Super-Coulombic atom-atom interactionsin hyperbolic media

Cristian L. Cortes1,2, Zubin Jacob1,2∗

Department of Electrical and Computer Engineering,University of Alberta, Edmonton, AB T6G 2V4, Canada andBirck Nanotechnology Center and Purdue Quantum Center,

School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN 47906, U.S.A.

Dipole-dipole interactions which govern phenomena like cooperative Lamb shifts, superradiantdecay rates, Van der Waals forces, as well as resonance energy transfer rates are conventionallylimited to the Coulombic near-field. Here, we reveal a class of real-photon and virtual-photon long-range quantum electrodynamic (QED) interactions that have a singularity in media with hyperbolicdispersion. The singularity in the dipole-dipole coupling, referred to as a Super-Coulombic interac-tion, is a result of an effective interaction distance that goes to zero in the ideal limit irrespective ofthe physical distance. We investigate the entire landscape of atom-atom interactions in hyperbolicmedia and propose practical implementations with phonon-polaritonic hexagonal boron nitride inthe infrared spectral range and plasmonic super-lattice structures in the visible range. Our workpaves the way for the control of cold atoms in hyperbolic media and the study of many-body atomicstates where optical phonons mediate quantum interactions.

Dipole-dipole interactions (DDI) are instrumental inmediating entanglement and super-radiance in coldatoms [1, 2], coherent coupling between single moleculesor single atoms [3–6], virtual photon exchange and fre-quency shifts in circuit QED [7, 8], and Forster resonanceenergy transfer transfer between dye molecules or quan-tum dots [9, 10]. There are two fundamental ways ofcontrolling the strength and length scales of dipole-dipoleinteractions. The first method involves the tuning of in-trinsic atomic properties such as transition dipole mo-ments and transition frequencies with highly-excited Ry-dberg atoms and superconducting qubits [6, 11, 12]. Thesecond method involves the tuning of the quantum elec-trodynamic vacuum, achieved through cavities, waveg-uides and photonic bandgaps [13–16]. This opens theimportant question whether a homogeneous material ormetamaterial can also enhance dipole-dipole interactions.In this Letter, we reveal a class of divergent excited-state atom-atom interactions that can occur in naturaland artificial media with hyperbolic dispersion. Unlikethe above mentioned approaches which engineer radia-tive coupling, we show that the homogeneous hyperbolicmedium itself fundamentally alters the Coulombic near-field. The resultant singular long-range interaction, re-ferred to as a Super-Coulombic interaction, is describedby an effective interaction distance that goes to zero(re → 0) along a material-dependent resonance angle.We show that this interaction affects the entire landscapeof real photon and virtual photon phenomena such as thecooperative Lamb shift, the cooperative decay rate, reso-nance energy transfer rates and frequency shifts as well asresonant interatomic forces. While we find that the sin-gularity is curtailed by material absorption, it still allows

(a)

(c)

(b)

θR

φ

Figure 1. (a) Two dopant atoms in a hyperbolic medium(e.g. h-BN) will interact via a long range super-Coulombicdipole-dipole interaction (DDI). (b)-(c) The DDI occurs overa broad range of frequencies along the resonance cone angleof a hyperbolic medium and causes the effective interactiondistance to approach zero irrespective of the physical distance.

for interactions with much larger magnitudes and longerranges than those found in any conventional media. Wealso show that atoms in a hyperbolic medium will exhibita strong orientational dependence that can effectivelyswitch the dipolar interaction off or on, providing an ad-ditional degree of freedom to control DDI. Our investiga-tion reveals a marked contrast between ground-state andexcited-state interactions which can be used to distin-guish the Super-Coulombic effect in experiment. Finally,we propose practical broadband hyperbolic systems con-sisting of phonon-polaritons and plasmonic-media to ex-perimentally verify our predicted Super-coulombic effect.

