Raymer-Smith, Technika Fizyka
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Photon wave functions, wave-packet quantization of light,
and coherence theory
Brian J Smith
1,2
and M G Raymer
2
1
Clarendon Laboratory, Oxford University, Parks Road, Oxford, OX1 3PU, UK
2
Oregon Center for Optics and Department of Physics, University of Oregon, Eugene,
Oregon, 97403, USA
Email:
b.smith1@physics.ox.ac.uk
Abstract.
The monochromatic Dirac and polychromatic Titulaer-Glauber quantized field
theories (QFTs) of electromagnetism are derived from a photon-energy wave function in
much the same way that one derives QFT for electrons, that is, by quantization of a single-
particle wave function. The photon wave function and its equation of motion are established
from the Einstein energy-momentum-mass relation, assuming a local energy density. This
yields a theory of photon wave mechanics (PWM). The proper Lorentz-invariant single-
photon scalar product is found to be non-local in coordinate space, and is shown to correspond
to orthogonalization of the Titulaer-Glauber wave-packet modes. The wave functions of
PWM and mode functions of QFT are shown to be equivalent, evolving via identical
equations of motion, and completely describe photonic states. We generalize PWM to two or
more photons, and show how to switch between the PWM and QFT viewpoints. The second-
order coherence tensors of classical coherence theory and the two-photon wave functions are
shown to propagate equivalently. We give examples of beam-like states, which can be used as
photon wave functions in PWM, or modes in QFT. We propose a practical mode converter
based on spectral filtering to convert between wave packets and their corresponding
biorthogonal dual wave packets.
PACS:
03.65.Pm, 03.65.Ta, 03.70.+k, 42.50.Ar, 42.50.Dv, 42.50.-p, 11.30.Cp
Submitted to:
New J. Phys.
1. Introduction—Photon Wave Mechanics
2
2. From Monochromatic Modes to Wave-Packet Modes
4
3. Deriving the single-photon wave equation from Einstein kinematics
7
4. Quantization of the single-photon wave function
12
5. Modes versus states and the wave-function extraction rule
16
6. Two-photon and multiphoton wave mechanics
17
6.1. Two-photon wave mechanics
17
6.2. Two-time wave functions
19
6.3. Two-photon mixed states
19
6.4. Multiphoton wave mechanics
20
Photon wave functions, wave-packet quantization of light, and coherence theory
2
7. Relation of photon wave mechanics to the Wolf equations of classical coherence theory
20
8. Relation of PWM to two-photon detection amplitudes
22
9. Examples and applications
23
9.1. Example wave-packet modes
23
9.2. Conversion between modes and dual-modes
25
9.3. Decoherence of spatially-entangled photon pair by atmospheric turbulence
27
10. Conclusions
27
Acknowledgements
28
Appendix A. Dual-mode basis in terms of the wave-packet modes
28
Appendix B. Two possible momentum-space wave function normalizations
28
Appendix C. Lorentz invariance of scalar-product kernel function
29
Appendix D. Lorentz transformation properties of momentum-space wave functions
29
References
30
1. Introduction—Photon Wave Mechanics
There are still many puzzling aspects of the nature of light. A central point to understand is the
distinction between a corpuscular viewpoint and a field viewpoint of light. In the old days, this was
called wave-particle duality, but this phrase does little justice to the subtle issues involved. A key
question is, can we view light as being comprised of particles called
photons
, or must one view light
as a field, and the “number of photons” only as the name we give to quantum states of the
electromagnetic field [1]? Certainly, one can create single-photon wave packets, which are more or
less localized in space-time, and we can describe them using standard quantum field theory (QFT) [2-
4]. In some papers, authors ask questions such as, which path did the photon take through my
interferometer? What slit did the photon go through? With sophistication, such questions can properly
refer to alternative quantum amplitudes that contribute to the final amplitude for detection. These
questions, however, presuppose that we know how to write down formulas for photon wave functions
to represent these alternatives, along with a proper quantum wave equation for the photon wave
function. Most papers fail to do that, yet many continue to use the photon-as-particle language loosely
or sometimes even sloppily. In cases where we care only about which beam a photon follows as it
traces its way through an interferometer made of beam splitters and mirrors, such a coarse language is
probably fine [5, 6]. On the other hand, when a problem involves light diffraction, ultrashort pulses,
or other spatially complex phenomena, one needs to use a more refined theory based on a photon
wave equation.
