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S2j. Coherent states of light as ensembles
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Let us look in some detail at the setting of a weak laser switched on
at time t_0 and switched off again at time t_1. The time T:=t_1-t_0
that the laser is switched on is a variable that we can choose at will.
Conventionally one models the light produced by a laser by coherent
states. If one tests the photon contents at the end of the beam by a
photodetector, one measures a series of clicks indicating (according to
tradition) the presence of single photons. Each click is conventionally
regarded as the measurement of a single photon; hence one measures an
ensemble of photons. Without this interpretation, much of the talk
about photons in quantum optics would not make sense.
Technically, and completely precisely, one has an ensemble of photons
in an indefinite photon number state. (Even a superposition of states
describes an ensemble, in the conventional interpretation.)
In a weak coherent state, the multiparticle contents is negligible;
one has essentially a superposition of the vacuum and the single
particle state. Conventionally (as for all somewhat rare events),
the vacuum part is ignored - one just restricts attention to the
times where a particle is present. This leaves a single particle state.
Thus, at least for weak coherent states, it is a good approximation
to say that a coherent state of definite frequency is an ensemble
of single-particle systems.
More formally, in the usual abbreviated form, a weak coherent state of
a stationary monochromatic beam has the form
|psi> = (1-eps||0> + eps|1> + O(eps^2), (*)
with eps<<1, and
= = = eps^2 + O(eps^3)
is not a mean photon number, but a mean rate - the mean intensity.
More precisely, each coherent state has a mode A=A(p); the modes are in
1-1 correspondence with creation operators a^*(A). They create,
in field theory language, one photon in this mode. So far, these
photons are only constructs on paper, used to be able to write down
multiparticle states, and have not yet an observable meaning.
An N-particle state of mode A is defined recursively from the vacuum
state by
|1,A> := a^*(A)|vac>, |N,A>: = a^*(A)|N-1,,A> for N>0,
and coherent states with mode A have the form
|z,A>> := const* sum_N z^N/sqrt{N!} |N,A>
with a complex amplitude z. and satisfies
a(A)|z,A>> = z|z,A>>.
The mean photon number associated with the coherent state is
Nbar := = = <>
= <> = z^*z <> = z^*z,
hence
Nbar = |z|^2,
independent of the time T.
The events are the clicks, and there is exactly one click per event
in a weak signal (for strong signals, one cannot separate the events).
But the events happen randomly in time, with a rate proportinal to eps.
It is conventional to regard each click as evidence for the presence
of a single photon - this more or less defines the experimental notion
of a photon. (See also the discussion in the section
''What is a photon?'' of this FAQ.)
Note that two photons arriving at different times cannot be considered
as being part of a N-particle state with N>1, since states are
considered at a fixed time! Also, the fact that the weak coherent
state has a negligible contribution of doubly excited states
means that N-particle state with N>1 are here completely irrelevant.
Thus one has an ensemble of single photons.
Clearly, the number of observable photons (in the sense
of detector clicks) is proportional to T. This shows that the
formal photon number operator in Fock space, N = a^*(A)a(A), has
nothing to do with the photon number as defined by the number of clicks;
instead its expectation is proportional to the mean rate of clicks
per unit time.
Thus (*) describes an ensemble of O(T*eps^2) single photons, where
$T$ is the duration of the experiment.
In particular, plane monochromatic light in the form of a coherent state
(three mathematical idealizations involved here) is an endless stream
of infinitely many photons passing with the speed of light through a
particular position on the beam. The rate of emission of photons is
proportional to the intensity of the incident beam. But the fact that
the model is an approximation only and that for real preparations,
observations are bounded in space and time does not change the results
of this analysis.
On the other hand, it is clear that a coherent state is not a 1-photon
state but a state with an indefinite number of photons (i.e., not an
eigenstate of the number operator). Thus there seems to be a conflict
in terminology - weak laser light is describerd by a coherent state
without definite number contents, but it behaves experimentally as
an ensemble of single photons.
This shows that the concept of a photon is somewhat ambiguous.
Different people mean different and often quite vague things by
''photon'', if they bother to spell out the meaning in some detail
(which is usually not done). This can be seen from the diverging
explanations given in a recent special issue on this topic:
The Nature of Light: What Is a Photon?
