# Lamb shift as a witness for quantum noninertial effects

Navdeep Arya<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education & Research (IISER) Mohali, Sector 81 SAS Nagar,
Manauli PO 140306 Punjab India.  Sandeep K. Goyal<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of Science Education &
Research (IISER) Mohali, Sector 81 SAS Nagar, Manauli PO 140306 Punjab India.

###### Abstract

The sustained intense experimental activity around atomic spectroscopy and the
resulting high-precision measurements of atomic spectral lines attracts
interest in Lamb shift as a witness for noninertial effects in quantum
systems. We investigate the Lamb shift in a two-level system, undergoing
uniform circular motion, coupled to a quantum electromagnetic field inside a
cavity. We show that when the separation between different cavity modes is
large compared to the width of each cavity mode, both the inertial and purely-
noninertial contributions to the Lamb shift are convergent. In addition, we
find that the purely-noninertial Lamb shift maximizes away from the atomic
resonance by an amount decided by the angular frequency of the circulating
atom, lending itself to efficient enhancement by suitably tuning the cavity
parameters. We argue that the purely-noninertial contribution becomes
detectable at accelerations $\sim 10^{14}~{}\mathrm{m/s^{2}}$.

††preprint: APS/123-QED

##  I Introduction

In several studies, the noninertial motion has been shown to modify various
properties of quantum systems, ranging from thermal signature in transition
rates [1] and acceleration-induced transparency [2] to potentially detectable
noninertial contributions to the geometric phase [3, 4]. Moreover, interesting
results have been obtained concerning the relationship between entanglement
and noninertial motion [5, *Alsing2006, 7, 8, 9, 10, 11, 12, 13], for example,
rotation can lead to the generation of entanglement [13]. These studies extend
our understanding of quantum physics beyond the well-understood domain of
inertial reference frames into the domain of noninertial reference frames,
thus laying the groundwork for ultimately investigating novel phenomena at the
interface of quantum physics and gravity [14].

Atoms, in such studies, are usually modeled as two-level systems coupling
locally to a quantum field [1, 15]. The response of such systems is controlled
by the field correlation functions. The field correlators perceived by the
atom are sensitive to its state of motion [16, 1, 17, 18, 19] or the presence
of gravity, leading to a distinct noninertial or gravitational [20, 21]
contribution to the atomic response. The resulting effects are usually very
weak and require extreme acceleration or gravitational field for a detectable
signature.

Different studies [22, 23, 24, 25, 26, 27, 28, *Barshay1980, *Kharzeev2006,
31, *bell1987, *unruh1998, 34, 35, 36, 37] have investigated various
properties of noninertial quantum systems under varied conditions, seeking
appreciable noninertial signatures and ease of measurement in laboratory
settings. An observable of interest in this context is the Lamb shift, or the
radiative energy shift in general (though in this work we will use the two
terms interchangeably).

In this work, we are interested in the correction to the radiative energy
shifts originating from the atom’s noninertial motion. The total radiative
energy shift in an atom on a noninertial trajectory has two contributions:
inertial and purely-noninertial. The additional purely-noninertial
contribution comes due to the acceleration of the atom. The atom is assumed to
be coupled to a quantum electromagnetic (EM) field inside a cavity. In
particular, we are interested in the effect of the modified density of field
states inside the cavity on the inertial and purely-noninertial contributions
to the Lamb shift.

The Lamb shift is a shift in the energy levels of an atom due to the atomic
electron’s coupling to a quantum electromagnetic field [38, 39]. The Lamb
shift in inertial atoms has been measured with great precision using different
experimental methods [40, 41, 42, 43]. As already mentioned, the response of
atoms coupled to a quantum field is determined by the field correlators, which
depend on the atom’s trajectory and therefore lead to a noninertial signature
in the radiative energy shifts. The energy shift can be an observable of
interest for the detection of the effects of acceleration [44, 45, 46] and
gravity [47, 48, 49, 50, 51] owing to the theoretical and experimental
advances that atomic spectroscopy has made [52, 53, 54, 55, 43, 56].
Particularly, transitions in Hydrogen atom have been measured with precision
in the $10^{-11}-10^{-12}$ range for optical transitions and $10^{-5}-10^{-6}$
range for microwave transitions [52, 57]. Such precise measurements of
transition frequencies enable the determination of various corrections to the
spectral lines, including those coming from the atomic electron’s interaction
with a quantum electromagnetic field.

The Lamb shift in atomic systems is predominantly a nonrelativistic phenomenon
in the sense that a major contribution to the Lamb shift comes from the atomic
electron’s coupling to the field modes with energy less than the electron’s
rest mass energy [58, 59]. Therefore, we will focus on such a nonrelativistic
treatment. The Lamb shift in a two-level atom on a stationary worldline [60],
coupled to a quantum scalar field in free space, has a logarithmically
divergent inertial contribution and a finite correction coming due to the
atom’s noninertial motion [44, 45]. The logarithmically divergent inertial
contribution necessitates the introduction of an ultraviolet cutoff [59]. Such
cutoffs employed in the nonrelativistic calculations of the radiative
corrections to matter properties lead to cutoff-sensitive results. See Ref.
[61] for a typical example related to the calculation of anomalous magnetic
moment of an electron.

The quantum electrodynamical properties of an atom interacting with an EM
field get modified if the density of field states is changed, for example, by
introducing conducting mirrors [62, *Barton1979, *Barton1987]. The EM cavities
have been fruitfully studied in the context of amplification and isolation of
the noninertial quantum field theoretic effects [65, 4, 26]. The radiative
energy shift, in particular, depends sensitively on the density of field modes
[66]. Using an EM cavity, the transition rates of an atom can be either
amplified or inhibited [67]. Therefore, as the Lamb shift arises due to the
absorption and emission of virtual photons by the atomic electron [68], better
control over the energy shifts can be obtained using an EM cavity [69]. The
radiative energy shifts in an inertial atom in the presence of a mirror [70],
in an atom placed between parallel metal plates [71], and in an atom placed
inside a confocal resonator [72] have been measured in various experiments
which, if the purely-noninertial contribution is appreciable, can conceivably
be extended to atoms on noninertial trajectories.

Here, we study the Lamb shift in an atom undergoing uniform circular
acceleration and coupled to an electromagnetic field inside a cavity.
Specifically, we focus on the behavior of the inertial and noninertial
contributions to the total Lamb shift as a function of the cavity’s normal
frequency and discuss the detectability of the noninertial contribution. We
show that when the separation between different cavity modes is large as
compared to the width of each cavity mode, both the inertial and purely-
noninertial contributions to the Lamb shift are convergent, leading to cutoff-
independent results. In addition, we find that the purely-noninertial Lamb
shift maximizes away from the atomic resonance by an amount decided by the
angular frequency of the circulating atom, lending itself to efficient
enhancement by suitably tuning the cavity parameters. We argue that the
purely-noninertial contribution can be detected at accelerations $\sim
10^{14}~{}\mathrm{m/s^{2}}$.

This paper is organized as follows. In Sec. II we discuss the Lamb shift in an
open quantum system from the perspective of the Lindblad master equation
formalism. In Sec. III, we detail the atom-cavity setup employed in this work.
In Sec. IV, we use the Lindblad master equation formalism to obtain the Lamb
shift in a two-level system on a circular trajectory inside an EM cavity.
Finally, in Sec. V we discuss the results and conclude with a discussion on
the outlook for the study carried out in this paper.

