# A Reinforcement Learning based Path Planning Approach in 3D Environment

Geesara Kulathunga  Innopolis University

###### Abstract

Optimal motion planning involves obstacles avoidance where path planning is
the key to success in optimal motion planning. Due to the computational
demands, most of the path planning algorithms can not be employed for real-
time based applications. Model-based reinforcement learning approaches for
path planning have received certain success in the recent past. Yet, most of
such approaches do not have deterministic output due to the randomness. We
analyzed several types of reinforcement learning-based approaches for path
planning. One of them is a deterministic tree-based approach and other two
approaches are based on Q-learning and approximate policy gradient,
respectively. We tested preceding approaches on two different simulators, each
of which consists of a set of random obstacles that can be changed or moved
dynamically. After analysing the result and computation time, we concluded
that the deterministic tree search approach provides highly stable result.
However, the computational time is considerably higher than the other two
approaches. Finally, the comparative results are provided in terms of accuracy
and computational time as evidence.

##  I Introduction

Over the past decade, deep learning-enabled approaches have been obtained
satisfactory results in various domains such as optimal motion planning,
hyperspectral image classification, and path planning. In comparison to
typical path planning algorithms, reinforcement learning (RL) based approaches
were received significant attention in the recent past due to the success of
the deep learning and computer vision. However, RL-based approaches without
incorporating deep learning and computer vision were there for decades without
proper success.

Most RL-based approaches consist of V (value function) or Q (function), whose
policies are defined in deterministic ways. However, the recent development of
deep learning can help to incorporate sophisticated learning to formulate V or
Q functions appropriately. RL-based path planning can be classified in two
ways: model-base, model free. Model-based path panning approaches have several
advantages over model-free approaches, for example, learning dynamics of
quadrotor (model-free) is rather complicated because using a model-based
approach does guarantee the dynamic feasibility since such approaches depend
on the actual motion model.

Typical graph-based or tree-based search algorithms, e.g., A*, Dijkstra and
Min-Max, can be applied for planning, scheduling, discrete optimization, and
games. In such search algorithms, in general, the algorithm describes by a set
of states and actions that correspond to each state, which can be described
explicitly in most situations. The objective of these algorithms is to find
the best action at each state by expanding the tree or graph to minimize the
cost or maximize the reward. However, tree or graph expansion becomes
computationally expensive when state-space (states and actions combinations)
complexity grows due to the enamours look ahead information. For example,
state-space complexity of Reversi game is $10^{28}$, for Tic-tac-toe it is
$10^{3}$, and for Go it is $10^{171}$ [1].

Our main contribution of the work is to explore the potentials of both RL-
based path planning and deterministic-based path planning and how can
incorporate the best of both to develop a fast and robust path planning
approach, especially for quadrotors. For that, initially, we formulated the
path planning problem for a deterministic system (actions are discrete and
known) in which the optimal action in each state was selected by using the
Optimal Planning of Deterministic Systems (OPD) algorithm [2].

Figure 1: The improved tree based RL path planner. $x_{s}$ and $x_{g}$ denote
start and goal positions, respectively. In blue colored trajectory depict the
planned paths whose the start position kept the same position while changing
the goal position

##  II Related work

Motion planning can be carried out several different ways [3]: search-based,
sampling-based, and optimization-based. In search-based approaches, search
space (or state space) is discretized into partitions, namely lattices and a
path between the initial position and the end position are connected through a
sequences of straight lines yields the shortest path between initial and end
poses. Dijkstra’s algorithm [4], A* [5], and D* [6] are a few widely used
search-based algorithms. In most of the situations, real-time constraints are
not met with search-based approaches due to high computational demands and
memory footprints for state space representation. Sampling-based approaches do
address state representation by incrementally drawing the state space. RRT*
[7] and PRM [8] are the mostly used ones. Optimization-based motion planning
approaches can alleviate most of the problems preceding two approaches have.
There are various techniques are proposed for this kind. Minimum-snap [9] one
of the first proposed techniques for optimal motion planing for Quadrotors.

In motion planning, agent (robot) is described by its state ($s_{t}\in S$) for
a given time index where $S$ is the state space. Based on the reasoning
approach, agent can take several different actions, e.g., $a_{t}\in A$ which
lead agent to move to $s_{t+1}\in S$. In other word, let $P\in M(S)^{S\times
A}$ be the system dynamics, $s_{t+1}$ is drawn based on a conditional
probability, i.e., $P(s_{t+1}\mid s_{t},a_{t})$, where $M(S)$ denotes the a
set of states within $S$. Moreover, control policy $\pi\in M(A)^{S}$, which
agent’s actions are drawn from, can be defined as $\pi(a_{t}\mid s_{t})$.
Hence, finding an optimal control policy $\pi^{*}$ is the main objective of
agent at each time index t. There are various approaches can be applied for
finding an $\pi^{*}$. Yet this is open research problem due to several
reasons, including high computational demands, execution time, and the nature
of the problem: linear or non-linear. In Reinforcement Learning (RL), finding
an optimal control policy can be designed as a leaning-based decision making
process, namely Marko Decision Process (MDP): ($S,A,P,R,\gamma$). Agent gets
rewards $R(s_{t},a_{t})$, where $R\in[0,1]^{S\times A}$ when agent moving from
$s_{t}$ to $s_{t+1}$. The ultimate objective is to maximize the rewards at
each t, which eventually gives $\pi^{*}$. Thus, total reward is estimated by
$R^{\pi}=\sum_{t=0}^{N}\gamma^{t}R(s_{t},a_{t})$, where N number of steps
agent moves along a trajectory
$\Pi=((s_{0},a_{0}),(s_{1},a_{1}),(s_{2},a_{3}),...,(s_{N},a_{N}))$. There
could be exist multiple trajectories that lead to optimal solution. Hence,
performance of a trajectory (policy $\pi$) is estimated through a value
function, namely $V^{\pi}(s)=\mathbb{E}[R^{\pi}\mid s_{0}=s]$. Thus, objective
is to find $V^{*}(s)=\max_{\pi}V^{\pi}(s),\;\forall s\in S$.

Model-based and model-free are the main families in RL. Either of such
families can be used for finding $\pi^{*}$. Yet, some of the work well over
others due to problem characteristics. Model-based learning is preferred when
the model dynamics are simple ([10, 11, 12]), and but $\pi$ estimation is
complex. On the contrary, model-free learning is preferred for model with
complex dynamics([13, 14]) yet $\pi$ estimation relatively simple. Dynamic
Programming (DP) approaches (e.g., value iteration [15], policy iteration
[16]) ensure the computational complexity around
$O(\left|S\right|\left|A\right|log_{\gamma}(\epsilon(1-\gamma)))$ for discrete
systems whose action-space is defined by $S\times A$. On the contrary, when S
becomes continuous, Approximate Dynamic Programming (ADP) gives feasible
solution over exact DP, where value function or policy is approximated within
a provided hypothesis. Other than ADP, sampling-based optimization(or black-
box optimizations) (e.g., Cross Entropy Method (CEM)[17], Covariance Matrix
Adaptation Evolution Strateg (CEM-ES)[18]) are used when S is large or
continuous. Such approaches does not depends on the complete knowledge of MDP,
but rather depends on the generator model, i.e., simulator, which can produce
next state $s^{`}\sim P(s^{`}\mid s,a)$ and corresponding reword $R(s,a)$.

Monte Carlo Tree Search (MCTS) [19] is preferred when the action space is
discrete. MCTS has look-ahead three that starts from the current state and
look for a optimal action to be taken as each step progressively expanding by
sampling the trajectories using the generative model. The basic MCTS algorithm
consists of four steps: selection, expansion, simulation and backpropagation.
Selection function is applied to end of the current tree and one tree node is
created in the expansion step. Take possible actions from such a expanded tree
node and simulate till look-ahead node reach to its terminal state. In the
backpropagation step, sum the total reward from the expansion node to terminal
node is calculated by $v_{i}+C\cdot\sqrt{\frac{lnN}{n_{i}}}$, where $C$ is a
exploration factor, $v_{i}$ mean value of node $n_{i}$, i.e., success ratio,
$n_{i}$ is the number of times $n_{i}$ was visited and the total simulations
count is given by $N$. The optimal or near-optimal action is identified using
Probability Approximation Correct (PAC) [20]. However, applicability for real-
time applications is limited due to the high computational demands. In [2]
proposed one of the first polynomial regret bound algorithms which is quite
faster, but applicability limited for deterministic dynamic and rewards. Later
was extended [21] for supporting stochastic rewards and dynamics in a open-
loop setting, i.e, sequence of actions. The preceding algorithm was further
improved in [22]. The modified version constraints on upper-confidence which
helps to improve the performance considerably. For upper-bounds identification
was determined with the help of Kullback-Leibler algorithm [23]. On the
contrary, graph-based optimistic algorithms can capture across the states,
which does not supports in the preceding algorithms (tree based algorithms).
Such a algorithm was initially proposed by Silver et al. [12] where states
values are derived from a shared Neural Network. Yet, authors of [24] proposed
state-space partitioning into smaller sets and aggregating similar states.

The proximal policy optimization (PPO) [25] is a supervised learning
technique, where policy is defined by a deep neural network, e.g., multi layer
perceptron, dueling network, convolutional network, ego attention network, and
variational autoencoders [26], depending on the situation.

##  III Methodology

Figure 2: Other than true interaction between agent and environment, planning
node simulates possible trajectories and pick the optimal action

The deterministic control problem is defined as follows: $(X,A,f,r$, where
action space and state space are denotes by A, X, respectively. Transition
dynamics is given by $f:X\times A\xleftarrow[]{}X$, and r denotes the reward
function, which lies in the interval [0,1]. state space can be large yet
action space should be deterministic, i.e., K. For any policy
$\pi:X\xleftarrow{}A$, value function and Q-value function are defined by:

$\displaystyle V^{\pi}(x)=\sum_{t\geq
0}\gamma^{t}r(x_{t},\pi(x_{t}))\;0\leq\gamma<1$ (1) $\displaystyle
Q^{\pi}(x,a)=r(x,a)+\gamma V^{\pi}(f(x,a))$

Thus, corresponding optimal V and Q functions can be determined as:

$\displaystyle V^{*}(x)=\max_{a\in A}r(x,a)+\gamma V^{*}(f(x,a))$ (2)
$\displaystyle Q^{*}(x,a)=r(x,a)+\gamma\max_{b\in A}Q^{*}(f(x,a),b)$

Let $A(n)$ be the action selection at each state, then regret yields selecting
action from $A(n)$ over selecting optimal action, which can be formulated as:

$R_{A}(n)=\max_{a\in A}Q^{*}(x,a)-Q^{*}(x,A(n))$ (3)

As listed in the Related Work section, a variety of tree search algorithms for
calculating optimal controlling polices have been suggested. However, due to
high computational demands, most of them are not a feasible solution for real-
time applications such as trajectory planning for quadrotors. For real-time
planning, control policy must be recalculated at least at 5Hz to obtain a
smooth flight experience. Hren and Munos [2] proposed an algorithm, namely,
OPD, that focuses on the the accuracy and computational demands at the same
time for deterministic systems. The OPD uses the best possible resources (CPU
time, memory) to solve the sequential decision making process. If execution is
limited by time, the algorithm returns sub-optimal policy, otherwise, it
returns higher accuracy policy. The OPD constructs a look-ahead tree at time
instance t. In general, in order to explore a look-ahead tree, it is needed to
get through all possible reachable states from the current state $x_{t}$.
However, OPD proposed an optimal strategy for exploring the look-ahead tree
from the current state $x_{t}$; thus, OPD works faster, which makes it more
applicable for real-time applications. Three approaches of OPD application are
Uniform look-ahead tree searching, Deep Q-Network, and Proximal Policy
Optimization. They are detailed in further sections.

