Dipole-dipole interaction in cavity QED: The weak-coupling, nondegenerate regime

PHYSICAL REVIEW A 96, 042714 (2017) Dipole-dipole interaction in cavity QED: The weak-coupling, nondegenerate regime M. Donaire,1,2,*J. M. Muñoz-Castañeda,3and L. M. Nieto1 1Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain 2Facultad de Educación de Palencia, Campus La Yutera, Universidad de Valladolid, Avd. Madrid, 44, 34004 Palencia, Spain 3Departamento de Física, ETSIAE, Universidad Politécnica de Madrid, 28049 Madrid, Spain (Received 31 January 2017; revised manuscript received 7 September 2017; published 24 October 2017) We compute the energies of the interaction between two atoms placed in the middle of a perfectly reflecting planar cavity, in the weak-coupling nondegenerate regime. Both inhibition and enhancement of the interactions can be obtained by varying the size of the cavity. We derive exact expressions for the dyadic Green’s function of the cavity field which mediates the interactions and apply time-dependent quantum perturbation theory in the adiabatic approximation. We provide explicit expressions for the van der Waals potentials of two polarizable atomic dipoles and the electrostatic potential of two induced dipoles. We compute the van der Waals potentials in three different scenarios: two atoms in their ground states, two atoms excited, and two dissimilar atoms with one of them excited. In addition, we calculate the phase-shift rate of the two-atom wave function in each case. The effect of the two-dimensional confinement of the electromagnetic field on the dipole-dipole interactions is analyzed. This effect depends on the atomic polarization. For dipole moments oriented parallel to the cavity plates, both the electrostatic and the van der Waals interactions are exponentially suppressed for values of the cavity width much less than the interatomic distance, whereas for values of the width close to the interatomic distance, the strength of both interactions is higher than their values in the absence of cavity. For dipole moments perpendicular to the plates, the strength of the van der Waals interaction decreases for values of the cavity width close to the interatomic distance, while it increases for values of the width much less than the interatomic distance with respect to its strength in the absence of cavity. We illustrate these effects by computing the dipole-dipole interactions between two alkali atoms in circular Rydberg states. DOI: 10.1103/PhysRevA.96.042714 I. INTRODUCTION Modifying the interaction of atoms with the electromagnetic (EM) field by means of a cavity is at the origin of cavity QED [1,2]. In the first place, a perfectly reflecting cavity reduces the density of EM states accessible to the spontaneous emission of a single atom. This results in an enhancement of the atomic lifetime as well as in a shift of the atomic levels. On the other hand, the strong coupling between the cavity modes and the atomic charges drives the coherent exchange of excitations between the atom and the cavity field [3]. For the case of a perfectly reflecting planar cavity, the atomic level shifts [4–8] and the modified lifetimes of an excited atom have been well studied theoretically [9–11] and measured experimentally [1,2,12]. Ultimately, these effects make possible the coherent manipulation of quantum states, the entanglement between separated quantum systems [13,14], and the storage of quantum information [15–17]. Considering the cavity as a macroscopic system hardly affected by the presence of the atoms inside, the interaction of the free EM field with the cavity plates can be integrated out in an effective Hamiltonian. The resultant EM interactions of the atoms are commonly referred to as cavity-assisted interactions [13,18,19]. This is a good approximation, as long as the absorption and emission spectra of the atoms and the cavity material do not overlap, and as long as the time resolution of observation is much larger than the time *On leave from Laboratoire Kastler Brossel, UPMC-Sorbonnes Universités, CNRS, ENS-PSL Research University, Collège de France; [email protected]; [email protected] of flight of photons from the atoms to the cavity plates. Under these conditions, transient transitions average out and the effective cavity-assisted interactions become stationary. The net result is that photon states get dressed by multiplescattering processes with the cavity plates, and so does the photon propagator. It is the modes of the dressed EM field which are commonly referred to as cavity modes of the cavity field. On the other hand, dispersion forces between neutral atoms and between