Vacuum source-field correlations and advanced
waves in quantum optics
Adam Stokes
Photon Science Institute, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
January 17, 2018
The solution to the wave equation as a Cauchy
problem with prescribed fields at an initial time
t = 0 is purely retarded. Similarly, in the
quantum theory of radiation the specification of
Heisenberg picture photon annihilation and cre-
ation operators at time t > 0 in terms of opera-
tors at t = 0 automatically yields purely retarded
source-fields. However, we show that two-
time quantum correlations between the retarded
source-fields of a stationary dipole and the quan-
tum vacuum-field possess advanced wave-like
contributions. Despite their advanced nature,
these correlations are perfectly consistent with
Einstein causality. It is shown that while they do
not significantly contribute to photo-detection
amplitudes in the vacuum state, they do effect
the statistics of measurements involving the ra-
diative force experienced by a point charge in
the field of the dipole. Specifically, the disper-
sion in the charge’s momentum is found to in-
crease with time. This entails the possibility of
obtaining direct experimental evidence for the
existence of advanced waves in physical reality,
and provides yet another signature of the quan-
tum nature of the vacuum.
1 Introduction
Electrodynamics describes the production, propagation
and absorption of electromagnetic waves. The problem
most commonly encountered in classical electrodynam-
ics is that in which the source is specified at some ini-
tial time, and then the electromagnetic fields are sought
at later times. The unique solution to this problem is
fully retarded, which means that the electromagnetic
source-fields depend in a causal way on the state of the
source in the past [1, 12, 36]. Advanced source-fields,
which depend on the source in the future are obtained
when Maxwell’s equations are solved running into the
past [1, 31]. These solutions are not viewed as physical
within the context of an initial value problem.
It is well known that in quantum optics the com-
mutators of the free electromagnetic fields involve both
retarded and advanced green’s functions for the wave-
operator [6]. On the other hand the subject of advanced
waves in the presence of sources has received more lim-
ited attention in quantum theory. The reason for this
may be that the problem of finding the Heisenberg pic-
ture source-fields at time t > 0 in terms of Schr¨odinger
picture operators at time t = 0 is an initial value prob-
lem whose solution entails purely retarded source-fields.
However, due to differences between quantum and clas-
sical theories the complete absence of advanced contri-
butions in quantum optics is not immediate. The fluc-
tuating quantum vacuum offers one of the most striking
examples of such a difference. The theories also differ
in their treatment of electromagnetic correlations.
The quantum vacuum is used to interpret various
phenomena in quantum optics including spontaneous
emission [18], atomic energy-shifts [24, 35], and Casimir
forces [5]. Together with the laws of space and time
the quantum theory of the vacuum predicts intriguing
physical phenomena such as Hawking radiation asso-
ciated with black holes [11] and the Unruh-Davies ef-
fect whereby an accelerated detector in the vacuum
of Minkowski spacetime registers thermal excitations
[8, 34]. The Unruh-Davies effect and related effects [20]
can be understood in terms of correlations between the
quantum vacuum at timelike separated points in space-
time. Vacuum induced correlations across spacelike sep-
arations have also received attention [25].
In atomic and optical physics many effects attributed
to the quantum vacuum can often also be understood, at
least partially, in terms of radiation reaction [7, 13, 17].
Thus, identifying specific signatures associated with the
quantum vacuum remains of broad interest in quantum
optics. Recently, direct observation of vacuum fluctu-
ations has been achieved [19, 26, 27], which opens up
new possibilities for investigating the interplay between
vacuum fluctuations and the electromagnetic fields pro-
duced by charged sources.
In this paper we consider the fields produced by a
quantum dipole. We show that the presence of the
Accepted in Quantum 2017-12-15, click title to verify 1
arXiv:1710.06207v3 [quant-ph] 16 Jan 2018
quantum vacuum and the distinct nature of quantum
correlations imply that certain physical predictions in-
volve advanced waves. More specifically, we show that
the combination of purely retarded source-fields and
quantum vacuum fields result in advanced-wave con-
tributions to two-time quantum correlations.
There are seven sections to this paper. We begin in
section 2 by briefly reviewing the interpretation of waves
in classical physics. In section 3 we introduce a model of
the dipole-field system and derive the electromagnetic
source-fields of the dipole. In section 4 we discern the
role of interference between vacuum and source-fields
within the process of spontaneous emission. In section
5 we consider general two-time quantum correlations
functions of the electromagnetic field, and show that
vacuum-source correlations receive advanced-wave con-
tributions. Our subsequent aim is to determine whether
such correlations might be physically measurable. In
section 6 we briefly verify that vacuum-source correla-
tions do not significantly contribute to photo-detection
amplitudes. In section 7 we consider the radiative force
experienced by a test charge in the field of a quantum
dipole. By considering a charge with relatively slow dy-
namics, we are able to develop a quantum description
of the free charge, which is analogous to the multipolar
description of bound charge systems. This description
enables us to properly identify the relevant vacuum and
source-fields. We then show that vacuum-source corre-
lations make signficant contributions to the statistical
predictions involving the force experienced by the test
charge. Finally in section 8 we summarise our findings.
2 Classical solutions of the wave equa-
tion
We begin by briefly reviewing the interpretation of so-
lutions to the wave equation in classical physics. This
interpretation may be contrasted with the interpreta-
tion given to the advanced-wave like contributions to
quantum correlations that are reported in subsequent
sections.
The inhomogeneous wave equation for a scalar field
ψ over Minkowski spacetime is
∂
2
t
− ∇
2
ψ(t, x) =
f(t, x), where we have assumed units such that the
speed of the wave is one and where f (t, x) describes
the sources of the waves. A well-posed problem is one
in which boundary conditions and the source data f are
specified, along with the fields ψ(t
0
, x) and ψ
t
0
(t
0
, x) =
∂
t
0
ψ(t
0
, x) at some time t
0
[1]. The fields at some other
time t are then sought.
Methods of solving the wave equation in classical elec-
trodynamics invariably involve green’s functions for the
wave-operator ∂
2
t
− ∇
2
[12, 31, 36]. The retarded and
advanced green’s functions G
+
and G
−
admit the rep-
resentations
G
±
(t, x|t
0
, x
0
) =
1
4π|x − x
0
|
δ (t
0
− t ± |x − x
0
|) . (1)
In conjunction with either initial or final conditions the
wave equation is easily solved in reciprocal space [31].
This method of solution is closest to that used in quan-
tum theory. The quantum electric field associated with
a point source within the Coulomb gauge is found using
essentially this method in appendix 9.4.
If initial conditions are specified such that before any
sources are activated at t = 0 there is a known input
configuration ψ
in
, which satisfies the homogenous equa-
tion
∂
2
t
− ∇
2
ψ
in
(t, x) = 0, the field at time t > 0 is
given by [31]
ψ(t, x) =ψ
in
(t, x)
+
Z
d
3
x
0
Z
t
0
dt
0
f(t
0
, x)G
+
(t, x|t
0
, x
0
). (2)
Significantly, the advanced green’s function G
−
does not
contribute to the solution for t > 0 of the initial value
problem. Moreover, when initial conditions are speci-
fied for the source function f , the homogeneous com-
ponent ψ
in
can be taken to vanish [36]. Therefore, in
classical electrodynamics only the retarded source-field
plays a role in the physical predictions associated with
a source that is initially fully specified. The retarded
solution is replaced by the advanced solution when fi-
nal conditions are specified instead of initial conditions,
and the Maxwell equations are solved running into the
past [1, 12, 31].
In contrast to the classical situation, in quantum the-
ory, even when all sources are fully specified the vac-
uum plays an important role in physical phenomena
[17]. We will show further that although all source-
fields remain fully retarded, certain predictions feature
advanced waves associated with the quantum vacuum.
3 Dipole-field Hamiltonian and electric
and magnetic source-fields
3.1 Dipole-field Hamiltonian
Consider a single atom coupled to the electromagnetic
field. Within the electric dipole approximation the
Hamiltonian describing this system is H = H
0
+V with
H
0
=
X
n
ω
n
|nihn| +
Z
d
3
k
X
λ
ω
a
†
λ
(k)a
λ
(k) +
1
2
= H
d
+ H
f
V = −
ˆ
d · D
T
(0) (3)
Accepted in Quantum 2017-12-15, click title to verify 2
where |ni denotes an energy eigenstate of the dipole
with associated energy ω
n
, a
λ
(k) denotes the annihila-
tion operator for a photon with momentum k and polar-
isation λ,
ˆ
d denotes the dipole moment operator, and
D
T
denotes the transverse electric displacement field.
The dipole is assumed to be located at the origin 0.
In terms of photonic operators a
λ
(k) the transverse
displacement field admits the mode expansion
D
T
(t, x) = D
(+)
T
(t, x) + D
(−)
T
(t, x),
D
(+)
T
(t, x) = i
Z
d
3
k
X
λ
r
ω
2(2π)
3
e
λ
(k)a
λ
(t, k)e
ik·x
= D
(−)
T
(t, x)
†
(4)
where the e
1,2
(k) are mutually orthogonal unit polarisa-
tion vectors orthogonal to k, and ω = |k|. The vacuum
(free) components of the electric field and transverse
displacement field are identical for all x and are given
by
D
(+)
T,0
(t, x) = E
(+)
0
(t, x),
= i
Z
d
3
k
X
λ
r
ω
2(2π)
3
e
λ
(k)a
λ
(k)e
−iωt+ik·x
.
(5)
For x 6= 0 the source components also coincide
D
T,s
= E
s
. In terms of photonic operators the elec-
tric field for x 6= 0 is therefore given by
D
T
(t, x) = E(t, x) = E
(+)
(t, x) + E
(−)
(t, x)
E
(+)
(t, x) = D
(+)
T
(t, x). (6)
The situation is different at the position of the dipole
x = 0. This is most easily seen by noting that
D(x) − E(x) = P(x) where P(x) = dδ(x) in the elec-
tric dipole approximation. Thus, at the position of the
dipole E differs from D by an infinite field. Since radi-
ation is detected by an absorbing atom at the position
of the atom, it is important to recognise the distinc-
tion between the electric field and the displacement field
at this position. The magnetic field admits the mode-
expansion
B(t, x) = B
(+)
(t, x) + B
(−)
(t, x),
B
(+)
(t, x)
= i
Z
d
3
k
X
λ
r
ω
2(2π)
3
ˆ
k × e
λ
(k) a
λ
(t, k)e
ik·x
(7)
and similarly to the transverse displacement field it can
be split into vacuum and source components via the
corresponding partitioning of a
λ
(t, k).
3.2 Electric and magnetic source-fields
The equation of motion for the photon annihilation op-
erator is found using the Hamiltonian (3) and once for-
mally integrated reads
a
λ
(t, k) = a
λ
(k)e
−iωt
+
Z
t
0
dt
0
e
−iω(t−t
0
)
r
ω
2(2π)
3
e
λ
(k) ·
ˆ
d(t
0
).
(8)
Away from the dipole, the electric and magnetic dipole
source-fields are found using Eqs. (6) and (80) and are
exactly analogous to classical fields of an oscillating
dipole;
E
s
(t, x) =
1
4πx
3
n
3
ˆ
x
h
ˆ
x ·
ˆ
d(t
r
)
i
−
ˆ
d(t
r
)
o
+
1
4πx
2
n
3
ˆ
x
h
ˆ
x ·
˙
ˆ
d(t
r
)
i
−
˙
ˆ
d(t
r
)
o
+
1
4πx
n
ˆ
x
h
ˆ
x ·
¨
ˆ
d(t
r
)
i
−
¨
ˆ
d(t
r
)
o
,
B
s
(t, x) = −
1
4πx
ˆ
x ×
¨
ˆ
d(t
r
) −
1
4πx
2
ˆ
x ×
˙
ˆ
d(t
r
) (9)
where t
r
= t−x > 0. For t < x the source-fields vanish.