##  II Background : Lamb Shift

In this section, we discuss the Lamb shift in a small system $S$ due to its
coupling to a large reservoir $B$. The intrinsic dynamics of $S$ and $B$ are
governed by the Hamiltonians $H_{S}$ and $H_{B}$, respectively. The
Schrodinger picture interaction Hamiltonian between the system and the
reservoir can be written as

$H_{I}=\sum_{\alpha}A_{\alpha}\otimes\tilde{B}_{\alpha},$ (1)

where $A_{\alpha}=A^{\dagger}_{\alpha}$ and
$\tilde{B}_{\alpha}=\tilde{B}^{\dagger}_{\alpha}$ are the system and the
reservoir operators, respectively. The Lindblad (interaction picture) master
equation governing the dynamics of the system $S$ is given by [73]

$\derivative{\rho_{S}(\tau)}{\tau}=-\frac{i}{\hbar}\commutator{H_{\text{LS}}}{\rho_{S}(\tau)}+\mathcal{D}\left(\rho_{S}(\tau)\right),$
(2)

where $\tau$ is the proper time, $\rho_{S}(\tau)$ is the system’s density
operator and $\mathcal{D}(\rho_{S}(\tau))$ is called the dissipator of the
master equation as it controls the dissipation and decoherence in the system.
$H_{\text{LS}}$ is known as the Lamb shift Hamiltonian as it leads to a
renormalization of the unperturbed energy levels induced by the system-
reservoir coupling.

The Lamb shift Hamiltonian is given by

$H_{\text{LS}}=\sum_{\nu}\sum_{\alpha,\beta}\hbar
S_{\alpha\beta}(\nu)A_{\alpha}^{\dagger}(\nu)A_{\beta}(\nu),$ (3)

where

$S_{\alpha\beta}(\nu)\equiv\frac{1}{2\pi}\int_{-\infty}^{\infty}\differential{\lambda}\mathcal{G}_{\alpha\beta}(\lambda)\text{P.V.}\left(\frac{1}{\nu-\lambda}\right),$
(4)

with

$\mathcal{G}_{\alpha\beta}(\lambda)\equiv\frac{1}{\hbar^{2}}\int_{-\infty}^{\infty}\differential{\tau_{-}}e^{i\lambda\tau_{-}}\tr_{\text{B}}\left(\tilde{B}_{\alpha}(\tau_{2})\tilde{B}_{\beta}(\tau_{1})\right),$
(5)

$\tau_{-}\equiv\tau_{2}-\tau_{1}$, P.V. denoting the Cauchy Principal value
integral, and $\tr_{\text{B}}(\cdot)$ denoting the trace over the reservoir
degrees of freedom. Note that $\mathcal{G}_{\alpha\beta}(\lambda)$ is the
Fourier transform of the two-point reservoir correlation function. Further,
the $\sum_{\nu}$, where $\nu\equiv\varepsilon^{\prime}-\varepsilon$, is
extended over all eigenvalues $\varepsilon$ and $\varepsilon^{\prime}$ of
$H_{\text{S}}$ with a fixed energy difference $\nu$. The $A_{\beta}(\nu)$ are
the eigenoperators of the system Hamiltonian $H_{S}$ and are defined as
$A_{\alpha}(\nu)\equiv\sum_{\nu}\Pi(\varepsilon)A_{\alpha}\Pi(\varepsilon^{\prime})$,
where $\Pi(\varepsilon)$ is the projector on the eigenspace belonging to the
eigenvalue $\varepsilon$ [79, 73]. The $A_{\alpha}(\nu)$ are also known as the
Lindblad operators.

Now, consider $S$ to be a two-level atom with the excited state $\ket{e}$ and
the ground state $\ket{g}$, interacting with a quantum electromagnetic field
inside an electromagnetic cavity. The proper frequency gap between the two
atomic levels is $\Omega_{0}$ and the atom carries an electric dipole moment
four-vector $\hat{d}^{\prime\mu}=(\hat{d}^{\prime
0},\hat{\mathbf{d}}^{\prime})$. In the interaction picture, the dipole moment
operator $\hat{\mathbf{d}}^{\prime}(\tau)$ is given by
$\hat{\mathbf{d}}^{\prime}(\tau)=\mathbf{d}^{\prime}\sigma_{-}\exp(-i\Omega_{0}\tau)+\mathbf{d}^{\prime*}\sigma_{+}\exp(i\Omega_{0}\tau)$,
where
$\mathbf{d}^{\prime}\equiv\bra{g}\hat{\mathbf{d}}^{\prime}(\tau=0)\ket{e}$,
and $\sigma_{+}=\sigma^{\dagger}_{-}=\ket{e}\bra{g}$ is the step-up operator
for the atomic states. The Lindblad operators for $S$ are given by [73]

$\mathbf{A}(\Omega_{0})=\mathbf{d}^{\prime}\sigma_{-},~{}\mathbf{A}(-\Omega_{0})=\mathbf{d}^{\prime*}\sigma_{+}.$
(6)

The electromagnetic field is assumed to be in the inertial vacuum state
$\ket{0}$. The interaction Hamiltonian between the atom and the
electromagnetic field is given by $H_{I}=-\hat{d}^{\mu}E_{\mu}$ [74], where
$E_{\mu}\equiv F_{\mu\nu}u^{\nu}$, $F_{\mu\nu}$ is the electromagnetic field
strength tensor and $u^{\nu}$ is the four-velocity of the atom. The
interaction Hamiltonian takes the form
$H_{I}=-\hat{\mathbf{d}}^{\prime}\cdot\mathbf{E^{\prime}}$ in the rest frame
of the atom, where $\mathbf{E}^{\prime}$ is the electric field 3-vector as
seen by the atom. Throughout this paper, primed quantities correspond to the
atom’s rest frame.

From Eqs. (3) and (6), for the two-level system we have

$H_{\text{LS}}=\hbar\sum_{\alpha,\beta}\Big{(}S_{\alpha\beta}(\Omega_{0})(d^{\prime}_{\alpha}\sigma_{-})^{\dagger}d^{\prime}_{\beta}\sigma_{-}\\\
+S_{\alpha\beta}(-\Omega_{0})(d^{\prime*}_{\alpha}\sigma_{+})^{\dagger}d^{\prime*}_{\beta}\sigma_{+}\Big{)}.$
(7)

For simplicity, we assume that $\mathbf{d}^{\prime}=(0,d^{\prime},0)$, and
obtain

$H_{\text{LS}}=\hbar\absolutevalue{\mathbf{d}^{\prime}}^{2}\left(S_{22}(\Omega_{0})\sigma_{+}\sigma_{-}+S_{22}(-\Omega_{0})\sigma_{-}\sigma_{+}\right).$
(8)

As is clear from the form of $H_{\mathrm{LS}}$ in above equation, it induces
transitions between the two atomic levels mediated by virtual photons. The
Lamb shift, $\Delta$, is obtained as

$\begin{split}\hbar\Delta&\equiv\expectationvalue{H_{\text{LS}}}{e}-\expectationvalue{H_{\text{LS}}}{g}\\\
&=\hbar\absolutevalue{\mathbf{d}^{\prime}}^{2}\left(S_{22}(\Omega_{0})-S_{22}(-\Omega_{0})\right),\end{split}$
(9)

that is,

$\Delta=\frac{\absolutevalue{\mathbf{d}^{\prime}}^{2}}{2\pi}\int_{-\infty}^{\infty}\differential{\nu^{\prime}}\mathcal{G}^{\prime}_{22}(\nu^{\prime})\text{P.V.}\left(\frac{1}{\nu^{\prime}+\Omega_{0}}-\frac{1}{\nu^{\prime}-\Omega_{0}}\right),$
(10)

where

$\mathcal{G}^{\prime}_{22}(\nu^{\prime})=\frac{1}{\hbar^{2}}\int_{-\infty}^{\infty}\differential{\tau_{-}}e^{i\nu^{\prime}\tau_{-}}G^{\prime+}_{22}(\tau_{-}),$
(11)

with
$G^{\prime+}_{22}(\tau_{-})\equiv\expectationvalue{E^{\prime}_{y}(\tau_{2})E^{\prime}_{y}(\tau_{1})}{0}$
being the positive frequency vacuum Wightman function.