###  III-A Uniform look-ahead tree searching

The foremost objective of Uniform look-ahead tree searching is to retrieve an
optimal action given a state $x$. In each state $x$, all possible reachable
states are considered applying available actions. Such expanding new states
are the children $c(i),i=0,...,K$ of the current state, when initial state is
denoted as the root. The same process is applied recursively on each child to
construct the next level of the tree. The order of the tree, d, is the number
of the levels the tree $\beta$ grows. Hence, the cost of reaching any child
$v_{i}$ of the tree is the supersum $v_{i}=max_{j\in C(i)}v_{j}$ and the
optimal value of the root $v*=max_{i\in\beta}v_{i}=max\\{v_{1},...,v_{K}\\}$.
At any iteration, $\beta_{n}$ denotes the nodes that already expanded and
$\beta_{s}=\beta-\beta_{n}$ denotes they have not expanded yet. Such
unexpanded nodes are the possible nodes that can be expanded in the next
iteration.

Figure 3: The regions $\beta_{n}$ and $\beta_{s}$ belong to expanded and to be
expanded nodes. The number of children K is 2 at each node and tree depth d is
3

###  III-B Deep Q-Network

Deep Q-Network (DQN) is the first deep reinforcement learning method proposed
by DeepMind [27]. DQN can learn the control policies from the sensor data
directly. Control policy is approximated from a deep neural network, e.g.,
convolutional neural network, whose input is constituted with the raw image
and whose output is estimated using a value function, i.e., the reward
function; the value function calculates the future action and uses the said
action to estimate the control policy. Afterwards, the obtained control policy
is learned by training a variant of Q-learning. The input for the network
depends on several factors: robot environment, robot behaviour, and sensing
capabilities of the robot. For example, in this research, we experimentally
evaluated three different types of deep neural networks: multilayer
perceptron, dueling network, and ego attention network for learning the
control policy. For all the listed types, input for the network was obtained
by the observations and distances to close-in obstacle positions after the
normalization. Finally, the network output in DQN provides the probable action
to be applied to the system.

###  III-C Proximal Policy Optimization

The proximal policy optimization (PPO) [25] is a supervised learning
technique, which fill-in sampling data through interaction with the
environment, that uses policy gradient methods for updating gradients, i.e,
stochastic gradient ascent. Due the novel objective, i.e.,
$L^{clip}(\theta)=\hat{E}_{t}[min(r_{t}(\theta)\hat{A}_{t}),clip(r_{t}(\theta),1-\epsilon,1+\epsilon)\hat{A}_{t}]$,
where $\theta$ is the policy parameter, $\hat{E}_{t}$ and $\hat{A}_{t}$ are
the estimated expectation over timesteps and advantage, respectively,
$\epsilon$ is a hyperparameter, and $r_{t}$ is the ratio of probability of new
and old policies, that enable minibatch updates on multiple epochs. Moreover,
PPO is able to update the trust region using trust region policy optimization
(TRPO). Sampling complexity quite low due the preceding novel objective
function.

###  III-D The problem formulation

Despite the recent achievements in reinforcement learning, e.g., [12], most of
the proposed solutions are not yet applied to industrial applications due to
the undesirable behaviour. However, model-based learning approaches can
alleviate such undesirable behaviours often better than their model-free
counterparts. Elouard [28] introduced an arbitrary reward function R which can
be maximized to shortages that come in critical setting where mistakes that
are made from the model to be minimized. More precisely, consider the
following linear system

$\displaystyle\dot{x}(t)=A(\theta)x(t)+Bu(t)+D\epsilon(t),\;\epsilon(t)\geq
0,$ (4) $\displaystyle y(t)=\dot{x}(t)+Cz(t)$

where $\theta$ in the state matrix $A(\theta)\in\mathbb{R}^{n_{x}\times
n_{x}}\subset\Psi\in\mathbb{R}^{d}$, control matrix
$B\in\mathbb{R}^{n_{x}\times n_{u}}$, disturbance matrix
$D\in\mathbb{R}^{n_{x}\times n_{u}}$. $x\in\mathbb{R}^{n}_{x}$,
$u\in\mathbb{R}^{n_{u}}$ and $\epsilon\in\mathbb{R}^{n_{r}}$ are state,
control and disturbances, respectively. The system observation is given by
$y(t)$, where $z(t)\in\mathbb{R}^{n_{n}}$ and $C\in\mathbb{R}^{n_{x}\times
n_{n}}$. B, C, D are known matrices with respect to the considered system.
Afterwards, they defined a robust control objective aims to maximize the
worse-case next state output with respect to a confident region $C_{n,\delta}$

$\max_{u\in\mathbb{R}^{\mathbb{N}}}V^{r}(\mathbf{u})$ (5)

, where $V^{r}(\mathbf{u})=\underset{\theta\in
C_{N,\delta}}{\inf}[\Sigma_{n=N+1}^{\infty}\gamma^{n}R(x_{n}(\mathbf{u},\omega))],\;\gamma\in(0,1)$.
Since system is linear A is known, $A(\theta)$ is estimated as follows:

$A(\theta)=A+\Sigma_{i=1}^{d}\theta_{i}\pi_{i}$ (6)

More information regarding the $\theta$ estimation [28]. Simulator we
developed for avoiding the obstacles in 3D space from start position to
desired goal position. The system is described by
$\mathbf{x}=[p_{x}\;p_{y}\;p_{z}\;v_{x}\;v_{y}\;v_{z}]^{T}$, where
$p_{\mu},v_{\mu},\mu=x,y,z$ are position and velocity along the $\mu$
direction. Action space $u=(u_{x},u_{y},u_{z})\in[-1,1]^{3}$ is divided into 8
directions: forward, backward, left, right, top, and down, as follows:

$\displaystyle A=\\{(0,0,0),(1,0,0),(-1,0,0),(0,1,0),(0,-1,0),$
$\displaystyle(0,0,1),(0,0,-1)]\\}$

The reward encodes cost towards to desired goal state $x_{g}$, while avoiding
collisions with obstacles:

$\displaystyle R(x)=\delta(x)/(1+\|x-x_{g}\|_{2}),$ (7)
$\displaystyle\delta(x)=\left\\{\begin{matrix}0&\delta(x)<obs_{d}\\\
\delta(x)&otherwise\end{matrix}\right.,$

where $obs_{d}$ denotes the minimum acceptance distance, i.e., 1m. The system
dynamics consist with friction parameters
$(\theta_{x},\theta_{y},\theta_{z})$:

$\begin{bmatrix}\dot{p_{x}}\\\ \dot{p_{y}}\\\ \dot{p_{z}}\\\ \dot{v_{x}}\\\
\dot{v_{y}}\\\ \dot{v_{z}}\\\ \end{bmatrix}=\begin{bmatrix}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ 0&0&0&-\theta_{x}&0&0\\\
0&0&0&0&-\theta_{y}&0\\\ 0&0&0&0&0&-\theta_{z}\\\
\end{bmatrix}\begin{bmatrix}{p_{x}}\\\ {p_{y}}\\\ {p_{z}}\\\ {v_{x}}\\\
{v_{y}}\\\ {v_{z}}\\\ \end{bmatrix}+\beta\begin{bmatrix}0\\\ 0\\\ 0\\\
{u_{x}}\\\ {u_{y}}\\\ {u_{z}}\\\ \end{bmatrix},$

where $0<\beta\leq 1$ is a scaling parameter that control the direction of the
action.

Figure 4: Back and red spheres denote the goal positions and obstacles
positions, respectively. The quadrotor starts at the origin (0,0,0) moving
towards to goal position while avoiding the obstacles

##  IV Experimental Results

We have modified the approach that proposed in [28], supporting for 3D
environment. In the improved version we incorporated OctoMap based map
representation where obstacle-free distance can be calculated from a desired
position. Such a distance estimation was helped to faster convergence of look-
ahead tree search, while keeping the almost the same accuracy. We have ran
theirs with the proposed over 100 episodes and calculated mean execution time
over as given in Table I. Furthermore, we have tried with Proximal Policy
Optimization with three different control policy generation, i.e.,
Convolutional Neural Network, Variational Autoencoder, and Dueling Neural
Network. However, we have not achieved high accurate trajectory when we used
Proximal Policy Optimization for predicting of next optimal action. Success
fraction is ratio of the number of successful trajectories to total number of
tries, i.e., 100 times. Mean execution time is calculated total time is
defined as ratio of time utilized for the total trajectory calculation to
number of steps. With reference to Table. I, it is clear that the proposed
approach has the optimal in comparison to the other approach that have
compared.

##  V Conclusion and Discussion

Accuracy, computation time, and memory consumption make path planning a
challenging task for real time application. Current advancement in the field
do not allow mobile robots, e.g., drones and autonomous ground vehicles, to
handle dynamic obstacles robustly. Two main classes of approaches that have
been used to address the said constrains are RL-based and deterministic-based
approaches. In this work, we have compared three RL-based approaches and one
deterministic approach for the same path planning task, i.e., finding an
obstacle-free path between start and goal position. Then, we proposed a hybrid
approach that combines Monticalo tree search and RL-based approach to solve
the same task. In terms of reaching the goal position, the accuracy of the
proposed hybrid approach is considerably higher than the other three
approaches: Proximal policy optimization, deep Q-network, and uniform tree
search; this ensures the safety and dynamic feasibility. However,
computational demands of the proposed hybrid approach are sightly higher
compared to the aforementioned three approaches, which constitutes a
limitation of the proposed hybrid approach because such demands yield
ineffability to apply in real-world motion planning applications. Thus, the
next stage of this research project focuses on incorporating Proximal Policy
Optimization-based planning to further reduction of computational time while
keeping the high accuracy. Further research could also focus on memory footage
of the algorithm.

