Collective spontaneous emission of two entangled atoms near an oscillating mirror

Marta Reina1∗, Michelangelo Domina2†, Alessandro Ferreri3‡, GiuseppeFiscelli1§, Antonio Noto1¶, Roberto Passante1,4∗∗, and Lucia Rizzuto1,4††

1 Dipartimento di Fisica e Chimica - Emilio Segrè,Università degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy2 School of Physics and CRANN Institute, Trinity College, Dublin 2, Ireland

3 Department of Physics, Paderborn University, Warburger Strasse 100, D-33098 Paderborn, Germany and4 INFN, Laboratori Nazionali del Sud, I-95123 Catania, Italy

We consider the cooperative spontaneous emission of a system of two identical atoms, interactingwith the electromagnetic field in the vacuum state and in the presence of an oscillating mirror. Weassume that the two atoms, one in the ground state and the other in the excited state, are preparedin a correlated (symmetric or antisymmetric) Bell-type state. We also suppose that the perfectlyreflecting plate oscillates adiabatically, with the field modes satisfying the boundary conditions atthe mirror surface at any given instant, so that the time-dependence of the interaction Hamiltonianis entirely enclosed in the instantaneous atoms-wall distance. Using time-dependent perturbationtheory, we investigate the spectrum of the radiation emitted by the two-atom system, showing howthe oscillation of the boundary modifies the features of the emitted spectrum, which exhibits twolateral peaks not present in the case of a static boundary. We also evaluate the transition rate to thecollective ground state of the two-atom system in both cases of the superradiant (symmetric) andsubradiant (antisymmetric) state. We show that it is modulated in time, and that the presence ofthe oscillating mirror can enhance or inhibit the decay rate compared to the case of atoms in vacuumspace or near a static boundary. Our results thus suggest that a dynamical (i.e. time-modulated)environment can give new possibilities to control and manipulate radiative processes of atoms ormolecules nearby, such as the cooperative decay, and strongly indicate a similar possibility for otherradiative processes, for example the resonance interaction and the energy transfer between atomsor molecules.

I. INTRODUCTION

Quantum electrodynamics predicts that an excitedatom, interacting with the quantum electromagnetic fieldin the vacuum state, spontaneously decays to its groundstate by emitting a photon. The emission probability forunit time is found to be

A =4

3

ω3eg | µeg |2

~c3, (1)

where µeg is the matrix element of the atomic dipole mo-ment operator between the atomic excited and groundstates, and ωeg is the transition frequency between thetwo atomic levels [1]. This result can be generalized tothe case of N atoms incoherently coupled to the quantumelectromagnetic field: in this case, the N atoms decay in-dependently, and the intensity of the emitted radiationis proportional to N . Dicke in 1954 [2] showed that thisconclusion is not valid in general: when N identical atomsare confined within a volume V λ3, where λ is thewavelength of emitted radiation, the assumption of un-

∗marta.reina@institutoptique.fr†dominam@tcd.ie‡alessandro.ferreri@uni-paderborn.de§giuseppe.fiscelli@unipa.it¶antonio.noto@unipa.it∗∗roberto.passante@unipa.it††lucia.rizzuto@unipa.it

correlated emitters is no longer valid and a closer recon-sideration of the problem is necessary. It was shown thatan ensemble of atoms coherently coupled to the quantumelectromagnetic field, acts as a single quantum emitter,with a decay rate equal to NA, and an intensity of theemitted radiation proportional to N2 [3, 4]. This en-hanced single-photon emission is known as superradiance,and its physical origin is in the correlation (symmetricstate) between the atomic dipoles, leading to a construc-tive interference in the emission of radiation.

The counterpart of superradiance is the so-called sub-radiance [2, 5], that occurs when the ensemble of atomsis prepared in a correlated antisymmetric state. In thiscase, a suppression of the emission intensity occurs, andthe decay is totally inhibited. Contrarily to superradi-ance, subradiance arises from anticorrelations betweenthe atomic dipoles, leading to a destructive interferencein the emission of radiation. While superradiant statesare affected by decoherence, subradiant states are free-decoherence robust states, and for these reasons they areconsidered promising for realization of high-performancequantum processors in quantum information technolo-gies [6].

Superradiance and subradiance have been investigatedin a variety of systems, including atoms [7, 8], trappedions [9], quantum dots [10] coupled to various environ-ments, such as cavities [11, 12], waveguides [13, 14], andphotonic crystals [15].

Very recently, the influence of a perfect reflector on thecooperative spontaneous emission process of two atomslocated nearby has been discussed [16]. The effect of a

arX

iv:2

010.

0321

5v2

[qu

ant-

ph]

11

Feb

2021

2

surface or a structured environment, or of an externalstatic electric field on other radiative processes, such asdispersion or resonance interactions between atoms, havebeen recently studied [17–23].

Most of these studies concern with a static environ-ment. In this paper, we consider a different and moregeneral situation, specifically we discuss the influence ofa dynamical (i.e. time-dependent) environment on thecooperative emission of two correlated identical atomslocated nearby.

Generally speaking, a dynamical environment can berealized by changing periodically the magneto-dielectricproperties of the material or by a mechanical motion ofmacroscopic objects, such as a reflecting mirror or thecavity walls. These systems, for example vibrating cavi-ties or oscillating mirrors, have been extensively exploredin connection with the dynamical Casimir and Casimir-Polder effect [24–27]. Also, dynamical cavities have beensimulated in circuit QED [28].

Recent investigations have shown that the presence ofa dynamical environment can give additional possibili-ties (not present in the case of a static environment) tomanipulate and control radiative properties of atoms ormolecules coupled to a quantum field. For example thespontaneous emission of an excited atom located near aperfectly reflecting plate that oscillates adiabatically hasbeen recently discussed [29, 30], and it has been shownthat the motion of the mirror significantly affects theatomic decay rate, as well as the spectrum of the emit-ted radiation, exhibiting the presence of two lateral andalmost symmetrical peaks, not present in the case of astatic boundary [29]. Similar results were also obtainedin the case of an excited atom embedded in a dynamicalphotonic crystal, when its transition frequency is closeto the photonic band edge of the photonic crystal [31].Here, the presence of a time-modulated photonic band-gap gives rise to two lateral peaks in the spectrum of radi-ation emitted. These lateral peaks are asymmetric due tothe rapidly varying local density of states at the edge ofthe gap. Furthermore, the time-dependent resonance in-teraction between atoms, the dynamical Casimir-Polderinteraction between atoms or between an atom and amirror, have been investigated during the dynamical self-dressing process of the system, starting from a nonequi-librium configuration; it has been shown that forces usu-ally attractive can become repulsive in non-equilibriumsituations [32–38]. These results show the striking poten-tialities of time-dependent environments and nonequilib-rium configurations for manipulating a variety of radia-tive processes.