atoms and macroscopic surfaces are the result of the coupling of the quantum fluctuations of the EM field, the atomic charges, and the surface currents [9,19]. The interaction between an atom and a macroscopic surface is known as Casimir-Polder interaction [19–23], whereas the interaction betweentwoneutralatomsisknowngenerallyasvanderWaals interaction [9,19,24–30]—referred to as London interaction for interatomic distances smaller than the resonant wavelength [31]. In this article we combine both kinds of interactions in a three-body problem. We address the cavity-assisted dipole-dipole interaction between two atoms placed in the middle of a perfectly reflecting cavity, at zero temperature, see Fig. 1. This is a common setup in the generation of quantum entanglement with Rydberg atoms [13,32], also in the absence of a cavity [33]. In particular, we study the modification of the atom-atom interactions varying the size of the cavity. We compute several quantities of interest in the weak-coupling, nondegenerate regime. These are the van der Waals (vdW) potentials for the case that both atoms are in their ground states, for the case that both atoms are excited, and for the case that one atom is excited while the other, of a different kind, is in its ground state. In addition, we calculate the electrostatic potential between two induced atomic dipoles. We calculate in 2469-9926/2017/96(4)/042714(16) 042714-1 ©2017 American Physical Society DONAIRE, MUÑOZ-CASTAÑEDA, AND NIETO PHYSICAL REVIEW A 96, 042714 (2017) FIG. 1. Sketch of the setup of the problem. each case the corresponding phase-shift rate of the two-atom wave function. Physically, the vdW and electrostatic potentials can be observed through the forces experienced by each atom when placed inside harmonic traps, which are proportional to the displacements of the atoms with respect to their equilibrium positions in the absence of interaction. On the other hand, the phase shift of the two-atom wave function can be observed using atom interferometry. For instance, it is the shift that is observed in the binary interaction of Rydberg atoms through the measurement of population probabilities [14,34]. Concerning our approach, we apply time-dependent quantum perturbation theory, up to fourth order, in the electric dipole approximation [35]. Except for the degenerate interaction addressed at the end of Sec. V, all the calculations are carried out in the perturbative nondegenerate regime, where the energy difference between the intermediate and initial atomic states is much greater than the interaction energy and the atom-cavity coupling is weak. In particular, the latter implies that we cannot restrict ourselves to the one-cavity mode approximation of the Jaynes-Cummings Hamiltonian [2]. For the case of the interaction between excited atoms, the excitation is assumed adiabatic with respect to the detuning between the atomic species, which is a situation commonly encountered in experiments. The main achievements of this work are, in the first place, thecomputationoftheGreenfunctionforthecavity field which mediates the interaction between two atomic dipoles placed in the middle of the cavity. The effects of the two-dimensional confinement of the EM field are revealed. Second, we find out explicit expressions for the resonant components of the vdW potentials of each atom when excited. The difference between each atom’s potentials as well as their discrepancy with the phase-shift rate of the wave function are exposed. The paper is organized as follows. In Sec. II we describe the setup of the problem and compute the dyadic Green’s function of the cavity field. In Sec. III we calculate the vdW potentials and the phase shifts on two-atom systems in the nondegenerate regime. In Sec. IV we compute the electrostatic potential of two induced dipoles. All the formulas derived in the previous sections are applied in Sec. Vto the calculation of the dipole-dipole interactions between two circular Rydberg atoms in a cavity. We summarize the conclusions in Sec. VI. II. GREEN’S DYADIC OF THE CAVITY FIELD We aim at computing the cavity-assisted dipole-dipole interactions between two atoms, Aand B, placed in the middle of a perfectly reflecting planar cavity of thickness dand separated by a distance ralong an axis parallel to the cavity plates (see Fig. 1). These interactions are mediated by virtual photons created and annihilated at the position of each atom. Therefore the relevant quantity to be computed is the Green function of the cavity field that the atomic dipoles induce at the position of each other. To this end, we use the effective semiclassical approach outlined in the Introduction. The plates of the cavity are treated as passive and semiclassical objects which reflect photons with no losses. The free Hamiltonian of the atoms and the EM cavity field is H0=HA+HB+HEM, with HA= i ¯hωA i|AiAi|,H B= i ¯hωB i|BiBi|, HEM =0 2d3R[|E(R)|2+c2|B(R)|2] = k, ¯hω(a† k,ak,+1/2).