We emphasize that the materials platform we intro-duce in this paper to enhance dipole-dipole interactionsis fundamentally different from the cavity QED [17, 18]

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or waveguide QED regimes [4, 7, 19, 20]. We do notrely on atom confinement [2, 4–6, 18], cavity resonancesor modal effects such as the quasi-TEM mode in circuitQED [7], the band-edge slow light as in PhC waveg-uides [4, 5, 21], low mode volume plasmonic waveguides[20, 22] or infinite phase velocity at a cut-off frequencyas in ENZ waveguides [13, 23]. We also stress thatthe Super-Coulombic effect engineers the conventionalnon-radiative fields as opposed to radiative modes andwill occur over a broad range of frequencies due to thebroadband nature of the hyperbolic dispersion relation[24, 25]. Figure 1 depicts a schematic of the proposedSuper-Coulombic dipole-dipole interaction using hexago-nal Boron Nitride (h-BN) [26] and two dopant atoms. Inthe infrared spectral range, h-BN is a uniaxial materialthat supports ordinary waves (polarization perpendicularto the optic axis) and extraordinary waves (polarizationalong the optic axis). Extraordinary waves satisfy the hy-perbolic dispersion relation k2

x/εz + k2z/εx = ω2/c2 when

εxεz < 0.We begin by formulating the QED theory [27] of dipole-dipole interactions between two neutral, non-magneticatoms in a hyperbolic medium. We focus on dipolar in-teractions where the electrodynamic field is initially pre-pared in the vacuum state |{0}〉. Using the multipolarHamiltonian, the interaction of two neutral atoms [posi-tions rj , transition frequencies ωj and transition electric

dipole moments dj (j = A,B)] is specified by the in-

teraction Hamiltonian Hint = −∑j

∫∞0dω[djE(rj , ω) +

h.c.], where h.c. stands for the Hermitian conju-gate. The matter-assisted electric field is given byE(r, ω) = i

√hω/πεoc

∫d3r′

√ε′′(r′, ω)G(r, r′, ω)f(r′;ω)

where G(r, r′;ω) is the classical dyadic Green functionthat satisfies the macroscopic Maxwell equations. Here,f†(r′, ω) and f(r′, ω) are bosonic field operators whichplay the role of the creation and annihilation operatorsof the matter-assisted electromagnetic (polaritonic) field.The unique interaction properties are a direct result ofthe dispersion relation of the hyperbolic polariton, as op-posed to the photonic dispersion relation, ω = ck, seenin vacuum. The electric field is defined so that it rig-orously satisfies the equal-time commutation relationsand fluctuation-dissipation theorem [27]. We use conven-tional perturbation theory to calculate the various dipo-lar interactions in a hyperbolic medium. We emphasizethat the QED theory captures both ground state-groundstate interactions and excited state-ground state interac-tions which a semiclassical approach cannot.If the initial state of the atomic system is prepared in thesymmetric or anti-symmetric state, |i〉 = 1√

2(|eA〉 |gB〉 ±

|gA〉 |eB〉), then one can show that the resonant dipole-dipole interaction (RDDI) [see Supp. Info.] is given by

Vdd = h(Jdd − i

γdd2

)= − ω2

A

εoc2dB ·G(rB , rA;ωA) · dA.

(1)

where dj = 〈gj |dj |ej〉 is the transition dipole moment ofatom j, assumed to be real. Jdd is the cooperative Lambshift (also known as the virtual photon exchange inter-action) and γdd is the cooperative decay rate commonlyassociated with superradiant or subradiant effects.

