# Nonlocality of the energy density of a spontaneously emitted single-photon
from a Hydrogen atom

M. Federico and H.R. Jauslin<EMAIL_ADDRESS>Laboratoire
Interdisciplinaire Carnot de Bourgogne - ICB, Université de Bourgogne, CNRS
UMR 6303, BP 47870, 21078 Dijon, France

###### Abstract

We analyze through the expectation value of the energy density the spatial
nonlocality of single photons emitted by the spontaneous decay of a Hydrogen
atom. By using a minimal coupling between the quantized electromagnetic field
and the atom, we compute the state of the photon under the assumption that
only a single-photon is produced. The calculations are thus performed in the
subspace of single-photon states which is essentially equivalent to the
rotating wave approximation. We obtain a characterization of the spatial decay
of the energy density. We compute the asymptotic limit of large distances from
the atom at each given time, and find an algebraic behavior of $1/r^{6}$. This
result confirms that the energy density of single-photon states is nonlocal
and the algebraic decay is far from the maximal quasiexponential localization
predicted by the theory.

††preprint: APS/123-QED

##  I Introduction

The description of the spontaneous decay of a Hydrogen atom is a standard
calculation that can be done using either the Weisskopf-Wigner theory [1, 2,
3] or time-dependent perturbation theory, e.g., with the Fermi golden rule
[3]. However, these approaches in general consider a dipolar coupling between
the field and the atom, taken as a two-level system, which prevents from
computing the state of the emitted photons since there is a frequency Lamb
shift given by a diverging intergral. The treatment of this divergence could
be done by a renormalization procedure, as suggested in [4]. This problem was
discussed, e.g., in [5, 6, 7, 8, 9, 10] and a way out was found by considering
a minimal coupling instead of the dipole approximation, which provides an
intrinsic ultraviolet cutoff function that avoids the need of a
renormalization. In this article we analyze the spatial properties of the
single-photon states that would be produced in this process. In view of the
intrinsic spatial nonlocality of single-photons, as discussed, e.g., in [11,
12, 13, 14, 15, 16, 17, 18], we would like to characterize this nonlocality
from the perspective of spontaneous emission. To do so, we will consider the
approach that uses the energy density to probe the nonlocality of photons, as
it was originally done in [16, 17] and recently extended to all single-photon
states in [18]. In this article it was shown that for any single-photon state,
the expectation value of the energy density in a small volume is different
from zero anywhere in space. This result leads to several natural questions if
we consider the photons produced by some concrete source: (i) If one assumes
that what is produced are single-photon states, what would be their spatial
localization properties? (ii) How does the mean value of the energy density
depend on the distance between the source and the detector? We analyze these
questions with the example of the spontaneous emission of a Hydrogen atom
prepared in its first excited state. We start with the complete Hamiltonian of
the atom interacting with the quantized electromagnetic field. The hypothesis
that only a single-photon is produced is implemented by projecting the model
to the single-photon subspace. Furthermore we keep only the fundamental and
the first excited state of the atom, corresponding to the Lyman-$\alpha$
transition. The time evolution in the Schrödinger picture can be represented
in terms of a system of Weisskopf-Wigner equations that can be solved exactly
in terms of some integrals [2]. This provides the spatio-temporal description
of the photon state, which we analyze at each fixed time as a function of the
distance from the source. The expectation value of the electromagnetic energy
density is non-zero in any finite volume at arbitrary distances $r$ from the
source, as it was proven for all single-photon states in [18]. The asymptotic
analysis for large distances leads to the result that at each fixed time the
energy density is proportional to $1/r^{6}$.

We conclude that the spontaneous emission cannot produce a strictly single-
photon state since it would contradict causality: At any fixed time $t$ there
would be a non-zero probability of detecting the photon with an instrument
located at an arbitrary distance $r$ from the source, in particular one could
take $r>ct$, where $c$ is the speed of light [19, 20, 21]. We remark that we
made the analysis for the situation where the atom is in the first excited
state at $t=0$, but a similar argument can be made if at $t=0$ the atom is in
a linear combination of the ground state and the first excited state. This
could correspond to the state prepared with a $\pi$-pulse of finite duration.

Several works have already addressed related questions in the context of
spontaneous emission. In particular, causality properties have been analyzed
in [22, 23, 24, 25], where different regimes, in the near- mid- and far-field
regions were described in terms of wave functions constructed by Glauber’s
extraction principle from a first order correlation function. This wave
function is identical, up to a global constant, with the Białynicki-Birula
representation of single-photon states [26], which is in turn equivalent to
the representation of the states that we will use in this work (Landau-
Peierls) [27]. However, we note that the formulas (25) in Ref. [24] do not
provide the asymptotic behavior for large distances
$r=\absolutevalue*{\vec{x}}$, since they still contain integrals over $k$
involving functions of $\absolutevalue*{\vec{x}}$ that will modify the
asymptotic form for large $\absolutevalue*{\vec{x}}$. Moreover, we stress that
our spatial analysis will not be performed for the state of the field but for
its energy density expectation value. This is a major difference since it has
been shown [18] that in some particular situations, the state function can be
spatially localized even though the corresponding energy density features a
nonlocality. The nonlocality property is thus not a property of the states
alone but of the joint representation of the states and the corresponding
local observables.

Causality properties were also analyzed in [28] where it was shown using the
electric energy density observable that the nonresonant terms are needed for
the spontaneous emission to be causal. This result is in agreement with the
characterization we perform in this work for the resonant part, for which we
also provide an expilcit asymptotic behavior. Such an asymptotic analysis,
performed through perturbation theory calculations, was also done in [29] but
for the dressed vacuum only, i.e., atom in its ground state and zero photons.

Finally, as it was noted by Shirokov [30], since the eigenfunctions of the
Hydrogen atom decrease exponentially in the radial variable, they are nonzero
everywhere. Therefore the emision of photons cannot be expected to be local,
but should have at least an exponential tail. This effect has been described
as a blurring of the causal light cone. However, this is an exponential decay
which is a relatively small effect compared to the nonlocality properties
described below.

To analyze the spatial properties of the photon states, we will use throughout
this work a position space formulation of the quantum theory of the
electromagnetic field called the Landau-Peierls representation (see Appendix A
for its definition). Any other position space representation could in
principle be used and the result is independent of this choice [27].