(1) Here, |Ai,|Bidenote the ith states of atoms A,B, with energies ¯hωA iand ¯hωB i, respectively. The operators a† k,and ak,are the creation and annihilation operators of photons of frequency ω=ck, momentum ¯hk, and polarization vector , respectively, in terms of which the electric field operator reads E(R)= k E(−) k(R)+E(+) k(R) =i k,¯hck 2V0 [akeik·R−∗a† ke−ik·R],(2) with Vbeing a volume of quantization. The magnetic field operator relates to E(R) through Maxwell’s equation, ∇× E(R)=−∂B(R)/∂t. Our semiclassical approximation with regard to the interaction between the EM field and the cavity plates consists of assuming that the virtual photons created at the location of one of the atoms reflect off the plates any number of times before being absorbed either by itself or by the other atom. In each reflection process the dynamical excitation of the plates is discarded, and so is the time of flight of photons between any pair of scattering processes. Under these conditions, the intermediate photonic states and the EM vacuum can be considered as dressed by multiple reflection processes with the cavity plates [7,23]. Equivalently, the dressing can be assigned to the electric field operator within the framework of macroscopic QED [18,19]. The net resultistheeffectivediscretizationofthemodesoftheEMfield within the cavity. Mathematically, this is achieved by taking the cavity volume as the volume of quantization in Eq. (2), V= VCav, or equivalently, by imposing ideal boundary conditions in Maxwell’s equations for the electric and magnetic field. Those conditions consist in the components of the electric field parallel to the plates and of the associated magnetic field perpendicular to the plates going to zero [19,36,37]. Alternatively, we can write Maxwell’s equation for the dyadic Green’s function Gof the electric field induced at a point Rby a nonpolarizable electric dipole of frequency ω=ck placed at R, [ω2I−∇×∇×]G(R;R;k)=δ(3)(R−R)I,(3) 042714-2 DIPOLE-DIPOLE INTERACTION IN CAVITY QED: THE . . . PHYSICAL REVIEW A 96, 042714 (2017) and impose the boundary conditions at the plates [37], ˆ n×G(R;R;k)|Ron the plates =0, (ˆ n·∇)·[G(R;R;k)−k2δ(3)(R−R)]|Ron the plates =0, (4) where ˆ nis a unitary vector, outward-pointing and orthogonal to the plates. By doing so, the resultant components of the dyadic Green’s function for the electric field induced at a point (r,d/2) by a nonpolarizable electric dipole of frequency ω=ck placed at (0,d/2) are [37] G(r,d/2; k)=d2q 2(2π)2eiq·rq2 −k2 k2ρsin ρd (1 −cos ρd), (5) G⊥⊥(r,d/2; k)=d2q 2(2π)2eiq·rq2 ⊥−k2 k2ρsin ρd (1 −cos ρd), (6) G00(r,d/2; k)=d2q 2(2π)2eiq·rq2 k2ρsin ρd (1 +cos ρd), (7) where qis a two-dimensional reciprocal vector parallel to the plates and ρ≡k2−q2. The quantization axis is taken perpendicular to the plates, which is denoted by the index 0, while the indices and ⊥refer to the axes parallel to the plates, which are parallel and perpendicular to r, respectively (cf. Fig. 1). The off-diagonal components are all null. The photons mediating the interactions between the two atoms are created at the position of the atoms by the EM field operators which enter the interaction Hamiltonian, W. It reads, in the electric dipole approximation, W=WA+WB, with WA,B =−dA,B ·E(RA,B). Here dA,B are the electric dipole moment operators of each atom, Eis the electric field operator, andRA,B aretheclassicalpositionvectorsofthe atomic centers of mass, with r=RA−RB.Wis considered as a perturbation to H0. Higher multipole orders in the interaction would be relevant for values of ror din the order of a few times the atomic radius [36]. Throughout this work we restrict ourselves to values of rand dat least an order of magnitude larger than the atomic radii, such that the electric dipole approximation suffices. According to the fluctuation-dissipation theorem, the quadratic fluctuations of the electric field in the dressed vacuum, |˜ 0γ, read at zero temperature: ˜ 0γ|E(0,d/2; k)E†(r,d/2; k)|˜ 0γ=−¯hk2 π0 Im[G(r,d/2; k)]. (8) It will be useful in the calculations to use the identity sin−1ρd =−2i(1 +eiρd )∞ m=1eiρmd in order to write Gas an infinite power series. In doing so, it is possible to ascribe a simple physical meaning to each term of the resultant series. That is, the term of order m,sayG(m)∼eimdρ , accounts for the contribution of mreflections