The source-fields can be split into radiative and non-
radiative components. The radiation fields are re-
sponsible for irreversible energy loss from the dipole.
They vary as x
−1
, are orthogonal to x and depend on
the dipole moment acceleration. They are related via
E
rad,s
(t, x) = −
ˆ
x × B
rad,s
(t, x). The remaining parts
of the source-fields are non-radiative. The non-radiative
electric field includes both velocity-dependent and static
Coulomb-like parts whereas the non-radiative magnetic
field is entirely velocity-dependent.
The creation and annihilation source-fields E
(±)
s
are
not causal, that is, they do not generally vanish for
t < x. However, as in references [4, 16] we can perform
both a rotating-wave approximation and an associated
Markov approximation, which yield for t > x
E
(+)
s
(t, x) =
X
n,m
n<m
{3
ˆ
x (
ˆ
x · d
nm
) − d
nm
}
iω
nm
4πx
2
+
1
4πx
3
+
ω
2
nm
4πx
{d
nm
−
ˆ
x(
ˆ
x · d
nm
)}
σ
nm
(t
r
),
B
(+)
s
(t, x) =
X
n,m
n<m
ˆ
x × d
nm
ω
2
nm
4πx
−
iω
nm
4πx
2
σ
nm
(t
r
)
(10)
where σ
nm
(t) = e
iHt
|nihm|e
−iHt
. For t < x the cre-
ation and annihilation fields vanish within the approx-
imations made. These approximations also ensure that
Accepted in Quantum 2017-12-15, click title to verify 3
the positive and negatve frequency fields are associated
with dipole lowering and raising operators respectively.
Eq. (10) can be further simplified through the ap-
proximation σ
nm
(t) ≈ σ
nm
(0)e
iω
nm
t
. The latter is ap-
propriate for perturbative evaluations of quadratic field
functionals correct to second order. The steps leading
to the final result in Eq. (10) are given in appendix 9.1.
In later sections we we will focus on the case of a two-
level dipole with ground and excited states denoted |gi
and |ei respectively. The two-level transition frequency
is denoted ω
0
, and the transition dipole moment, which
is assumed to be real, is denoted d. We also introduce
the two-level operators
σ
+
= |eihg|, σ
−
= (σ
+
)
†
, σ
z
= [σ
+
, σ
−
] (11)
in terms of which the source-fields can be written
E
(+)
s
(t, x) = E
∗
(x)σ
−
(t
r
),
B
(+)
s
(t, x) = B
∗
(x)σ
−
(t
r
) (12)
wherein for convenience we have defined
E(x) =
iω
0
4πx
2
+
1
4πx
3
{3
ˆ
x (
ˆ
x · d) − d}
+
ω
2
0
4πx
{d −
ˆ
x(
ˆ
x · d)},
B(x) =
ω
2
0
4πx
−
iω
0
4πx
2
ˆ
x × d. (13)
We remark finally that it is not obvious that the
equal-time commutation relations between the fields
whose source components are given in Eq. (12) are the
same as those for the fields in Eq. (6), because approx-
imations have been made. That the commutation rela-
tions do remain the same is however necessary in order
that the resulting theory is formally consistent. We
prove that the equal-time commutators are in fact pre-
served in appendix 9.3.
4 Vacuum-source correlations and radi-
ation intensity
Before considering general quadratic functionals (corre-
lation functions) of the electromagnetic fields we con-
sider in detail a particular example - the radiation in-
tensity. Our aim is to identify the contribution made by
the quantum vacuum, and in particular, the role played
by vacuum source-field interference. The treatment in
this section extends previous perturbative treatments
found in [21, 28, 29, 32].
4.1 Radiated power
In order to identify the radiation intensity, which is the
radiated power per unit area, we begin by calculating
the radiated power due to the spontaneous photon emis-
sion of an initially excited dipole. The S-matrix element
representing the probability amplitude for the sponta-
neous emission process |e, 0i → |m, kλi with m < e
and |kλi the state of a single photon, can be calculated
easily using second order perturbation theory. The as-
sociated rate at which this process occurs is found using
Fermi’s golden rule to be
Γ
em
=
ω
3
em
|d
em
|
2
3π
(14)
where ω
em
= ω
e
− ω
m
and d
em
= he|d |mi. The S-
matrix consists of probability amplitudes for processes
that conserve the bare energy H
0
. Thus, in spontaneous
emission the energy of the resulting photon is precisely
the energy of the atomic transition through which it
was emitted. The energy flux in the process |e, 0i →
|m, kλi, m < e is therefore ω
em
Γ
em
, which implies that
the total radiated power is to second order given by
P =
X
m<e
ω
em
Γ
em
. (15)
The above derivation of P relies entirely on the calcula-
tion of matrix elements of the interaction Hamiltonian,
so it does not involve using any electromagnetic fields
away from the source at 0. In fact, since D
T
(0) 6= E(0),
the electric and magnetic fields do not directly feature
anywhere in the above calculation.
4.2 Derivation using electromagnetic fields
An alternative expression for the radiated power that
does involve the electromagnetic fields, can be obtained
from Poynting’s theorem. The energy flux radiated by
the dipole located at the centre of a sphere with radius
x is given by
P = lim
x→∞
Z
dΩ x
2
I(t, x) =
Z
dΩ x
2
I
rad
(t, x) (16)
where the integration is performed over all directions.
Eq. (16) is definitive of the radiation intensity I
rad
,
which is the radiated power per unit area. The intensity
component I
rad
is a quadratic functional of the radiation
fields, which vary as x
−1
.
We now proceed to determine the radiated power us-
ing the fields in section 3.1. In order to compare the
result of Eq. (16) with Eq. (15), which was obtained
using second order perturbation theory we make use of
Eq. (10) coupled with the perturbative approximation
σ
nm
(t) ≈ σ
nm
(0)e
iω
nm
t
. First let us suppose that the
intensity I
rad
in Eq. (16) is of the Glauber type
I
G
(t, x) = hE
(−)
rad
(t, x) · E
(+)
rad
(t, x)i
0,e
= hE
(−)
rad,s
(t, x) · E
(+)
rad,s
(t, x)i
e;0
(17)
Accepted in Quantum 2017-12-15, click title to verify 4
where the average is taken in the state |e, 0i, and we
have retained only the radiative contribution dependent
on E
rad
= E
0
+ E
rad,s
. Letting σ
nm
(t) ≈ σ
nm
(0)e
iω
nm
t
in Eq. (10) we find that
I
G
(t, x) =
1
4πx
2
X
m<e
ω
4
em
[d
em
−
ˆ
x(
ˆ
x · d
em
)]
2
. (18)
Using Eq. (17), the corresponding radiated power is
P
G
=
Z
dΩ x
2
I
G
(t, x) =
1
2
X
m<e
ω
em
Γ
em
, (19)
which is only half of the power given in Eq. (15). The
correct power is evidently obtained from the intensity
2I
G
rather than I
G
itself
4.3 General definition of the radiation Intensity
In order to motivate a general definition of the radiation
intensity let us first consider the case of a free field con-
fined to a volume V . The vacuum energy H
vac
and the
associated density H
vac
/V are not measurable. Only
differences in energy-density from the vacuum value
are measurable. The integral of the measurable local
energy-density over all space should therefore yield the
energy difference H
f
− H
vac
[3, 30].
Quite generally, the vacuum energy density can be
expressed as
U
vac
=
H
vac
V
= hE
(+)
0
(t, x) · E
(−)
0
(t, x)i
0
= [E
(+)
0,i
(t, x), E
(−)
0,i
(t, x)]
= [B
(+)
0,i
(t, x), B
(−)
0,i
(t, x)]. (20)
Thus, in the general case in which sources are present
the average electromagnetic energy-density can be writ-
ten
U
em
(t, x) − U
vac
≡
1
2
E(t, x)
2
+ B(t, x)
2
− [E
(+)
0,i
(t, x), E
(−)
0,i
(t, x)]. (21)
Since the total fields E
(±)
and the vacuum fields E
(±)
0
satisfy the same equal-time commutation relations (cf
appendix 9.3),
[E
(+)
0,i
(t, x), E
(−)
0,j
(t, x
0
)] = [E
(+)
i
(t, x), E
(−)
j
(t, x
0
)]
= [B
(+)
i
(t, x), B
(−)
j
(t, x
0
)] = [B
(+)
0,i
(t, x), B
(−)
0,j
(t, x
0
)],
(22)
for the total energy density U = U
em
+ U
matter
, where
U
matter
is the energy density of sources, we have
U(t, x) − U
vac
= : U (t, x) : (23)
where : · : denotes normal-ordering of the creation and
annihilation operators within U
em
.
By Poynting’s theorem the local energy-flux can be
written in terms of the Hermitian Poynting vector
S(t, x) =
1
2
[E(t, x) × B(t, x) + H.c.] . (24)
Analogously to the vacuum energy-density we define the
vacuum Poynting vector by
S
vac,i
(t, x)
= hS
0,i
(t, x)i
0
=
ijk
[E
(+)
0,j
(t, x), B
(−)
0,k
(t, x)]
=
ijk
[E
(+)
j
(t, x), B
(−)
k
(t, x)] (25)
where
ijk
denotes the Levi-Civita symbol. Since U
vac
and S
vac
are independent of time and space Poynting’s
theorem can be written in terms of : U(t, x) : = U(t, x)−
U
vac
and : S(t, x) : = S(t, x) − S
vac
as
∂
t
h: U(t, x) :i = −∇·h: S(t, x) :i. (26)
The practice of normally-ordering operator products in
this way constitutes a basic application of Wick’s theo-
rem.
For a dipole at 0 the radiated power can be obtained
by integrating the right-hand-side of Eq. (26) over a
sphere centered at 0 with radius x. Using the divergence
theorem and subsequently taking the limit x → ∞ the
associated intensity is then seen to be
I
rad
(t, x) =
ˆ
x · hS
rad
(t, x) − S
vac
i,
S
rad
=
1
2
[E
rad
× B
rad
+ H.c.] (27)
where E
rad
(t, x) = E
0
(t, x) + E
rad,s
(t, x) and
B
rad
(t, x) = B
0
(t, x)+B
rad,s
(t, x). Assuming the initial
state |e, 0i and making use of the identities
ˆ
x · E
rad,s
(t, x) = 0 =
ˆ
x · B
rad,s
(t, x),
ˆ
x × E
rad,s
(t, x) = B
rad,s
(t, x),
ˆ
x × B
rad,s
(t, x) = −E
rad,s
(t, x),
B
2
rad
= E
2
rad
(28)
it is straightforward to show that the radiation intensity
can also be written
I
rad
(t, x) = hE
rad
(t, x)
2
− E
0
(t, x)
2
i
e;0
= 2I
G
(t, x)
(29)
where the second equality follows within the rotating-
wave approximation. This explains why 2I
G
rather than
I
G
yields the correct radiated power. The anti-normally
ordered combination E
(+)
rad
(t, x)·E
(−)
rad
(t, x), which is such
that h: E
(+)
rad
(t, x)·E
(−)
rad
(t, x) :i = hE
(−)
rad
(t, x)·E
(+)
rad
(t, x)i,
is slowly-varying (resonant), so it is not eliminated by
the rotating-wave and Markov approximations.