The structure and the main results of the article are as follows. In Section
II we formulate the Weisskopf-Wigner model with the
$\vec{\hat{A}}\cdot\vec{\hat{p}}$ interaction and we use the vector spherical
harmonics and helicity basis for single-photon states, as proposed in [5, 6,
7, 8, 9, 10]. This basis has the advantage that it leads easily to exact
selection rules for the interaction with Hydrogen atoms. In Section III we
write the Weisskopf-Wigner solution in terms of integrals. This solution is
exact under the condition that the spectrum of the associated Friedrichs
operator is absolutely continuous, which is expected in the small coupling
regime. Some of these integrals can be calculated explicitly if one makes a
resonant approximation which has essentially the same effect as the Weisskopf-
Wigner approximation, but without extending an integral to non-physical
negative frequencies. With the coupled atom-electromagnetic field state, we
calculate in Section IV the expectation value of the energy density at an
arbitrary position in $\mathbb{R}^{3}$. We then analyze its dependence on the
radial distance, in particular for the far-field, which characterizes the
global extension of the nonlocality. We obtain the result that the expectation
value of the energy density is proportional to $1/r^{6}$ for large $r$, i.e.,
it has an algebraic decay, which is far from the theoretically maximal
quasiexponential decay predicted in [16].

##  II Weisskopf-Wigner model

We consider a non-relativistic Hydrogen atom represented by a two-level system
corresponding to the Lyman-$\alpha$ transition, i.e., with ground state
$\ket{n_{g}=1;j_{g}=0;m_{g}=0}\equiv\ket{g}$ and excited state
$\ket{n_{e}=2;j_{e}=1;m_{e}=0,\pm 1}\equiv\ket{e}$. This transition is
relevant since it can be prepared experimentally by applying, e.g., a resonant
$\pi$-pulse on the ground state with a linearly polarized laser light for
$m_{e}=0$ and a circularly polarized laser light for $m_{e}=\pm 1$. The
wavefunctions associated to the ground and excited states are, respectively,

$\displaystyle\ket{g}$
$\displaystyle=\varphi_{g}(r,\vartheta,\varphi)=\frac{1}{\sqrt{\pi
r_{\text{B}}^{3}}}e^{-\frac{r}{r_{\text{B}}}},$ (1a) $\displaystyle\ket{e}$
$\displaystyle=\varphi_{e}(r,\vartheta,\varphi)=\beta_{m_{e}}(\vartheta,\varphi)\frac{r}{r_{\text{B}}}e^{-\frac{r}{2r_{\text{B}}}},$
(1b)

where $r_{\text{B}}=\hbar/(\alpha mc)$ is the Bohr radius expressed here in
terms of the fine structure constant $\alpha$, the electron mass $m$ and the
speed of light $c$. The functions $\beta_{m_{e}}(\vartheta,\varphi)$ depend on
the choice that is made for $m_{e}=\\{0,\pm 1\\}$ and read

$\displaystyle\beta_{0}(\vartheta,\varphi)$
$\displaystyle=\frac{1}{4\sqrt{2\pi r_{\text{B}}^{3}}}\cos\vartheta,$ (2)
$\displaystyle\beta_{\pm 1}(\vartheta,\varphi)$
$\displaystyle=\mp\frac{1}{8\sqrt{\pi r_{\text{B}}^{3}}}\sin\vartheta\ e^{\pm
i\varphi}.$ (3)

The free Hamiltonian of the two-level atom can be expressed as

$\hat{H}_{\text{at}}=E_{g}\ket{g}\bra{g}+E_{e}\ket{e}\bra{e},$ (4)

where $E_{g}$ and $E_{e}$ are the energies of the ground and excited states,
respectively. To describe the spontaneous emission, the two-level Hydrogen
atom is interacting with the quantized electromagnetic field which we write
using the basis $\ket{k,J,M,\lambda}\equiv\vec{\psi}_{k,J,M}^{(\lambda)}$ of
helicity vector spherical harmonics (see Appendix B for more details about
these functions)

$\hat{H}_{\text{em}}=\int_{0}^{\infty}dk\sum_{J,M}\sum_{\lambda=\pm}\hbar\omega_{k}\hat{B}^{\dagger}_{\vec{\psi}_{k,J,M}^{(\lambda)}}\hat{B}_{\vec{\psi}_{k,J,M}^{(\lambda)}}.$
(5)

For the interaction part, and following [5], we take an operator of the form

$\hat{H}_{\text{int}}=-\frac{e}{m}\vec{\hat{p}}\cdot\vec{\hat{A}}(\vec{x}),$
(6)

where $\vec{\hat{p}}=-i\hbar\nabla$ is the momentum operator of the electron
and $\vec{\hat{A}}(\vec{x})$ is the vector potential operator in the Coulomb
gauge. We have neglected the $\hat{A}^{2}$ term coming from the minimal
coupling since it has been shown by Moses [5] (see also [8, 9]) that taking an
interaction of the form $\vec{\hat{p}}\cdot\vec{\hat{A}}$ is enough to have a
consistent model allowing to compute the photon state. Moreover, the typical
order of magnitude for the effects relating to the $\hat{A}^{2}$ term have
been estimated to be much smaller than those from the
$\vec{\hat{p}}\cdot\vec{\hat{A}}$ term [31]. The resulting total Hamiltonian
is finally

$\displaystyle\hat{H}$
$\displaystyle=\hat{H}_{\text{at}}+\hat{H}_{\text{em}}+\hat{H}_{\text{int}}$
(7a) $\displaystyle=E_{g}\ket{g}\bra{g}+E_{e}\ket{e}\bra{e}$
$\displaystyle\quad+\int_{0}^{\infty}dk\sum_{J,M}\sum_{\lambda=\pm}\hbar\omega_{k}\hat{B}^{\dagger}_{\vec{\psi}_{k,J,M}^{(\lambda)}}\hat{B}_{\vec{\psi}_{k,J,M}^{(\lambda)}}$
$\displaystyle\quad\quad-\frac{e}{m}\vec{\hat{p}}\cdot\vec{\hat{A}}(\vec{x}).$
(7b)

In the following, since we are interested in the characterization of single
photons, we project the system into the subspace generated by the vacuum and
the single-photon states, which corresponds in the tensor product Hilbert
space to the subspace generated by

$\\{\ket{e;\varnothing},\ket*{g;\vec{\psi}_{k,J,M}^{(\lambda)}}\\},$

i.e., the atom in its excited state and zero photons and the atom in its
ground state and one photon. This approximation is essentially equivalent to
the rotating wave approximation that is performed in general for the dipole
coupling. Within this subspace, the interaction coefficient takes the form [5]

$\displaystyle\rho(k,J,M,\lambda)$
$\displaystyle=-\frac{e}{m}\bra*{g;\vec{\psi}_{k,J,M}^{(\lambda)}}\vec{\hat{p}}\cdot\vec{\hat{A}}(\vec{x})\ket*{e;\varnothing}$
(8a)
$\displaystyle=\left(\frac{2}{3}\right)^{7/2}\sqrt{\frac{\alpha^{5}}{\pi}}mc^{2}\frac{k/K}{\sqrt{k}}\frac{\delta_{J,1}\delta_{M,m_{e}}}{\left[1+\left(\frac{k}{K}\right)^{2}\right]^{2}}$
(8b) $\displaystyle\equiv\rho(k)\delta_{J,1}\delta_{M,m_{e}},$ (8c)

where $K=3/(2r_{\text{B}})$. Here, we have chosen a slightly different
convention for the global phases of the eigenfunctions
$\vec{\psi}_{k,J,M}^{(\lambda)}$ than the one of [5]. Our convention has the
advantage that $\rho$ is real and independent of the helicity $\lambda$.