In this paper, we consider two identical atoms, onein the ground state and the other in the excited state,prepared in a correlated Bell-type state (symmetric orantisymmetric), while the electromagnetic field is in itsvacuum state. In the Dicke model, these states are thewell-known superradiant and subradiant states, respec-tively [2]. We assume that the two atoms are locatednear a perfectly reflecting mirror that oscillates adia-

batically along a prescribed trajectory, and we investi-gate the effects of the mirror’s motion on the cooper-ative spontaneous decay, the spectrum emitted by thetwo quantum emitters, and their decay rate. We supposethat the reflecting plate oscillates adiabatically along asinusoidal trajectory. Under these assumptions, the fieldmode functions, satisfying the boundary conditions atthe mirror surface at any time, are time-dependent. Us-ing time-dependent perturbation theory, we investigatethe spectrum of the emitted radiation, and the cooper-ative decay rate of the two-atom system. We show thatthe adiabatic motion of the mirror modifies the physicalfeatures of the spectrum of the radiation emitted. In par-ticular, we find the presence of two symmetric side peaksin the spectrum, not present in the case of a static mirror,and separated by the central peak by the mirror’s oscil-lation frequency. We also evaluate the transition rate tothe collective ground state of the two-atom system, inboth cases of the superradiant (symmetric) and subra-diant (antisymmetric) state, and show that it dependson the interatomic separation and the time-dependentatom-plate distances. We also find that the motion ofthe mirror can cause a significant enhancement or sup-pression of superradiance of the two quantum emitters,depending from the specific configuration of the system,with respect to the cases of a mirror at rest or atomsin the unbounded space. These results show how a dy-namical environment can influence the physical featuresof the superradiant and subradiant emission by the twocorrelated atoms, that can be enhanced or inhibited com-pared to the case of atoms in the vacuum space or neara static boundary. In general, this further confirms thata dynamical (i.e. time-modulated) environment can givenew possibilities to control, manipulate and also acti-vate or inhibit radiative processes of atoms and moleculesnearby, such as the cooperative spontaneous emission bytwo correlated atoms. It suggests that also other radia-tive processes, such as the resonance interaction and theenergy transfer between atoms or molecules, can be tai-lored exploiting a dynamical environment.

The paper is organized as follows. In Section II, weintroduce our system, and investigate the spectrum of theradiation emitted by the two-atom system, and discussits main physical features (some technical points on ourmodel are in the Appendix). In Section III we investigatethe collective decay rate of the two quantum emittersin the presence of the oscillating mirror. Section IV isdevoted to our concluding remarks.

II. SPECTRUM OF THE RADIATIONEMITTED BY TWO ENTANGLED ATOMS

NEAR AN OSCILLATING MIRROR

Let us consider two atoms, labeled as A and B, locatedin the half-space z > 0 near an infinite perfectly con-ducting plate, modeled as two-level systems with atomictransition frequency ω0, and interacting with the electro-

3

magnetic field in the vacuum state. We suppose that themirror oscillates with an angular frequency ωp, along thez direction with the trajectory a(t) = a sin(ωpt), where ais the oscillation amplitude of the plate around its aver-age position z = 0.

Let us suppose that the two identical two-level atomsare initially prepared in a symmetric or antisymmetricentangled state, i.e.

|φ〉± =1√2

(|eA, gB〉 ± |gA, eB〉) , (2)

and that the quantum field is in its vacuum state. Thus,the initial state of the system at time t = 0 is

|i〉± = |φ〉±|vac〉 . (3)

The sign ± in (2) refers to the symmetric or antisym-metric state respectively, |vac〉 is the vacuum state of theelectromagnetic field, while |eA(B)〉 (|gA(B)〉) indicatesthe excited (ground) state of atom A(B). In the states(2) the excitation is delocalized between the two atoms.In the Dicke model, these states are the so-called super-radiant and subradiant states, respectively. They can berealized experimentally with actual techniques [39, 40].Symmetric (antisymmetric) states are also at the originof the resonant interaction energy, which is a second-order interaction between correlated atoms [41].

Our physical system is displayed in Figure 1.

|"⟩w0 |$⟩

atomA

atomB

%&

%'

( ) = ( sin ./)

mirror

z

FIG. 1: Sketch of the system: two atoms, modeled as two-levelsystems with transition frequency ω0, are placed in front of anoscillating mirror. The atomic dipole moment of each atomcan be oriented parallel or perpendicular to the oscillatingreflecting plate.

We assume that the oscillation frequency ωp of theplate is much smaller than the atomic transition fre-quency ω0 of both atoms, and of the inverse of the timetaken by the photon emitted by one of the two atoms,to reach the other atom after reflection on the mirror(ωp c/rA, c/rB , c/(rA + rB), where rA/B is the aver-age atom-plate distance of each atom from the mirror).

Under these assumptions, we can neglect real photonsemission by dynamical Casimir effect, and investigatethe collective spontaneous emission by the two correlatedatoms in the adiabatic approximation. These assump-tions are fully verified by typical values of the relevantparameters of the system, for example ωp ∼ 109 s−1,ω0 ∼ 1015 s−1, and an atom-plate average distance ofthe order of 10−6 m, achievable in the laboratory. Westress that such a system is experimentally feasible, usinga dynamical mirror, that is a slab of semiconductor ma-terial whose dielectric properties are modulated in timefor simulating the oscillating mirror [25, 27], and keepingthe atoms at a fixed position exploting atomic trappingtechniques [42].

We write the Hamiltonian of our system in theCoulomb gauge and in the multipolar coupling scheme,within the dipole approximation [41, 43–45]:

H = ~ω0(SAz + SBz ) +∑kj

~ωka†kjakj +HI , (4)

where Sz = 12 (|e〉〈e|−|g〉〈g|) is the pseudospin atomic op-

erator, akj (a†kj) are the bosonic annihilation (creation)operators for photons with wave vector k and polariza-tion j, and HI is the interaction Hamiltonian, given by

HI = −(SA+ +SA−)µegA ·E(rA)−(SB+ +SB− )µegB ·E(rB). (5)

Here, S+ = |e〉〈g|, S− = |g〉〈e| are atomic pseudospinoperators, and µegA(B) = 〈eA(B)|µA(B)|gA(B)〉 are the ma-trix elements, assumed real, of the atomic dipole momentoperator µA(B) of atom A (B) between its excited andground state. E(rA(B)) is the electric field operator atthe atomic position rA(B); it includes a time dependenceas a consequence of the motion of the conducting wall,as discussed in detail in the Appendix.

The orientation of the atomic dipole moment is deter-mined by the specific atomic states |e〉 and |g〉 taken inour two-level model. For example, if the excited state isone of the three degenerate states with n = 2, l = 1, m =0,±1 of the hydrogen atom, and the ground state is thestate with n = 1, l = 0, m = 0 (n is the principal quan-tum number, l the orbital quantum number, and m themagnetic quantum number), then the excited state withm = 0 gives a dipole matrix element along z (perpen-dicular to the wall), while specific linear combinations ofthe states with m = 1 and m = −1 give a dipole along xor y, that is parallel to the wall.