It is known that one can describe single-photon states of light using a photon-as-particle
viewpoint, specifying the photon wave function (PWF). We call this approach
photon wave
mechanics
(PWM). Nevertheless, not all quantum optics researchers are well versed in the techniques
for handling single-photon wave packets and photon wave functions. Also, some of the deeper
connections between photon wave functions, quantized-field wave packets, and optical coherence
theory have not been previously reported. These concerns motivate the present paper.
A main theme of this article is that there is utility in being able to switch correctly between a
photon viewpoint and a field viewpoint. We begin by reviewing and extending the QFT of photon
wave packets introduced by Titulaer and Glauber (T-G) [7]. We then briefly review the theory of the
“energy-density photon wave function” in coordinate space, which has developed over the past dozen
years [8-11], and we extend it in several ways. We show that both the quantized field theory of light
developed by Dirac [12], based on monochromatic modes, as well as its generalization to non-
monochromatic modes developed by Titulaer and Glauber, can be derived directly from the photon-
Photon wave functions, wave-packet quantization of light, and coherence theory
3
. Therefore, as well argued by Bialynicki-Birula [8-10]
and by Sipe [11], for photons it is best to adopt a wave function whose modulus squared is the
photon’s mean energy density, rather than being a position probability density, as is the case for
electrons. We call this the
mean-energy-density wave function
or the Bialynicki-Birula-Sipe (BB-S)
wave function and its equation of motion the
photon wave equation
. A connection of PWM to
experiments can be seen in the authors’ determination of transverse spatial PWFs at the single-photon
level [16].
The wave functions (states) of a single photon, when treated as a particle-like object, are found to
be equivalent to the mode functions of the quantized electromagnetic field. This field is conveniently
expressed in terms of the complex electromagnetic field,
. This is also called the Riemann-
Silberstein (RS) vector field [10]. Conversely, the QFT for light is constructed by quantizing the
single-photon wave function. When the connections between the different formalisms are understood,
it can be seen that there are at least two ways of interpreting the photon-wave-function theory. One
interpretation is as a theory of quantum particles [17]. The other is to interpret PWF theory as an
alternative means for describing states and dynamics of the quantum field in the case that we can
restrict the description to a fixed subspace of the larger Fock space of photon numbers. Then, there is
a clear relation between the PWF and the mode functions that appear in the state description of the
quantum field. A subtlety arises when treating the Hilbert-space scalar product for PWFs. We present
the Lorentz-invariant scalar product of the photon wave function, which is non-local in the coordinate
representation. This form of scalar product can also be used to better interpret, understand, and utilize
the Titulaer-Glauber wave-packet quantization formalism.
After showing how to switch from a photon viewpoint to a field viewpoint by quantization of the
single-photon wave mechanics theory, we show how to switch in the opposite direction. That is,
starting from conventional Dirac-Titulaer-Glauber QFT for light, we can extract the correct forms for
photon wave functions and their equations of motion in coordinate space. This connects to similar
treatments by Muthukrishnan et al, Lapaire and Sipe, and by Eberly et al. [3, 4, 18, 19]. This result
shows how to incorporate interactions between light and matter in the photons viewpoint. A related
example is that of Fini et al, who formulated the propagation of intense solitons in a Kerr third-order
nonlinear optical medium in terms of the coordinate-space wave functions of many photons [20].
Single-photon spatial-temporal states can be used for a non-monochromatic wave-packet basis set
in which to expand the electromagnetic field in QFT. As an example of this, we introduce a basis set
of beam-like wave-packet modes with broad spectra. Such wave-packet modes are not orthogonal
under the standard, local, coordinate-space scalar product, but are orthogonal under the non-local
scalar product arising in photon wave mechanics. The closely related concept of the dual-mode basis
is discussed in terms of orthogonalization of the T-G wave-packet modes, and leads directly to the
non-local scalar product. We propose an experimental scheme to convert between wave-packet modes
and their dual modes through a spectral-filtering scheme similar to a pulse shaper.
