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What is a photon?
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According to quantum electrodynamics, the most accurately verified
theory in physics, a photon is a single-particle excitation of the
free quantum electromagnetic field. More formally, it is a state of
the free electromagnetic field which is an eigenstate of the photon
number operator with eigenvalue 1.
The pure states of the free quantum electromagnetic field
are elements of a Fock space constructed from 1-photon states.
A general n-photon state vector is an arbitrary linear combinations
of tensor products of n 1-photon state vectors; and a general pure
state of the free quantum electromagnetic field is a sum of n-photon
state vectors, one for each n. If only the 0-photon term contributes,
we have the dark state, usually called the vacuum; if only the
1-photon term contributes, we have a single photon.
A single photon has the same degrees of freedom as a classical vacuum
radiation field. Its shape is characterized by an arbitrary nonzero
real 4-potential A(x) satisfying the free Maxwell equations, which in
the Lorentz gauge take the form
nabla dot nabla A(x) = 0,
nabla dot A(x) = 0,
expressing the zero mass and the transversality of photons. Thus for
every such A there is a corresponding pure photon state |A>.
Here A(x) is _not_ a field operator but a photon amplitude;
photons whose amplitude differ by an x-independent phase factor are
the same. For a photon in the normalized state |A>, the observable
electromagnetic field expectations are given by the usual formulas
relating the 4-potential and the fields,
<\E(x)> =
= - partial \A(x)/partial x_0 - c nabla_\x A_0(x),
and
<\B(x)> = = nabla_\x x \A(x)
[hmmm. check if the latter really is the case.]
Here \x (fat x) and x_0 are the space part and the time part of a
relativistic 4-vector, \E(x), \B(x) are the electromagnetic
field operators (related to the operator 4-potential by analogous
formulas), and c is the speed of light. Amplitudes A(x) producing
the same \E(x) and \B(x) are equivalent and related by a gauge
transformation, and describe the same photon.
In momentum space (frequently but not always the appropriate choice),
single photon states have the form
|A> = integral d\p^3/p_0 A(\p)|\p>,
where |\p> is a single particle state with definite 3-momentum
\p (fat p), p_0=|\p| is the corresponding photon energy divided by c,
and the photon amplitide A(\p) is a polarization 4-vector.
Thus a general photon is a superposition of monochromatic waves with
arbitrary polarizations, frequencies and directions.
(The Fourier transform of A(\p) is the so-called analytic signal
A^(+)(x), and by adding its complex conjugate one gets the real
4-potential A(x) in the Lorentz gauge.)
The photon amplitude A(\p) can be regarded as the photon's
wave function in momentum space. Since photons are not localizable
(though they are localizable approximately), there is no
meaningful photon wave function in coordinate space; see the
next entry in this FAQ. One could regard the 4-potential A(x) as
coordinate space wave function, but because of its gauge dependence,
this is not really useful.
This is second quantized notation, as appropriate for quantum fields.
This is how things always look in second quantization, even for a
harmonic oscillator. The wave function psi(x) or psi(p) in standard
(first quantized) quantum mechanics becomes the state vector
psi = integral dx psi(x) |x> or integral dp psi(p) |p>
in Fock space; the wave function at x or p turns into the coefficient
of |x> or |p>. In quantum field theory, x, A (the photon amplitude), and
E(x) (the electric field operator) correspond to k (a component of the
momentum), x, and p_k. Thus the coordinate index k is inflated to the
spacetime position x, the argument of the wave function is inflated to
a solution of the free Maxwell equations, the momentum operator is
inflated to a field operator, and the integral over x becomes a
functional integral over photon amplitudes,
psi = integral dA psi(A) |A>.
Here psi(A) is the most general state vector in Fock space; for a
single photon, psi depends linearly on A,
psi(A) = integral d\p^3/p_0 A(\p)|\p> = |A>.
Observable electromagnetic fields are obtained as expectation values
of the field operators \E(x) and \B(x) constructed by differentiation of
the textbook field operator A(x). As the observed components of
the mean momentum, say, in ordinary quantum mechanics are
= integral dx psi(x)^* p_k psi(x),
so the observed values of the electromagnetic field are
<\E(x)> = = integral dA psi(A)^* \E(x) psi(A).
