Re: The time it takes to emit one photon

Paul Danaher wrote:
> We have ("Mossbauer effect and retarded interactions") a fairly subtle
> debate going on which an outsider with some mathematics but not nearly
> enough physics feels should be capable of a direct answer. One of the
> questions posed in this is, how long does it take to emit one photon. I
> would really welcome a "simple" answer.

To be really safe, the answer has to be infinity. Of course, we see
photons emitted all the time and these events happen rather more
quickly than that. Of course, this just means that the photon emission
time scale is very long compared to some other characteristic length,
yet still very short compared to the time scales we are used to. To
identify what each of these time scales is, we need to look at how
photon emission is observed and theoretically analyzed.

First lets look at how photons appear theoretically. Take Maxwell's
equations. They form a system of linear translation invariant
equations. A Fourier transform decouples the fields into independently
vibrating modes, each with a characteristic frequency. In this form,
quantization is trivial. Each mode is treated as an independent
harmonic oscillator.

The spectrum of stationary states (those with definite energy) of a
harmonic oscillator is well known. It is discrete and equally spaced,
with the spacing given by the frequency. Each state is labeled by an
integer n. The energy of the state is basically a multiple of this
number. We like to say that the state n contains n quanta, each with
energy proportional to the frequency of the oscillator. These quanta
are called phonons. Each stationary state of the electromagnetic field
can be described by a linear superposition of states with definite
finite photon number in each vibration mode.

So far we have only described stationary states of the EM field and
given them an interpretation in terms of photons. Obviously this does
not cover non-stationary states, which describe photon emission for
example. If we know all the stationary states and their energies, we
can in principle describe all time dependent states of the EM field as
well, but *only* of the EM field. To model interesting physical
situations, we have to introduce other matter or fields and
interactions with them. Unfortunately, for most interacting systems, we
can't write down a closed form solution for its state spectrum.
Calculations are usually done in perturbation theory. The crucial
assumption is that we can approximate the state at the beginning and
end of the experiment by stationary states of the non-interacting EM
field + matter system. The interaction can then be turned on and off in
between. By necessity, for the initial and final states to be well
approximated by stationary ones, the on/off switching of the
interaction must be slow, i.e. adiabatic. That is why we formally place
the initial state at time -oo and the final state at time +oo.

So, with this set up, how do we characterize processes in which photons
are emitted? It's rather simple really. Count the photons in the
initial state and count the photons in the final state. If the latter
number is larger than the former, then we say that some photons were
emitted. Note however, that in the only answer we can give to how long
it to took for the emission to take place is infinity, which is the
elapsed time between the initial and final states.

Ok, now we know what theorists mean whey they speak of photons. How
about experimentalists? If you look at the experimental eveidence that
is pointed to when speaking of photons, it always comes back to
quantization of energy, for example the photoelectric effect and
Planck's radiation spectrum. But to measure energy we need to observe
the system for a sufficiently long time. In other words, the
experimental setup is such that the system's state can be approximated
by a stationary one. But that's exactly where the theoretical
description of photons comes in. Everyone is on the same page here.

Finally, we come to observation of emission. The experimental procedure
is to prepare the system in some known state (again most likely
approximated by some stationary state), wait for the emission event and
for the detector to register the emission (wait for the detector to
also settle close to a stationary state). This matches the theoretical
description as well. But then we face the same uncertainty about the
time needed for emission.

However, now we have at least some bounds the needed time just by the
amount of time needed to setup the experiment and wait for the resutls.
With better technology, the time needed to perform the experiment can
be shortened. Can this be done until the least upper bound on the
emission duration is reached? Here, quantum mechanics says that you can
get close, but not quite. If you don't allow enough time for the system
to come close to a stationary state, the theoretical approximation and
the photon description break down. You will still measure photons
absorbed or emitted. But the results of the experiment will become more
and more erratic. In other words, the uncertainty in the number of
photons absorbed or emitted grows the less time you devote to setup of
the experiment and to observation. This is nothing but the energy time
uncertainty relation.

So, at the end of this perhaps overly verbose explanation, the simple
answer is given by the Heisenberg uncertainty principle. If the energy
of the photon is E, the time needed to emit a photon of this energy is
greater than hbar/E. The longer you wait, the more sure you are how
many quanta were emitted. The less you wait, the less certainty there
is about the number of photons emitted, which could also be zero.

Hope this helps.

Igor

.

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