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Dispersion for es in ma ros opi quantum

ele trodynami s

Stefan Yoshi Buhmann

∗,1, Dirk-Gunnar Wels h

1

Theoretis h-Physikalis hes Institut, Friedri h-S hiller-Universität Jena,

Max-Wien-Platz 1, 07743 Jena, Germany

Abstra t

The des ription of dispersion for es within the framework of ma ros opi quantum

ele trodynami s in linear, dispersing and absorbing media ombines the benets of

approa hes based on normal-mode te hniques of standard quantum ele trodynami s

and methods based on linear-response theory in a natural way. It renders generally

valid expressions for both the for es between bodies and the for es on atoms in the

presen e of bodies while showing very learly the intimate relation between the dif-

ferent types of dispersion for es. By onsidering examples, the inuen e of various

fa tors like form, size, ele tri and magneti properties, or intervening media on the

for es is addressed. Sin e the approa h based on ma ros opi quantum ele trody-

nami s does not only apply to equilibrium systems, it an be used to investigate

dynami al ee ts su h as the temporal evolution of for es on arbitrarily ex ited

atoms.

Key words: dispersion for e, ma ros opi quantum ele trodynami s, Casimir

ee t, van der Waals for e, atomsurfa e intera tion, intermole ular potential,

atomi polarizability, magneto-ele tri medium, multilayer stru ture, spontaneous

de ay, weak atomeld oupling, strong atomeld oupling, Rabi os illations

PACS: 12.20.-m, 42.50.Ct, 34.50.Dy, 42.50.Nn

Contents

1 Introdu tion 2

1.1 Dispersion for es 3

1.2 Experimental observations 4

∗Corresponding author.

Email address: s.buhmanntpi.uni-jena.de (Stefan Yoshi Buhmann).

URL: www.tpi.uni-jena.de/tpi/qophysi s/qo.html (Dirk-Gunnar Wels h).

1This work was supported by the Deuts he Fors hungsgemeins haft.

1.3 Appli ations 8

1.4 Theoreti al approa hes 9

2 Elements of QED in linearly responding media 19

2.1 The medium-assisted ele tromagneti eld 19

2.2 Atomeld intera tion 25

3 For es on bodies 32

3.1 Casimir stress in planar stru tures 36

3.2 Ma ro- and mi ro-obje ts 39

4 For es on atoms 45

4.1 Ground-state atoms 45

4.2 Ex ited atoms 61

5 Con luding remarks 75

A knowledgements 77

A Overview over s enarios 78

B Green tensors 80

Referen es 81

1 Introdu tion

Dispersion for es originate from the ele tromagneti intera tion between ele -

tri ally neutral obje ts whi h do not arry permanent ele tri and magneti

moments. They have been of in reasing interest be ause of their important

impa t on many areas of s ien e. In parti ular, the extremely miniaturized

omponents in nanote hnology an be strongly ae ted by dispersion for es.

Re ent progress in experimental te hniques has led to a urate measurements

of dispersion for es whi h have onrmed some of the theoreti al predi tions

while posing new questions at the same time.

2

1.1 Dispersion for es

The predi tion of dispersion for es is one of the most prominent a hievements

of quantum ele trodynami s (QED) where they an be regarded as being a

onsequen e of quantum ground-state u tuations. In order to understand how

quantum u tuations lead to dispersion for es, it may be helpful to rst re all

the orresponding lassi al situation. A ording to lassi al ele trodynami s,

ele tri ally neutral, unpolarized material obje ts will not intera t with ea h

other, even if they are polarizable. An intera tion an only o ur if (i) at least

one of the obje ts is polarized or (ii) an ele tromagneti eld is applied to at

least one of the obje ts. In the former ase the obje t's polarization will give

rise to an ele tromagneti eld whi h an indu e a polarization of the other

polarizable obje t(s); in the latter ase the applied eld indu es a polarization

of the obje t whi h in turn gives rise to an ele tromagneti eld a ting on the

other obje t(s). Both ases result in polarized obje ts intera ting with ea h

other via an ele tromagneti eld, the intera tion and the resulting attra tive

for es between them being a onsequen e of the departure from the lassi al

ground stateunpolarized obje ts and vanishing ele tromagneti eld.

In QED, the state that most losely orresponds to the lassi al ground state

is given by the material obje ts being in their (unpolarized) quantum ground

states and the ele tromagneti eld being in its va uum state, su h that both

the ele tromagneti eld and the polarization of all obje ts vanish on the

quantum average. At rst glan e, one ould hen e expe t the absen e of any

intera tion between the obje ts. However, the Heisenberg un ertainty prin iple

ne essarily implies the existen e of ground state u tuations, i.e., both (i) u -

tuating polarizations of the obje ts and (ii) a u tuating ele tromagneti eld

will always be present. These u tuations give rise to an intera tion between

the obje tsa purely quantum ee t whi h is manifested in the dispersion

for es a ting on them. At nite temperatures, additional thermal u tuations

ome into play.

Thus, dispersion for esalso known as Casimir or van der Waals for es

2are

ever-present long-range for es between atoms and/or ma ros opi bodies, i.e.,

they exist even if the intera ting obje ts are ele tri ally neutral and do not

arry ele tri or magneti moments. Naturally, dispersion for es have many

important onsequen es. On a mi ros opi level, they inuen e, e.g., the prop-

erties of weakly bound mole ules [1,2. A prominent ma ros opi signature of

dispersion for es is the well-known orre tion to the equation of state of an

ideal gas, leading to the more general van der Waals equation.

3But disper-

2Often, the term Casimir for e is used to denote dispersion for es on a ma ros opi

level whereas dispersion for es on a mi ros opi level are referred to as van der Waals

for es.

3In fa t, it was in this ontext that the existen e of dispersion for es was rst

3

sion for es also inuen e the ma ros opi properties of liquids and solids su h

as the anomalies of water [4, the magneti , thermal and opti al properties of

solid oxygen [5 or the melting behavior of weakly bound rystals [6.

The inuen e of dispersion for es be omes even more pronoun ed in the pres-

en e of interfa es between dierent phases and/or media. Atomsurfa e dis-

persion intera tions drive the adsorption of inert gas atoms to solid surfa es

[79, inuen e the wetting properties of liquids on su h surfa es [8,10,11 and

lead to the phenomenon of apillarity [12. The mutual dispersion attra tions

of olloidal parti les suspended in a liquid [13 inuen e the stability of su h

suspensions [14,15; unless su iently balan ed by repulsive for es, they lead

to a lustering of the parti les, ommonly known as o ulation [16,17.

The above mentioned relevan e of dispersion for es to material s ien es and

physi al hemistry being rather obvious, it is perhaps more surprising to note

that they also play a role in astrophysi s and biology. Thus, dispersion for es

initiate the preplanetary dust aggregation leading to the formation of planets

around a star [18. Furthermore, they are needed for an understanding of the

intera tion of mole ules with ell membranes [19,20 and of ell adhesion driven

by mutual ell-membrane intera tions [19,21. Re ently, dispersion for es have

been found to be responsible for the remarkable abilities of some ge ko [22

and spider spe ies [23 to limb smooth, dry surfa es.

1.2 Experimental observations

A for e between two ma ros opi bodies an most easily be measured in a

quasi-stati way where the bodies are brought lose together and the for e to

be measured is ompensated by a for e of known magnitude. In the rst use of

this idea for measuring dispersion for es, the ompensating for e was provided

by a Hookean spring and the distan e of the bodies was measured by means of

opti al interferometry [24. In this way, for es between two diele tri [2428

and metal plates [2932 were investigated. Di ulties in aligning the plates

were over ome by using alternative setups of a plate intera ting with a spher-

i al lens [28,3335, two intera ting spheres [36, and rossed ylinders [36,37.

Magneti for es generated by ele tri urrents were also used to ompensate

dispersion for es where via feedba k, the for e values ould be inferred from

urrent measurements [3841. The typi al out ome of all these experiments

was the observation of attra tive for es whi h follow 1/z4 and 1/z3 power

laws for the plateplate and platesphere

4geometries, respe tively, with z

denoting the obje tobje t separation (see Ref. [42 for a review). However,

owing to the rather low a ura ythe main limitations being the presen e

of ele trostati for es due to residual harges, the roughness of the samples,

predi ted, for a histori al review, see Ref. [3.

4Note that the platesphere separation was mu h smaller than the sphere radius.

4

the la k of a urate position ontrol and the low resolution of the a tual for e

measurementsthe early results remained ontroversial. This is best illus-

trated by the fa t that diering power laws [24,25 and even signs [29 were

found.

Substantial progress was made by using a torsion-balan e s heme [43 where

very smooth bodies, piezoele tri devi es for position ontrol, and apa itive

for e dete tion, were used to measure the for e between a metal sphere and

plate with high a ura y, thereby onrming an attra tive for e proportional

to 1/z3. This experiment has been followed by a number of experiments whi h

have proted by re ent developments in nanote hnology (for an overview, see

Refs. [44,45). By atta hing a mi rosphere to the antilever of an atomi for e

mi ros ope (playing the role of the spring) and monitoring the dispersion-for e

indu ed bending of the antilever via dee tion of an opti al beam, the for e

between the sphere and a nearby surfa e was measured to obtain very a u-

rate results for various metals [4650 and/or diele tri s [51,52, in luding the

inuen e of nite ondu tivity [46,47,49 as well as surfa e roughness [48,50.

It was further demonstrated that one-dimensional periodi surfa e orruga-

tions an lead to a sinusoidally varying tangential for e in addition to the

attra tive normal Casimir for e [50,53,54. A similar experimental setup was

used to measure the for e between two diele tri ylinders [55 and spheres

[56. Dispersion for es have also been measured by means of a mi roma hined

torsion-balan e s heme where a small plate suspended by two thin rods ro-

tates in response to the Casimir for e exerted by a nearby mi rosphere and

this tilting is monitored by apa itive measurements. The s heme was used

to study the for e between dissimilar metals [57,58 and to demonstrate its

dependen e on the thi kness of the intera ting obje ts [59,60. Changing the

obje ts' ree tivity in the visible region by means of hydrogen deposition was

found to have no observable inuen e on the for e [60,61, indi ating that it

should depend on the frequen y-dependent body properties in some integral

form.

Dispersion for es an also be measured in a dynami al setup, based on the idea

that any intera tion will ae t the relative motion of two obje ts. In the rst

experimental realization of this idea, a spheri al lens was mounted on a loud

speaker and periodi ally driven, thereby indu ingby means of the Casimir

for ea similar motion of a nearby plate mounted on a mi rophone [62,63.

Dete tion of the amplitude of the indu ed os illations led to a urate for e

measurements. This idea has been applied in modern experiments to infer the

Casimir for e from the periodi motion of an obje t su h as a mi rosphere os-

illating at the tip of an atomi for e mi ros ope antilever while intera ting

with a surfa e [64,65 or an os illating mi roma hined torsion balan e inter-

a ting with a rigidly mounted sphere [57,58,66,67. Dynami al measurements

of this kind an also provide high-pre ision results for atombody dispersion

intera tions, whi h was demonstrated by observing the hange of the os il-

5

latory motion of a single ex ited ion trapped in a standing ele tromagneti

wave [68 and of old gases of (ground-state) atoms onned in a magneti

trap [69 or an opti al latti e [70, indu ed by their dispersion intera tion

with a nearby surfa e. In the latter experiment, a temperature dependen e of

dispersion for es was observed.

Controlled dynami al measurements of dispersion for es on atoms have only

be ome feasible re ently due to the availability of e ient te hniques to ool

and trap single atoms. In the early experiments, s attering te hniques were

employed whi h are of ourse mu h simpler to implement. Atomatom inter-

a tions have been observed by s attering a beam of ground-state atoms with

known narrow velo ity prole o a se ond beam of atoms with thermal velo -

ity distribution [7173 or a stationary target gas [7477 where an attra tive

1/z7 for e, was found. S attering te hniques have also been employed to probe

the intera tions of atoms with anisotropi mole ules [7880 and even the in-

tera tion of ex ited atoms with ground-state atoms [81 where in the latter

ase a strong enhan ement of the for e, was observed. Eviden e of atombody

for es was rst found by observing the dee tion of a beam of ground-state

atoms or mole ules passing near the surfa e of a metal or diele tri ylinder,

the results suggesting an attra tive 1/z4 for e [8285. 5 In a similar s heme,

the dee tion of atoms passing between two metal plates was monitored by

observing the atom ux losses due to the sti king of atoms to the plates. In

doing so, a strong enhan ement of the for e on ex ited atoms was observed

[86. It was further found [87,88 that the distan e dependen e of the ground-

state for e hanges from a 1/z4 power law for small atomsurfa e separations

(non-retarded regime) to the more rapidly de reasing 1/z5 power law as soon

as the separations ex eed the relevant transition wavelengths of the atoms and

the bodies (retarded regime

6).

It has turned out that introdu ing a ontrollable ompensating for e is also

useful in the ontext of atom-s attering experiments; this is the entral idea of

the evanes ent-wave mirror: A laser beam is in ident on the surfa e of a diele -

tri from the inside at a su iently shallow angle, su h that total ree tion

leads to an exponentially de aying ele tri eld at its exterior. An atom pla ed

in the vi inity of the body will intera t with this evanes ent eld, leading to

an opti al potential. If the laser frequen y is larger than the relevant atomi

transition frequen y (blue detuning), then this potential is repulsive; thus re-

ating the required ompensating for e whi h an be ontrolled by varying the

laser frequen y and intensity. In this way, dispersion for es on ground-state

atoms an be measured by monitoring the ree tion of the atoms in ident on

5In the experiments, the minimum atomsurfa e separation was so small that to a

good approximation, the ylinder surfa es an be regarded as planar.

6Note that the above mentioned measurements of Casimir for es between ma ro-

s opi bodies typi ally operate in the retarded regime.

6

evanes ent-wave mirrors [8991. Alternatively, ompensating for es an be

provided by the magneti elds reated by magneti lms, the strength being

ontrolled by varying the lm thi kness [92.

Ee ts due to the wave nature of the atomi motion be ome relevant for small

values of the ( enter-of-mass) momentum su h that the atomi de Broglie

wavelength be omes su iently large. In this ase quantum ree tion of an

atom from the potential asso iated with the atombody for e may o ur [93.

Quantum ree tion of ground-state [9497 and ex ited atoms [98,99 in ident

on the surfa e of diele tri bodies was observed in various experiments where a

detailed measurement of the atomsurfa e dispersion potential, was a hieved

by re ording the ree tivities at dierent normal velo ities. Another prominent

wave phenomenon that an be exploited for the measurement of atombody

potentials is the dira tion of an atomi wave in ident on a transmission

grating forming a periodi array of parallel slits. When passing the slits (whi h

may be regarded as small planar avities), the atomi matter wave a quires

a phase shift due to the dispersion potential whi h ae ts the interferen e

pattern forming behind the slits. By omparing the experimental observations

with theoreti al simulations, the intera tion of ground-state [100103 as well

as ex ited atoms [104106 with diele tri s in the non-retarded regime has

been measured.

Spe tros opi measurements provide a powerful indire t method for studying

atombody dispersion intera tions. Here, the fa t is exploited that the dis-

persion potential of an atom an be identied with the position-dependent

shift of the respe tive atomi energy level [107. The resulting shifts of the

atomi transition frequen ies an be observed by spe tros opi means. As the

shifts are usually mu h more noti eable for ex ited levels, this approa h yields

good estimates of the dispersion potentials of atoms in ex ited energy eigen-

states. This was demonstrated in experiments measuring dispersion potentials

of atoms inside planar [108110 and spheri al metal avities [111,112, near a

diele tri half spa e [113,114, and of an ion near a metal plate [115. In this

ontext, sele tive ree tion spe tros opy of atomi gases has proven to be a

parti ularly powerful method [116. It is based on the fa t that the ree tion of

a laser beam in ident on a gas ell is modied due to the laser-indu ed polar-

ization of the gas atoms whi h in turn is strongly inuen ed by the dispersion

intera tion of the atoms with the walls of the ell. By omparing measured

ree tivity spe tra with theoreti ally omputed ones, very a urate informa-

tion on the non-retarded dispersion intera tion of atoms with diele tri plates

[117122 was obtained, in luding the potentials of atoms in very short-lived

ex ited energy eigenstates whi h are di ult to study by s attering methods.

Note that the dispersion intera tion with metal plates is mu h more di ult

to observe via sele tive ree tion spe tros opy [123. As a major a hievement,

the method has shown that the dispersion for es on ex ited atoms an be

repulsive [124,125.

7

1.3 Appli ations

Taking advantage of the substantially improved sensitivity of dispersion-for e

measurements, omparison of the experimental results with theoreti al pre-

di tions an nowadays even be used to pla e onstraints on other short-s ale

intera tions of fundamental interest, su h as non-standard gravitational for es

[44,58,69,126129. In addition, dispersion for es have be ome of in reasing im-

portan e in applied s ien e su h as nanote hnology and related elds. While

providing a powerful tool for surfa e ontrol, e.g., in near-eld s anning mi-

ros opy [130,131, they an also be a disturbing fa tor whose inuen e will

be ome more and more pronoun ed with pro eeding miniaturization. In par-

ti ular, they an lead to an undesired and permanent sti king of (small) ob-

je ts to surfa es [132134. A similarly disturbing ee t is observed when atom

traps are operated near surfa es where dispersion for es an diminish the depth

of magneto-opti al traps, thereby imposing limits upon the near-surfa e op-

eration of su h traps [135,136. Traps based on evanes ent waves [137141

ne essarily operate in the near-surfa e regime so that dispersion for es au-

tomati ally ome into play. The inuen e of dispersion for es also needs to

be taken into a ount when onstru ting evanes ent-wave based elements for

atom guiding [142146.

