Yan Peng, Shijing Chengand Wenting Zhou

1 Department of Physics,Synergetic Innovation Center for Quantum Effects and Applications and Institute of Interdisciplinary Studies, Hunan Normal University, Changsha 410081, China

2 School of Physics and Information Engineering, Shanxi Normal University, Taiyuan 030031, China

3 Department of Physics,School of Physical Science and Technology,Ningbo University,Ningbo 315211,China

Abstract We study the interaction potential of two nonidentical ground-state atoms coupled to a scalar field in a vacuum by separately calculating the contributions of vacuum fluctuations and those of the radiation reaction of the atoms.Both cases of atoms in a free space and in parallel or vertical alignment to a reflecting boundary are considered.For the former case, we find that the leading-order interaction potential in the region λA ?L ?λB exhibits the same separationdependence as that in the region L ?λA ?λB, where L, λA and λB are respectively the interatomic separation and the transition wavelengths of two atoms with λA ?λB.For the latter case, we find that boundary-induced modifications are very remarkable when L ?z, with z characterizing the separation between the two-atom system and the boundary.Particularly,when L further satisfies L ?λA and L ?λB, the interaction potential in the parallel- and the verticalalignment cases respectively scales as z4L-7 and z2L-5,the L-dependence of which is one order higher than those of two atoms in regions where L ?z and meanwhile L ?λA or/and L ?λB.Our results suggest that retardation for the interaction potential of two nonidentical atoms with remarkably distinctive transition frequencies happens only when the interatomic separation is much greater than the transition wavelengths of both atoms.

Keywords: boundary effects, interatomic interaction potential, nonidentical atoms with remarkably distinctive transition frequencies

1.Introduction

In a quantum sense, vacuum fluctuates all the time and vacuum fluctuations are responsible for various well-known phenomena, for instance, the interaction potential between two neutral polarizable atoms (molecules) [1-3].When an atom is perturbed by electromagnetic vacuum fluctuations,an instantaneous electric dipole is induced which then emits a radiative field acting on the other atom, and thus the two atoms are correlated and an interaction potential is resulted.It is now textbook knowledge that the interaction between two inertial atoms in a vacuum is always attractive, and its separation-dependence in the far regionL?λ is, due to the effect of retardation, one order higher than that in the near regionL?λ,whereLand λ are respectively the interatomic separation and the atomic transition wavelength [2].Usually,the interatomic interactions in the two regionsL?λ andL?λ are referred to as the van der Waals (vdW) and the Casimir-Polder (CP) interactions, respectively.

During the past decades, the interatomic interaction has been extensively studied in various circumstances, such as for atoms in a thermal bath [4-12], in synchronous accelerated motion in a vacuum[13,14],and in the vicinity of boundaries[10,15,16].In almost all aforementioned studies,the potential is analyzed for atoms with almost the same transition frequency and so the role of different frequencies of the atoms is rarely much studied.Although the situation of natural atoms with remarkably different principal transition wavelengths is not very common, there in principle exist some special cases in which the atomic transition wavelengths differ obviously, for example, a hydrogen atom in the vicinity of a Cesium atom,whose first transition wavelengths are respectively ～1218 ? and ～8946 ?.Let us note that when two nonidentical atoms with dramatically different transition frequencies ωA?ωBare considered,three rather than two length parameters single out,i.e.Landλξ=2πω-ξ1with ξ=A,B,so that the full region of the interatomic separationLcan be divided into three typical regionsL?λA?λB, λA?L?λB, and λA?λB?L.Then interesting questions arise as to how the interatomic interaction will behave in different regions.Intuitively,one may anticipate that the interaction in regionsL?λA?λBand λA?λB?Lbehaves similarly to their counterparts in regionsL?λ andL?λ in the two-identical-atom case.But for the intermediate region λA?L?λBwhich does not exist in the two-identicalatom case,one may not be so sure about the answer since there seems to be at least three possibilities: it may exhibit a vdWlike behavior becauseL?λB, or a CP-like behavior since λA?L, or a new behavior completely distinctive from both the vdW and the CP interactions sinceLis not simultaneously much smaller or larger than both λAand λB.

