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Journal of Modern Optics

Exact solution to laser rate equations: three-level laser as a Morse-like oscillator

Roberto de J. León‐Montiel, H. M. Moya-Cessa

Published Feb 6, 2016• 2 citations
Publisher

Abstract

It is shown how the rate equations that model a three-level laser can be cast into a single second-order differential equation, whose form describes a time-dependent harmonic oscillator. Using this result, we demonstrate that the resulting equation can be identified as a Schrödinger equation for a Morse-like potential, thus allowing us to derive exact closed-form expressions for the dynamics of the number of photons inside the laser cavity, as well as the atomic population inversion.

DOI
10.1080/09500340.2016.1144824

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Piela replies: The Morse oscillator is a single point mass subject to the Morse original potential cited in Ilya Kaplan’s book, equation 5.22. Contrary to what Donald Truhlar writes, the Morse oscillator does not represent two point masses with a spring, not to mention a diatomic molecule. Therefore, Kaplan’s equation 5.23 is an exact solution of the Schrödinger equation for the Morse oscillator. The same solution is, of course, an approximate one for the Schrödinger equation for two point masses with a Morse-like spring or any real diatomic molecule.Truhlar could literally repeat his arguments for the harmonic oscillator, instead of the Morse one. His conclusion in such a case would mean that the widely known solution to the Schrödinger equation for the harmonic oscillator is not exact. 1 1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. Google Scholar© 2008 American Institute of Physics.

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Physical Review•Sep 9, 1966•47 citations
It is shown how the rate equations that model a three-level laser can be cast into a single second-order differential equation, whose form describes a time-dependent harmonic oscillator. Using this result, we demonstrate that the resulting equation can be identified as a Schrödinger equation for a Morse-like potential, thus allowing us to derive exact closed-form expressions for the dynamics of the number of photons inside the laser cavity, as well as the atomic population inversion.
Piela replies: The Morse oscillator is a single point mass subject to the Morse original potential cited in Ilya Kaplan’s book, equation 5.22. Contrary to what Donald Truhlar writes, the Morse oscillator does not represent two point masses with a spring, not to mention a diatomic molecule. Therefore, Kaplan’s equation 5.23 is an exact solution of the Schrödinger equation for the Morse oscillator. The same solution is, of course, an approximate one for the Schrödinger equation for two point masses with a Morse-like spring or any real diatomic molecule.Truhlar could literally repeat his arguments for the harmonic oscillator, instead of the Morse one. His conclusion in such a case would mean that the widely known solution to the Schrödinger equation for the harmonic oscillator is not exact. 1 1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. Google Scholar© 2008 American Institute of Physics.
Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.