Dispersion for es are indispensable in atom opti s [147 where mirrors and

beam splitters for atomi matter waves, have been onstru ted based on the

dispersion intera tions of atoms with at surfa es and transmission gratings,

respe tively. Transmission gratings an be used to realize Ma hZehnder-type

interferometers for atoms [102. Flat quantum ree tive mirrors provide a fo-

ussing me hanism when dispersion and gravitational for es are ombined with

in an appropriate way [148. In addition, by lo ally enhan ing the ree tivity

of the mirrors via a Fresnel ree tion stru ture [149, ree tion holograms for

atomi matter waves an be realized [150. The e ien y of atomi mirrors an

also be enhan ed by using evanes ent-wave mirrors whi h an even operate

quantum-state sele tively [151. As re ently reported, the quantum ree tion

of ultra old gases at diele tri surfa es gives rise to interesting phenomena,

su h as the ex itation of solitons and vortex stru tures [152.

Further impa t on the appli ation of dispersion for es has been made by the

re ent proposal [153 and subsequent fabri ation [154 of materials with tai-

lored magneto-ele tri properties, also known as metamaterials.

7Metamate-

rials displaying simultaneous negative permittivity and permeability in some

frequen y range, allow for the existen e of traveling ele tromagneti waves

whose ele tri -eld, magneti -eld and wave ve tor, form a left-handed triad

[160

8, leading to a number of unusual ee ts. It is yet an open question

7For the urrent state-of-art of metamaterial fabri ation, see, e.g., Refs. [155159.

8For this reason materials with these properties are ommonly referred to as left-

8

whether left-handed properties an lead to interesting phenomena in the on-

text of dispersion for es and to what extent metamaterials an be exploited

to tailor the shape and sign of these for es. An interesting behavior of disper-

sion for es may also o ur in onjun tion with soft-magneti alloys, su h as

permalloy or Mumetal [161,162. After heating and rapid ooling (a pro ess

alled annealing), these materials are in a state of extremely high permeability;

values of more than 5× 104 have been reported for Mumetal [163.

1.4 Theoreti al approa hes

As already mentioned, dispersion for es arise from quantum zero-point u tu-

ations, namely the u tuating harge and urrent distributions of the intera t-

ing obje ts and the va uum u tuations of the (transverse) ele tromagneti

eld. If the separation of the obje ts is smaller than the wavelengths of the

relevant eld u tuations, then the latter an be disregarded, allowing for a

simplied treatment of dispersion for es. In this non-retarded regime, disper-

sion for es are dominated by the Coulomb intera tion of u tuating harge

distributions. In parti ular, the Coulomb intera tion between two atoms may

within a leading-order multipole expansion be regarded as the intera tion of

two ele tri dipoles d and d′,

V =d·d′ − 3dzd

′z

4πε0z3. (1)

This approa h was rst used by London in onjun tion with leading-order

perturbation theory to derive the potential energy of two isotropi ground-

state atoms to be

U(z) = −Cz6, C =

1

24π2ε20

∑

kk′

∣∣∣〈0|d|k〉∣∣∣2 ∣∣∣〈0′|d′|k′〉

∣∣∣2

Ek + Ek′ − (E0 + E0′)(2)

with |k(′)〉 and Ek(′) denoting the eigenstates and -energies of the unperturbed

atoms [164. The London potential implies an attra tive for e proportional to

1/z7. The idea of deriving dispersion for es from dipoledipole intera tions by

means of perturbation theory was later applied to three- [165171, four- [172

and N-atom intera tion potentials [173176. Lennard-Jones showed [177 that

the intera tion of an atom with a perfe tly ondu ting plate an be treated on

an equal footing, by using the image- harge method. Considering the dipole

dipole intera tion of the atom with its own image in the plate (instead of a

handed materials.

9

se ond atom) within rst-order perturbation theory, he found a potential

U(z) = −〈0|d2|0〉

48πε0z3, (3)

implying an attra tive 1/z4 for e. The inuen e of u tuating magneti dipoles

[178180, ele tri quadrupoles [181 and higher multipoles [182,183 as well as

permanent ele tri [164,168,184,185 and magneti dipoles [184 on the atom

atom intera tion has also been dis ussed. In this way, it was found that the

for e between a magnetizable and a polarizable atom is repulsive and propor-

tional to 1/z5, in ontrast to the attra tive 1/z7 for e between two polarizable

atoms. Furthermore, studying the intera tion of atoms prepared in ex ited

energy eigenstates showed that the ontributions to the for e whi h arise from

real, resonant transitions an be attra tive or repulsive [168,179,184 (for fur-

ther reading regarding the atomatom intera tion, refer to Refs. [186188).

Similar extensions have been a omplished regarding the intera tion of atoms

with (planar) bodies. Quadrupole [189 and higher-order multipole atomi mo-

ments [190192 were in luded in the intera tion of ground-state atoms with

perfe tly ondu ting plates and extensions of the image- harge method to the

intera tion of ground-state [193,194 and atoms in ex ited energy eigenstates

[195 with planar diele tri bodies were given.

The method an be further improved by des ribing the atom and the body on

an equal footing in terms of their harge densities and expressing the resulting

intera tion potential in terms of ele trostati linear response fun tions

9of the

two systems. This was rst demonstrated for a ground-state atom intera ting

with a realisti ele tri

10half spa e exhibiting non-lo al properties [199. The

approa h was demonstrated to lead to a nite value of the intera tion potential

in the limit z → 0 [190,200203.

11For su iently large values of z, the

potential for an atom in front of a half spa e an be given by an asymptoti

power series in 1/z [199,206209

U(z) = − ~

16π2ε0z3

∫ ∞

0dξ α(iξ)

ε(iξ)− 1

ε(iξ) + 1+O(1/z4) (4)

[α(ω), dipole polarizability of the atom; ε(ω), lo al permittivity of the half

spa e where the leading-order term orresponds to an attra tive 1/z4 for e

9In ontrast to the quantities appearing in earlier attempts to treat ondu tors in

a more realisti way [196198, the two response fun tions are dire tly a essible to

measurements.

10The term ele tri is used where no expli it distin tion is made between metals

( ondu tors) and diele tri s (insulators). Likewise, the notion magneto-ele tri is

used to refer to metals or diele tri s possessing non-trivial magneti properties.

11For the dispersion intera tion between two diele tri half spa es, this is shown in

Refs. [204,205.

10

and oin ides with the perfe t ondu tor result (3) in the limit of innite per-

mittivity. Next-order orre tions are due to the atomi quadrupole polarizabil-

ity on the one hand [203 and the leading-order non-lo al diele tri response

on the other hand [199. The response-fun tion approa h has been used to

study the intera tion of various ground-state obje ts with half spa es of dif-

ferent kinds, su h as the for es on an ion [210 and a permanently polarized

atom [211,212 in front of a metal half spa e, an anisotropi mole ule in front

of an ele tri half spa e [213 as well as the intera tion of two atoms in front

of a metal [214,215 and an ele tri half spa e [216. Extensions in lude the

intera tion of an atom in an ex ited energy eigenstate with an ele tri [217

and a birefringent diele tri half spa e [218; non-perturbative ee ts [219;

ee ts due to a onstant external magneti eld [220; and the intera tion

of single ground-state atoms/mole ules with bodies of various shapes where

perfe tly ondu ting [221, non-lo al metalli [222,223 and ele tri spheres

[221, non-lo al metalli [224,225 and ele tri ylinders [226, perfe tly on-

du ting planar [227 and non-lo al metalli spheri al avities [228,229, have

been onsidered.

The intera tion of two ma ros opi bodies B and B′was rst treated by

pairwise summation over the mi ros opi London potentials (2) between the

atoms onstituting the bodies [230,231,

U(z) = −∑

r∈B

∑

r′∈B′

C

|r− r′|6 , (5)

yielding an attra tive 1/z3 for e between two diele tri half spa es [230,232.

Though appli able to bodies of various shapes ( f. also Refs. [200,233), the

method ould only yield approximate results due to the restri tion to two-

atom intera tions. By modeling the body atoms by harmoni os illators, the

intera tion energy of the bodies ould be shown to be a sum of all possible

many-atom intera tion potentials [230,232,234. Appli ations to the intera -

tion of two half spa es [175 and two spheres [235 were studied. Mi ros opi

al ulations of the dispersion intera tion between bodies were soon realized to

be very umbersome, in parti ular for more involved geometries. In an alter-

native approa h based on ma ros opi ele trostati s, the intera tion energy

an be derived from the eigenmodes of the ele trostati Coulomb potential

whi h are subje t to the boundary onditions imposed by the surfa es of dis-

ontinuity [236,237 (for an overview, f. Ref. [45). The method was used

to al ulate Casimir for es between ele tri spheres [237; ele tri spheri al

avities [238; metal half spa es exhibiting non-lo al properties [208; rough

ele tri half spa es [239,240; and ele trolyti half spa es separated by a di-

ele tri [241.

Even though ele trostati methods have been developed into a sophisti ated

theory overing various aspe ts of dispersion for es, they an only render ap-

proximate results valid in the non-retarded limit where the obje t separa-

tions are su iently small so that the inuen e of the transverse ele tromag-

11

neti eld an be disregarded. This was rst demonstrated by Casimir and

Polder [107,242. Using a normal-mode expansion of the quantized ele tro-

magneti eld inside a planar avity bounded by perfe tly ondu ting plates,

they showed that the for e between the plates an be derived from the total

zero-point energy of the modes

E =∑

k

12~ωk. (6)

The di ulty that this energy is divergent was over ome by subtra ting the re-

spe tive diverging energy orresponding to innite plate separation, the nite

result implying a for e per unit area

F =π2~c

240

1

z4. (7)

In a similar way, they obtained the for e on an atom near one of su h plates

and the for e between two atoms in free spa e from the ground-state energy

of the respe tive system in leading-order perturbation theory. They re overed

the results of the non-retarded limit, Eqs. (2) and (3), and found that in the

retarded limit the atomatom and atomplate potentials are given by

U(z) = −23~cα(0)α′(0)

64π3ε20z7

(8)

and

U(z) = − 3~cα(0)

32π2ε0z4, (9)

respe tively, whi h orrespond to attra tive 1/z8 and 1/z5 for es that de reasemore rapidly than the ones in the non-retarded limit. Casimir and Polder had

thus developed a unied theory to des ribe dispersion intera tions over a large

range of distan es.

Normal-mode te hniques have sin e been widely used to study dispersion

intera tions. The two-atom intera tion has been onrmed in various ways

[243257, inter alia by basing the al ulations on the multipolar- oupling

s heme [243,245,254 in pla e of the minimal- oupling s heme originally used

by Casimir and Polder, and relativisti orre tions have been onsidered [258.

Extensions in lude the intera tion of three [247,255,259262 or more atoms

[263,264, the inuen e of higher-order multipole moments [265 and perma-

nent dipole moments [266 on the two-atom for e, the intera tion between

anisotropi ally polarizable atoms [267 and that between a polarizable and a

magnetizable atom [268274 (for an overview, f. Ref. [275). In parti ular,

it was found that in the retarded limit the for e between a polarizable and

a magnetizable atom is repulsive as in the non-retarded limit, but follows

the same 1/z8 power law as that between two polarizable atoms. Further-

more, the intera tion of atoms in ex ited energy eigenstates [276282 and

the inuen e of external onditions su h as nite temperature [273,283286,

12

applied ele tromagneti elds [285, or additional bodies [267,287289 on the

atomatom intera tion have been studied. In parti ular, when the interatomi

separation ex eeds the thermal wavelength, the for e de reases more slowly

(∼ 1/z7) than in the zero-temperature limit. Similarly, the CasimirPolder re-

sult for the atomplate intera tion has been onrmed [247250,252,269,290,

atoms that arry permanent ele tri dipole moments [291 or are magnetizable

[292 have been onsidered and the inuen e of nite temperature [284,293

as well as for e u tuations [294 has been studied. In lose analogy to the

atomatom intera tion, it was found that the intera tion between a magne-

tizable atom and a perfe tly ondu ting plate is repulsive and that the for e

de reases more slowly (∼ 1/z4) than in the zero-temperature limit as soon as

the atomplate separation ex eeds the thermal wavelength. In ontrast to the

atomatom intera tion, the atomplate potential for an atom in an ex ited

energy eigenstate was found to show an os illatory behavior in the retarded

limit [251,287,295,296, thereby making the ee t of the transverse ele tro-

magneti eld more expli it. In addition, atoms intera ting with bodies of

dierent shapes and materials have been onsidered, su h as: Perfe tly on-

du ting planar [291,297301 and paraboli avities [302,303; metal [304, ele -

tri [305308 and magneto-ele tri half spa es [309; ele tri planar [310,311

and spheri al avities [312.

Needless to say that the pioneering work of Casimir and Polder on disper-

sion for es has also stimulated further studies of the problem of bodybody

intera tions (for reviews see Refs. [44,313). Apart from onrming and in-

terpreting the original results [314316, normal-mode te hniques have been

employed to in lude ee ts that arise from nite temperatures [317, surfa e

roughness [318320, the presen e of a diele tri medium between the plates

[321 and even virtual ele tron-positron pairs [322,323 (where the latter were

found to be negligibly small). As in the ase of atombody intera tions, vari-

ous other geometries and materials have been onsidered su h as: Two ele tri

[324327, diele tri [328, lo ally [329331 and non-lo ally responding metal

plates [332; two plates that are polarizable and magnetizable [333339; the

fa es of a perfe tly ondu ting re tangular avity [340,341; two ele tri multi-

layer sta ks [342; a perfe tly ondu ting plate and ylinder [343; two ele tri

spheres [344; a perfe tly ondu ting plate and a small ele tri sphere [345;

a sphere and a surrounding spheri al avity [346. The results qualitatively

resemble the ndings for the atomatom and atombody intera tions. In par-

ti ular, retardation was found to lead to a stronger asymptoti de rease of

the for es whi h is softened due to thermal ee ts as soon as the separations

ex eed the thermal wavelengths; and the for e between a polarizable obje t

and a magnetizable one was found to be repulsive. Perhaps a more surpris-

ing result is the fa t that two birefringent plates may exert a non-vanishing

dispersion torque on ea h other [347,348. Moreover, the problem of Casimir

energies of single bodies

12has been addressed, motivated by a onje ture

12The Casimir energy of a single body an be dened as the geometry-dependent

13

made by Casimir [349, a ording to whi h an attra tive Casimir energy of an

ele tron (modeled as a small perfe tly ondu ting sphere) should be able to

ounterbalan e the repulsive self-energy of the ele tron harge and thus ex-

plain its stability.

13However, the energy of a perfe tly ondu ting sphere was

found to be repulsive [350,351, with similar ndings for a weakly diele tri

sphere [352355. On the ontrary, the Casimir energy of a weakly diele tri

ylinder was found to be attra tive, in agreement with expe tations [353,356.

The physi al signi an e of Casimir energies of single obje ts is yet un lear;

in parti ular it was shown by pairwise summation over mi ros opi dispersion

energies that dispersion energies of ma ros opi obje ts are in fa t dominated

by the always attra tive volume and surfa e energies and may hen e never be

observed [357. In standard al ulations of Casimir energies, these volume and

surfa e energies are either not onsidered from the very beginning or dis arded

during regularization pro edures [355.

Normal-mode te hniques have proved to be a powerful tool for studying disper-

sion for es ( f. also Refs. [3,358). Nevertheless, some prin ipal limitations of

the approa h have be ome apparent re ently, in parti ular in view of the new

hallenges in onne tion with re ent improvements on the experimental side.

So, normal-mode al ulations an be ome extremely umbersome when ap-

plied to obje t geometries relevant to pra ti e or when a realisti des ription of

the ele tromagneti properties of the intera ting obje ts is required. The lim-

itations are also illustrated by the ontroversy regarding the low-temperature

behavior of dispersion for es on bodies (for a re ent a ount of the debate, see

Ref. [359 and referen es therein). The answer to this question requires detailed

knowledge of the ompli ated interplay of positional, thermal and spe tral fa -

tors. To see this, one has to bear in mind that, in general, a large range of

frequen ies ontributes to the for es where the relative inuen e of dierent

frequen y intervals is determined by the obje tobje t separation, tempera-

ture and the frequen y dependen e of the obje t properties. As a onsequen e,

approximations su h as long-/short-range, high-/low-temperature or perfe t-

ree tivity limits be ome intrinsi ally intertwined. A typi al material property

relevant to dispersion for es is the permittivity whi h is a omplex fun tion of

frequen y, with the real and the imaginary part being responsible for disper-

sion and absorption, respe tively. In parti ular, absorption whi h introdu es

additional noise into a system, inhibits the appli ation of normal-mode expan-

sion on a ma ros opi level. This point was rst taken into a ount by Lifshitz

in his al ulation of the dispersion for e between two ele tri half spa es at

nite temperature [360,361 where he derived the for e from the average of the

stress tensor of the u tuating ele tromagneti eld at the surfa es of the half

spa es, with the sour e of the eld being the u tuating noise urrent within

part of the total ele tromagneti energy where the notion geometry-dependent part

is subje t to ambiguities, f. the dis ussion below. For further reading on the Casimir

energy of a single body, f. Ref. [313.