In general, the presence of time-dependent boundaryconditions leads to introducing new creation and annihi-lation field operators, related to the old ones by a Bogoli-ubov transformation [24], and to time-dependent modefunctions, satisfying the appropriate time-dependentboundary conditions. However, in the present case ofan adiabatic motion of the mirror as defined above, thefield operators instantaneously follow the mirror’s mo-tion, and the creation and annihilation operators are thesame of the static-wall case; also, we can set the usual

4

boundary conditions for the electromagnetic field in thereference where the wall is instantaneously at rest, andthen go back to the laboratory frame by the appropriatetime-dependent space translation. Thus, the field anni-hilation and creation operators remain the same as inthe static case, and the mirror’s motion is entirely in-cluded in the field modes (ωpa/c 1). The expressionfor the field operator appearing in (5) with the adiabat-ically moving mirror, as well as the instantaneous fieldmodes and relevant expressions for the sum over polar-izations, are discussed in the Appendix.

Thus, the time dependence of the interaction Hamil-tonian will be made explicit in the mode functions only(see the Appendix), while, as mentioned, the field annihi-lation and creation operators are the same as in the staticcase; in other words, in our adiabatic approximation, theatoms locally interact with the vacuum field fluctuationsthat instantaneously follow the motion of the mirror. Ingeneral, in dealing with our system, we can adopt twodifferent points of view: with respect to the laboratoryframe, where both atoms are at rest and the plate os-cillates along a prescribed trajectory, or in the referenceframe comoving with the mirror. In this paper, we willadopt the laboratory frame. Taking into account our hy-pothesis of an adiabatic motion of the mirror, the fieldvacuum state in our initial state (3), at t = 0, is that rel-ative to the instantaneous position of the mirror at that

time, and it is independent of its previous motion (in fact,nonadiabatic effects such as photon emission by dynam-ical Casimir effect, or atomic excitation by dynamicalCasimir-Polder effect [27], are negligible in our hypothe-sis). Mathematically, it is defined by akj |vac〉 = 0, wherethe annihilation operator akj does not depend from time,because all time dependence relative to the wall’s motionis embedded in the field modes.

Using the interaction Hamiltonian (5), a straightfor-ward application of first-order time-dependent perturba-tions theory yields the transition amplitude from the ini-tial entangled state (3) to the state |gA, eB , 1k,j〉 (bothatoms in their ground state, and one photon emitted inthe mode (k, j))

c(kj, t) =

√πck

~V

∫ t

0

dt′ei(ωk−ω0)t′

× [µegA ·E(rA, t′)± µegB ·E(rB , t

′)] , (6)

where the ± sign refers to the superradiant or subradiantstate of Eq. (2).

The probability that the system, initially prepared inthe correlated state (3), decays at time t to the collectiveatomic ground-state, emitting a photon with wavevectork and polarization j, is then given by

|c(kj, t)|2 =πck

~V

∫ t

0

∫ t

0

dt′dt′′µegA · fkj(rA, t

′)µegA · fkj(rA, t′′) + µegB · fkj(rB , t

′)µegB · fkj(rB , t′′)

±[µegA · fkj(rA, t

′)µegB · fkj(rB , t′′) + µegA · fkj(rA, t

′′)µegB · fkj(rB , t′)]ei(ωk−ω0)(t′′−t′). (7)

The first two terms in the right-hand side of Eq. (7) arerelated to the probability that each atom independentlydecays by emitting a photon; on the contrary, the contri-bution inside the square bracket is an interference term,and it is responsible of the superradiant or subradiantbehavior of the two-atom system.

From Eq. (7) we can obtain the frequency spectrumof the radiation emitted by the two atoms, that is theemission probability for unit frequency, by taking the sumover polarization and the integration over the directionsof k as

P (ωk, t) =V

(2π)3

ωk2

c3

∫dΩ∑j

|c(kj, t)|2, (8)

where V is the quantization volume, and Ω the solidangle. The integration over the directions of k that wewill explicitly perform in the following is on the full 4πsolid angle, because our field modes, given in the Ap-pendix, allow positive and negative values of the compo-

nents of k.We perform the sum over polarizations j = 1, 2, using

the relation (A.8) given in the Appendix, that extends toour adiabatic dynamical case the expression obtained in[46] for the static case. For convenience, we report herethis relation∫

dΩ∑j

[fkj(ru, t′)]`[fkj(rv, t

′′)]m

→∫dΩ<

(δ`m − k`km)eik·(ru−rv)

−σ`p(δpm − kpkm)eik·(ru(t′)−σrv(t′′)), (9)

where < indicates the real part.We stress that the relation (9) is valid only in our

adiabatic approximation in the laboratory frame, that iswhen the electromagnetic field operators instantaneouslyfollow the motion of the plate. The first term in (9)is a free-space contribution, and it is time-independentbecause the two atoms are fixed in space. On the con-

5

trary, the second term takes into account the presenceof the oscillating mirror through the reflection matrixσ = diag(1, 1,−1) introduced in (A.7), and, in our adia-batic approximation, depends on the instantaneous time-dependent atom-mirror and atom-image distances (seethe presence of the σ reflection matrix).

The second term in Eq. (9) can be written as

eik·(ru(t′)−σrv(t′′)) = eik·Ruv−ik·a[sin(ωpt′)+sin(ωpt

′′)], (10)

where Ruv = ru−σrv. For a single atom, say A, u = v =A, and RA = rA − σrA represents the distance of atomA from its image through the mirror; on the other hand,RAB = rA−σrB , is the distance of one atom (say A) fromthe image of the other atom (say B). For small oscillationamplitudes, such that a RA(B), RA(B), RAB , we canperform a series expansion of the exponential function in(10) in powers of a, obtaining

eik·(ru(t′)−σrv(t′′)) ' eik·Ruv

[1− i(k · n)a

×(sin(ωpt′) + sin(ωpt

′′))− 1

2(k · n)2a2

×(sin(ωpt′) + sin(ωpt

′′))2 + ....

