Qualitative description of stimulated emission & population inversion

[rtareen]

TL;DR Summary

Attached is section 39.4 of the Sears and Zemansky "University Physics" 14th edition. I'm self studying and want to understand what they are attempting to explain.

I'm having trouble understanding stimulated emission and population inversion, and how they work together to make a laser work. I pretty much need this explained completely.

1. Spontaneous emission, they say, is when an atom absorbs and then later emits a photon. Isn't that just regular emission?

2. For stimulated emission they say that the atom is already excited when the photon is absorbed, so it emits two photons? Is that right? I don't understand exactly what the book is saying. And these two photons are in phase so the amplitude is doubled, right?

3. Next they say we have to distinguish between energy levels and states, but they never go on to explain what that is. They just say there aren't enough atoms in an excited state in a gas for stimulated emission to occur.

4. Next, what about stimulated emission guarantees that all the photons will be emitted in the same direction?

5. Now for population inversion. They say that we need to stimulate a majority of atoms into the metastable state, which they also say is the state where no photons can be emitted. Why then, if you can't emit a photon from it? They say E_2 is the metastable state in this case.

6. Is it the energy level that prevents the photon in the metastable state from being emitted, or is it about the state (how it got to that energy level)? Again, they say we have to distinguish, but then don't go on to distinguish it themseelves.

7. They completely lost me when they started talking about the process itself.

Please keep in mind not to go out of scope and to keep your explanations within the range of what the book attempts to explain. I don't want to get confused or overloaded with info I won't need for the course.

 

[PeterDonis]

1. Spontaneous emission, they say, is when an atom absorbs and then later emits a photon. Isn't that just regular emission?

"Regular emission" is not a standard technical term in this area of physics. That's why your reference isn't using it. It's using the standard technical terms for the two possible types of emission from an individual atom: spontaneous emission and stimulated emission. Spontaneous emission means there are no other photons present and the atom just emits at a random time; stimulated emission means there are other photons present and the atom emits a photon into the same state as the photons that are already present.

2. For stimulated emission they say that the atom is already excited when the photon is absorbed, so it emits two photons?

No. An atom in the ground state would not emit photons at all, either spontaneous or stimulated. So the atom has to be already excited when spontaneous emission occurs, just like stimulated emission. When the atom absorbs a photon, as in your description of spontaneous emission above, it transitions to an excited state. See part (a) of Fig. 39.28 in your attachment.

what about stimulated emission guarantees that all the photons will be emitted in the same direction?

The fact that stimulated emission means the photon emitted by the atom must be in the same state as the other photons that are already present (there have to be photons already present for stimulated emission to occur).

They say that we need to stimulate a majority of atoms into the metastable state, which they also say is the state where no photons can be emitted.

It's not that no photons can be emitted from the metastable state; it's just that that state has a much lower probability per unit time to spontaneously emit a photon than the state the book is calling the "excited" states (E1 and E3). That means the metastable state E2 has a much longer lifetime than the E1 and E3 states.

 

[rtareen]

No. An atom in the ground state would not emit photons at all, either spontaneous or stimulated. So the atom has to be already excited when spontaneous emission occurs, just like stimulated emission. When the atom absorbs a photon, as in your description of spontaneous emission above, it transitions to an excited state. See part (a) of Fig. 39.28 in your attachment.

As far as I know, if an atom is in the ground state, and it then if it absorbs the right frequency photon, it will get excited. The lifetime of an excited state is very small, so it will go back to ground level quickly and so then it will emit a photon. And that is what spontaneous emission is. Is that right? So then what's the deal with stimulated emission?

Also, can you explain this quote? Where does the second photon come from?

"
In stimulated emission (Fig. 39.28c), each incident photon encounters a previously
excited atom. A kind of resonance effect induces each atom to emit a second photon with the same frequency, direction, phase, and polarization as the incident photon, which is not changed by the process. For each atom there is one photon before a stimulated emission and two photons after—thus the name light amplification. Because the two photons have the same phase, they emerge together as coherent radiation. The laser makes use of stimulated emission to produce a beam consisting of a large number of such coherent photons
"

 

[DaveE]

I think you should split this problem into two pieces to understand it easily.