off the plates. In the following, we will use either formulation according to its mathematical manageability. Lastly, it is also useful to write the components of Gin the spherical basis, with components {0,+,−},in order to trace the polarization of the photons which mediate the corresponding atomic transitions, {π,σ−,σ+}, respectively. The change of basis yields the following relationships: G+− = G−+ =(G +G⊥⊥)/2, G++ =G−− =(G −G⊥⊥)/2. The imaginary parts of G,G⊥⊥, and G00 derive from the poles of Eqs. (5), (6), and (7), respectively: Im[G(r,d/2; k)] = Int( kd π)  n=1 (−1)n−1 4dk2n2π2 d2J0(rk2−n2π2/d2)+k2−n2π2/d2 rJ1(rk2−n2π2/d2),(9) Im[G⊥⊥(r,d/2; k)] = Int( kd π)  n=1 (−1)n−1 4dk2k2J0(rk2−n2π2/d2)−k2−n2π2/d2 rJ1(rk2−n2π2/d2),(10) Im[G00(r,d/2; k)] =−1 4dJ0(kr)−1 2k2d Int( kd 2π)  n=1 [k2−4π2n2/d2]J0(rk2−4n2π2/d2),(11) where J0and J1are the Bessel functions of the first kind of orders 0 and 1, respectively. As for the real parts of G, making use of the Kramers-Kronig relationship, k2Re[G(k)] =2 π∞ 0dkk3Im[G(k)]/(k2−k2), we obtain Re[G(r,d/2; k)] =− Int( kd π)  n=1 (−1)n−1 4dk2n2π2 d2Y0(rk2−n2π2/d2)+k2−n2π2/d2 rY1(rk2−n2π2/d2) +∞  n=Int( kd π)+1 (−1)n−1 2πdk2n2π2 d2K0(rn2π2/d2−k2)+n2π2/d2−k2 rK1(rn2π2/d2−k2),(12) 042714-3 DONAIRE, MUÑOZ-CASTAÑEDA, AND NIETO PHYSICAL REVIEW A 96, 042714 (2017) Re[G⊥⊥(r,d/2; k)] =− Int( kd π)  n=1 (−1)n−1 4dk2k2Y0(rk2−n2π2/d2)−k2−n2π2/d2 rY1(rk2−n2π2/d2) +∞  n=Int( kd π)+1 (−1)n−1 2πdk2k2K0(rn2π2/d2−k2)−n2π2/d2−k2 rK1(rn2π2/d2−k2),(13) Re[G00(r,d/2; k)] =1 4dY0(kr)+1 2k2d Int( kd 2π)  n=1 [k2−4π2n2/d2]Y0(rk2−4n2π2/d2)−1 πk2d ∞  n=Int( kd 2π)+1 [k2−4π2n2/d2] ×K0(r4n2π2/d2−k2),(14) where Y0,1and K0,1are the Bessel functions (Y) and modified Bessel functions (K) of the second kind, of orders 0 and 1, respectively. Alternatively, the above expressions can be written as a series in powers of the number of reflections off the plates, G(r,d/2; k)=−eikr 4πk22 r3−2ik r2−∞  n=1 (−1)nik 2π1 0 dζ eiζknd1+ζ2 kr1−ζ2J1(kr1−ζ2)−ζ2J2(kr1−ζ2) −∞  n=1 (−1)nk 2π∞ 0 dζ e−ζknd1−ζ2 kr1+ζ2J1(kr1+ζ2)+ζ2J2(kr1+ζ2),(15) G⊥⊥(r,d/2; k)=−eikr 4πk2−1 r3+ik r2+k2 r−∞  n=1 (−1)nik 2π1 0 dζ eiζknd1+ζ2 kr1−ζ2J1(kr1−ζ2)−J2(kr1−ζ2) −∞  n=1 (−1)nk 2π∞ 0 dζ e−ζknd1−ζ2 kr1+ζ2J1(kr1+ζ2)−J2(kr1+ζ2),(16) G00(r,d/2; k)=−eikr 4πk2−1 r3+ik r2+k2 r−∞  n=1 (−1)nik 2π1 0 dζ eiζknd(1 −ζ2)J0(kr1−ζ2) −∞  n=1 (−1)nk 2π∞ 0 dζ e−ζknd(1 +ζ2)J0(kr1+ζ2),(17) where the dimensionless variable of integration ζis such that k1−ζ2is the norm of the two-dimensional reciprocal vector parallel to the plates. In Appendix Awe give the expressions of the Green function components in the spherical basis in the asymptotic limits dr,k−1[i.e., three-dimensional (3D) limit] and dr,k−1[i.e., two-dimensional (2D) limit]. III. VAN DER WAALS POTENTIALS At leading order in time-dependent perturbation theory, i.e., order four in W, 24 processes contribute to the vdW potentials of each atom in which two photons are exchanged between the two atoms in all possible orders in time, two terms for each of the diagrams in Figs. 2and 3[28,38,39]. In terms of the vdW potentials, WA,B/2, the forces on each atom are FA,B =∓∇rWA,B/2, respectively, with r=RA−RB[38]. In every case, i.e., either for groundor for excited-state atoms, virtual transitions between atomic levels are accompanied by the exchange of off-resonant photons of frequency ωc/r. Their contribution to the vdW potentials are referred to as off-resonant vdW potentials. In addition, for the case that one or both atoms are excited, transitions to lower-energy atomic levels proceed through the exchange of photons which resonate with the transitions. Their contribution to the vdW potentials are referred to as resonant vdW potentials [18,23]. Interestingly, while the off-resonant potentials of each atom are equivalent, their resonant potentials differ [38,39]. For the sake of illustration we give below the expression of diagram (1) in Fig. 2,  l,j 1 ¯h3∞ 0 Vk2dk (2π)3∞ 0 Vk2dk (2π)34π 0 d4π 0 dia,b,˜ 0γ|ei(ωa+ωb)T|a,b,˜ 0γT −∞ dt t −∞ dtt −∞ dt ×eη(t+t+t)a,b,˜ 0γ|dA·E(−) k(RA)|l,b,˜ 1ke−i(ω+ωl+ωb)(T−t)l,b,˜ 1k|dB·E(+) k(RB)|l,j,˜ 0γ ×e−i(ωl+ωj)(t−t)l,j,˜ 0γ|dB·E(−) k(RB)|l,b,˜ 1ke−i(ω+ωl+ωb)(t−t)l,b,˜ 1k|dA·E(+) k(RA)|a,b,˜ 0γe−i(ωa+ωb)t +[k↔k]†,η→0+,(18) 042714-4 DIPOLE-DIPOLE INTERACTION IN CAVITY QED: THE . . . PHYSICAL REVIEW A 96, 042714 (2017) FIG. 2. Diagrammatic representation of the 24 terms which contribute to WA(T), two for each of the 12 diagrams. Thick straight lines stand for propagators of atomic states, while wavy lines stand for photon propagators. In diagram (1), the initial states a,b, and the generic intermediate states of each atom l,jare indicated. Although not shown, the