], (11)

where n = (0, 0, 1) is the unit vector orthogonal to theoscillating plate. We can now substitute the relations(11) and (9) into (7), and integrate over time. Takinginto account only terms up to the second order in theoscillation amplitude a, after some algebra we get

∫dΩ∑j

|c(kj, t)|2 ' gA(ωk, t) + gB(ωk, t)

±gAB(ωk, t) (12)

where

gA(B)(ωk, t) =πck

~V(µegA(B))`(µ

egA(B))m<

∫dΩ[(δ`m − k`km)h0(ωk − ω0, t)− σ`p(δpm − kpkm)eik·RA(B)

×(h0(ωk − ω0, t)− i(k · n)ah1(ωk − ω0, ωp, t)− (k · n)2 a

2

2(h2(ωk − ω0, ωp, t) + h3(ωk − ω0, ωp, t))

)](13)

are the single-atom contributions (< indicates the real part), and

gAB(ωk, t) =4πck

~V(µegA )`(µ

egB )m<

∫dΩ[(δ`m − k`km)eik·RABh0(ωk − ω0, t)− σ`p(δpm − kpkm)eik·RAB

×(h0(ωk − ω0, t)− i(k · n)a h1(ωk − ω0, ωp, t)− (k · n)2 a

2

2(h2(ωk − ω0, ωp, t) + h3(ωk − ω0, ωp, t))

)]. (14)

is the interference term, yielding the cooperative effects.In the expressions (13) and (14), we have introduced thefollowing functions

h0(ωk − ω0, t) =sin2((ωk − ω0)t/2)

((ωk − ω0)/2)2, (15)

6

h1(ωk − ω0, ωp, t) = sin(ωpt/2)sin[(ωk − ω0)t/2]

(ωk − ω0)/2

(sin[(ωk − ω0 + ωp)t/2]

(ωk − ω0 + ωp)/2+

sin[(ωk − ω0 − ωp)t/2]

(ωk − ω0 − ωp)/2

), (16)

h2(ωk − ω0, ωp, t) =sin2[(ωk − ω0 + ωp)t/2])

(ωk − ω0 + ωp)2/2+

sin2[(ωk − ω0 − ωp)t/2]

(ωk − ω0 − ωp)2/2− cos(ωpt)

× sin[(ωk − ω0 + ωp)t/2] sin[(ωk − ω0 − ωp)t/2)]

(ωk − ω0 + ωp)(ωk − ω0 − ωp)/4, (17)

h3(ωk − ω0, ωp, t) =sin2[(ωk − ω0)t/2]

[(ωk − ω0)/2]2− 2 cos(ωpt)

sin[(ωk − ω0)t/2]

ωk − ω0

(sin[(ωk − ω0 + 2ωp)t/2]

ωk − ω0 + 2ωp

+sin[(ωk − ω0 − 2ωp)t/2]

ωk − ω0 − 2ωp

). (18)

These functions give the behaviour of the emitted spec-trum by the two-atom system, as a function of the mir-ror’s oscillation frequency ωp and the atomic transitionfrequency ω0. They are responsible of the qualitativefeatures and changes (with respect to the fixed-mirrorcase) of the spectrum of the radiation emitted, due tothe motion of the boundary. In fact, inspection of (15)-(18) clearly shows that, in addition to the usual centralpeak at ωk = ω0 (present also in the case of a static mir-ror), new lateral peaks at ωk = ω0 ± ωp appear in thespectrum, due to the presence of energy denominators asωk − ω0 ± ωp in Eqs. (16-18). These contributions areclearly related to the motion of the mirror, and vanish inthe limit of a static boundary, namely when a and/or ωp

vanish.Substituting Eqs. (12-14) into (8), and separating the

terms according to the order of the plate’s oscillationamplitude a, some straightforward algebra finally yieldsthe expression of the emission spectrum in the form

P (ωk, t) = P (0)(ωk, t) + P (1)(ωk, t) + P (2)(ωk, t), (19)

where P (0)(ωk, t) is the 0-th order contribution, whileP (1)(ωk, t) and P (2)(ωk, t) give respectively the first- andsecond-order (in the mirror’s oscillation amplitude a)modification to the spectrum consequent to the adiabaticmotion of the mirror. Such contributions are

P (0)(ωk, t) =k3

2π~

B∑u=A

(µegu )`(µegu )m

[2

3δ`m − σ`pF Ru

mp

sin(kRu)

k3Ru

]sin2((ωk − ω0)t/2)

((ωk − ω0)/2)2

±ck3

π~(µegA )`(µ

egB )m

[FRAB

`m

sin(kRAB)

k3RAB− σ`pF RAB

mp

sin(kRAB))

k3RAB

]sin2((ωk − ω0)t/2)

((ωk − ω0)/2)2, (20)

P (1)(ωk, t) =k3

2π~aσ`p

[ B∑u=A

(µegu )`(µegu )m(n · ∇Ru)F Ru

mp

sin(kRu)

k3Ru± 2(µegA )`(µ

egB )m(n · ∇RAB )F RAB

mp

sin(kRAB)

k3RAB

]×h1(ωk − ω0, ωp, t), (21)

P (2)(ωk, t) = − k3

2π~a2

2σ`p

[ B∑u=A

(µegu )`(µegu )m(n · ∇Ru)2F Ru

mp

sin(kRu)

k3Ru± 2(µegA )`(µ

egB )m(n · ∇RAB )2F RAB

mp

sin(kRAB)

k3RAB

]×[h2(ωk − ω0, ωp, t) + h3(ωk − ω0, ωp, t)

]. (22)

Here

F r`m = (−δ`m∇2 +∇`∇m)r (23)

is a differential operator acting on variable r, RAB =

7

|rA − rB |, RA(B) = |rA(B) − σrA(B)|, RAB = |rA − σrB |,and rA, rB respectively being the positions of atoms Aand B.

A comparison of these expressions with the analogousquantity for the static-mirror case, shows that the maindifference is the presence of terms related to the oscilla-tion frequency of the mirror, specifically two new lateralpeaks in the spectrum at frequencies ωk = ω0±ωp. Theirrelative intensities are of the order of a/Ri (see Eq. (21)),and (a/Ri)

2 (see Eq. (22)), and give a qualitative changeof the spectrum. We wish to point out that secondarylateral peaks at frequency ωk = ω0±2ωp, stemming fromsecond-order terms in the expansion in a, are also present(as Eq. (18) shows). They represent, at the order consid-ered, a sort of nonlinear effect; however, within the rangeof validity of our approximations, they give a quite smallcontribution to the overall spectrum.

Our expression for P (ωk, t) is valid for a generic geo-metric configuration of the two atoms with respect to theoscillating plate.

-5.×107 0 5.×1071.×10-8

2.×10-8

-2 ×109 -1 ×109 0 1 ×109 2 ×1090

2.×10-10

4.×10-10

6.×10-10

8.×10-10

1.×10-9

ωk -ω0 [s-1]

P(ωk,t)

[s]

FIG. 2: Spectrum (scaled with respect to the total emissionprobability) emitted by the two-atom system, prepared in thecorrelated symmetric state, as a function of the detuning ωk−ω0, both in the static case (green line) and in the dynamicalcase, with the two lateral peaks (red line). The atoms arealigned perpendicularly to the mirror, with dipole momentsalong the x-axis (parallel to the plate). The figure clearlyshows that the presence of a dynamical mirror produces twolateral peaks (red line) shifted from the central peak by themirror’s modulation frequency. The inset shows a zoom ofthe central peak in the two cases considered. Parameters arechosen such that a = 2 × 10−7 m, z0A = 10−6 m, z0B = 1.1 ×10−6 m, ωp = 1.5 × 109 s−1, ω0 = 1015 s−1, t = 1.6 × 10−7 s,µ ∼ 10−30C·m.