##  III Atom-cavity setup

In this section, we discuss the atom-cavity setup employed in this study. The
atomic transition rates depend on the field spectral density in the resonant
mode, that is, in the mode with frequency $\Omega_{0}$. From Eq. (10),
however, note that since P.V.$\left(1/(\nu\pm\Omega_{0})\right)$ vanishes for
$\nu=\pm\Omega_{0}$ and behaves as $1/(\nu\pm\Omega_{0})$ away from
$\nu=\pm\Omega_{0}$, the radiative energy shift depends on the field spectral
density in all the modes, except the resonant mode, with a weight that falls
off away from the resonant mode [75]. If we consider a cavity such that the
frequency separation between different cavity modes is large as compared to
the width of each cavity mode, then due to the presence of the factor
P.V.$\left(1/(\nu\pm\Omega_{0})\right)$ in Eq. (10) the dominant contribution
to the energy shift will come from the modes in the vicinity of the cavity’s
normal frequency $\omega_{\rm c}$, as compared to the contribution of the
higher frequencies supported by the cavity at $n\omega_{\rm c},n>1$. We assume
that the cavity modes are separated in frequency by much more than their width
(i.e., a cavity with a high quality-factor). If the inertial rate at which the
radiation deposited in the cavity is damped is $\omega_{c}/Q$, where $Q$ is
the quality factor of the cavity, then the density of field states inside the
cavity is given by [76, *Lewenstein1993, 72]

$\rho(\omega_{k})=\frac{1}{\pi}\frac{(\omega_{c}/Q)}{(\omega_{c}/Q)^{2}+(\omega_{k}-\omega_{c})^{2}}.$
(12)

Further, to ensure the validity of the Markovian approximation [73], which is
inherent in the derivation of the master equation (2), we will work in the
bad-cavity regime, that is,

$g\ll\kappa,$ (13)

where, for a cavity of volume $V$,
$g\equiv\absolutevalue{\mathbf{d}^{\prime}}\sqrt{\omega_{c}/(2\hbar\epsilon_{0}V)}$
is the atom-cavity field coupling constant and $\kappa\equiv\omega_{c}/Q$ is
the cavity-field decay rate [78]. For a given atom and a given normal
frequency $\omega_{c}$ of the cavity, the condition $g\ll\kappa$ will decide
the allowed values of $V$ and $Q$ consistent with the Markovian approximation.
See the caption of Fig. (1).

The variation of P.V.$\left(1/(\nu\pm\Omega_{0})\right)$ and the density of
field modes $\rho(\nu)$, entering Eq. (10) through the field correlation
function, decide the magnitude of the radiative energy shift.

##  IV Results

Figure 1: Fig. (1) plots $\Delta_{0}/\eta$ as a function of the cavity’s
normal frequency [See Eq. (29)], where we have written
$\omega_{c}=10^{7}+\epsilon$. We see that the Lamb shift in an inertial atom
maximizes when the cavity is tuned to a frequency in the vicinity of the
atomic resonance and falls off on both sides of this region. Since we are
considering a two-level system with $\Omega_{0}=10^{7}~{}\mathrm{Hz}$, we take
$\omega_{c}\sim 10^{7}~{}\mathrm{Hz}$. For a cavity with $Q\sim 10^{7}$, we
take $V\sim 10^{-5}~{}\mathrm{m^{3}}$ to ensure consistency with Eq. (13).
Therefore $\eta\sim 10^{-9}~{}\mathrm{Hz}$
($\eta\equiv\absolutevalue{\mathbf{d}^{\prime}}^{2}/3\pi\hbar\epsilon_{0}V$),
which leads to $\Delta_{0}\sim 10^{-3}~{}\mathrm{Hz}$. For an atom on a
circular orbit with angular velocity $\omega$, however, there are additional
peaks in the neighborhood of $\omega_{c}=\omega\pm\bar{\Omega}_{0}$. See Fig.
2 as well. Fig. (1) gives a plot of $\Delta_{\omega}/\eta$, defined in Eq.
(31), as a function of the cavity’s normal frequency for $\omega=5\times
10^{11}~{}\mathrm{Hz}$. In this case, $\omega_{c}\sim\omega\pm\Omega_{0}$ and
$\omega\gg\Omega_{0}$. Therefore, for $Q\sim 10^{7}$, we take $V\sim
10^{-9}~{}\mathrm{m^{3}}$, and $R=10^{-5}~{}\mathrm{m}$. This corresponds to
$\eta\sim 10^{-5}~{}\mathrm{Hz}$, and an average acceleration
$a=\omega^{2}R\sim 10^{18}~{}\mathrm{m/s^{2}}$ and leads to a purely-
noninertial Lamb shift $\Delta_{\omega}\sim 10^{-3}~{}\mathrm{Hz}$. Such
accelerations can be achieved with electrons in storage rings [31].

We now consider a two-level atom on a circular trajectory of angular frequency
$\omega$ and radius $R$ inside an electromagnetic cavity of volume $V$.

In the following subsection, we determine the Fourier transform of the two-
point vacuum Wightman function as given in Eq. (11) and then use Eq. (10) to
obtain the Lamb shift in the subsequent subsection.

###  IV.1 Fourier transform of the field correlation function

We consider an atom on a circular trajectory of radius $R$ and angular
frequency $\omega$ such that its position four-vector in the lab frame is
given by

$\begin{split}&x^{\mu}(\tau)=(t(\tau),x(\tau),y(\tau),z(\tau))\\\
&=\left(c\gamma\tau,x_{0}+R\cos(\omega\gamma\tau),0,z_{0}+R\sin(\omega\gamma\tau)\right),\end{split}$
(14)

where $\gamma\equiv(1-\zeta(\omega))^{-1/2}$, and
$\zeta(\omega)\equiv\omega^{2}R^{2}/c^{2}$. To compute
$\mathcal{G}^{\prime}_{22}(\nu)$, we need the positive-frequency vacuum
Wightman function $G^{\prime+}_{22}(\tau_{-})$ in the atom’s frame. We start
by noting that
$G^{\prime+}_{\mu\nu}\equiv\expectationvalue{E^{\prime}_{\mu}(\tau_{2})E^{\prime}_{\nu}(\tau_{1})}{0}$
can be obtained from its counterpart
$G^{+}_{\mu\nu}\equiv\expectationvalue{E_{\mu}(x^{\lambda}_{2})E_{\nu}(x^{\lambda}_{1})}{0}$
in the lab frame using the tensor transformation between the two frames.
Consequently, we have

$G^{\prime+}_{22}=\sum_{\alpha\beta}\partialderivative{x^{\alpha}}{x^{\prime
2}}\partialderivative{x^{\beta}}{x^{\prime 2}}G^{+}_{\alpha\beta}=G^{+}_{22},$
(15)

where we have used the fact that the rotating
$(\tau,x^{\prime},y^{\prime},z^{\prime})$ and the inertial coordinates
($t,x,y,z$) are related as

$\displaystyle x^{\prime}$ $\displaystyle=x-x_{0}-R\cos\omega t,$
$\displaystyle y^{\prime}$ $\displaystyle=y,$ $\displaystyle z^{\prime}$
$\displaystyle=z-z_{0}-R\sin\omega t,$ $\displaystyle\tau$
$\displaystyle=\left(1-\omega^{2}R^{2}/c^{2}\right)^{1/2}t,$ (16)

with $(x_{0},0,z_{0})$ being the center of the circular trajectory.

The electric field perceived by the atom and as reported by the inertial
observer is given by $E_{\mu}(x^{\lambda})=F_{\mu\nu}(x^{\lambda})u^{\nu}$,
where $u^{\nu}\equiv\derivative*{x^{\nu}(\tau)}{\tau}$ is the atomic four-
velocity in the lab frame. Therefore, using Eq. (14) we have

$\begin{split}E_{2}=\gamma\left[E_{y}-\omega
R\left\\{B_{z}\sin(\omega\gamma\tau)-B_{x}\cos(\omega\gamma\tau)\right\\}\right],\end{split}$
(17)

which we will use for the computation of $G^{+}_{22}$ below. The calculation
of $\mathcal{G}^{\prime}_{22}(\nu)$ can be further simplified by noting that
under the Markovian approximation, we have the identity [79]

