waves in quantum optics
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-
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
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
+
and G
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
t+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-
a
λ
(t, k) = a
λ
(k)e
t
+
Z
t
0
dt
0
e
(tt
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
= tx > 0. For t < x the source-ﬁelds vanish.
The source-ﬁelds can be split into radiative and non-
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
(t, x) =
ˆ
x × B
(t, x). The remaining parts
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
}
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
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
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) =
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
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.
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].
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
d x
2
I(t, x) =
Z
d x
2
I
(t, x) (16)
where the integration is performed over all directions.
Eq. (16) is deﬁnitive of the radiation intensity I
,
which is the radiated power per unit area. The intensity
component I
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
nm
t
. First let us suppose that the
intensity I
in Eq. (16) is of the Glauber type
I
G
(t, x) = hE
()
(t, x) · E
(+)
(t, x)i
0,e
= hE
()
(t, x) · E
(+)
(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
= E
0
+ E
. Letting σ
nm
(t) σ
nm
(0)e
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
d 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
(t, x) =
ˆ
x · hS
(t, x) S
vac
i,
S
=
1
2
[E
× B
+ H.c.] (27)
where E
(t, x) = E
0
(t, x) + E
(t, x) and
B
(t, x) = B
0
(t, x)+B
(t, x). Assuming the initial
state |e, 0i and making use of the identities
ˆ
x · E
(t, x) = 0 =
ˆ
x · B
(t, x),
ˆ
x × E
(t, x) = B
(t, x),
ˆ
x × B
(t, x) = E
(t, x),
B
2
= E
2
(28)
it is straightforward to show that the radiation intensity
can also be written
I
(t, x) = hE
(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
(+)
(t, x)·E
()
(t, x), which is such
that h: E
(+)
(t, x)·E
()
(t, x) :i = hE
()
(t, x)·E
(+)
(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
(t, x) =hE
(t, x)
2
i
e,0
+ hE
0
(t, x) · E
(t, x)
+ E
(t, x) · E
0
(t, x)i
e,0
. (30)
First we calculate the anti-normally ordered combina-
tion hE
(+)
(t, x) · E
()
(t, x)i
e;0
, which is easily found
using Eq. (10) to be
hE
(+)
(t, x) · E
()
(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
in Eq. (30) to the radiated power to second
order as
P
s
=
Z
d x
2
hE
(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
()
(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
vacs
=
Z
d x
2
hE
0
(t, x) · E
(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
(
0
Γ/2)t
σ
+
+
Z
t
0
dt
0
e
(
0
Γ/2)(tt
0
)
σ
+
vac
(t
0
)
(36)
σ
z
(t) + 1 = e
Γt
[σ
z
+ 1] +
Z
t
0
dt
0
e
Γ(tt
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
(+)
(t, x) · E
()
(t, x)i
e;0
to be
hE
(+)
(t, x) · E
()
(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
()
(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
vacs
(t) = ω
0
Γ
e
Γt
r
1
2
. (45)
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.
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-
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
/2](t
r
+t
0
r
)
e
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
(
0
Γ/2)t
0
r
e
(
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
(
0
Γ/2)t
r
e
(
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
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
0
(tt
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
0
(tt
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
±
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
0
(tt
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
0
(tt
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
0
(tt
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
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
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.
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