Accepted in Quantum 2017-12-15, click title to verify 5
4.4 Vacuum source-field correlations
We now turn our attention to the contribution of vac-
uum source-correlations to the radiation intensity. In
references [21, 28, 29, 32] Eq. (27) is used to pertur-
batively calculate the radiated power. We provide a
simpler derivation of the same result, and in section 4.5
we provide a non-perturbative extension. The physical
intensity is determined relative to the vacuum value,
so assuming the initial state |e, 0i the intensity can be
written according to Eq. (29) as
I
rad
(t, x) =hE
rad,s
(t, x)
2
i
e,0
+ hE
0
(t, x) · E
rad,s
(t, x)
+ E
rad,s
(t, x) · E
0
(t, x)i
e,0
. (30)
First we calculate the anti-normally ordered combina-
tion hE
(+)
rad,s
(t, x) · E
(−)
rad,s
(t, x)i
e;0
, which is easily found
using Eq. (10) to be
hE
(+)
rad,s
(t, x) · E
(−)
rad,s
(t, x)i
e;0
=
1
4πx
2
X
m>e
ω
4
em
[d
em
−
ˆ
x(
ˆ
x · d
em
)]
2
(31)
to second order in the transition dipole moments. By
substituting Eqs. (18) and (31) into Eq. (16) we obtain,
within the rotating-wave approximation, the contribu-
tion of E
2
rad,s
in Eq. (30) to the radiated power to second
order as
P
s
=
Z
dΩ x
2
hE
rad,s
(t, x)
2
i =
1
2
X
m
ω
em
Γ
em
(32)
where all dipole levels m are included within the sum-
mation.
It remains to calculate the contribution of the
vacuum-source correlations on the first line in Eq. (30).
This can be done a number of ways, with varying com-
plexity. The authors [21, 28, 29] use the full source-
field soultions of the electric and magnetic fields and
separate out time-independent and oscillatory contri-
butions to the power. The time-independent contri-
butions give our final result Eq. (34) below, while the
time-dependent contributions vanish on average, and
can be interpreted as virtual transients. Reference [32]
gives a simplified derivation of Eq. (34), which involves
using only the radiation source-fields rather than the
full source-fields. Here we give a yet simpler derivation
which makes use of the Markov and rotating-wave ap-
proximations. The latter eliminate virtual photon con-
tributions, and therefore yield Eq. (34) directly, without
requiring a time-average.
The vacuum-source correlation terms within (30), are
found to second order to be (appendix 9.2)
hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0;e
+ c.c. =
1
(4πx)
2
X
m
sgn(ω
em
)ω
4
em
[d
em
−
ˆ
x(
ˆ
x · d
em
)]
2
(33)
in agreement with references [21, 28, 29, 32]. The cor-
responding contribution to the radiated power is
P
vac−s
=
Z
dΩ x
2
hE
0
(t, x) · E
rad,s
(t, x)i + c.c.
=
1
2
X
m
sgn(ω
em
)ω
em
Γ
em
. (34)
By adding the two contributions Eqs. (32) and (34) one
obtains the correct result Eq. (15). Vacuum-source cor-
relations are seen to be responsible for supplying half
of the total radiated power, as well as for eliminat-
ing, in the final result, the contributions of energy non-
conserving virtual transitions for which m > e.
4.5 Non-perturbative extension
The results of section 4.4 can be extended beyond the
use of second order perturbation theory. For this pur-
pose we restrict our attention to a two-level emitter.
Judicious use of the Markov and rotating-wave approx-
imations within the Heisenberg equations, leads to a
tractable system of coupled Heisenberg-Langevin equa-
tions for the dipole operators, which can then be for-
mally integrated.
We begin by using the solution (80), and the rotating-
wave and Markov approximations to calculate the
dipole’s own reaction field as
D
(+)
T,s
(t, 0) = i
Z
d
3
k
r
ω
2(2π)
3
X
λ
e
λ
(k)a
λ,s
(t, k)
= i
Γ
2
ˆ
d
d
σ
−
(t). (35)
Substituting Eq. (100) into the Heisenberg equation for
each of σ
+
and σ
z
, making the rotating-wave approx-
imation, and formally integrating the result yields the
solutions
σ
+
(t) = e
(iω
0
−Γ/2)t
σ
+
+
Z
t
0
dt
0
e
(iω
0
−Γ/2)(t−t
0
)
σ
+
vac
(t
0
)
(36)
σ
z
(t) + 1 = e
−Γt
[σ
z
+ 1] +
Z
t
0
dt
0
e
−Γ(t−t
0
)
σ
z
vac
(t
0
).
(37)
Accepted in Quantum 2017-12-15, click title to verify 6
where
σ
+
vac
(t) = i[d · D
(−)
T,0
(t, 0)]σ
z
(t) (38)
σ
z
vac
(t) = −2i[d · D
(−)
T,0
(t, 0)]σ
−
(t) + H.c.. (39)
In addition to the solutions to the Bloch equations
(without driving), the solutions (36) and (37) also con-
tain contributions from the quantum vacuum field.
Using Eqs. (36), (37) and (12) we find the Glauber
intensity function in the state |0, ei to be
I
G
(t) =
ω
2
0
4πx
[d −
ˆ
x(
ˆ
x · d)]
2
e
−Γt
r
(40)
The corresponding power is
P
G
(t) =
1
2
ω
0
Γe
−Γt
r
(41)
whose short-time limit coincides with the power in
Eq. (19) restricted to two levels.
Next, following the same method as in section
4.4 we find the anti-normally ordered combination
hE
(+)
rad,s
(t, x) · E
(−)
rad,s
(t, x)i
e;0
to be
hE
(+)
rad,s
(t, x) · E
(−)
rad,s
(t, x)i
e;0
=
ω
2
0
4πx
[d −
ˆ
x(
ˆ
x · d)]
2
(1 − e
−Γt
r
), (42)
which when combined with Eq. (40) gives the total con-
tribution of the source-field to the radiated power as
P
s
(t) =
1
2
ω
0
Γ. (43)
Finally, the vacuum-source correlations are found using
Eqs. (36), (37), (12) and (5) to be (appendix 9.2)
hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0,e
+ c.c.
=
ω
2
0
4πx
[d −
ˆ
x(
ˆ
x · d)]
2
2e
−Γt
r
− 1
. (44)
The corresponding contribution to the power is
P
vac−s
(t) = ω
0
Γ
e
−Γt
r
−
1
2
. (45)
By adding Eqs. (43) and (45) we obtain the total radi-
ated power
P (t) = ω
0
Γe
−Γt
r
= 2P
G
(t) (46)
whose short time limit coincides with the power in
Eq. (15) for a two-level system.
These results mirror the previous perturbative results
in that neither Eq. (41) nor Eq. (43) yield the correct ra-
diated power. Furthermore, since the pure source power
in Eq. (43) does not decay, it is unphysical for long times
when considered on its own. As with the previous per-
turbative results only the combination of the individu-
ally non-physical contributions (43) and (45), yields the
correct physical power.
5 Vacuum-source correlations and ad-
vanced waves
In section 4 we showed that a specific quadratic field
functional, namely, the intensity functional is the dif-
ference between the quantum version of the classical
definition and the quantum vacuum value. Accounting
for vacuum-source correlations was essential in obtain-
ing the correct physical results.
The most general quadratic field functionals are cor-
relation functions defined in terms of the components
of the electromagnetic field tensor at different points in
spacetime. In a designated (laboratory) inertial frame
all such correlations have the form hX
i
(t, x)Y
j
(t
0
, x
0
)i
where X, Y = E, B. In the rotating-wave
approximation the terms hX
(+)
i
(t, x)Y
(+)
j
(t
0
, x
0
)i and
hX
(−)
i
(t, x)Y
(−)
j
(t
0
, x
0
)i are taken as rapidly oscillating
and are negligible. The corresponding Glauber-type
correlation function hX
(−)
i
(t, x)Y
(+)
j
(t
0
, x
0
)i is therefore
related to hX
i
(t, x)Y
j
(t
0
, x
0
)i by
G
X
i
Y
j
(t, x|t
0
, x
0
) = 2hX
(−)
i
(t, x)Y
(+)
j
(t
0
, x
0
)i
≈ hX
i
(t, x)Y
j
(t
0
, x
0
) − [X
(+)
i
(t, x), Y
(−)
j
(t
0
, x
0
)]i. (47)
In the vacuum state the commutator
[X
(+)
i
(t, x), Y
(−)
j
(t
0
, x
0
)], which is subtracted from
hX
i
(t, x)Y
(
j
t
0
, x
0
)i in Eq. (47) includes the pure
vacuum value hX
0,i
(t, x)Y
0,j
(t
0
, x
0
)i
0
as well as source-
dependent components. If instead we subtract only the
vacuum value from hX
i
(t, x)Y
j
(t
0
, x
0
)i we obtain the
alternative correlation function
C
X
i
Y
j
(t, x|t
0
, x
0
)
= hX
i
(t, x)Y
j
(t
0
, x
0
) − [X
(+)
0,i
(t, x), Y
(−)
0,j
(t
0
, x
0
)]i. (48)
For t 6= t
0
the commutators between the total fields E
(±)
and B
(±)
are not equal to their counterparts defined in
terms of the vacuum fields E
(±)
0
and B
(±)
0
. As a re-
sult even within the rotating-wave approximation G
X
i
Y
j
and C
X
i
Y
j
do not generally coincide. Unlike G
X
i
Y
j
the
vacuum-fields contribute to C
X
i
Y
j
within the vacuum
state.
We define the difference operator
∆
X
i
Y
j
(t, x|t
0
, x
0
) =[X
(+)
i
(t, x), Y
(−)
j
(t
0
, x
0
)]
− [X
(+)
0,i
(t, x), Y
(−)
0,j
(t
0
, x
0
)] (49)
in terms of which, in the rotating-wave approximation,
we have
C
X
i
Y
j
= G
X
i
Y
j
+ h∆
X
i
Y
j
i. (50)
A perturbative calculation of certain correlation func-
tions of the form given in Eq. (48) in the particular
Accepted in Quantum 2017-12-15, click title to verify 7
case that t = t
0
has been carried out without using
the rotating-wave or Markov approximations within the
non-relativistic QED framework by Power and Thiruna-
machandran [21, 22] (see also [28, 29, 32]). Their results
are consistent with the results presented in section 4.4,
and differ only by the inclusion of oscillatory transient
contributions. The results in references [21, 22] are also
consistent with the short-time limiting behaviour of the
non-perturbative results given in section 4.5.
Here we consider the the more general case in which
t and t
0
are arbitrary, which is fundamentally different,
because ∆
X
i
Y
j
(t, x|t
0
, x
0
) is generally non-vanishing.
While we utilise the rotating-wave and Markov approx-
imations our approach is not perturbative and is simi-
lar to that used in section 4.5. More precisely, we make
use of the Heisenberg-Lengavin equations for a two-level
dipole. Details of the calculations leading to the results
below are given in appendix 9.3.
Using Eq. (12) we write X
(−)
s
(t, x) = X (x)σ
+
(t − x)
where X = E, B and correspondingly X = E , B. The
G
X
i
Y
j
are easily found to be
G
X
i
Y
j
(t, x|t
0
, x
0
) =
2X
i
(x)Y
∗
j
(x
0
)θ(t
r
)θ(t
0
r
)hσ
+
(t
r
)σ
−
(t
0
r
)i. (51)
where t
r
= t − x and t
0
r
= t
0
− x
0
and where θ denotes
the Heaviside step-function. Assuming the initial state
|e, 0i we obtain
G
X
i
Y
j
(t, x|t
0
, x
0
) =
2X
i
(x)Y
∗
j
(x
0
)θ(t
r
)θ(t
0
r
)e
−[Γ/2](t
r
+t
0
r
)
e
iω
0
(t
r
−t
0
r
)
. (52)
Next we consider ∆
X
i
Y
j
, which can be partitioned into
three terms as
∆
X
i
Y
j
(t, x|t
0
, x
0
) = [X
(+)
s,i
(t, x), Y
(−)
s,j
(t
0
, x
0
)]
+ [X
(+)
0,i
(t, x), Y
(−)
s,j
(t
0
, x
0
)] + [X
(+)
s,i
(t, x), Y
(−)
0,j
(t
0
, x
0
)].