<\B(x)> = = integral dA psi(A)^* \B(x) psi(A).
]
In a frequently used interpretation (valid only approximately),
the term A(\p)|\p> represents the one-photon part of a monochromatic
beam with frequency nu=cp_0/h, direction \n(\p)=\p/p_0, and
polarization determined by A(\p). Here h = 2 pi hbar, where hbar is
Planck's number; omega=cp_0/hbar is the angular frequency.
The polarization 4-vector A(\p) is orthogonal to the 4-momentum p
composed of p_0 and \p, obtained by a Fourier transform of the
4-potential A(x) in the Lorentz gauge. (The wave equation translates
into the condition p_0^2=\p^2, causality requires p_0>0, hence
p_0=|\p|, and orthogonality p dot A(\p) = 0 expresses the Lorentz
gauge condition. For massless particles, there remains the additional
gauge freedom to shift A(\p) by a multiple of the 4-momentum p, which
can be used to fix A_0=0.)
A(\p) is usually written (in the gauge with vanishing time component) as
a linear combination of two specific polarization vectors eps^+(p) and
eps^-(p) for circularly polarized light (corresponding to helicities +1
and -1), forming together with the direction vector \n(\p) an
orthonormal basis of complex 3-space. In particular,
eps^+(p) eps^+(p)^* + eps^-(p)eps^-(p)^* + \n(\p)\n(\p)^* = 1
is the 3x3 identity matrix. (This is used in sums over helicities for
Feynman rules.) Specifically, eps^+(p) and eps^-(p) can be obtained by
finding normalized eigenvectors for the eigenvalue problem
[check. The original eigenvalue problem is p dot J eps = lambda eps.]
p x eps = lambda eps
with lambda = +-i|p|. For example, if p is in z-direction then
eps^+(p) = (1, -i, 0)/sqrt(2),
eps^-(p) = (i, -1, 0)/sqrt(2),
and the general case can be obtained by a suitable rotation.
An explicit calculation gives almost everywhere
eps^+(p) = u(p)/p_0
where p_0=|p| and
u_1(p) = p_3 - i p_2 p'/p'',
u_2(p) = -i p_3 - i p_1 p'/p''
u_3(p) = p'
with
p' = p_1+ip_2,
p''= p_3+p_0.
[what is eps^-(p)?]
These formulas become singular along the negative p_3-axis,
so several charts are needed to cover
For experiments one usually uses nearly monochromatic light bundled
into narrow beams. If one also ignores the directions (which are
usually fixed by the experimental setting, hence carry no extra
information), then only the helicity degrees of freedom remain,
and the 1-photon part of the beam behaves like a 2-level quantum
system ('a single spin').
A general monochromatic beam with fixed direction in a pure state is
given by a second-quantized state vector, which is a superposition of
arbitrary multiphoton states in the Bosonic Fock space generated by
the two helicity degrees of freedom. This is the basis for most
quantum optics experiments probing the foundations of quantum
mechanics.
The simplest state of light (generated for example by
lasers) is a coherent state, with state vector proportional to
e(A) = |vac> + |A> + 1/sqrt(2!) |A> tensor |A>
+ 1/sqrt(3!) |A> tensor |A> tensor |A> + ...
where |A> is a one-photon state. Thus coherent states also have the
same degrees of freedom as classical electromagnetic radiation.
Indeed, light in coherent states behaves classically in most respects.
At low intensity, the higher order terms in the expansion are
negligible, and since the vacuum part is not directly observable,
a low intensity coherent states resembles a single photon state.
On the other hand, ordinary light is essentially never, and high-tech
light almost never, describable by single photons. True single photon
states are very hard to produce to good accuracy, and were created
experimentally only recently:
B.T.H. Varcoe, S. Brattke, M. Weidinger and H. Walther,
Preparing pure photon number states of the radiation field,
Nature 403, 743--746 (2000).
see also
http://www.qis.ucalgary.ca/quantech/fock.html
A good informal discussion of what a photon is from a more practical
perspective was given by Paul Kinsler in
http://www.lns.cornell.edu/spr/2000-02/msg0022377.html
But this does not tell the whole story. An interesting collection of
articles explaining different current views is in
The Nature of Light: What Is a Photon?
