Re: wave trains

[John Denker <jsd@MONMOUTH.COM>]

At 11:24 AM 9/2/00 -0400, Ludwik Kowalski wrote:
> The "wave train" of a laser is often very long, for example,
> 100 meters or more.

It's not very precise to speak of the "length" of a wave train.  To one way
of looking at it, the wave train for a CW laser that has been turned on for
one second is 300,000 kilometers long.

I assume Ludwik meant to say that the coherence length is 100 meters.

> Obviously, it can not be identified with a single photon from one atomic
> transition.

Huh?  Two ideas are mixed together in that sentence.
 -- Lasers, almost by definition, don't involve single-atom transitions.
 -- Whether or not there is more than one photon per coherence-length
depends on intensity.

> It is a sequence of photons matching each other perfectly.

Huh?  Again, there are multiple ideas mixed together.
 -- The 300,000 kilometer CW beam can (loosely speaking) be considered
    a succession of photons, but they are not matched together perfectly.
 -- On the shorter length scale (the coherence length) things are
    matched together, but they are not a sequence.

> Suppose that the train is made from 1000 photons and
> that one of them, from somewhere in the middle, happens
> to ionize an atom. What happens to the train? Is it divided
> into two uncorrelated trains of shorter length?

Strictly speaking, the photon-number operator is not compatible with the
position operator.  It is a violation of basic QM principles to ask about
the position and the photon-number of the same state.

More generally, one can construct wave packets which are a compromise,
having not unduly much uncertainty in photon-number (if you don't look too
closely) and not unduly much uncertainty in position (if you don't look too
closely).  One can, loosely speaking, apply the word "photon" to such a
construction, although this risks spreading confusion among the
non-experts.  Still, even in this generalized case, asking what is
happening to the (approximate) number operator on a length scale less than
the coherence length is a violation of the commutation relations.

==================

To say it in more physical terms:  Let's assume Ludwik was asking about a
beam so dim that there is only one photon per coherence length.  Then any
process that is so localized in time and space that it could tell the
difference between the "middle" of a coherence-length and any other part
would be such a violent process that the results would have to be discussed
in its own terms, not in terms of the insipid wave-packets coming from the
laser.