Quantum Optics IV: Coherence & LWI

In lesson 1 we talked about lasers. We said that to have a laser you need something called “population inversion”, meaning you need more atoms that have too much energy than you have atoms in the lowest energy state. That way, we can “stimulate emission” of this extra energy in the form of photons that give us laser light. Based on this idea, it might seem silly for someone to think that maybe they could make a laser without this population inversion. But they did. And it worked. In fact, there are now many papers that discuss new ways to achieve lasing without inversion in various ways. Many of them are just theoretical models, but some have been proven experimentally. In this lesson we’ll consider a simple and rather general system that allows us to lase without inversion. First, we need to learn a new term.


Coherence

Coherence is a very important word in quantum optics, and while we talk about coherence on a daily basis and assume coherent light for many calculations, we spend little time actually defining what coherence is. There are many different ways to think about it, and it took me reading all of them (and more than those introduced here) before it clicked. I’ll present many statements about “coherence” and hopefully one or two of them will make sense to you. Even if they don’t, meh, you’ll be okay.


You've probably heard something about Heisenberg’s uncertainty principle before… it gives you a limit on how well you can know certain things about a quantum system (insert joke about telling an officer you don’t know how fast you were going but you know where you are…).


--Well, a coherent wavepacket (collection of many light waves traveling together) always has minimum uncertainty. So while the uncertainty principle will be stated as something is “greater than or equal to” something else, coherent wavepackets go with the “equal to”.

--The uncertainty relationship for an electromagnetic wave is between the phase and amplitude. So if the field was E=Acos(wt+ϕ), There would be some limit on how well we know the variation of A and the variation of ϕ. For a coherent field, these variations are equal.

--A coherent wave packet resembles a classical light field as closely as possible. So the quantum and classical descriptions should be almost exactly the same.

--A coherent wave will oscillate back and forth in the harmonic oscillator potential (see figure in lesson 2) without ever changing shape. It sticks together; it “coheres”.

--I’ll throw this one out there just for fun, for those of you that know quantum: The coherent state vector is an eigenstate of the field annihilation operator. (The “a” operator we met in lesson 2 that destroys photons!)


Although we won’t bother with the math that illustrates this principle, it is because of quantum coherence that we get lasing without inversion, electromagnetically induced transparency, and many other cool phenomena. Without coherence, none of this would work. So back to the point at hand now…



Lasing Without Inversion

For LWI we need a special 3 energy level system. The one we’ll talk about is actually a very common system we call the “lambda configuration”. It’s shown below:


+ Show Spoiler +


The frequencies ν1 and ν2 are the transition frequencies between the levels.

Now, the derivations behind the following equations are wayyy beyond the scope of this series. They literally take pages and pages of math to get through…I’ll present the results and follow it up with an explanation of what they mean conceptually.

If you want the real math, PM me or comment and I can scan a hand written derivation and post it, but I won’t have time to offer extremely detailed explanations of each step. Or check out the book "Quantum Optics" by Scully and Zubairy.


Basically, we apply electromagnetic fields to this system of atoms with the energy levels depicted in the figure above. The fields we apply have the same frequencies as the transitions between levels (ν1 and ν2). Then we solve the master laser equations that tells you how many atoms are in each energy level and what the electromagnetic field looks like.

We want an increase in the field with time in order to get a laser (we want more photons contributing to the light beam so the light amplifies), and we want to be able to do this with fewer atoms in the upper state than in the lower states (no inversion).


Let’s assume we prepare the system such that the populations in each of the 3 levels are distributed such that:

Ca(0)=0 Cb(0)=1/sqrt(2) Cc(0)=1/sqrt(2)*exp(-i*ϕ)

The C’s are how many atoms are in each level at time 0. Nothing is in a; the atoms are split evenly between levels b and c, except we put a phase shift between them (the exponent term). This just changes how they move with respect to each other.


If we solved the governing equations for this system with those initial conditions, we would get probabilities of an atom moving between each energy level. The probability of transitioning up to level a is given by the following expression:

Pa= Ct^2 * (1+cos(φ1-φ2-ϕ))

φ1, φ2 are phases associated with the transitions between a & b and a & c.


So if φ1-φ2-ϕ =±π, this is 0. There is no absorption! This is good if we want to lase, because absorption steals photons we would like to stimulate emission with. Since we aren’t doing the derivations here, I’ll just state that you can prove stimulated emission to either b or c is nonzero for any phase. So, if we know the values for φ1 and φ2, we can pick ϕ so that absorption is eliminated.


Alternatively…


An easy way to think of this concept: Levels b and c are independent of each other. Level a is coupled to both b and c. The probability of going from b to a and c to a is the sum of two independent probabilities. The probability of going from a to b and a to c is not a simple sum of the probabilities; there is an interference term because the transition that happens is dependent on both states b and c. This interference term can therefore make the total probability of going up to a from both states =0.

(Like the interference pattern in the double slit experiments, we’re creating constructive and destructive interference in this quantum system using properties of coherence. That’s why coherence is important, and if we’d gone through the painstaking math we would've used coherent properties explicitly.)

This suppression of absorption is what allows us to lase without exciting everything into the upper level first. As long as a few atoms are already there (which a few will be in any realistic system) then we can stimulate emission and the signal will amplify, because none of these photons are lost to absorption. We don’t need MORE atoms in the excited state than in the ground state, we just need a few.

These properties of coherence, interference, and suppressed absorption are also what allow us to have electromagnetically induced transparency (EIT). In EIT, we can prepare a system that is normally opaque (you can’t see through it) and then successfully shine a light of a certain wavelength through it as if it were transparent. That's the magic behind a lot of the "invisible" object stuff you might have run across on the internet.

These phenomena were actually some of the first discoveries that really gave rise to the field of modern quantum optics. LWI is something our lab group holds close to heart, as it was first demonstrated experimentally in our laboratory. Carrying out these experiments is not a trivial task; you don't need population inversion, but you do have to prepare your system into a very particular initial state and apply fields in a very specific manner to get everything to work out right. You can't get something from nothing.

At any rate, LWI/EIT are pretty cool from a science-for-the-sake-of-science perspective.