In order to get a clear physical insight it is helpfulto analyze P (ωk, t) in the specific case of atoms alignedalong the z axis (i.e. perpendicularly to the mirror), forexample when rA = (0, 0, z0

A) and rB = (0, 0, z0B). Figure

2 shows the spectrum (scaled with respect to the totalemission probability) in the symmetric case and in thelimit of long times, as a function of the detuning ωk−ω0:the red line shows the dynamical case, while the greenline shows the static-mirror case. As the figure shows,the presence of the dynamical mirror determines the two

-5.×107 0 5.×1071.5 ×10-8

2.5 ×10-8

-2 ×109 -1 ×109 0 1 ×109 2 ×1090

5.×10-11

1.×10-10

1.5 ×10-10

2.×10-10

2.5 ×10-10

ωk -ω0 [s-1]

P(ωk,t)

[s]

FIG. 3: Spectrum (scaled with respect to the total emissionprobability) emitted by the two-atom system, prepared inthe correlated symmetric state, as a function of the detun-ing ωk − ω0, both in the static case (green line) and in thedynamical case with the two lateral peaks (blue line). Thedipole moments are perpendicular to the plate (along the z-axis). As before, the presence of a dynamical mirror producestwo lateral peaks (blue line) shifted from the central peak bythe mirror’s modulation frequency. The inset shows a zoom ofthe central peak in the two cases considered (the two curvespractically overlap each other) . The numerical values of theparameters are the same as in the plot in Fig. 2.

-3.×107 0 3.×1071.5 ×10-8

2.×10-8

2.5 ×10-8

-2 ×109 -1 ×109 0 1 ×109 2 ×1090

2.×10-10

4.×10-10

6.×10-10

8.×10-10

1.×10-9

ωk -ω0 [s-1]

P(ωk,t)

[s]

FIG. 4: Comparison between the emitted spectra (scaled withrespect to the total emission probability) by the two-atom sys-tem, when the dipole moments are aligned parallel (red line)and perpendicular (blue line) to the plate. The figure showsthat the lateral peaks in the emitted spectrum by dipole mo-ments aligned along the z-axis are smaller than those obtainedin the case of dipole moments oriented parallel to the mirror(along the x-axis). The inset shows a zoom of the centralpeak in the two cases considered. The numerical values of theparameters are the same as in Fig. 2.

symmetric lateral peaks shifted from the central peak bythe modulation frequency. These two lateral peaks aresymmetric with respect to the central peak, because thephotonic density of states is essentially the same at thetwo frequencies. Analogous lateral peaks were found fora single two-level atom located near an oscillating mir-ror [29]. A similar result is obtained for dipole momentsaligned perpendicularly to the mirror, as figure 3 shows.Interestingly, although the image dipole of µ⊥ is still

8

µ⊥, and a constructive interference between the atomicdipoles and their mirror images is expected, the inten-sity of the two lateral peaks in the emitted spectrum issmaller than that obtained in the case of dipole momentsoriented parallel to the plate, as shown in figure 4. Thiseffect seems to suggest that the oscillation of the mir-ror can induce a sort of destructive interference betweenthe atomic dipoles and their images, oriented along thez-direction, parallel to the motion of the plate.

We have also considered the emitted spectrum by thetwo-atom system at different times. The results obtainedare illustrated in Fig. 5, for two atoms prepared in asymmetric superposition with dipole moments orientedparallel to the oscillating mirror. The figure shows thatthe central and the lateral peaks increase with time, asexpected.

01.×10-9

5.×10-9

1.×10-8

1.5 ×1090

1.×10-10

2.×10-10

-2 ×109 -1 ×109 0 1 ×109 2 ×1090

5.0 ×10-10

1.0 ×10-9

1.5 ×10-9

2.0 ×10-9

ωk -ω0 [s-1]

P(ωk,t)

[s]

FIG. 5: Comparison between the emitted spectra (scaled withrespect to the total emission probability) by the two-atomsystem, at different times. The dipole moments are alignedparallel to the plate. Continuous green line, blue line, and redline refer to the emitted spectrum at times t = 10−8s (contin-uous green), t = 3 × 10−8s (continuous blue), t = 6 × 10−8s(continuous red), respectively. The figure shows that the cen-tral peak (see inset on the left) and the two lateral peaks (seeinset on the right) in the spectrum increase with time, as ex-pected. The numerical values of the other parameters are thesame as in Fig. 2.

Finally, we wish to stress that our results are in prin-ciple detectable using current experimental techniques;for example, for two hydrogen atoms and typical opti-cal transitions, the natural linewidth is of the order of∼ 108 s−1; thus an oscillation frequency in the rangeωp ∼ 109 − 1010 s−1, that can be currently obtained ex-ploiting the technique of dynamical mirrors [25, 27], issufficient to resolve the lateral lines in the emitted spec-trum.

III. COLLECTIVE SPONTANEOUS DECAYRATE OF THE TWO-ATOM SYSTEM

We now evaluate the decay rate of the two-atom sys-tem to the ground state. This is obtained by integratingP (ωk, t) over k, and then taking its time derivative,

Γ(t) =d

dt

∫ ∞0

dωkP (ωk, t). (24)

Since the functions h0(ωk − ω0, t) and hi(ωk − ω0, ωp, t)(i = 1, 2, 3) are strongly peaked at ωk ∼ ω0 and ωk ∼ω0 ± ωp, we can approximate the space-dependent func-tions in (20)-(22), by their expressions in k0 (taking alsointo account that ωp ck0) and then take them out ofthe integrals. Taking into account only terms up to thefirst order in the expansion (11) on the mirror’s oscilla-tion amplitude, a straightforward calculation gives

Γ(t) = ΓA(t) + ΓB(t)± ΓAB(t), (25)

where

ΓA(B)(t) =k3

0

~(µegA(B))`(µ

egA(B))m

[2

3δ`m − σ`pF

RA(B)mp

sin(k0RA(B))

k30RA(B)

+ 2a sin(ωpt)σ`p(n · ∇RA(B))FRA(B)mp

sin(k0RA(B))

k30RA(B)

],(26)

ΓAB(t) =2k3

0

~(µegA )`(µ

egB )m

[FRAB

`m

sin(k0RAB)

k30RAB

− σ`pF RABmp

sin(k0RAB)

k30RAB

+ 2a sin(ωpt)σ`p(n · ∇RAB )F RABmp

sin(k0RAB)

k30RAB

].(27)