13For a further a dis ussion of this idea, f. Ref. [3.

14

the diele tri matter. The required average was obtained by noting that the

urrent u tuations are linked to the imaginary part of the permittivity via

the u tuationdissipation theorem. In this way, Lifshitz ould express the

for e per unit area in terms of the permittivities ε(ω), ε′(ω) of the two half

spa es where in parti ular, in the non-retarded (zero-temperature) limit the

for e per unit area, was obtained to be

F =~

8π2z3

∫ ∞

0dξ

ε(iξ)− 1

ε(iξ) + 1

ε′(iξ)− 1

ε′(iξ) + 1. (10)

The Lifshitz theory has been applied and extended by a number of authors

(for an overview, see Refs. [44,313), who studied the inuen e of dierent

frequen y ranges [362365, ee ts of nite temperatures [366369 and sur-

fa e roughness [370, and other planar stru tures su h as ele trolyti half

spa es separated by a diele tri [371, magneto-ele tri half spa es [372, metal

plates of nite thi kness [373, metal half spa es exhibiting non-lo al proper-

ties [368,374 and ele tri multilayer systems [375 (for further reading, f.,

e.g., Ref. [233). A typi al approximation for treating small deviations from

planar stru tures (like a sphere that is su iently lose by a plate [330) is

the proximity for e approximation where it is assumed that the intera tion of

two obje ts with gently urved surfa es an be obtained by simply integrating

the (Lifshitz) for e per unit area along the surfa es [376.

14While the debate

regarding the temperature dependen e of the for e between realisti metal

half spa es still seems unsettled, general onsensus is rea hed that in lusion

or negle t of material absorption (i.e., use of a Drude-type or a plasma-type

permittivity) leads to the disagreeing results [359. It is worth noting that

the for es in a planar stru ture an be reexpressed in terms of (frequen y-

dependent) ree tion oe ients dire tly a essible from experiments. This

formulation of the theory has been applied to metal [378380 and ele tri

half spa es [381, metal half spa es with non-lo al properties [382, ele tri

multilayer sta ks [383,384 and, in some approximation, to rough perfe tly

ondu ting [385,386 and metal half spa es [387,388 where the surfa e rough-

ness an give rise to a tangential for e omponent [389 and a torque [390.

Lifshitz's idea of expressing dispersion for es in terms of response fun tions

is of ourse not restri ted to planar systems, but an be extended to arbi-

trary geometries. This an be a hieved by expressing the results obtained by

normal-mode expansion in terms of the Green tensor of the ( lassi al) ele -

tromagneti eld [391393. Alternatively, the Green tensor whi h ontains

all the ne essary information on the shape and the relevant ele tromagneti

properties of the obje ts, an be introdu ed by applying path-integral te h-

niques [394,395 or employing the u tuationdissipation theorem [396. The

14Re ently, validity limits for the proximity for e approximation in the ase of per-

fe tly ondu ting obje ts have been dis ussed on the basis of numeri al al ulations

[377.

15

theory has been used to study the for es between two perfe tly ondu ting

plates [391,397,398, a perfe tly ondu ting plate and a perfe tly permeable

plate [393, two diele tri half spa es [328,399, two ele tri plates [396 and, in

some approximation, two perfe tly ondu ting spheres [393,397, a perfe tly

ondu ting sphere and a perfe tly ondu ting plate [397,398; the for e on a

re tangular piston [400; and the for e and the torque between weakly diele -

tri obje ts of arbitrary shapes [392. In parti ular, it was shown that the

for e between two mirror-symmetri ele tri obje ts is always attra tive [401.

Casimir energies of a perfe tly ondu ting sphere [391, a diele tri sphere

[395, a magneto-diele tri sphere [402 and a perfe tly ondu ting ylinder

[403 have also been studied in this way.

The on ept of linear-response theory has also been widely used to study

dispersion for es on atoms. In parti ular, it an be shown that the (position-

dependent part of the) intera tion energy between a ground-state atom and the

(body-assisted) ele tromagneti va uum in leading-order perturbation theory

an be expressed in terms of the linear response fun tions of the two systems,

U(r) =~µ0

2π

∫ ∞

0dξ ξ2α(iξ) TrG (1)(r, r, iξ), (11)

i.e., the atomi polarizability α(ω) on the one hand and the s attering Green

tensor G

(1)(r, r, ω) of the ele tromagneti eld on the other, thus rendering

a general expression for the for e on an atom in the presen e of arbitrary

bodies [404407

15(for an alternative, semi lassi al approa h based on nd-

ing the eigenenergies of the lassi al ele tromagneti eld intera ting with a

harmoni -os illator atom, see Ref. [200). The method whi h an easily be ex-

tended to thermal elds [408410, also applies to the dispersion intera tion of

two ground-state atoms in free spa e [404,408 or in the presen e of bodies [216

( f. also Refs. [200,411). However, it annot dire tly be applied to atoms in

ex ited energy eigenstates where it is ne essary to again start from the leading-

order intera tion energy and only express the eld ontribution in terms of the

respe tive response fun tion [412,413. Linear response theory has been used

to study the dispersion intera tion of a single ground-state atom with a multi-

tude of bodies su h as: Perfe tly ondu ting plates [404,405,407,408; diele tri

[414, ele tri [405,407,409,415 and magneto-ele tri half spa es [416; metal

half spa es exhibiting non-lo al properties [406,417 and/or surfa e roughness

[406,418 or being overed by a thin overlayer [419; perfe tly ondu ting [420

and diele tri spheres [395,414,421,422; diele tri ylinders [414,423425; per-

fe tly ondu ting and ele tri planar avities [426; diele tri spheri al [427,

ylindri al [424 and perfe tly ondu ting wedge-shaped avities [395,422.

Moreover, the for e on an atom in an ex ited energy eigenstate in front of

15Note that the method is the natural extension of the approa h based on the

ele trostati response fun tion whi h is now repla ed by the response fun tion for

the omplete ele tromagneti eld in luding its transverse part.

16

a perfe tly ondu ting [412,413, diele tri [413 and birefringent diele tri

half spa e [428 has been onsidered; and the intera tion of two ground-state

atoms embedded in a non-lo ally responding ele trolyte [429, pla ed near a

perfe tly ondu ting [430 and a ele tri half spa e [415,431 or inside a per-

fe tly ondu ting [411,430 and diele tri planar avity [432 has been studied.

Finally, relations between mi ros opi and ma ros opi dispersion for es have

been established whose validity is no longer restri ted to the non-retarded

limit. Modeling ma ros opi bodies as olle tions of harmoni -os illator atoms

intera ting with the ele tromagneti eld and al ulating the total energy

of the intera ting system, both the the for e on a single ground-state atom

in the presen e of a diele tri half spa e and the for e between two diele -

tri half spa es were derived from mi ros opi atomatom intera tions [433

where the former result was later extended beyond the harmoni -os illator

model [434. A harmoni -os illator model of atoms with the atoms being ou-

pled to a heat bath, was used to derive the for e between absorbing diele tri

half spa es, onrming the result of Lifshitz theory [435. The mi ros opi -

model al ulations show that only in the limit of weakly polarizable bodies,

i.e., small values of the sus eptibilities, a pairwise sum over two-atom for es

is su ient to obtain the total for e, stressing on e more the importan e of

many-atom intera tions in the ontext of body-assisted dispersion for es. Pair-

wise summation of two-atom intera tions an be used to obtain an approx-

imate des ription of the intera tion between intri ately shaped obje ts, e.g.,

bodies with rough surfa es [436,

16or of atombody/bodybody intera tions

involving ex ited atoms and/or bodies [281,282,439. Conversely, from well-

known formulas for the bodybody intera tion, formulas for the atombody

intera tion [360,361,394,395,440446 as well as the atomatom intera tion

[286,360,361,443,446,447 an be obtained in the limit of the respe tive sus-

eptibilities being asymptoti ally small.

As we have seen, various on epts have been developed to des ribe disper-

sion for esan overview over the dierent s enarios whi h have been studied

theoreti ally, is given in App. A in tabular form. These on epts, to some

extent, are based on dierent basi assumptions and hen e impose dierent

limitations upon the appli ability. The QED on epts based on normal-mode

expansion of the quantized ele tromagneti eld typi ally suer from the fa t

that when ma ros opi bodies ome into play, these bodies should be regarded

as non-absorbing and hen e also non-dispersing. To over ome this di ulty,

arguments from other theories, su h as the u tuationdissipation theorem

of statisti al physi s, must be borrowed. On the ontrary, the on epts based

on linear response theory abandon an expli it eld quantization and employ

16For bodies with small deviations from the planar geometry, the result of pairwise

summation an be improved by introdu ing a orre tion fa tor obtained from Lifshitz

theory [437,438.

17

the u tuationdissipation theorem from the beginning. However, the appli-

ability of methods that make use of the u tuationdissipation theorem by

some means or other is limited to equilibrium systems a disadvantage when

dynami al aspe ts of ex ited atoms are to be onsidered. All on epts have in

ommon that ma ros opi bodies are typi ally des ribed in terms of ma ro-

s opi ele trodynami s, i.e., boundary onditions at surfa es of dis ontinuity

and/or onstitutive relations.

In this arti le we show that by following the formalism of ma ros opi QED in

media (as developed, e.g., in Refs. [448450) from the very beginning, one an

obtain a unied approa h to dispersion for es whi h does not only ombine

the benets of normal-mode QED and linear-response theory in a natural way,

but also a entuates the ommon origin of and intimate relations between the

dierent types of for es and extends the range of appli ation. In parti ular, the

approa h an be used to study dispersion for es for a wide lass of dierent

s enarios, in luding many of those listed in the above overview whi h have

originally been studied by means of a variety of dierent methods.

The further ontents of the arti le are organized as follows. In Se . 2, the

main features of the quantization of the ele tromagneti eld in linear, dis-

persing and absorbing media and the intera tion of the medium-assisted eld

with atoms is outlined, with spe ial emphasis on magneto-ele tri media de-

s ribed in terms of spatially varying permittivities and permeabilities whi h

are omplex fun tions of frequen y. On this basis, in Se . 3, very general for-

mulas for the for e on a ma ros opi body due to its intera tion with other

ma ros opi bodies are presented whi h are valid for arbitrarily shaped bod-

ies as all the relevant properties of the bodies are fully expressed in terms

of the Green tensor of the asso iated ma ros opi Maxwell equations. Both

Casimir stress and Casimir for e are introdu ed, and a very general relation

to many-atom van der Waals for es is established. In parti ular, it is shown

that both the for e on a single ground-state atom intera ting with a body and

the for e between two ground-state atoms, an be obtained as limiting ases

of the general formulas. In Se . 4, for es on individual atoms intera ting with

the body-assisted ele tromagneti eld are studied in more detail, with spe ial

emphasis on expli itly solving the orresponding quantum-me hani al inter-

a tion problem. It is demonstrated how the for e on one or two ground-state

atoms in the presen e of magneto-ele tri bodies an be al ulated by leading-

order perturbation theory, the results agreeing with those obtained in Se . 3.

A number of examples is studied where it is shown that dispersion for es are

often given by simple asymptoti power laws in the retarded and non-retarded

limits. The for e on a single atom initially prepared in an arbitrary ex ited

quantum state is al ulated by solving the atomeld dynami s, leading to

expli itly time-dependent results. Some on luding are given in Se . 5.

18

2 Elements of QED in linearly responding media

It is well known that the properties of the ele tromagneti eld in media an

signi antly dier from those observed in free spa e, and hen e, the intera tion

of the eld with atoms an strongly be inuen ed by the presen e of media.

In lassi al ele trodynami s, linear media are ommonly des ribed in terms of

phenomenologi ally introdu ed ma ros opi ele tri and magneti sus eptibil-

ities (or permittivities and permeabilities, respe tively) available from mea-

surable data. This on ept whi h an be transferred to quantum ele trody-

nami s, has the benet of being universally valid be ause it uses only very

general physi al properties, without the need for spe i mi ros opi matter

models and involved ab initio al ulations.

2.1 The medium-assisted ele tromagneti eld

The medium-assisted ele tromagneti eld in the absen e of free harges or

urrents obeys the ma ros opi Maxwell equations whi h in the Fourier do-

main read

∇·B(r, ω) = 0, (12)

∇×E(r, ω)− iωB(r, ω) = 0, (13)

ε0∇·E(r, ω) = ρin(r, ω), (14)

κ0∇×B(r, ω) + iωε0E(r, ω) = jin(r, ω) (15)

(κ0 = µ−10 ) where the internal harge and urrent densities of the magneto-

ele tri media ρin(r, ω) and j

in(r, ω) are the sour es for the ele tri eld E(r, ω)

and the indu tion eld B(r, ω). Note that the pi ture-independent Fourier

omponents O(r, ω) of an operator eld O(r) are dened a ording to

O(r) =∫ ∞

0dω O(r, ω) + H.c. (16)

so that O(r, ω, t) = e−iω(t−t′)O(r, ω, t′) in the Heisenberg pi ture. Sin e the

internal harge and urrent densities are subje t to the ontinuity equation

−iωρin(r, ω) +∇· j

in(r, ω) = 0, (17)

they may be related to polarization and magnetization elds P(r, ω) and

M(r, ω) as follows:

ρin(r, ω) =−∇·P(r, ω), (18)

jin(r, ω) =− iωP(r, ω) +∇×M(r, ω). (19)

19

Upon introdu ing the displa ement eld

D(r, ω) = ε0E(r, ω) + P(r, ω) (20)

and the magneti eld

H(r, ω) = κ0B(r, ω)− M(r, ω), (21)

the inhomogeneous Maxwell equations (14) and (15) an hen e be written in

the familiar equivalent form

∇·D(r, ω) = 0, (22)

∇×H(r, ω) + iωD(r, ω) = 0 (23)

where the sour e terms asso iated with the internal harge and urrent den-

sities are now ontained in the displa ement and magneti elds.

In parti ular in the ase of linearly and lo ally responding magneto-ele tri

media, Eqs. (20) and (21) take the form

P(r, ω) = ε0[ε(r, ω)− 1]E(r, ω) + PN(r, ω), (24)

M(r, ω) = κ0[1− κ(r, ω)]B(r, ω) + MN(r, ω) (25)

[κ(r, ω) = µ−1(r, ω) with ε(r, ω) and µ(r, ω) being the (relative) ele tri per-

mittivity and magneti permeability of the media, respe tively. Causality im-

plies that ε(r, ω) and µ(r, ω) whi h vary with spa e in general, are omplex-

valued fun tions of frequen y with the KramersKronig relations being satis-

ed [451.

17A ording to the u tuationdissipation theorem, PN(r, ω) and

MN(r, ω) are the (linear) noise polarization and magnetization, respe tively,

asso iated with the (linear) absorption des ribed by the imaginary parts of

ε(r, ω) [Im ε(r, ω)> 0 and µ(r, ω) [Imµ(r, ω)> 0. For simpli ity, in Eqs. (24)

and (25) the material is assumed to be isotropi .

18

Substituting Eqs. (20), (21), (24) and (25) into Eq. (23) and making use

of Eq. (13), one an verify that the ele tri eld obeys the inhomogeneous

17Note that both metals and diele tri s an be des ribed in terms of their permit-

tivity with the main dieren e being that the permittivity of a diele tri is analyti

in the whole upper half of the omplex frequen y plane whereas that of a metal is

ommonly assumed to exhibit a simple pole at ω=0 [451.

18The theory an be extended to arbitrary media, by starting from the general linear

response relation between the urrent density and the ele tri eld. Formulas in this

arti le whi h do not expli itly refer to material properties (but solely via the Green

tensor) are valid for arbitrary linear media [452.