In this work, we try to answer these questions by examining the interaction potential of two ground-state atoms with remarkably distinctive transition frequencies.For this purpose, we first consider the potential of two atoms in interaction with a scalar field in a free space and then that of two atoms near an infinite reflecting boundary.Concerning the boundary effects, it is worth re-mentioning that although there have already been some related discussions on the interaction potential of two ground-state atoms in a vacuum near a boundary, i.e.[15, 16], they are incomplete, since the interaction potentials were considered in only a few regions,let alone the role of atomic distinctive transition frequency.In our present work, we will fill this gap by giving a complete discussion for the interaction potential in full regions.As we shall demonstrate, we find some unique behaviors for the interaction potential in both the two-identical-atom and the two-nonidentical-atom cases.To the best of our knowledge,even those results for the two-identical-atom case have never been reported before.We exploit the model of atoms in interaction with a scalar field,which although less realistic as compared with the model of atoms in interaction with an electromagnetic field, helps to reduce, to a large extent, the complexity of calculation and meanwhile still preserves the essential physics as the underlying mechanism of atomic radiative properties in both the scalar and electromagnetic field cases is similar.That is also the reason why it was frequently utilized for the study of atom-field interaction[13, 14, 17-23].

We evaluate the interaction potential with an approach distinctive from those exploited in [15, 16], i.e.the DDC formalism(the formalism proposed by Dalibard,Dupont-Roc and Cohen-Tannoudji) [17, 24, 25], which is a powerful method for the study of radiative properties of atoms in interaction with various quantum fields.The original DDC formalism was established for the study of evolution of a small quantum system in interaction with a large reservoir[24, 25], and then it was widely exploited to study various atomic radiative phenomena, such as the excitation rate and energy shifts of a noninertial atom [18-22, 26, 27] (just to name a few), which are all second-order perturbation effects.Recently, it was generalized from the second order to the fourth order in an attempt to deal with the interaction between two ground-state atoms which is of the fourth order [17].It was found that the interaction potential is generally ascribed to the contributions of vacuum fluctuations and those of the radiation reaction of the atoms, i.e.the contributions of the free field which exists even when there is no coupling between the atoms and the field and those of the source field which is induced by the atom-field interaction.With this approach, we will show how the interaction potential of two nonidentical atoms has resulted from joint contributions of the vacuum fluctuations and the radiation reaction of atoms,discuss how remarkable discrepancy in the atomic transition frequency affects the interatomic interaction potential, and demonstrate the boundary effects.

The paper is organized as follows.Section 2 is devoted to the interaction potential of two atoms in a free space,for which we first derive general expressions and then analyze in detail the potential in three typical regions.Then in section 3, we demonstrate the boundary effects by concretely considering the potential of two nonidentical atoms in parallel or vertical alignment to the boundary.The effects of large discrepancy in the atomic transition frequency will also be analyzed detailedly.We give a summary of our work in section 4.Throughout the paper, we exploit units that ?=c=1.

2.Interatomic interaction potential in a free space

We now consider the interaction potential of two static groundstate atomsAandBin a free space.Assume that the atoms are two-level ones and their ground and excited states are denoted by |gξ〉 and |eξ〉 respectively with ξ=A,B, and then the Hamiltonian of the two-atom system can be expressed as

withtthe coordinate time,ωξthe transition frequency of atom ξ,andIn a vacuum, the atoms are inevitably perturbed by the fluctuating field

where k is the wave vector of field modes,akdenotes the annihilation operator with momentum k,and‘H.c.’ represents the Hermitian conjugate, respectively.Then the Hamiltonian of the scalar field reads

and that describing the atom-field interaction can be depicted by

where μ is a small coupling constant,is the monopole moment operator of atom ξ withR+ξ= ∣eξ〉〈gξ∣andR-ξ= ∣gξ〉〈eξ∣.

As a result of the interaction between the atoms and the field,each atom is endowed with a monopole and meanwhile emits a radiative field superimposed on the free field which exists even when there is no coupling between the atoms and the field,and thus both the free and the atomic radiative fields contribute to the interatomic interaction.As we have mentioned in the introduction, the fourth-order DDC formalism[17]provides a general framework to evaluate the interaction potential by separately calculating the contributions of the free field and those of the atomic radiative field.Here we evaluate the interatomic interaction potential in that way.

