Energy levels in a time dependent Hamiltonian If we have a time dependent hamiltonian, for example a 2 level system (say an atom) in the presence of an EM field. Let's assume for simplicity that the only relevant (off-diagonal) term is the dipole interaction of the form $-dE$, where $d$ is the dipole moment of the system and $E = E_0cos(\omega t)$ is the electric field. Let's say that when there is no field present, the eigenstates $|0>$ and $|1>$ have energies $0$ and $E$. Can we still talk about energy levels once we account for the time dependent interaction? Of course we can still diagonalize the new hamiltonian, get the new eigenvalues and write the new eigenstates in terms of the old ones, but are these actually the new energy levels of the new combined atom-EM field system? Can we define energy levels at all in the case of a time dependent hamiltonian? Thank you!
 A: When you have a time dependent hamiltonian you can always define the instantaneaous eigenstates and eigenvalues (in literature sometimes are called adiabatic eigenstates).
The question "can we still define eigenvalues and eigenstates" is a bit confused because of course you can but differently from the time-independant hamiltonian those are not static meaning that evolving in time those states do not change only for a phase.
Anyhow very early in the history of Quantum Mechanics a concept known as "Adiabatic theorem" has been introduced. The adiabatic theorem states that if a system under a time-dependant hamiltonian is in an instantaneous eigenstates, the perturbation is slow and there is a gap in the spectrum with the eigenvalues that comes before and after it, then the system remain in its instantaneous eigenstate.
Of course one has to define what it is meant by "slow", in numerical simulations like DFT where electrons modes are always faster than atomic-core modes this is always used as formulated by the Born-Oppenheimer Approximation. A good explanation can be found at this link:
https://ocw.mit.edu/courses/physics/8-06-quantum-physics-iii-spring-2018/lecture-notes/MIT8_06S18ch6.pdf
here you can also see an elegant connection with Berry phases.
