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# Excited-State Lifetime and Natural Linewidth

The excited state of an atom will have an intrinsic lifetime due to radiative decay given by:

where Nj is the population in the excited state (j) and the Aji's are the Einstein spontaneous emission coefficients for all of the radiative transitions originating from level j. The negative sign arises because the rate decreases with time. Intergrating this equation produces:

where Nj(t) is the excited-state population at any time t, Nj(0) is the initial excited-state population at t=0, and j is the radiative lifetime defined as:

Strong atomic transitions have Aji's of 108 to 109 s-1, so lifetimes are 1 to 10 ns. The above expressions give only the radiative lifetime. Lifetimes can be shortened by collisions or stimulated emission. The following plot shows an emission decay curve after populating the excited state with an excitation pulse at time = 0. The lifetime is 2 ns.

## Natural Linewidth of Atomic Transitions

The natural linewidth (the intrinsic linewidth in the absence of external influences) of an energy level is determined by the lifetime due to the Heisenberg uncertainty principle:

`E*t  h / 2`
So the natural width of an energy level is:
```           h
Ej =  ------
2 j```
or
```        h (i<j) Aji
Ej =  ------------
2```
Since E = h
` = (i<j) Aji`
where is the linewidth in frequency units of a transition between an excited state and the ground state. Since the ground state has an essentially infinite lifetime, the transition linewidth is governed by the width of the excited state.

The lineshape of a transition with only natural broadening is a Lorentzian.

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