20

Helmholtz equation

[∇×κ(r, ω)∇×− ω

2

c2ε(r, ω)

]E(r, ω) = iωµ0jN(r, ω) (26)

where the sour e term is determined by the noise urrent density

jN(r, ω) = −iωPN(r, ω) +∇×MN(r, ω). (27)

Note that noise urrent density and noise harge density

ρN(r, ω) = −∇·PN(r, ω) (28)

fulll the ontinuity equation

−iωρN(r, ω) +∇· j

N(r, ω) = 0 (29)

[re all Eqs. (17)(19). The solution to Eq. (26) an be given in the form

E(r, ω) = iωµ0

∫d3r′G (r, r′, ω)· j

N(r′, ω) (30)

whi h, a ording to Eq. (13), implies that

B(r, ω) = µ0

∫d3r′ ∇×G(r, r′, ω)· j

N(r′, ω). (31)

Here, G(r, r′, ω) is the ( lassi al) Green tensor whi h is dened by the equation

[∇×κ(r, ω)∇×− ω

2

c2ε(r, ω)

]G(r, r′, ω) = δ(r− r′)I (32)

(I : unit tensor) together with the boundary ondition

G(r, r′, ω)→ 0 for |r− r′| → ∞. (33)

It should be pointed out that the Green tensor is uniquely dened by Eqs. (32)

and (33) provided that the stri t inequalities Im ε(r, ω)>0 and Imµ(r, ω)> 0hold. Note that it is an analyti fun tion of ω in the upper omplex half plane

and and has the following useful properties (A

T

ij =Aji):

G

∗(r, r′, ω) = G(r, r′,−ω∗), (34)

G(r, r′, ω) = G

T(r′, r, ω), (35)

∫d3s

−Im κ(s, ω)

[∇s×G(s, r, ω)

]T ·[∇s×G ∗(s, r′, ω)

]

+ω2

c2Im ε(s, ω)G(r, s, ω)·G∗(s, r′, ω)

= ImG (r, r′, ω). (36)

21

Noise polarization and magnetization and hen e noise urrent density, an be

related to dynami al variables fλ(r, ω) and f†λ(r, ω) (λ∈e,m) of the system

whi h onsists of the ele tromagneti eld and the magneto-ele tri matter, in-

luding the dissipative system responsible for absorption, as follows [448,449:

PN(r, ω) = i

√~ε0π

Im ε(r, ω) fe(r, ω), (37)

MN(r, ω) =

√

−~κ0π

Imκ(r, ω) fm(r, ω) =

√√√√ ~

πµ0

Imµ(r, ω)

|µ(r, ω)|2 fm(r, ω) (38)

with the fλ(r, ω) and f†λ(r, ω) being attributed to the olle tive Bosoni ex i-

tations of the system,

[fλi(r, ω), f

†λ′i′(r

′, ω′)]= δλλ′δii′δ(r− r′)δ(ω − ω′), (39)

[fλi(r, ω), fλ′i′(r

′, ω′)]= 0. (40)

By substituting Eqs. (37) and (38) into Eq. (30), on re alling Eq. (27), we may

express the medium-assisted ele tri eld in terms of the dynami al variables

fλ(r, ω) and f†λ(r, ω) to obtain

E(r, ω) =∑

λ=e,m

∫d3r′Gλ(r, r

′, ω)· fλ(r′, ω) (41)

where

G e(r, r′, ω) = i

ω2

c2

√~

πε0Im ε(r′, ω) G(r, r′, ω), (42)

Gm(r, r′, ω) = i

ω

c

√

− ~

πε0Imκ(r′, ω) [∇′×G (r′, r, ω)]

T, (43)

so that, a ording to Eq. (16),

E(r) =∫ ∞

0dω E(r, ω) + H.c.

=∑

λ=e,m

∫d3r′

∫ ∞

0dωGλ(r, r

′, ω)· fλ(r′, ω) + H.c.. (44)

Note that the relation (36) an be written in the more ompa t form

∑

λ=e,m

∫d3sGλ(r, s, ω)·G∗T

λ (r′, s, ω) =~µ0

πω2ImG(r, r′, ω). (45)

By starting from Eq. (41) and making use of the Maxwell equations in the

Fourier domain, Eqs. (13) and (23) together with Eqs. (20), (21), (24), (25),

(37) and (38), the other ele tromagneti -eld quantities su h as B(r), D(r)

22

and H(r), an be expressed in terms of the dynami al variables fλ(r, ω) andf†λ(r, ω) in a straightforward way. In parti ular, one derives

B(r, ω) =1

iω

∑

λ=e,m

∫d3r′ ∇×Gλ(r, r

′, ω)· fλ(r′, ω), (46)

and hen e,

B(r) =∫ ∞

0dω B(r, ω) + H.c.

=∑

λ=e,m

∫d3r′

∫ ∞

0

dω

iω∇×Gλ(r, r

′, ω)· fλ(r′, ω) + H.c. (47)

In view of the treatment of the intera tion of the medium-assisted ele tro-

magneti eld with atoms, it may be expedient to express the ele tri and

indu tion elds in terms of potentials,

E(r) = −∇ϕ(r)− ˙A(r), (48)

B(r) = ∇×A(r). (49)

In Coulomb gauge, ∇ ·A(r) = 0, the rst and se ond terms on the r.h.s. of

Eq. (48) are equal to the longitudinal (‖) and transverse (⊥) parts of the

ele tri eld, respe tively. From Eq. (44) it then follows that ∇ϕ(r) and A(r) an be expressed in terms of the dynami al variables fλ(r, ω) and f

†λ(r, ω) as

∇ϕ(r) = −E‖(r) = −∑

λ=e,m

∫d3r′

∫ ∞

0dω ‖

Gλ(r, r′, ω)· fλ(r′, ω) + H.c., (50)

A(r) =∑

λ=e,m

∫d3r′

∫ ∞

0

dω

iω⊥Gλ(r, r

′, ω)· fλ(r′, ω) + H.c. (51)

Note that the longitudinal (transverse) part of a ve tor eld is given by

F‖(⊥)(r) =∫

d3r′ δ‖(⊥)(r− r′)·F(r′) (52)

where

δ‖(r) = −∇∇

(1

4πr

), δ⊥(r) = δ(r)I − δ‖(r) (53)

(and for a tensor eld a ordingly).

The ommutation relations for the ele tromagneti elds an be dedu ed from

the ommutation relations for the dynami al variables fλ(r, ω) and f†λ(r, ω) as

given by Eqs. (39) and (40). In parti ular, it an be shown [448,449 that the

ele tri and indu tion elds obey the well-known (equal-time) ommutation

23

relations

[Ei(r), Bi′(r

′)]= −i~ε−1

0 ǫii′k∂kδ(r− r′), (54)

[Ei(r), Ei′(r

′)]= 0 =

[Bi(r), Bi′(r

′)]. (55)

Introdu ing the anoni al momentum eld asso iated with the (transverse)

ve tor potential,

Π(r) = −ε0E⊥(r), (56)

one an also prove that the (equal-time) ommutation relations

[Ai(r), Πi′(r

′)]= i~δ⊥ii′(r− r′), (57)

[Ai(r), Ai′(r

′)]= 0 =

[Πi(r), Πi′(r

′)]

(58)

are fullled.

It is an almost trivial onsequen e of the quantization s heme that upon hoos-

ing the Hamiltonian of the ombined system to be

Hmf =∑

λ=e,m

∫d3r

∫ ∞

0dω ~ω f

†λ(r, ω)· fλ(r, ω), (59)

the Heisenberg equation of motion

˙O =

i

~

[Hmf , O

](60)

generates the orre t Maxwell equations in the time domain,

∇×E(r) + ˙B(r) = 0, (61)

∇×H(r)− ˙D(r) = 0. (62)

Note that the Maxwell equations ∇·B(r)=0 and ∇·D(r)=0 are fullled by

onstru tion.

The Hilbert spa e an be spanned by Fo k states obtained in the usual way

by repeated appli ation of the reation operators f†λ(r, ω) on the ground state

|0〉 whi h is dened by

fλ(r, ω)|0〉 = 0 ∀ λ, r, ω. (63)

Note that the ground state refers to the ombined system of the ele tromag-

neti eld and the medium. In parti ular, from Eq. (41) together with the

ommutation relations (39) and (40) one derives, on using the integral rela-

tion (45),

〈0|Ei(r, ω)E†

i′(r′, ω′)|0〉 = π−1

~µ0ω2ImG ii′(r, r

′, ω)δ(ω − ω′), (64)

24

whi h reveals that the ground-state u tuations of the ele tri eld are deter-

mined by the imaginary part of the Green tensorin full agreement with the

u tuationdissipation theorem.

It should be stressed that Im ε(r, ω) > 0 and Imµ(r, ω) > 0 are assumed to

hold everywhere. Even in almost empty regions or regions where absorption

is very small and an be negle ted in pra ti e, the imaginary parts of the

permittivity and permeability must not be set equal to zero in the integrands

of expressions of the type (44). To allow for empty-spa e regions, the limits

Im ε(r, ω)→ 0 and Imµ(r, ω)→0may be performed a posteriori, i.e., after tak-

ing the desired expe tation values and having arried out all spatial integrals.

In this sense, the theory provides the quantized ele tromagneti eld in the

presen e of an arbitrary arrangement of linear, ausal magneto-ele tri bodies

hara terized by their permittivities and permeabilities where Im ε(r, ω)≥ 0and Imµ(r, ω)≥ 0.

As outlined above, quantization of the ele tromagneti eld in the presen e of

dispersing and absorbing magneto-ele tri bodies an be performed by start-

ing from the ma ros opi Maxwell equations in luding noise terms asso iated

with absorption, expressing the ele tromagneti eld in terms of these noise

terms and relating them to Bosoni dynami al variables in an appropriate

way ( f. also Refs. [453455). Alternatively, absorption an be a ounted for

by expressing the ele tromagneti eld in terms of auxiliary elds with the

dynami s of the auxiliary elds being su h that the Maxwell equations are

fullled [456. It has been shown that the two approa hes are equivalent [457.

It is worth noting that the quantization s heme is in full agreement with the

results of (quasi-)mi ros opi models of diele tri matter where the polariza-

tion is modeled by harmoni -os illator elds and damping is a ounted for

by introdu ing a bath of additional harmoni os illators [458. After a Fano

diagonalization [459 of the total Hamiltonian of the system (whi h onsists of

the ele tromagneti eld, the polarization and the bath), an expression of the

form (59) is obtained. A model of this type was rst developed for homoge-

neous diele tri s [458 and later extended to inhomogeneous diele tri bodies

[460462, in luding bodies exhibiting non-lo al properties [463. In parti ular

in the latter ase, the dierential equation (32) for the Green tensor obviously

hanges to an integro-dierential equation and Eqs. (37) and (38) must be

modied a ordingly.

2.2 Atomeld intera tion

Let us onsider a system of nonrelativisti parti les of masses mα and harges

qα whi h form an atomi system, e.g., an atom or a mole ule, intera ting with

the medium-assisted ele tromagneti eld. The Hamiltonian governing the

25

dynami s of the atomi system (briey referred to as atom in the following) in

the absen e of the medium-assisted ele tromagneti eld is ommonly given

in the form

Hat =∑

α

p2α

2mα

+ 12

∫d3r ρat(r)ϕat(r) (65)

where ρat(r) and ϕat(r), respe tively, are the harge density and the s alar

potential whi h are attributed to the atom,

ρat(r) =∑

α

qαδ(r− rα), (66)

ϕat(r) =∫d3r′

ρat(r′)

4πε0|r− r′| =∑

α

qα4πε0|r− rα|

, (67)

and the standard ommutation relations

[rαi, pα′i′] = i~δαα′δii′ , (68)

[rαi, rα′i′] = 0 = [pαi, pα′i′] (69)

hold. Obviously, ϕat(r) and ρat(r) obey the Poisson equation

ε0∆ϕat(r) = −ρat(r), (70)

and the ontinuity equation

˙ρat(r) +∇· jat(r) = 0 (71)

is fullled where the atomi urrent density jat(r) reads

jat(r) =12

∑

α

qα[˙rαδ(r− rα) + δ(r− rα) ˙rα

]. (72)

It may be useful [464,465 to introdu e enter-of-mass and relative oordinates

rA =∑

α

mα

mArα, rα = rα − rA (73)

(mA =∑

αmα) with the asso iated momenta being

pA =∑

α

pα, pα = pα −mα

mA

pA. (74)

Combining Eqs. (65) and (74), the atomi Hamiltonian may be written in the

form

Hat =p2A

2mA+∑

α

p2α

2mα+ 1

2

∫d3r ρat(r)ϕat(r)

=p2A

2mA

+∑

n

En|n〉〈n| (75)

26

where En and |n〉 are the eigenenergies and eigenstates of the internal Hamil-

tonian. From the ommutation relations (68) and (69) it then follows that the

non-vanishing ommutators of the new variables are

[rAi, pAi′] = i~δii′ , (76)

[rαi, pα′i′

]= i~δii′

(δαα′ − mα′

mA

). (77)

In parti ular, when mα′/mA≪ 1, then

[rαi, pα′i′

]≃ i~δii′δαα′ . (78)

Further atomi quantities that will be of interest are the atomi polarization

Pat(r) and magnetization Mat(r) [466,

Pat(r) =∑

α

qαrα

∫ 1

0dσ δ

(r− rA − σrα

), (79)

Mat(r) =12

∑

α

qα

∫ 1

0dσ σ

[δ(r−rA−σrα

)rα× ˙

rα − ˙rα×rαδ

(r−rA−σrα

)];

(80)

it an be shown that for neutral atoms, the atomi harge and urrent densities

are related to the atomi polarization and magnetization a ording to

ρat(r) = −∇·Pat(r) (81)

and

jat(r) =˙Pat(r) +∇×Mat(r) + jr(r) (82)

where

jr(r) =12∇×

[Pat(r)× ˙rA − ˙rA×Pat(r)

](83)

whi h is due to the enter-of-mass motion, is known as the Röntgen urrent

density [254,464. Note that Eqs. (70) and (81) imply

ε0∇ϕat(r) = P‖at(r). (84)

Expanding the delta fun tions in Eqs. (79) and (80) in powers of the rela-

tive oordinates rα, we see that the leading-order terms are the ele tri and

magneti dipole densities asso iated with the atom,

Pat(r) = dδ(r− rA), (85)

Mat(r) = mδ(r− rA) (86)

27

where the ele tri and magneti atomi dipole moments read

d =∑

α

qαrα =∑

α

qαrα, (87)

m = 12

∑

α

qαrα× ˙rα. (88)

Note that the se ond equality in Eq. (87) only holds for neutral atoms. Using

the atomi Hamiltonian (75) together with the ommutation relation (77) and

the denition (74), one an easily verify the useful relation

∑

α

qαmα〈m|pα|n〉 = iωmndmn (89)

[ωmn = (Em − En)/~, dmn = 〈m|d|n〉 whi h in turn implies the well-known

sum rule

1

2~

∑

m

ωmn(dnmdmn + dmndnm) =∑

α

q2α2mα

I . (90)

2.2.1 Minimal oupling

Having established the Hamiltonians of the medium-assisted eld and the

atom, we next onsider the atomeld intera tion. A ording to the mini-

mal oupling s heme ( f., e.g., Ref. [254), this may be done by making the

repla ement pα 7→ pα − qαA(rα) in the atomi Hamiltonian (65), summing

the Hamiltonians of the medium-assisted eld and the atom and adding the

Coulomb intera tion of the atom with the medium-assisted eld, leading to

[448,465

H =∑

λ=e,m

∫d3r

∫ ∞

0dω ~ω f

†λ(r, ω)· fλ(r, ω) + 1

2

∑

α

m−1α

[pα − qαA(rα)

]2

+ 12

∫d3r ρat(r)ϕat(r) +

∫d3r ρat(r)ϕ(r)

= Hmf + Hat + Hint (91)

where ϕ(r) and A(r) must be thought of as being expressed in terms of the

dynami al variables fλ(r, ω) and f†λ(r, ω), a ording to Eqs. (50) and (51).

Hen e, the atomeld intera tion energy reads

Hint =∑

α

qαϕ(rα)−∑

α

qαmα

pα ·A(rα) +∑

α

q2α2mα

A2(rα). (92)

Note that the s alar produ t of pα and A(rα) ommutes in the Coulomb gauge

used.

28

The total ele tromagneti eld in the presen e of the atom reads

E(r) = E(r)−∇ϕat(r), B(r) = B(r), (93)

D(r) = D(r)− ε0∇ϕat(r), H(r) = H(r). (94)

Obviously, B(r) and D(r) obey the Maxwell equations

∇·B(r) = 0, (95)

∇·D(r) = ρat(r), (96)

and it is a straightforward al ulation [448,465 to verify that the Hamiltonian

(91) generates the remaining two Maxwell equations

∇×E(r) + ˙B(r) = 0, (97)

∇×H(r)− ˙D(r) = jat(r) (98)

and the Newton equations of motion for the harged parti les,

mα¨rα = qαE(rα) +

12qα[˙rα×B(rα)− B(rα)× ˙rα

](99)

where

˙rα = m−1α

[pα − qαA(rα)

]. (100)

In many ases of pra ti al interest one may assume that the atom is small

ompared to the wavelength of the relevant ele tromagneti eld. It is hen e

useful to employ enter-of-mass and relative oordinates [Eqs. (73) and (74)

and apply the long-wavelength approximation by performing a leading-order

expansion of the intera tion Hamiltonian (92) in terms of the relative parti le

oordinates rα. Considering a neutral atom and re alling Eq. (50), one nds

Hint = −d·E‖(rA)−∑

α

qαmα

pα ·A(rA) +∑

α

q2α2mα

A2(rA). (101)

Note that the last term on the r.h.s. of Eq. (101) is independent of the relative

parti le oordinates and hen e does not a t on the internal state of the atom.

When onsidering pro esses aused by strong resonant transitions between

dierent internal states of the atom, it may therefore be negle ted.