We extend the single-photon wave mechanics by developing two-photon and
n
-photon wave
mechanics [21]. We show that there are deep connections between this extended photon-wave-
mechanics theory and optical coherence theory—both quantum and classical. These connections are
shown to be related to standard photodetection theory and to the standard wave-function collapse
hypothesis. We also present a description of entanglement in the state of two photons, and the correct
method for reduction of a pure two-photon wave-function state to a single-photon density matrix
written in terms of single-photon wave functions. Here the non-local scalar product plays a crucial
role in eliminating information about the traced-out photon.
as-particle viewpoint. This actually provides a “derivation” of the Maxwell equations, starting from
fundamental principles. One does this by considering the relativistic particle kinematics of a single
photon, finding a formulation for a single photon that is analogous to the Dirac equation for an
electron, and then quantizing that single-photon theory to create a quantum field theory of light. The
derivation parallels that of Dirac for the electron and its quantum field [13, 14]. A key difference
between the electron and photon derivations has to do with the famous localization problem for the
photon [15]. Whereas (non-relativistic) electrons can be in a position eigenstate, at least in principle, a
photon cannot. On the other hand, the energy density of the electromagnetic field in free space can be
expressed as a local quantity,
E
2
(
x
)+
c
2
B
2
(
x
)
E
+
ic
B
Photon wave functions, wave-packet quantization of light, and coherence theory
4
2. From Monochromatic Modes to Wave-Packet Modes
We first develop the theory of photon wave packets (WP) in terms of QFT. The quantized
electromagnetic theory developed by Dirac [12] starts from the classical Maxwell theory of the
electromagnetic field, which is canonically quantized in terms of monochromatic modes. In free
space, the electric and magnetic-induction field operators obey the Maxwell equations, in SI units,
1
t
ˆ
Ex
,
(
=
c
2
ˆ
Bx
,
t
t
(
,
ˆ
Ex
,
t
(
= 0,
(1)
t
ˆ
Bx
,
(
=
ˆ
Ex
,
t
t
(
,
ˆ
Bx
,
t
(
= 0.
The positive-frequency parts of the fields may be expanded using monochromatic modes as [12, 22-
25]
d
3
k
(2)
3
ck
2
0
1/2
E
()
x
,
(
=
i
t
a
k
,
u
k
,
()
exp
i
k
t
( )
,
(2)
d
3
k
(2)
3
ck
2
0
1/2
ck
u
k
,
()
B
()
x
,
t
(
=
i
a
k
,
exp
i
k
t
( )
,
(3)
where
k
=
ck
=
c
k
,
c
is the vacuum speed of light, and
0
is the permittivity of the vacuum. The
monochromatic, orthonormal, plane-wave modes are
u
k
,
()
=
e
k
,
exp
i
k
( )
,
(4)
where the
e
k
,
are unit polarization vectors. The sum is over the two mode-polarization indices
. It turns out to be advantageous to assume circular polarization for modes (corresponding to
positive and negative helicity), so we do this throughout this paper. For circular polarization
k
e
k
,
=
, (5)
although we prefer not to invoke this here, in order to keep (3) general. An advantage to using the
plane-wave modes
i
k e
k
,
u
k
,
()
is that they are orthogonal under the standard definition of the scalar
( )
product, which hereafter we call the
overlap integral
, and denote it by
. This is given by
( )
=
u
k
,
()
u
k
,
()
d
3
x
= 2
()
3
()
k
( )
,
.
(6)
(
, (7)
where the negative-frequency parts are given by Hermitian conjugates of the positive-frequency
operators,
E
()
x
,
(
=
E
()
x
,
t
(
+
E
()
x
,
t
(
,
ˆ
Bx
,
t
(
=
B
()
x
,
t
(
+
B
()
x
,
t
(
=
E
()
x
,
t
()
†
and
B
()
x
,
(
=
B
()
x
,
t
t
()
†
. The monochromatic annihilation
and creation operators
a
k
,
and
a
k
,
†
obey the bosonic commutation relations
a
k
,
,
a
k
,
†
= (2)
3
()
(
k
k
)
,
.