Our result for the resonant dipole-dipole interaction ina hyperbolic medium (ε = diag[εx, εx, εz]) is

Vdd =eikore

4πεo√εxr3

e

dB ·[(1− ikore)κnf− k2

or2eκff

]·dA+V eodd

(2)valid when rA 6= rB . The first term arises exclusivelyfrom extraordinary waves following a hyperbolic disper-sion while the second term V eodd arises from a combinationof ordinary and extraordinary waves [SI]. Here, we havedefined the near-field and far-field dipole orientation ma-trix factors κnf = εxεz(ε

−1−3(ε−1 ·r)(ε−1 ·r)/(r·ε−1 ·r))and κff = εxεz(ε

−1 − (ε−1 · r)(ε−1 · r)/(r · ε−1 · r)) re-spectively. Equation (2) reduces to the vacuum RDDIexpression when εx = εz = 1, which is applicable bothin the retarded (r � λ) and non-retarded (r � λ)regimes. The most unique aspect of dipole-dipole interac-tions in uniaxial media is the divergence that is predictedfrom the first term only when the hyperbolic condition(εxεz < 0) is satisfied. In the ideal lossless limit, wefind that the effective interaction distance between twoatoms, re =

√εxεz(r · ε−1 · r) = r

√εz sin2 θ + εx cos2 θ,

tends towards the limit

re → 0 as θ → θR = tan−1√−εx/εz. (3)

This super–coulombic effect results in the divergence ofthe dipole-dipole interaction strength |Vdd|/h along theresonance angle θR, defined with respect to the optic axis.

Atoms in a hyperbolic medium will then have an as-sociated cooperative Lamb shift (CLS) and cooperativedecay rate (CDR)

Jdd ≈√εxεz

4πhεor3e

dB ·[ε−1− 3(ε−1 · r)(ε−1 · r)

r · ε−1 · r]· dA (4)

γdd ≈ω3A

√εxεz

3πhεoc3dB ·ε−1 ·dA (5)

in the limit θ → θR. Equations (4) and (5) are the domi-nant factors of the extraordinary wave contribution only.

This scaling behavior mediated by hyperbolic modesis in striking contrast to the behavior of the cooperativeLamb shift in vacuum. In vacuum, the CLS dependscrucially on the interatomic distance. For separation dis-tances much larger than the transition wavelength, theCLS scales as Jdd ∼ γo cos(kor)/(kor) and becomes muchsmaller than the free-space spontaneous emission rate γo.For distances much smaller than the wavelength, the CLSscales as Jdd ∼ γo/(kor)

3, which implies that it can be-come much larger than the spontaneous emission rate.In contrast, the cooperative Lamb shift in a hyperbolic

3

medium is dependent on r−3e , Jdd ∼ γo/(kore)3, for all in-

teratomic distances. The material-dependent factor 1/r3e

diverges in the lossless case and therefore results in giantdipole-dipole interactions for short and large interatomicdistances.

This contrast is also revealed in the cooperative de-cay rate. At large distances, the CDR in vacuum scalesas γdd ∼ γo sin(kor)/(kor), therefore becoming weak fordistances much larger than the wavelength. For dis-tances much smaller than the wavelength, the CDR be-comes independent of position, γdd ∼ γo, and remainson the order of the free space spontaneous emission rate.In contrast, the cooperative decay rate in a hyperbolicmedium along the resonance angle is not dependent onthe effective interaction distance re, and instead it de-pends crucially on the orientation angle φ of the dipoles,γdd ∼ γo(εz/

√εx sin2 φ+

√εx cos2 φ). When both dipoles

are oriented perpendicular to the optic axis (φ = π/2),there exists a unique wavelength when the medium canachieve an anisotropic epsilon-near-zero (ENZ) medium(εx → 0 and εz 6= 0) resulting in a divergent cooper-ative decay rate, independent of interatomic distance.When both dipoles are parallel to the optic axis (φ = 0),the same anisotropic ENZ condition gives a null CDRbetween the two atoms, independent of interatomic dis-tance.