2.2.2 Multipolar oupling

An equivalent des ription of the atomeld intera tion that is widely used is

based on the multipolar- ouplingHamiltonian.

19For a neutral atom, the tran-

19For an extension of the formulas given below to the ase of more than one atoms,

see Ref. [467.

29

sition from the minimal- oupling Hamiltonian (91) to the multipolar- oupling

Hamiltonian is a anoni al transformation of the dynami al variables, orre-

sponding to a unitary transformation with the transformation operator being

given by

U = exp[i

~

∫d3r Pat(r)·A(r)

](102)

where A(r) and Pat(r) are dened by Eqs. (51) and (79), respe tively. This

transformation is ommonly known as the PowerZienauWoolley transfor-

mation [468,469; obviously it does not hange rα,

r′α = U rαU† = rα (103)

and a straightforward al ulation yields [448,465

p′α = U pαU

† = pα − qαA(rα)−∫

d3r Ξα(r)×B(r) (104)

and

f ′λ(r, ω) = U fλ(r, ω)U† = fλ(r, ω) +

1

~ω

∫d3r′ P⊥

at(r′)·G∗

λ(r′, r, ω) (105)

where

Ξα(r) = qαrα

∫ 1

0dσ σδ

(r− rA − σrα

)

− mα

mA

∑

β

qβ rβ

∫ 1

0dσ σδ

(r− rA − σrβ

)+mα

mA

Pat(r). (106)

Now we may express the minimal- oupling Hamiltonian (91) in terms of the

transformed variables to obtain the multipolar- oupling Hamiltonian in the

form

H =∑

λ=e,m

∫d3r

∫ ∞

0dω ~ωf ′†λ (r, ω)· f ′λ(r, ω) +

1

2ε0

∫d3r P′2

at(r)

+∑

α

1

2mα

[p′α +

∫d3r Ξ′

α(r)×B′(r)]2−∫d3r P′

at(r)·E′(r). (107)

Here, E′(r) and B′(r), respe tively, are given by Eqs. (44) and (47) with

fλ(r, ω) [f†λ(r, ω) being repla ed with f ′λ(r, ω) [f

′†λ (r, ω). Note that r′α = rα,

r′α= rα, r′A= rA, P

′at(r)= Pat(r), Ξ

′α(r)= Ξα(r), B

′(r)= B(r), but

E′(r) = E(r) + ε−10 P⊥

at(r) (108)

whi h means that the transformed (medium-assisted) ele tri eld E′(r) hasthe physi al meaning of a displa ement eld, in ontrast to E(r) whi h has

the physi al meaning of an ele tri eld.

30

Hamiltonian (107) an be de omposed into three parts,

H = Hmf ′ + Hat′ + Hint′ (109)

where Hmf ′ is given by Eq. (59) with the primed variables in pla e of the

unprimed ones,

Hmf ′ =∑

λ=e,m

∫d3r

∫ ∞

0dω ~ω f

′†λ (r, ω)· f ′λ(r, ω), (110)

Hat′ is the atomi Hamiltonian,

Hat′ =p′2A

2mA+∑

α

p′2α

2mα+

1

2ε0

∫d3r P′2

at(r)

=p′2A

2mA

+∑

n

E ′n|n′〉〈n′| (111)

and Hint′ is the oupling term,

Hint′ = −∫

d3r P′at(r)·E′(r)−

∫d3r

ˆM′

at(r)·B′(r)

+∑

α

1

2mα

[∫d3r Ξ′

α(r)×B′(r)]2

+1

mA

∫d3r p′

A ·P′at(r)×B′(r) (112)

where

ˆM′

at(r) =∑

α

qα2mα

∫ 1

0dσ σ

[δ(r−r′A−σr′α

)r′α×p′

α − p′α×r′αδ

(r−r′A−σr′α

)].

(113)

Note that in ontrast to the physi al magnetization Mat(r) [Eq. (80),ˆMat(r)

is dened in terms of the anoni ally onjugated momenta rather than the

velo ities, as is required in a anoni al formalism. The Hamiltonian (107)

implies the relation

mα˙r′α = p′

α +∫d3r Ξ′

α(r)×B′(r) (114)

and it is not di ult to see [re all Eqs. (100) and (104) that mα˙r′α=mα

˙rα. Itshould be pointed out that the eigenenergies E ′

n of the internal Hamiltonian

in Eq. (111) may be dierent from the orresponding ones of the internal

Hamiltonian in Eq. (75), be ause of the additional term ontained in

1

2ε0

∫d3r P′2

at(r) =12

∫dr ρ′at(r)ϕ

′at(r) +

1

2ε0

∫d3r

[P′⊥

at (r)]2. (115)

A ordingly, the eigenstates of the two internal Hamiltonians are not related

to ea h other via the unitary transformation U [Eq. (102) in general.

31

One of the advantages of the multipolar oupling s heme is the fa t that it al-

lows for a systemati expansion in terms of the ele tri and magneti multipole

moments of the atom. In parti ular, in the long-wavelength approximation, by

retaining only the leading-order terms in the relative oordinates r′α, the in-

tera tion energy (112) reads

Hint′ = − d′ ·E′(r′A)− ˆm′ ·B′(r′A) +∑

α

q2α8mα

[ˆr′α×B′(r′A)

]2

+3

8mA

[d′×B′(rA)

]2+

1

mA

p′A ·d′×B′(r′A) (116)

where

ˆm′ =∑

α

qα2mα

r′α×p′α. (117)

Note that, in ontrast to m [Eq. (88),

ˆm is dened in terms of the anoni al

momenta. The rst two terms on the r.h.s. of Eq. (116) represent ele tri and

magneti dipole intera tions, respe tively; the next two terms des ribe the

(generalized) diamagneti intera tion; and the last term is the Röntgen inter-

a tion due to the enter-of-mass motion. For non-magneti atoms, Eq. (117)

redu es to the intera tion Hamiltonian in ele tri -dipole approximation,

Hint′ = −d′ ·E′(r′A) +p′A

mA·d′×B′(r′A) (118)

whi h in ases where the inuen e of the enter-of-mass motion on the atom

eld intera tion does not need to be taken into a ount, redu es to

Hint′ = −d′ ·E′(r′A). (119)

3 For es on bodies

Ele tromagneti for es are Lorentz for es. As known, the total Lorentz for e

FL a ting on the matter ontained in a volume V is given by

FL =∫

Vd3r

[ρ(r)E(r) + j(r)×B(r)

]. (120)

Here, the ele tromagneti eld a ts on the the total harge and urrent den-

sities ρ(r) and j(r), respe tively, whi h in general in lude the internal harge

and urrent densities ρin(r) and jin(r), respe tively, whi h are attributed to a

medium [Eqs. (18) and (19) as well as those due to the presen e of additional

sour es, su h as ρat(r) and jat(r) [Eqs. (66) and (72). With the help of the

Maxwell equations [as given by Eqs. (12)(15) with ρin(r) 7→ ρ(r), jin(r) 7→ j(r)

32

one easily nds

ρ(r)E(r) + j(r)×B(r) = ∇·T (r)− ε0∂

∂t

[E(r)×B(r)

](121)

so that

FL =∫

∂Vda·T (r)− ε0

d

dt

∫

Vd3r E(r)×B(r) (122)

where the Maxwell stress tensor

T (r) = ε0E(r)E(r) + µ−10 B(r)B(r)− 1

2

[ε0E

2(r) + µ−10 B2(r)

]I (123)

has been introdu ed. In parti ular, if the volume integral in the se ond term

on the r.h.s. of Eq. (122) does not depend on time, then the total for e redu es

to the surfa e integral

FL =∫

∂VdFL (124)

where

dFL = da·T (r) = T (r)·da (125)

may be regarded as the innitesimal for e element a ting on an innitesimal

surfa e element da. Note that a onstant term in the stress tensor does not

ontribute to the integral in Eq. (124) and an therefore be omitted.

If the Minkowski stress tensor

T

(M)(r) = D(r)E(r) + H(r)B(r)− 12

[D(r)·E(r) + H(r)·B(r)

]I

= T (r) + P(r)E(r)− M(r)B(r)− 12

[P(r)·E(r)− M(r)·B(r)

]I (126)

(whi h agrees with Abraham's stress tensor [470) is used in Eq. (124) [together

with Eq. (125) instead of the Maxwell stress tensor T (r) to al ulate the for e,one nds

dF(M) = da·T (M)(r) = dFL

+ da·P(r)E(r)− M(r)B(r)− 1

2

[P(r)·E(r)− M(r)·B(r)

]I

, (127)

and it is seen that in general

dFL 6= da·T (M)(r). (128)

That is to say, the use of the Minkowski stress tensor is expe ted not to

yield the Lorentz for e, in general. Indeed, a areful analysis and interpreta-

tion of lassi al ele tromagneti for e experiments [471478 shows that the

(energymomentum four-tensor asso iated with the) Lorentz for e passes the

theoreti al and experimental tests and qualies for a orre t des ription of

the energymomentum properties of the ele tromagneti eld in ma ros opi

ele trodynami s [479 (also see Se s. 3.1 and 3.2).

33

In lassi al ele trodynami s, ele tri ally neutral material bodies at zero tem-

perature whi h do not arry a permanent polarization and/or magnetization

are not subje t to a Lorentz for e in the absen e of external ele tromagneti

elds. As already noted in Se . 1.1, the situation hanges in quantum ele -

trodynami s, sin e the ground-state u tuations of the body-assisted ele tro-

magneti eld and the body's polarization/magnetization harge and urrent

densities an give rise to a non-vanishing ground-state expe tation value of

the Lorentz for ethe Casimir for e [480

F =∫

Vd3r

〈0|

[ρ(r)E(r′) + j(r)×B(r′)

]|0〉

r′→r

. (129)

Here, the oin iden e limit r′ → r must be performed in su h a way that

unphysi al (divergent) self-for e ontributions are dis arded after the va uum

expe tation value has been al ulated for r′ 6= r, an expli it pres ription will

be given below Eq. (134).

To al ulate the Casimir for e, let us onsider linear media that lo ally respond

to the ele tromagneti eld and an be hara terized by a spatially varying

omplex permittivity ε(r, ω) and a spatially varying omplex permeability

µ(r, ω). Following Se . 2.1, we may write the medium-assisted ele tri and

indu tion elds in the form of Eqs. (44) and (47). Provided that the volume of

interest V does not ontain any additional harges or urrents, the harge and

urrent densities that are subje t to the Lorentz for e (120) are the internal

ones, ρ(r) = ρin(r) and j(r) = jin(r) [re all Eqs. (18) and (19). Making use of

Eqs. (24) and (25) together with Eqs. (28)(32), one an easily see that

ρ(r, ω) = −ε0∇·[ε(r, ω)− 1]E(r, ω)

+ (iω)−1

∇· jN(r, ω) (130)

and

j(r, ω) = − iωε0[ε(r, ω)− 1]E(r, ω)

+ ∇×κ0[1− κ(r, ω)]B(r, ω)

+ j

N(r, ω). (131)

Taking into a ount that E(r, ω) and B(r, ω) an be given in the forms (30)

and (46), respe tively, and that the Green tensor G(r, r′, ω) obeys the dier-ential equation (32), one may perform Eqs. (130) and (131) to obtain

ρ(r, ω) =iω

c2

∫d3r′ ∇·G(r, r′, ω)· j

N(r′, ω), (132)

j(r, ω) =∫

d3r′[∇×∇×−ω

2

c2

]G(r, r′, ω)· j

N(r′, ω). (133)

In this way, the elds ρ(r, ω), j(r, ω), E(r, ω) [Eq. (30) and B(r, ω) [Eq. (31)

are expressed in terms of the noise urrent density jN(r, ω). Making use of

34

Eq. (27) together with Eqs. (37) and (38) and re alling the ommutation rela-

tions (39) and (40), one an easily al ulate the ground-state orrelation fun -

tion 〈0|jN(r, ω)j

†

N(r′, ω′)|0〉 whi h an then be used, on re alling Eqs. (30),

(31), (132) and (133), to al ulate all the orrelation fun tions relevant to the

Casimir for e, as given by Eq. (129). The result is [481

F =~

π

∫

Vd3r

∫ ∞

0dω

(ω2

c2∇·ImG (r, r′, ω)

+ Tr

I×

[∇×∇×−ω

2

c2

]ImG (r, r′, ω)×←−∇′

)

r′→r

= − ~

π

∫

Vd3r

∫ ∞

0dξ

(ξ2

c2∇·G(r, r′, iξ)

− Tr

I×

[∇×∇×+

ξ2

c2

]G (r, r′, iξ)×←−∇′

)

r′→r

(134)

[(TrT )j=T ljl,←−∇ introdu es dierentiation to the left where it is now appar-

ent that in the oin iden e limit r′ → r the Green tensor has to be repla ed

with its s attering part at ea h spa e point. In parti ular, when the mate-

rial in the spa e region V is homogeneous, then the Green tensor therein an

be globally de omposed into a bulk part G

(0)(r, r′, ω) and a s attering part

G

(1)(r, r′, ω),

G(r, r′, ω) = G

(0)(r, r′, ω) +G

(1)(r, r′, ω) (r ∈ V ). (135)

In this ase, the oin iden e limit r′ → r simply means that the Green ten-

sor G(r, r′, ω) an be globally repla ed by its well-behaved s attering part

G

(1)(r, r′, ω).

A ording to Eqs. (123)(125), the Casimir for e an be equivalently rewritten

as a surfa e integral over a stress tensor [480,

F =∫

∂Vda·T (r, r′)r′→r (136)

where

T (r, r′) = 〈0|ε0E(r)E(r

′) + µ−10 B(r)B(r′)

− 12

[ε0E(r)·E(r′) + µ−1

0 B(r)·B(r′)]I

|0〉 (137)

whi h leads to

T (r, r′) = S(r, r′)− 12

[TrS(r, r′)

]I (138)

35

with

S(r, r′) =~

π

∫ ∞

0dω

[ω2

c2ImG(r, r′, ω)−∇×ImG(r, r′, ω)×←−∇′

]

= − ~

π

∫ ∞

0dξ

[ξ2

c2G (r, r′, iξ) +∇×G(r, r′, iξ)×←−∇′

]. (139)

Both Eq. (134) and Eq. (136) [together with Eqs. (138) and (139) are valid

for arbitrary bodies that linearly respond to the ele tromagneti eld, sin e

the for e is fully determined by the Green tensor of the lassi al, ma ros opi

Maxwell equations with the material properties entering the for e formulas

only via the Green tensor. Moreover, Eqs. (134) and (139) reveal that the

for e is proportional to ~ and hen e represents a pure quantum ee t. The

results an be generalized to nite temperatures T in a straightforward way,

by averaging in Eqs. (129) and (137) over the thermal state instead of the

va uum state. As a onsequen e, the r.h.s. of the rst equalities of Eqs. (134)

and (139) are modied a ording to [480

∫ ∞

0dω . . . 7→

∫ ∞

0dω coth

(~ω

2kBT

). . . , (140)

(kB, Boltzmann onstant) so that the nal forms of these equations hange as

~

π

∫ ∞

0dξ f(iξ) 7→ 2kBT

∞∑

n=0

(1− 1

2δn0)f(iξn) (141)

with

ξn =2πkBT

~n (142)

being the Matsubara frequen ies.

3.1 Casimir stress in planar stru tures

Let us apply the theory to a planar magneto-ele tri stru ture dened a ord-

ing to

ε(r, ω) =

ε−(z, ω) z < 0,

ε(ω) 0 < z < d,

ε+(z, ω) z > d,

(143)

µ(r, ω) =

µ−(z, ω) z < 0,

µ(ω) 0 < z < d,

µ+(z, ω) z > d

(144)

36

and restri t our attention to the zero-temperature limit. To determine the

Casimir stress in the interspa e 0< z < d, we need the s attering part of the

Green tensor in Eq. (139) for both spatial arguments within the interspa e

(0 < z = z′ < d). Sin e the omponent q of the wave ve tor parallel to the

interfa es is onserved and the polarizations σ = s, p de ouple, the required

Green tensor (as given in App. B) an be expressed in terms of ree tion

oe ients rσ± = rσ±(ω, q) (q = |q|) referring to ree tion of waves at the

right (+) and left (−) wall, respe tively, as seen from the interspa e. Expli it

(re urren e) expressions for the ree tion oe ients are available if the walls

are multi-slab magneto-ele tri s, f. Eqs. (B.7) and (B.8).

20In the simplest

ase of two homogeneous, semi-innite half spa es, the oe ients rσ± redu e

to the well-known Fresnel amplitudes, Eq. (B.10).