$\int_{-\infty}^{\infty}\differential{u}e^{i\omega
u}\expectationvalue{E_{j}(u)E_{k}(0)}{0}\\\
=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\differential{\tau_{2}}\int_{0}^{T}\differential{\tau_{1}}e^{i\omega(\tau_{2}-\tau_{1})}\expectationvalue{E_{j}(\tau_{2})E_{k}(\tau_{1})}{0},$
(18)

where $j,k=1,2,3$. Using Eq. (18) in the atom’s comoving frame we can write

$\begin{split}&\mathcal{G}^{\prime}_{22}(\nu)=\frac{1}{\hbar^{2}}\int_{-\infty}^{\infty}\differential{\tau_{-}}e^{i\nu\tau_{-}}G^{\prime+}_{22}(\tau_{-})\\\
&=\lim_{T\rightarrow\infty}\frac{1}{\hbar^{2}T}\int_{0}^{T}\differential{\tau_{2}}\int_{0}^{T}\differential{\tau_{1}}e^{i\nu(\tau_{2}-\tau_{1})}\expectationvalue{E^{\prime}_{2}(\tau_{2})E^{\prime}_{2}(\tau_{1})}{0},\end{split}$
(19)

and further using Eq. (15) and $\differential{\tau}=\differential{t}/\gamma$,
we have

$\begin{split}\mathcal{G}^{\prime}_{22}(\nu)&=\lim_{T\rightarrow\infty}\frac{1}{\gamma^{2}\hbar^{2}T}\int_{0}^{\gamma
T}\differential{t_{2}}\int_{0}^{\gamma
T}\differential{t_{1}}e^{i\bar{\nu}(t_{2}-t_{1})}\\\
&\times\expectationvalue{E_{2}(x^{\mu}_{2})E_{2}(x^{\mu}_{1})}{0},\end{split}$
(20)

where $\bar{\nu}\equiv\nu/\gamma$.

As shown in Appendix A, form Eq. (20) we can obtain

$\mathcal{G}^{\prime}_{22}(\bar{\nu})=\frac{\gamma}{\hbar^{2}}\int_{-\infty}^{\infty}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\Big{[}\bra{0}E^{y}(x^{\mu}_{2})E^{y}(x^{\mu}_{1})\ket{0}\\\
+\frac{R^{2}\omega^{2}}{2}\cos(\omega
t_{-})\bra{0}B^{z}(x^{\mu}_{2})B^{z}(x^{\mu}_{1})\ket{0}\\\
+\frac{R^{2}\omega^{2}}{2}\cos(\omega
t_{-})\bra{0}B^{x}(x^{\mu}_{2})B^{x}(x^{\mu}_{1})\ket{0}\Big{]},$ (21)

where we have also explicitly indicated the dependence of
$\mathcal{G}^{\prime}_{22}$ on $\bar{\nu}$.

The inertial frame electric and magnetic field operators in Coulomb gauge
inside a quantization volume $V$ are given by [80]

$\mathbf{E}[x(\tau)]=i\sum_{\mathbf{k},\lambda}\mathcal{E}_{k}\epsilon_{\mathbf{k},\lambda}\left(a_{\mathbf{k},\lambda}e^{-i\left(\omega_{k}t(\tau)-\mathbf{k}.\mathbf{x}(\tau)\right)}-\text{h.c.}\right),$
(22a)
$\mathbf{B}[x(\tau)]=\frac{i}{c}\sum_{\mathbf{k}\lambda}\mathcal{E}_{k}\left(\hat{\mathbf{k}}\times\epsilon_{\mathbf{k}\lambda}\right)\left(a_{\mathbf{k}\lambda}e^{-i(\omega_{k}t(\tau)-\mathbf{k\cdot
x(\tau)})}-\text{h.c.}\right),$ (22b)

respectively, where
$\mathcal{E}_{k}\equiv\sqrt{\hbar\omega_{k}/(2\epsilon_{0}V)}$;
$\epsilon_{\mathbf{k},\lambda}$ with $\lambda=1,2$ are the two orthogonal
polarization vectors;
$[a_{\mathbf{k}\lambda},a^{\dagger}_{\mathbf{k}^{\prime}\lambda^{\prime}}]=\delta_{\mathbf{kk^{\prime}}}\delta_{\lambda\lambda^{\prime}}$;
and h.c. denotes Hermitian conjugate.

Using Eqs. (22) we can evaluate the different field correlators [65] appearing
in Eq. (21) and, to first order in $\zeta(\omega)$, obtain

$\mathcal{G}^{\prime}_{22}(\bar{\nu})=\frac{\gamma}{3\pi\hbar\epsilon_{0}V}\int_{0}^{\infty}\differential{\omega_{k}}\rho(\omega_{k})\omega_{k}\Big{[}\delta(\bar{\nu}-\omega_{k})\\\
+\frac{R^{2}\omega^{2}}{2c^{2}}\frac{1}{2}\left[\delta(\bar{\nu}-\omega_{k}+\omega)+\delta(\bar{\nu}-\omega_{k}-\omega)\right]-\frac{2\omega_{k}^{2}R^{2}}{5c^{2}}\times\\\
\Big{\\{}\delta(\bar{\nu}-\omega_{k})-\frac{1}{2}\left(\delta(\bar{\nu}+\omega-\omega_{k})+\delta(\bar{\nu}-\omega-\omega_{k})\right)\Big{\\}}\Big{]},$
(23)

where we have used

$\sum_{\mathbf{k}}\to\int\differential{\Omega_{\mathbf{k}}}\int\frac{\differential{k}\rho(k)}{(2\pi)^{3}},$
(24)

and $\rho(k)\differential{k}=\rho(\omega_{k})\differential{\omega_{k}}$, with
$\rho(\omega_{k})$ being the density of field modes inside the cavity as given
in Eq. (12). Also note that to obtain Eq. (23) we have assumed a closed
cavity. If we had assumed, for example, a concentric resonator instead of a
closed cavity, the angular integrals in the field correlators appearing in Eq.
(21) would have split into two parts, one with the free space field mode
density $\rho_{\text{free}}(\omega_{k})=\omega^{2}_{k}$, and the other with
the field mode density modified by the resonator [72]:

$\rho_{\text{cav}}(\omega_{k})=\begin{cases}\rho(\omega_{k}),&\mathrm{for~{}}\mathbf{k}\text{~{}in~{}}\Delta\Omega_{\text{cav}},\\\
\rho_{\text{free}}(\omega_{k}),&\mathrm{for~{}}\mathbf{k}\text{~{}in~{}}\Delta\Omega_{\text{free}}\end{cases},$
(25)

where $\rho(\omega_{k})$ is given by Eq. (12), and $\Delta\Omega_{\text{cav}}$
is the solid angle subtended by the resonator mirrors at the center of the
cavity. And, the continuum limit on the free space part would have been
obtained as
$(1/V)\sum_{\mathbf{k}}\to(2\pi)^{-3}\int\differential{\Omega_{\mathbf{k}}}\int\differential{\omega_{k}}\rho_{\mathrm{free}}(\omega_{k})$.

The qualitative features of interest to us, that is, the variation of the Lamb
shift with cavity detuning, will not change if we consider a concentric
resonator in place of the closed cavity. Therefore, for convenience, we work
with a closed cavity.

Evaluating the integral in Eq. (23), we obtain

$\begin{split}\mathcal{G}_{22}(\bar{\nu})&=\frac{\gamma}{3\pi\hbar\epsilon_{0}V}\Big{[}\rho(\bar{\nu})\bar{\nu}\Theta(\bar{\nu})\\\
&+\frac{\zeta(\omega)}{4}\Big{[}(\bar{\nu}+\omega)\rho(\bar{\nu}+\omega)\Theta(\bar{\nu}+\omega)\\\
&+(\bar{\nu}-\omega)\rho(\bar{\nu}-\omega)\Theta(\bar{\nu}-\omega)\Big{]}-\frac{2}{5}\Big{\\{}\frac{\bar{\nu}^{2}R^{2}}{c^{2}}\bar{\nu}\rho(\bar{\nu})\Theta(\bar{\nu})\\\
&-\frac{1}{2}\Big{(}\frac{(\bar{\nu}+\omega)^{2}R^{2}}{c^{2}}(\bar{\nu}+\omega)\rho(\bar{\nu}+\omega)\Theta(\bar{\nu}+\omega)\\\
&+\frac{(\bar{\nu}-\omega)^{2}R^{2}}{c^{2}}(\bar{\nu}-\omega)\rho(\bar{\nu}-\omega)\Theta(\bar{\nu}-\omega)\Big{)}\Big{\\}}\Big{]},\end{split}$
(26)

where $\Theta(x)$ is the Heaviside theta function.