TABLE I: Comparison the proposed approach with a few other variants Approach | Control Policy Estimation
---|---
[28] | Deterministic Tree Search
Improved version of [28] |  | Uniform look-ahead
---
Tree Search
Proximal Policy Optimization |  | Convolutional Neural
---
Network
Proximal Policy Optimization | Variational Autoencoder
Proximal Policy Optimization | Dueling Neural Network
Mean Execution Time (s) | Success Fraction
2.234 | 0.967
1.230 | 0.912
0.345 | 0.235
0.334 | 0.342
0.315 | 0.453

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# Motion induced excitation and electromagnetic radiation from an atom facing
a thin mirror

César D. Fosco 1<EMAIL_ADDRESS>Fernando C. Lombardo2
<EMAIL_ADDRESS>Francisco D. Mazzitelli1<EMAIL_ADDRESS>1
Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía
Atómica, R8402AGP Bariloche, Argentina  2 Departamento de Física Juan José
Giambiagi, FCEyN UBA and IFIBA CONICET-UBA, Facultad de Ciencias Exactas y
Naturales, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina

(today; today; August 27, 2024)

###### Abstract

We evaluate the probability of (de-)excitation and photon emission from a
neutral, moving, non-relativistic atom, coupled to the quantum electromagnetic
field and in the presence of a thin, perfectly conducting plane (“mirror”).
These results extend, to a more realistic model, the ones we had presented for
a scalar model, where the would-be electron was described by a scalar
variable, coupled to an (also scalar) vacuum field. The latter was subjected
to either Dirichlet or Neumann conditions on a plane. In our evaluation of the
spontaneous emission rate produced when the accelerated atom is initially in
an excited state, we pay attention to its comparison with the somewhat
opposite situation, namely, an atom at rest facing a moving mirror.

##  I Introduction

Several important effects are associated with the quantum vacuum fluctuations
of the electromagnetic (EM) field, ranging from the microscopic realm
(spontaneous emission by excited atoms, the Lamb shift, anomalous magnetic
moments of elementary particles, Van der Waals interactions) to the Casimir
interaction between neutral macroscopic bodies booksCasimir . The Casimir-
Polder force correspond to a hybrid situation, since it involves an atom and a
macroscopic medium. Different manifestations of the fluctuations of the EM
field in analogous situations correspond to the effect of a change in boundary
conditions on the probability of spontaneous emission from an atom. This has
been studied for an atom in the presence of a perfectly conducting plane, or
inside a cavity cavityqed .

New effects arise when one introduces time dependence; for instance, when the
atom or the macroscopic media are in motion: photon creation by accelerated
mirrors (Dynamical Casimir effect) and quantum friction for an atom and a
surface (or between two surfaces) in relative motion at constant velocity
reviewsdce . In the present work we are concerned with a dynamical situation,
focussing on the changes in the decay probability (of an initially excited
atom) as well as on the possibility of excitation of an atom which is
initially in its ground state. Note that, for this kind of system, one can
even have the production of photon pairs without any change in the atomic
state, what is the microscopic analog of the dynamical Casimir effect NetoMic
.

In Ref. Fulling1 , authors considered an atom at rest near an accelerating
mirror shown that uniformly accelerated motion of the mirror yields excitation
of a static two-level atom with simultaneous emission of a real photon. They
also found that the excitation probability oscillates as a function of the
atomic position because of interference between contributions from the waves
incident on and reflected from the mirror. In Fulling2  an atom accelerating
near a mirror is considered and a radiative effect is reported. From an
inertial point of view, the process arises from a collision of the negative
vacuum energy of Rindler space with the mirror. There is a qualitative
symmetry under interchange of accelerated and inertial subsystems, but it
hinges on the accelerated detector’s being initially in its own Rindler
vacuum.

In a previous work PaperI , we presented a study on the excitation and decay
probabilities for a moving atom in front of a planar mirror, in a simplified
model: the “atom” was endowed with a scalar variable describing the electron,
and it was coupled to a real quantum scalar field. Perfect conductor boundary
conditions were replaced with Neumann and Dirichlet boundary conditions on the
mirror’s plane.

We paid particular attention to two different processes, both taking place to
the lowest order in the coupling constant: transition of the atom from the
ground state to the first excited state, with simultaneous emission of a
photon, and spontaneous emission of an initially excited atom. We considered a
small-amplitude motion of the atom, and analysed the spectral and directional
dependence of the radiation on the motion.

In this paper we generalise those results to the more realistic case of an
atom coupled to the quantum electromagnetic field, taking into account the
$\mathbf{v}\times\mathbf{B}$ interaction between the moving atom and the
magnetic field, the so called Röntgen term, which is a consequence of the
vector character of the electromagnetic field and is crucial to maintain
Lorentz covariance Wilkens  to the relevant order in the velocity.

This paper is organized as follows. In Sect. II we describe our model in terms
of its classical action. In Sect. III we compute the vacuum persistence
amplitude from the imaginary part of the effective action. After recovering
known results for an atom oscillating in free space, we also present the
calculation for the case of the atom oscillating in front of a planar perfect
mirror. In Sect. IV, we compute the transition probabilities for two
processes: decay of an excited atom and excitation of an atom initially in its
ground state. We compare the results for a moving atom in front of a static
mirror with those for an oscillating mirror and a static atom Zoller ;
Passante , tracing the differences in terms of the Röntgen current. Section V
contains the conclusions of our work.

##  II The model and its classical action

Our starting point to define the model, shall be the action
${\mathcal{S}}_{a}$, for an atom coupled to the EM field, in the electric
dipole approximation. This approximation, to be unambiguous, must be
formulated on a comoving system. Indeed, let us assume that in the system
where the atom is at rest there is an electric dipole moment
${\mathbf{d}}_{0}$, and a vanishing magnetic dipole: ${\mathbf{m}}_{0}=0$.
Then, in the Lab system, and to the first order in the velocity of the atom
(in our conventions, the speed of light $c\equiv 1$) we shall have:

$\displaystyle{\mathbf{d}}(t)$
$\displaystyle={\mathbf{d}}_{0}(t)+{\mathbf{v}}(t)\times{\mathbf{m}}_{0}(t)={\mathbf{d}}_{0}(t)+{\mathbf{v}}(t)\times\,0={\mathbf{d}}_{0}(t)$
$\displaystyle{\mathbf{m}}(t)$
$\displaystyle={\mathbf{m}}_{0}(t)\,-\,{\mathbf{v}}(t)\times{\mathbf{d}}_{0}(t)=-{\mathbf{v}}(t)\times{\mathbf{d}}_{0}(t)$
$\displaystyle=-{\mathbf{v}}(t)\times{\mathbf{d}}(t).$ (1)

Therefore, the action ${\mathcal{S}}_{a}$ (in the Lab system) must also
include a coupling, to the magnetic field, of the motion-induced magnetic
dipole:

$\displaystyle{\mathcal{S}}_{a}$ $\displaystyle=\;\int
dt\,\Big{[}\frac{m}{2}\dot{\mathbf{x}}^{2}(t)\,-\,V({\mathbf{x}}(t))\,+\,{\mathbf{d}}(t)\cdot{\mathbf{E}}(t,{\mathbf{r}}(t))$
$\displaystyle\,+\,{\mathbf{m}}(t)\cdot{\mathbf{B}}(t,{\mathbf{r}}(t))\Big{]}$
$\displaystyle=\;\int
dt\,\Big{\\{}\frac{m}{2}\dot{\mathbf{x}}^{2}(t)\,-\,V({\mathbf{x}}(t))$
$\displaystyle\,+\,e\,{\mathbf{x}}(t)\cdot\big{[}{\mathbf{E}}(t,{\mathbf{r}}(t))\,+\,\dot{\mathbf{r}}(t)\times{\mathbf{B}}(t,{\mathbf{r}}(t))\big{]}\Big{\\}}\;,$
(2)

where ${\mathbf{x}}(t)$ denotes the position of the electron with respect to
the (center of mass of the) atom while ${\mathbf{r}}(t)$ does so for the atom
with respect to the origin of the Lab system. Regarding the potential $V$
binding the electron, for the sake of simplicity, we shall assume here that it
has a harmonic oscillator form: $V=\frac{m}{2}\Omega^{2}{\mathbf{x}}^{2}$.

On the other hand, the free EM field action ${\mathcal{S}}_{\rm em}(A)$ is
given by

${\mathcal{S}}_{\rm em}(A)\;=\;\int
d^{4}x\,\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,+\,{\mathcal{L}}_{\rm
g.f.}(A)\right]\;,$ (3)

with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, which includes
a gauge-fixing term ${\mathcal{L}}_{\rm g.f.}$. We want to consider the cases
of an atom moving either in free space or in the presence of a perfect mirror.
The second case shall be dealt with when integrating out the EM field
fluctuations. Not unexpectedly, the outcome will turn out to be the sum of the
free space result plus a “reflected” contribution (in the method of images
sense).

##  III Effective action and its imaginary part

As a first step in the derivation of the effective action
$\Gamma[{\mathbf{r}}(t)]$, which will only depend on the atom’s trajectory, we
first integrate out the electron’s degrees of freedom, ${\mathbf{x}}(t)$, to
obtain an intermediate effective action ${\mathcal{S}}_{\rm
eff}(A;{\mathbf{r}})$.