First just focus on the 2 state scenario, with energy levels E1<E2. If the electron (or atom, molecule,...) is initially in energy state E1 and a photon interacts with it that has energy E2-E1, it can be absorbed and move the systems to level E2.

An atom/molecule at state E2 will stay there until one of two things happens:
1) Stimulated emission: a photon of energy E2-E1 interacts with it and and causes a second photon with the same state (energy, direction, phase) to be emitted. You start with an excited atom/molecule and a photon and you end with a less excited atom and two identical photons.
2) After some time (the mean value is typically called the "upper state lifetime") the atom/molecule may spontaneously emit a photon with energy E2-E1 at a random time in a random direction. This, of course is much more common in the real world, so you can call it "normal", but a physicist won't really understand exactly what "normal" is.

So the point of the population inversion is this: Suppose a photon travels through a laser gain medium with energy E2-E1, is it more likely to be absorbed by an atom/molecule in state E1, or is it more likely to duplicate itself by interacting with an atom in state E2 (spontaneous emission). The reason it's called population inversion is that normally atoms/molecules have more low energy states than high energy states. The reason it's called population inversion is that normally atoms/molecules are more likely to be in low energy states than high energy states. Only some materials at some energy levels can create a population inversion (more high energy states than low energy states). Spontaneous emission is part of the solution here. Good laser materials have a long upper state lifetime.

So the second piece: How do you put lots of atoms/molecules into the higher energy state. That's the part about metastable states. There are many possible energy states; some have large U.S. lifetimes, most don't. A common way to get a bunch of atoms into level E2, the metastable state, is to put energy into a higher state that will decay quickly (spontaneous emission) to that more stable state. That is pretty much the whole story, not counting about a million complex details like how CO₂, Ar⁺, and NdYAG materials are used (all common laser materials) and what their particular energy states are.

edit: Opps! see post #6. Fixed now

 

[PeterDonis]

if an atom is in the ground state, and it then if it absorbs the right frequency photon, it will get excited.

Yes. And there will be more than one excited state available, so more than one frequency of photon that it can absorb.

The lifetime of an excited state is very small

That is true for some excited states, but not all. The lifetimes of excited states can vary very widely. The "metastable" state your book is talking about is one with a very long lifetime compared to other excited states.

it will go back to ground level quickly and so then it will emit a photon. And that is what spontaneous emission is.

Emission of a photon (whether spontaneous or stimulated) does not have to take the atom back to the ground state. It can take the atom to any lower energy state, including an excited state that is lower energy than the one the atom is currently in.

Where does the second photon come from?

From the atom. One photon is already present; it stimulates the emission of a second photon from the atom, with the same state as the one already present. So after the emission there are two photons.

 

[PeterDonis]

normally atoms/molecules have more low energy states than high energy states

No, normally a substance with a lot of atoms/molecules in it will have more atoms/molecules in lower energy states than higher energy states. The point is the number of atoms/molecules in a given energy state, not the number of states.

 

[DaveE]

No, normally a substance with a lot of atoms/molecules in it will have more atoms/molecules in lower energy states than higher energy states. The point is the number of atoms/molecules in a given energy state, not the number of states.

Yes, I said that completely wrong. It's the probability that you encounter a "good" atom or a "bad" atom that matters

 

[rtareen]

An atom/molecule at state E2 will stay there until one of two things happens:
1) Stimulated emission: a photon of energy E2-E1 interacts with it and and causes a second photon with the same state (energy, direction, phase) to be emitted. You start with an excited atom/molecule and a photon and you end with a less excited atom and two identical photons.

What do you mean by "interacts" with the atom? So the beginning setup is that we have photon with just the right energy, and an excited atom. This photon is not absorbed like usual but "stimulates" the atom so that it emits an identical photon? Is that right?