Figure 2: The $\Delta_{\omega}/\eta$ versus $\omega_{\rm c}$ and
$\Delta_{\omega}/\Delta_{0}$ versus $\omega_{\rm c}$ plots for a two-level
system on a circular trajectory of $R=10^{-5}~{}\mathrm{m}$ with
$\omega=5\times 10^{9}~{}\mathrm{Hz}$ (for plots (a) and (b)) and
$\omega=5\times 10^{10}~{}\mathrm{Hz}$ (for plots (c) and (d)). For plots (a)
and (c), we have written $\omega_{\rm c}=\omega+\bar{\Omega}_{0}+\delta\times
10^{5}$. For values of cavity’s normal frequency in the vicinity of
$\omega\pm\bar{\Omega}_{0}$, large spikes in both $\Delta_{\omega}/\eta$ and
$\Delta_{\omega}/\Delta_{0}$ are recorded. For clarity, the plots show only
the spike at $\omega+\bar{\Omega}_{0}$ [see Fig. (1)]. The plots (a) and (b)
correspond to an average accelerations of $a=\omega^{2}R\sim
10^{14}~{}\mathrm{m/s^{2}}$, while the plots (c) and (d) correspond to an
average acceleration of $a=\omega^{2}R\sim 10^{16}~{}\mathrm{m/s^{2}}$. For
the plots we have taken $Q\sim 10^{7}$, and $V\sim 10^{-8}~{}\mathrm{m^{3}}$
which gives $\eta\sim 10^{-6}~{}\mathrm{Hz}$. See the discussion following Eq.
(13) on how to chose cavity parameters consistent with the bad-cavity regime.

###  IV.2 Lamb shift in the circulating atom

The Lamb shift in the circulating atom can be obtained by combining Eqs. (10)
and (26). To obtain the Lamb shift we first recast Eq. (10) using
$\bar{\nu}=\nu/\gamma$ and $\bar{\Omega}_{0}=\Omega_{0}/\gamma$, as

$\begin{split}\Delta&=\frac{\absolutevalue{\mathbf{d}^{\prime}}^{2}}{2\pi}\int_{-\infty}^{\infty}\differential{\bar{\nu}}\mathcal{G}_{22}(\bar{\nu})\mathrm{P.V.}\left(\frac{1}{\bar{\nu}+\bar{\Omega}_{0}}-\frac{1}{\bar{\nu}-\bar{\Omega}_{0}}\right),\end{split}$
(27)

which, using Eq. (26), leads to the total Lamb shift given by

$\Delta=\frac{\gamma\eta}{2\pi}\int_{-\infty}^{\infty}\differential{\bar{\nu}}\Bigg{[}\rho(\bar{\nu})\bar{\nu}\Theta(\bar{\nu})+\frac{\zeta(\omega)}{4}\left[(\bar{\nu}+\omega)\rho(\bar{\nu}+\omega)\Theta(\bar{\nu}+\omega)+(\bar{\nu}-\omega)\rho(\bar{\nu}-\omega)\Theta(\bar{\nu}-\omega)\right]\\\
-\frac{2}{5}\left\\{\zeta(\bar{\nu})\bar{\nu}\rho(\bar{\nu})\Theta(\bar{\nu})-\frac{1}{2}\left(\zeta(\bar{\nu}+\omega)(\bar{\nu}+\omega)\rho(\bar{\nu}+\omega)\Theta(\bar{\nu}+\omega)+\zeta(\bar{\nu}-\omega)(\bar{\nu}-\omega)\rho(\bar{\nu}-\omega)\Theta(\bar{\nu}-\omega)\right)\right\\}\Bigg{]}\\\
\times\mathrm{P.V.}\left(\frac{1}{\bar{\nu}+\bar{\Omega}_{0}}-\frac{1}{\bar{\nu}-\bar{\Omega}_{0}}\right),$
(28)

Figure 3: Plot (a) shows the decay rate
$\Gamma_{0}=\eta\rho(\Omega_{0})\Omega_{0}$ [65, 4], and Lamb shift
$\Delta_{0}$ [Eq. (29)] of an inertial atom with $\Omega_{0}=10$MHz, as a
function of the cavity’s normal frequency $\omega_{\rm
c}=\Omega_{0}+\epsilon$. The general behavior of the two quantities w.r.t
cavity detuning agrees well with experimental results [72]. Similarly, plot
(b) shows the purely-noninertial decay rate $\Gamma_{\omega}^{\downarrow}$
[Eq. (32)] and Lamb shift $\Delta_{\omega}$ [Eq. (31)] of a circulating atom
(with $\Omega_{0}=10$MHz and $\omega=50$GHz) as a function of the cavity’s
normal frequency $\omega_{\rm c}=\omega+\Omega_{0}+\delta\times 10^{5}$. In
the plots, a black dot marks the point at which the Lamb shift vanishes, and
orange dots mark the points at which the Lamb shift attains its minimum or
maximum value.

where
$\eta\equiv\absolutevalue{\mathbf{d}^{\prime}}^{2}/(3\pi\hbar\epsilon_{0}V)$,
$\zeta(\nu)\equiv\nu^{2}R^{2}/c^{2}$. Eq. (28) shows the dependence of the
Lamb shift on the atom’s acceleration $a=\omega^{2}R$ through the parameter
$\zeta(\omega)\equiv\omega^{2}R^{2}/c^{2}$. The parameter $\eta$ has the
dimensions of inverse time and can be expressed in terms of the atom-cavity
coupling constant
$g\equiv\absolutevalue{\mathbf{d}^{\prime}}\sqrt{\omega_{c}/(2\hbar\epsilon_{0}V)}$,
and cavity’s normal frequency $\omega_{\rm c}$ as $\eta=2g^{2}/3\pi\omega_{\rm
c}$. The volume dependence of $\Delta$ is entirely contained in $\eta$. To
obtain plots independent of a specific choice of the mode volume $V$ of the
cavity, we are plotting $\Delta_{0}/\eta$ and $\Delta_{\omega}/\eta$ as a
function of the cavity’s normal frequency $\omega_{\rm c}$. For different
values of the atom’s angular frequency, we take different values of $V$ for an
optimum signal while maintaining consistency with the bad-cavity regime
mentioned in Eq. (13). The captions of Figs. (1) and (2) mention the
corresponding values of $V$ and $\eta$. These $\eta$ values are then used to
determine the purely-noninertial and inertial contributions to the Lamb shift.

From Eq. (28), the Lamb shift of an inertial atom can be obtained in the limit
$\omega\rightarrow 0$. For the inertial Lamb shift we obtain

$\Delta_{0}=\frac{\eta}{2\pi}\int_{-\infty}^{\infty}\differential{\nu}\rho(\nu)\nu\Theta(\nu)\mathrm{P.V.}\left(\frac{1}{\nu+\Omega_{0}}-\frac{1}{\nu-\Omega}_{0}\right).$
(29)

The Cauchy Principal value integral can be evaluated exactly. For a high-$Q$
cavity, specifically, we obtain a simpler expression given by [see Appendix B]

$\Delta_{0}\approx-\eta\Omega_{0}\left[\frac{\omega_{c}(\Omega_{0}^{2}+\omega_{c}^{2})\log(\omega_{c}/\Omega_{0})}{2\pi^{2}Q\left(\omega_{c}^{2}-\Omega_{0}^{2}\right)^{2}}+\frac{\omega_{c}}{\pi\left(\omega_{c}^{2}-\Omega_{0}^{2}\right)}\right],$
(30)

for $\omega_{c}\neq\Omega_{0}$. As can be seen in Fig. (1), $\Delta_{0}$
maximizes when the cavity is tuned near, but not exactly at, the atomic
resonance $\Omega_{0}$. The purely-noninertial contribution to the Lamb shift
can be obtained as

$\Delta_{\omega}=\Delta-\Delta_{0}.$ (31)

The three integrals in the first line of Eq. (28) are manifestly convergent.
The three integrals in the second line, however, are individually
logarithmically divergent but the total Lamb shift is convergent as shown in
Appendix C. One can obtain a closed form expression for $\Delta_{\omega}$,
however, the expression is cumbersome without offering additional insights.
Therefore, we instead resort to the plots [Figs. (1) and (2)] to illustrate
the main features.