Optics and Photonics News, October 2003
http://www.osa-opn.org/Content/ViewFile.aspx?Id=3185
which presents five mutually incompatible views,
* Light reconsidered (Arthur Zajonc)
* What is a photon? (Rodney Loudon)
* What is a photon? (David Finkelstein)
* The concept of the photon - revisited (Ashok Muthukrishnan,
Marlan O. Scully, and M. Suhail Zubairy)
* A photon viewed from Wigner phase space (Holger Mack and
Wolfgang P. Schleich)
In QED, a ''one-photon state'' is a well-defined object, but ''one
photon'' in an experiment is not (unless one identifies it with a
detector click - which leaves unsaid what an undetected photon would
be). The relation between the two is quite indirect, and there is no
agreement in the literature on the precise relation.
My own views (not mainstream, but consistent with experiment) are:
1. that clicks have nothing at all to do with photons, they are just
a stochastic measure of intensity, and arise also if the incindent
field is modelled completely classical;
2. that what is typically called a photon is not an arbitrary single
particle state of the electromagnetic field (in particular, never
an approximately plane wave) but a state of the electromagnetic field
that at each time is localized in space, whose energy contents is
that of hbar*omega. Otherwise, the idea of producing photons of
demands makes no sense.
3. It is the field of the incident beam that counts; the talk about
photons in the incoming beam is not very meaningful and only blurs
the picture; the right language is that of field theory.
Indeed, a theoretical model of a photo-detector excited by an external
classical monochromatic e/m field contains no photons, but in this
model the detector responds by clicking randomly according to a Poisson
statistics; see Chapter 9 of the book
L. Mandel and E. Wolf,
Optical Coherence and Quantum Optics,
Cambridge University Press, 1995.
Thus a precise meaning of ''photon'' is not needed to defend
statement 1.
No matter which view one takes with regard to statement 1., the
question is how one relates a 1-photon state to what one actually
prepares in a beam of light. What does it mean in experimental terms
to have prepared _one_ photon in this state?
Reading the details of preparation schemes for photons on demands
as discussed (with references to the original literature) in
http://www.mat.univie.ac.at/~neum/ms/lightslides.pdf
http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
one finds that no clear answer can be given to this question, but
that the evidence points to statement 2. ov my view presented above.
In this view, the difference between the preparation of a coherent
state and that of a single photon is that a weak coherent state
generates an infinitely long random sequence of Poisson-distributed
clicks, while a single photon (in the above sense of a space-localized
field) generates (in an ideal detector) a single click only.
The practice seems to be that one silently ignores the vacuum
contribution in (*) and obtains after rescaling to a normalized state
a state
psi' = |1> + O(eps) (*')
which, with perfect right, can be considered to be an approximate
1-photon state. Indeed, most photon states produced in the laboratory
are superposition with the vacuum, and still people speak of photons.
This also holds for other systems than simple laser light. For example,
entanglement studies are typically made with squeezed states,
which differ from coherent states only in that they have instead
of (*) a representation
psi = (1-eps||0> + eps|2> + O(eps2), (**)
and everyone refers to (**) as an ensemble of 2-photon states.
Indeed, parametric down conversion is well-known to produce an
ensemble of 2-photon states, but if one looks closer at the models
one finds that they actually produce states of the form (**)
that produce endless streams of photon pairs.
While photons on demand are based on exciting single atoms,
the only way of reliably creating single photons was for a long time
to use a source in a state of the form (**), where the photon pairs
are entangled pairs of photons with different momentum vectors (hence
located on different beams). Then one observes photons (clicks) on the
left beam with a detector, and knows from general principles that at
the same time a photon is underway in the other beam. Thus one can know
about the presence of single photons without having them observed yet.
This interpretation again explains away the vacuum part of the state
in (**). One restricts attention to the 2-photon sector of (**) by
ignoring the times where nothing but the vacuum part is observed, and
focuses on the times when something - and then by the form of (**)
the 2-photon part - is observed. This is the sense in which one
interprets as an ensemble of 2-photon states.
Then one observes the part of the 2-photon system in one beam, to know
when a photon is present in the other beam. Bot of course, although
this is the way talked about the situation, in reality one still has
the superposition with the vacuum, except that one chooses to ignore
the times where nothing happens to get rid of the vacuum.