In order to determine the Casimir for e, it is lear for symmetry reasons that

one requires the z omponent Tzz(r)≡ Tzz(r, r)r′→r of the stress tensor in the

interspa e 0<z<d whi h, upon using the Green tensor from App. B, an be

given in the form [480

Tzz(r) = −~

8π2

∫ ∞

0dξ∫ ∞

0dq

q

bµ(iξ)g(z, iξ, q) (145)

where the fun tion g(z, ξ, q) is dened by

g(z, iξ, q) = − 2[b2(1 + n−2) + q2(1− n−2)

]e−2bd rs+rs−D

−1s

− 2[b2(1 + n−2)− q2(1− n−2)

]e−2bd rp+rp−D

−1p

+ (b2 − q2)(1− n−2)[e−2bzrs− + e−2b(d−z)rs+

]D−1

s

− (b2 − q2)(1− n−2)[e−2bzrp− + e−2b(d−z)rp+

]D−1

p (146)

with

n = n(iξ) =√ε(iξ)µ(iξ) , (147)

b = b(iξ, q) =

√

n2(iξ)ξ2

c2+ q2 , (148)

Dσ = Dσ(iξ, q) = 1− rσ+rσ−e−2bd. (149)

A ording to Eq. (136), Eq. (145) [together with Eqs. (146)(149) gives the

for e per unit area between two arbitrary planar multilayer sta ks of (lo ally

responding) dispersing and absorbing magneto-ele tri material where the in-

terspa e between them may ontain an additional magneto-ele tri medium.

It is worth noting that many spe i planar systems that an be addressed

by means of Eq. (145) have been studied by using alternative methods: The

most prominent example is the ase of two ele tri half spa es separated by

20For ontinuous wall proles, Ri ati-type equations have to be solved [482.

37

va uum, as rst onsidered by Lifshitz [360,361 and later readdressed [362

365,378,381, inter alia based on normal-mode QED [325327,329. Extended

s enarios range from ele tri half spa es separated by an ele tri medium (as

studied by means of ele trostati theory [204,205,236,241, normal-mode QED

[321 and an extended Lifshitz theory [371) over ele tri plates of nite thi k-

ness (as addressed on the basis of the Lifshitz theory [373 and linear-response

theory [396), ele tri multilayer sta ks (as treated by generalizing the results

of the Lifshitz theory [375,383,384 as well as normal-mode QED [342) to

magneto-ele tri half spa es (as studied by means of Lifshitz theory [393 as

well as normal-mode QED [333337,372). For purely ele tri multilayer sys-

tems, various ee ts not being taken into a ount by Eq. (145), have also

been addressed in a number of works, su h as the inuen e on the for e of -

nite temperature [324,328,330332,335,337,360,361,366369,379,380,391, sur-

fa e roughness [318320,239,240,370,385389 and non-lo ally responding ma-

terials [208,332,368,374. As outlined above, nite temperature an be easily

in luded in Eq. (145) by applying Eqs. (141) and (142) [480, whereas surfa e

roughness as well as non-lo al material response an be taken into a ount

by returning to the more general formula (136) [together with Eqs. (138) and

(139) and spe ifying the Green tensor appropriately.

To further (numeri ally) evaluate Eq. (145), knowledge of the ξ- and q-depen-den e of the ree tion oe ients whi h depend on the respe tive planar sys-

tem (see, e.g., the examples studied in Refs. [445,446,483485), is required.

Let us here restri t our attention to the limit of perfe tly ondu ting surfa es,

i.e., rp±=−rs±=1 and assume that the wall separation d is su iently large,

so that the permittivity and the permeability of the medium in the interspa e

an be repla ed by their stati values ε≡ ε(0) and µ≡ µ(0). It is then not

di ult to al ulate the simplied integrals in Eq. (145) analyti ally to obtain

the attra tive Casimir for e per unit area, F =Tzz(d) [re all Eq. (136) whi ha ts between two perfe tly ree ting plates as

21

F =π2~c

240

õ

ε

(2

3+

1

3εµ

)1

d4. (150)

In parti ular, when the interspa e is empty (ε = 1, µ = 1), then Eq. (150)

redu es to

F =π2~c

240

1

d4, (151)

in agreement with the famous result (7) rst obtained by Casimir on the basis

21Note that the terms proportional to e−2b(d−z)

in Eq. (146) give rise to divergent

integrals in Eq. (145) in the limit z→ d whi h obviously results from the assumptions

of frequen y-independent response of the plates and the intervening medium and

innite lateral extension of the plates. Sin e in a realisti system these ontributions

are an eled by similar ontributions o urring at the ba ks of the plates [480, they

an be dis arded.

38

of a normal-mode expansion [242 and subsequently rederived [315,317, inter

alia based on lassi al orbits [393,397 or dimensional arguments [314.

In the same approximation, the for e that a ts on a perfe tly ondu ting

plate in a planar avity bounded by perfe tly ondu ting walls and lled with

a magneto-ele tri medium obviously reads

F =π2~c

240

õ

ε

(2

3+

1

3εµ

)(1

d4r− 1

d4l

)(152)

where dl (dr) is the left (right) platewall separation. On the ontrary, use of

the Minkowski stress tensor (126) leads, for µ=1, to [483

F (M) =π2~c

240

1√ε

(1

d4r− 1

d4l

). (153)

Comparing the two results for µ=1, we see that |F |≤ |F (M)|. Introdu tion of

a (polarizable) medium into the interspa e between the plate and the avity

walls is obviously asso iated with some s reening, thereby redu ing the for e

a ting on the plate. Sin e the internal harges and urrents of the interspa e

medium are fully in luded only in the Lorentz for e [re all Eqs. (120)(125),

the for e based on the Minkowski stress tensor or an equivalent quantity un-

derestimates the s reening ee t and is hen e larger than the Lorentz for e in

general.

3.2 Ma ro- and mi ro-obje ts

Let us return to the general formula (134) and onsider the Casimir for e

a ting on ele tri matter of sus eptibility χ(r, ω)= ε(r, ω)−1 in some parti -

ular spa e region V in the presen e of arbitrary linearly responding bodies

(outside V ) in more detail. If G (r, r′, ω) and G(r, r′, ω), respe tively, denotethe Green tensors in the absen e and presen e of the ele tri matter in Vwith both of them taking into a ount the bodies in the remaining spa e, the

dierential equation (32) for G(r, r′, ω) an be onverted into the Dyson-type

integral equation

G(r, r′, ω) = G(r, r′, ω) +ω2

c2

∫

Vd3s χ(s, ω)G(r, s, ω)·G(s, r′, ω) (154)

where, for r ∈ V , the Green tensor G(r, r′, ω) satises the same dierential

equation as the free-spa e Green tensor,

[∇×∇×−ω

2

c2

]G(r, r′, ω) = δ(r− r′)I , (155)

39

from whi h it follows that

ω2

c2∇·G(r, r′, ω) = −∇δ(r − r′) (156)

for r∈V . Equations (154) and (156) imply (r ∈ V , ω real)

∇·ImG(r, r′, ω) = −∇·Im[χ(r, ω)G(r, r′, ω)]. (157)

In a similar way, one nds that (r ∈ V , ω real)

[∇×∇×−ω

2

c2

]ImG (r, r′, ω)×←−∇′ =

ω2

c2Im[χ(r, ω)G(r, r′, ω)]×←−∇′. (158)

Substituting Eqs. (157) and (158) into Eq. (134) one an then show that the

Casimir for e a ting on an ele tri body of volume VI whi h is an inner part

of a larger ele tri body (o upying volume V ) reads [481

F = − ~

2π

∫ ∞

0dξ

ξ2

c2

∫

VI

d3r χ(r, iξ)∇Tr[G(r, r′, iξ)]r′→r

− 2∫

∂VI

da·χ(r, iξ)[G(r, r′, iξ)]r′→r

. (159)

In parti ular, in the ase of an isolated body, i.e., when the region V ⊃ VI isempty apart from the ele tri matter ontained in VI, then the surfa e integral

an be dropped, hen e

F = − ~

2π

∫

VI

d3r∫ ∞

0dξ

ξ2

c2χ(r, iξ)∇Tr[G (r, r′, iξ)]r′→r. (160)

Let us briey ompare Eq. (159) with the equation obtained on the basis of

the Minkowski stress tensor,

F(M) =~

2π

∫

VI

d3r∫ ∞

0dξ

ξ2

c2[∇χ(r, iξ)] Tr[G(r, r, iξ)]r′→r. (161)

It diers from Eq. (159) by a surfa e integral, in general [481. Hen e, the two

for e formulas agree in the ase of an isolated body where the surfa e integrals

do not ontribute to the for e and both equations redu e to Eq. (160). In

ontrast to Eq. (159), appli ation of Eq. (161) to any inner, homogeneous part

of a body leads to the paradoxi al result that the for e identi ally vanishes,

be ause of ∇χ(r, iξ)=0. In other words, the only atoms that are subje t to a

for e are those at the surfa e of the body. On the ontrary, it is known that

the van der Waals for es on all atoms of a body ontribute to the Casimir

for e (Se . 3.2.1).

We have seen that within the framework of ma ros opi QED, Casimir for es

on linearly responding bodies an be expressed in terms of the respe tive

40

Green tensor of the Maxwell equations for the body-assisted ele tromagneti

eld. Hen e, the main problem to be solved in pra ti e is the determination

of the Green tensors for the spe i systems of interest. Sin e losed formulas

for Green tensors are only available for highly symmetri systems (see, e.g.,

Ref. [482), approximative and numeri al methods are required. For exam-

ple, one an start from an appropriately hosen Green tensor as zeroth-order

approximation to the exa t one and perform a Born expansion of the exa t

Green tensor by iteratively solving the orresponding Dyson-type equation. In

parti ular, iteratively solving Eq. (154) yields the Born series

G(r, r′, ω) = G(r, r′, ω) +∞∑

K=1

(ω

c

)2K

K∏

J=1

∫

Vd3sJ χ(sJ , ω)

×G (r, s1, ω)·G(s1, s2, ω) · · ·G(sK , r′, ω). (162)

3.2.1 Weakly polarizable bodies, mi ro-obje ts and atoms

The for e formulas (159) and (160) whi h follow from ma ros opi QED with-

out involved mi ros opi onsiderations, do not only apply to ele tri ma ro-

obje ts but also to mi ro-obje ts. Moreover they also allow for studying the

limiting ase of individual atoms and determining in this way even the disper-

sion for es with whi h bodies a t on atoms and atoms a t on ea h other in the

presen e of bodies. To see this, let us onsider diele tri bodies whi h may be

typi ally thought of as onsisting of distinguishable (ele tri ally neutral but

polarizable) mi ro- onstituents (again briey referred to as atoms), so that

the ClausiusMossotti relation [451,486

χ(r, ω) = ε−10 η(r)α(ω)[1− η(r)α(ω)/(3ε0)]−1

= ε−10 η(r)α(ω) [1 + χ(r, ω)/3] (163)

may be assumed to be valid where α(ω) is the atomi polarizability and η(r)their number density.

22Let VI be the volume of an isolated diele tri body

of sus eptibility χ(r, ω). The Born series (162) and the ClausiusMossotti

relation (163) imply that when the body is su iently small and/or weakly

polarizable, then the for e, as given by Eq. (160), is essentially determined by

the leading-order term proportional to χ(r, ω)≃ε−10 η(r)α(ω), so that Eq. (160)

approximates to

23

F = −~µ0

2π

∫

VI

d3r η(r)∫ ∞

0dξ ξ2α(iξ)∇TrG (1)(r, r, iξ) (164)

22Note that Eq. (163) is onsistent with the requirement that both α(ω) and χ(r, ω)

be Fourier transforms of response fun tions i η(r)α(0)/(3ε0)< 1.23For a small and/or weakly polarizable body that is an inner part of a larger body,

see Refs. [443,481,487.

41

where, a ording to the de omposition

G(r, r′, ω) = G

(0)(r, r′, ω) +G

(1)(r, r′, ω) (r ∈ V ) (165)

[ f. Eq. (135), G

(1)(r, r′, ω) is simply the s attering part of the Green tensor

G(r, r′, ω) of the system without the diele tri body under onsideration.

It an be easily seen that Eq. (164) may be rewritten as [481

F =∫

VI

d3r η(r)F(r) (166)

where

F(r) = −∇U(r) (167)

with

U(r) =~µ0

2π

∫ ∞

0dξ ξ2α(iξ) TrG (1)(r, r, iξ) (168)

being nothing but the van der Waals potential of a single ground-state atom

of polarizability α(ω) at position r in the presen e of arbitrary linearly re-

sponding bodies at zero temperature (Se . 4). Note that in the limiting ase

when VI → 0 and η → ∞ but ηVI = 1, su h that VI overs a single atom at

position rA, then F redu es to F(rA)the for e a ting on a single atom. Note

that a relation of the kind (166) was already used by Lifshitz to dedu e the

dispersion for e between a single atom and an ele tri half spa e from that

between a diele tri and an ele tri half spa e [360,361.

Equation (166) reveals that the for e a ting on a weakly polarizable diele tri

body is the sum of the for es a ting on all body atoms due to their intera tion

with other bodies giving rise to the Green tensor G(r, r′, ω). Let us onsiderin more detail the intera tion of a weakly polarizable body with a se ond iso-

lated diele tri body of volume V ′I whi h is also weakly polarizable [χ′(r, ω)≃

ε−10 η′(r)α′(ω). Denoting the Green tensor asso iated with all remaining bodies

ex ept for the two under onsideration by G (r, r′, ω), expanding G(r, r′, ω), by

starting from G(r, r′, ω) in the Born series [i.e., using Eq. (162) with G 7→G ,

G 7→ G and V 7→ V ′I and again omitting terms of higher than linear order in

the sus eptibility, Eq. (164) leads to [481

F =∫

VI

d3r η(r)∫

V ′I

d3r′ η′(r)F(r, r′) (169)

where

F(r, r′) = −∇U(r, r′), (170)

is the for e with whi h an atom of polarizability α′(ω) at position r′ a ts on

an atom of polarizability α(ω) at position r with

U(r, r′) = −~µ20

2π

∫ ∞

0dξ ξ4α(iξ)α′(iξ)Tr

[G(r, r′, iξ)·G(r′, r, iξ)

](171)

42

being the two-atom van der Waals potential [467,488. Equation (169) learly

shows that the Casimir for e is a volume for e and not a surfa e for e as

ould be suggested on the basis of the Minkowski stress tensor. A ording

to Eq. (169), the Casimir for e between weakly polarizable diele tri bodies

is the sum of all two-atom van der Waals for es between the body atoms

a result whi h was already obtained by Lifshitz for the spe ial ase of two

diele tri half spa es [360,361. In fa t, su h a relation formed the basis of early

al ulations of dispersion for es between bodies [230,231 and it is still used for

treating bodies exhibiting surfa e roughness [436 or ontaining ex ited media

[281,282.

The for e between a polarizable atom and a magnetizable one an be obtained

in a similar way [489. For this purpose, we onsider the intera tion of polar-

izable atoms ontained in the rst, weakly polarizable body (volume VI) witha se ond, weakly magnetizable body of volume V ′

I and magneti sus eptibility

ζ(r, ω)=µ(r, ω)−1=µ0η′(r)β ′(ω) where β ′(ω) denotes the magnetizability of

the atoms ontained in V ′I . Again expanding the Green tensor G(r, r′, ω) asso-

iated with all the bodies ex ept for the rst one by starting from the Green

tensor G(r, r′, ω) (asso iated with all the bodies ex ept for the two under

onsideration) and retaining only the linear order in ζ(r, ω), one obtains

G(r, r′, ω) = G(r, r′, ω)−∫

V ′I

d3s ζ(s, ω)[G(r, s, ω)×←−∇s

]·∇s×G(s, r′, ω).

(172)

Upon substitution of Eq. (172), the for e on the rst body (164) an again

be written as a sum over two-atom for es, Eqs. (169) and (170) where the

potential of a polarizable atom intera ting with a magnetizable one reads

[489

U(r, r′) = −~µ20

2π

∫ ∞

0dξ ξ2α(iξ)β ′(iξ)

× Tr[G(r, s, iξ)×←−∇r

]·∇s×G (s, r, iξ)

s=r′

. (173)

3.2.2 Many-atom van der Waals intera tions

In general, not only two-atom intera tions but all many-atom intera tions

must be taken into a ount to obtain exa t dispersion for es involving ma ro-

s opi bodies. To illustrate this point, let us return to Eq. (168) and onsider

the intera tion of a single atom with a diele tri body (volume V ), whose sus- eptibility χ(r, ω) is given by the ClausiusMossotti relation (163). Re all from

Se . 3.2.1 that G (r, r′, ω) denotes the Green tensor of the whole arrangement

of bodies and G(r, r′, ω) is the Green tensor of all (ba kground) bodies ex ept

for the one under onsideration. Substituting the Born series (162) [G 7→G ,

43

G 7→ G into Eq. (168), one an write the atombody potential in the form

U(r) =∞∑

K=0

UK(r) (174)

where

U0(r) =~µ0

2π

∫ ∞

0dξ ξ2α(iξ)TrG

(1)(r, r, iξ) (175)

is the atomi potential due to the ba kground bodies

24and (K≥ 1)

UK(r) =(−1)K~µ0

2πc2K

K∏

J=1

∫

Vd3sJ χ(sJ , iξ)

∫ ∞

0dξ ξ2K+2α(iξ)

× Tr[G(r, s1, iξ)·G(s1, s2, iξ) · · · G(sK , r, iξ)

](176)

is the ontribution to the potential that is of Kth order in the sus eptibility

of the diele tri body. To further treat the sum on the right-hand side of

Eq. (176), the Green tensors therein are de omposed into singular and regular

parts a ording to

G (r, r′, ω) = −13

(c

ω

)2

δ(r− r′)I + G

′(r, r′, ω) (177)

and use is made of the ClausiusMossotti relation (163). A somewhat lengthy

al ulation then leads to the result that [488,490

U(r) = U0(r) +∞∑

K=1

1

K!

