I assume you are already familiar with Einstein coefficients(https://en.wikipedia.org/wiki/Einstein_coefficients), stimulated emission/absorption and spontaneous emission.

In a regular one photon experiment, because the absorption wave length is different from emission wave length, the excitation process is through a laser but fluorescence process is spontaneous.

When we talk about lifetimes, we always refer to lifetime of spontaneous emission process, because lifetime varies with laser intensity when stimulated emission is involved. Therefore you are comparing spontaneous emission with stimulated absorption, which is not a fair comparison. There is really nothing to be compared to spontaneous emission, since there cannot be any absorption when no photon is present.

The inverse process CAN be made extremely fast with stimulated emission in a Laser Induced-Fluorescence (https://en.wikipedia.org/wiki/Laser-induced_fluorescence) experiment, where a second laser is used to induce the emission.

Dear Xiaolei,

Thanks for your consideration and instructive comment,

I want to know where I could find the quantum theoretical calculation for different process life-times

thanks again for your time and answer.

Dear Yavar,

You can look up most of the required math on the page of Einstein coefficients(https://en.wikipedia.org/wiki/Einstein_coefficients).

It might be noteworthy that the timescale of light absorption/emission process is different from the lifetime of excited or unexcited species. Lifetime refers the (mean) time it takes for one species to disappear.

Lifetime is calculated from the rate of decay. For spontaneous emission, mean lifetime is simply the reciprocal of Einstein coefficient A: t = 1/A

Value of all Einstein's coefficients can be quite straightforwardly calculated from oscillator strength f ( https://en.wikipedia.org/wiki/Oscillator_strength ;Also see the Einstein coefficient wikipedia page). Fermi's Golden rule can be applied to get an approximation of oscillator strength (the dipole component), which is calculated from Transition Dipole Moments (https://en.wikipedia.org/wiki/Transition_dipole_moment). To get transition dipole moment, one would need the wave functions of both of the electronic states, which requires the solution of Schrodinger's equation. This often requires a computer.

For stimulated emission and absorption, mean life time is the reciprocal of the product of Einstein's coefficient B and spectral energy density u: t = 1/(B*u)

The spectral energy density depend on the strength of the light source, so it is somewhat dependent on the set up of an experiment. However, if a laser is used, u is extremely large and the lifetime are therefore usually short. However, in a natural process, u can be extremely small and the lifetime can be quite long for the unexcited.

Dear Xiaolei,

I have followed the path in your answer,

thanks for very complete and detailed answer.

Bests.

Yavar

The "lifetime of the first excited state" can be calculated from Einstein coefficient. With oscillator strength and first excitation energy ( from TDDFT calculations) , the lifetime of the first excited state can be obtained. Therefore when we have excitation energy of each state (from TDDFT), lifetime of different states can be calculated. I have a PPt file in this regard.

I do not understand your statement that absorption is "fast". How fast?

Absorption is "stimulated", as can emission be, and the probabilities are determined by the oscillator strength (Einstein co-efficients) and the radiation field. As such they do not have a specific lifetime. Spontaneous emisison is independent of the radiation field and is associated with a probability/lifetime characteristic of the material, although this can also be stringly influenced by local fielf effects

Dear Professor Brink,

I think my question was problematic, Fast means femtosecond absorption of dyes under sunlight radiation.

Thanks for your consideration and HAPPY NEW YEAR,

Bests

Dear Professor Byrne,

Thanks for your instructive comment,

Happy new year,

Bests

Hugh summarises it nicely. The apparent relative slowness of emission is because most of the time we do not stimulate emission, but this is a routine procedure carried out in a range of spectroscopies. When stimulated, the emission occurs on comparative timescales to absorption. Without stimulation of the emission the decay will depend on the relative stability of the excited state and the routes available for energy dissipation, so the rate of the decay can vary enormously depending.

The full theory of time dependence of transitions was treated in detail in S.S. Penner's "Quantitative Spectroscopy and Gas Emissivities", Addison Wesley, London, pp 115-176, 1959 as well as a nice perturbation treatment by W. Lick in J. Chem. Phys. v47, p2438, 1967. I combined the derivation with Lick's examples in "Electronic Structure Modeling, Connections Between Theory and Software" by Trindle and Shillady, CRC Press, 2008 on pp 202-210. There is also a clear discussion of Einstein transitions in the famous text by Pauling and Wilson, "Introduction to Quantum Mechanics with Applications to Chemistry", McGraw-Hill, 1935. pp299-319. Molecular transitions between states consist of electronic, rotational and vibrational wavefunctions. Spin is also a consideration and transitions occur most easily when spin is conserved in vertical transitions but if the upper state decays to say a triplet spin the system may "hang" in an upper state leading to phosphorescence. Thus one needs to consider spin transitions as well as the group theory of symmetry of upper and lower states. A text which combines all these considerations is "Symmetry and Spectroscopy, an Introduction to Vibrational and Electronic Specroscopy" by Harris and Bertolucci, Oxford University Press, New York, 1978. I have taught from the Harris-Bertolucci text a number of times and it is far and away the clearest and most complete treatment at the level of a first year graduate student.

Don Shillady, Emeritus Professor of Chemistrry, Virginia Commonwealth University

Donald

I like Barrow, and it is free on the web. do you know it?

Also Puling and Wilson is free on the web, I like it as a good basis for Quantum Physics but less for spectroscopy