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I have a question about the upper laser level (the metastable level) in a 3-level laser system. I will call the ground level of the 3-level laser system by "g" and the metastable level by "m".

The metastable level m has a long life time, meaning that it decays only slowly via spontaneous emission. I understand that this is because of the angular momentum selection rules which forbid transitions between the metastable level m and the ground level g through radiation. If a transition is possible between two energy levels, the emitted photon carries away an amount of angular momentum, "which for the photon must be at least 1, since it is a vector particle" (quote from Wikipedia/selection_rules). So I assume that in a forbidden transition (like the one from m to g), the difference in angular momentum between the states m and g does not match the amount of angular momentum that can be carried away by a photon.

Metastable states are necessary in laser systems in order to achieve population inversion between levels m and g. If we pump the 3-level system, the electrons will get excited and eventually accumulate in the state m and will not decay to the state g via spontaneous emission (because m is metastable). However, by an incoming photon of the right frequency v, stimulated emission will happen, taking an electron from m to g, while releasing a second photon with frequency v. This process then cascades through the system, de-exciting the electrons from m to g, and producing the desired coherent optical amplification effect.

My question is: If the metastable state forbids the spontaneous emission because of the selection rules of angular momentum, then why is a de-excitation via stimulated emission possible? Wouldn't this imply that the second (emitted) photon would have to carry an angular momentum forbidden by the selection rules?

I would be glad for an explanation. Or, if there is a logical mistake in my reasoning, I would be glad to find out where.

Thanks a lot in advance, A.F.

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Good question indeed. I will rather join your question, than answer it in this post. There is only one remark I would like to make. Are you sure spontaneous emission is impossible from a metastable state? From what I have met, metastable states have there own lifetimes, after which radiation occurs. Like, for Neon it is approximately 13 s. –  Peyruz Oct 23 '13 at 22:16
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My question is: If the metastable state forbids the spontaneous emission because of the selection rules of angular momentum, then why is a de-excitation via stimulated emission possible? Wouldn't this imply that the second (emitted) photon would have to carry an angular momentum forbidden by the selection rules?

The quantum mechanical metastable levels are not metastable because of absolute conservation rules, like angular momentum conservation,but of selection rules. They are metastable because of the probabilities of overlapping the orbital of the ground state given by the transition moment integral.

This depends on the functional form of the solution and the symmetries displayed by them, and will have a lifetime that can be much longer than the typical relaxation lifetime of excites states. That is how we have luminescence, with different time constants depending on the material. The radiation though is incoherent.

The lasing technique is to use the possibility of coherence between emitted photons with carefully chosen materials that do have metastable states and the geometry of the container. The existence of metastability in the material allows to accumulate electrons in higher orbitals.

The photons that arrive to stimulate the emission change the probability functions, that define the orbitals, by their interaction with the electron at that energy level. This will destroy the metastability by changing the symmetry of the transitions moment integral and change the probabilities in favor of the usual de-excitation life times. So it is not angular momentum conservation that defines how the system evolves but the solutions of the quantum mechanical equations for the system.

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