Since we are assuming a harmonic oscillator form for $V$ in (2), the
functional integral over ${\mathbf{x}}$ becomes a Gaussian. The result of such
an integral is (modulo an irrelevant constant) tantamount to replacing in the
action the integrated variable in terms of its source, using the classical
equation of motion for ${\mathbf{x}}$. The latter corresponds to a harmonic
oscillator forced by a time-dependent force, which is the Lorentz force acting
on ${\mathbf{r}}(t)$ (not on ${\mathbf{x}}$) . Solving for ${\mathbf{x}}$ in
terms of that force, and recalling that Feynman conditions are to be imposed
on the time dependence of that solution, we find:

${\mathcal{S}}_{\rm eff}(A;{\mathbf{r}})={\mathcal{S}}_{\rm
em}(A)+{\mathcal{S}}_{\rm I}^{(a)}(A;{\mathbf{r}})\;,$ (4)

with

$\displaystyle{\mathcal{S}}_{\rm
I}^{(a)}(A,{\mathbf{r}})=\frac{ie^{2}}{2m}\int_{t,t^{\prime}}\Delta_{\Omega}(t-t^{\prime})\big{[}{\mathbf{E}}(t,{\mathbf{r}}(t))$
(5)
$\displaystyle+\dot{\mathbf{r}}(t)\times{\mathbf{B}}(t,{\mathbf{r}}(t))\big{]}\cdot\big{[}{\mathbf{E}}(t^{\prime},{\mathbf{r}}(t^{\prime}))+\dot{\mathbf{r}}(t^{\prime})\times{\mathbf{B}}(t^{\prime},{\mathbf{r}}(t^{\prime}))\big{]}$

where we have used a shorthand notation for the integration over time, for
example:
$\int_{t,t^{\prime}}\ldots\equiv\int_{-\infty}^{+\infty}dt\int_{-\infty}^{+\infty}dt^{\prime}\ldots$,
and:

$\displaystyle\Delta_{\Omega}(t-t^{\prime})$ $\displaystyle=$
$\displaystyle\int\frac{d\nu}{2\pi}e^{-i\nu(t-t^{\prime})}\widetilde{\Delta}_{\Omega}(\nu)$
$\displaystyle\widetilde{\Delta}_{\Omega}(\nu)$ $\displaystyle=$
$\displaystyle\frac{i}{\nu^{2}-\Omega^{2}+i\epsilon}.$ (6)

We can produce an explicit expression for $\Delta_{\Omega}(t-t^{\prime})$,
which will turn out to be quite useful:

$\Delta_{\Omega}(t-t^{\prime})\;=\;\frac{1}{2\Omega}\,\big{[}\theta(t-t^{\prime})\,e^{-i\Omega(t-t^{\prime})}\,+\,\theta(t^{\prime}-t)\,e^{i\Omega(t-t^{\prime})}\big{]}\;.$
(7)

The final form of the effective action of the system,
$\Gamma[{\mathbf{r}}(t)]$, is obtained by including the EM field fluctuations.
Namely,

$e^{i\Gamma[{\mathbf{r}}(t)]}\;=\;\frac{\int{\mathcal{D}}A\,e^{i{\mathcal{S}}_{\rm
eff}(A;{\mathbf{r}})}}{\int{\mathcal{D}}A\,e^{i{\mathcal{S}}_{\rm
eff}(A;{\mathbf{r}}_{0})}}\;,$ (8)

where ${\mathbf{r}}_{0}$ is the average position of the atom, which we will
assume to be time independent. Then, up to first order in $e^{2}$ we obtain

$\displaystyle\Gamma[{\mathbf{r}}(t)]$
$\displaystyle=\;\frac{ie^{2}}{2m}\;\int_{t,t^{\prime}}\;\Delta_{\Omega}(t-t^{\prime})\,\Big{[}\big{\langle}{\mathbf{E}}(t,{\mathbf{r}}(t))\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}(t^{\prime}))\rangle$
$\displaystyle+\,2\,\big{\langle}{\mathbf{E}}(t,{\mathbf{r}}(t))\cdot\dot{\mathbf{r}}(t^{\prime})\times{\mathbf{B}}(t^{\prime},{\mathbf{r}}(t^{\prime}))\big{\rangle}$
$\displaystyle+\,\,\big{\langle}\dot{\mathbf{r}}(t)\times{\mathbf{B}}(t,{\mathbf{r}}(t))\cdot\dot{\mathbf{r}}(t^{\prime})\times{\mathbf{B}}(t^{\prime},{\mathbf{r}}(t^{\prime}))\big{\rangle}$
$\displaystyle-\,\big{\langle}{\mathbf{E}}(t,{\mathbf{r}}_{0})\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}_{0})\rangle\Big{]}\;,$
(9)

where the symbol $\big{\langle}\ldots\big{\rangle}$ denotes the functional
averaging

$\big{\langle}\ldots\big{\rangle}\;=\;\frac{\int{\mathcal{D}}A\;\ldots\;\exp\\{i\big{[}{\mathcal{S}}_{\rm
em}(A)\big{]}\Big{\\}}}{\int{\mathcal{D}}A\exp\\{i\big{[}{\mathcal{S}}_{\rm
em}(A)\big{]}\Big{\\}}}\;.$ (10)

The presence of the mirror may be introduced in more than one way; in the
previous definition of the functional averages, since the action is the one
for free space, we have implicitly assumed that it is dealt with by a proper
definition of the integration measure. Namely, the integral is over fields
satisfying perfect boundary conditions on the mirror. Our choice of
coordinates is such that the mirror occupies the $x_{3}=0$ plane (for the sake
of simplicity, we shall make no distinction between lower and upper indices,
from now on. Thus, $x_{3}\equiv x^{3}\equiv z$). From (III), we see that we
just need to perform functional averages for pairs of fields (each factor
involves derivatives of the gauge field). Therefore, we shall only need the
gauge field propagator with perfect conductor boundary conditions on the
mirror.

Just before inserting the explicit expressions for the EM field correlators,
it is convenient to perform an expansion in powers of the departures about the
average position of the atom, ${\mathbf{r}}_{0}$. To that end, we set
${\mathbf{r}}(t)\;=\;{\mathbf{r}}_{0}+{\mathbf{y}}(t)$, expand up to the
second order in ${\mathbf{y}}(t)$, discard terms which, by their very
structure, cannot contribute to the imaginary part of the effective action.

Thus, with this in mind, we may present the expression for the effective
action, expanded to the second order in ${\mathbf{y}}(t)$, as follows:

$\Gamma\;=\;\Gamma_{EE}\,+\,\Gamma_{EB}\,+\,\Gamma_{BB}\;,$ (11)

where

$\displaystyle\Gamma_{EE}$
$\displaystyle=\,\frac{ie^{2}}{2m}\;\int_{t,t^{\prime}}\;y_{i}(t)y_{j}(t^{\prime})\;\Delta_{\Omega}(t-t^{\prime})$
$\displaystyle\times\Big{(}\frac{\partial^{2}}{\partial
r_{i}\partial{r^{\prime}}_{j}}\langle{\mathbf{E}}(t,{\mathbf{r}})\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}^{\prime})\rangle\Big{)}\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}\;,$
(12) $\displaystyle\Gamma_{EB}$
$\displaystyle=\,\frac{ie^{2}}{m}\;\int_{t,t^{\prime}}\;y_{i}(t)\dot{y}_{j}(t^{\prime})\;\Delta_{\Omega}(t-t^{\prime})\,\varepsilon_{kjl}\,$
$\displaystyle\times\Big{(}\frac{\partial}{\partial r_{i}}\langle
E_{k}(t,{\mathbf{r}})\,B_{l}(t^{\prime},{\mathbf{r}}^{\prime})\rangle\Big{)}\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}\;,$
(13) $\displaystyle\Gamma_{BB}$
$\displaystyle=\,\frac{ie^{2}}{2m}\;\int_{t,t^{\prime}}\;\dot{y}_{i}(t)\dot{y}_{j}(t^{\prime})\;\Delta_{\Omega}(t-t^{\prime})\,\varepsilon_{kil}\varepsilon_{kjm}$
$\displaystyle\times\langle
B_{l}(t,{\mathbf{r}})\,B_{m}(t^{\prime},{\mathbf{r}}^{\prime})\rangle\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}\;.$
(14)

The previous formulae, valid in free space, hold true when a mirror is
present, the difference between those two situations being the form of the EM
field correlation functions. Let us first consider the free space case.

###  III.1 Free space

We evaluate each one of the three terms into which we have decomposed $\Gamma$
in (11) in turn. They involve different correlation functions between
components of the EM field in free space. Note that any time-local term
appearing in those correlation functions (namely, a polynomial in
$\delta(t-t^{\prime})$ and its derivatives) will not contribute to the
imaginary part and we shall therefore discard them. We shall use a $(0)$ to
denote the free space version of an object, to distinguish it from the one
when the perfectly conducting plane is present.

For $\Gamma_{EE}^{(0)}$ we have:

$\displaystyle\langle
E_{i}(t,{\mathbf{r}}(t))E_{j}(t^{\prime},{\mathbf{r}}(t^{\prime}))\rangle^{(0)}$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}e^{i{\mathbf{k}}\cdot({\mathbf{r}}-{\mathbf{r}^{\prime}})}$
$\displaystyle\times$ $\displaystyle
k^{2}(\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}})\Delta_{k}(t-t^{\prime}),$

where $\langle\ldots\rangle^{(0)}$ denotes correlation functions in free
space. Here, $\Delta_{k}$ is defined as in (III), with $k\equiv|{\mathbf{k}}|$
playing the role of $\Omega$:
$\Delta_{k}(t-t^{\prime})\equiv[\Delta_{\Omega}(t-t^{\prime})]_{\Omega\to k}$.
Therefore, from (III.1) we derive:

$\displaystyle\Big{(}\frac{\partial^{2}}{\partial r_{i}\partial
r^{\prime}_{j}}\langle{\mathbf{E}}(t,{\mathbf{r}})$
$\displaystyle\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}^{\prime})\rangle^{(0)}\Big{)}\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}$
$\displaystyle=2\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}\,{\mathbf{k}}^{2}\,k_{i}k_{j}\,\Delta_{k}(t-t^{\prime})$
$\displaystyle=\;\frac{2}{3}\,\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}\,k^{4}\,\delta_{ij}\,\Delta_{k}(t-t^{\prime})\;.$
(16)

Inserting this into $\Gamma_{EE}^{(0)}$, we note that the resulting expression
will contain the product $\Delta_{\Omega}\Delta_{k}$. This product may be
simplified by using the property, valid for any pair $\Omega_{1}$,
$\Omega_{2}$:

$\Delta_{\Omega_{1}}(t-t^{\prime})\,\Delta_{\Omega_{2}}(t-t^{\prime})\;=\;\frac{\Omega_{1}+\Omega_{2}}{2\,\Omega_{1}\Omega_{2}}\,\Delta_{\Omega_{1}+\Omega_{2}}(t-t^{\prime})\;.$
(17)

In our case, this leads to:

$\Gamma_{EE}^{(0)}=\frac{ie^{2}}{6m\Omega}\int_{t,t^{\prime}}y_{i}(t)y_{i}(t^{\prime})\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}(\Omega+k)k^{3}\Delta_{\Omega+k}(t-t^{\prime})$
(18)

Writing then $\Delta_{\Omega+k}$ in terms of its Fourier transform, and
Fourier transforming the departures, we obtain

$\Gamma_{EE}^{(0)}=\frac{ie^{2}}{6m\Omega}\int\frac{d\nu}{2\pi}\tilde{y}^{*}_{i}(\nu)\tilde{y}_{i}(\nu)\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}(\Omega+k)k^{3}\widetilde{\Delta}_{\Omega+k}(\nu).$
(19)

The imaginary part of $\Gamma_{EE}$ is then straightforwardly obtained from
the one of $\widetilde{\Delta}_{\Omega+k}$:

$\displaystyle{\rm Im}\big{[}\Gamma_{EE}^{(0)}\big{]}=\frac{\pi
e^{2}}{6m\Omega}$
$\displaystyle\int\frac{d\nu}{2\pi}\tilde{y}^{*}_{i}(\nu)\tilde{y}_{i}(\nu)$
$\displaystyle\,\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}(\Omega+k)$
$\displaystyle\times k^{3}\delta\big{[}\nu^{2}-(\Omega+k)^{2}\big{]}.$

Performing the integration over ${\mathbf{k}}$,

$\displaystyle{\rm Im}\big{[}\Gamma_{EE}^{(0)}\big{]}$ $\displaystyle=$
$\displaystyle\int\frac{d\nu}{2\pi}\;m_{EE}^{(0)}(\nu)|\tilde{\mathbf{y}}(\nu)|^{2}$
$\displaystyle=$ $\displaystyle\frac{e^{2}}{24\pi
m\Omega}\,\int\frac{d\nu}{2\pi}\;\theta(|\nu|-\Omega)\,(|\nu|-\Omega)^{5}\;|\tilde{\mathbf{y}}(\nu)|^{2}.$

For the computation of $\Gamma_{EB}^{(0)}$ we start from the correlation
function:

$\displaystyle\langle
E_{j}(t,{\mathbf{r}})B_{l}(t^{\prime},{\mathbf{r}}^{\prime})\rangle^{(0)}$
$\displaystyle=$ $\displaystyle
i\epsilon_{jlm}\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}e^{i{\mathbf{k}}\cdot({\mathbf{r}}-{\mathbf{r}^{\prime}})}$
$\displaystyle\times k_{m}\partial_{t}\Delta_{k}(t-t^{\prime}).$

Upon insertion of this into the expression for $\Gamma_{EB}^{(0)}$, the
product $\Delta_{\Omega}\partial_{t}\Delta_{k}$ arises. For this object we use
the property:

$\Delta_{\Omega_{1}}(t-t^{\prime})\,\partial_{t}\Delta_{\Omega_{2}}(t-t^{\prime})\;=\;\frac{1}{2\Omega_{1}}\,\partial_{t}\Delta_{\Omega_{1}+\Omega_{2}}(t-t^{\prime})\;,$
(23)

to get:

$\Gamma_{EB}^{(0)}\;=\;\frac{ie^{2}}{3m\Omega}\,\int_{t,t^{\prime}}\,y_{i}(t)\dot{y}_{i}(t^{\prime})\,\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}\;k^{2}\;\partial_{t}\Delta_{\Omega+k}(t-t^{\prime})\;.$
(24)

Integrating by parts and Fourier transforming,

$\Gamma_{EB}^{(0)}=-\frac{ie^{2}}{3m\Omega}\int\frac{d\nu}{2\pi}\tilde{y}^{*}_{i}(\nu)\tilde{y}_{i}(\nu)\nu^{2}\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}k^{2}\widetilde{\Delta}_{\Omega+k}(\nu),$
(25)

whence the imaginary part then becomes:

$\displaystyle{\rm Im}$
$\displaystyle\big{[}\Gamma_{EB}^{(0)}\big{]}=\int\frac{d\nu}{2\pi}|\tilde{\mathbf{y}}(\nu)|^{2}m_{EB}^{(0)}(\nu)$
$\displaystyle=$ $\displaystyle-\frac{e^{2}}{12\pi
m\Omega}\int\frac{d\nu}{2\pi}|\tilde{\mathbf{y}}(\nu)|^{2}\theta(|\nu|-\Omega)|\nu|(|\nu|-\Omega)^{4}.$

Finally, to evaluate $\Gamma_{BB}^{(0)}$, we need the correlator of two
magnetic fields. It is straightforward to see that:

$\displaystyle\varepsilon_{kil}\varepsilon_{kjm}\langle
B_{l}(t,{\mathbf{r}})\,B_{m}(t^{\prime},{\mathbf{r}}^{\prime})\rangle^{(0)}$
$\displaystyle=$
$\displaystyle\frac{4}{3}\,\delta_{ij}\,\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}e^{i{\mathbf{k}}\cdot({\mathbf{r}}-{\mathbf{r}^{\prime}})}$
(27) $\displaystyle\times k^{2}\,\Delta_{k}(t-t^{\prime}).$

Therefore,

$\Gamma_{BB}^{(0)}=\frac{2ie^{2}}{3m}\int_{t,t^{\prime}}\dot{y}_{i}(t)\dot{y}_{i}(t^{\prime})\Delta_{\Omega}(t-t^{\prime})\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}k^{2}\Delta_{k}(t-t^{\prime}).$
(28)

Proceeding in an analogous fashion as for the previous two terms, we find:

$\displaystyle{\rm Im}\big{[}\Gamma_{BB}^{(0)}\big{]}$ $\displaystyle=$
$\displaystyle\int\frac{d\nu}{2\pi}\;|\tilde{\mathbf{y}}(\nu)|^{2}\;m_{BB}^{(0)}(\nu)$
$\displaystyle=$ $\displaystyle\frac{e^{2}}{12\pi
m\Omega}\int\frac{d\nu}{2\pi}|\tilde{\mathbf{y}}(\nu)|^{2}\;\theta(|\nu|-\Omega)\nu^{2}(|\nu|-\Omega)^{3}.$

Adding the three contributions to the imaginary part of the effective action,
we get:

$\displaystyle{\rm Im}[\Gamma^{(0)}]=\;{\rm Im}[\Gamma_{EE}^{(0)}]\,+\,{\rm
Im}[\Gamma_{EB}^{(0)}]\,+\,{\rm Im}[\Gamma_{BB}^{(0)}]$
$\displaystyle=\;\frac{e^{2}}{24\pi
m\Omega}\,\int_{-\infty}^{+\infty}\frac{d\nu}{2\pi}\;|\tilde{\mathbf{y}}(\nu)|^{2}\;\theta(|\nu|-\Omega)$
$\displaystyle\times(|\nu|-\Omega)^{3}\,(\nu^{2}+\Omega^{2}).$ (30)

This coincides with the result obtained in Ref.NetoMic , if one performs the
angular integration of the probability distribution obtained there.

###  III.2 Perfect mirror

We now evaluate the different terms contributing to the imaginary part of the
effective action when a perfect mirror is present. The difference is in the
form of the EM field correlation functions. They may be obtained from the ones
of the gauge field, which in turn can be constructed, for example, by using
the method of images. To that end, it is convenient to introduce first a
special notation to distinguish among spacetime coordinates. We shall use:
$x\,=\,(x_{\shortparallel},x_{3})$, with $x_{\shortparallel}$ denoting
$x_{0},\,x_{1},\,x_{2}$, the coordinates for which there is translation
invariance. When using indices, the ones from the beginning of the Greek
alphabet: $\alpha,\,\beta,\ldots$, will be implicitly assumed to run over the
values $0$, $1$ and $2$. Besides, $a,\,b,\,\ldots$ will take the values $1$
and $2$ (these appear when dealing with spatial coordinates on the plane).

Then, the correlator in the presence of the mirror,

$\langle
A_{\mu}(x)A_{\nu}(y)\rangle\;\equiv\;D_{\mu\nu}(x_{\shortparallel}-y_{\shortparallel};x_{3},y_{3})\;,$
(31)

may be written as follows:

$\displaystyle D_{\mu\nu}(x_{\shortparallel}-y_{\parallel};x_{3},y_{3})$
$\displaystyle=$ $\displaystyle
D^{(0)}_{\mu\nu}(x_{\shortparallel}-y_{\shortparallel};x_{3},y_{3})$ (32)
$\displaystyle+$ $\displaystyle
D^{(R)}_{\mu\nu}(x_{\shortparallel}-y_{\shortparallel};x_{3},y_{3})\;,$

where $D^{(0)}_{\mu\nu}$ is the gauge-field propagator in free space, and
$D^{(R)}_{\mu\nu}$ is the reflected contribution:

$D^{(R)}_{\mu\nu}\;=\;-\,g_{\mu}^{\alpha}\,g_{\nu}^{\beta}\;D^{(0)}_{\alpha\beta}(x_{\shortparallel}-y_{\shortparallel};x_{3},-y_{3})\;.$
(33)

Because of the fact that the EM field correlators will be the sum of two
terms, the first one identical to the free space one, and the second a
“reflection” ($R$) term, also the effective action and its imaginary part will
share this property. Namely,

$\displaystyle\Gamma$ $\displaystyle=\;\Gamma^{(0)}\,+\,\Gamma^{(R)}\;,$
$\displaystyle\Gamma^{(R)}$
$\displaystyle=\;\Gamma_{EE}^{(R)}\,+\,\Gamma_{EB}^{(R)}\,+\,\Gamma_{BB}^{(R)}\;.$
(34)

We now evaluate each one of the three reflection terms above, having in mind
that they are to be added to the free space terms; namely, they have no
meaning by themselves and in particular their imaginary parts could be
negative.

$\Gamma_{EE}^{(R)}$ is obtained by using the reflection term instead of the
free space correlator in the analogous formula we have already used for free
space. Indeed,

$\displaystyle\Gamma_{EE}^{(R)}\,=\,\frac{ie^{2}}{2m}\;$
$\displaystyle\int_{t,t^{\prime}}\;$ $\displaystyle
y_{i}(t)y_{j}(t^{\prime})\;\Delta_{\Omega}(t-t^{\prime})$
$\displaystyle\times$ $\displaystyle\Big{(}\frac{\partial^{2}}{\partial
r_{i}\partial{r^{\prime}}_{j}}\langle{\mathbf{E}}(t,{\mathbf{r}})\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}^{\prime})\rangle^{(R)}\Big{)}\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}$

Now, because of the presence of the mirror, three-dimensional rotation
symmetry is lost. We will, as a consequence, have different contributions to
$\Gamma_{EE}^{(R)}$ (and its imaginary part) depending on whether the motion
is parallel or normal to the plane. It is rather straightforward to see that
$\Gamma_{EE}^{(R)}$ becomes the sum of two independent contributions, one for
each kind of motion:

$\Gamma_{EE}^{(R)}[{\mathbf{y}}(t)]\,=\,\Gamma_{EE,\shortparallel}^{(R)}[{\mathbf{y}_{\shortparallel}}(t)]\,+\,\Gamma_{EE,\perp}^{(R)}[y_{3}(t)]\;,$
(36)

since:

$\displaystyle\Gamma_{EE}^{(R)}[{\mathbf{y}}(t)]\,$
$\displaystyle=\,\frac{ie^{2}}{2m}\;\int_{t,t^{\prime}}\;\Big{\\{}\frac{1}{2}\,{\mathbf{y}}_{\shortparallel}(t)\cdot{\mathbf{y}}_{\shortparallel}(t^{\prime})\;\Delta_{\Omega}(t-t^{\prime})$
(37) $\displaystyle\times\Big{(}\frac{\partial^{2}}{\partial
r_{a}\partial{r^{\prime}}_{a}}\langle{\mathbf{E}}(t,{\mathbf{r}})\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}^{\prime})\rangle^{(R)}\Big{)}\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}$
$\displaystyle+\,y_{3}(t)y_{3}(t^{\prime})\;\Delta_{\Omega}(t-t^{\prime})$
$\displaystyle\times\Big{(}\frac{\partial^{2}}{\partial r_{3}\partial
r^{\prime}_{3}}\langle{\mathbf{E}}(t,{\mathbf{r}})\cdot{\mathbf{E}}(t^{\prime},{\mathbf{r}}^{\prime})\rangle^{(R)}\Big{)}\Big{|}_{{\mathbf{r}}={\mathbf{r}^{\prime}}={\mathbf{r}}_{0}}\Big{\\}}\;.$

In particular, for parallel motion, we find:

$\Gamma_{EE,\shortparallel}^{(R)}[{\mathbf{y}_{\shortparallel}}(t)]=-\frac{ie^{2}}{4m}\int_{t,t^{\prime}}{\mathbf{y}_{\shortparallel}}(t){\mathbf{y}_{\shortparallel}}(t^{\prime})\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}{\rm
cos}(2k_{3}a){\mathbf{k}_{\shortparallel}}^{2}({\mathbf{k}_{\shortparallel}}^{2}+3k_{3}^{2})\Delta_{\Omega}(t-t^{\prime})\Delta_{k}(t-t^{\prime})\,,$
(38)

while for motion along the perpendicular, $x_{3}$ direction:

$\displaystyle\Gamma_{EE,\perp}^{(R)}[y_{3}(t)]$ $\displaystyle=$
$\displaystyle\frac{ie^{2}}{2m}\int_{t,t^{\prime}}y_{3}(t)y_{3}(t^{\prime})\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}{\rm
cos}(2k_{3}a)k_{3}^{2}$ (39) $\displaystyle\times$
$\displaystyle({\mathbf{k}_{\shortparallel}}^{2}+3k_{3}^{2})\Delta_{\Omega}(t-t^{\prime})\Delta_{k}(t-t^{\prime})\,.$

By using an entirely analogous procedure to the one for the free space part,
we find the respective imaginary parts. Note that both can, and will, depend
on $a$, the distance of ${\mathbf{r}}_{0}$ to the mirror:

${\rm
Im}\big{[}\Gamma_{EE,\shortparallel}^{(R)}\big{]}=\int\frac{d\nu}{2\pi}\,|\tilde{\mathbf{y}}_{\shortparallel}(\nu)|^{2}\;m_{EE}^{\parallel}(\nu),$
(40)

where

$m_{EE}^{\parallel}(\nu)=-\frac{e^{2}}{32\pi
m\Omega}\,\theta(|\nu|-\Omega)(|\nu|-\Omega)^{5}\,f_{1}[(|\nu|-\Omega)a],$
(41)

with:

$f_{1}(x)\,=\;3(\frac{1}{x^{4}}-\frac{1}{2x^{2}}){\rm
cos}(2x)\,+\,\frac{1}{2}(-\frac{3}{x^{5}}+\frac{11}{2x^{3}}){\rm sin}(2x)\,.$
(42)

For the perpendicular case, we have

${\rm
Im}\big{[}\Gamma_{EE,\perp}^{(R)}\big{]}=\int\frac{d\nu}{2\pi}\,|\tilde{y}_{3}(\nu)|^{2}\;m_{EE}^{\perp}(\nu),$
(43)

where

$m_{EE}^{\perp}(\nu)=\frac{e^{2}}{16\pi
m\Omega}\,\int\frac{d\nu}{2\pi}\,\theta(|\nu|-\Omega)(|\nu|-\Omega)^{5}\,f_{2}[(|\nu|-\Omega)a],$
(44)

with:

$f_{2}(x)\,=\;(-\frac{3}{x^{4}}+\frac{5}{2x^{2}}){\rm
cos}(2x)\,+\,\frac{1}{4}(\frac{6}{x^{5}}-\frac{13}{x^{3}}+\frac{6}{x}){\rm
sin}(2x)\,.$ (45)

The term which involves the mixed correlator between the electric and magnetic
fields, may be written as follows

$\displaystyle\Gamma_{EB}^{(R)}$ $\displaystyle=$
$\displaystyle\frac{ie^{2}}{m}\;\int_{t,t^{\prime}}\;\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}e^{2ik_{3}a}\,\Delta_{\Omega}(t-t^{\prime})\partial_{t}\Delta_{k}(t-t^{\prime})$
(46) $\displaystyle\times$ $\displaystyle
y_{j}(t)\dot{y}_{l}(t^{\prime})\;(k_{j}\,\delta_{la}k_{a}-2k_{j}\delta_{l3}k_{3}).$

The imaginary parts for parallel and normal motion become

${\rm
Im}\big{[}\Gamma_{EB,\shortparallel}^{(R)}\big{]}=\int\frac{d\nu}{2\pi}\,|\tilde{\mathbf{y}}_{\shortparallel}(\nu)|^{2}\;m_{EB}^{\parallel}(\nu),$
(47)

where

$m_{EB}^{\parallel}(\nu)=-\frac{e^{2}}{8\pi
m\Omega}\theta(|\nu|-\Omega)\,|\nu|(|\nu|-\Omega)^{4}\,f_{3}[(|\nu|-\Omega)a]\;,$
(48)

with:

$f_{3}(x)\,=\;-\frac{1}{4x^{2}}{\rm cos}(2x)\,+\,\frac{1}{8x^{3}}\,{\rm
sin}(2x)\,,$ (49)

and

${\rm
Im}\big{[}\Gamma_{EB,\perp}^{(R)}\big{]}=\int\frac{d\nu}{2\pi}\,|\tilde{y}_{3}(\nu)|^{2}\;m_{EB}^{\perp}(\nu),$
(50)

where,

$m_{EB}^{\perp}(\nu)=\frac{e^{2}}{8\pi
m\Omega}\,\theta(|\nu|-\Omega)\,|\nu|(|\nu|-\Omega)^{4}\,f_{4}[(|\nu|-\Omega)a],$
(51)

with:

$f_{4}(x)\,=\;\frac{1}{x^{2}}\,{\rm
cos}(2x)\,-\,(\frac{1}{2x^{3}}-\frac{1}{x}){\rm sin}(2x)\,.$ (52)

Finally, for $\Gamma_{BB}^{(R)}$ we have:

$\displaystyle\Gamma_{BB}^{(R)}$
$\displaystyle=\,-\frac{ie^{2}}{2m}\;\int_{t,t^{\prime}}\;\int\frac{d^{3}{\mathbf{k}}}{(2\pi)^{3}}e^{2ik_{3}a}\,\Delta_{\Omega}(t-t^{\prime})\Delta_{k}(t-t^{\prime})$
$\displaystyle\times\Big{\\{}\,\dot{y}_{a}(t)\dot{y}_{a}(t^{\prime})\;({\mathbf{k}_{\shortparallel}}^{2}-k_{3}^{2})-2\,k_{3}^{2}\,\dot{y}_{3}(t)\dot{y}_{3}(t^{\prime})\;\Big{\\}}\;.$
(53)

and the respective imaginary parts for parallel and normal motion,

${\rm
Im}\big{[}\Gamma_{BB,\shortparallel}^{(R)}\big{]}=\int\frac{d\nu}{2\pi}\,|\tilde{\mathbf{y}}_{\shortparallel}(\nu)|^{2}m_{BB}^{\parallel}(\nu),$
(54)

where

$m_{BB}^{\parallel}(\nu)=-\frac{e^{2}}{16\pi
m\Omega}\theta(|\nu|-\Omega)\nu^{2}(|\nu|-\Omega)^{3}\,f_{5}[(|\nu|-\Omega)a],$
(55)

with:

$f_{5}(x)\,=\;-\frac{1}{x^{2}}{\rm
cos}(2x)\,+\,\frac{1}{2}(\frac{1}{x^{3}}-\frac{1}{x}){\rm sin}(2x)\,,$ (56)

and

${\rm
Im}\big{[}\Gamma_{BB,\perp}^{(R)}\big{]}=\int\frac{d\nu}{2\pi}\,|\tilde{y}_{3}(\nu)|^{2}\;m_{BB}^{\perp}(\nu),$
(57)

where

$m_{BB}^{\perp}(\nu)=\frac{e^{2}}{16\pi
m\Omega}\theta(|\nu|-\Omega)\,\nu^{2}(|\nu|-\Omega)^{3}\,f_{6}[(|\nu|-\Omega)a]\;,$
(58)

with:

$f_{6}(x)\,=\;\frac{1}{x^{2}}\,{\rm
cos}(2x)\,+\,(-\frac{1}{2x^{3}}+\frac{1}{x}){\rm sin}(2x)\,.$ (59)

In Fig. 1 we plot $m_{1}=1+m_{EE}^{\parallel}/m_{EE}^{(0)}$ for the parallel
motion and $m_{2}=1+m_{EE}^{\perp}/m_{EE}^{(0)}$ for the normal one, both as
functions of the dimensionless variable $x=a(|\nu|-\Omega)$. In both cases,
the contribution from $m_{EE}^{\parallel,\perp}$ goes to zero as
$a|\nu|\rightarrow\infty$. In the other limit, i.e., when $x\rightarrow 0$, we
get $m_{EE}^{\parallel}/m_{EE}^{(0)}=-7/10$ and
$m_{EE}^{\perp}/m_{EE}^{(0)}=11/10$. This case appears as qualitatively
similar to the Dirichlet contribution reported in Ref. PaperI , (see comment
in footnote ). Figs. 2 and 3 show similar behaviour than the previous one. The
rates goes to zero in the large limit and
$m_{EB}^{\parallel}/m_{EB}^{(0)}=1/2$, $m_{EB}^{\perp}/m_{EB}^{(0)}=-1$, and
$m_{BB}^{\parallel}/m_{BB}^{(0)}=-1/4$ and $m_{BB}^{\perp}/m_{BB}^{(0)}=1/2$
when $x\rightarrow 0$. These limits show a different behaviour with respect to
the Dirichlet and Neumann cases reported in PaperI .

Figure 1: We plot $m_{1}=1+m_{EE}^{\parallel}/m_{EE}^{(0)}$ and
$m_{2}=1+m_{EE}^{\perp}/m_{EE}^{(0)}$ as a function of the dimensionless
$x=a(|\nu|-\Omega)$. Figure 2: $m_{1}=1+m_{EB}^{\parallel}/m_{EB}^{(0)}$ and
$m_{2}=1+m_{EB}^{\perp}/m_{EB}^{(0)}$ as a function of the dimensionless
$x=a(|\nu|-\Omega)$. Figure 3: Ratios
$m_{1}=1+m_{BB}^{\parallel}/m_{BB}^{(0)}$ and
$m_{2}=1+m_{BB}^{\perp}/m_{BB}^{(0)}$ as a function of the dimensionless
$x=a(|\nu|-\Omega)$.

