# Stimulated Emission in quantum mechanics

As I explained in the last post, about superradiance, we are gearing up to discuss Peter Hagelstein’s spin-boson model” theory of cold fusion. The theory relies very heavily on two effects that make it easier to transfer energy to an oscillation mode: Dicke superradiance and stimulated emission. This post goes over how stimulated emission works in quantum mechanics. It’s much simpler than the last post, don’t worry. As before, experts can skip this post.

What is stimulated emission?

Let’s say we have a certain photon mode (imagine a “standing wave” where light is bouncing back and forth between two mirrors). The frequency of the mode is ω, and the energy of each photon in the mode is $E=\hbar \omega$.

(I’m using a photon mode as an example, but the math works the same for other kinds of “oscillations” like phonons or plasmons.)

Now let’s say there is a process that can add a new photon into this mode. For example, imagine that there is an atom in an excited state with excitation energy E, which can relax to its ground state while emitting a photon into this mode. Stimulated emission says that when there are some photons that are already in the mode, any process that adds yet another photon to the mode becomes faster and more likely. The more photons already present, the easier it is to add another. The photons are extroverts I guess!

Fun facts about stimulated emission: (A) One of Albert Einstein’s many claims to fame is the prediction of stimulated emission in a 1917 paper. (B) Stimulated emission is how lasers work! In fact, “LASER” is an acronym, and the third and forth letters stand for “stimulated emission”. The whole acronym is “Learning About Stimulated Emission is Really cool.” Or something like that.

How to describe stimulated emission in quantum mechanics

As above, we pick out a particular photon mode. This photon mode can have many different quantum states. There can be 0 photons in the mode: We call this state $|0\rangle$. Or, there can be 1 photon in the mode: We call this state $|1\rangle$. The other states are $|2\rangle$, $|3\rangle$, etc. (The mode can also be in a quantum superposition of two or more of these states. Then it would be uncertain how many photons are in the mode.)

Now, there is an “annihilation operator” a which annihilates a photon and a “creation operator” $a^\dagger$ which creates a photon. These operators are what show up in the equations describing light-matter interaction. Here’s everything you need to know about absorption and emission:

$a |n \rangle = \sqrt{n} |n-1\rangle, \quad a^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle$

(You may remember a pair of equations like this from the quantum harmonic oscillator that you studied in introductory quantum mechanics. It’s no coincidence, the math is basically all the same.)

If you square the coefficient in front of the ket, you get a number telling you how fast this process occurs (more or less). That means:

• If a ground-state atom is absorbing photons (the a operator is acting), the speed that it absorbs a photon is proportional to n, the number of photons available to be absorbed. No surprise, it’s just what you would expect.
• If there is an excited atom which can emit photons (the $a^\dagger$ operator is acting), and there are n photons present, then the speed that it emits a photon is proportional to n+1. If we split up the expression “n+1″…
• The “+1” part, which ensures that emission happens even when n=0, is called spontaneous emission.
• The “n” part, which ensures that the process occurs faster and faster as you increase the number of photons, is stimulated emission.