K∏

J=1

∫d3sJ η(sJ)

U(r, s1, . . . , sK) (178)

where

U(r1, r2, . . . , rN) =(−1)N−1

~µN0

(1 + δ2N )π

∫ ∞

0dξ ξ2Nα1(iξ) . . . αN (iξ)

× S Tr[G

′(r1, r2, iξ)·G ′(r2, r3, iξ) · · · G ′(rN , r1, iξ)]

(179)

is the N-atom van der Waals potential in the presen e of arbitrary linearly

responding (ba kground) bodies at zero temperature and hen e generalizes the

free-spa e result given in Refs. [263,264. In Eq. (179), the symbol S introdu es

the symmetrization pres ription

Sf(r1, r2, . . . , rN) =1

(2− δ2N )N∑

Π∈P (N)

f(rΠ(1), rΠ(2), . . . , rΠ(N)) (180)

where P (N) denotes the permutation group of the numbers 1, 2, . . . , N . Note

that G

′(r, r′, ω) = G (r, r′, ω) for r 6= r′. Clearly, for N = 2, Eq. (179) redu esto the two-atom potential already given, Eq. (171).

24Note that U0(r)=0 for free-spa e ba kground, i.e., if there are no further bodies.

44

Equation (178) reveals that under very general onditions the intera tion of a

single ground-state atom with a diele tri body of ClausiusMossotti sus epti-

bility may be regarded as being due to all many-atom intera tions of the atom

in question with the body atoms. A relation of this type was rst derived for

the spe ial ase of a homogeneous half spa e lled with harmoni -os illator

atoms [174,175,433 and later extended to homogeneous diele tri bodies of

arbitrary shapes with va uum ba kground by means of the EwaldOseen ex-

tin tion theorem [434. Note that in lose analogy to Eq. (178), the dispersion

for e between two diele tri bodies is due to all many-atom intera tions of

atoms in the rst body with atoms in the se ond one; this was expli itly ver-

ied for the ases of two homogeneous half spa es [175,433 and spheres [235

and was also shown for two homogeneous bodies of arbitrary shapes [234.

4 For es on atoms

As demonstrated in Se s. 3.2.1 and 3.2.2, for es on individual ground-state

atoms in the presen e of linearly responding bodies an be dedu ed from

the for es between diele tri bodies of ClausiusMossotti type in the limiting

ase of the bodies being weakly polarizable. Alternatively, these for es an be

derived by expli itly studying the intera tion of atoms with the body-assisted

ele tromagneti eld a ording to Se . 2.2. This approa h to dispersion for es

on atoms allows for studying the inuen e of the internal atomi dynami s on

the for es; in parti ular, ex ited atoms an also be onsidered.

4.1 Ground-state atoms

Atoms initially prepared in their ground state will remain in this state provided

that the body-assisted eld is also initially prepared in the ground state. In

addition, the atomeld intera tion in this ase will involve only virtual, o-

resonant transitions of the atoms and the body-assisted eld. Consequently,

dispersion for es on ground-state atoms may adequately be des ribed within

the framework of time-independent perturbation theory.

4.1.1 Single-atom for e

Following the idea of Casimir and Polder [107, one an derive the for e on a

single atom at zero temperature from the shift ∆E of the system's ground-

state energy E for given enter-of-mass position of the atom whi h arises from

the atomeld oupling. The potential U(rA), whose negative gradient gives

the sought van der Waals for e, is the position-dependent part of this energy

45

shift,

25

∆E = ∆(0)E + U(rA). (181)

It an be seen that the ee tive enter-of-mass Hamiltonian

Heff =p2A

2mA

+ U(rA) (182)

leads to the equation of motion

F(rA) = mA¨rA = − 1

~2

[Heff ,

[Heff , mArA

]]= −∇AU(rA). (183)

Making use of the intera tion Hamiltonian (119), one obtains the ground-

state energy shift for a non-magneti atom in se ond-order (i.e., leading-order)

perturbation theory

26as

∆E =∑

k

∑

λ=e,m

∫d3rP

∫ ∞

0dω

∣∣∣〈0|〈0| −d·E(rA)|1λ(r, ω)〉|k〉∣∣∣2

−~(ωk0 + ω)(184)

[P, prin ipal part; |1λ(r, ω)〉= f†λ(r, ω)|0〉. Using Eqs. (39)(41), (44) and

(45), one derives

∆E = −µ0

π

∑

k

P∫ ∞

0

dω

ωk0 + ωω2d0k ·ImG (rA, rA, ω)·dk0 (185)

where G(r, r′, ω) is the Green tensor of the body onguration onsidered.

By dis arding the position-independent ontribution ∆E(0)asso iated with

G

(0)(rA, rA, ω) [re all Eqs. (135) and (181) whi h may be thought of as being

already in luded in the unperturbed energy, the van der Waals potential an

be written in the form [465,491,492

U(rA) =~µ0

2π

∫ ∞

0dξ ξ2Tr

[α(iξ)·G (1)(rA, rA, iξ)

](186)

where

α(ω) = limǫ→0

2

~

∑

k

ωk0d0kdk0

ω2k0 − ω2 − iωǫ

(187)

25The position-independent part ∆(0)E is a ontribution to the Lamb shift in free

spa e; for a dis ussion of the Lamb shift within the multipolar oupling s heme,

see, e.g., Ref. [468.

26In the following, the multipolar oupling s heme will be employed and the primes

dis riminating the respe tive atomi and eld variables from the minimal oupling

ones will be dropped, for notational onvenien e. Equation (119) an be employed,

be ause the se ond term in Eq. (118) gives rise to a ontribution of order v/c (v, enter-of-mass speed) [465 whi h an be negle ted.

46

is the ground-state polarizability of the atom. For isotropi atoms, it simplies

to

α(ω) = α(ω)I = limǫ→0

2

3~

∑

k

ωk0|d0k|2ω2k0 − ω2 − iωǫ

I , (188)

so the potential simplies to

U(rA) =~µ0

2π

∫ ∞

0dξ ξ2α(iξ) TrG (1)(rA, rA, iξ). (189)

The perturbative result hen e agrees with what has been inferred from the

for e on weakly polarizable bodies and renders an expli it expression for the

polarizability. Note that the s attering Green tensor G

(1)in Eq. (189) has

exa tly the same meaning as G in Eq. (168). From Eq. (189) together with

Eq. (188) it an be seen that the potential an be given in the equivalent form

U(rA) = −~µ0

2π

∫ ∞

0dω ω2Im

[α(ω)TrG (1)(rA, rA, ω)

](190)

whi h allows for a simple physi al interpretation of the for e as being due to

orrelation of the u tuating ele tromagneti eld with the orresponding in-

du ed ele tri dipole of the atom plus the orrelation of the u tuating ele tri

dipole with its indu ed ele tri eld [409. It should be pointed out that an

analogous treatment based on the minimal- oupling Hamiltonian (101) leads

to the formally same result [465 where of ourse the unperturbed eigenstates

and energies o urring in the polarizability (187) are now determined by the

atomi Hamiltonian (65) in pla e of (111). Needless to say that both results

are approximations to the same Hamiltonian of the total system. Bearing in

mind that the ground-state energy shift is entirely due to virtual, o-resonant

transitions, it is ru ial to retain the A2term in the minimal oupling s heme

whi h ontributes to the ground-state energy shift already in rst-order per-

turbation theory.

Equation (186) gives the potential of a single ground-state atom in the presen e

of an arbitrary arrangement of linearly responding bodies at zero temperature

in terms of the polarizability of the atom and the Green tensor of the body-

assisted ele tromagneti eld. A relation of this kind was rst derived for arbi-

trary ele tri bodies on the basis of linear-response theory [404,405,407, re all

Eq. (11); alternatively, it was obtained from a QED path-integral approa h

[394 and semi lassi al onsiderations [200. A perturbative derivation based

on ma ros opi QED very similar to that presented here is given in Refs. [493

495. The linear-response approa h has also been applied to magneto-ele tri

bodies [416 and nite temperatures [408410. Note that in lose analogy to

the ase of the Casimir for e, the single-atom potential at nite temperature

an be obtained from the zero-temperature result (186) or (190) by making

the repla ements (141) or (140), respe tively.

47

4.1.2 Two-atom for e

The intera tion potential of two polarizable ground-state atoms in the pres-

en e of linearly responding bodies giving rise to the Green tensor G(r, r′, ω) an also be obtained by means of time-independent perturbation theory. The

leading ontribution is now of fourth order in the atomeld intera tion and

a somewhat lengthy al ulation yields [467,488

U(rA, rB) = −~µ2

0

2π

∫ ∞

0dξ ξ4

× Tr[αA(iξ)·G(rA, rB, iξ)·αB(iξ)·G(rB, rA, iξ)] (191)

whi h for isotropi atoms redu es to

U(rA, rB) = −~µ2

0

2π

∫ ∞

0dξ ξ4αA(iξ)αB(iξ)

× Tr[G(rA, rB, iξ)·G(rB, rA, iξ)]. (192)

Note that Eq. (192) agrees with Eq. (171) [G

′(r1, r2, ω) 7→ G (rA, rB, ω) for

r1 6= r2. The total for e a ting on atom A (B) an be obtained by supple-

menting the single-atom for e F(rA(B)) with the two-atom for e

F(rA(B), rB(A)) = −∇A(B)U(rA, rB) (193)

where in general F(rA, rB) 6= −F(rB, rA), due to the presen e of the bodies.

Equations (191)(193) form a general basis for al ulating two-atom poten-

tials in the presen e of linearly responding bodies at zero temperature. An

equation of the type (192) was rst derived for the ase of ele tri bodies

by means of linear-response theory [216. Derivations based on semi lassi al

models of harmoni -os illator atoms intera ting in the presen e of perfe tly

ondu ting [411 and ele tri bodies [200 were given and extended to allow

for nite temperatures [430. Equation (192) has been used to study the inter-

a tion of two atoms embedded in an ele trolyte [429, situated near perfe tly

ondu ting [430, non-lo al metalli [215, ele tri [216,415,431 and magneto-

ele tri half spa es [467,489, pla ed inside perfe tly ondu ting [411,430 and

ele tri planar avities [432.

Let us onsider the simplest ase of two atoms in free spa e where the single-

atom for e identi ally vanishes and the two-atom potential (192) an be al u-

lated by using the free-spa e Green tensor G (r, r′, ω)=G free(r, r′, ω) (App. B),

leading to (r= |rA− rB|)

U(rA, rB) = −~

32π3ε20r6

∫ ∞

0dξ αA(iξ)αB(iξ)g(ξr/c) (194)

48

where

g(x) = 2e−2x(3 + 6x+ 5x2 + 2x3 + x4

), (195)

whi h shows that the intera tion between two polarizable ground-state atoms

is always attra tive. Equations (194) and (195) are in agreement with the

famous result of Casimir and Polder [107, whose derivation was based on

fourth-order perturbation theory and normal-mode QED. The problem has

been re onsidered many times, inter alia within the frameworks of normal-

mode QED [243,244,246,250252,256,257 and linear-response theory [404;

and it has even be ome a ommon textbook example [245,253,254.

In the retarded limit where r≫ c/ωmin (ωmin denoting the minimum of the

relevant resonan e frequen ies of atoms A and B), the fun tion g(ξr/c) ee -tively limits the ξ integral in Eq. (194) to a range where αA(B)(iξ)≃αA(B)(0),so the potential approa hes

U(rA, rB) = −23~cαA(0)αB(0)

64π3ε20r7

, (196)

as already pointed out in Ref. [107, f. Eq. (8). In the non-retarded limit where

r≪ c/ωmax (ωmax denoting the maximum of the relevant resonan e frequen ies

of atoms A and B), the integral is limited by the polarizabilities αA(B)(iξ) toa range where g(iξ)≃ g(0)= 6, leading to

U(rA, rB) = −3~

16π3ε20r6

∫ ∞

0dξ αA(iξ)αB(iξ). (197)

Upon re alling Eq. (188), one may easily verify that this non-retarded asymp-

tote is nothing but the well-known London potential (2) whi h was origi-

nally obtained from a perturbative treatment of the Coulomb intera tion [164

( f. Refs. [169,170 for similar derivations).

It is illustrative to ompare the potential between two polarizable atoms with

the potential between a polarizable atom A of polarizability αA(ω) and a

magnetizable atom B of magnetizability βB(ω) whi h, a ording to Eq. (173),is given by

U(rA, rB) = −~µ2

0

2π

∫ ∞

0dξ ξ2αA(iξ)βB(iξ)

× Tr[G(rA, r, iξ)×

←−∇

]·∇×G(r, rA, iξ)

r=rB

. (198)

When the two atoms are in free spa e so that G (r, r′, ω)=G free(r, r′, ω), then

Eq. (198) reads

U(rA, rB) =~µ2

0

32π3r4

∫ ∞

0dξ ξ2αA(iξ)βB(iξ)h(ξr/c) (199)

49

where

h(x) = 2e−2x(1 + 2x+ x2

)(200)

whi h is in agreement with results found on the basis of normal-mode QED

[270,274. In ontrast to the attra tive intera tion between two polarizable

atoms, the intera tion between a polarizable and a magnetizable atom is al-

ways repulsive, as an be easily seen from Eq. (199) together with Eq. (200).

In parti ular in the retarded and non-retarded limits, respe tively, Eq. (199)

redu es to

U(rA, rB) =7~cµ0αA(0)βB(0)

64π3ε0r7(201)

and

U(rA, rB) =~µ2

0

16π3r4

∫ ∞

0dξ ξ2αA(iξ)βB(iξ) (202)

whi h was already given in Refs. [268,269 and [178,180.

Comparing Eqs. (196) and (201), we see that in the retarded limit the ab-

solute value of the for e between two polarizable atoms and that between a

polarizable and a magnetizable atom follow the same 1/r8 power law with

the strength being weaker in the latter ase by a fa tor 7/23. Comparison of

Eqs. (197) and (202) shows that in the non-retarded limit the absolute value

of the for e between a polarizable and a magnetizable atom whi h follows a

1/r5 power law, is more weakly diverging than that between two polarizable

atoms whi h follows a 1/r7 power law.

4.1.3 Atom in a planar stru ture

Let us return to Eq. (186) for the potential of a single ground-state atom

in the presen e of an arbitrary arrangement of linearly responding bodies

at zero temperature. It has been used to al ulate the potential for a va-

riety of parti ular geometries, in luding dierent planar stru tures (see be-

low) as well as perfe tly ondu ting [221,420, non-lo al metalli [222,223,

diele tri [221,395,414,421,422,496 and magneto-ele tri spheres [488; diele -

tri [226,414,423,424 and non-lo al metalli ylinders [224,225; magneto-

diele tri rings [490; ele tri ylindri al shells [493495; ele tri [427 and

non-lo al metalli spheri al avities [229; ele tri ylindri al avities [424

and perfe tly ondu ting wedge-shaped avities [395,422.

To illustrate the theory, let us onsider an atom pla ed in a free-spa e region

between two planar magneto-ele tri walls, as dened by Eqs. (143) and (144)

with ε(ω)=µ(ω)=1. Inserting the Green tensor for this system (App. B) into

50

Eq. (189) leads to the single-atom potential [491,497

U(zA) =~µ0

8π2

∫ ∞

0dξ ξ2α(iξ)

∫ ∞

0dq

q

b

e−2bzA

[rs−Ds−(1 + 2

q2c2

ξ2

)rp−Dp

]

+ e−2b(d−zA)

[rs+Ds

−(1 + 2

q2c2

ξ2

)rp+Dp

](203)

where zA is the separation of the atom from the left wall, d is the separation ofthe two walls, b and Dσ are given by Eqs. (148) and (149), respe tively, with

n(iξ) = 1, and rσ± = rσ±(ξ, q) are again the ree tion oe ients asso iated

with the left/right walls. If the atom is pla ed near a single wall, say the right

wall is missing, Eq. (203) redu es to (rσ≡ rσ−)

U(zA) =~µ0

8π2

∫ ∞

0dξ ξ2α(iξ)

∫ ∞

0dq

q

be−2bzA

[rs −

(1 + 2

q2c2

ξ2

)rp

]. (204)

4.1.3.1 Perfe tly ree ting plate Consider rst an atom pla ed near

a perfe tly ree ting ele tri (i.e., perfe tly ondu ting) plate, rp =−rs = 1.By hanging the integration variable in Eq. (204) from q to b [Eq. (148), theresulting integral an be performed to obtain

U(zA) = −~

16π2ε0z3A

∫ ∞

0dξ α(iξ) e−2ξzA/c

1 + 2

(ξzAc

)+ 2

(ξzAc

)2, (205)

in agreement with the result found by Casimir and Polder on the basis of

normal-mode QED [107 ( f. also Refs. [247,250) whi h has been reprodu ed

by means of linear-response theory [404,405,407 and dynami al image-dipole

treatments [290. In the retarded limit, zA≫ c/ωmin, the exponential e−2ξzA/c

ee tively limits the ξ integral to the region where α(iξ)≃ α(0), so the poten-tial approa hes

U(zA) = −3~cα(0)

32π2ε0z4A

, (206)

as already noted in Refs. [107,245,249, f. Eq. (9). In the non-retarded limit,

zA ≪ c/ωmax, the polarizability α(iξ) restri ts the integration to the region

where ξzA/c≃ 0, leading to

U(zA) = −1

48πε0z3A

∑

k

|d0k|2 = −〈0|d2|0〉48πε0z3A

, (207)

in agreement with the result (3) obtained by Lennard-Jones on the basis of

an ele trostati al ulation [177 [re all Eq. (188).