##  IV Transition amplitudes

Let us now study the transition amplitudes and probabilities for the EM field
model. The first-order transition matrix will now be given by:

$T_{fi}\;\equiv\;e\,\int dt\,\langle
f|{\mathbf{x}}(t)\cdot[{\mathbf{E}}(t,{\mathbf{r}}(t))+\dot{\mathbf{r}}(t)\times{\mathbf{B}}(t,{\mathbf{r}}(t))]|i\rangle\;,$
(60)

where

$|i\rangle=|i_{a}\rangle\otimes|i_{EM}\rangle\quad\quad|f\rangle=|f_{a}\rangle\otimes|f_{EM}\rangle\,\,,$
(61)

with the "$a$" and "$EM$" indices denoting the atom and electromagnetic field
states.

For the electron’s degrees of freedom we have, in the interaction picture

${\mathbf{x}}(t)\;=\;\frac{1}{\sqrt{2m\,\Omega}}\left({\mathbf{a}}e^{-i\Omega
t}\,+\,{\mathbf{a}}^{\dagger}e^{i\Omega t}\right)\;.$ (62)

Here, ${\mathbf{a}}=\sum_{l=1}^{3}a_{l}\;\widehat{\mathbf{e}}_{l}$, where
$\widehat{\mathbf{e}}_{l}$ are three orthonormal vectors, since the
Hamiltonian for the electron is essentially a three-dimensional harmonic
oscillator. This implies that, when considering a transition from the vacuum
to an excited state, that process will introduce a spatial direction, in other
words, a polarization. On the other hand, for the gauge field in the Coulomb
gauge we use the expansion:

${\mathbf{A}}(x)=\int
d^{2}{\mathbf{k}}_{\shortparallel}\int_{0}^{\infty}dk_{z}\sum_{\lambda}\left[{\alpha}_{\lambda}({\mathbf{k}}){\mathbf{f}}^{(\lambda)}_{\mathbf{k}}(x)+{\mbox{h}.c.}\right],$
(63)

where $\lambda$ sums over the two independent modes for each ${\mathbf{k}}$,
which are consistent with the perfect conductor condition at $z=0$. In this
gauge, that amounts to a vanishing, on that plane, of the components of the
vector potential which are parallel to that surface.

Including a global factor to normalize the states, we may write those modes as
follows (see, for example modes ):

$\displaystyle{\mathbf{f}}^{(1)}_{\mathbf{k}}(x)$
$\displaystyle=\;e^{-ikt}\sqrt{\frac{2}{(2\pi)^{3}k}}\;\big{(}\widehat{\mathbf{k}}_{\shortparallel}\times\widehat{\mathbf{z}}\big{)}\sin(k_{z}z)\,e^{i{\mathbf{k}}_{\shortparallel}\cdot{\mathbf{x}}_{\shortparallel}}$
$\displaystyle{\mathbf{f}}^{(2)}_{\mathbf{k}}(x)$
$\displaystyle=\;e^{-ikt}\sqrt{\frac{2}{(2\pi)^{3}k}}\;k^{-1}\,\big{[}\widehat{\mathbf{z}}\;|{\mathbf{k}}_{\shortparallel}|\cos(k_{z}z)$
$\displaystyle-\,i\,\widehat{\mathbf{k}}_{\shortparallel}\,k_{z}\,\sin(k_{z}z)\big{]}e^{i{\mathbf{k}}_{\shortparallel}\cdot{\mathbf{x}}_{\shortparallel}}\;,$
(64)

($\widehat{\mathbf{k}}_{\shortparallel}$ and $\widehat{\mathbf{z}}$ denote
unit vectors). The notation

${\mathbf{f}}^{(\lambda)}_{\mathbf{k}}(x)=\;N_{k}\,e^{-ikt}{\mathbf{g}}^{(\lambda)}_{\mathbf{k}}(x)\,\;,\,\;N_{k}=\sqrt{\frac{2}{(2\pi)^{3}k}}$
(65)

will be useful in what follows.

###  IV.1 The decay process

Let us now consider a decay process, in which the initial state of the atom is
an excited state and the EM field is in vacuum

$|i_{a}\rangle=a_{l}^{\dagger}|0_{a}\rangle\quad\quad\,|i_{EM}\rangle=|0_{EM}\rangle,$
(66)

while final states are

$|f_{a}\rangle=|0_{a}\rangle\quad\quad\,|f_{EM}\rangle={\alpha}^{\dagger}_{\lambda}({\mathbf{k}})|0_{EM}\rangle.$
(67)

Note that for the electronic transition we have in principle three independent
polarizations (not necessarily along the three coordinate axis), so we have to
choose a polarization for the excited state of the electron, and also for the
final state of the EM field.

The matrix elements for the decay process then read

$\displaystyle T_{fi}^{\rm{(dec)}}({\mathbf{k}},l,\lambda)\;$ $\displaystyle=$
$\displaystyle\;\frac{e}{\sqrt{2m\,\Omega}}\;\int_{-\infty}^{+\infty}dt\,e^{-it\Omega}\,\langle
0_{EM}|\alpha_{\lambda}({\mathbf{k}})\,\widehat{\mathbf{e}}_{l}\,\cdot[{\mathbf{E}}(t,{\mathbf{r}}(t))+\dot{\mathbf{r}}(t)\times{\mathbf{B}}(t,{\mathbf{r}}(t))]|0_{EM}\rangle$
(68) $\displaystyle=$
$\displaystyle\;\frac{e}{\sqrt{2m\,\Omega}}\;\int_{-\infty}^{+\infty}dt\,e^{-it\Omega}\,\widehat{\mathbf{e}}_{l}\,\cdot\left[-\partial_{t}{\mathbf{f}}_{\mathbf{k}}^{(\lambda)*}(x)+\dot{\mathbf{r}}(t)\times\left({\mathbf{\nabla}}\times{\mathbf{f}}_{\mathbf{k}}^{(\lambda)*}(x)\right)\right]_{x=(t,{\mathbf{r}}(t))}.$

We now expand the results up to the second order in $\mathbf{y}(t)$, which is
the departure from $\mathbf{r}_{0}=(0,0,a)$. Note that, as the matrix elements
have a contribution at zeroth order, it is necessary to expand them up to the
second order, to compute consistently the decay probabilities beyond the
static case. We denote the different orders by

$T_{fi}^{\rm{(dec)}}=T_{fi}^{\rm{(dec,0)}}+T_{fi}^{\rm{(dec,1)}}+T_{fi}^{\rm{(dec,2)}}\,.$
(69)

Performing the expansion we obtain

$\displaystyle T_{fi}^{\rm{(dec,0)}}$
$\displaystyle({\mathbf{k}},l,\lambda)=\frac{-2\pi
ieN_{k}k}{\sqrt{2m\,\Omega}}\delta(\Omega-k)\widehat{\mathbf{e}}_{l}\cdot\mathbf{g}^{(\lambda)*}_{\mathbf{k}}(\mathbf{x})|_{\mathbf{x}=\mathbf{r}_{0}}$
(70) $\displaystyle T_{fi}^{\rm{(dec,1)}}$
$\displaystyle({\mathbf{k}},l,\lambda)=\frac{eN_{k}}{\sqrt{2m\,\Omega}}\widehat{\mathbf{e}}_{l}\cdot[(-ik)\widetilde{y}_{j}(k-\Omega)\partial_{j}\mathbf{g}^{(\lambda)*}_{\mathbf{k}}(\mathbf{x})$
$\displaystyle-i(k-\Omega)\tilde{\mathbf{y}}(k-\Omega)\times(\mathbf{\nabla}\times\mathbf{g}^{(\lambda)*}_{\mathbf{k}}(\mathbf{x}))]|_{\mathbf{x}=\mathbf{r}_{0}}$
$\displaystyle T_{fi}^{\rm{(dec,2)}}$
$\displaystyle({\mathbf{k}},l,\lambda)=A_{fi}({\mathbf{k}},l,\lambda)+B_{fi}({\mathbf{k}},l,\lambda)\,,$

with

$\displaystyle A_{fi}({\mathbf{k}},l,\lambda)$ $\displaystyle=$
$\displaystyle\frac{-ieN_{k}k}{2\sqrt{2m\,\Omega}}\int_{\\-\infty}^{\infty}dte^{-it(\Omega-k)}y_{i}(t)y_{j}(t)$
$\displaystyle\widehat{\mathbf{e}}_{l}\cdot\partial_{i}\partial_{j}\mathbf{g}^{(\lambda)*}_{\mathbf{k}}(\mathbf{x})|_{\mathbf{x}=\mathbf{r}_{0}}$
$\displaystyle B_{fi}({\mathbf{k}},l,\lambda)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\lambda}ieN_{k}k}{\sqrt{2m\,\Omega}}\int_{\\-\infty}^{\infty}dte^{-it(\Omega-k)}y_{j}(t)\dot{\mathbf{y}}(t)$
(71)
$\displaystyle\cdot(\widehat{\mathbf{e}}_{l}\times\partial_{j}\mathbf{g}^{(\lambda^{\prime})*}_{\mathbf{k}}(\mathbf{x}_{\parallel},z-\frac{\pi}{2k_{z}})|_{\mathbf{x}=\mathbf{r}_{0}}\,.$

Here we used the notation $\lambda^{\prime}=1$ for $\lambda=2$ and vice versa.

In practice, we expect the experiments not to detect the polarization state of
the excited state of the atom; therefore, it makes sense to consider the sum
over the $3$ posible values of $l$ when evaluating the probabilities. Namely,
we obtain results for the probabilities which depend only the polarization of
the photon. Therefore, for an unpolarized initial state, the total decay
probability reads

$dP_{fi}^{\rm dec}(\mathbf{k})=dP_{fi}^{\rm dec}(\mathbf{k},1)+dP_{fi}^{\rm
dec}(\mathbf{k},2)$ (72)

with

$dP_{fi}^{\rm
dec}(\mathbf{k},\lambda)=\frac{1}{3}d^{3}\mathbf{k}\sum_{l=1}^{3}|T_{fi}^{\rm
dec}(\mathbf{k},l,\lambda)|^{2}\,.$ (73)

Note that, when computing $|T_{fi}^{\rm dec}(\mathbf{k},l,\lambda)|^{2}$,
there will be a contribution of zeroth order that gives the emission
probability for a static atom. By energy conservation, this probability is
proportional to $\delta(\Omega-k)$. It is corrected by the second order
contribution to the matrix element. On the other hand, the first order
contribution to the transition amplitude produces an emission probability
that, for a center of mass oscillation with frequency $\Omega_{\rm cm}$, has
lateral peaks at $k=\Omega\pm\Omega_{\rm cm}$. This is the main qualitative
change induced by the center of mass motion on the spectrum of emitted
photons.