2.1.General results

Following [17] and as we have detailedly demonstrated in appendix A.1, we first derive the symmetric correlation function and linear susceptibility of the field at the two positions of the atoms as well as the two statistical functions of the atoms,use them in the basic formulas describing the vfand rr-contributions (equations (64) and (66) of [17]), do some simplifications, and then obtain

which are the contributions of the free field in vacuum (vfcontribution) and those of the radiation reaction of the atoms(rr-contribution), respectively.The total interaction potential is then given by the sum of the two contributions

The above equation is the general expression of the total interaction potential of two nonidentical atoms with an arbitrary separationLin a free space while equations (5) and (6)are those of the vf- and rr-contribution to it.Obviously,although both the vf- and the rr-contributions contain oscillatory terms with respect toL, the total interaction potential exhibits a monotonic decreasing behavior with the increase ofL,as a result of the perfect cancellation of oscillatory terms inIn addition, equation (7) suggests that the total interaction potential is always negative.

Further simplifications of these results with general values ofLare formidable,however,we can still obtain some simple analytical results in some limiting cases which are of particular physical interest, i.e.regions where the interatomic separationLis much smaller or larger than the transition wavelengths of the atomsIn the following, we assume ωA?ωBand then the three characteristic lengthsL,λAand λBin this problem naturally divide the full region ofLinto three typical regions, i.e.L?λA?λB, λA?L?λB,and λA?λB?L.We next first analyze the interaction potentials in the two regionsL?λA?λBand λA?λB?Lwhich reduce to the vdW regionL?λ and the CP regionL?λ for two identical atoms with the same transition wavelength λ, and then the region λA?L?λBwhich does not exist for two identical atoms.

2.2.Interaction potential in limiting cases

WhenL?λA?λB, we find that equations (5) and (6) are approximated by

Comparing the absolute values of the aboveandwe find thatwhich is qualitatively similar to those of two identical atoms in a free space,and thus

which exhibits theL-2-dependence and isc2times of that of two identical ones with the transition frequency ω.

When λA?λB?L, equations (5) and (6) reduce to

Figure 1.Sketch of two static atoms in parallel or vertical alignment with respect to an infinite boundary.

which shows theL-3-dependence one order higher than that in the near regionL?λA?λB.This phenomenon is a result of retardation and it also happens in the two-identical-atom case, i.e.the interaction potential of two identical atoms in regionsL?λ andL?λ also respectively displays theL-2-andL-3-dependence.However, despite this similarity, let us note that the strength of this potential for two nonidentical atoms, as compared with that of two identical ones, is modified by the coefficientc3characterizing the discrepancy in the atomic transition frequency.

Now let us turn our attention to the interaction potential in the region λA?L?λBwhich does not exist for the twoidentical-atom case.We find that when λA?L?λB,equations (5) and (6) are respectively simplified into

Obviously, the separation-dependence of these results is very different from those in the far region λA?λB?L;while they resemble, to some extent, those in the near regionL?λA?λB(equations(8)and(9))in the sense that both the vf-contribution in the present region and that inL?λA?λBlead to a positive potential although the separation-dependences in two regions are not identical,the leading terms of the rr-contributions in both regions are identical, and the rrcontribution dominates over the vf-contribution.As a result,the total potential which mainly comes from the rr-contribution is approximated by

This potential in the leading order is identical to that in the near regionL?λA?λB, and distinctions in the total interaction potential in these two regions appear from the subleading order (see equations (8) and (9)).Further comparing this result with equations (10) and (13), it is obvious that retardation for the interaction potential of two nonidentical atoms never appears in the region λA?L?λB;it shows up only when the interatomic separation is much larger than the transition wavelengths of both atoms.

So far,we have analyzed the interaction potential of two nonidentical atoms in a free space.We next focus on that of two atoms near a boundary.

3.Interatomic interaction potential near a boundary

As shown in figure 1, we now consider the interaction potential of two nonidentical atomsAandBfixed near a perfectly reflecting boundary in two different configurations,i.e.atoms aligned with a separationLparallel (‘‖’) or perpendicular (‘⊥’) to the boundary.