We will now consider the role of material absorption(εx = ε′x + iε′′x and εz = ε′z + iε′′z ) on atom-atom in-teractions in a hyperbolic medium. We find that theeffective interaction distance tends to the finite value

|re| → |r|[ε′′z |ε

′x|+ε

′′x |ε

′z|

|ε′x|+|ε′z|

]1/2as θ → θR, which curtails

the singularity of the hyperbolic dipolar interaction butnevertheless allows for very large interaction strengthscompared to conventional media whenever |re|/|r| < 1is satisfied. The second important consequence of mate-rial absorption results from the mixing of terms in thereal and imaginary components of the RDDI in eqn.(2). Both the cooperative Lamb shift and decay ratescale as r−3

e thus becoming large-yet-finite values in thesmall interatomic distance limit. The third consequenceof material absorption on resonant dipole-dipole inter-actions is in the transition from non-retarded (r−3) toretarded (r−1) interactions. In vacuum, the transitionoccurs when the interatomic separation distance is onthe order of wavelength,(ω/c)r ∼ 1. Along the reso-nance angle of a hyperbolic medium, the transition isexpected to occur approximately when (ω/c)|re| ∼ 1. Itshould be noted that in the lossless limit, this condi-tion is never satisfied since re → 0 for ideal hyperbolicmedia. Therefore we find that RDDI should scale withthe characteristic power law of near-field non-radiativeinteractions (r−3) for all interatomic distances. Oncematerial absorption is included, dipolar interactions willtransition from the power law (r−3

e ) to the exponentialscaling law (e−(ω/c)Im[re]) valid at large interatomic dis-

Separation Distance0 λ 2λ 3λ 4λ 5λ

-1

0

1

2

3

vacuum vacuumhyperbolic

medium

hyperbolic

medium

(d)(c)

(a) (b)

Separation Distance

-1

0

1

2

3

0 λ 2λ 3λ 4λ 5λ

J dd

/ γ

o

θ

-2

-1

0

1

2

θR

0 20 40 60 80

J dd

/ γ

o

1

-1

0

1

-1

0

θ

-6

-4

-2

0

2

4

Hyperbolic Medium

Lossy Dielectric

Vacuum

0 20 40 60 80

θR

θ

γ dd

/ γ

oγ d

d /

γo

2λ2λ

2λ

-2λ -2λ

2λ

-2λ 2λ -2λ 2λ

Figure 2. Angular dependence of (a) cooperative Lamb shift(CLS) Jdd and (b) cooperative decay rate (CDR) γdd for twoz-oriented dipoles in a lossy hyperbolic medium, lossy dielec-tric, and vacuum. The CLS and CDR have large peaks nearthe resonance angle of the hyperbolic medium indicative ofthe super-Coulombic interaction, even for distances of a wave-length. Comparison of (c) CLS and (d) CDR at the resonanceangle versus interatomic separation distance. The CLS andCDR both obey a 1/r3 power law dependence in the near-fielddue to the inclusion of absorption in the hyperbolic medium.Note that the giant interactions start occuring at distanceson the order of a wavelength (arrows) even in the presence ofmaterial absorption which is in stark contrast to vacuum. Theinsets show the contrasting spatially-resolved (c) CLS and (d)CDR for vacuum and for a hyperbolic medium.

tances. Figure (2) shows the result of the cooperativeLamb shift and decay rate for two z-oriented dipoles ina hyperbolic medium that includes material absorption.We compare the resonant dipole-dipole interactions withthe conventional results of a lossy dielectric and vacuum.Note that the RDDI peaks near the resonance angle θRas predicted theoretically. The spatial field plots in theinsets clearly demonstrate the distinguishing features ofthe RDDI in a hyperbolic medium compared to vacuum.Fig (2c) and (2d) demonstrate the r−3

e Super-Coulombicspatial dependence along the resonance angle. Note thatthe sign of the interaction is dependent on the orienta-tion of the dipoles as well as the relative position of thedipoles within the hyperbolic medium.