We now assume a normal motion for the center of mass of the atom, with
$\tilde{y}_{j}=\tilde{y}_{\perp}\delta_{j3}$. The zeroth order contribution
$T_{fi}^{(dec,0)}$ in Eq.(70) generates the spontaneous emission probability
for a static atom. It reads

$dP_{fi}^{(\rm
dec,0)}(\mathbf{k})=\frac{Te^{2}N_{k}^{2}}{2m\Omega}2\pi\delta(\Omega-k)k^{2}\sum_{\lambda}|{\mathbf{g}}^{(\lambda)}_{\mathbf{k}}(\mathbf{r}_{0})|^{2}\,d^{3}\mathbf{k}\,.$
(74)

The second order contribution in Eq.(70), when multiplied by the zeroth order,
produces a correction to the static probability in Eq.(74) that is given by

$\displaystyle dP_{fi}^{(\rm sta,2)}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}N_{k}^{2}}{2m\Omega}2\pi\delta(\Omega-k)k^{2}k_{z}^{2}\sum_{\lambda}|{\mathbf{g}}^{(\lambda)}_{\mathbf{k}}(\mathbf{r}_{0})|^{2}$
(75)
$\displaystyle\times\left(\int_{-\infty}^{\infty}dt\,y_{\perp}^{2}(t)\right)\,d^{3}\mathbf{k}\,.$

Note that the position of the peak is independent of the center of mass motion
of the atom.

We now consider the novel contribution to the emission probability coming from
$T_{fi}^{(dec,1)}$ in Eq.(70). It is given by

$\displaystyle dP_{fi}^{\rm(dyn,2)}(\mathbf{k},1)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{12\pi^{2}m\Omega}\,|\tilde{y}_{\perp}(k-\Omega)|^{2}\Omega^{2}k^{3}\cos^{2}\left(ka\cos\theta)\right)\cos^{2}\theta\sin\theta\,d\theta\,dk$
(76) $\displaystyle\equiv$
$\displaystyle\frac{e^{2}}{12\pi^{2}m\Omega}\,|\tilde{y}_{\perp}(k-\Omega)|^{2}k^{3}\,p_{1}(ka,\Omega
a,\theta)\,\sin\theta\,d\theta\,dk$

and

$\displaystyle dP_{fi}^{\rm(dyn,2)}(\mathbf{k},2)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{12\pi^{2}m\Omega}\,|\tilde{y}_{\perp}(k-\Omega)|^{2}k^{3}[k^{2}\sin^{2}\theta\cos^{2}\theta\sin^{2}\left(ka\cos\theta\right)$
(77) $\displaystyle+$
$\displaystyle(\Omega-k\sin^{2}\theta)^{2}\cos^{2}\left(ka\cos\theta\right)]\sin\theta\,d\theta\,dk$
$\displaystyle\equiv$
$\displaystyle\frac{e^{2}}{12\pi^{2}m\Omega}\,|\tilde{y}_{\perp}(k-\Omega)|^{2}k^{3}\,p_{2}(ka,\Omega
a,\theta)\,\sin\theta\,d\theta\,dk.$

In both equations we have used spherical coordinates in $\mathbf{k}$-space and
integrated over the angle $\varphi$ (by symmetry, the results do not depend on
this angle). As the correction in Eq.(75), these are of course contributions
quadratic in the amplitude of the center of mass motion. In Fig. 4 we plot the
total contribution to the emission probability $p_{1}(ka,\Omega
a,\theta)+p_{2}(ka,\Omega a,\theta)$ per unit of solid angle for two different
values of $ka$ at a fixed value of $\Omega a=10$.

(a) $ka=9,\Omega a=10$

(b) $ka=11,\Omega a=10$

Figure 4: Total contribution to the emission probability $p_{1}(ka,\Omega
a,\theta)+p_{2}(ka,\Omega a,\theta)$ per unit of solid angle as a function of
spherical angle $\theta$. We show two different values of $ka$ at a fixed
value of $\Omega a=10$.

The total decay probability can be obtained by summing Eqs.(76) and(77) and
integrating the $\theta$ angle. The result is

$\displaystyle\frac{dP_{fi}^{\rm(dyn,2)}}{dk}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{144\pi^{2}m\Omega
a^{5}}\,|\tilde{y}_{\perp}(k-\Omega)|^{2}\Big{\\{}8a^{5}k^{3}\left(k^{2}-2k\Omega+2\Omega^{2}\right)$
$\displaystyle+6ak\left(a^{2}(k+\Omega)^{2}-6\right)\cos(2ak)+3\left(4a^{4}k^{2}\Omega^{2}-a^{2}\left(9k^{2}+2k\Omega+\Omega^{2}\right)+6\right)\sin(2ak)\Big{\\}}\,.$

If we assume that the normal displacement $y_{\perp}(t)$ is an oscillatory
function $y_{\perp}(t)=y_{\perp}^{0}\sin(\Omega_{\rm cm}t)$, where
$\Omega_{\rm cm}$ is the frequency of the center of mass, the spectrum of the
emitted photons has peaks at $\Omega,\Omega\pm\Omega_{\rm cm}$. In Fig.4 we
choose two different values of $ka$ ($ka=9$ in Fig.4(a) and $ka=11$ in
Fig.4(b)) in order to show that just beyond adiabatic approximation
($ka\approx\Omega a$) the emission probability is non-symmetric with respect
to the central emission peak.

In a recent work Passante , the decay probability of an atom in front of an
oscillating mirror has been computed using the adiabatic approximation
$\Omega\gg\Omega_{\rm cm}$. In this limit, our results for the moving atom
have the same structure: the spectrum of the emitted photons has the above
mentioned peaks, the decay probability decreases with the distance to the
plane, and it shows oscillations with a frequency $2ka\simeq 2\Omega a$. There
is a disagreement, however, between the coefficients of the terms appearing in
our result and the ones of Passante . We have verified that this difference
comes from the Röntgen term. Indeed, omitting this contribution both results
coincide. For the case of a moving atom, it is well known that this
interaction term is crucial for Lorentz covariance. It would be interesting to
check if it also appears for the case of a moving mirror and static atom, in
the next to leading order of the adiabatic approximation. Note that the
boundary conditions for the electromagnetic field on a moving perfect mirror
has a velocity-dependent term PMN96 .

The adiabatic approximation is justified when $\Omega\gg\Omega_{\rm cm}$. It
is noteworthy that, for some physical systems, this inequality may be
violated: for Rydberg or artificial atoms may have $\Omega$ of the order of
GHz, and mechanical resonators may attain such frequencies. In this situation,
the decay probabilities may be qualitatively different from those obtained in
the adiabatic approximation.

###  IV.2 Excitation process

We now consider the probability of excitation of an atom that is initially in
its ground state. This excitation is accompanied by the emission of a photon.
The initial states read

$|i_{a}\rangle=|0_{a}\rangle\quad\quad\,|i_{EM}\rangle=|0_{EM}\rangle,$ (79)

while final states are

$|f_{a}\rangle=a_{l}^{\dagger}|0_{a}\rangle\quad\quad\,|f_{EM}\rangle={\alpha}^{\dagger}_{\lambda}({\mathbf{k}})|0_{EM}\rangle.$
(80)

It is not necessary to repeat all calculations. Indeed, the matrix elements
for the excitation process can be obtained from those of the decay process
just changing the sign of the frequency $\Omega$, that takes into account the
changes in the initial and final states of the atom. This change of the sign
produces the expected threshold for the center of mass frequency, and the
excitation occurs only above it. Therefore, the zeroth order in the transition
amplitudes is absent for this process.

From Eqs.(76) and (77) we obtain

$dP_{fi}^{\rm
exc}(\mathbf{k},1)=\frac{e^{2}}{12\pi^{2}m\Omega}\,|\tilde{y}_{\perp}(k+\Omega)|^{2}\Omega^{2}k^{3}\cos^{2}\left(ka\cos\theta)\right)\cos^{2}\theta\sin\theta\,d\theta\,dk\,$
(81)

and

$\displaystyle dP_{fi}^{\rm exc}(\mathbf{k},2)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{12\pi^{2}m\Omega}\,|\tilde{y}_{\perp}(k+\Omega)|^{2}k^{3}[k^{2}\sin^{2}\theta\cos^{2}\theta\sin^{2}\left(ka\cos\theta\right)$
(82) $\displaystyle+$
$\displaystyle(\Omega+k\sin^{2}\theta)^{2}\cos^{2}\left(ka\cos\theta\right)]\sin\theta\,d\theta\,dk\,$

As before, in both equations we have used spherical coordinates in
$\mathbf{k}$-space and integrated over the angle $\varphi$.

##  V Conclusions

In this paper we considered the interaction between an accelerated atom near a
perfect mirror and the vacuum fluctuations of the electromagnetic field. We
first computed the vacuum persistence probability, and then the probabilities
for excitation and decay, for an atom that is initially in its ground or first
excited state, respectively. The results generalize our previous work in which
we studied, as a toy model, a quantum scalar field instead of the full
electromagnetic field.

We have compared our results for an atom in perpendicular motion with respect
to the mirror, with those in which the atom is at rest and the mirror is
oscillating. Up to the lowest order adiabatic approximation, the Röntgen
current does not appear for a moving mirror Passante , and this is a source of
discrepancy between the results for both situations. It would be interesting
to check whether the next to leading order adiabatic correction for the case
of a moving mirror restores the equivalence between these two different
physical situations or not.

Our results for the moving atom are valid beyond the adiabatic approximation,
and we have pointed out that, for artificial or Rydberg atoms, this
approximation may be violated. Therefore, one could observe signs of non-
adiabaticity in the spectrum of emitted particles

If the atom has a center of mass motion parallel to the mirror, the excitation
and de-excitation will depend both on the acceleration and the distance to the
mirror. Although in this paper we have not presented an analysis of the
transition amplitudes for the parallel motion, the presence of dissipative
effects is clear from the computation of the imaginary part of the effective
action. These effects have no analogs for a static atom in front of a moving
(perfect) mirror.

## Acknowledgments

This research was supported by Agencia Nacional de Promoción Científica y
Tecnológica (ANPCyT), Consejo Nacional de Investigaciones Científicas y
Técnicas (CONICET), Universidad de Buenos Aires (UBA) and Universidad Nacional
de Cuyo (UNCuyo). The research of F.C.L. was supported in part by the National
Science Foundation under Grant No. PHY-1748958.

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