Can two photons excite an electron consecutively? I know that a photons energy is quantized and that it can excite a bounded electron from one energy state to the other whic depends upon the energy the photon carries my question is that can two photons consequtively excite an electron from an initial state (say E1) to E2 and then E3 without the electron falling from E2 to E1 in between.
 A: Short answer : yes.
Why can't it? An electron can always continue to gain energy from compatible photons and move to higher and higher energy levels (till it finally leaves the atom itself!) as long as the time between the consecutive excitations is not sufficient enough for the electron to drop back to its ground state.
Relaxation (moving from an excited state to a lower energy state) happens through three pathways:

*

*Spontaneous emission (time : $\sim 10^{-8}$s)

*Stimulated emission

*non-radiative decay

All of these take some time and do not happen instantaneously. So if the electron is re-excited before it relaxes to a lower energy state, then for sure it will go to an even higher energy state $E_n, \ n>1$.
A: To add to the answer by @CrazyGoblin. Every state can be characterized by a lifetime, $\tau_i$, which characterizes the rate of relaxation $\Gamma_{i\rightarrow 0}=1/\tau_i$ to the ground state. Inducing an absorption to a higher energy state requires that the absorption happens faster than the relaxation. In other words, the rate of absorption, given by the Fermi Golden rule, $\Gamma_{i\rightarrow j}$, should be higher than the rate of relaxation to the ground state (preferably much higher, for the effect to be clearly observable):
$$
\Gamma_{i\rightarrow j} \gg \Gamma_{i\rightarrow 0}.
$$
Since the absorption rate is proportional to the square of the matrix element, i.e., to the square of the field inducing the absorption, this usually requires very strong optical fields, and as such belongs to the domain of nonlinear optics. The absorption from the ground state obviously has the advantage that the ground state has infinite lifetime, and the absorption can be easily observed even for relatively low fields.
Finally, let me note that given strong fields and appropriate non-dipolar couplings, one can observed two- and multi-photon absorption even with no intermediate levels present.
