Photoelectric emission at frequency less then threshold frequency If I shine an EM radiation of frequency $\nu$ on a metal surface which has threshold frequency of $\nu_o$, where $\nu < \nu_o$ then, will the emission occur by multi photon absorption?
My reasoning is this

Since there are quantised energy levels in metals forming a band, so it may happen that the incident photon excites the electron (even though the photon has not enough energy to knock the electron out) from lower energy level to higher one and when the second photon is absorbed by that excited electron, it is knocked out.

Please tell me where am I wrong in my reasoning.
Thanks:)
 A: Let me first remark that one have to distinguish between cascade processes (such as upconversion) and multiphoton processes (such as two-photon absorption)scattering).
Cascade process is, e.g., when an electron makes a transition from one level to a higher one by absorbing a photon, and then moves to an even higher level b y absorbing a second photon - in essence, two absorption processes occur in a short sequence, before the excited electron relaxes. The probability of such processes is very low, since the second photon should be absorbed very quickly, faster than relaxation time. Thus, they require very high photon density and rarely (or maybe never) observed in metals, although thy do happen in semiconductors or materials doped with, e.g., rare earth elements.
Multiphoton process does not require existence of an intermediate level (although one would often speak of a virtual level) - it typically occurs due to the existence of higher order coupling terms between material and the EM field and/or simply as a higher-order process for basic coupling (such as Raman scattering). Again, the special nature or the higher order of the process make it less probable and therefore difficult to detect.
Both types of processes take place in actual photoelectric effect, leading to the existence of a rather complex structure of the absorption edge (see K-edge). In the quotes below below the cascade and multiphoton processes are referred to respectively as electric-dipole allowed and quadrupole transitions (emphasis is mine):

Pre-edge
The K-edge of an open shell transition metal ion displays a weak pre-edge 1s-to-valence-metal-d transition at a lower energy than the intense edge jump. This dipole-forbidden transition gains intensity through a quadrupole mechanism and/or through 4p mixing into the final state.


Rising-edge
A rising-edge follows the pre-edge, and may consist of several overlapping transitions that are hard to resolve. The energy position of the rising-edge contains information about the oxidation state of the metal.
In the case of copper complexes, the rising-edge consists of intense transitions, which provide information about bonding. For CuI species, this transition is a distinct shoulder and arises from intense electric-dipole-allowed 1s→4p transitions.

A: This is the experiment that established the photoelectric effect:

Within experimental errors there are no electrons coming out when the frequency is below the threshold for the metal.

so it may happen that the incident photon excites the electron (even though the photon has not enough energy to knock the electron out) from lower energy level to higher one and when the second photon is absorbed by that excited electron, it is knocked out.

where you are  wrong is in considering the probability of this to happen. The experiment shows it does not happen, so , even if this reaction has a probability to happen, the number of excited electrons is too small and the probability that a second photon will impact the atom with the excited electron is very small , as seen in the experiment. The electromagnetic coupling is 1/137 and where multiple interaction enter it prohibits observing such a phenomenon.The reaction is possible to be computed with the diagrams needed, but the small coupling makes the probability small, for the first absorption, which leaves a very much much smaller number of target atoms than the  atoms found by photons that give the photoelectric effect.