The expressions (25)-(27) are general, valid for ageneric configuration of the two atoms with respect tothe plate, and show oscillations of the decay rate withtime, directly related to the adiabatic motion of the mir-ror. In fact, the emission rate of our system shows a

term that oscillates in time by following the mirror’s lawof motion, of course. This is strictly related to our hy-pothesis of adiabatic motion of the boundary. In orderto discuss in more detail this result, similarly to what wedid in the previous section, we analyze the specific case

9

of atoms aligned along the z-direction, i.e. perpendicularto the reflecting plate. In this case of a perpendicular

orientation we obtain

ΓA(t) =k3

0

~(µegA )`(µ

egA )m

2

3δ`m − σ`p

[−(δpm − 3( ˆRA)p(

ˆRA)m

)( sin k0RAk3

0R3A

− cos k0RAk2

0R2A

)+(δpm − ( ˆRA)p(

ˆRA)m

) sin k0RAk0RA

]+

2a sin(ωpt)

RAσ`p

[(δpm − ( ˆRA)p(

ˆRA)m

)cos k0RA

−2(δpm − 3( ˆRA)p(

ˆRA)m

) sin k0RAk0RA

+ 3(δpm − 5( ˆRA)p(

ˆRA)m

)( sin k0RAk3

0R3A

− cos k0RAk2

0R2A

)−(δmz(

ˆRA)p + δpz(ˆRA)m

)( sin k0RAk0RA

+ 3cos k0RAk2

0R2A

− 3sin k0RAk3

0R3A

)], (28)

ΓB(t) = (ΓA(t) withA→ B), (29)

ΓAB(t) =2k3

0

~(µegA )`(µ

egB )m

[−(δ`m − 3(RAB)`(RAB)m

)( sin k0RABk3

0R3AB

− cos k0RABk2

0R2AB

)+(δ`m − (RAB)`(RAB)m

) sin k0RABk0RAB

]+ σ`p

[(δpm − 3( ˆRAB)p(

ˆRAB)m

)( sin k0RABk3

0R3AB

− cos k0RABk2

0R2AB

)−(δpm − ( ˆRA)p(

ˆRAB)m

) sin k0RABk0RAB

]+

2a sin(ωpt)

RABσ`p

[(δpm − ( ˆRAB)p(

ˆRAB)m

)cos k0RAB

−2(δpm − 3( ˆRAB)p(

ˆRAB)m

) sin k0RABk0RAB

+ 3(δpm − 5( ˆRAB)p(

ˆRAB)m

)( sin k0RABk3

0R3AB

− cos k0RABk2

0R2AB

)−(δmz(

ˆRAB)p + δpz(ˆRAB)m

)( sin k0RABk0RAB

+ 3cos k0RABk2

0R2AB

− 3sin k0RABk3

0R3AB

)]. (30)

Expressions (28-30) show that the motion of the mir-ror yields new time-dependent terms of the order ofa/RA/B and a/RAB . We have neglected second-orderterms in the perturbative expansion; this approxima-tion is valid for small oscillation amplitudes with respectto other relevant length scales in the system, that isfor a RA/B , RAB and a k−1

0 . For example, fork0 ∼ 107 m−1, RA/B ∼ 10−6 m, and a = 10−8 m, wehave a/RA/B , a/RAB ∼ 10−1, k0a ∼ 10−1, and weneglect the second-order term proportional to a2. Theconditions above are within reach of currently achievableexperimental techniques.

Figure 6 shows the scaled (with respect to Einsteincoefficient A, given in (1)) collective decay rate at dif-ferent times, as a function of the distance of atom Bfrom the mirror’s average position, when atom A is ata fixed position. The two atoms are aligned orthogonalto the mirror and prepared in the symmetric state. The

dipole moments are parallel to the mirror. The figureshows that the decay rate oscillates in time, and that,at a given time, in specific distance ranges it can be in-creased (in the figure, see the red continuous line, bluedashed line and orange dotted line) with respect to thestatic-mirror case (green dot-dashed line); in other dis-tance ranges, the opposite occurs. Analogous results areobtained in the case of atoms prepared in an antisym-metric configuration, as shown in figure 7.

In conclusion, our results show that the spectrum ofthe emitted radiation can be qualitatively changed ex-ploiting the oscillation of the plate, and that the collec-tive spontaneous emission can be controlled (enhancedor suppressed) by modulating in time the position of themirror. This suggest the possibility to control also otherradiative processes by modulated (time-dependent) en-vironments, for example the resonance energy transferbetween atoms or molecules.

10

2.×10-6 4.×10-6 6.×10-6 8.×10-6

0

0.5

1.0

1.5

2.0

zB [m]

Γ

A

FIG. 6: Plot of the collective decay rate for two atoms pre-pared in the correlated symmetric state, at different times, asa function of the distance of atom B from the mirror, when theatom A is kept fixed (RA = zA = 1.25× 10−6m). The atomsare aligned along the z direction, with dipole moments parallelto the mirror. Continuous red line, blue dashed line, orangedotted line refer to two atoms near the oscillating mirror, attimes t = 2 × 10−7s (continuous red), t = 2.3 × 10−7s (bluedashed), t = 2.4 × 10−7s (orange dotted), respectively. Dot-dashed green line refers to two atoms in the presence of a staticmirror. We have also used a = 2×10−7 m, ωp = 1.5×109 s−1,ω0 = 1015 s−1.

2.×10-6 4.×10-6 6.×10-6 8.×10-6

0

0.5

1.0

1.5

zB [m]

Γ

A

FIG. 7: Plot of the collective decay rate for two atoms pre-pared in the correlated antisymmetric state, at different times,as a function of the distance of atom B from the mirror, whenthe atom A is kept fixed (RA = zA = 1.25 × 10−6m). Theatoms are aligned along the z direction, with dipole momentsparallel to the mirror, Continuous red line, blue dashed line,orange dotted line refer to two atoms near the oscillating mir-ror, at times t = 2× 10−7s (continuous red), t = 2.3× 10−7s(blue dashed), t = 2.4 × 10−7s (orange dotted), respectively.The green dot-dashed line refers to two atoms near a staticmirror. The other relevant parameters are the same of Fig.6.

IV. CONCLUSION

In this paper, we have discussed the collective sponta-neous decay of a system of two identical two-level atomsprepared in a correlated (symmetric or antisymmetric)Bell-type state, and located near an oscillating perfectly

reflecting plate, in the adiabatic regime. We have firstdiscussed in detail the effect of the motion of the mir-ror on the spectrum of the radiation emitted by the twoatoms, and then their collective spontaneous decay rate.We have shown that the motion of the mirror stronglyaffects the features of the spectrum, which exhibits, inaddition to the usual peak at ω = ω0, two new lateralpeaks separated from the atomic transition frequency bythe oscillation frequency of the plate, similarly to previ-ous results for the single-atom decay [29]. We have alsofound that the decay rate to the collective ground-stateis modulated in time, and can be increased or decreased,compared with the static-boundary case, according totime and atoms-wall distances, by exploiting the oscil-lating boundary. Our results show that modulated en-vironments can give additional possibilities, with respectto fixed boundaries, to manipulate and tailor atomic ra-diative processes such as the cooperative spontaneousemission; also, they strongly indicate a similar possibilityfor other relevant radiative processes such as the energytransfer between two atoms, or the resonance interactionenergy between correlated atoms. We will consider thesephysical systems in a future publication.