##  III Solution in terms of integrals

Starting with a general state in the subspace of interest

$\ket{\Psi}=c_{0}\ket{e;\varnothing}+\int_{0}^{\infty}dk\sum_{J,M,\lambda}c_{k,J,M,\lambda}\ket*{g;\vec{\psi}_{k,J,M}^{(\lambda)}},$
(9)

we want to solve the Schrödinger equation

$i\hbar\frac{\partial}{\partial t}\ket{\Psi}=\hat{H}\ket{\Psi},$ (10)

with the initial condition $c_{0}(t=0)=1$ and $c_{k,J,M,\lambda}(t=0)=0$, for
any $k,J,M$ and $\lambda$. Expressed for the nonzero coefficients, we arrive
at the set of equations

$\displaystyle i\hbar\dot{c}_{0}(t)$
$\displaystyle=E_{e}c_{0}(t)+\int_{0}^{\infty}dk\sum_{\lambda=\pm}\rho(k)c_{k,\lambda}(t),$
(11a) $\displaystyle i\hbar\dot{c}_{k,\lambda}(t)$
$\displaystyle=\rho(k)c_{0}(t)+\left(E_{g}+\hbar\omega_{k}\right)c_{k,\lambda}(t),$
(11b)

where we have introduced the shorter notation $c_{k,J=1,M=m_{e},\lambda}\equiv
c_{k,\lambda}$. Moreover, since $\rho(k)$ is $\lambda$-independent, the last
equation is also $\lambda$-independent and thus $c_{k,+}(t)=c_{k,-}(t)$ for
all $t$ and one can define $d_{k}(t)=\sqrt{2}c_{k,\lambda}(t)$, to rewrite the
set of dynamical equations as

$\displaystyle i\dot{c}_{0}(t)$
$\displaystyle=\omega_{\text{a}}c_{0}(t)+\int_{0}^{\infty}dk\frac{\sqrt{2}}{\hbar}\rho(k)d_{k}(t),$
(12a) $\displaystyle i\dot{d}_{k}(t)$
$\displaystyle=\frac{\sqrt{2}}{\hbar}\rho(k)c_{0}(t)+\omega_{k}d_{k}(t),$
(12b)

where we have set the energy of the ground state to $E_{g}=0$, and introduced
the Bohr frequency of the two-level system
$\omega_{\text{a}}=(E_{e}-E_{g})/\hbar$. Expressed in terms of
$\omega_{k}=ck$, equations (12) can be written in a matrix form

$i\begin{bmatrix}\dot{c}_{0}\\\
\dot{D}_{\omega_{k}}\end{bmatrix}=\begin{bmatrix}\omega_{\text{a}}&\bra{\tilde{\rho}}\ket{\cdot}\\\
\tilde{\rho}&\omega_{k}\end{bmatrix}\begin{bmatrix}c_{0}\\\
D_{\omega_{k}}\end{bmatrix},$ (13)

where $D_{\omega_{k}}=d_{\omega_{k}}/\sqrt{c}$ and

$\tilde{\rho}(\omega_{k})=\sqrt{\frac{2}{c\hbar^{2}}}\rho(\omega_{k}).$ (14)

This formulation is known as the Friedrichs-Lee model [32, 33] and the exact
solution in terms of integrals can be formulated as [2]

$\displaystyle c_{0}(t)$ $\displaystyle=\int_{0}^{\infty}d\omega\
g(\omega)e^{-i\omega t},$ (15a) $\displaystyle D_{\omega_{k}}(t)$
$\displaystyle=-ie^{-i\omega_{k}t}\tilde{\rho}(\omega_{k})\int_{0}^{t}dt^{\prime}\
e^{i\omega_{k}t^{\prime}}c_{0}(t^{\prime}),$ (15b)

with

$g(\omega)=\frac{1}{2\pi}\frac{\Gamma(\omega)}{(\omega-\omega_{e}-\Delta(\omega))^{2}+\frac{\Gamma(\omega)^{2}}{4}},$
(16)

and

$\displaystyle\Gamma(\omega)$
$\displaystyle=2\pi\absolutevalue*{\tilde{\rho}(\omega)}^{2},$ (17a)
$\displaystyle\Delta(\omega)$
$\displaystyle=\text{pv}\int_{0}^{\infty}d\omega_{k}\
\frac{\absolutevalue*{\tilde{\rho}(\omega_{k})}^{2}}{\omega-\omega_{k}},$
(17b)

where pv is the principal value. In the weak coupling regime, $g(\omega)$ has
a high peak centered at $\omega=\omega_{\text{a}}$ and one can thus make the
following approximation [2]

$\displaystyle g(\omega)\simeq g_{\text{w}}(\omega)$
$\displaystyle=\frac{1}{2\pi}\frac{\Gamma(\omega_{\text{a}})}{(\omega-\omega_{\text{a}}-\Delta(\omega_{\text{a}}))^{2}+\frac{\Gamma(\omega_{\text{a}})^{2}}{4}}$
(18a)
$\displaystyle=\frac{1}{2\pi}\frac{\Gamma_{\text{a}}}{(\omega-\omega_{\text{a}}-\Delta_{\text{a}})^{2}+\frac{\Gamma_{\text{a}}^{2}}{4}},$
(18b)

where $\Gamma_{\text{a}}=\Gamma(\omega_{\text{a}})$ and
$\Delta_{\text{a}}=\Delta(\omega_{\text{a}})$. This approximation allows to
express $c_{0}(t)$ as

$c_{0}(t)\simeq\int_{0}^{\infty}d\omega\ g_{\text{w}}(\omega)e^{-i\omega
t}=e^{-\frac{\Gamma_{\text{a}}}{2}t}e^{-i(\omega_{\text{a}}+\Delta_{\text{a}})t},$
(19)

yielding a result that is essentially equivalent to the Weisskopf-Wigner
approximation. From this, one can further compute

