quantum mechanics - Is energy exchange quantized?

In the photoelectric effect there is a threshold frequency that must be exceeded, to observe any electron emission, I have two questions about this.

I) Lower than threshold: What happen with lesser frequency/energy photons? I mean there is no energy transferred? If some energy were transferred, (and it must be transfered quantized), how can be experimentally known it was quantized, if there is no electron emission?

II) Intensity dependence: There is any known dependency with intensity? (I mean a dependency not about the number of electrons (I am already aware of this) but about the energy of them) I thought energy of electron emited was independent with intensity, but then I found this link dependency with intensity (is a paper to buy that I haven't read) but relates energy with intensity.

Answer

Energy exchange is quantized when moving a electron from one bound state to another bound state.

This isn't because the exchange is inherently quantized, but because the states the electron may occupy are quantized.

Thus the standard photo-electric effect in which a photon can not excite an atom unless it has a minimum energy.

---

However,...

There are multi-photon processes by which sub-threshold light can excite transitions.

Cross-section for them go by intensity-squared (or worse) and are very small for any reasonable light intensity. To study or employ them you get powerful, short pulsed laser systems. Where short pulsed means nano-second or faster pulses and powerful means "Do not look into beam with remaining eye". Even then you don't get a lot of rate.

These processes are utterly negligible for the kind of benchtop experiment we use to teach the photoelectric effect: you just can't get enough intensity. (See below for how negligible.)

The conceptual model here is that the first photon bumps the electron to a short-lived, unstable state without well defined quantum numbers, and the second comes along before that state decays and finishes the job.

We're currently exploring the application of such a process to calibrating light yields, opacities in a large volume of scintillating material.

---

From New J.Phys.12:113024, 2010:

For gases the one-photon absorption cross-section $\sigma_1$ is typically of the order of $10^{−17}\text{ cm}^2$, whereas the two-photon and the three-photon cross-sections are of the order of $\sigma_s = W/F_2 \approx 10^{−50}\text{ cm}^4\text{ s}$ and $\sigma_3 = W/F_3 \approx 10^{−83}\text{ cm}^6\text{ s}^2$, respectively.

Where $F$ is intensity in photons/second and W is excitation rate in reciprocal seconds.