(53)
The pure source component is easily found using
Eq. (12) to be
[X
(+)
s,i
(t, x), Y
(−)
s,j
(t
0
, x
0
)]
= X
∗
i
(x)Y
j
(x
0
)θ(t
r
)θ(t
0
r
)[σ
−
(t
r
), σ
+
(t
0
r
)]. (54)
The remaining components involve both the vacuum
and source-fields, and they each consist of both retarded
and advanced parts. After fairly lengthy calculations we
obtain the compact expression (appendix 9.3)
[X
(+)
0,i
(t, x), Y
(−)
s,j
(t
0
, x
0
)] =
− X
∗
i
(x)Y
j
(x
0
)θ(t
0
r
− t
r
)θ(t
r
)θ(t
0
r
)[σ
−
(t
r
), σ
+
(t
0
r
)]
± X
i
(x)Y
j
(x
0
)θ(t
0
r
− t
a
)θ(t
0
r
)[σ
−
(t
a
), σ
+
(t
0
r
)], (55)
1
(t, x)
(t
0
, x
0
)
t
0
r
t
a
time
(0, 0)
Figure 1: The world line of the dipole, represented by the cen-
tral vertical arrow, passes through the intersection of the back-
ward lightcone of the event (t
0
, x
0
) and the forward lightcone
of (t, x). The commutator [X
(+)
0,i
(t, x), Y
(−)
s,j
(t
0
, x
0
)] contains a
component that depends on the source operator σ
−
(t
a
) where
t
a
is the advanced time associated with the event (t, x) at
which the vacuum field X
(+)
0,i
(t, x) is evaluated.
where t
a
= t + x and t
0
a
= t
0
+ x
0
, and the notation ±
means that + holds for the case X = E while − holds
when X = B. Similarly we find that
[X
(+)
s,i
(t, x), Y
(−)
0,j
(t
0
, x
0
)] =
− X
∗
i
(x)Y
j
(x
0
)θ(t
r
− t
0
r
)θ(t
r
)θ(t
0
r
)[σ
−
(t
r
), σ
+
(t
0
r
)]
± X
∗
i
(x)Y
∗
j
(x
0
)θ(t
r
− t
0
a
)θ(t
r
)[σ
−
(t
r
), σ
+
(t
0
a
)] (56)
where + holds for the case Y = E while − holds when
Y = B. The sum of the purely retarded parts from
Eqs. (55) and (56) exactly cancels the pure source com-
ponent given in Eq. (54). We therefore obtain the final
result
∆
X
i
Y
j
(t, x|t
0
, x
0
) =
(−1)
α
X
i
(x)Y
j
(x
0
)θ(t
0
r
− t
a
)θ(t
0
r
)[σ
−
(t
a
), σ
+
(t
0
r
)]
+(−1)
β
X
∗
i
(x)Y
∗
j
(x
0
)θ(t
r
− t
0
a
)θ(t
r
)[σ
−
(t
r
), σ
+
(t
0
a
)].
(57)
where α = 0 for X = E and α = 1 for X = B
while β = 0 for Y = E and β = 1 for Y = B.
The Heaviside functions in this expression imply that
∆
X
i
Y
j
(t, x|t, x
0
) = 0 as required. However, for t 6= t
0
,
h∆
X
i
Y
j
(t, x|t
0
, x
0
)i generally consists of non-vanishing
two-time correlations, which are due to the advanced
components of the commutators of the free and source-
fields in Eqs. (55) and (56). Assuming the initial state
Accepted in Quantum 2017-12-15, click title to verify 8
|e, 0i, with some work we obtain (appendix 9.3)
h∆
X
i
Y
j
(t, x|t
0
, x
0
)i
e,0
=
(−1)
α
X
i
(x)Y
j
(x
0
)θ(t
0
r
− t
a
)θ(t
0
r
)θ(t
a
)
× e
(iω
0
−Γ/2)t
0
r
e
(−iω
0
−Γ/2)t
a
(e
Γt
a
− 2)
+ (−1)
β
X
∗
i
(x)Y
∗
j
(x
0
)θ(t
r
− t
0
a
)θ(t
r
)θ(t
0
a
)
× e
(−iω
0
−Γ/2)t
r
e
(iω
0
−Γ/2)t
0
a
(e
Γt
0
a
− 2). (58)
Since ∆
X
i
,Y
j
(t, x|t
0
, x
0
) is non-local in both space
and time the presence of advanced contributions is not
acausal. Supposing for definiteness that t
0
> t then
all fields at events within the backward lightcone of
(t
0
, x
0
) may contribute to ∆
X
i
,Y
j
(t, x|t
0
, x
0
). As figure
1 illustrates, provided a portion of the dipole’s world
line belongs to the intersection of the backward light-
cone of (t
0
, x
0
) and the forward lightcone of (t, x) we
have t
0
r
≥ t
a
. In this case ∆
X
i
,Y
j
(t, x|t
0
, x
0
) receives a
non-zero advanced contribution coming from the first
line in Eq. (57).
6 Photo-detection theory
The theory of photo-detection is fundamental within
quantum optics and was initiated some time ago [10,
14, 15]. In this section we briefly verify that when
source and detector are taken as identical two-level
dipoles, advanced wave correlations do not contribute
to photo-detection amplitudes within the rotating-wave
and Markov approximations.
The source is located at the origin 0 and the detector
is located at a position x with x ω
−1
0
. The interaction
Hamiltonian for the model is given by
V = −
ˆ
d
s
· D
T
(0) −
ˆ
d
d
· D
T
(x) (59)
where the subscripts s and d stand for source and de-
tector respectively. The state of the composite system
in which the source is in the excited state, the detector
is in the ground state, and the field contains no photons
is denoted |ψi. The rate of excitation of the detector is
found using the equation of motion for hσ
z
d
(t)i
ψ
to be
˙p
d
(t) = − d
i
d
j
Z
t
0
dt
0
e
−iω
0
(t−t
0
)
× hD
T,i
(t, x)D
T,j
(t
0
, x)σ
z
d
(t
0
)i
ψ
+ c.c. . (60)
Since actual photo-detection events depend on photo-
ionisation of an electron within the continuum of final
detector states, it is argued in [16] that the rate of stimu-
lated emission from an excited state should be assumed
to be much smaller than the rate of absorption from the
ground state. This assumption allows us to make the
perturbative approximation σ
z
d
(t
0
) ≈ σ
z
d
(0) in Eq. (60),
which for a detector in the ground state yields
˙p
d
(t) =d
i
d
j
Z
t
0
dt
0
hD
T,i
(t, x)D
T,j
(t
0
, x)i
ψ
e
−iω
0
(t−t
0
)
+ c.c. . (61)
The total transverse displacement field at the position
of the detector includes the detectors own reaction field,
and is given by
D
T
(t, x) = E
0
(t, x) + E
s
(t, x) + D
T,d
(t, x). (62)
Using again the perturbative approximation σ
±
d
(t) ≈
σ
±
d
e
±iω
0
t
for the detector operators, and the rotating-
wave approximation, all terms in Eq. (61) that involve
the detector’s own reaction field are seen to vanish un-
der the assumption that the detector is in the ground
state. Moreover, while the pure vacuum correlation
function hE
(+)
0,i
(t, x)E
(−)
0,j
(t
0
, x)i
ψ
can be shown to con-
tribute to the detector level-shifts, it does not contribute
within the rotating-wave and Markov approximations to
the detectors excitation [16]. We therefore obtain
˙p
d
(t) = d
i
d
j
Z
t
0
dt
0
C
E
i
E
j
(t, x|t
0
, x)e
−iω
0
(t−t
0
)
+ c.c.
(63)
The advanced contributions to this expression are found
using Eq. (58) with x = x
0
. Since t
0
takes values from
0 upto t > 0 the first term on the right-hand-side of
Eq. (58) does not contribute. It now suffices to note that
the second term on the right-hand-side of Eq. (58) pos-
sesses an oscillatory dependence of the form e
−iω
0
(t−t
0
)
,
so that when substituted into Eq. (63) this term gives
only rapidly oscillating contributions to ˙p
d
(t), which
are negligible using the rotating-wave approximation.
Within the rotating wave approximation we therefore
have that
˙p
d
(t) = d
i
d
j
Z
t
0
dt
0
G
E
i
E
j
(t, x|t
0
, x)e
−iω
0
(t−t
0
)
+ c.c.,
(64)
which is essentially the same result originally derived
by Glauber [10].
A few concluding remarks are in order regarding
the final result Eq. (64). We note that the replace-
ment of C
E
i
E
j
by G
E
i
E
j
in the last step is a result of
the rotating-wave approximation. Therefore, advanced
wave components of C
E
i
E
j
may in principle contribute
to the detector excitation, but such contributions are no
larger than conventional counter-rotating contributions
involving, for example, the detector’s own reaction field.
The rotating-wave approximation is generally con-
sidered to be valid for optical frequencies and realis-
tic detector response times [16]. On the other hand,
Accepted in Quantum 2017-12-15, click title to verify 9
in the more realistic case of multi-level non-identical
dipoles, the validity of the rotating-wave approximation
is less clear. One could also consider detection on much
shorter time scales. However, it is not entirely clear how
regimes which require moving beyond the rotating-wave
approximation should be modelled. In such regimes
a retention of the conventional ontology consisting of
atomic excitations and photons defined in terms of a
bare energy H
0
, which is not even approximately con-
served, seems physically dubious. Little more can be
said presently about the role of advanced waves within
such regimes. We remark that the techniques found in
[2, 9, 33], which allow one to eliminate counter-rotating
terms from the dipole-field interaction Hamiltonian may
offer a possible recourse.
7 Radiative forces
In this section we show that certain measurement statis-
tics involving the radiative force acting on an elemen-
tary quantum point charge q due to the field of a quan-
tum dipole, depend on the two-time correlation func-
tions found in section 5. The point charge has position
and momentum operators r and p. The Lorentz force
F(t) = m
¨
r(t) experienced by the charge due to an elec-
tromagnetic field is
m
¨
r(t) =qE(t, r(t))
+
q
2
[˙r(t) × B(t, r(t)) − B(t, r(t)) × ˙r(t)].
(65)
The momentum π = m
˙
r is found by integrating
Eq. (65). As an example of a statistical quantity that
depends on two-time correlations in the external fields
we consider the rate of change of the dispersion of the
momentum, i.e., the momentum diffusion, which is
d
dt
∆π(t) = (hπ(0) · F(t)i + c.c.) − 2hπ(0)i · hF(t)i
+
Z
t
0
dt
0
[hF(t) · F(t
0
)i + c.c.] − 2hF(t)i · hF(t
0
)i
.
(66)
Substitution of the Lorentz force given by Eq. (65) into
Eq. (66), leads to a dependence of the momentum diffu-
sion on integrated two-time correlation functions of the
electromagnetic field. Unlike in photo-detection the-
ory, these correlation functions are not accompanied by
oscillatory factors due to the detector itself. As a re-
sult the advanced components of the relevant correla-
tion functions are not generally negligible.
We now derive an explicit expression for the momen-
tum diffusion based on a series of simplifying assump-
tions and approximations. We begin by noting that a
free charge q can be described using the Coulomb-gauge
Hamiltonian
H
0
=
1
2m
[p − qA
T
(r)]
2
+ V
self
(r) + H
f
(67)
where A
T
denotes the transverse vector potential, V
self
denotes the Coulomb self-energy of the charge, and
the Coulomb gauge free field Hamiltonian is defined in
terms of the transverse electric field E
T
= −
˙
A
T
as
H
f
=
1
2
Z
d
3
x [E
T
(x)
2
+ B(x)
2
]. (68)
The total electric field is found by adding to E
T
the
longitudinal field E
L
= −∇φ
coul
where φ
coul
is the
Coulomb potential of the point charge. The vacuum and
source components of the transverse electric field E
T,0
and E
T,s
, which are defined by E
T
= E
T,0
+ E
T,s
are
such that changes in E
T,0
are generated by the Hamilto-
nian H
f
alone, i.e.,
˙
E
T,0
(t, x) = −i[E
T,0
(t, x), H
f
(0)].
