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.