$D_{\omega_{k}}(t)\simeq-
ie^{-i\omega_{k}t}\tilde{\rho}(\omega_{k})\int_{0}^{t}dt^{\prime}\
e^{-\frac{\Gamma_{\text{a}}}{2}t^{\prime}}e^{-i(\omega_{\text{a}}-\omega_{k}+\Delta_{\text{a}})t^{\prime}}.$
(20)

##  IV Energy density expectation value

Following [16, 18], we want to characterize the localization of the emitted
single-photon state using the energy density operator

$\hat{\mathcal{E}}_{\text{em}}(\vec{x})=\frac{\varepsilon_{0}}{2}:\left(\vec{\hat{E}}^{2}(\vec{x})+c^{2}\vec{\hat{B}}^{2}(\vec{x})\right):,$
(21)

for which the expectation value for a single-photon state $\ket*{\vec{\psi}}$
in the Landau-Peierls (LP) representation (see Appendix A) reads [16, 18]

$\langle\hat{\mathcal{E}}_{\text{em}}(\vec{x})\rangle=\hbar\absolutevalue{\Omega^{1/2}\vec{\psi}^{(h+)}(\vec{x})}^{2}+\hbar\absolutevalue{\Omega^{1/2}\vec{\psi}^{(h-)}(\vec{x})}^{2}.$
(22)

The exponents (h±) refer here to the positive and negative helicity parts of
the LP representation (see Appendix A). This splitting into helicity
components is an important feature of the result since it leads to the
nonlocality of all single-photon states [18] that we want to characterize for
spontaneous emission.

Using the results of the preceding section, we are going to express the state
of the photon that is spontaneously emitted by the Lyman-$\alpha$ transition
of the Hydrogen atom in the single-photon approximation, and then compute the
associated expectation value of the electromagnetic energy density. We
consider here only the photon part of the state, i.e.,

$\vec{\psi}_{\text{ph}}(r,\vartheta,\varphi)=\int_{0}^{\infty}dk\sum_{\lambda=\pm}c_{k,\lambda}(t)\vec{\psi}^{(\lambda)}_{k}(r,\vartheta,\varphi),$
(23)

where we use the abridged notation
$\vec{\psi}^{(\lambda)}_{k}\equiv\vec{\psi}^{(\lambda)}_{k,J=1,M=m_{e}}$. The
positive and negative helicity parts of the state can be written as

$\vec{\psi}_{\text{ph}}^{(h\pm)}(r,\vartheta,\varphi)=\int_{0}^{\infty}dk\frac{d_{k}(t)}{\sqrt{2}}\vec{\psi}^{(\pm)}_{k}(r,\vartheta,\varphi).$
(24)

Therefore, to compute the mean value, one needs to express

$\absolutevalue{\Omega^{1/2}\vec{\psi}_{\text{ph}}^{(h\lambda)}(r,\vartheta,\varphi)}^{2}=\absolutevalue{\int_{0}^{\infty}dk\sqrt{\frac{\omega_{k}}{2}}d_{k}(t)\vec{\psi}^{(\lambda)}_{k}(r,\vartheta,\varphi)}^{2}$
(25)

for both helicities. To do so, we use the expression of
$\vec{\psi}_{k}^{(\lambda)}$ in terms of vector spherical harmonics and Bessel
functions, as given in Appendix B, to obtain

$\absolutevalue{\Omega^{1/2}\vec{\psi}_{\text{ph}}^{(h\lambda)}}^{2}=\sum_{L=0}^{2}F_{L}(r,t)\vec{Y}_{1,m_{e}}^{L}(\vartheta,\varphi),$
(26)

where the $F_{L}$ functions are

$\displaystyle F_{0}(r,t)$
$\displaystyle=i\sqrt{\frac{c}{3\pi}}\int_{0}^{\infty}dk\
k^{3/2}d_{k}(t)j_{0}(kr),$ (27a) $\displaystyle F_{1}(r,t)$
$\displaystyle=\lambda\sqrt{\frac{c}{2\pi}}\int_{0}^{\infty}dk\
k^{3/2}d_{k}(t)j_{1}(kr),$ (27b) $\displaystyle F_{2}(r,t)$
$\displaystyle=-i\sqrt{\frac{c}{6\pi}}\int_{0}^{\infty}dk\
k^{3/2}d_{k}(t)j_{2}(kr).$ (27c)

In order to simplify the expression of the mean value, we perform an
asymptotic analysis of the $F_{L}$ functions for distances $r$ far from the
atom. The details of this analysis are given in the Appendix C. We obtain that
the leading term is $F_{1}$ which behaves as

$F_{1}(r,t)\underset{r\to\infty}{\sim}-i\lambda\left(\frac{2}{3}\right)^{11/2}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}r_{\text{B}}^{2}}{\pi\hbar}\mathcal{T}(t)\frac{1}{r^{3}},$
(28)

where $\mathcal{T}(t)$ is a dimensionless function of time given in equation
(64). The expectation value of the energy density becomes thus in the
asymptotic limit of large $r$

$\absolutevalue{\Omega^{1/2}\vec{\psi}_{\text{ph}}^{(h\lambda)}(\vec{x})}^{2}\underset{r\to\infty}{\sim}\absolutevalue*{F_{1}(r,t)}^{2}\vec{Y}_{1,m_{e}}^{1\star}\cdot\vec{Y}_{1,m_{e}}^{1},$
(29)

which can be computed for different values of $m_{e}$.

Figure 1: Plot of the angular distribution
$\gamma_{\absolutevalue*{m_{e}}}(\vartheta)$ of the photon state emitted by
spontaneous emission of the Lyman-$\alpha$ transition of a Hydrogen atom,
obtained in the asymptotic limit of large distances. The green solid line is
the distribution for an excited state with $m_{e}=0$. The red dashed line is
the distribution for an excited state with $\absolutevalue*{m_{e}}=1$.