The source component E
T,s
(t, x) includes two separate
acausal components (appendix 9.4). The first is equal
to −E
L
(t, x), which is independent of the definition of
H
f
and which is cancelled out within the source-field
E
s
(t, x). The second instantaneous component depends
on the source at the initial time t = 0, and arises be-
cause in the Coulomb gauge H
f
is defined in terms of E
T
rather than E itself. Due to this component the electric
source-field E
s
is not causal within the Coulomb gauge.
A complete derivation of the Coulomb gauge electric
source-field is given in appendix 9.4.
The occurrence of a static precursor within the elec-
tric source-field also arises in the Coulomb gauge treat-
ment of a dipole system [23]. On the other hand, in the
multipolar gauge the free field Hamiltonian H
f
is de-
fined in terms of the transverse displacement field D
T
,
which possesses a fully retarded source component and
is equal to the total electric field away from the dipole
[6, 23]. Using superscripts C and m to refer to the
Coulomb and multipolar gauges respectively, we have
that although E
C
0
6= E
m
0
and E
C
s
6= E
m
s
it can be shown
that E
C
0
− E
m
0
= E
m
s
− E
C
s
, so that the total field E is
indeed gauge-invariant [23]. Only the multipolar gauge
however, correctly identifies the physical vacuum and
source components E
0
and E
s
, such that E
s
is fully
causal and is the quantised version of the classical field
of an oscillating dipole as given in Eq. (9).
To correctly identify the vacuum and source compo-
nents of the electromagnetic fields when considering the
free charge q, we seek a description analogous to the
multipolar description of the dipole. To this end we
consider a heavy charge with relatively slow dynam-
ics characterised by the time scale t
q
Γ
−1
. We
assume a state of the charge such that initially the
spread in position is small compared with an optical
Accepted in Quantum 2017-12-15, click title to verify 10
wavelength
√
∆r ω
−1
0
, and such that the spread in
Doppler shifts is small compared to the dipole’s decay
rate; ω
0
p
∆p/m Γ. The initial state of the charge is
therefore well-localised in position and momentum.
Since the coupling of the charge to the field is weak
and t t
q
we have that A
T
(t, r(t)) ≈ A
T
(t, r(0) +
p(0)t/m). Furthermore since the atom is initially
well-localised in position and momentum we have
A
T
(t, r(0) + p(0)t/m) ≈ A
T
(t, r
0
+ p
0
t/m) where r
0
=
hr(0)i and p
0
= hp(0)i. Finally we restrict our at-
tention to a charge for which p
0
= 0, which implies
that A
T
(t, r(t)) ≈ A
T
(t, r
0
). In summary, for times
t t
q
the initial spatial localisation of the charge is
maintained, which means that hr(t)i−r
0
remains small
compared with ω
−1
0
. For times t t
q
the kinetic energy
of the charge is therefore
H
q
(t) ≈
1
2m
[p(t) − qA
T
(t, r
0
)]
2
. (69)
In this Hamiltonian the charge’s initial wave-packet cen-
tre r
0
acts like the reference centre of a dipole, which
allows us to construct a description of the charge analo-
gous to the multipolar description of a dipole . Specifi-
cally, we are now able to transform out the vector poten-
tial dependence of H
q
(t) via a unitary transformation
e
−iqr (t)·A
T
(t,r
0
)
.
We transform the Coulomb gauge Hamiltonian for the
whole system consisting of the charge, dipole and field
using the unitary e
−iqr(t)·A
T
(t,r
0
)
e
−id(t)·A
T
(t,0)
. We
then make the two-level and rotating-wave approxima-
tions for the dipole and in the usual way, we absorb all
dipole and charge self-energy terms into the definitions
of their respective bare energies. This yields the final
Hamiltonian H = H
0
+ V where
H
0
= ω
0
σ
+
σ
−
+
p
2
2m
+
Z
d
3
k
X
λ
ω
a
†
λ
(k)a
λ
(k) +
1
2
,
V = d · Π
(−)
T
(0)σ
−
+ σ
+
d · Π
(+)
T
(0) + qr · Π
T
(r
0
)
(70)
with
Π
(+)
T
(t, x)
= −i
Z
d
3
k
X
λ
r
ω
2(2π)
3
e
λ
(k)a
λ
(t, k)e
ik·x
. (71)
The field canonical momentum Π
T
coincides with mi-
nus the total electric field away from the dipole and the
charge; Π
T
(x) = −E(x), x 6= 0, r
0
. In the descrip-
tion that uses the Hamiltonian in Eq. (70) the electric
source-field is a properly causal field, and moreover, the
charge’s canonical momentum p coincides with the me-
chanical momentum m
˙
r.
Since we can replace π(0) by p
0
= 0 in the first term
in Eq. (66) this term is seen to vanish. To obtain the
momentum diffusion correct to second order in q we
require the force
˙
p(t) correct to first order in q, which
is found using the Hamiltonian (70) to be
˙
p(t) = qE
e
(t, r
0
), E
e
= E
0
+ E
s
, (72)
where E
0
and E
s
are given in Eqs. (5) and (12) respec-
tively. By only considering terms upto order q we find
that the charge does not influence the dipole source op-
erators on which the external field in Eq. (72) depends.
We may therefore also use Eqs. (36) and (37). Assuming
the initial state |e, 0i for the dipole and the radiation
field, the momentum diffusion is to O(q
2
) found using
Eq. (72) to be
d
dt
∆p(t) = q
2
Z
t
0
dt
0
hE
e
(t, r
0
) · E
e
(t
0
, r
0
)i
0,e
+ c.c.
(73)
The correlation function in this expression includes an
infinite pure-vacuum contribution which is indepen-
dent of the dipole. A similar contribution was found
in the context of photo-detection theory in section 6
where it was found to be equivalent to the detectors
own radiation-reaction. This can be understood as
a fluctuation-dissipation relation [16]. Neglecting the
pure-vacuum contribution in Eq. (73) while retaining
all components that depend on the source dipole we
obtain
d
dt
∆p(t) = q
2
Z
t
0
dt
0
C
E
i
E
i
(t, r
0
|t
0
, r
0
) + c.c. (74)
where C
E
i
E
i
is defined in Eq. (48). Using Eq. (50)
we write Eq. (74) as the sum of two contributions;
d∆p/dt = d∆p
s
/dt + d∆p
vac−s
/dt. Assuming the ini-
tial state |e, 0i and restricting our attention to the ra-
diation zone x ω
−1
0
, the first contribution is found
using Eq. (52) to be
d
dt
∆p
s
(t) = q
2
Z
t
0
dt
0
G
E
i
E
i
(t, r
0
|t
0
, r
0
) + c.c.
=
2θ(t
r
)
N
×
e
−
Γ
2
t
r
(Γ cos[ω
0
t
r
] + 2ω
0
sin[ω
0
t
r
]) − Γe
−Γt
r
(75)
where
1
N
=
q
2
E
rad
(r
0
)
2
(Γ/2)
2
+ ω
2
0
(76)
with E
rad
the radiative component of E given in
Eq. (13). What remains is the advanced-wave contri-
bution arising due to vacuum source-field correlations.
Accepted in Quantum 2017-12-15, click title to verify 11
0
10
2r
0
c
6
Γt
N [∆p
vac−s
(t) − ∆p
vac−s
]
N [∆p
s
(t) − ∆p
s
]
Figure 2: The contributions N [∆p
s
(t) − ∆p
s
] and
N [∆p
vac−s
(t)−∆p
vac−s
] are plotted. The decay rate Γ = 10
8
is chosen in the optical regime. To clearly illustrate the oscil-
latory character the transition frequency has been chosen as
ω
0
= 10Γ. The time r
0
/c where c denotes the speed of light
in the vacuum has been chosen as 1/(3Γ).
This contribution is found using Eq. (58) to be
d
dt
∆p
vac−s
(t) = q
2
Z
t
0
dt
0
h∆
E
i
E
i
(t, r
0
|t
0
, r
0
)i + c.c.
=
1
N
θ(t
r
− r
0
)
Γ(2e
−Γt
r
+ 1)
− e
−
Γ
2
t
Γ(e
Γr
0
+ 2) cos[ω
0
(t − 2r
0
)]
− 2ω
0
(e
Γr
0
− 2) sin[ω
0
(t − 2r
0
)]
. (77)
Comparing Eqs. (75) and (77) we see that the contribu-
tion from G
E
i
E
i
is non-zero for times t ≥ r
0
while the
contribution from h∆
E
i
E
i
i is only non-zero for times
t ≥ 2r
0
. This allows the two contributions to be clearly
distinguished. For t < 2r
0
the world line of the dipole
does not pass through the intersection of the backward
lightcone of (t, r
0
) and the forward lightcone of any
point on the charge’s world line with time coordinate
in the interval [0, t]. In this case there are no advanced
contributions to C
E
i
E
i
(t, r
0
|t
0
, r
0
).
The change in momentum dispersion is
∆p(t) − ∆p =
Z
t
0
dt
0
d
dt
0
∆p(t
0
), (78)
which can be partitioned in the same way as the mo-
mentum diffusion as the sum of two terms ∆p
s
(t)−∆p
s
and ∆p
vac−s
(t)−∆p
vac−s
. These two contributions are
shown separately in figures 2 and 3. Both contributions
have an oscillatory character, which is increasingly sup-
pressed as time increases.
0
10
2r
0
c
4
Γt
N [∆p
vac−s
(t) − ∆p
vac−s
]
N [∆p
s
(t) − ∆p
s
]
Figure 3: The contributions N [∆p
s
(t) − ∆p
s
] and
N [∆p
vac−s
(t) − ∆p
vac−s
] are plotted as in figure 2, but with
the larger transition frequency ω
0
= 100Γ. These parameters
are well within the Markovian regime ω
0
Γ. Comparing fig-
ures 2 and 3 we see that for fixed Γ increasing ω
0
/Γ results
in more rapid oscillations, but the same overall behavior with
increasing time.
It is instructive to compare this scenario with that
of a free particle whose initial Gaussian wave-packet
spreads in time, but whose momentum dispersion is
conserved. We see that since the particle is not free
the momentum dispersion is oscillatory and is not con-
served. However, if we retain only the contribution
∆p
s
(t) − ∆p
s
that comes from G
E
i
E
i
then in the long
time limit when the dipole has completely decayed the
charge’s momentum dispersion reaches a steady value,
effectively behaving like that of a free particle possess-
ing non-zero average momentum. Of course, even if
initially saturated the uncertainty principle for a free
charge is highly unsaturated for times on the order of
t
q
due to the spreading of the initial wave-packet.
If the contribution ∆p
vac−s
(t)−∆p
vac−s
coming from
∆
E
i
E
i
is also included we see that the behaviour of the
momentum dispersion is quite different. In contrast to
the case of a free particle the momentum dispersion
increases linearly in time in the long-time limit, despite
the decay of the initially excited dipole. The change
in the dispersion of the position, which quantifies the
spreading of the wave-packet is found via integration of
Eq. (72). Assuming the initial dipole-field state |e, 0i,
Accepted in Quantum 2017-12-15, click title to verify 12
0
13
2r
0
c
4
N [∆p(t) − ∆p]
Γt
Figure 4: The complete change in momentum dispersion
N [∆p(t) − ∆p] is plotted with all parameters chosen as in
figure 3. Due to the advanced-wave contribution coming
from vacuum-source correlations the momentum dispersion in-
creases with time. For large t a good fit for the behaviour of
N [∆p(t) − ∆p] is N [∆p(t) − ∆p] = k + Γt where k is a
constant.
it is given by
∆r(t) − ∆r =
t
2
2m
2
∆p +
q
2
m
2
×
Z
t
0
dt
1
Z
t
0
dt
2
Z
t
1
0
dt
3
Z
t
2
0
dt
4
C
E
i
E
i
(t
3
, r
0
|t
4
, r
0
)
+ c.c. (79)
where we have assumed that there are no initial cor-
relations between the position and momentum of the
charge, and where we have again neglected the pure
vacuum contribution. As well as the free component
t
2
∆p/m
2
that results in the usual spreading of the wave
packet, there is also a dipole-dependent contribution in
Eq. (79), which oscillates in time. In the radiation
zone r
0
ω
−1
0
the free component dominates, because
the remaining contribution decreases as r
−2
0
. As in the
case of a free charge the product ∆r(t)∆p(t) increases
with time such that the uncertainty principle becomes
increasingly unsaturated.