Following similar procedures as we demonstrated in section 2 and appendix A.2, we find that the vf- and rrcontributions to the interaction potential of the two atoms near the boundary can be expressed as a sum of their counterparts in a free spaceand other terms crucially dependent on the relative position of the two-atom system to the boundary and denoted by, i.e.

with

Here α=‖ or ⊥ is introduced to identify the two distinctive alignments,,R⊥=(L+2z), and

Hereafter,zdenotes the separation between the boundary and the atoms in the parallel-alignment case or the separation between the boundary and the atom closer to the boundary(atomA) in the vertical-alignment case.Then the total interaction potential follows

boundary-induced modifications.Now one can easily judge that there are four characteristic lengths in this problem,i.e.L,λξ=2πω-ξ1with ξ=A,B,andz.Since the above expressions are generally difficult to simplify, we next analyze them in some limited cases.Similar to the previous section, we still assume that λA?λB,then the four parametersL,λA,λB,andzdivide the full region ofLinto twelve regions.As we shall demonstrate, the potentials in some of these regions, as compared with that in a free space,are slightly modified while those in the other regions are severely altered.

3.1.Weak effects of the boundary

We find that the total interaction potential of the two nonidentical atoms as well as the vf-and rr-contributions to it are slightly modified in both the parallel- and vertical-alignment cases whenL?z, i.e.when the interatomic separation is much smaller thanzcharacterizing the relative position of the two-atom system to the boundary.Particularly whenzand λξfurther satisfyz?λξ, the modifications in the total interaction potential equally come from the vf-and rr-contributions;while ifzis much smaller than at least one of λAand λB,they mainly come from the rr-contribution.

3.1.1.Slight modifications equally from the vf- and rrcontribution.WhenL?λA?λB?zor λA?L?λB?zor λA?λB?L?z, the boundary induces slight modifications for both the vf- and rr-contributions to the interaction potential of atoms in both the parallel- and vertical-alignment cases, i.e.And to the leading order, they are identical in the two alignment cases,i.e.

which manifest obvious oscillation behaviors with respect toz.Since the amplitudes of oscillatory terms in these two equations are much greater than the non-oscillatory terms,bothcan be positive or negative and even null, depending on the concrete values of ρ andz,suggesting that both the vf-and rr-contributions can either be enhanced or weakened or even unaltered by the presence of the boundary.However, their joint contribution leads to a non-oscillatory modification, i.e.

Noteworthily, this modification is positive, while the interaction potential in a free spaceis definitely negative with the absolute value much larger (see equations (10), (13) and (16)), and thus the presence of the boundary in these regions only slightly weakens the interaction between the two nonidentical atoms.

The above results are about the interaction potential of two nonidentical atoms with remarkably distinct frequencies.Then does there exist any essential differences between this potential and that of two identical atoms?Note that the region λA?L?λB?zmakes no sense for two identical atoms,while the other two regionsL?λA?λB?zand λA?λB?L?zfor two nonidentical atoms respectively correspond toL?λ ?zand λ ?L?zfor two identical ones,in which the boundary-induced modifications for the vfand rr-contributions are

Adding the above two equations up, we obtain

which is the boundary-induced modification for the total interaction potential of two identical atoms, exhibiting the same monotonic separation-dependence ofL-1as that of the interaction potential of two nonidentical atoms in a free space(see equation (25)).And in comparison, boundary-induced modifications for the total interaction potential of two nonidentical atoms in regionL?λA?λB?zor λA?L?λB?zor λA?λB?L?z, as demonstrated by equation (25), istimes of that of two identical ones with frequency ω in regionsL?λ ?zand λ ?L?z, i.e.remarkable discrepancy in the atomic transition frequency rescales the strength of boundary-induced modifications for the total interaction potential but never changes its separationdependence.

3.1.2.Slight modifications mainly from the rr-contribution.We find that whenL?λA?z?λBorL?z?λA?λBor λA?L?z?λB, the vf-contributions in the leading order behave quite the same in both the parallel- or verticalalignment cases and they are allz-dependent:

which in regionsL?λA?z?λBand λA?L?z?λBare partly fromand partly from, while in regionL?z?λA?λBis actually the leading-orderz-dependent term ofwhich is singled out since those strongerzindependent terms inandcancel out completely.The rr-contribution to the leading order is,however, not altered by the presence of the boundary, i.e., and boundary-induced modifications for it in the three regions are identical in the leading order with distinctions showing up at the subleading order:

Notice that here the second term is the leading-order, which behaves quite the same in three regions, in sharp contrast to the leading-orderbehaving distinctively in the three regions (see equation (29)).