For $m_{e}=0$, the angular part reads

$\vec{Y}_{1,0}^{1\star}\cdot\vec{Y}_{1,0}^{1}=\frac{3}{8\pi}\sin^{2}\vartheta,$
(30)

and for $\absolutevalue*{m_{e}}=1$

$\vec{Y}_{1,\pm 1}^{1\star}\cdot\vec{Y}_{1,\pm
1}^{1}=\frac{3}{16\pi}\left(1+\cos^{2}\vartheta\right).$ (31)

In the end, the expectation value as a funtion of $m_{e}$ can be expressed as

$\langle\hat{\mathcal{E}}_{\text{em}}\rangle=\left(\frac{2}{3}\right)^{10}\frac{m^{2}c^{3}r_{\text{B}}^{4}\alpha^{5}}{2\pi^{3}\hbar}\gamma_{\absolutevalue*{m_{e}}}(\vartheta)\absolutevalue{\mathcal{T}(t)}^{2}\frac{1}{r^{6}}+o\left(\frac{1}{r^{6}}\right),$
(32)

where the angular distribution is given by the functions
$\gamma_{\absolutevalue*{m_{e}}}$ which read for the different values of
$m_{e}$

$\displaystyle\gamma_{0}(\vartheta)$ $\displaystyle=\sin^{2}\vartheta,$ (33a)
$\displaystyle\gamma_{1}(\vartheta)$
$\displaystyle=\frac{1}{4}(1+\cos^{2}\vartheta),$ (33b)

and are represented in FIG. 1. We point out that the angular distribution we
have found matches that of a classical dipole that is either oscillating
linearly for $\gamma_{0}$ or circularly for $\gamma_{1}$ [34, 35]. We further
remark that in the $m_{e}=0$ case, the leading term of the asymptotics is zero
for $\vartheta=0$ due to the shape of the angular distribution. However, we
emphasize that the spontaneous emission in this direction is not zero but
features a faster falloff.

The result obtained in equation (32) exhibits an algebraic decrease for the
energy density which is therefore far from the quasiexponential limit
introduced in [16], and cannot be attributed only to the blurring of the light
cone as described in [30]. This behavior is an illustration of the intrinsic
nonlocality of the energy density of single-photon states, as shown for
particular cases in [17] and for all single-photon states in [18]. It is also
an other demonstration of the importance of multiphoton components that are
predicted by the coupling to guarantee a causal emission [28].

## Acknowledgments

We thank Jonas Lampart for fruitful discussions. This work was supported by
the “Investissements d’Avenir” program, project ISITE-BFC/IQUINS
(ANR-15-IDEX-03), QUACO-PRC (ANR-17-CE40-0007-01) and the EUR-EIPHI Graduate
School (17-EURE-0002). We also acknowledge support from the European Union’s
Horizon 2020 research and innovation program under the Marie Sklodowska-Curie
Grant Agreement No. 765075 (LIMQUET) and from France 2030 (ANR-22-CMAS-0001)
QuanTEdu-France

##  Appendix A Position space representations and related operators

To analyze the spatial properties of photons, one needs to use a position
space representation of the states. This can be done using several equivalent
choices [27]. In this work, we have chosen to work with the Landau-Peierls
(LP) representation which is constructed as follows

$\vec{\psi}(\vec{x})=\sqrt{\frac{\varepsilon_{0}}{2\hbar}}\left[\Omega^{1/2}\vec{A}(\vec{x})-i\Omega^{-1/2}\vec{E}(\vec{x})\right],$
(34)

where $\vec{A}$ is the classical potential vector in the Coulomb gauge and
$\vec{E}$ is the classical electric field. The operators $\Omega^{\pm 1/2}$
are self-adjoint operators constructed from the frequency operator
$\Omega=c(-\Delta)^{1/2}$ for the Laplacian $\Delta$. These operators are well
defined and Refs [36, 27, 18] give the more details of their construction. The
LP representation is made in the Hilbert space

$\mathcal{H}_{LP}=\left\\{\vec{\psi}\
\Big{|}\nabla\cdot\vec{\psi}=0,\bra*{\vec{\psi}}\ket*{\vec{\psi}^{\prime}}_{LP}<\infty\right\\},$
(35)

with the scalar product

$\bra*{\vec{\psi}}\ket*{\vec{\psi}^{\prime}}_{LP}=\int_{\mathbb{R}^{3}}d^{3}x\
\vec{\psi}^{\star}\cdot\vec{\psi}^{\prime},$ (36)

and Maxwell’s equations written in this representation read

$i\frac{\partial\vec{\psi}}{\partial
t}=\Omega\vec{\psi},\quad\nabla\cdot\vec{\psi}=0.$ (37)

This Hilbert space can then be used to construct the bosonic Fock space of
states as

$\mathbb{F}^{\mathfrak{B}}(\mathcal{H}_{LP})=\bigoplus_{n=0}^{\infty}\mathcal{H}_{LP}^{\otimes_{S}n},$
(38)

where $\mathcal{H}_{LP}^{\otimes_{S}n}$ is the symmetrized $n$-times tensor
product of $\mathcal{H}_{LP}$ with itself. Annihilation and creation operators
directly constructed on arbitrary pulse-shaped functions can then be used to
write states of the fields from the vacuum $\ket{\varnothing}$:

$\displaystyle\hat{B}_{\vec{\psi}}\ket{\varnothing}$ $\displaystyle=0,$ (39a)
$\displaystyle\hat{B}^{\dagger}_{\vec{\psi}}\ket{\varnothing}$
$\displaystyle=\ket*{\vec{\psi}}.$ (39b)

They satisfy the general bosonic commutation relations [37, 38, 27]

$\left[\hat{B}_{\vec{\psi}},\hat{B}^{\dagger}_{\vec{\psi}^{\prime}}\right]=\bra*{\vec{\psi}}\ket*{\vec{\psi}^{\prime}}_{LP}.$
(40)

Another important tool that is used is the helicity operator, defined as

$\Lambda=c\Omega^{-1}\nabla\times.$ (41)

For transverse fields, its spectrum is $\\{\pm 1\\}$ and one can thus
decompose any transverse field $\vec{v}$ into a sum of a positive- and a
negative-helicity part $\vec{v}^{(h\pm)}$,

$\vec{v}=\vec{v}^{(h+)}+\vec{v}^{(h-)},$ (42)

where $\Lambda\vec{v}^{(h\pm)}=\pm\vec{v}^{(h\pm)}$. This splitting is an
important property for the energy density mean value (22) since it leads to
the nonlocality of all single-photon sates [18].