Note the presence of
$\zeta(\bar{\nu}\pm\omega)(\bar{\nu}\pm\omega)\rho(\bar{\nu}\pm\omega)\Theta(\bar{\nu}\pm\omega)$
terms in Eq. (28). The plots show that the purely-noninertial contribution to
the Lamb shift, in contrast with the inertial contribution, maximizes away
from the atomic resonance $\Omega_{0}$, in the vicinity of frequencies
$\omega\pm\bar{\Omega}_{0}$. This allows us to isolate and enhance the purely-
noninertial contribution relative to the inertial contribution by
appropriately tuning the cavity.

At this point, recall that the radiative shift to the energy levels of an atom
occurs through processes mediated by virtual photons. For these processes, the
state of the radiation field remains unchanged in the initial and final states
of the atom-field composite system, and changes only in the intermediate
composite state [68]. These processes complement the processes in which the
emission or absorption of a real photon is involved which lead to transitions
between different energy levels of the atom. Fig. 3(a) shows the emission rate
and Lamb shift in an inertial atom as a function of the cavity’s normal
frequency $\omega_{\rm c}$. An inertial two-level atom undergoes transitions
dominantly at $\Omega_{0}$, while the Lamb shift in its energy levels
maximizes when the cavity is tuned to a frequency in the vicinity of
$\Omega_{0}$ as shown in Fig. 1. As was reported and exploited in refs. [65,
4], a circulating two-level atom undergoes transitions at field frequencies
$\omega_{k}=\absolutevalue{\omega\pm\bar{\Omega}_{0}}$, in addition to the
transitions at the inertial resonant frequency $\Omega_{0}$. The purely-
noninertial decay rate of the circulating atom is given to the first order in
$\zeta(\omega)$ by [65, 4]

$\begin{split}\Gamma_{\omega}^{\downarrow}&=\frac{\eta\zeta(\omega)\Omega_{0}}{2}\Bigg{[}-\Omega_{0}\rho^{\prime}(\Omega_{0})+\frac{9}{10}\frac{\omega+\bar{\Omega}_{0}}{\Omega_{0}}\rho(\omega+\bar{\Omega}_{0})\Bigg{]}\end{split}$
(32)

where $\rho^{\prime}(\omega_{k})=\partial\rho/\partial\omega_{k}$. Fig. 3(b)
depicts $\Gamma^{\downarrow}_{\omega}$ and $\Delta_{\omega}$ as a function of
the cavity’s normal frequency in the neighborhood of
$\omega+\bar{\Omega}_{0}$.

Thus we see that for a microwave two-level system, weakly coupled to EM field
inside a bad-cavity, when the cavity is tuned at the atomic resonance, a
maximum inertial Lamb shift of the order of $10^{-3}~{}\mathrm{Hz}$ is
obtained. Further, by tuning the cavity in the vicinity of either
$\omega+\bar{\Omega}_{0}$ or $\omega-\bar{\Omega}_{0}$, the purely-noninertial
contribution can be made dominant [see Figs. (1) and (2)]. For example, at an
average acceleration $a=\omega^{2}R\sim 10^{14}~{}\mathrm{m/s^{2}}$, a
noninertial contribution $\Delta_{\omega}\sim 10^{-8}~{}\mathrm{Hz}$, and for
$a\sim 10^{16}~{}\mathrm{m/s^{2}}$, a purely-noninertial contribution
$\Delta_{\omega}\sim 10^{-6}~{}\mathrm{Hz}$ to the Lamb shift can be obtained.
Further, at accelerations achievable with electrons inside storage rings [31],
a purely-noninertial contribution as large as $\sim 10^{-3}~{}\mathrm{Hz}$ can
be obtained. The plots 2 and 2 show the enhancement
$\Delta_{\omega}/\Delta_{0}$ of the purely-noninertial contribution to the
Lamb shift over the inertial contribution. This enhancement signifies the
degree of decontamination of the Lamb shift from the inertial contribution in
the sense that the interest of some prospective experiment would be in the
purely-noninertial contribution. Given that the Lamb shift in Hydrogen atom
has been measured to seven significant digits [43], the purely-noninertial
contribution to the Lamb shift can be observed with current [81] or near-
future technology.

##  V Discussion and Conclusions

In this work, we have studied the radiative energy shifts in a first-quantized
two-level system on a uniform circular trajectory due to its interaction with
a quantum electromagnetic field inside a cavity. We have argued that the
radiative energy shift in atomic levels is an observable of interest for the
detection of noninertial effects. We have shown that the inertial contribution
to the Lamb shift in a circulating atom maximizes in the vicinity of the
atomic resonance whereas the purely-noninertial contribution maximizes away
from the atomic resonance at frequencies decided by the atom’s angular
frequency. By suitably tuning the cavity parameters, an observable purely-
noninertial contribution $\sim 10^{-8}-10^{-6}~{}\mathrm{Hz}$ can be obtained
for average accelerations $\sim 10^{14}-10^{16}~{}\mathrm{m/s^{2}}$.

It is instructive to compare the current theoretical proposal with other
proposals aiming at the detection of the noninertial effects. In free space,
the Unruh effect demands acceleration of the order of
$10^{21}\mathrm{m/s^{2}}$ if the detector transition rate [1] or radiative
energy shift [44] is observed, and $10^{17}\mathrm{m/s^{2}}$ if the observed
quantity is geometric phase [3]. Inside a long cylindrical cavity, however, it
has been argued that the Unruh effect can be detected at accelerations as low
as $\sim 10^{9}\mathrm{m/s^{2}}$ by observing the atom’s spontaneous emission.
Similarly, detection of noninertial effects due to uniform circular motion of
an atom inside an electromagnetic cavity, by observing the atomic spontaneous
decay rate, requires an acceleration of the order of $10^{14}\mathrm{m/s^{2}}$
[65]. Whereas, if the geometric phase is observed, such noninertial effects
may become detectable at accelerations as low as $\sim 10^{7}\mathrm{m/s^{2}}$
[4]. This comparison illustrates the role of the density of field states in
relation to the amplification of the noninertial effects. In addition, this
comparison points to the fact that different system properties might differ in
their ability to capture such effects, which can possibly guide the selection
and design of suitable experiments. For example, since the spectroscopic
techniques are much well-established as compared to the mixed-state geometric
phase measurements, the Lamb shift proposal might be easier to implement than
the geometric phase proposal.

Furthermore, the current work can be generalized in several directions. For
example, we have considered only a two-level atom but real atoms have more
than two levels which can lead to new features [70]. Also note that we have
worked in the weak-coupling regime. An analysis involving strong coupling
between the atom and the field can possibly lead to even higher energy shifts.

Additionally, the results obtained here encourage one to investigate the Lamb
shift for the detection of the Unruh effect [1]. A cavity with a cylindrical
geometry suits the requirement of uniform linear acceleration in the case of
the Unruh effect and has been argued to possibly facilitate its detection at
low accelerations when the atomic transition rates are observed [26]. The
problem of the possibility of detecting the Unruh effect through the radiative
energy shifts in an atom undergoing uniform linear acceleration inside a
cylindrical cavity will be taken up in a subsequent work.

###### Acknowledgements.

N.A. acknowledges financial support from the University Grants Commission
(UGC), Government of India, in the form of a research fellowship (Sr. No.
2061651285). N.A. thanks Kinjalk Lochan and Vikash Mittal for carefully
reading the manuscript and making useful suggestions.