Comparing the absolute value of equation (30) with that of equation (29), one finds that the former is much greater,and thus the total interaction potential is approximated by

So, as a result of the presence of the boundary, the interaction potential of two nonidentical atoms in regionsL?λA?z?λB,L?z?λA?λB, and λA?L?z?λBis slightly weakened with modifications mainly from the rrcontribution.

At the end of this subsection,it is worth pointing out that although the leading-orderandfor two nonidentical atoms, as we have shown in equations (23),(24), (29), and (30), are identical in every region we have sofar considered for both the parallel- and vertical-alignment cases, i.e.regionsL?λA?λB?z, λA?L?λB?z,λA?λB?L?z,L?λA?z?λB,L?z?λA?λBand λA?L?z?λB,they actually differ in higher orders.As we shall show in the next subsection, things become quite different in the other six regions where boundary-induced modifications become very important, and distinctions inandin two alignment cases exist even in the leading order.

3.2.Strong effects of the boundary

We find that whenz?L, i.e.when the separation between the two-atom system and the boundary is much smaller than the interatomic separation, the boundary induces remarkable modifications for the interaction potential.As a result, the total potential in the leading order isz-dependent, and it should be mainly attributed to the rr-contribution ifLis further much smaller than at least one of λAand λB,and to equal vf- and rr-contributions ifLis much larger than both λAand λB.

3.2.1.Remarkable modifications mainly from the rrcontribution.When the atoms are in parallel alignment to the boundary in regionz?L?λA?λBor λA?z?L?λBorz?λA?L?λB, the vf-contribution in the leading order behaves differently in three regions as

and it is much greater than(δE)‖vf.As a result, the interaction potential in these three regions should be mainly ascribed to the rr-contribution, and thus we have

The above results are about the interaction potential of two atoms in parallel alignment in regionz?L?λA?λBor λA?z?L?λBorz?λA?L?λB.Then how about the results in these regions in the vertical-alignment case?We find that both the leading-order vf- and rr-contributions as well as the total interaction potential in the vertical-alignment case are accuratelyz-2L2times of their counterparts in the parallel-alignment case (equations (32)-(34)), and thus characters of the interaction potential in both cases are qualitatively the same.However, sincez-2L2?1, the interatomic interaction potential in the vertical-alignment case scales asz2L-4and it is much stronger than that in the parallel-alignment case.

3.2.2.Remarkable modifications equally from the vf- and rrcontributions.When the two atoms are located in regions whereL?zand meanwhileL?λAandL?λB, i.e.

λA?λB?z?Lor λA?z?λB?Lorz?λA?λB?L,boundary-induced modifications are also very remarkable, but the interaction potential is no longer mainly contributed by the radiation reaction of the atoms as we have demonstrated in the previous sub-subsection,but equally contributed by the vacuum fluctuations and the radiation reaction of the atoms.To be specific, if the atoms are in parallel alignment to the boundary,the vf- and rr-contributions are respectively

which solely oscillate with the interatomic separationL.However, their summation gives rise to a total interaction potential scaling monotonically withLas

Note that thisL-7-dependence of the interaction potential is sharply different from the ～L-3-dependence in a free space(equation (13)), and as compared with the interaction potential in regionz?L?λA?λBor λA?z?L?λBorz?λA?L?λBdiscussed in the previous sub-subsection,it is one order higher, and this is retardation.In addition, let us also point out that here we have only preserved in equation (37) the leading-order interaction potential which is identical in three regions,and actually distinctions appear in higher orders.