##  Appendix B Helicity vector spherical harmonics

To compute the state of the emitted photon, we have used the basis of helicity
vector spherical harmonics
$\ket*{k,J,M,\lambda}\equiv\vec{\psi}_{k,J,M}^{(\lambda)}(\vec{x})$ that are
eigenfunctions of the following operators [5, 39, 40]

$\displaystyle-\Delta\vec{\psi}_{k,J,M}^{(\lambda)}$
$\displaystyle=k^{2}\vec{\psi}_{k,J,M}^{(\lambda)},\hskip
34.14322ptk\in(0,\infty),$ (43a)
$\displaystyle\hat{J}^{2}\vec{\psi}_{k,J,M}^{(\lambda)}$
$\displaystyle=J(J+1)\vec{\psi}_{k,J,M}^{(\lambda)},\hskip
5.69046ptJ\in\\{0,1,2,\dots\\},$ (43b)
$\displaystyle\hat{J}_{3}\vec{\psi}_{k,J,M}^{(\lambda)}$
$\displaystyle=M\vec{\psi}_{k,J,M}^{(\lambda)},\hskip
28.16838ptM\in\\{-J,\dots,J\\},$ (43c)
$\displaystyle\Lambda\vec{\psi}_{k,J,M}^{(\lambda)}$
$\displaystyle=\lambda\vec{\psi}_{k,J,M}^{(\lambda)}\hskip
44.10185pt\lambda=\pm 1,$ (43d)

where

$\displaystyle\vec{\hat{J}}$ $\displaystyle=\vec{\hat{L}}+\vec{\hat{S}},$
(44a) $\displaystyle\vec{\hat{L}}$
$\displaystyle=\nabla\times\vec{x}=\vec{x}\times\nabla,$ (44b)
$\displaystyle\vec{\hat{S}}$
$\displaystyle=\left(\hat{S}_{1},\hat{S}_{2},\hat{S}_{3}\right)^{T},\quad[\hat{S}_{l}]_{m,n}=-i\epsilon_{lmn},$
(44c)

where $l,m,n\in\\{1,2,3\\}$. The helicity vector spherical eigenfunctions can
be written in spherical coordinates $(r,\vartheta,\varphi)$ as

$\displaystyle\vec{\psi}_{k,J,M}^{(\lambda)}(r,\vartheta,\varphi)=\frac{i}{\sqrt{2}}$
$\displaystyle\Bigg{[}\sqrt{\frac{J+1}{2J+1}}\vec{\psi}_{k,J,M}^{J-1}(r,\vartheta,\varphi)$
$\displaystyle-\sqrt{\frac{J}{2J+1}}\vec{\psi}_{k,J,M}^{J+1}(r,\vartheta,\varphi)$
$\displaystyle\hskip
14.22636pt-i\lambda\vec{\psi}_{k,J,M}^{J}(r,\vartheta,\varphi)\Bigg{]},$ (45)

where the functions $\vec{\psi}_{k,J,M}^{L}$ can be expressed in terms of
vector spherical harmonics $\vec{Y}_{J,M}^{L}$, i.e., eigenfunctions of
$\hat{J}^{2}$, $\hat{J}_{3}$, $\hat{L}^{2}$ and $\hat{S}^{2}$ but not of
$\Lambda$, and spherical Bessel functions $j_{L}(kr)$ as

$\vec{\psi}_{k,J,M}^{L}(r,\vartheta,\varphi)=\sqrt{\frac{2}{\pi}}k\
j_{L}(kr)\vec{Y}_{J,M}^{L}(\vartheta,\varphi).$ (46)

We remark that by constructing the functions $\vec{\psi}_{k,J,M}^{(\lambda)}$
from $\vec{\psi}_{k,J,M}^{L}$ we obtain eigenfunctions of $-\Delta$,
$\hat{J}^{2}$, $\hat{J}_{3}$ and $\Lambda$ but not of $\hat{L}^{2}$ while
$\vec{\psi}_{k,J,M}^{L}$ is an eigenfunction of $-\Delta$, $\hat{J}^{2}$,
$\hat{J}_{3}$ and $\hat{L}^{2}$ but not of $\Lambda$.

The helicty spherical vector eigenfunctions
$\\{\vec{\psi}_{k,J,M}^{(\lambda)}\\}$ are transverse,
$\nabla\cdot\vec{\psi}_{k,J,M}^{(\lambda)}=0$, orthonormal

$\displaystyle\int_{0}^{\infty}dr\ r^{2}$
$\displaystyle\int_{0}^{\pi}d\vartheta\ \sin\vartheta\int_{0}^{2\pi}d\varphi\
\vec{\psi}_{k,J,M}^{\lambda\star}\cdot\vec{\psi}_{k^{\prime},J^{\prime},M^{\prime}}^{\lambda^{\prime}}=$
$\displaystyle\delta(k-k^{\prime})\delta_{J,J^{\prime}}\delta_{M,M^{\prime}}\delta_{\lambda,\lambda^{\prime}},$
(47)

and form a complete set of the subspace of transverse field since the set of
function
$\\{\vec{\psi}_{k,J,M}^{J-1},\vec{\psi}_{k,J,M}^{J},\vec{\psi}_{k,J,M}^{J+1}\\}$
from which they are defined is a complete set with

$\displaystyle\int_{0}^{\infty}d^{3}k$
$\displaystyle\sum_{J,M,L}\left[\vec{\psi}_{k,J,M}^{L}(r^{\prime},\vartheta^{\prime},\varphi^{\prime})\right]_{l}\left[\vec{\psi}_{k,J,M}^{L\star}(r,\vartheta,\varphi)\right]_{n}=$
$\displaystyle\frac{1}{r^{2}\sin\vartheta}\delta_{l,n}\delta(r-r^{\prime})\delta(\vartheta-\vartheta^{\prime})\delta(\varphi-\varphi^{\prime}).$
(48)

The last equation is a consequence of the completeness of the Bessel functions
$j_{L}(kr)$ and vector spherical harmonics $\vec{Y}_{J,M}^{L}$ [39].

##  Appendix C Asymptotic analysis of radial integrals

In this appendix, we compute the asymptotic behavior of the radial integrals
that appear in the calculation of the energy density of a spontaneously
emitted photon from the Lyman-$\alpha$ transition of a Hydrogen atom. The
integrals have the following forms

$\displaystyle F_{0}(r,t)$
$\displaystyle=\left(\frac{2}{3}\right)^{3}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{q^{2}}{\left(1+q^{2}\right)^{2}}j_{0}(qr^{\prime}),$
(49a) $\displaystyle F_{1}(r,t)$
$\displaystyle=-i\lambda\left(\frac{2}{3}\right)^{5/2}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{q^{2}}{\left(1+q^{2}\right)^{2}}j_{1}(qr^{\prime}),$
(49b) $\displaystyle F_{2}(r,t)$
$\displaystyle=-\left(\frac{2}{3}\right)^{3}\sqrt{\frac{\alpha^{5}}{2c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{q^{2}}{\left(1+q^{2}\right)^{2}}\left(\frac{3}{qr^{\prime}}j_{1}(qr^{\prime})-j_{0}(qr^{\prime})\right),$
(49c)

where we have introduced the dimensionless parameters

$\displaystyle
A=\frac{\tilde{\Gamma}_{\text{a}}}{cK}=\frac{\Gamma_{\text{a}}}{2cK},$
$\displaystyle\quad
B=\frac{\Omega_{\text{a}}}{cK}=\frac{\omega_{\text{a}}+\Delta_{\text{a}}}{cK},$
(50a) $\displaystyle q=\frac{k}{K},$ $\displaystyle\quad p=cKt,\quad
r^{\prime}=Kr.$ (50b)