8 Conclusions
In this paper we have focused on quadratic function-
als of the electromagnetic field associated with a single
stationary dipole. We have extended previous pertur-
bative results, which demonstrate that vacuum-source-
field correlations provide significant contributions to
the radiation intensity of the dipole. We have de-
rived general non-perturbative expressions for arbitrary
quadratic field functionals using standard optical ap-
proximations. Due to correlations between the vacuum-
field and the source-fields of the dipole, contributions
coming from the advanced green’s function for the wave-
operator are generally non-vanishing within unequal-
time field correlation functions. This lies in marked con-
trast to the derivation of the source-fields themselves.
The contribution of vacuum-source correlations to
photo-detection amplitudes was shown to be insignif-
icant. However, by developing a description of a free
charge q analogous to the multipolar description of a
dipole, it was shown that vacuum-source correlations
yield significant contributions to statistical predictions
involving the force experienced by a free charge in the
field of a dipole.
In classical electrodynamics, while advanced solutions
to the wave equation can be used as a theoretical tool,
advanced waves are not usually thought to posses any
basis in physical reality. Our results indicate that ad-
vanced waves associated with the quantum vacuum do
exist, and that it should in principle be possible to ver-
ify this using an experiment. These results offer yet
another signature of the quantum nature of the vac-
uum and open up interesting prospects for further in-
vestigation of advanced-wave like correlations in various
branches of quantum optics.
Acknowledgement. This work was supported by the
UK engineering and physical sciences research council
grant number EP/N008154/1. I thank Dr. A. Nazir for
useful discussions relating to this work.
Accepted in Quantum 2017-12-15, click title to verify 13
9 Appendix
9.1 Derivation of source-fields
The equation of motion for the photon annihilation operator is found using the Hamiltonian (3) and once formally
integrated reads
a
λ
(t, k) = a
λ
(k)e
−iωt
+
Z
t
0
dt
0
e
−iω(t−t
0
)
r
ω
2(2π)
3
X
nm
e
λ
(k) · d
nm
σ
nm
(t
0
) ≡ a
λ,0
(t, k) + a
λ,s
(t, k). (80)
where σ
nm
(t) = e
iHt
|nihm|e
−iHt
. To go further we express Eq. (80) in terms of interaction picture dipole operators
˜σ
nm
(t) = σ
nm
(t)e
−iω
nm
t
and perform a rotating-wave approximation, which neglects terms oscillating rapidly at
frequencies ω + ω
nm
, n > m. Substituting the resulting expression into Eq. (6), and evaluating the angular integral
and polarisation summation yields
E
(+)
rad,s,i
(t, x) =
i
4π
2
X
n<m
Z
∞
0
dω ω
3
τ
ij
(ωx)d
j
nm
Z
t
0
dt
0
e
−i(ω+ω
nm
)(t−t
0
)
˜σ
nm
(t
0
)e
iω
nm
t
(81)
where
τ
ij
(ωx) = (δ
ij
− ˆx
i
ˆx
j
)
sin(ωx)
ωx
+ (δ
ij
− 3ˆx
i
ˆx
j
)
cos(ωx)
(ωx)
2
−
sin(ωx)
(ωx)
3
(82)
The near, intermediate and far zone components of expression (81) are evaluated separately. Using the identity
sin x = (e
ix
− e
−ix
)/2i the far zone component can be written
E
(+)
rad,s
(t, x) =
1
8π
2
x
X
n<m
[d
nm
−
ˆ
x(
ˆ
x · d
nm
)]
Z
t
0
dt
0
˜σ
nm
(t
0
)
Z
∞
0
dω ω
2
e
i(ω+ω
nm
)t
0
e
−iωt
r
− e
−iωt
a
(83)
where t
r
= t − x and t
a
= t + x. This expression involves both retarded and advanced waves but, it will be seen
that the advanced waves do not contribute. The integrand in Eq. (83) is dominated by the resonant contribution
at ω = ω
mn
> 0. We can therefore make use of the following Markov approximation
Z
∞
0
dω f(ω)e
iωt
0
e
−iωt
r
± e
−iωt
a
≈ f(ω
mn
)
Z
∞
−∞
dω e
iωt
0
e
−iωt
r
± e
−iωt
a
= 2πf(ω
mn
)[δ(t
0
− t
r
) ± δ(t
0
− t
a
)], (84)
where f is a suitably behaved function, to obtain the radiation source-field
E
rad,s
(t, x) = E
(+)
rad,s
(t, x) + E
(−)
rad,s
(t, x), E
(+)
rad,s
(t, x) =
1
4πx
X
n<m
ω
2
nm
[d
nm
−
ˆ
x(
ˆ
x · d
nm
)]σ
nm
(t
r
). (85)
The derivations of the near and intermediate-zone components of the electric source-field, and the magnetic source-
field do not involve any essentially different steps to those above. The final result including both full source-fields
is given by Eq. (10).
9.2 Contribution of vacuum source-field correlations to radiated power
9.2.1 Perturbative calculation of vacuum-source-field correlations and associated radiated power
The equation of motion for the operator σ
nm
(t) can be integrated to give
˜σ
nm
(t) = σ
nm
+ i
Z
t
0
dt
0
X
p
n
[d
mp
· D
T
(t
0
, 0)]˜σ
np
(t
0
)e
iω
mp
t
0
− [d
pn
· D
T
(t
0
, 0)]˜σ
pm
(t
0
)e
iω
pn
t
0
o
. (86)
where
D
T
(t, 0) = i
Z
d
3
k
X
λ
r
ω
2(2π)
3
e
λ
(k)
h
a
λ
(t, k) − a
†
λ
(t, k)
i
. (87)
Accepted in Quantum 2017-12-15, click title to verify 14
The first order component of Eq. (86) is
˜σ
(1)
nm
(t) = i
Z
t
0
dt
0
X
p
n
[d
mp
· D
T,0
(t
0
, 0)]σ
np
e
iω
mp
t
0
− [d
pn
· D
T,0
(t
0
, 0)]σ
pm
e
iω
pn
t
0
o
(88)
where
D
T,0
(t, 0) = i
Z
d
3
k
X
λ
r
ω
2(2π)
3
e
λ
(k)
h
a
λ
(k)e
−iωt
− a
†
λ
(k)e
iωt
i
. (89)
Using Eqs. (85) and (88) the radiation source-field correct to second order can now be completely expressed in
terms of operators at t = 0;
E
(2)
rad,s
(t, x) =
1
4πx
X
n,m
ω
2
nm
[d
nm
−
ˆ
x(
ˆ
x · d
nm
)]˜σ
(1)
nm
(t
r
)e
iω
nm
t
r
. (90)
Meanwhile the vacuum electric field is
E
0
(t, x) = i
Z
d
3
k
X
λ
r
ω
2(2π)
3
e
λ
(k)a
λ
(k)e
−iωt+ik·x
+ H.c.. (91)
Using Eqs. (90) and (91) we obtain
hE
0
(t, x) · E
(2)
rad,s
(t, x)i
0;e
=
i
(8π
2
)
2
x
Z
d
3
k ω e
ik·x
X
n
ω
2
en
[d
en
· e
λ
(k)] e
λ
(k) · [d
ne
−
ˆ
x(
ˆ
x · d
ne
)]
×
Z
t
r
0
dt
0
e
iω(t
0
−t)
h
e
iω
en
(t
0
−t
r
)
− e
−iω
en
(t
0
−t
r
)
i
. (92)
We now evaluate the angular integral and sum over polarisations, and we retain only terms which vary as x
−2
, to
give
hE
0
(t, x) · E
(2)
rad,s
(t, x)i
0;e
= −
1
4(2π)
3
x
2
Z
t
r
0
dt
0
X
n
ω
2
en
[d
en
−
ˆ
x(
ˆ
x · d
en
)]
2
e
iω
en
(t
0
−t
r
)
− e
−iω
en
(t
0
−t
r
)
Z
∞
0
dω ω
2
e
iω(t
0
−t
r
)
− e
iω(t
0
−t
a
)
. (93)
Next we perform a rotating wave-approximation as in the derivation of Eq. (83), which yields
hE
0
(t, x) · E
(2)
rad,s
(t, x)i
0;e
= −
1
4(2π)
3
x
2
Z
t
r
0
dt
0
Z
∞
0
dω ω
2
e
iω(t
0
−t
r
)
− e
iω(t
0
−t
a
)
×
"
X
n>e
ω
2
en
[d
en
−
ˆ
x(
ˆ
x · d
en
)]
2
e
iω
en
(t
0
−t
r
)
−
X
n<e
ω
2
en
[d
en
−
ˆ
x(
ˆ
x · d
en
)]
2
e
−iω
en
(t
0
−t
r
)
#
.
(94)
Since the integrand is now dominated by positive resonant frequencies ω = ω
en
, n < e and ω = −ω
en
, n > e, we
can perform the Markov approximation, Eq. (84), to obtain
hE
0
(t, x) · E
(2)
rad,s
(t, x)i
0;e
=
1
2
1
(4πx)
2
X
n
sgn(ω
en
)ω
4
en
[d
en
−
ˆ
x(
ˆ
x · d
en
)]
2
(95)
where we have used
Z
t
0
dt
0
δ(t
0
− t)f(t
0
) =
1
2
f(t). (96)
From Eq. (95) we obtain the corresponding contribution to the radiated power;
P
vac−s
=
Z
dΩ x
2
hE
0
(t, x) · E
(2)
rad,s
(t, x)i
0;e
+ c.c.
=
1
2
X
n
sgn(ω
en
)ω
en
Γ
en
. (97)
which is Eq. (34).
Accepted in Quantum 2017-12-15, click title to verify 15
9.2.2 Radiation reaction: derivation of the Heisenberg-Langevin equations
Restricting our attention to a two-level dipole with ground state |gi and excited state |ei, we define the dipole
operators
σ
+
= |eihg|, σ
−
|gihe|, σ
z
= [σ
+
, σ
−
]. (98)
The two-level transition frequency, dipole moment and decay rate are denoted ω
0
, d and Γ respectively. From the
solution (80) we calculate the dipole’s own reaction field as
D
(+)
T,s
(t, 0) = i
Z
d
3
k
r
ω
2(2π)
3
X
λ
e
λ
(k)a
λ,s
(t, k)
= i
Γ
2π
ˆ
d
d
Z
∞
0
dω
ω
ω
0
3
Z
t
0
dt
0
e
iω(t
0
−t)
h
˜σ
+
(t
0
)e
iω
0
t
0
+ ˜σ
−
(t
0
)e
−iω
0
t
0
i
. (99)
where we have evaluated the angular integral and polarisation summation. We now perform the rotating-wave
approximation and then the Markov approximation, Eq. (84), which give
D
(+)
T,s
(t, 0) = i
Z
d
3
k
r
ω
2(2π)
3
X
λ
e
λ
(k)a
λ,s
(t, k) = i
Γ
2
ˆ
d
d
σ
−
(t) (100)
where we have used used Eq. (96). Substituting Eq. (100) into the Heisenberg equation for each of σ
+
and σ
z
yields
the following equations of motion
˙σ
+
(t) =
iω
0
−
Γ
2
σ
+
(t) +
Γ
2
σ
−
(t) + σ
0+
vac
(t) (101)
˙σ
z
(t) = −Γ[σ
z
(t) + 1] + σ
0z
vac
(t) (102)
where
σ
0+
vac
(t) =
Z
d
3
k
X
λ
r
ω
2(2π)
2
d · e
λ
(k)
h
a
†
λ
(k)σ
z
(t)e
iωt
− σ
z
(t)a
λ
(k)e
−iωt
i
(103)
σ
0z
vac
(t) =
Z
d
3
k
X
λ
s
2ω
(2π)
2
d · e
λ
(k)
h
a
†
λ
(k)[σ
+
(t) − σ
−
(t)]e
iωt
− [σ
+
(t) − σ
−
(t)]a
λ
(k)e
−iωt
i
. (104)
Making the rotating-wave approximation in Eqs. (101) and (102) yields optical Heisenberg-Langevin type equations
(with no external driving). Formally integrating these equations then yields
σ
+
(t) = e
(iω
0
−Γ/2)t
σ
+
+
Z
t
0
dt
0
e
(iω
0
−Γ/2)(t−t
0
)
σ
+
vac
(t
0
) (105)
σ
z
(t) + 1 = e
−Γt
[σ
z
+ 1] +
Z
t
0
dt
0
e
−Γ(t−t
0
)
σ
z
vac
(t
0
). (106)
where
σ
+
vac
(t) =
Z
d
3
k
X
λ
r
ω
2(2π)
2
d · e
λ
(k)a
†
λ
(k)σ
z
(t)e
iωt
= i[d · D
(−)
T,0
(t, 0)]σ
z
(t) (107)
σ
z
vac
(t) = −
Z
d
3
k
X
λ
s
2ω
(2π)
2
d · e
λ
(k)
h
a
†
λ
(k)σ
−
(t)e
iωt
+ σ
+
(t)a
λ
(k)e
−iωt
i
= −2i[d · D
(−)
T,0
(t, 0)]σ
−
(t) + H.c.