Appendix: Field operators and modes with theadiabatically moving mirror

For a static mirror located at z = 0, the electric fieldoperator is given by

E(r) = i∑kj

√2π~ckV

fkj(r)(a†kj − akj), (A.1)

where akj and a†kj are annihilation and creation operators

obeying usual boson commutation relations, and fkj(r)are the mode functions satisfying the appropriate bound-ary condition at the mirror’s surface. The annihilationand creation operators in (A.1) are time-independent, be-cause we are working in the Schrödinger representation.We start considering a cubic cavity of side L with wallsat x = ±L/2, y = ±L/2, z = 0, z = L; in this case, wehave [29, 46]

[fkj(r)]x =√

8(ekj)x cos

[kx

(x+

L

2

)]× sin

[ky

(y +

L

2

)]sin [kzz] , (A.2)

[fkj(r)]y =√

8(ekj)y sin

[kx

(x+

L

2

)]× cos

[ky

(y +

L

2

)]sin [kzz] , (A.3)

[fkj(r)]z =√

8(ekj)z sin

[kx

(x+

L

2

)]× sin

[ky

(y +

L

2

)]cos [kzz] , (A.4)

11

where ekj (j = 1, 2) are unit polarization vectors, as-sumed real, satisfying ekj · ekj′ = δjj′ and ekj ·k = 0. Inthe limit L → ∞, the case of a single mirror at z = 0 isrecovered. The field modes (A.2-A.4) are normalized as

∫d3r fkj(r) · fk′j′(r) = V δkk′δjj′ , (A.5)

where V is the quantization volume.The polarization sum can be conveniently done exploit-

ing the following relation [46]∫dΩ∑j

[fkj(r)]`[fkj(r′)]m

=

∫dΩ<

(δ`m − k`km

)eik·(r−r

′)

−σ`p(δpm − kpkm

)eik·(r−σr

′), (A.6)

where

σ =

1 0 00 1 00 0 −1

, (A.7)

is the reflection matrix on the reflecting plate placed atz = 0. For the validity of the relation (A.6), it is un-derstood that all other quantities present inside the an-gular integration are invariant under the transformationk→ −k.

We now extend the relations above to the time-dependent case of an oscillating plate, under the adia-batic approximations defined in section II. We choosethe laboratory frame, where the atoms are at rest andthe mirror is moving. The mirror is moving along the zdirection with amplitude a and angular frequency ωp, asdiscussed in section II. We indicate with fkj(r, t) the in-stantaneous modes of the form (A.2-A.4), relative to timet (they change according to the wall’s position oscillatingaround z = 0), to be used in the electric field operator

(A.1) for our dynamical case. The relation we use in ourcalculations in section II is (A.6), appropriately gener-alized to our (adiabatic) dynamical case. This is doneby taking into account that in the right-hand-side, thequantity r − σr′ in the second term is the distance be-tween the point r and the image of the point r′ reflectedon the mirror; this is a time-dependent quantity becausethe position of the mirror changes with time. Thus weuse the following relation∫

dΩ∑j

[fkj(ru, t′)]`[fkj(rv, t

′′)]m

=

∫dΩ<

(δ`m − k`km)eik·(ru−rv)

−σ`p(δpm − kpkm)eik·(ru(t′)−σrv(t′′))

(A.8)

(`,m, p = x, y, z). In (A.8), ru(v) (u, v = A,B) is theposition vector of atom A or B, ru(t) = ru − a sin(ωpt)is the instantaneous time-dependent atom-wall distance,and we have defined the vector a = (0, 0, a).

Relations (A.6) and (A.8) are obtained in the limitL → ∞, where the case of a single mirror is recovered.For the validity of (A.8), the conditions mentioned after(A.6) should be verified, as well as our adiabatic approx-imation.

We wish to point out that we are not including atime dependence of the eigenfrequencies ωk, as one couldin principle expect for a cavity with a time-dependentlength, because at the end we take the limit L→∞, andin this limit the field eigenfrequencies have a continuousand time-independent spectrum.

Acknowledgements

The authors gratefully acknowledge financial supportfrom the Julian Schwinger Foundation and MIUR.

[1] P. A. M. Dirac, The quantum theory of the emission andabsorption of radiation, Proc. R. Soc. Lond. A 114, 243(1927).

[2] R. H. Dicke, Coherence in Spontaneous Radiation Pro-cesses, Phys. Rev. 93, 99 (1954).

[3] C. Leonardi, F. Persico, and G. Vetri, Dicke modeland the theory of driven and spontaneous emission, Riv.Nuovo Cimento 9, 1 (1986).

[4] M. G. Benedict, A. M. Ermolaev, V. A. Malyshev,I. V. Sokolov, and E. D. Trifonov, Super-radiance: Mul-tiatomic Coherent Emission, (Taylor Francis Group, NewYork 1996).

[5] M. Gross and S. Haroche, Superradiance: An essay onthe theory of collective spontaneous emission, Phys. Rep.93, 304 (1982).

[6] D. Petrosyan and G. Kurizki, Scalable solid-state quan-tum processor using subradiant two-atom states, Phys.Rev. Lett. 89, 207902 (2002).

[7] D. Pavolini, A. Crubellier, P. Pillet, L. Cabaret, and S.Liberman, Experimental Evidence for Subradiance, Phys.Rev. Lett. 54, 1917 (1985).

[8] C. Braggio, F. Chiossi, G. Carugno, A. Ortolan, and G.Ruoso, Spontaneous formation of a macroscopically ex-tended coherent state, Phys. Rev. Res. 2, 033059 (2020).

[9] R. G. DeVoe and R. G. Brewer, Observation of Super-radiant and Subradiant Spontaneous Emission of TwoTrapped Ions, Phys. Rev. Lett. 76, 2049 (1996).

[10] M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G.Bacher, T. Passow, and D. Hommel, Superradiance ofquantum dots, Nat Phys. 3, 106 (2007).

12

[11] V. V. Temnov and U. Woggon, Superradiance and Sub-radiance in an Inhomogeneously Broadened Ensemble ofTwo-Level Systems Coupled to a Low-Q Cavity, Phys.Rev. Lett. 95, 243602 (2005).

[12] J. Pan, S. Sandhu, Y. Huo, N. Stuhrmann, M. L.Povinelli, J. S. Harris, M. M. Fejer, and S. Fan, Exper-imental demonstration of an all-optical analogue to thesuperradiance effect in an on-chip photonic crystal res-onator system, Phys. Rev. B 81, 041101 (2010).

