously and obtaining a variety of other shape-preserving steady-state solutions. These new solutions, however, are neither factorable nor simply periodic. In this connection we discuss the implications of the factorization assumption, which appears to be the only condition leading to closed-form analytic solutions.
We study phenomenon of electromagnetically induced absorption in the Hanle configuration by solving time-dependent optical Bloch equations for the case of the closed multilevel F g = 1 → F e = 2 transition.Our model gives optical Bloch equations as a non-homogeneous system of ordinary linear differential equations.For weak laser fields (Ω Γ i.e.Rabi frequency small compared to spontaneous emission rate), a perturbative method to solve linear differential equations can be applied.Perturbative method is realized by solving (in the time-domain) higher-order corrections to the density matrix which in the sum converge to the exact solution of optical Bloch equations.By its form, each successive correction is also system of ordinary linear differential equations which depends on the solution of previous ones.Corrections are partitioned such that odd give corrections to optical coherences, while even give corrections to populations and Zeeman coherences.We present numerical results for the behavior of density matrix elements with successive corrections, and compare them with exact solution of optical Bloch equations.Electromagnetically induced absorption is observed as a 4th and higher (even) correction to populations, when behavior in respect to both time and magnetic field is viewed.Since in our method each correction depends on the solution of previous ones, we can analyze how (through mechanism of transfer of coherences and transfer of populations between Zeeman sublevels) electromagnetically induced absorption is formed.We also discuss qualitative differences in the behavior (with respect to time) of certain density matrix elements for magnetic fields "inside" and "outside" electromagnetically induced absorption resonance.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
A focus of recent research in quantum computing has been on developing quantum algorithms for solving differential equations using variational methods on near-term quantum devices. A promising approach involves variational algorithms, which combine classical Runge-Kutta methods with quantum computations. However, a rigorous error analysis, essential for assessing real-world feasibility, has so far been lacking. In this paper, we provide an extensive analysis of error sources and determine the resource requirements needed to achieve specific target errors. In particular, we derive analytical error and resource estimates for scenarios with and without shot noise, examining shot noise in quantum measurements and truncation errors in Runge-Kutta methods. Our analysis does not take into account representation errors and hardware noise, as these are specific to the instance and the used device. We evaluate the implications of our results by applying them to two scenarios: classically solving a 1D ordinary differential equation and solving an option pricing linear partial differential equation with the variational algorithm, showing that the most resource-efficient methods are of order 4 and 2, respectively. This work provides a framework for optimizing quantum resources when applying Runge-Kutta methods, enhancing their efficiency and accuracy in both solving differential equations and simulating quantum systems.