As for population inversion, we want the atoms in the metastable state so that they can interact with the photon which will cause stimulated emission?

So really its all about this "interaction" between an excited atom and the photon. Can you explain?

 

[rtareen]

From the atom. One photon is already present; it stimulates the emission of a second photon from the atom, with the same state as the one already present. So after the emission there are two photons.

How does the photon not get absorbed but rather stimulate emission of an identical photon? Do photons have some other property that has not yet been discussed?

 

[DaveE]

What do you mean by "interacts" with the atom?

I just mean that in causes a change in the atoms energy state somehow. There are also photons that can pass by without doing anything to the atom.

 

[rtareen]

I just mean that in causes a change in the atoms energy state somehow. There are also photons that can pass by without doing anything to the atom.

Oh wait it wouldn't necessarily get absorbed because it would need to have an energy between the metastable state and the next highest state. So this interaction is because the photon has an energy equal to the difference between the metastable state and a lower energy state?

 

[rtareen]

How does the photon not get absorbed but rather stimulate emission of an identical photon? Do photons have some other property that has not yet been discussed?

Never mind I just realized it does not necessarily have to get absorbed. See my above post. But I'd still like to know how the interaction works

 

[PeterDonis]

this interaction is because the photon has an energy equal to the metastable state and a lower energy state?

To the difference in energy between those states, yes.

I'd still like to know how the interaction works

The only answer to that is "because those are the laws of quantum mechanics".

 

[PeterDonis]

I just mean that in causes a change in the atoms energy state somehow.

One can be a little more specific than that. The Hamiltonian of the overall system includes an interaction term that couples the atom to the electromagnetic field. This interaction term in the Hamiltonian (combined with other quantum mechanical laws) is what determines the probabilities for stimulated emission, as well as spontaneous emission and absorption.

 

[rtareen]

To the difference in energy between those states, yes.
The only answer to that is "because those are the laws of quantum mechanics".

Ok. Got it. Thank you and thanks to Dave too!

 

[DaveE]

One can be a little more specific than that. The Hamiltonian of the overall system includes an interaction term that couples the atom to the electromagnetic field. This interaction term in the Hamiltonian (combined with other quantum mechanical laws) is what determines the probabilities for stimulated emission, as well as spontaneous emission and absorption.

OK, you can be more specific, if you like.
I meant it in a very general weaselly sort of way. Sometime it happens sometimes it doesn't. The detailed features of those probabilistic events aren't an important feature in a description at this level.

 

[PeterDonis]

The detailed features of those probabilistic events aren't an important feature in a description at this level.

Yes, agreed, but I think it's still worth pointing out that "interaction" does refer to a specific feature of the quantum mechanical model; it's not just vague handwaving.

 

[DaveE]

Yes, agreed, but I think it's still worth pointing out that "interaction" does refer to a specific feature of the quantum mechanical model; it's not just vague handwaving.

Except when it is vague handwaving. Glad you weren't my Ph2 professor, I wasn't ready for Hamiltonians when I first learned about energy levels and photons.

 

[vanhees71]

As far as I know, if an atom is in the ground state, and it then if it absorbs the right frequency photon, it will get excited. The lifetime of an excited state is very small, so it will go back to ground level quickly and so then it will emit a photon. And that is what spontaneous emission is. Is that right? So then what's the deal with stimulated emission?

Also, can you explain this quote? Where does the second photon come from?

Spontaneous emission is process like
$$\text{atom}^* \rightarrow \text{atom} + \gamma$$
(where the star indicates an higher excited atom state).

Stimulated emission is process like
$$\text{atom}^* + \gamma \rightarrow \text{atom} + \gamma+\gamma.$$
While this also works in the semiclassical approximation, i.e., when you don't quantize the electromagnetic field for spontaneous emission you need field quantization. In this sense spontaneous emission is the most simple effect of field quantization.