(8)
Excitation-number operators are
n
k
,
=
a
k
,
†
a
k
,
. (We refrain from using the word
photon
here.) The
electromagnetic-field Hamiltonian operator is expressed in terms of the annihilation and creation
operators as
H
=
d
3
k
(2)
3
k
a
k
,
†
a
k
,
,
(9)
where we follow the common practice of neglecting the infinite vacuum energy term. The interaction
of the quantized electromagnetic field with atomic systems can be introduced through an interaction
term (usually the electric-dipole interaction in non-relativistic treatments) in the atom-field
Hamiltonian.
The free-space field operators in (2) and (3) are expressed in terms of plane waves, which serve
well for simple models. However, when localized space-time interactions are considered, such as
1
We “derive” the Maxwell equations in Section 4.
=±1
u
k
,
u
k
,
The Hermitian field operators are
ˆ
Ex
,
t
t
Photon wave functions, wave-packet quantization of light, and coherence theory
5
[7]. Such modes
were used to study transient Raman scattering [26], and have been recently discussed in terms of the
transverse spatial modes of light [27]. These wave-packet modes are related to the orthogonal,
monochromatic, plane-wave modes through the non-unitary transformation
v
j
,
x
,
()
t
v
j
,
x
,
t
(
=
i
c
2
0
1/2
d
3
k
(2)
3
k U
j
()
()
u
k
,
()
exp
i
k
t
( )
,
(10)
is a unitary transformation “matrix.” Equation (4) shows that this relation is a
Fourier transform, which can be inverted to give
U
j
()
()
e
k
,
=
2
0
c
1/2
( )
ik
d
3
x
(
exp
i
k
( )
.
exp
i
k
t
v
j
,
x
,
t
(11)
We call
v
j
,
x
,
()
t
the wave-packet (WP) modes. The “matrix”
U
j
()
()
is unitary, that is, for fixed
value of
,
j
U
j
()*
()
U
j
()
()
= (2)
3
()
(
k
k
),
d
3
k
(2)
3
U
j
()
()
U
j
()
()
=
j
j
.
(12)
Nevertheless, the
r
elation between the WP modes and the monochromatic modes is not unitary
because of the
k
factor in (10). This turns out to be a crucial point.
The annihilation and creation operators are changed by the unitary transformation, leading to new
annihilation and creation operators
b
j
,
and
b
j
,
†
given by
b
j
,
=
d
3
k
(2)
3
U
j
()
()
a
k
,
,
(13)
. This means that one can
construct states of definite excitation number
N
in a particular WP mode by applying the creation
operator to the vacuum state:
=
j
,
m
,
(
b
j
,
†
)
N
vacuum =
N
j
,
. The inverse of (13) is
a
k
,
=
j
U
j
()
()
b
j
,
.
(14)
In terms of the WP modes, the positive-frequency parts of the electric and magnetic field operators
are
E
()
x
,
(
=
t
j
,
b
j
,
v
j
,
x
,
()
t
,
(15)
B
()
x
,
(
=
t
j
,
b
j
,
k
j
v
j
,
x
,
()
t
.
(16)
c
are orthogonal under the standard definition
of the scalar product, that is the overlap integral, (6). In contrast, the WP modes
u
k
,
()
()
, do not
generally form an orthogonal set under a scalar product defined by the overlap integral.
2
That is,
v
j
,
x
,
t
( )
=
v
j
,
()
v
m
,
x
,
t
t
()
d
3
x
=
c
2
0
d
3
k
(2)
3
kU
j
()
()
U
()
()
jm
.
(17)
The n
on
-orthogonality arises from different weightings given to different frequency components by
the
k
factor in (10). In fact, the WP modes are overcomplete, which might appear to be a
2
The wave-packet modes are also non-orthogonal under the integral
d
3
x
dt
.
spontaneous emission from an atom, [4, 11], the plane-wave description becomes inefficient. In such
a case, Titulaer and Glauber (T-G) showed that one may expand the electric and magnetic fields in
terms of non-orthogonal, non-monochromatic, spatial-temporal modes
in which
U
j
()
()
which obey bosonic commutation relations
b
j
,
,
b
m
,
†
k
j
The monochromatic plane-wave basis functions
v
j
,
v
m
,
x
,
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