(108)
Making the rotating-wave approximation yields the solutions in Eqs (36) and (37).
Accepted in Quantum 2017-12-15, click title to verify 16
9.2.3 Non-perturbative calculation of vacuum-source-field correlations and associated radiated power
We now use the solutions (36) and (37) to calculate the radiated power. We note that although Markov and rotating
wave approximations have been employed we have not used perturbation theory. The pure source-field contribution
to the intensity is immediately found using the solution (85) restricted to two dipole levels;
hE
rad,s
(t, x) · E
rad,s
(t, x)i
0,e
=
ω
2
0
4πx
[d −
ˆ
x(d ·
ˆ
x)]
2
hσ
+
(t
r
)σ
−
(t
r
) + σ
−
(t
r
)σ
+
(t
r
)i
0;e
=
ω
2
0
4πx
[d −
ˆ
x(d ·
ˆ
x)]
2
. (109)
The corresponding contribution to the power is then easily obtained as P
s
= ω
0
Γ/2. The vacuum source-field
correlation function is found using Eqs. (91) and (85) to be
hE
0
(t, x) · E
rad,s
(t, x)i
0,e
= hE
(+)
0
(t, x) · E
rad,s
(t, x)i
0,e
=
ω
2
0
4πx
[d −
ˆ
x(d ·
ˆ
x)] · hE
(+)
0
(t, x)[σ
+
(t
r
) + σ
−
(t
r
)]i
0;e
.
(110)
From Eqs. (36) and (38) it follows that
hE
(+)
0
(t, x)[σ
+
(t
r
) + σ
−
(t
r
)]i
0;e
= hE
(+)
0
(t, x)σ
+
(t
r
)i
0;e
= i
Z
d
3
k
r
ω
2(2π)
3
X
λ
e
λ
(k)ha
λ
(k)σ
+
(t
r
)i
0;e
e
−iωt+ik·x
= i
Z
d
3
k
X
λ
ω
2(2π)
3
e
λ
(k)[e
λ
(k) · d]e
ik·x
Z
t
r
0
dt
0
hσ
z
(t
0
)i
0;e
e
−iω(t−t
0
)
e
(iω
0
−Γ/2)(t
r
−t
0
)
(111)
Substituting this result into Eq. (110) we obtain
hE
(+)
0
(t, x) · E
rad,s
(t, x)i
0,e
= hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0,e
=
iω
2
0
4πx
Z
d
3
k
X
λ
ω
2(2π)
3
(e
λ
(k) · d)(e
λ
(k) · [d −
ˆ
x(d ·
ˆ
x)])e
ik·x
Z
t
r
0
dt
0
(2e
−Γt
0
− 1)e
(iω
0
−Γ/2)(t
r
−t
0
)
e
−iω(t−t
0
)
(112)
where we have used hσ
z
(t)i
0;e
= 2e
−Γt
−1, which follows from Eq. (37). We see from Eq. (112) that the rotating-wave
and Markov approximations do not eliminate the anti-normally ordered contribution hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0,e
,
which is a slowly varying resonant contribution. It is nonetheless not included in the correlation function hE
(−)
(t, x)·
E
(+)
(t, x)i
0,e
. Evaluating the angular integral and polarisation summation now yields
hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0,e
=
ω
0
4πx
[d −
ˆ
x(d ·
ˆ
x)]
2
Z
t
r
0
dt
0
2e
−Γt
0
− 1
e
(iω
0
−Γ/2)(t
r
−t
0
)
1
2π
Z
∞
0
dω ω
2
h
e
−iω(t
r
−t
0
)
− e
−iω(t
a
−t
0
)
i
. (113)
We now make use of the Markov approximation, Eq. (84), which gives
hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0,e
=
ω
2
0
4πx
[d −
ˆ
x(d ·
ˆ
x)]
2
e
−Γt
r
−
1
2
(114)
where we have used Eq. (96). From Eq. (114) we obtain the corresponding contribution to the radiated power;
P
vac−s
=
Z
dΩ x
2
hE
(+)
0
(t, x) · E
(−)
rad,s
(t, x)i
0;e
+ c.c.
= ω
0
Γ
e
−Γt
r
−
1
2
, (115)
which is Eq. (45).
Accepted in Quantum 2017-12-15, click title to verify 17
9.3 Unequal-time commutators of electromagnetic fields and advanced-wave correlations
9.3.1 Unequal-time commutators
We assume that dipole operators σ
±
(t) and σ
z
(t) commute with the field operators a
λ
(t, k) and a
†
λ
(t, k). We
demonstrate here how various commutation relations between the fields E
±
, B
±
at arbitrary space-time points can
be calculated. As our example we take the electric field commutator [E
(+)
i
(t, x), E
(−)
j
(t
0
, x
0
)], which can be written
as the sum of four terms;
[E
(+)
i
(t, x), E
(−)
j
(t
0
, x
0
)]
= [E
(+)
0,i
(t, x), E
(−)
0,j
(t
0
, x
0
)] + [E
(+)
0,i
(t, x), E
(−)
s,j
(t
0
, x
0
)] + [E
(+)
s,i
(t, x), E
(−)
0,j
(t
0
, x
0
)] + [E
(+)
s,i
(t, x), E
(−)
s,j
(t
0
, x
0
)]. (116)
The pure vacuum term is easily found using Eq. (5) to be
[E
(+)
0,i
(t, x), E
(−)
0,j
(t
0
, x
0
)] =
Z
d
3
k
X
λ
ω
2(2π)
3
e
−iω(t−t
0
)+ik·(x−x
0
)
. (117)
The pure source term is also easily found with the help of Eq. (12) to be
[E
(+)
s,i
(t, x), E
(−)
s,j
(t
0
, x
0
)] = E
∗
i
(x)E
j
(x
0
)θ(t
r
)θ(t
0
r
)[σ
−
(t
r
), σ
+
(t
0
r
)] (118)
where t
r
= t − x and t
0
r
= t
0
− x
0
.
Simplification of the remaining two terms on the right-hand-side of Eq. (116), which involve both the vacuum
and source-fields requires more work. We begin by noting that by rearranging the rotating-wave approximated
integrated Heisenberg equation for the operator a
λ
(t, k), we can express a
λ
(k) as
a
λ
(k) = e
iωt
a
λ
(t, k) −
Z
t
0
dt
0
e
iωt
0
r
ω
2(2π)
3
[e
λ
(k) · d]σ
−
(t
0
). (119)
Noting that E
(−)
s,j
(t
0
, x
0
) is a function of σ
+
(t
0
r
) we use Eq. (119) to express the vacuum field as a function of a
λ
(t
0
r
, k),
which allows us to write the term [E
(+)
0,i
(t, x), E
(−)
s,j
(t
0
, x
0
)] as
[E
(+)
0,i
(t, x), E
(−)
s,j
(t
0
, x
0
)] = −iE
j
(x
0
)θ(t
0
r
)
Z
d
3
k
X
λ
ω
2(2π)
3
e
λ,i
(k)[e
λ
(k) · d]e
ik·x
Z
t
0
r
0
ds e
−iω(t−s)
[σ
−
(s), σ
+
(t
0
r
)]
(120)
where we have used the fact that equal-time dipole and field operators commute. We now evaluate the polarisation
summation and integration over solid angle to obtain
[E
(+)
0,i
(t, x), E
(−)
s,j
(t
0
, x
0
)] = −
i
4π
2
E
j
(x
0
)d
k
θ(t
0
r
)
Z
∞
0
dω ω
3
τ
ik
(ωx)
Z
t
0
r
0
ds e
−iω(t−s)
[σ
−
(s), σ
+
(t
0
r
)] (121)
where τ
ik
is defined in Eq. (82). Using Eq. (82) yields
[E
(+)
0,i
(t, x), E
(−)
s,j
(t
0
, x
0
)]
= −
1
8π
2
E
j
(x
0
)d
k
θ(t
0
r
)
Z
∞
0
dω
Z
t
0
r
0
ds [σ
−
(s), σ
+
(t
0
r
)]
ω
2
x
(δ
ik
− ˆx
i
ˆx
k
)
e
iω(s−t
r
)
− e
iω(s−t
a
)
+
iω
x
2
(δ
ik
− 3ˆx
i
ˆx
k
)
e
iω(s−t
r
)
+ e
iω(s−t
a
)
−
1
x
3
(δ
ik
− 3ˆx
i
ˆx
k
)
e
iω(s−t
r
)
− e
iω(s−t
a
)
, (122)
Accepted in Quantum 2017-12-15, click title to verify 18
and using the Markov approximation we obtain
[E
(+)
0,i
(t, x), E
(−)
s,j
(t
0
, x
0
)]
≈ −
1
4π
E
j
(x
0
)d
k
θ(t
0
r
)
Z
t
0
r
0
ds [σ
−
(s), σ
+
(t
0
r
)]
ω
2
0
x
(δ
ik
− ˆx
i
ˆx
k
) [δ(s − t
r
) − δ(s − t
a
)]
+
iω
0
x
2
(δ
ik
− 3ˆx
i
ˆx
k
) [δ(s − t
r
) + δ(s − t
a
)] −
1
x
3
(δ
ik
− 3ˆx
i
ˆx
k
) [δ(s − t
r
) − δ(s − t
a
)]
= −
1
4π
E
j
(x
0
)d
k
θ(t
0
r
)θ(t
r
)θ(t
0
r
− t
r
)[σ
−
(t
r
), σ
+
(t
0
r
)]
ω
2
0
x
(δ
ik
− ˆx
i
ˆx
k
) +
iω
0
x
2
−
1
x
3
(δ
ik
− 3ˆx
i
ˆx
k
)
−
1
4π
E
j
(x
0
)d
k
θ(t
a
)θ(t
0
r
− t
a
)[σ
−
(t
a
), σ
+
(t
0
r
)]
−
ω
2
0
x
(δ
ik
− ˆx
i
ˆx
k
) +
iω
0
x
2
+
1
x
3
(δ
ik
− 3ˆx
i
ˆx
k
)
= − E
∗
i
(x)E
j
(x
0
)θ(t
0
r
)θ(t
r
)θ(t
0
r
− t
r
)[σ
−
(t
r
), σ
+
(t
0
r
)] + E
i
(x)E
j
(x
0
)θ(t
0
r
)θ(t
a
)θ(t
0
r
− t
a
)[σ
−
(t
a
), σ
+
(t
0
r
)].