Now a comparison between this interaction potential of two nonidentical atoms and that of two identical ones is in order.When two identical atoms are considered, the two regions λA?λB?z?Landz?λA?λB?Lof two nonidentical atoms reduce to λ ?z?Landz?λ ?L,where the vf- and rr-contributions are respectively approximated by

Then how about the interaction potential of two nonidentical atoms in vertical alignment to the boundary in the three regions λA?λB?z?L, λA?z?λB?Landz?λA?λB?L? The answer is that they are qualitatively quite similar to what we have just demonstrated for those of atoms in the parallel-alignment case, i.e.both the vf- and rrcontributions oscillate withLand they solely correspond to an either attractive or repulsive and even vanishing interaction force but jointly give rise to a total interaction potential scaling monotonically withL.However,quantitatively speaking,the interaction potentials in these two alignment cases are distinctive, since

which is proportional toz2L-5while that in the parallelalignment case is proportional toz4L-7(see equation(37)),and the former is much stronger.Note also that thisL-dependence of the interaction potential in regions λA?λB?z?L,λA?z?λB?L, andz?λA?λB?Lin the vertical-alignment case is one order higher than those in regionsz?L?λA?λBor λA?z?L?λBorz?λA?L?λBwhere the interatomic separationLis much smaller than at least one of λAand λB, and this is retardation.This conclusion together with that drawn below equation (37) indicates that retardation which happens in a free space also occurs when two atoms are located near a boundary,if the interatomic separationLis much larger than the transition wavelengths of both atoms λAand λB.

4.Summary

In this paper, we studied the interaction potential of two nonidentical static ground-state atomsAandBwith a constant separationLin a free space or in the vicinity of a completely reflecting boundary by calculating the separate contributions of the vacuum fluctuations and the radiation reaction of the atoms.Besides deriving general expressions for the interaction potential of the two atoms in a free space and those of two atoms in parallel or vertical alignment with respect to the boundary, we discussed the interaction potential in detail in various typical regions with special attention paid to the roles of a discrepancy in the atomic transition frequency as well as the presence of the boundary.

Assuming that λA?λB,i.e.the transition wavelength of atomAis much smaller than that of atomB,then for the two atoms in a free space,three characteristic length parametersL,λA, and λBdivide the full region ofLinto three typical regionsL?λA?λB, λA?L?λBand λA?λB?L.We found that the separation-dependence of both the vf- and rrcontributions as well as the total interaction potential of the two atoms in regionL?λA?λBare the same as their counterparts of two identical atoms in the regionL?λ, and the discrepancy in the atomic transition frequency only modifies the strength of the interaction.As a result, the total interaction potential mainly comes from the rr-contribution scales asL-2.In contrast,both the vf-and rr-contributions to the interaction potential of the two atoms in region λA?λB?Lwhich solely oscillate with the interatomic separationLare obviously altered as compared with that of two identical atoms in region λ ?L.The total interaction potential which equally comes from the vf- and rr-contributions, however, displays the sameL-3-dependence as that of two identical atoms in regionL?λ,and it is one order higher than that in the regionL?λA?λB.So in this region, the dramatic discrepancy in the atomic transition frequency also only rescales the strength of the total interatomic interaction but never alters its separation-dependence.In the region λA?L?λBwhich does not exist for two identical atoms,the interaction potential which mainly comes from the rrcontribution behaves in the leading order quite the same as that in the regionL?λA?λB, though the leading-order vfcontribution and the subleading-order rr-contribution are obviously distinctive from their counterparts in the regionL?λA?λB.These results suggest that retardation for the interaction potential of two nonidentical atoms in a free space appears only when the interatomic separation is much greater than the transition wavelengths of both atoms.

For two atoms near a boundary,we found that boundaryinduced modifications are negligible whenz?Land remarkable whenz?L, withzcharacterizing the atomboundary separation in the parallel-alignment case and the separation between the boundary and the atom closer to the boundary (atomA) in the vertical-alignment case.Similar to that in a free space,although a remarkable discrepancy in the atomic transition frequency may lead to obvious change for the sole separation-dependence of the vf-and rr-contribution,it only rescales the strength of the total potential but never alters its separation-dependence.

To be specific,whenz?L,the four length parametersL,z,λA,and λBwith λA?λBdivide the full region ofLinto six regions which we classify into two sets with the first set including regionsL?λA?λB?z, λA?L?λB?z, and λA?λB?L?z, and the second set includingL?λA?z?λB,L?z?λA?λB,and λA?L?z?λB.In the first set of regions which are in common thatz?Land meanwhile,zis much larger than both λAand λB, boundaryinduced modifications for both the vf- and rr-contributions exhibit obvious distinctive oscillation behaviors from those in regionsL?λ ?zand λ ?L?zin the two-identical-atom case, and they equally slightly weaken the interaction potential,as compared with those in a free space.While in the second set of regions which are in common thatz?Landzis much smaller than at least one of λAand λB, boundaryinduced modifications for the rr-contribution are much larger than those for the vf-contribution, and as a result, the interatomic interaction potential is also slightly weakened as compared with that in a free space.These conclusions are valid for both the parallel- and vertical-alignment cases.

