I’d like to pose the following question for this week’s blog: Can you kill a person with a table tennis ball (without eating the ball and dying from poison in the celluloid)? There Well, with the right physics knowledge you might, but I’ll leave you to decide for yourself:
Here’s what a world-class table tennis player can do (which is still pretty cool):
Suppose you’re on a game show and there are three doors, behind two of which there is a goat, and the remaining one a car. Your objective is to guess which door has the car. You’re allowed two guesses except after the first guess the host of the game show (who knows exactly which door has a goat and which has a car) will open one of the other two doors, always revealing a goat. You’re now given the opportunity to switch doors. The question is what should you do? Switch, or stick with the same door as your first choice (may be because you think the host is trying to play reverse psychology on you). If you’re a probability enthusiast then you’ll know that this is the well-known Monty-Hall problem, and if you’re a not, then (I hope) that you think the answer is very intuitive – namely that it doesn’t make any difference whether you switch doors or not, the probability of choosing the door with the car is 50%. Well, when I first met this problem some years ago I learnt that this intuition is wrong. It completely shattered any confidence I had in probabilities. As a graduate student I learnt what the answer* is and understood the reasoning behind it but the Monty-Hall problem still bothers me somewhat even now. It still bothers me not because I have trouble recalling the solution, but because I found it difficult to resist the temptation to follow the 50/50 intuition on a recent revisit of the problem. It seems that most of us have trouble resisting this temptation (which is why the Monty-Hall problem became famous). To me this means that we just don’t have a robust intuition about probabilities. As the famous polymath John von Neumann said “in mathematics you don’t understand things. You just get used to them.”
*I’m not going to tell you the reason why this is wrong (because it’s readily available on the internet and also because I don’t want to spoil the fun of solving the problem for those who like to challenge).
Most quantum optics textbooks will spend a chapter or two on the theory of open quantum systems. The typical exposition consists of deriving the master equation in the Born-Markov approximation. This is a complicated integro-differential equation governing the evolution of the system state and further simplification requires the specification a concrete model Hamiltonian. The textbook will usually proceed to
give example Hamiltonians and derives the explicit form of the Born-Markov master equation. If you’ve had any exposure to open systems theory from a quantum optician’s perspective then you’ll know that all this is pretty standard. However, just because
it’s in a textbook (or “standard”) doesn’t mean it’s easy to understand. The Born-Markov approximation is rather abstract and explanations of it are often brief. Thus it leaves the reader a lot to swallow, or just simply stuck only after a couple of pages into the
chapter on open systems. Aside from getting the physical principles behind the approximation, the most difficult thing to swallow is that after making the approximation one is still left with a very complicated equation which is supposed to describe valid Markovian evolution for the system, and this is not at all obvious. Just seeing the Born-Markov master equation should make one lose heart that it really works. This is when knowing a bit of maths helps. We fight the abstract with the abstract. I’m referring to what is known as the Lindblad theorem which too often gets swept under the carpet in quantum optics texts. In brief this theorem states what the most general form of a Markovian master equation must look like, often called the Lindblad form*. Knowing this, the natural question is then whether the Born-Markov master equation leads to master equations in the Lindblad form? Lo and behold it does! So it appears then that one should just learn Lindblad’s theory and ditch Born and Markov since Lindblad’s result is much more general. Well, no, because Lindblad’s theorem is too general, mostly just formal mathematics. Lindblad’s theorem tells us what the correct answer has to look like but it doesn’t tell us how to get to the answer i.e. Lindblad’s theorem doesn’t provide a recipe for how to derive a Markovian master equation from model Hamiltonians (which is what physicists want). Learning Lindblad’s result does however settle any doubt that one had about the Born-Markov approximation when it was first encountered and this is great. Furthermore, it leads to a greater appreciation of the Born-Markov approximation for the reason that it actually points out the physics leading** to valid Markovian evolution. It is, in my opinion a remarkable fact that the Born-Markov master equation can be used to churn out a master equation in the Lindblad form. Similarly we can appreciate Lindblad’s work a lot more when we question the validity of the Born-Markov approximation. The moral of the story is that
Born-Markov and Lindblad complement each other and only by knowing both results do we get a complete picture of quantum Markovian dynamics. I believe therefore quantum
optics textbooks should present a more balanced approach to open systems theory by stating Lindblad’s theorem because this is when knowing a mathematical theorem actually allows one to appreciate the physics more.
*The Lindblad form should also be credited to Gorini, Kossakowski, and Sudarshan but I’ll refer to it as Lindblad just for convenience.
** I use the word “leads” because the master equation in the Born-Markov approximation alone does will not directly give us an equation in the Lindblad form.
In quantum optics one usually need to make a further approximation called the rotating-wave approximation to obtain the Lindblad form.