##  Appendix A Derivation of Eq. (21)

In Eq. (20), introducing a change of integration variables
$(t_{1},t_{2})\mapsto(t_{+},t_{-})$ with

$\displaystyle t_{+}$ $\displaystyle\equiv(t_{1}+t_{2})/2,$ (33a)
$\displaystyle t_{-}$ $\displaystyle\equiv t_{2}-t_{1},$ (33b)

and using Eq. (17), we obtain

$\begin{split}&\mathcal{G}^{\prime}_{22}(\nu)=\lim_{T\rightarrow\infty}\frac{1}{\hbar^{2}T}\int_{0}^{\gamma
T}\differential{t_{+}}\int_{-\gamma T}^{\gamma
T}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\times\\\
&\bra{0}\left\\{E^{y}(x^{\mu}_{2})-R\omega\left(\sin(\omega
t_{2})B^{z}(x^{\mu}_{2})-\cos(\omega
t_{2})B^{x}(x^{\mu}_{2})\right)\right\\}\\\
&\times\left\\{E^{y}(x^{\mu}_{1})-R\omega\left(\sin(\omega
t_{1})B^{z}(x^{\mu}_{1})-\cos(\omega
t_{1})B^{x}(x^{\mu}_{1})\right)\right\\}\ket{0}.\end{split}$ (34)

Next, we show that in Eq. (34) the contribution of the terms containing cross-
correlations of the form $\bra{0}E^{i}(x^{\mu}_{2})B^{j}(x^{\mu}_{1})\ket{0}$
and $\bra{0}B^{k}(x^{\mu}_{2})B^{l}(x^{\mu}_{1})\ket{0},k\neq l$; vanishes.
Collecting the cross-terms in (34), and denoting the collection by
$\mathcal{G}_{22}^{{}^{\prime}\text{CT}}$, we get

$\begin{split}\mathcal{G}^{{}^{\prime}\text{CT}}_{22}(\nu)&=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{\gamma
T}\differential{t_{+}}\int_{-\gamma T}^{\gamma
T}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\\\
&\Big{[}-R\omega\Big{\\{}\sin(\omega
t_{2})\bra{0}B^{z}(x^{\mu}_{2})E^{y}(x^{\mu}_{1})\ket{0}\\\ &+\sin(\omega
t_{1})\bra{0}E^{y}(x^{\mu}_{2})B^{z}(x^{\mu}_{1})\ket{0}\\\ &+\cos(\omega
t_{2})\bra{0}B^{x}(x^{\mu}_{2})E^{y}(x^{\mu}_{1})\ket{0}\\\ &+\cos(\omega
t_{1})\bra{0}E^{y}(x^{\mu}_{2})B^{x}(x^{\mu}_{1})\ket{0}\Big{\\}}\\\
&+R^{2}\omega^{2}\Big{\\{}\cos(\omega t_{2})\sin(\omega
t_{1})\bra{0}B^{x}(x^{\mu}_{2})B^{z}(x^{\mu}_{1})\ket{0}\\\ &+\sin(\omega
t_{2})\cos(\omega
t_{1})\bra{0}B^{z}(x^{\mu}_{2})B^{x}(x^{\mu}_{1})\ket{0}\Big{\\}}\Big{]}\\\
&\equiv I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}.\end{split}$ (35)

It is straightforward to conclude that $I_{1}+I_{2}$ and $I_{3}+I_{4}$ vanish
under $t_{+}-$integration. For the remaining two terms, we have

$\begin{split}I_{5}+I_{6}&\equiv
R^{2}\omega^{2}\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{\gamma
T}\differential{t_{+}}\int_{-\gamma T}^{\gamma
T}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\\\ &\times\Bigg{[}\cos(\omega
t_{2})\sin(\omega t_{1})\bra{0}B^{x}(x_{2})B^{z}(x_{1})\ket{0}\\\
&+\sin(\omega t_{2})\cos(\omega
t_{1})\bra{0}B^{z}(x_{2})B^{x}(x_{1})\ket{0}\Bigg{]}\\\
&=\frac{R^{2}\omega^{2}}{2}\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{\gamma
T}\differential{t_{+}}\Bigg{[}-\int_{-\gamma T}^{\gamma
T}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\\\ &\times\sin(\omega
t_{-})\bra{0}B^{x}(x_{2})B^{z}(x_{1})\ket{0}+\int_{-\gamma T}^{\gamma
T}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\\\ &\times\sin(\omega
t_{-})\bra{0}B^{z}(x_{2})B^{x}(x_{1})\ket{0}\Bigg{]},\end{split}$ (36)

where we have used Eqs. (33) and the terms containing $\sin(2\omega t_{+})$
vanished under the $t_{+}-$integration. Further, Eq. (22b) and

$\sum_{\lambda}\left(\hat{\mathbf{k}}\times\epsilon_{\mathbf{k}\lambda}\right)^{i}\left(\hat{\mathbf{k}}\times\epsilon_{\mathbf{k}\lambda}\right)^{j}=\delta^{ij}-\frac{k^{i}k^{j}}{k^{2}},$
(37)

lead to

$\begin{split}\bra{0}{B}^{i}(x^{\mu}_{2}){B}^{j}(x^{\mu}_{1})\ket{0}&=\frac{1}{c^{2}}\sum_{\mathbf{k}}\frac{\hbar\omega_{k}}{2\epsilon_{0}V}\left(1-\frac{k^{i}k^{j}}{k^{2}}\right)\\\
&\times
e^{-i\omega_{k}(t_{2}-t_{1})}e^{i\mathbf{k}\cdot(\mathbf{x}(t_{2})-\mathbf{x}(t_{1}))}.\end{split}$
(38)

From Eq. (38) we have
$\bra{0}{B}^{i}(x^{\mu}_{2}){B}^{j}(x^{\mu}_{1})\ket{0}=\bra{0}{B}^{j}(x^{\mu}_{2}){B}^{i}(x^{\mu}_{1})\ket{0}$
which leads to $I_{5}+I_{6}=0$. Thus, we obtain

$\begin{split}\mathcal{G}^{\prime}_{22}(\nu)&=\lim_{T\rightarrow\infty}\frac{1}{\hbar^{2}T}\int_{0}^{\gamma
T}\differential{t_{+}}\int_{-\gamma T}^{\gamma
T}\differential{t_{-}}e^{i\bar{\nu}t_{-}}\\\
&\times\Big{[}\bra{0}E^{y}(x^{\mu}_{2})E^{y}(x^{\mu}_{1})\ket{0}\\\
&+\frac{R^{2}\omega^{2}}{2}\cos(\omega
t_{-})\bra{0}B^{z}(x^{\mu}_{2})B^{z}(x^{\mu}_{1})\ket{0}\\\
&+\frac{R^{2}\omega^{2}}{2}\cos(\omega
t_{-})\bra{0}B^{x}(x^{\mu}_{2})B^{x}(x^{\mu}_{1})\ket{0}\Big{]}.\end{split}$
(39)

Finally, performing the $t_{+}$-integral leads to Eq. (21).

##  Appendix B Inertial Lamb shift

Performing the integral in Eq. (29), we obtain

$\begin{split}&\Delta_{0}=\frac{\eta}{2\pi^{2}\left(8Q^{2}\left(1-4Q^{2}\right)\Omega_{0}^{2}\omega_{c}^{2}+\left(4Q^{2}+1\right)^{2}\omega_{c}^{4}+16Q^{4}\Omega_{0}^{4}\right)}\\\
&\times\Bigg{[}8Q^{3}\Omega_{0}^{3}\omega_{c}\Bigg{\\{}\log\left(\frac{4Q^{2}}{\left(4Q^{2}+1\right)\omega_{c}^{2}}\right)+2Q\left(\pi+2\tan^{-1}(2Q)\right)\\\
&+2\log\left(\Omega_{0}\right)\Bigg{\\}}-2Q\left(4Q^{2}+1\right)\Omega_{0}\omega_{c}^{3}\Bigg{\\{}2\log\left(\omega_{c}\right)\\\
&+\log\left(\frac{1}{4Q^{2}}+1\right)+2Q\left(\pi+2\tan^{-1}(2Q)\right)-2\log\left(\Omega_{0}\right)\Bigg{\\}}\Bigg{]}.\end{split}$
(40)

For $Q\gg 1$, we can ignore $1$ in comparison to $Q$. Also, using
$\lim_{Q\to\infty}\tan^{-1}Q=\pi/2$, we can write

$\Delta_{0}\approx\frac{\Omega_{0}\omega_{c}\eta}{4\pi^{2}Q\left(\omega_{c}^{2}-\Omega_{0}^{2}\right)^{2}}\Bigg{[}-(\Omega_{0}^{2}+\omega_{c}^{2})\log(\omega^{2}_{c})\\\
+4\pi
Q(\omega_{0}^{2}-\omega_{c}^{2})+(\Omega_{0}^{2}+\omega_{c}^{2})\log(\Omega^{2}_{0})\Bigg{]}.$
(41)

Further rearrangements and simplifications lead to

$\Delta_{0}\approx-\eta\Omega_{0}\left[\frac{\omega_{c}(\Omega_{0}^{2}+\omega_{c}^{2})\log(\omega_{c}/\Omega_{0})}{2\pi^{2}Q\left(\omega_{c}^{2}-\Omega_{0}^{2}\right)^{2}}+\frac{\omega_{c}}{\pi\left(\omega_{c}^{2}-\Omega_{0}^{2}\right)}\right],$
(42)

for $\omega_{c}\neq\Omega_{0}$.

