Approximating evolution as occurring in a two-dimensional subspace Suppose you have a quantum system with a Hamiltonian having some number (greater than 2, possibly infinite) of eigenfunctions, and that the system is prepared in the ground state.
When can you approximate it as by two-level system (using just the ground state and first excited state)? Is there some property that will make it better approximated by a two-level system (e.g., something like a bigger energy gap between the second and third energy levels)?
 A: It depends very much on the hamiltonian and the external potential. There is a huge class of possible situations and of possible behaviours, and many of those do lend themselves quite often to two-level approximations.
The most common, I think, is when you have a weak perturbation which oscillates at the right frequency to couple only two levels and leave out the others. In this scheme, you have a hamiltonian with a discrete spectrum (e.g. an atom), and you start off in the ground state. Then, if 


*

*your perturbation is weak, so only single-photon transitions can occur, 

*your perturbation is tuned exactly to the energy difference between the ground and some excited state (not necessarily the first), and most importantly

*its bandwidth is smaller than the distance to any neighbouring states, so it has essentially no power at those frequency components,


you can essentially ignore all other levels and simply treat your system as having two levels. This scheme, or variations to include more levels with suitably-tuned additional lasers, is essentially everywhere in quantum optics and any quantum information processing with matter systems.
Another possible scheme is to have a perturbation that can be strong enough to appreciably shift the energy levels of your initial hamiltonian, but that varies slowly enough. In the adiabatic limit of infinitely slow variations, you will stay in the "dressed" ground state, which is the ground state of the total hamiltonian at each particular instant. As you slowly increase the rate of change of the perturbation, you begin to get Landau-Zener transitions to the first excited state at the points where the two levels are closest. If the second excited state is far away enough, then you can increase the nonadiabaticity strongly enough that you get significant transfer of population between the ground and first excited states, while still ignoring all other levels.
Further than that there are, of course, more situations where you can reduce the dimensionality of interest of your problem to two or only a few states, but they get more and more specific. In general, when one is faced a difficult quantum mechanical problem, a large part of the solution process is finding the subspaces where most of the population is, and coming up with schemes to reduce the dimensionality to something more manageable; once you do that you're essentially in a position to easily solve whatever's left to do.
A: Two-level approximations are quite common in quantum optics. Suppose, for instance, you have an atom or similar in its ground state and you subject it to an external electromagnetic field. Then, if the field is weak enough and in the right frequency range, and the transition is optically allowed, then the first excited state can be excited. But the probability for the excitation of higher states is usually very small and also the frequency of the field is usually not right. Then the two-level approximation makes sense.
With intense fields, for instance using powerful lasers, the situation can be completely different. Multi-photon processes become possible and also ionisation can occur. Here we can encounter the interesting phenomenon of above threshold ionisation.