Optics and Photonics News, October 2003
http://www.osa-opn.org/Content/ViewFile.aspx?Id=3185
Further discussion is given in the section ''Coherent states of light
as ensembles'' of the present FAQ, and in my slides
http://www.mat.univie.ac.at/~neum/ms/lightslides.pdf
http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
The standard reference for quantum optics is
L. Mandel and E. Wolf,
Optical Coherence and Quantum Optics,
Cambridge University Press, 1995.
Mandel and Wolf write (in the context of localizing photons),
about the temptation to associate with the clicks of a photodetector
a concept of photon particles. [If there is interest, I can try to
recover the details.] The wording suggests that one should resist the
temptation, although this advice is usually not heeded. However,
the advice is sound since a photodetector clicks even when it
detects only classical light! This follows from the standard analysis
of a photodetector, which treats the light classically and only
quantizes the detector. Thus the clicks are an artifact of
photodetection caused by the quantum nature of matter, rather than
a proof of photons arriving!!!
A coherent light source (laser) produces a coherent state of light,
which is a superposition of the vacuum state, a 1-photon state,
a 2-photon state, etc, with squared amplitudes given by a Poisson
distribution. At low intensity, this is misinterpreted in practice
as random single photons arriving at the end of the beam in a
random Poisson process, because the photodetector produces clicks
according to this distribution.
Incoherent light sources usually consist of thermal mixtures and
produce other distributions, but otherwise the description (and
misinterpretation) is the same.
Nevertheless, one must understand this misinterpretation in order
to follow much of the literature on quantum optics.
Thus the talk about photons is usually done inconsistently;
almost everything said in the literature about photons should be taken
with a grain of salt.
There are even people like the Nobel prize winner Willis E. Lamb
(the discoverer of the Lamb shift) who maintain that photons don't
exist. See towards the end of
http://web.archive.org/web/20040203032630/www.aro.army.mil/phys/proceed.htm
The reference mentioned there at the end appeared as
W.E Lamb, Jr.,
Anti-Photon,
Applied Physics B 60 (1995), 77--84
This, together with the other reference mentioned by Lamb, is reprinted
in
W.E Lamb, Jr.,
The interpretation of quantum mechanics,
Rinton Press, Princeton 2001.
I think the most apt interpretation of an 'observed' photon as used
in practice (in contrast to the photon formally defined as above) is
as a low intensity coherent state, cut arbitrarily into time slices
carrying an energy of h*nu = hbar*omega, the energy of a photon at
frequency nu and angular frequency omega.
Such a state consists mostly of the vacuum (which is not directly
observable hence can usually be neglected), and the contributions of
the multiphoton states are negligible compared to the single photon
contribution.
With such a notion of photon, most of the actual experiments done make
sense, though it does not explain the quantum randomness of the
detection process (which comes from the quantized electrons in the
detector).
A nonclassical description of the electromagnetic field where states of
light other than coherent states are required is necessary mainly for
special experiments involving recombining split beams, squeezed
state amplification, parametric down-conversion, and similar
arrangements where entangled photons make their appearance.
There is a nice booklet on this kind of optics:
U. Leonhardt,
Measuring the Quantum State of Light,
Cambridge, 1997.
Nonclassical electromagnetic fields are also relevant in the
scattering of light, where there are quantum corrections
due to multiphoton scattering. These give rise to important effects
such as the Lamb shift, which very accurately confirm the quantum
nature of the electromagnetic field. They involve no observable
photon states, but only virtual photon states, hence they are unrelated
to experiments involving photons. Indeed, there is no way to observe
virtual particles, and their name was chosen to reflect this.
(Observed particles are always onshell, hence massless for photons,
whereas it is an easy exercise that the virtual photon mediating
electromagnetic interaction of two electrons in the tree approximation
is never onshell.)