To obtain the asymptotic behavior of the $F_{L}$ integrals, and due to the
expression of the spherical Bessel functions

$j_{0}(qr^{\prime})=\frac{\sin(qr^{\prime})}{qr^{\prime}},\quad
j_{1}(qr^{\prime})=\frac{\sin(qr^{\prime})}{q^{2}r^{\prime
2}}-\frac{\cos(qr^{\prime})}{qr^{\prime}},$ (51)

one needs to analyze integrals of the form

$\int_{0}^{\infty}dqR(q)\sin(qr^{\prime}),\quad\int_{0}^{\infty}dqR(q)\cos(qr^{\prime}).$
(52)

Here, $R(q)$ is a function with the properties that for some $N_{s}\geq 0$
even or $N_{c}\geq 0$ odd,

$\displaystyle\left.\frac{d^{n}R(q)}{dq^{n}}\right|_{q=0}$
$\displaystyle=\left\\{\begin{array}[]{ll}0&\text{for $n<N_{c,s}$, $n$ same
parity as $N_{c,s}$},\\\ \text{cst}\neq 0&\text{for $n=N_{c,s}$, $n$ same
parity as $N_{c,s}$},\end{array}\right.$ (53c)
$\displaystyle\lim_{q\to\infty}\frac{d^{n}R(q)}{dq^{n}}$ $\displaystyle=0,$
(53d)

and

$\int_{0}^{\infty}dq\absolutevalue{\frac{d^{n}R(q)}{dq^{n}}}<\infty,\quad\text{for
all {\hbox{n<N_{c,s}}}}.$ (54)

The indices $c,s$, are needed to treat the cosine and sine integrals in (52),
respectively. Using these properties and some integrations by parts the sine
intergral becomes

$\displaystyle\int_{0}^{\infty}dqR(q)\sin(qr^{\prime})$
$\displaystyle=-\frac{1}{r^{\prime}}\Big{[}R(q)\cos(qr^{\prime})\Big{]}_{0}^{\infty}+\frac{1}{r^{\prime}}\int_{0}^{\infty}dq\frac{dR(q)}{dq}\cos(qr^{\prime})$
(55a)
$\displaystyle=\frac{1}{r^{\prime}}R(0)-\frac{1}{r^{\prime}}\int_{0}^{\infty}dq\frac{dR(q)}{dq}\cos(qr^{\prime})$
(55b)
$\displaystyle=\frac{1}{r^{\prime}}R(0)+o\left(\frac{1}{r^{\prime}}\right).$
(55c)

The last equality is obtained by the Riemann-Lebesgue lemma [41] which states
that

$\absolutevalue{\int_{0}^{\infty}dq\frac{dR(q)}{dq}\cos(qr^{\prime})}\underset{r^{\prime}\rightarrow+\infty}{\longrightarrow}0,$
(56)

if $\int_{0}^{\infty}dq\absolutevalue{\frac{dR(q)}{dq}}<\infty$, which is true
according to (54). Therefore, the first term of (55c) gives the asymptotics
for large $r^{\prime}$ if $N_{s}=0$, otherwise, one has to iterate the
integration by parts as

$\displaystyle\int_{0}^{\infty}dqR(q)\sin(qr^{\prime})$
$\displaystyle=\frac{1}{r^{\prime
2}}\left[\frac{dR}{dq}\sin(qr^{\prime})\right]_{0}^{\infty}-\frac{1}{r^{\prime
2}}\int_{0}^{\infty}dq\frac{d^{2}R(q)}{dq^{2}}\sin(qr^{\prime})$ (57a)
$\displaystyle=-\frac{1}{r^{\prime
2}}\int_{0}^{\infty}dq\frac{d^{2}R(q)}{dq^{2}}\sin(qr^{\prime})$ (57b)
$\displaystyle=\frac{1}{r^{\prime
3}}\left[\frac{d^{2}R}{dq^{2}}\cos(qr^{\prime})\right]_{0}^{\infty}+\frac{1}{r^{\prime
3}}\int_{0}^{\infty}dq\frac{d^{3}R(q)}{dq^{3}}\cos(qr^{\prime})$ (57c)
$\displaystyle=\frac{1}{r^{\prime
3}}\left.\frac{d^{2}R}{dq^{2}}\right|_{q=0}+\frac{1}{r^{\prime
3}}\int_{0}^{\infty}dq\frac{d^{3}R(q)}{dq^{3}}\cos(qr^{\prime})$ (57d)
$\displaystyle=\frac{1}{r^{\prime
3}}\left.\frac{d^{2}R}{dq^{2}}\right|_{q=0}+o\left(\frac{1}{r^{\prime
3}}\right),$ (57e)

and again, the first term gives the asymptotics by the Riemann-Lebesgue lemma
if $N_{s}=2$. Otherwise, the same process can be continued until one reaches
$N_{s}$. A similar process can be done for the cosine integral and gives after
one step

$\displaystyle\int_{0}^{\infty}dqR(q)\cos(qr^{\prime})$
$\displaystyle=-\frac{1}{r^{\prime
2}}\left.\frac{dR}{dq}\right|_{q=0}+\frac{1}{r^{\prime
2}}\int_{0}^{\infty}dq\frac{d^{2}R(q)}{dq^{2}}\cos(qr^{\prime})$ (58a)
$\displaystyle=-\frac{1}{r^{\prime
2}}\left.\frac{dR}{dq}\right|_{q=0}+o\left(\frac{1}{r^{\prime 2}}\right),$
(58b)

by the Riemann-Lebesgue lemma. This step is enough if $N_{c}=1$ but others can
be performed if not.