same states appear in the rest of the diagrams in analogoussites.TheatomsAandBareseparated bya distanceralong the horizontal direction, whereas time runs along the vertical. The gray circles on the left of each diagram stand for the insertion of the Schrödinger operator WAwhose expectation value is computed. Each diagram contributes with two terms, one from each of the operators WAinserted. They are sandwiched between two time propagators, U(T)andU †(T) (depicted by vertical arrows), which evolve the initial state |a,b,˜ 0γtowards the observation time at which WAapplies. where |a,b,˜ 0γis the quasistationary (initial) two-atom–zerophoton state, |land |jare intermediate atomic states of the atoms Aand B, respectively, |˜ 1kis a one-cavity photon state of momentum kand frequency ω=ck, the complex time exponentials are the result of the application of the free timeevolution operator U0(t)=e−i¯h−1H0tbetween the interaction vertices WA,B, and the real time exponential eη(t+t+t)takes care of the adiabatic approximation. After integrating in time and solid angles, and using the fluctuation-dissipation relation of Eq. (8), one arrives at  l,j 8αfc3 π0e2 dm ladn jbds bj dp al ωa+ωb−ωl−ωj Re ∞ 0 dk k2ImGmn(r,k) ω−ωa+ωl−iη ×∞ 0 dkk2ImGsp(r,k) ω−ωa+ωl−iη,η→0+,(19) where αfis the fine-structure constant and summation over repeated tensor indices is implicit. Generally, the rest of the diagrams yield integrands analogous to the one in Eq. (19), which may contain poles at the transition frequencies. Those poles are slightly shifted along the imaginary axis as a consequence of the adiabaticity parameter η→0+. When performing the integral over kand kin the complex plane along the quarter-circle of infinite radius in the first quadrant, FIG. 3. Diagrammatic representation of the 24 terms which contribute to WB(T), two for each of the 12 diagrams. Thick straight lines stand for propagators of atomic states, while wavy lines stand for photon propagators. In diagram (1), the initial states a,b and the generic intermediate states of each atom l,jare indicated. Although not shown, the same states appear in the rest of diagrams in analogous sites. The atoms Aand Bare separated by a distance r along the horizontal direction, whereas time runs along the vertical. The white circles on the right of each diagram stand for the insertion of the Schrödinger operator WBwhose expectation value is to be computed. Each diagram contributes with two terms, one from each of the operators WBinserted. They are sandwiched between two time propagators, U(T)andU †(T) (depicted by vertical arrows), which evolve the initial state |a,b,˜ 0γtowards the observation time at which WBapplies. one can separate the integral along the positive imaginary axis from the sum over residues. The former is part of the off-resonant vdW interaction, while the latter is part of the resonant vdW interaction [19,23,40,41]. We compile in Appendix Bthe equations of the contributions of some diagrams to the resonant components of the vdW potentials and phase shifts. A. Off-resonant van der Waals potentials and off-resonant phase-shift rate The off-resonant component of the vdW potentials is present in the interaction between any pair of atoms, regardless of whether they are in excited or ground states. It includes upwards and downwards virtual transitions to any intermediate atomic levels in the 12 diagrams of Figs. 2and 3. In the calculation, the imaginary shifts of the poles in equations analogous to Eq. (19), with η→0+in the adiabatic approximation, play no role, and the off-resonant potentials can be also computed within the framework of stationary perturbation theory [18,19,23–25]. This explains also the fact that the off-resonant potentials of each atom coincide, WA/2off =WB/2off, and so does the associated phase-shift rate of the two-atom wave function, δEoff =WA,B/2off [38]. Let us denote by aand bthe states of the atoms Aand B, respectively. By adding up equations analogous to that in 042714-5 DONAIRE, MUÑOZ-CASTAÑEDA, AND NIETO PHYSICAL REVIEW A 96, 042714 (2017) Eq. (19) coming from the 12 diagrams of Figs. 2,3,or5, one obtains the following integral along imaginary frequencies [18,24,25], k=iu: WA,B/2off =−2 π¯h2 0c3 i,j ∞ 0 du u4ωiaωjb u2+k2 iau2+k2 jb ×dai ·G(r;iu)·djb dbj ·G(r;iu)·dia =δEoff,(20) with ωia =ωi−ωa,kia =ωia/c,ωjb =ωj−ωb,kjb = ωjb/c,dai =a|dA|i, and dbj =b|dB|j. As it stands, it suffices to substitute the expressions of the Green function components in order to calculate the off-resonant vdW potential for any particular case. Evaluating Eqs. (15)–(17) at imaginary frequencies and performing the summation over any number of