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S1i. Modes and wave functions of laser beams
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The physical state described by a typical laser beam is a state with
an indeterminate number of photons, since it is usually not an
eigenstate of the photon number operator. This essentially means that
in a beam, a certain number of photons cannot be meaningfully asserted;
instead, one has a meaningful photon density, referred to as the beam
intensity.
Thus the traditional N-particle picture does not apply.
Instead one has to work in a suitable Fock space.
The Maxwell-Fock space is obtained by 'second quantization' of the mode
space H_photon, consisting of all mode functions, i.e., solutions A(x)
of the free Maxwell equations, describing a classical background
electromagnetic field in vacuum. H_photon may be thought of as the
single photon Hilbert space, in analogy to the single electron Hilbert
space of solutions of the Dirac equation. (However, following up on
this analogy and calling A(x) a wave function leads to confusion later
on, and is best avoided.)
Actually, because of gauge invariance, the situation is slightly more
complicasted, and best described in momentum space. The Maxwell
equations reduce in Lorentz gauge, partial dot A(x) = 0, to
partial^2 A(x)=0, whence the Fourier transform of A(x) has the form
delta(p^2) Ahat(p), and Ahat(p) must satisfy the transversality
condition
p dot Ahat(p) = 0.
By gauge invariance, only the coset of Ahat(p) obtained by adding
arbitrary multiples of p has a physical meaning, reflecting the
transversal nature of the free electromagnetic field.
This coset construction is needed to turn the space of modes
into a Hilbert space H_photon with invariant inner product
= integral Ahat(p) dot Bhat(p) Dp,
where
Dp = d\p/p_0 = dp_1 dp_2 dp_3/p_0,
is the Lorentz invariant measure on the photon mass shell,
0 < p_0 = |\p| = sqrt(p_1^2+p_2^2+p_3^2)
(negative frequencies are discarded to get an irreducible
representation of the Poincare group).
Indeed, without the coset construction, the inner product is only
positive semidefinite, hence gives only a pre-Hilbert space.
Each (sufficiently nice) mode function A(x) gives rise to a coherent
state ||A>> in the Maxwell-Fock space, to an associated annihilation
operator
a(A) = integral Ahat(p) a(p) Dp,
where a(p) is the QED annihilation operator for a photon with
momentum p, and to the corresponding creation operator a^*(A) = a(A)^*.
The annihilation and creation operators a(A) and a^*(A) produce a
single-mode Fock subspace consisting of all |A,psi>, where psi is the
unnormalized wave function of a harmonic oscillator; |psi|^2 is the
intensity of the beam.
The coherent state itself corresponds to the normalized vacuum state
of the harmonic oscillator, ||A>> = |A,vac>. If psi is a Hermite
polynomial H_k, |A,psi> is an eigenstate of the photon number operator
with eigenvalue k, and one has a k-photon state.
The Maxwell-Fock space is the closure of the space spanned by all
the |A,psi> together (and indeed, already the closure of the space
spanned by all ||A>>). This space is the pure electromagnetic field
sector of QED, describing a physical vacuum, i.e., a region of the
universe where matter is absent though radiation may be present.
In optics experiments, laser beams are often idealized by ignoring
their extension perpendicular to the transmission direction. Then each
beam can be described by some |A,psi>. In particular, for a
monochromatic beam, A is a plane wave, A(x)=A_0 exp(-i p dot x).
Of course, this matches the original approximation that we have a
beam only with a grain of salt, since a plane wave is not normalized.
A coherent pair of laser beams obtained by splitting is described by
a superposition |A_1,psi_1> + |A_2,psi_2> of the two beams.
Beams of thermal light (such as that from the sun) and pairs of
beams created by independent sources, cannot be described by wave
functions alone, but need a density formulation. A single light beam
is then described (in the same idealization) by a mode A and a density
matrix rho in a single-mode Fock space, while k light beams are
described by k modes A and a density matrix rho in a k-mode Fock space.
In many treatments, the modes are left implicit, so that one works
only in the k-mode Fock space. This simplifies the presentation, but
hides the connection to the more fundamental QED picture.
For a thorough study of the latter, see the bible on quantum optics,
L. Mandel and E. Wolf,
Optical Coherence and Quantum Optics,
Cambridge University Press, 1995.