(123)
This result coincides with the result given in Eq. (55) with X = E = Y. The remaining term [E
(+)
s,i
(t, x), E
(−)
0,j
(t
0
, x
0
)]
on the right-hand-side of Eq. (116) can be calculated in the same way as the term calculated above and is given
by the right-hand-side of Eq. (56) with X = E = Y. Thus, having calculated all source-dependent terms on the
right-hand-side of Eq. (116) we have using Eqs. (118) and (123) that
∆
E
i
E
j
(t, x|t
0
, x
0
) = [E
(+)
i
(t, x), E
(−)
j
(t
0
, x
0
)] − [E
(+)
0,i
(t, x), E
(−)
0,j
(t
0
, x
0
)]
= E
i
(x)E
j
(x
0
)θ(t
0
r
)θ(t
a
)θ(t
0
r
− t
a
)[σ
−
(t
a
), σ
+
(t
0
r
)] + E
∗
i
(x)E
∗
j
(x
0
)θ(t
r
)θ(t
0
a
)θ(t
r
− t
0
a
)[σ
−
(t
r
), σ
+
(t
0
a
)], (124)
which coincides with Eq. (57) in the case X = E = Y. Calculations of the same type as above can be used to find the
remaining unequal time commutators [X
(+)
i
(t, x), Y
(−)
j
(t
0
, x
0
)], X, Y = E, B, which eventually yields Eqs. (55), (56)
and (57). By setting t = t
0
in Eq. (57) the right-hand-side vanishes, which proves that [X
(+)
i
(t, x), Y
(−)
j
(t, x
0
)] =
[X
(+)
0,i
(t, x), Y
(−)
0,j
(t, x
0
)] within the Markov and rotating-wave approximations. Thus, the approximate theory is
formally consistent as claimed at the end of section 3.2.
9.3.2 Advanced-wave correlations
Supposing that the initial state of the dipole and field is |0, ei the average h∆
X
i
Y
j
(t, x|t
0
, x
0
)i
e,0
depends, according
to Eq. (57), on the averages h[σ
−
(t
a
), σ
+
(t
0
r
)]i
e,0
and h[σ
−
(t
r
), σ
+
(t
0
a
)]i
e,0
. These averages can be found using
Eqs. (36) and (37). The averages hσ
+
(t
0
r
)σ
−
(t
a
)i
e,0
and hσ
+
(t
0
a
)σ
−
(t
r
)i
e,0
are immediately found to be
hσ
+
(t
0
r
)σ
−
(t
a
)i
e,0
= e
(iω
0
−Γ/2)t
0
r
e
(−iω
0
−Γ/2)t
a
, hσ
+
(t
0
a
)σ
−
(t
r
)i
e,0
= e
(iω
0
−Γ/2)t
a
e
(−iω
0
−Γ/2)t
0
r
. (125)
The average hσ
−
(t
a
)σ
+
(t
0
r
)i
e,0
is
hσ
−
(t
a
)σ
+
(t
0
r
)i
e,0
= d
i
d
j
Z
t
a
0
ds
Z
t
0
r
0
ds
0
e
(−iω
0
−Γ/2)(t
a
−s)
e
(iω
0
−Γ/2)(t
0
r
−s
0
)
hσ
z
(s)D
(+)
T,0,i
(s, 0)D
(−)
T,0,j
(s
0
, 0)σ
z
(s
0
)i
e,0
.
(126)
Using the commutator [D
(+)
T,0,i
(s, 0), D
(−)
T,0,j
(s
0
, 0)] Eq. (126) can be written as the sum of two terms;
hσ
−
(t
a
)σ
+
(t
0
r
)i
e,0
=
Z
t
a
0
ds
Z
t
0
r
0
ds
0
Z
d
3
k
X
λ
ω
2(2π)
3
[e
λ
(k) · d]
2
e
(−iω
0
−Γ/2)(t
a
−s)
e
(iω
0
−Γ/2)(t
0
r
−s
0
)
e
−iω(s−s
0
)
hσ
z
(s)σ
z
(s
0
)i
e,0
+ d
i
d
j
Z
t
a
0
ds
Z
t
0
r
0
ds
0
e
(−iω
0
−Γ/2)(t
a
−s)
e
(iω
0
−Γ/2)(t
0
r
−s
0
)
hσ
z
(s)D
(−)
T,0,j
(s
0
, 0)D
(+)
T,0,i
(s, 0)σ
z
(s
0
)i
e,0
. (127)
Accepted in Quantum 2017-12-15, click title to verify 19
Performing the angular integration and polarisation summation and employing the Markov approximation in the
first term on the right-hand-side yields
Γ
Z
t
a
0
ds
Z
t
0
r
0
ds
0
δ(s − s
0
)e
(−iω
0
−Γ/2)(t
a
−s)
e
(iω
0
−Γ/2)(t
0
r
−s
0
)
hσ
z
(s)σ
z
(s
0
)i
e,0
. (128)
Since within ∆
X
i
Y
j
(t, x|t
0
, x
0
) this term is multiplied by θ(t
0
r
− t
a
) we can assume that t
0
r
≥ t
a
in the above.
Performing the s integral then yields
Γe
(−iω
0
−Γ/2)t
a
e
(iω
0
−Γ/2)t
0
r
Z
t
a
0
ds
0
e
Γs
0
= e
(−iω
0
−Γ/2)t
a
e
(iω
0
−Γ/2)t
0
r
(e
Γt
a
− 1). (129)
To show that the second term on the right-hand-side of Eq. (127) vanishes one can show that for s 6= s
0
it is
always possible to change the order of σ
z
(s)D
(−)
T,0,j
(s
0
, 0) or of D
(+)
T,0,i
(s, 0)σ
z
(s
0
) within the expectation value. The
integral in Eq. (127), which involves this expectation value must therefore vanish. First consider the commutator
[σ
z
(s), D
(−)
T,0,j
(s
0
, 0)]. Using Eq. (119) we can express this as
[σ
z
(s), D
(−)
T,0,j
(s
0
, 0)] = i
Z
d
3
k
X
λ
ω
2(2π)
3
e
λ,j
(k)[e
λ
(k) · d]
Z
s
0
ds
00
[σ
z
(s), σ
+
(s
00
)]e
iω(s
0
−s
00
)
(130)
Writing this expression in terms of the interaction picture operator ˜σ
+
(s
00
) allows us to use the Markov approxima-
tion, which gives
[σ
z
(s), D
(−)
T,0,j
(s
0
, 0)] = i
ˆ
d
j
d
Γ
Z
s
0
ds
00
δ(s
00
− s
0
)[σ
z
(s), σ
+
(s
00
)] = i
ˆ
d
j
d
Γ[σ
z
(s), σ
+
(s
0
)]θ(s − s
0
)θ(s
0
). (131)
Similarly the commutator [D
(+)
T,0,i
(s, 0), σ
z
(s
0
)] is found to be
i
ˆ
d
i
d
Γ[σ
+
(s), σ
z
(s
0
)]θ(s
0
− s)θ(s). (132)
Since hσ
z
(s)D
(−)
T,0,j
(s
0
, 0)D
(+)
T,0,i
(s, 0)σ
z
(s
0
)i
e,0
= h[σ
z
(s), D
(−)
T,0,j
(s
0
, 0)][D
(+)
T,0,i
(s, 0), σ
z
(s
0
)]i
e,0
using Eqs. (131) and
(132) the second term on the right-hand-side of Eq. (127) can be expressed in the form
Z
t
a
0
ds
Z
t
0
r
0
ds
0
f(s, s
0
)θ(s − s
0
)θ(s
0
− s) = 0. (133)
Thus, the second term on the right-hand-side of Eq. (127) vanishes within the Markov approximation. As a result
hσ
−
(t
a
)σ
+
(t
0
r
)i
e,0
is given by the right-hand-side of Eq. (129). Using Eq. (125) and (129) we therefore have
h[σ
−
(t
a
), σ
+
(t
0
r
)]i
e,0
= e
(−iω
0
−Γ/2)t
a
e
(iω
0
−Γ/2)t
0
r
(e
Γt
a
− 2) (134)
in agreement with Eq. (58). The remaining commutator h[σ
−
(t
r
), σ
+
(t
0
a
)]i
e,0
can be found in a similar fashion and
this yields the remaining part of Eq. (58).
9.4 Static precursors within Coulomb gauge electric source-field of a charge
The Hamiltonian (67) yields the following solution for the Coulomb gauge operator a
λ
(t, k)
a
λ
(t, k) = e
−iωt
a
λ
(k) + i
Z
t
0
dt
0
e
−iω(t−t
0
)
1
√
2ω
e
λ
(k) ·
˜
J(t
0
, k) (135)
where
˜
J denotes the Fourier transform of the current J;
˜
J(t, k) =
1
p
(2π)
3
Z
d
3
x J(t, x)e
−ik·x
, J(t, x) =
q
2
[
˙
r(t)δ(x − r(t)) + δ(x − r(t))
˙
r(t)] . (136)
Accepted in Quantum 2017-12-15, click title to verify 20
When substituted into the mode-expansion for the transverse electric field Eq. (135) yields the source-field
E
T,s
(t, x) = −
Z
d
3
k
Z
t
0
dt
0
1
2
p
(2π)
3
˜
J(t
0
, k) −
ˆ
k[
ˆ
k ·
˜
J(t
0
, k)]
e
ik·x
e
−iω(t−t
0
)
+ H.c. (137)
Using integration by parts and the continuity equation ˙ρ = −∇·J where ρ(t, x) = qδ(x −r(t)) is the charge density,
this can be written
E
T,s
(t, x) =
"
i
p
(2π)
3
Z
d
3
k
ˆ
k˜ρ(t
0
, k)
ω
cos[ω(t − t
0
)]e
ik·x
#
t
0
=t
t
0
=0
−
1
p
(2π)
3
Z
t
0
dt
0
d
dt
Z
d
3
k
˜
J(t
0
, k)
sin[ω(t − t
0
)]
ω
e
ik·x
−
∇
p
(2π)
3
Z
d
3
k
Z
t
0
dt
0
˜ρ(t
0
, k)
sin[ω(t − t
0
)]
ω
e
ik·x
. (138)
The function sin[ω(t−t
0
)]/
p
(2π)
3
ω is essentially the Fourier transform of the green’s function for the wave-operator.
Using the convolution theorem and Gauss’ law
˜
E
L
(t, k) = −i
ˆ
k˜ρ(t, k)/ω Eq. (138) can be written
E
T,s
(t, x) = − E
L
(t, x) +
d
dt
Z
d
3
x
0
E
L
(0, x
0
)G(t, x|0, x
0
)
−
d
dt
Z
d
3
x
0
Z
t
0
dt
0
J(t
0
, x
0
)G
+
(t, x|t
0
, x
0
) − ∇
Z
d
3
x
0
Z
t
0
dt
0
ρ(t
0
, x
0
)G
+
(t, x|t
0
, x
0
) (139)
where G = G
+
+ G
−
, and where we have used
−
i
p
(2π)
3
Z
d
3
k
ˆ
k˜ρ(0, k)
ω
cos[ωt]e
ik·x
=
1
p
(2π)
3
Z
d
3
k
˜
E
L
(0, k) cos[ωt]e
ik·x
=
d
dt
1
p
(2π)
3
Z
d
3
k
˜
E
L
(0, k)
sin[ωt]
ω
e
ik·x
=
d
dt
Z
d
3
x
0
E
L
(0, x
0
)G(t, x|0, x
0
). (140)
By adding the longitudinal field to E
T,s
(t, x) we obtain the total electric source-field E
s
(t, x). Along with the
well-known retarded source-field given by the second line in Eq. (139), the electric source-field also possesses a term
given in Eq. (140), which is dependent on the charge density at the initial time t = 0.
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Accepted in Quantum 2017-12-15, click title to verify 23