reflections, we find in the spherical basis, G+−(r;iu) =e−ur 8πu2[1/r3+u/r2−u2/r] +∞ 1 dζ 4π euζd −1 e2uζd −1u(1 +ζ2)J0(urζ2−1),(21) G++(r;iu) =e−ur 8πu2[3/r3+3u/r2+u2/r] +∞ 1 dζ 4π euζd −1 e2uζd −1u(1 −ζ2)J2(urζ2−1),(22) G00(r;iu) =e−ur −4πu2[1/r3+u/r2+u2/r] +∞ 1 dζ 2π euζd +1 e2uζd −1u(ζ2−1)J0(urζ2−1),(23) where the dependence of Gon d/2 has been omitted in its argument for brevity. In all the expressions above the first terms are the components of the Green function in free space, whereas the second terms result from multiple scattering of the cavity plates. As a consequence, the factor G2in the integrand of Eq. (20) contains terms with two free-space factors which decay exponentially from u≈2/r, terms with two multiplescattering factors which are exponentially suppressed from u≈2/d, and terms which combine free-space and scattering factors that decay exponentially from u≈min(1/r,1/d). The calculation of WA,B/2off requires the numerical integration of Eq. (20), which depends generally on the transition frequencies of both atoms. Nonetheless, assuming that those frequencies are of the same order, say K≃kia,kjb ∀i,j, the dependence of WA,B/2off on the particular values of transition frequencies and dipole moments can be factored out such that approximately universal potentials can be defined as functions of rand donly. This is, for instance, the case of the vdW interaction between circular Rydberg atoms (see Sec. V FIG. 4. Graphic representation of the three components of the dimensionless tensor potential Voff of Eq. (25) normalized by their values in free space, Vfree off . The interatomic distance is fixed at r= 1/5K, whereas dvaries between 0 and 1.4/K. below). That is, we can write WA,B/2off ≃−2K5 π¯h2 0c i,j Cij d0 i0d0 0j 2V00 off (r,d) +(|d+ iad+ bj |2+|d− iad− bj |2)V++ off (r,d) +(|d+ iad− bj |2+|d− iad+ bj |2)V+− off (r,d),(24) where Cij is a numerical factor of order unity whose sign is given by sgn(ωaiωjb); dp ia =Aa|dp A|Aiis the pth-vector component of the ith transition dipole moment of atom A, and likewise for atom B; and Vps off (r,d)=∞ 0 dχ χ4G2 ps(r;iKχ)/[K(χ2+1)]2,(25) with p,s ={+,−,0}, are the components in the spherical basis of the dimensionless off-resonant vdW tensor potential, which depend only on r,d. The dimensionless variable of integration χis the imaginary frequency uin units of K. The components of Voff are represented in Fig. 4as functions of dfor a fixed value of the interatomic distance, r= 1/5K, normalized by their values in free space. We observe that the effects of the cavity confinement become relevant as dapproaches r. Interestingly, three different behaviors are found. Whereas V++ off decreases monotonically to zero for dr, the component V+− off shows a bump around d≈r,after which it goes to zero as well. In contrast, V00 off gets minimum around d≈rand increases monotonically as dapproaches 0. The vanishing of the components V+− off and V++ off is an effect of the confinement of the lines of the electric field parallel to the plates. The decrease of both components is indeed exponential as d/r →0 and so is the decrease of their associated Green’s functions, see Eqs. (A3) and (A4). On the contrary, for d/r →0, the field lines perpendicular to the plates bounce infinite times off the plates when going from one atom to the other, augmenting the strength of V00 off as ∼1/d2.Asamatteroffact,G00 goes like 1/d as d/r →0, see Eq. (A2). 042714-6 DIPOLE-DIPOLE INTERACTION IN CAVITY QED: THE . . . PHYSICAL REVIEW A 96, 042714 (2017) B. Resonant van der Waals potentials and resonant phase-shift rate In contrast to the off-resonant interaction, part of the vdW interaction between excited atoms is mediated by virtual photons which resonate with the transitions of one or the other atom, which is referred to as resonant interaction [23]. Correspondingly, we refer to the resonant contributions to the potentials and to the phase shift as resonant vdW potentials (res) and resonant phase-shift rate, respectively. On the other hand, these resonant photons mediate also the periodic transfer of the excitation between both atoms. In the perturbative nondegenerate regime this transfer has a small probability proportionalto|W|/¯h|AB |1, where AB isthedetuning between the relevant transition frequencies of the atoms. It is due to this partial and periodic transfer, as well as to the finite lifetime of excited states, that the vdW potentials with excited atoms become dynamical and are to be computed within the framework of time-dependent perturbation theory [27–30,38,39]. Further, for the usual case that the excitation of the atoms be adiabatic with respect to the detuning AB [27,38], the calculation simplifies to assuming that in the far past the atoms are initially excited and the interaction potential Wis turned on adiabatically. 