General Adiabatic Theorem

Sufficiency Criterion for the Validity of the Adiabatic Approximation

D.M. Tong, K. Singh, L.C. Kwek, and C.H. Oh

PRL 98, 150402 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e150402

The authors take a close look at the adiabatic approximation. They find that the usual criterion presented in textbooks (for instance, Messiah) is valid for a restricted set of systems, and they show that two additional criteria are required for a general quantum system. This was a difficult read, but the result was worth the effort.

The adiabatic theorem sounds so sensible that I'm not surprised it was used without being proven for so many years: If you have a system that is in an eigenstate |n(0)>, then you modify the system very slowly, then it will remain in |n(t)>. For instance, if I have a magnetic dipole that points in the direction of an applied magnetic field, the adiabatic theorem says if I slowly rotate the magnetic field, the dipole continues to point along it. (For the quantum version, replace "magnetic dipole" with "spin.") Or, if I am in the ground state of a harmonic oscillator and I slowly change the spring constant, at any time, the system will be in the ground state of the current oscillator.

It turns out that there are cases where common sense is misleading. I read such a counterexample just a couple months ago. The system was designed to satisfy the requirement of the adiabatic approximation, yet violate its prediction. The authors of the current paper point out that the adiabatic theorem assumes the first time derivative is small and all higher derivatives are smaller. This is not the case in the counterexample.

Clearly the adiabatic theorem works a lot of the time. Otherwise, it wouldn't still be used and taught to graduate students. The question raised by the counterexample is, How can you tell if the adiabatic theorem will work or not? In a one and a half pages of straight-forward but tedious calculations, the authors derive three criteria that apply to any quantum system. (That's 1.5 PRL pages -- probably equivalent to 5 normal pages.) If these criteria are satisfied, then the system will be in |n(t)> at time t with a probability that approaches 1.

The first of these criteria is the industry standard. It applies to systems where the energy difference between states is a constant and the time evolution of the inner product of two states is a constant. The other two criteria involve integrals that can't be evaluated in general. However, an upper limit can be placed on the integrals by replacing the integrand with its largest value. This gives a product of the maximum value and the time interval that must be small, so it sets limits on how long you might expect the adiabatic approximation to be valid.

This could be quite useful. I've never seen a calculation that suggests how long you might expect the adiabatic theorem to hold. In my examples above, I used the term slowly. The authors have given theorists a way to quantify exactly what we mean by "slowly." They apply their criteria to a spin 1/2 system in a rotating magentic field and find that the adiabatic theorem will only be valid for a fixed number of periods. Even "slow evolution" isn't allowed to take forever!