We now turn to the unique orientational dependenceof the RDDI between two atoms positioned along the res-onance angle θR. In figure (3), we plot the normalizedcooperative Lamb shift of two atoms a full wavelengthapart (r = λ) as a function of dipole orientation angleφ. The cooperative Lamb shift has a minimum whenφ = θR and a maximum when φ = θR + π/2. Assum-ing that |ε′| = |ε′x| ≈ |ε′z|, ε′′ = ε′′x ≈ ε′′z , and ε′′ � |ε′|,we find that the ratio between the maximum and min-

imum is Jdd(φ=θR+π/2)Jdd(φ=θR) ≈ − 3

2

(ε′

ε′′

)2

, showing that it is

4

Orientation Angle φ

0 45

-1

0

1

2

3

4

θR+ 90θ

R

Hyperbolic Medium

Lossy Dielectric

Vacuum

φ =φ =

J dd

/ γ

oθR

θ

90 135 180

0.25

0

-0.25

Figure 3. Unique orientational dependence of RDDI in hy-perbolic media. The plot shows CLS versus orientation an-gle φ for two dipoles positioned along the resonance angle.The cooperative Lamb shift is minimized when the dipolesare collinear with the the resonance angle, and it is max-imized when the dipoles are perpendicular to the resonanceangle. The inset shows the asymmetric nature of the spatially-resolved Jdd/γo when the dipoles are orthogonal to the reso-nance angle.

proportional to the square of the figure of merit of thehyperbolic medium. In figure (3), we use the full Green’sfunction to calculate the orientational dependence of thedipolar interaction in a hyperbolic medium with materialabsorption, and find excellent agreement with the analyt-ical expression.

We now consider 2nd order super-coulombic QED in-teractions arising from initial state preparation consist-ing of atom A in its excited state and atom B in itsground state, |i〉 = |eA〉 |gB〉. In the weak-couplingregime, an irreversible resonance energy transfer takesplace transferring a photon from atom A to atom B.This process is Forster resonance energy transfer (FRET)and the transfer rate given by Fermi’s golden rule isΓ

ET= 2πh−1|Vdd|2δ(hωA − hωB). Along the resonance

angle, FRET is mediated by hyperbolic modes and therate is given by

ΓET≈ 2π

h

|dB · κnf · dA|2

(4πεo)2|εx||re|6δ(hωA − hωB) (6)

which shows a r−6e scaling dependence – the key sig-

nature of second order Super-Coulombic interactionsin hyperbolic media. In addition to the FRET rate,there is also a predicted frequency shift that comesfrom the initial state preparation |i〉 = |eA〉 |gB〉.This is the excited-state Casimir-Polder potential,Ueg(r) = Ureg(r) + Uoreg (r), composed of a resonantand off-resonant contribution. The resonant excited-state Casimir-Polder potential is of the form Ureg(r) =

− |dA|2ω4A

3ε2oc4 αB(ωA)Re{Tr[G(rB , rA;ωA)G(rA, rB ;ωA)]}

[28]. We therefore predict that the excited-state energypotential will also diverge with a r−6

e scaling dependence

Separation Distance

0.01λ 0.1λ 10λ

Ca

sim

ir-P

old

er

Po

ten

tia

l

10-20

10-10

100

1010

0.01λ 0.1λ 10λ

ET

Ra

te

10 -20

10 -10

10 0

10 10

λ

λ

Hyperbolic Medium

Isotropic Medium

Vacuum-Ugg

Ueg

Figure 4. Casimir-Polder interaction energy between twoground-state atoms (Ugg) and between an excited-state atomand ground-state atom (Ueg) show fundamental differenceswhen interacting in hyperbolic medium. Ueg >> Ugg sinceresonant interactions lie completely within the bandwidth ofhyperbolic dispersion and are strongly enhanced. The resultsare normalized to Ugg in vacuum, evaluated at the near-fieldinteratomic distance of ro = λ/100. The inset shows the gi-ant enhancement of the FRET rate, ΓET , as compared tovacuum. The FRET rate is normalized to the vacuum energytransfer rate evaluated at ro.

similar to the FRET rate. Figure (4) shows the fullnumerical results for the 2nd order dipole-dipole inter-actions in a lossy hyperbolic medium, a lossy dielectric,and vacuum. In the non-retarded regime (r � λ), weclearly see the effect of the Super-Coulombic interactionwhich results in a large enhancement of the dipolarinteractions Ueg and ΓET (shown in inset).