1. Two dissimilar atoms, one of them excited For the case that the atoms are of different kinds and only one of the them is excited, say atom Aat state a>0, while atom Bis in its ground state with b=0, only the diagrams (1) and(3)ofFigs.2and 3contribute to the resonant potentials of atoms Aand B, respectively, yielding [38,39] WA/2res = j,i<a 2ωj0k4 ai 2 0¯hω2 ai −ω2 j0dm aidn 0jdp j0ds ia ×{Re[Gmn(r,kai)]Re[Gps(r,kai)] −Im[Gmn(r,kai)]Im[Gps(r,kai)]},(26) WB/2res = j,i<a 2ωj0k4 ai 2 0¯hω2 ai −ω2 j0dm aidn 0jdp j0ds ia ×{Re[Gmn(r,kai)]Re[Gps(r,kai)] +Im[Gmn(r,kai)]Im[Gps(r,kai)]},(27) where the tensor indices are m,n,p,s ={+,−,0}in the spherical basis and summation over repeated tensor indices is implicit. For the sake of illustration, we write in Appendix Bthe explicit expressions of the contributions of diagrams Fig. 2(3) and Fig. 3(2) to WA/2res and WB/2res, respectively. Again, assuming that those frequencies are roughly of the same order, say K≃kai,kj0∀i,j,Eqs.(26) and (27) can be approximated by WA,B/2res ≃2K5 π¯h2 0c i,j C ij d0 iad0 0j 2V00 A,Bres(r,d) +(|d+ iad+ 0j|2+|d− iad− 0j|2)V++ A,Bres(r,d) +(|d+ iad− 0j|2+|d− iad+ 0j|2)V+− A,Bres(r,d),(28) FIG. 5. Diagrammatic representation of the 12 terms which contribute to the phase-shift rate of the two-atom wave function, δE. The explanation of the symbols is as in Figs. 2and 3. Diagrams (1, 3, 5, 7, 9, 12) contain gray circles on the left which stand for the insertion of a Schrödinger operator WA,whereasdiagrams(2,4,6,8, 10, 11) contain white circles on the right which stand for the insertion of an operator WB. Differently to the diagrams which contribute to WA(T)and WB(T)in Figs. 2and 3, respectively, as in diagrams (1) and (2), here the operators WAor WBare sandwiched between two time propagators, one of which is the free time-evolution operator, U0(T). The explanation of this can be found in Ref. [38]. where C ij is a numerical factor of order unity, of the same sign as ωai −ωj0, and the dimensionless potentials read Vpq Ares(r,d)={Re2[Gps(r,K)] −Im2[Gps(r,K)]}/K2, (29) Vpq Bres(r,d)={Re2[Gps(r,K)] +Im2[Gps(r,K)]}/K2, (30) with p,s ={+,−,0}. As for the resonant phase-shift rate, the addition of diagrams (1) and (3) of Fig. 5is in this case δEres =WA/2res [27,38]. As in free space, it is the discrepancy between the signs of the second terms on the right-hand side of Eqs. (29) and (30) that gives rise to a net force on the two-atom system [29,38,39]. The difference is, however, negligible in the nonretarded regime, rK 1. On the contrary, in the retarded regime, it was already found in Refs. [29,38] that, in free space, while the components of VAres oscillate in space changing sign periodically, the components of VBres decrease monotonically as rincreases. In the presence of a cavity the behavior of the potentials depend on the value of das well. The components of VAres and VBres are represented in Fig. 6as functions of the interatomic distance r, with rK > 1, for two different values of d,2/K (upper inset) and 20/K (lower inset). In contrast to the results in free space, for d<π/K, only the component V00 Ares oscillates in space, whereas V++ Ares and V+− Ares decrease monotonically and are equivalent to the potential components of atom B. The reason is that, for d<π/K, 042714-7 DONAIRE, MUÑOZ-CASTAÑEDA, AND NIETO PHYSICAL REVIEW A 96, 042714 (2017) FIG. 6. Graphic representation of the three components of the dimensionless potentials in Eqs. (29)and(30), as functions of rK, for two different values of d,2/K (upper inset) and 20/K (lower inset). only the oscillating terms of G00 entering Eqs. (29) and (30) are different from zero, (1/4d)[Y0(kr)−iJ 0(kr)]. On the contrary, for d>π/K (lower inset of Fig. 6) all the components of VAres oscillate along rand so do, although with longer periods, the components of VBres. The latter, however, do not change sign and constitute a sort of envelope of the oscillating curves of VAres. 