##  Appendix C Convergence of the Lamb shift

Here, we show that the Lamb shift given in Eq. (28) converges without
introducing any ultraviolet cutoff on $\bar{\nu}$. The first three integrals
in Eq. (28) are manifestly convergent. Consider the last three integrals:

$\begin{split}\mathcal{I}&\equiv\frac{1}{2}\Bigg{[}\int_{0}^{\infty}\differential{\bar{\nu}}\left\\{\frac{\bar{\nu}^{2}R^{2}}{c^{2}}\bar{\nu}\rho(\bar{\nu})-\frac{(\bar{\nu}+\omega)^{2}R^{2}}{c^{2}}(\bar{\nu}+\omega)\rho(\bar{\nu}+\omega)\right\\}\\\
&+\int_{0}^{\infty}\differential{\bar{\nu}}\left\\{\frac{\bar{\nu}^{2}R^{2}}{c^{2}}\bar{\nu}\rho(\bar{\nu})-\frac{(\bar{\nu}-\omega)^{2}R^{2}}{c^{2}}(\bar{\nu}-\omega)\rho(\bar{\nu}-\omega)\right\\}\\\
&-\int_{-\omega}^{0}\differential{\bar{\nu}}\frac{(\bar{\nu}+\omega)^{2}R^{2}}{c^{2}}(\bar{\nu}+\omega)\rho(\bar{\nu}+\omega)\\\
&-\int_{0}^{\omega}\differential{\bar{\nu}}\frac{(\bar{\nu}-\omega)^{2}R^{2}}{c^{2}}(\bar{\nu}-\omega)\rho(\bar{\nu}-\omega)\Bigg{]}\\\
&\times\mathrm{P.V.}\left(\frac{1}{\bar{\nu}+\bar{\Omega}_{0}}-\frac{1}{\bar{\nu}-\bar{\Omega}_{0}}\right)\\\
&\equiv\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4},\end{split}$
(43)

where we have grouped different integrals in a way that is helpful in
ascertaining their convergence. Noting that $\mathcal{I}_{3}$ and
$\mathcal{I}_{4}$ are manifestly convergent, we focus on $\mathcal{I}_{1}$ and
$\mathcal{I}_{2}$:

$\begin{split}\mathcal{I}_{1}&=\frac{1}{2}\int_{0}^{\infty}\differential{\bar{\nu}}\left\\{\frac{\bar{\nu}^{3}R^{2}}{c^{2}}\rho(\bar{\nu})-\frac{(\bar{\nu}+\omega)^{3}R^{2}}{c^{2}}\rho(\bar{\nu}+\omega)\right\\}\\\
&\times\mathrm{P.V.}\left(\frac{1}{\bar{\nu}+\bar{\Omega}_{0}}-\frac{1}{\bar{\nu}-\bar{\Omega}_{0}}\right)\\\
&\equiv\mathcal{I}_{1a}-\mathcal{I}_{1b},\end{split}$ (44)

where we have defined

$\begin{split}\mathcal{I}_{1a}&\equiv\frac{\mathrm{P.V.}}{2}\int_{0}^{\infty}\differential{\bar{\nu}}\frac{\bar{\nu}^{3}R^{2}}{c^{2}}\rho(\bar{\nu})\left(\frac{1}{\bar{\nu}+\bar{\Omega}_{0}}-\frac{1}{\bar{\nu}-\bar{\Omega}_{0}}\right)\\\
&\equiv\frac{R^{2}}{2c^{2}}\mathrm{P.V.}\int_{0}^{\infty}\differential{\bar{\nu}}f_{1a}(\bar{\nu},\bar{\Omega}_{0}),\end{split}$
(45)

and

$\begin{split}\mathcal{I}_{1b}&\equiv\frac{\mathrm{P.V.}}{2}\int_{0}^{\infty}\differential{\bar{\nu}}\frac{(\bar{\nu}+\omega)^{3}R^{2}}{c^{2}}\rho(\bar{\nu}+\omega)\left(\frac{1}{\bar{\nu}+\bar{\Omega}_{0}}-\frac{1}{\bar{\nu}-\bar{\Omega}_{0}}\right)\\\
&\equiv\frac{R^{2}}{2c^{2}}\mathrm{P.V.}\int_{0}^{\infty}\differential{\bar{\nu}}f_{1b}(\bar{\nu},\bar{\Omega}_{0}).\end{split}$
(46)

Let us first analyze $\mathcal{I}_{1a}$:

$\begin{split}&\mathcal{I}_{1a}=\frac{R^{2}}{2c^{2}}\mathrm{P.V.}\int_{0}^{\infty}\differential{\bar{\nu}}f_{1a}(\bar{\nu},\bar{\Omega}_{0})\\\
&\equiv\frac{R^{2}}{2c^{2}}\left[\mathrm{P.V.}\int_{0}^{\bar{\Omega}_{0}+\epsilon}\differential{\bar{\nu}}f_{1a}(\bar{\nu},\bar{\Omega}_{0})+\int_{\bar{\Omega}_{0}+\epsilon}^{\infty}\differential{\bar{\nu}}f_{1a}(\bar{\nu},\bar{\Omega}_{0})\right],\end{split}$
(47)

where $\epsilon$ is some positive number. Note that for
$\bar{\nu}\rightarrow\infty$, $f_{1a}(\bar{\nu},\bar{\Omega}_{0})$ behaves as

$f_{1a}(\bar{\nu},\bar{\Omega}_{0})\rightarrow-\frac{\bar{\Omega}_{0}\omega_{c}}{\pi
Q\bar{\nu}}-\frac{2\bar{\Omega}_{0}\omega^{2}_{c}}{\pi
Q\bar{\nu}^{2}}+\order{\frac{1}{\bar{\nu}^{3}}},$ (48)

which means that the integral
$\int_{\bar{\Omega}_{0}+\epsilon}^{\infty}\differential{\bar{\nu}}f_{1a}(\bar{\nu},\bar{\Omega}_{0})$
diverges logarithmically. Therefore, $\mathcal{I}_{1a}$ diverges
logarithmically. Similarly, since for $\bar{\nu}\rightarrow\infty$ the
integrand $f_{1b}(\bar{\nu},\bar{\Omega}_{0})$ of $\mathcal{I}_{1b}$ behaves
as

$f_{1b}(\bar{\nu},\bar{\Omega}_{0})\rightarrow-\frac{\bar{\Omega}_{0}\omega_{c}}{\pi
Q\bar{\nu}}-\frac{\bar{\Omega}_{0}\omega_{c}(\omega+2\omega_{c})}{\pi
Q\bar{\nu}^{2}}+\order{\frac{1}{\bar{\nu}^{3}}},$ (49)

$\mathcal{I}_{1b}$ diverges logarithmically as well. But $\mathcal{I}_{1}$,
being the difference of $\mathcal{I}_{1a}$ and $\mathcal{I}_{1b}$, tends to
zero faster than $1/\bar{\nu}$ and hence converges. Similarly we can deduce
that $\mathcal{I}_{2}$ also converges.

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