It is interesting that the dispersive Van der Waalsinteraction between two ground state atoms does notdiverge in a hyperbolic medium. Using fourth orderperturbation theory [29], the interaction energy be-tween two ground-state atoms is given by Ugg(rA, rB) =−hµ2

o

2π

∫∞0dηη4αA(iη)αB(iη)Tr[G(rB , rA; iη)G(rA, rB ; iη)],

where αA,B(ω) is the isotropic electric polarizability ofatom A or B. In the non-retarded limit, the dominantcontribution is given by

Ugg ≈ −h

32π3ε2o

∫ ∞0

dηTr[κ2

nf (iη)]

εx(iη)r6e(iη)

αA(iη)αB(iη) (7)

which reduces to the well known free-space non-retardedVan der Waals interaction energy when εx = εz = 1. It isimportant to note that the integral is performed over theentire range of positive imaginary frequencies (η = iω).Generally, the hyperbolic condition εxεz < 0 is only sat-isfied within a finite bandwidth of the electromagneticspectrum. We therefore expect that it would not alter thebroadband cumulative effect of the entire electromagneticspectrum, and as a result we predict that the ground-state ground-state interaction energy will not diverge ina hyperbolic medium. From fig. 4, it is also clear thatthe ground-state ground-state Casimir Polder potential

5

Wavelength (nm)

450 500 550 600-10

-5

0

5

10(a) (b)

Wavelength (nm)

450 500 550 600

-10

-5

0

5

10

J dd

/ γ

o

J dd

/ γ

o

xB (nm)100

-2

0

2

4

6

0 50

x

z

ˆ

xB

φ

Figure 5. Giant long-range Cooperative Lamb shift, Jdd,across a practical plasmonic super-lattice calculated for thecorresponding (a) effective hyperbolic model and (b) a 40-layer multilayer system taking into account dissipation, dis-persion and finite unit cell size. Atom A is 4 nm away fromtop interface, while atom B is adsorbed to bottom interfacewith a horizontal displacement of xB = 5 nm. The totalslab thickness is 100 nm. The inset shows the cooperativeLamb shift dependence on atom B’s horizontal displacement.The red and blue curves denote the two orientations of thetransition dipole moment of the atoms. Good agreement isseen between the EMT model and practical multilayer de-sign paving the way for an experimental demonstration of thesuper-Coulombic effect with cold atoms.

Ugg does not show any type of enhancement for the hy-perbolic medium, in agreement with our discussion. Notethat the scaling dependence in the non-retarded regions isin agreement with equations (6)-(7), as expected. In theretarded regime (r � λ), the excited-state interactionsUeg and Γ

ETdisplay an exponential damping behaviour

due to material absorption, while the ground-state inter-action Ugg displays the typical Casimir-Polder power lawdependence, r−7 (Fig. 4).

Fig. (5) proposes a practical plasmonic super-latticesystem to enhance atom-atom interactions taking intoaccount the role of dissipation, dispersion and finite unitcell size. We show the large enhancement of cooperativeLamb shift Jdd for an effective medium model and com-pare it to a 40-layer structure consisting of Ag and TiO2

with a total slab thickness of 100 nm. The two largepeaks seen in fig. (5) occur when the dispersive reso-nance angle θR(λ) is equal to the fixed separation angle,i.e. θR(λ) = θo in agreement with theory.

To conclude, we revealed a class of singular excited-state atom-atom interactions in hyperbolic media thatarises from a fundamental modification of the coulom-bic near-field. The experimental observation of such ef-fects will require careful isolation of the medium inducedcooperative interactions between atoms from the effectof single atoms interacting with the hyperbolic medium.Our work paves the way for studies of long-range entan-glement and self-organization [5] and is also a first steptowards cold atom studies with hyperbolic media exhibit-ing unique effects that are not found in photonic crystalsor cavities.

∗ zjacob@purdue.edu

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