2. Two dissimilar atoms excited When both atoms are excited, say on states a>0 and b>0, in addition to diagrams (1) and (3), also the diagrams (2), (4), and (5)–(10) of Figs. 2,3, and 5are relevant. The perturbative nondegenerate regime implies in this case that |W|  ¯h|ωai −ωjb|, for any pair of intermediate states i,j, with i<a,j>b, and |W|  ¯h|ωbj −ωai|for any i>a,j<b. For the sake of illustration, explicit expressions of the contributions of diagrams (9) and (10) of Figs. 2,3, and 5to WA/2res,WB/2res, and δEres, respectively, have been included in Appendix Bfor the case of two dissimilar atoms excited. Generically,wecandistinguishthree differentcontributions to WA,B/2res. These are, a first one in which the intermediate states satisfy i>a,j<b, a second one in which they satisfy i<a,j>b, and a third one for which i<a,j<b. Putting them all together we have WA/2res = j>b,i<a 2ωjbk4 ai 2 0¯hω2 ai −ω2 jbdm aidn bj dp jbds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] −Im[Gmn(r,kai)]Im[Gps(r,kai)]} + j<b,i>a 2ωiak4 bj 2 0¯hω2 bj −ω2 iadm bj dn aidp iads jb{Re[Gmn(r,kbj )]Re[Gps(r,kbj )] +Im[Gmn(r,kbj )]Im[Gps(r,kbj )]} − j<b,i<a 2ωbj k4 ai 2 0¯hω2 ai −ω2 bj dm aidn jbdp bj ds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] −Im[Gmn(r,kai)]Im[Gps(r,kai)]} + j<b,i<a 2ωaik4 bj 2 0¯hω2 ai −ω2 bj dm bj dn aidp iads jb{Re[Gmn(r,kbj )]Re[Gps(r,kbj )] +Im[Gmn(r,kbj )]Im[Gps(r,kbj )]},(31) WB/2res = j>b,i<a 2ωjbk4 ai 2 0¯hω2 ai −ω2 jbdm aidn bj dp jbds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] +Im[Gmn(r,kai)]Im[Gps(r,kai)]} + j<b,i>a 2ωiak4 bj 2 0¯hω2 bj −ω2 iadm bj dn aidp iads jb{Re[Gmn(r,kbj )]Re[Gps(r,kbj )] −Im[Gmn(r,kbj )]Im[Gps(r,kbj )]} 042714-8 DIPOLE-DIPOLE INTERACTION IN CAVITY QED: THE . . . PHYSICAL REVIEW A 96, 042714 (2017) − j<b,i<a 2ωbj k4 ai 2 0¯hω2 ai −ω2 bj dm aidn jbdp bj ds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] +Im[Gmn(r,kai)]Im[Gps(r,kai)]} + j<b,i<a 2ωaik4 bj 2 0¯hω2 ai −ω2 bj dm bj dn aidp iads jb{Re[Gmn(r,kbj )]Re[Gps(r,kbj )] −Im[Gmn(r,kbj )]Im[Gps(r,kbj )]},(32) where the tensor indices are m,n,p,s ={+,−,0}in the spherical basis and summation over repeated tensor indices is implicit. Note that analogous expressions were obtained by Barcellona et al. in Ref. [29] in free space. Lastly, as for the phase-shift rate of the two-atom wave function we find, δEres = j>b,i<a 2ωjbk4 ai 2 0¯hω2 ai −ω2 jbdm aidn bj dp jbds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] −Im[Gmn(r,kai)]Im[Gps(r,kai)]} + j<b,i>a 2ωiak4 bj 2 0¯hω2 bj −ω2 iadm bj dn aidp iads jb{Re[Gmn(r,kbj )]Re[Gps(r,kbj )] −Im[Gmn(r,kbj )]Im[Gps(r,kbj )]} − j<b,i<a 2ωbj k4 ai 2 0¯hω2 ai −ω2 bj dm aidn jbdp bj ds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] −Im[Gmn(r,kai)]Im[Gps(r,kai)]} + j<b,i<a 2ωaik4 bj 2 0¯hω2 ai −ω2 bj dm bj dn aidp iads jb{Re[Gmn(r,kbj )]Re[Gps(r,kbj )] −Im[Gmn(r,kbj )]Im[Gps(r,kbj )]}.(33) 3. Two identical atoms excited We consider next the case in which the two atoms are identical, A=B, and find in the same excited state a>0. The nondegenerate condition necessary for the calculation to be perturbative and nondegenerate reads in this case |W|  ¯h|ωai − ωja|, for any pair of intermediate states i,j, with i<a,j>a. In comparison to the case of dissimilar atoms, the only difference in the calculation is the presence of double poles when i=jin the frequency integrals which derive from the diagrams (3), (4), (9), and (10). Explicit expressions of the contributions of diagram (4) in Figs. 2and 5to WA/2res and δEres, respectively, have been included in Appendix Bfor this case. As for the resonant vdW potential, it reads, WA/2res = i<a,j=i 4ωjak4 ai 2 0¯hω2 ai −ω2 jadm aidn jadp aj ds iaRe[Gmn(r,kai)]Re[Gps(r,kai)] + i<a,j=i k2 ai 2 0c¯hdm aidn iadp aids ia ×kaiRe[Gmn(r,kai)]Re[Gps(r,kai)] −2Re[Gmn(r,kai)] ∂ ∂k[k2Re[Gps(r,k)]]k=kai ,(34) whereas the phase-shift rate of the two-atom wave function is δEres = i<a,j=i 4ωjak4 ai 2 0¯hω2 ai −ω2 jadm aidn jadp aj ds ia{Re[Gmn(r,kai)]Re[Gps(r,kai)] −Im[Gij (r,kai)]Im[Gps(r,kai)]} + i<a,j=i k2 ai 2 0c¯hdm aidn iadp aids iakaiRe[Gmn(r,kai)]Re[Gps(r,kai)] −kaiIm[Gmn(r,kai)]Im[Gps(r,kai)] −2Re[Gmn(r,kai)] ∂ ∂k[k2Re[Gps(r,k)]]k=kai .(35) IV. ELECTROSTATIC POTENTIAL BETWEEN INDUCED DIPOLES Another case of interest commonly encountered in experiments is that of the electrostatic interaction between two atomic dipoles induced by an external static field E0.The contribution of the 24 diagrams of Fig. 7reduces to the electrostatic interaction between two induced electric dipoles, Aand B, with moments αa A(0)E0and αb B(0)E0, where αa,b A,B(0) are the static polarizabilities of each atom in the states aand b, respectively. The interaction potentials of each atom coincide in this case, and so does the associated phase-shift rate of the two-atom wave function. If we denote the electrostatic potential by Vst AB (r), it holds WA=WB=δEst ≡Vst AB (r), and 042714-9 DONAIRE, MUÑOZ-CASTAÑEDA, AND NIETO PHYSICAL REVIEW A 96, 042714 (2017) [3] Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, Phys.Rev.Lett.51,1175 (1983). [4] G. Barton, Proc. Roy. Soc. Lond. A 367,117 (1979); 410,141 (1987). [5] A. A. Belov, Yu. E. 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