To apply this technique, we now need the concrete form of the functions $R(q)$
and thus we rewrite the $F_{L}$ integrals as

$\displaystyle F_{0}$
$\displaystyle=\left(\frac{2}{3}\right)^{3}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\frac{1}{r^{\prime}}\int_{0}^{\infty}dq\left(R^{(1)}_{+}(q)+R^{(1)}_{-}(q)\right)\sin(qr^{\prime}),$
(59a) $\displaystyle F_{1}$
$\displaystyle=-i\lambda\left(\frac{2}{3}\right)^{5/2}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\Bigg{\\{}\frac{1}{r^{\prime
2}}\int_{0}^{\infty}dq\left(R^{(0)}_{+}(q)+R^{(0)}_{-}(q)\right)\sin(qr^{\prime})-$
$\displaystyle\hskip
85.35826pt\frac{1}{r^{\prime}}\int_{0}^{\infty}dq\left(R^{(1)}_{+}(q)+R^{(1)}_{-}(q)\right)\cos(qr^{\prime})\Bigg{\\}},$
(59b)

where the $R$ functions are

$\displaystyle
R_{+}^{(1)}(q)=\frac{e^{-iqp}}{A+i(B-q)}\frac{q}{\left(1+q^{2}\right)^{2}},$
$\displaystyle\quad\quad\quad
R_{-}^{(1)}(q)=-\frac{e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{q}{\left(1+q^{2}\right)^{2}},$
(60a) $\displaystyle
R_{+}^{(0)}(q)=\frac{e^{-iqp}}{A+i(B-q)}\frac{1}{\left(1+q^{2}\right)^{2}},$
$\displaystyle\quad\quad\quad
R_{-}^{(0)}(q)=-\frac{e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{1}{\left(1+q^{2}\right)^{2}}.$
(60b)

The integral $F_{2}$ can be deduced rapidly from the other two as we will
discuss later. The asymptotics for $F_{0}$ and $F_{1}$ can be computed using
the iterative process and we obtain

$\displaystyle F_{0}(r,t)\underset{r^{\prime}\to\infty}{\sim}$
$\displaystyle\left(\frac{2}{3}\right)^{3}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\frac{1}{r^{\prime
4}}\left(\left.\frac{d^{2}R_{+}^{(1)}}{dq^{2}}\right|_{q=0}+\left.\frac{d^{2}R_{-}^{(1)}}{dq^{2}}\right|_{q=0}\right),$
(61a) $\displaystyle F_{1}(r,t)\underset{r^{\prime}\to\infty}{\sim}$
$\displaystyle-i\lambda\left(\frac{2}{3}\right)^{5/2}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\frac{1}{r^{\prime
3}}\left(R_{+}^{(0)}(0)+R_{-}^{(0)}(0)+\left.\frac{d^{2}R_{+}^{(1)}}{dq^{2}}\right|_{q=0}+\left.\frac{d^{2}R_{-}^{(1)}}{dq^{2}}\right|_{q=0}\right),$
(61b)

which can be further simplified using

$\displaystyle R^{(0)}_{+}(q=0)$ $\displaystyle=\frac{1}{A+iB},\hskip
113.81102ptR^{(0)}_{-}(q=0)=-\frac{e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+iB},$
(62a) $\displaystyle\left.\frac{d^{2}R_{+}^{(1)}}{dq^{2}}\right|_{q=0}$
$\displaystyle=\frac{i}{(A+iB)^{2}}-\frac{ip}{A+iB},\hskip
51.21504pt\left.\frac{d^{2}R^{(1)}_{-}}{dq^{2}}\right|_{q=0}=-i\frac{e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{(A+iB)^{2}}.$
(62b)

Since $F_{0}\sim F_{1}/r^{\prime}$, we compute only $F_{1}$ which will
dominate for large $r^{\prime}$. We obtain then the result used in Section IV
by inserting (62) in (61b)

$F_{1}(r^{\prime},t)\underset{r^{\prime}\to\infty}{\sim}-i\lambda\left(\frac{2}{3}\right)^{5/2}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\mathcal{T}(t)\frac{1}{r^{\prime 3}},$ (63)

where the dimensionless function of time reads

$\mathcal{T}(t)=\frac{1}{A+iB}\left(1-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}-2icKt+i\frac{1-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+iB}\right).$
(64)

Regarding $F_{2}$ we recall its expression

$F_{2}(r,t)=-\left(\frac{2}{3}\right)^{3}\sqrt{\frac{\alpha^{5}}{2c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{q^{2}}{\left(1+q^{2}\right)^{2}}\left(\frac{3}{qr^{\prime}}j_{1}(qr^{\prime})-j_{0}(qr^{\prime})\right),$
(65)

from which we see that the term involving $j_{0}$ will behave as $F_{0}$ and
can thus be neglected. For the term involving $j_{1}$, one has to be more
cautious since it does not have the same form as what we had for $F_{1}$. It
yields

$\displaystyle F_{2}(r^{\prime},t)\underset{r^{\prime}\to\infty}{\sim}$
$\displaystyle-\left(\frac{2}{3}\right)^{2}\sqrt{\frac{2\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\frac{1}{r^{\prime}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{q}{\left(1+q^{2}\right)^{2}}j_{1}(qr^{\prime})$
(66a)
$\displaystyle=-\left(\frac{2}{3}\right)^{2}\sqrt{\frac{2\alpha^{5}}{c}}\frac{mc^{2}}{\pi\hbar
r_{\text{B}}}\left\\{\frac{1}{r^{\prime
3}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{1}{q\left(1+q^{2}\right)^{2}}\sin(qr^{\prime})\right.$
$\displaystyle\>\>\>\>\>\>\left.-\frac{1}{r^{\prime
2}}\int_{0}^{\infty}dq\frac{e^{-iqp}-e^{-i(\Omega_{\text{a}}-i\tilde{\Gamma}_{\text{a}})t}}{A+i(B-q)}\frac{1}{\left(1+q^{2}\right)^{2}}\cos(qr^{\prime})\right\\},$
(66b)

from which one can already see that $F_{2}$ will decrease faster than $F_{1}$.
Indeed, the first term scales like $1/r^{\prime 3}$ before making the
iterative process to analyze the integral. Moreover, the function of $q$ to
integrate does not yield a constant when $q\rightarrow 0$ meaning that the
iterative process must be done more than one time. Thus, this term will
decrease faster than $1/r^{\prime 3}$ and can be neglected. The second term
involves a cosine for which the first iterative step (58) already scales as
$1/r^{\prime 2}$, so that the full term would scale like $1/r^{\prime 4}$,
decreasing faster than $F_{1}$. The leading term that is needed to compute the
mean value is thus given by $F_{1}$ and if one reinstates the units using
(50), it yields

$F_{1}(r,t)\underset{r\to\infty}{\sim}-i\lambda\left(\frac{2}{3}\right)^{11/2}\sqrt{\frac{\alpha^{5}}{c}}\frac{mc^{2}r_{\text{B}}^{2}}{\pi\hbar}\mathcal{T}(t)\frac{1}{r^{3}}.$
(67)

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