The term temperature is well defined only for quasi-static system averaged both over time and over space. In some cases, it is possible to neglect one of the averaging, e.g. a temperature of a molecule in ergodic gas can usefully be defined. In any case, the averaging time must be long compared to period of any thermal vibrations.

Femtosecond pulses are too short compared to molecular vibrations for using the term temperature in this time domain, fs-palses are used e.g. to study quantum beats of vibrational states. I'd say: Forget cooling rate at this time. Speak about energy flow with photons and with phonons... Or else be ready (perfectly ready!) to discuss negative absolute temperatures and all related stuff of non-stationary temperatures so popular soon after invention of lasers.

when I read some research papers, I saw that the value of laser heating rate is 10000 K/s, and cooling rate is 1000000 K/s. I don't know how we can get these values. In the of short pulse laser duration, is it still the same ?

Instantaneous ultrafast laser heating is a non-equilibrium process, but it is possible to obtain temperature data on very short time scales.

For example, the "two temperature" model decouples the temperature of the electron gas from that of the ion cores - a search will find many hits such as

http://cat.inist.fr/?aModele=afficheN&cpsidt=17227149

This model was developed in the 1950's, prior to the invention of the laser.

My dissertation was focused on building instrumentation to measure ultrafast responses to laser-matter interactions; see http://hdl.handle.net/2027.42/63758

I do not know what is the meaning of the term laser heating in those papers, but if it is about something like laser nuclear fusion, then it can be e.g. 1 microsec laser pulse heating the target to 1000K, and the heat rate is 10^9K/s. Note that a microsecond is quite many of femtoseconds...

@Peter: You definitely are ready to discuss all the stuff related to non-equilibrium systems, as obviously you have been using the second alternative I mentioned...

BTW, did you encounter any mentioning of negative absolute temperatures unrelated to lasers? I' did not, but I might be too much biased towards lasers.

I have read some paper about laser heating model. There are 2 kind of laser model are used in fs laser

1) Two temperature model (TTM) Which Peter told me

2) extend two temperature model (ETTM) which include the term of electron and phonon relaxation time.

Could you tell me when we use the second model ? which one is better ?

Btw, Is it TTM is applied for a very short time scale? In the case of longer time scale, can we use the single temperature model ?

@Vladimir - negative temperatures on the absolute scale are always due to non-equilibrium excited states, so they naturally appear with lasers during the generation process, which is very well controlled.

But I'm certain that they also occur in other situations, but normally as transient phenomena. If not part of a coherent process they will result in a rapid thermalization.

@Nguyen - the two temperature model applies on very short time scales as the heat capacity of an electron "gas" is very low, and depending upon the material, is usually measured in femtoseconds. Phonon decay is measured in picoseconds, so which model you use depends upon the relevant time scale.

You can "easily" measure the electron gas temperatures with optical probes - the effects include changes in conductivity and reflectivity.

@Nguyen - two (or multi) temperature models are applied quite often and in different areas, e.g. in plasma physics. Usually, when interaction between subsystems is somewhat limited. The important part of the approach is to understand what you want from the model. Since the normal properties of temperature is not directly applicable in the model (think of negative absolute T), you will need to prove correctness of any further results... Depending on your problem and your background, this might be easy or impossible.

And by the way, I agree with everything what Peter said.

I am a beginner working on the phase transformation of material after laser irradiation. Sorry if my question is stupid.

In the case of very short time scale, which temperature will cause transformation in materials ( electron or lattice temperature) ?

Let talk about plasma, Do the plasma absorb the energy of incident laser beam? If yes, plasma can reduce the energy delivery on the material surface, right? Why some other research said that plasma can enhance the temperature in material ?

@Nguen - The answers depend on what exactly you are talking about. E.g. for discharging (or recharging) an electret, it might be enough to heat only electrons. For lattice changes, at least some of lattice temperatures need a possibly temporal increase.

Plasma does absorb light. And reflects it, too. Therefore, for a transparent material plasma can be a mediator of energy transfer from light to heat in the material, like in water at visible light. And plasma can shield the material from light, like plasma at a metal surface...

Do not forget, that microsecond and femtosecond pulses are as different as a heart beat and a year season, and effects might be very different. E.g. in fs optics, self-focusing is rarely as big problem as with micro-nanosecond lasers.

@Nguyen - depending upon the fluence, your laser pulse can create a plasma at the material interface; the evolution of this plasma interface is critical for some processes as the plasma both absorbs and reflects light, depending only on the "plasma frequency", which depends on the electron density.

You can simplify your calculations if it is possible to avoid plasma generation.

"I am a beginner working on the phase transformation of material after laser irradiation."

If you are looking at phase transformations it is important to understand the material and the timescale of its transformation in relation to the laser. Things are much different if they happen at the surface vs. if they are constrained within a transparent solid. The one thing that is pretty clear is that most phenomena you will be looking at will be driven by the residual heating and shock delivered after the laser pulse-not the initial electron temperature. Shock waves and spallation may be as important (or more important) than the actual temperatures reached in the evolution of your material's phase and microstructure over time.

One of the ways would be to build a multi physics model and provide the appropriate boundary conditions, laser processing parameters such as input power and pulse duration, and material  thermophysical properties. To support these, experimental observations such as microstructure are important as well.