[13] D. Martín-Cano, L. Martín-Moreno, F.J. García-Vidal,and E. Moreno, Resonance Energy Transfer and Super-radiance Mediated by Plasmonic Nanowaveguides, NanoLett. 10, 3129 (2010).

[14] R. Fleury R and A. Alú, Enhanced superradiance inepsilon-near-zero plasmonic channels, Phys. Rev. B 87,201101 (2013).

[15] A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz,O. Painter, and H. J. Kimble, Superradiance for AtomsTrapped along a Photonic Crystal Waveguide, Phys. Rev.Lett. 115, 063601 (2015).

[16] R. Palacino, R. Passante, L. Rizzuto, P. Barcellona, andS. Y. Buhmann, Tuning the collective decay of two en-tangled emitters by means of a nearby surface, J. Phys.B, 50, 154001 (2017).

[17] R. Incardone, T. Fukuta, S. Tanaka, T. Petrosky, L. Riz-zuto, and R. Passante, Enhanced resonant force betweentwo entangled identical atoms in a photonic crystal, Phys.Rev. A 89, 062117 (2014).

[18] V. Notararigo, R. Passante, and L. Rizzuto, Resonanceinteraction energy between two entangled atoms in a pho-tonic bandgap environment, Sci. Rep. 8, 5193 (2018).

[19] G. Fiscelli, L. Rizzuto, R. Passante, Resonance energytransfer between two atoms in a conducting cylindricalwaveguide, Phys. Rev. A 98, 013849 (2018).

[20] F. Armata, S. Butera, G. Fiscelli, R. Incardone, V. No-tararigo, R. Palacino, R. Passante, L. Rizzuto, and S.Spagnolo, Effect of boundaries on vacuum field fluctua-tions and radiation-mediated interactions between atoms,J. Phys: Conf. Series 880, 012064 (2017).

[21] E. Shahmoon and G. Kurizki, Dispersion forces insidemetallic waveguides, Phys. Rev. A 87, 062105 (2013).

[22] E. Shahmoon, I. Mazets, and G. Kurizki, Giant vacuumforces via trasmission lines, PNAS 111, 10485 (2014).

[23] G. Fiscelli, L. Rizzuto, and R. Passante, Dispersion Inter-action between Two Hydrogen Atoms in a Static ElectricField, Phys. Rev. Lett. 124, 013604 (2020).

[24] V. V. Dodonov, Current status of the dynamical Casimireffect, Phys. Scr. 82, 038105 (2010).

[25] C. Braggio, G. Bressi, G. Carugno, C. Del Noce, G.Galeazzi, A. Lombardi, A. Palmieri, G. Ruoso, D.A.Zanello, A novel experimental approach for the detec-tion of the dynamic Casimir effect. Europhys. Lett. 70,754 (2005).

[26] V.V. Dodonov, Fifty Years of the Dynamical Casimir Ef-fect, Physics 2, 67 (2020).

[27] M. Antezza, C. Braggio, G. Carugno, A. Noto, R. Pas-sante, L. Rizzuto, G. Ruoso, and S. Spagnolo, Optome-chanical Rydberg-atom excitation via dynamic Casimir-Polder coupling, Phys. Rev. Lett. 113, 023601 (2014).

[28] S. Bosco, J. Lindkvist, and G. Johansson, Simulatingmoving cavities in superconducting circuits., Phys. Rev.

A 100, 023817 (2019).[29] A. Ferreri, M. Domina, L. Rizzuto, and R. Passante,

Spontaneous Emission of an Atom Near an OscillatingMirror, Symmetry 11, 1384 (2019).

[30] A. W. Glaetzle, K. Hammerer, A. J. Daley, R. Blatt, andP. Zoller, A single trapped atom in front of an oscillatingmirror, Opt. Commun. 283, 758 (2010).

[31] G. Calajo, L. Rizzuto, and R. Passante, Control of spon-taneous emission of a single quantum emitter througha time-modulated photonic-band-gap environment, Phys.Rev. A 96, 023802 (2017).

[32] L. Rizzuto, R. Passante, and F. Persico, DynamicalCasimir-Polder energy between an excited- and a ground-state atom, Phys. Rev. A 70, 012107 (2004).

[33] R. Vasile and R. Passante, Dynamical Casimir-Polder en-ergy between an atom and a conducting wall, Phys. Rev.A 78, 032108 (2008).

[34] R. Messina, R. Vasile, and R. Passante, DynamicalCasimir-Polder force on a partially dressed atom near aconducting wall, Phys. Rev. A 82, 062501 (2010).

[35] F. Armata, R. Vasile, P. Barcellona, S. Y. Buhmann,L. Rizzuto, and R. Passante, Dynamical Casimir-Polderforce between an excited atom and a conducting wall,Phys. Rev. A 94, 042511 (2016).

[36] P. Barcellona, R. Passante, L. Rizzuto, and S. Y. Buh-mann, Dynamical Casimir-Polder interaction between achiral molecule and a surface, Phys. Rev. A 93, 032508(2016).

[37] P. Barcellona P, R. Passante, L. Rizzuto, and S. Y. Buh-mann, van der Waals interactions between excited atomsin generic environments, Phys. Rev. A 94, 012705 (2016).

[38] H. R. Haakh, C. Henkel, S. Spagnolo, L. Rizzuto, and R.Passante, Dynamical Casimir-Polder interaction betweenan atom and surface plasmons, Phys. Rev. A 89, 022509(2014).

[39] S. Filipp, A.F. van Loo, M. Baur, L. Steffen, and A. Wall-raff, Preparation of subradiant states using local qubitcontrol in circuit QED, Phys. Rev. A 84, 061805 (2011).

[40] J. A. Mlynek, A. A. Abdumalikov, C. Eichler , and A.Wallraff, Observation of Dicke superradiance for two arti-ficial atoms in a cavity with high decay rate, Nat. Comm.5, 5186 (2014).

[41] D. P. Craig and T. Thirunamachandran, MolecularQuantum Electrodynamics (Dover Publ., Mineola, NY,1998).

[42] C. E. Wieman, D. E. Pritchard, and D. J. Wineland,Atom cooling, trapping, and quantum manipulation, Rev.Mod. Phys. 71, 253 (1999).

[43] G. Compagno, R. Passante, and F. Persico, Atom-fieldinteractions and dressed atoms (Cambridge UniversityPress, Cambridge, UK, 1995).

[44] A. Salam, Molecular quantum electrodynamics in theHeisenberg picture: A field theoretic viewpoint, Int. Rev.Phys. Chem. 27, 405 (2008).

[45] R. Passante, Dispersion interactions between neutralatoms and the quantum electrodynamical vacuum, Sym-metry 10, 735 (2018).

[46] E. A. Power and T. Thirunamachandran, Quantum elec-trodynamics in a cavity, Phys. Rev. A 25, 2473 (1982).