Whenz?L, the four parametersL,z, λA, and λBwith λA?λBalso divide the full region ofLinto two sets of typical regions withz?L?λA?λB,λA?z?L?λBandz?λA?L?λBthe first set and λA?λB?z?L,λA?z?λB?Landz?λA?λB?Lthe second set.We find that in both sets of regions, boundary-induced modifications for both the vf- and rr-contributions in the leading order arez-dependent.As a result,the total interaction potential in the first set of regions mainly comes from the rrcontribution scales asz4L-6in the parallel-alignment case and asz2L-4in the vertical-alignment case; while that in the second set which equally comes from the vf- and rr-contributions scales asz4L-7in the parallel-alignment case and asz2L-5in the vertical-alignment case.These results suggest that retardation happens for two nonidentical atoms only when the interatomic separation is much greater than both λAand λB.

Acknowledgments

We thank Professor Hongwei Yu for fruitful discussions and helpful suggestions.This work was supported in part by the NSFC under Grant Nos.11690034, 12075084, 11875172,12047551 and 12105061, and the KC Wong Magna Fund in Ningbo University.

Appendix.Derivations for the interatomic interaction potential

According to [17], the vf- and rr-contributions to the interaction potential of two static ground-state atoms in vacuum can be expressed as

and

whereCF(xA(t),xB(t＇))andχF(xA(t),xB(t＇))are respectively the symmetric correlation function and the linear susceptibility of the field defined as

with |0〉 denoting the vacuum state and

the free field operator withgk=[( 2π)32ωk]-12and ωk=k,and

are respectively the symmetric and antisymmetric statistical functions of atom ξ with

The two statistical functions equations (A6) and (A7),after the insertion of equation (A8), can be further simplified into

So to obtain the vf- and rr-contributions, we should first calculate the symmetric correlation function and the linear susceptibility of fieldCF(xA(t),xB(t＇))andχF(xA(t),xB(t＇)).

A.1.Interaction potential of two atoms in a free space:derivations for equations (5) and (6)

Using equations (A5) in (A3) and (A4) and doing some simplifications, we get

withLthe interatomic separation.Now insert equations(A9)-(A12)into(A1)and(A2),perform the triple integrations with respect tot3,t2andt1with the assumption of a sufficiently long time intervalt-t0→∞, and we obtain

which are respectively the vf- and rr-contributions to the interaction potential of two nonidentical atoms in a free space in a vacuum.Further performing the ω2-integration in these two equations, we then obtain equations (5) and (6) in section 2.

A.2.Interaction potential of two atoms near a boundary:derivations for equations (18) and (19)

For two atoms near the boundary, we choose coordinates such that the ‘xoy’ plane coincides with the boundary (see figure 1),and label the atom closer to the boundary in the vertical-alignment case byA.The interaction potential then can be derived by following similar procedures as we have demonstrated in appendix A.1.Here to derive the symmetric correlation function and the linear susceptibility of feildC F(xA(t),xB(t＇))andχF(xA(t),xB(t＇)), let us note that the two-point correlation function of the scalar feild 〈0∣φf(x)φf(x＇) ∣0〉at two arbitrary positionsxandx＇ near a boundary is [28]

where ? is a positive infinitesimal and∣Δx?∣=Now use this relation in equations(A3)and(A4),do the Fourier transform,and we arrive at

where the subscript α=‖, ⊥is introduced to identify the two distinctive alignments,andR⊥=L+2z.

Next,we insert equations(A16),(A17),(A9),(A10)into equations (A1) and (A2), perform the triple integrations with respect tot3,t2andt1with the assumptiont-t0→∞, and obtain

which are the vf- and rr-contributions to the interaction potential.After the evaluation of the ω2-integration,these two equations are then further reduced to