In ontrast, the potential of an atom in front of an innitely permeable plate

51

PSfrag repla ements

(a) (b)

d

d′

mm′

−zA−zA zA zAz z

− −

−

− + +++

Fig. 1. The image dipole onstru tion for an (a) ele tri (b) magneti dipole in front

of a perfe tly ree ting ele tri plate, is shown.

is repulsive, as an be seen by setting rp=−rs=−1 in Eq. (204),

U(zA) =~

16π2ε0z3A

∫ ∞

0dξ α(iξ)e−2ξzA/c

1 + 2

(ξzAc

)+ 2

(ξzAc

)2 . (208)

It approa hes

U(zA) =3~cα(0)

32π2ε0z4A(209)

in the retarded limit ( f. also Ref. [309) and

U(zA) =〈0|d2|0〉48πε0z3A

(210)

in the non-retarded limit.

The dierent signs of the non-retarded potentials (207) and (210) in the ases

of a perfe tly ree ting ele tri and a perfe tly ree ting magneti plate, re-

spe tively, an be understood from an image-dipole model [177. The non-

retarded potential an be regarded as being due to the Coulomb intera tion of

an ele tri dipole d= (dx, dy, dz) situated at distan e zA from the plate with

its image d′ = (−dx,−dy, dz) in the plate [Fig 1(a). The average intera tion

energy of the dipole and its image hen e reads [451

27

U(zA) =1

2

〈0|d·d′ − 3dzd′z|0〉4πε0(2zA)3

= −〈0|d2 + d2z|0〉

64πε0z3A= −〈0|d

2|0〉48πε0z3A

(211)

[〈0|d2x|0〉= 〈0|d2y|0〉= 〈0|d2z|0〉= (1/3)〈0|d2|0〉, in agreement with Eq. (207).

Sin e the intera tion of a polarizable atom with an innitely permeable plate is

equivalent to the intera tion of a magnetizable atom with a perfe tly ree ting

ele tri plate by virtue of the duality of ele tri and magneti elds, a magneti

27The fa tor 1/2 in Eq. (211) a ounts for the fa t that the se ond dipole is indu ed

by the rst one.

52

dipole m= (mx, my, mz) in front of a perfe tly ree ting ele tri plate an be

onsidered. A magneti dipole behaves like a pseudo-ve tor under ree tion,

so its image is given by m′= (mx, my,−mz) [Fig 1(b). The intera tion energyof the magneti dipole and its image reads

U(zA) =1

2

〈0|m·m′ − 3mzm′z|0〉

4πε0(2zA)3=〈0|m2|0〉48πε0z3A

(212)

whi h by means of a duality transformation is equivalent to Eq. (210). The

dierent signs of the potentials (207) and (210) an thus be understood from

the dierent ree tion behavior of ele tri and magneti dipoles.

4.1.3.2 Magneto-ele tri half spa e To be more realisti , let us next

onsider an atom in front of a semi-innite half spa e of given ε(ω) and µ(ω).Upon substitution of the Fresnel ree tion oe ients (B.10), Eq. (204) takes

the form [491,497

U(zA) =~µ0

8π2

∫ ∞

0dξ ξ2α(iξ)

∫ ∞

0dq

q

be−2bzA

[µ(iξ)b− b1µ(iξ)b+ b1

−(1 + 2

q2c2

ξ2

)ε(iξ)b− b1ε(iξ)b+ b1

](213)

with b1 ≡ b1− dened as in Eq. (B.9), in agreement with the result found

by means of linear-response theory [416. From Eq. (213), the results ob-

tained by means of normal-mode QED [252,306,309 and linear-response the-

ory [216,405,407 for an ele tri half spa e an also be re overed.

One an show that in the retarded limit zA ≫ c/ωmin (with ωmin being the

minimum of all relevant atom and medium resonan e frequen ies) the potential

takes the asymptoti form [491,497

28

U(zA) = −3~cα(0)

64π2ε0z4A

∫ ∞

1dv

[(2

v2− 1

v4

)ε(0)v −√ε(0)µ(0)− 1 + v2

ε(0)v +√ε(0)µ(0)− 1 + v2

− 1

v4µ(0)v −

√ε(0)µ(0)− 1 + v2

µ(0)v +√ε(0)µ(0)− 1 + v2

](214)

whi h an be attra tive or repulsive, depending on the strengths of the om-

peting magneti and ele tri properties of the half spa e. Figure 2 shows the

borderline between attra tive and repulsive potentials in the ε(0)µ(0)-plane.

28Obviously this limit does not apply for metals where the ondition an never be

fullled. Similarly, Eqs. (215), (220) and (221) only hold for diele tri s.

53

1

3

5

7

9

11

1 1.5 2 2.5 3

PSfrag repla ements

ε(0)

µ(0)

U(zA) > 0

U(zA) < 0

Fig. 2. Borderline of attra tive and repulsive retarded potentials of a ground-state

atom in front of a magneto-diele tri half spa e.

In parti ular, repulsion o urs i µ(0)−1>3.29[ε(0)−1] or µ(0)>5.11ε(0) forweak and strong magneto-diele tri properties, respe tively.

In the non-retarded limit, n(0)zA≪ c/ωmax [ωmax, maximum of all relevant

atom and medium resonan e frequen ies; n(0) =√ε(0)µ(0), the situation is

more omplex, be ause ele tri and magneti medium properties give rise to

potentials with dierent asymptoti power laws. In parti ular, the potential

approa hes

U(zA) = −~

16π2ε0z3A

∫ ∞

0dξ α(iξ)

ε(iξ)− 1

ε(iξ) + 1(215)

in the ase of a purely diele tri half spa ein agreement with the result (4)

found by onsidering the Coulomb intera tion and using image-dipole methods

[194 or linear-response theory [199and

U(zA) =~

32π2ε0zA

∫ ∞

0dξ

(ξ

c

)2

α(iξ)[µ(iξ)− 1][µ(iξ) + 3]

µ(iξ) + 1(216)

in the ase of a purely magneti one [491,497. Thus for a magneto-diele tri

half spa e the attra tive 1/z3A potential asso iated with the polarizability of

the half spa e will always dominate the repulsive 1/zA potential related to its

magnetizability. It should be noted that the non-retarded limit is in general

in ompatible with the limit of perfe t ree tivity onsidered in Se . 4.1.3.1.

So, Eq. (216) does not approa h Eq. (210) as µ(iξ) tends to innity. On the

ontrary, Eq. (215) onverges to Eq. (207) as ε(iξ) tends to innity [304.

Combining the observations from the retarded and non-retarded limits, one

may thus expe t a potential barrier, provided that the permeability of the

54

0

0

PSfrag repla ements

zAω10/c

U(z

A)1

2π2ε 0

c3/(

ω3 10d

2 01)

µ(0) = 1.00

µ(0) = 2.00

µ(0) = 3.25

µ(0) = 4.06

µ(0) = 5.00

µ(0) = 6.06

0 1 2 3 4 5 6 7

0

−0.001

0.001

0.002

0.003

Fig. 3. The potential of a ground-state two-level atom in front of a magneto-diele tri

half spa e, Eq. (213), is shown as a fun tion of the distan e between the atom and the

half spa e for dierent values of µ(0) (ωPe/ω10=0.75, ωTe/ω10=1.03, ωTm/ω10=1,γe/ω10 = γm/ω10 =0.001).

half spa e is su iently large [491,497. This is illustrated in Fig. 3 where

the potential of a two-level atom as a fun tion of its distan e from the half

spa e, is shown for various values of the (stati ) permeability. In the gure,

the permittivity and permeability of the half spa e have been assumed to be

of DrudeLorentz type,

ε(ω) = 1 +ω2Pe

ω2Te − ω2 − iωγe

, µ(ω) = 1 +ω2Pm

ω2Tm − ω2 − iωγm

. (217)

From the gure it is seen that with in reasing value of µ(0), a potential barrierbegins to form at intermediate distan es, as expe ted. It is shifted to smaller

distan es and in reases in height as the value of µ(0) is further in reased.

4.1.3.3 Plate of nite thi kness Consider now an atom in front of

magneto-ele tri plate of nite thi kness d. Evaluating the relevant ree tion oe ients (B.7) and (B.8) (d≡ d1−), one nds that Eq. (204) takes the form

55

PSfrag repla ements

zAω10/c

U(z

A)1

2π2ε 0

c3/(

ω3 10d

2 01)

dω10/c=0.01

dω10/c=0.10

dω10/c=0.55

dω10/c=1.50

dω10/c=4.00

dω10/c→∞

0

2 4 6 8

0.0012

0.0008

0.0004

Fig. 4. The potential of a ground-state two-level atom in front of a magneto-diele tri

plate, Eq. (218), is shown as a fun tion of the atomplate separation for dierent

values of the plate thi kness d [µ(0)= 5; whereas all other parameters are the same

as in Fig. 2.

[491,497

U(zA) =~µ0

8π2

∫ ∞

0dξ ξ2α(iξ)

∫ ∞

0dq

q

be−2bzA

×

[µ2(iξ)b2 − b21] tanh(b1d)2µ(iξ)bb1 + [µ2(iξ)b2 + b21] tanh(b1d)

−(1 + 2

q2c2

ξ2

)[ε2(iξ)b2 − b21] tanh(b1d)

2ε(iξ)bb1 + [ε2(iξ)b2 + b21] tanh(b1d)

(218)

whi h redu es to the result in Ref. [310 in the spe ial ase of an ele tri plate.

For an asymptoti ally thi k plate, d≫ zA, the exponential fa tor restri ts theintegral in Eq. (218) to a region where b1d∼ d/(2zA)≫ 1. One may hen e make

the approximation tanh(b1d)≃1, leading ba k to Eq. (213) whi h demonstrates

that the semi-innite half spa e is a good model provided that d≫ zA.

On the ontrary, in the limit of an asymptoti ally thin plate, n(0)d≪ zA, the

56

approximation b1d≪ 1 results in [491,497

U(zA) =~µ0d

8π2

∫ ∞

0dξ ξ2α(iξ)

∫ ∞

0dq

q

be−2bzA

[µ2(iξ)b2 − b21

2µ(iξ)b

−(1 + 2

q2c2

ξ2

)ε2(iξ)b2 − b21

2ε(iξ)b

](219)

whi h in the retarded limit approa hes

U(zA) = −~cα(0)d

160π2ε0z5A

[14ε2(0)− 9

ε(0)− 6µ2(0)− 1

µ(0)

]. (220)

In the non-retarded limit one an again distinguish between a purely diele tri

plate and a purely magneti plate. Equation (219) approa hes

U(zA) = −3~d

64π2ε0z4A

∫ ∞

0dξ α(iξ)

ε2(iξ)− 1

ε(iξ)(221)

in the former ase and

U(zA) =~d

64π2ε0z2A

∫ ∞

0dξ

(ξ

c

)2

α(iξ)[µ(iξ)− 1][3µ(iξ) + 1]

µ(iξ)(222)

in the latter ase. Comparing Eqs. (220)(222) with Eqs. (214)(216), we see

that the power laws hange from z−nA to z

−(n+1)A when the plate thi kness

hanges from being innitely large to being innitely small.

If the permeability is su iently big, a magneto-diele tri plate of nite thi k-

ness features a potential wall [491,497, as illustrated in Fig. 4 for a two-level

atom, with the permittivity and permeability of the plate being again given

by Eq. (217). It is seen that for a thin plate the barrier is very low. It raises

with in reasing thi kness of the plate, rea hes a maximal height for some in-

termediate thi kness and then lowers slowly towards the asymptoti half spa e

value as the thi kness is further in reased.

4.1.3.4 Planar avity When an atom is pla ed between two magneto-

ele tri plates, then the ompeting ee ts of attra tive and repulsive intera -

tion with the two plates an result in the formation of a trapping potential

[491,497. Let us model a magneto-ele tri planar avity by two identi al half

spa es of permittivity ε(ω) and permeability µ(ω) whi h are separated by

a distan e d. 29 Substitution of the Fresnel ree tion oe ients (B.10) into

29Purely ele tri planar avities have been modeled with various degrees of detail,

e.g., by two perfe tly ondu ting plates [227,290,293,298301,426, two ele tri half

spa es [426, or even two ele tri plates of nite thi kness [310,311.

57

0

0

PSfrag repla ements

zAω10/c

U(z

A)1

2π2ε 0

c3/(

ω3 10d

2 01)

(a)

(b)

( )

0 5 10 15

0

−0.002

−0.001

0.001

0.002

Fig. 5. The potential of a ground-state two-level atom pla ed between two (a) mag-

neto-diele tri (all parameters as in Fig. 3), (b) purely diele tri [µ(ω) = 1, otherparameters as in (a), and ( ) purely magneti [ε(ω) = 1, other parameters as in

(a) half spa es of separation d= 15c/ω10, Eq. (223), is shown as a fun tion of the

position of the atom.

Eq. (203) yields the potential of an atom pla ed within a avity bounded by

the half spa es:

U(zA) =~µ0

8π2

∫ ∞

0dξ ξ2α(iξ)

∫ ∞

0dq

q

b

[e−2bzA + e−2b(d−zA)

][ 1

Ds

µ(iξ)b− b1µ(iξ)b+ b1

−(1 + 2

q2c2

ξ2

)1

Dp

ε(iξ)b− b1ε(iξ)b+ b1

]. (223)

As expe ted, the potential is in general not the sum of the potentials asso iated

with the left and right plates separately, as a omparison of Eq. (223) with

Eq. (213) shows. Clearly, the dieren e is due to the ee t of multiple ree tion

between the two plates whi h gives rise to the denominators Dσ,

1

Dσ=

∞∑

n=0

(rσ−e

−bdrσ+e−bd)n, (224)

re all Eq. (149).

58

The formation of a potential well is illustrated in Fig. 5 where the potentials

of an atom pla ed between purely diele tri and purely magneti plates, are

also shown. It is seen that the attra tive (repulsive) potentials asso iated with

ea h of two purely diele tri (magneti ) plates ombine to an innite potential

wall (well) at the enter of the avity, while a potential well of nite depth

an be realized within the avity in the ase of two magneto-diele tri plates

of su iently large permeability.

4.1.4 Asymptoti power laws

As we have seen, the dispersive intera tion of polarizable/magnetizable obje ts

in their ground states an often be given in terms of simple asymptoti power

laws in the retarded and non-retarded limits. Typi al examples are given in

Tab. 1 where the asymptoti power laws for the dispersion for e on an atom

intera ting with another atom [Eqs. (196), (197), (201) and (202), a small

sphere [488, thin ring [490, a thin plate [Eqs. (220)(222) and a semi-innite

half spa e [Eqs. (214)(216), and for the for e per unit area between two half

spa es [372,485, are shown.

Comparing the dispersion for es between obje ts of dierent shapes and sizes,

it is seen that the signs are always the same, while the leading inverse powers

are the same or hanged by some global power when moving from one row of

the table to another. This an be understood by assuming that the leading

inverse power is determined by the ontribution to the for e whi h results

from the two-atom intera tion [row (a) by pairwise summation. Obviously,

integration of two-atom for es over the (nite) volumes of a small sphere (b)

or a thin ring ( ) does not hange the respe tive power law, while integration

over an innite volume lowers the leading inverse power a ording to the

number of innite dimensions. So, the leading inverse powers are lowered by

two and three for the intera tion of an atom with a thin plate of innite lateral

extension (d) and a half spa e (e), respe tively. The power laws for the for e

between two half spa es (f) an then be obtained from the atomhalf-spa e

for e (e) by integrating over the three innite dimensions where integration

over z lowers the leading inverse powers by one while the trivial integrations

over x and y yield an innite for e, i.e., a nite for e per unit area. It follows

from the table that many-atom intera tions do not hange the leading power

laws resulting from the summation of pairwise intera tions, but only modify

the proportionality fa tors.

All dispersion for es in Tab. 1 are seen to be attra tive for two polarizable

obje ts and repulsive for a polarizable obje t intera ting with a magnetizable

one. It an further be noted that in the retarded limit the for es de rease more

rapidly with in reasing distan e than might be expe ted from onsidering only

the non-retarded limit. Finally, the table shows that the retarded dispersion

59

Distan e → Retarded Nonretarded

Polarizability → p↔ p p↔ m p↔ p p↔ m

(a)

PSfrag repla ements

r

− 1

r8+

1

r8− 1

r7+

1

r5

(b)

PSfrag repla ements

rA

− 1

r8A+

1

r8A− 1

r7A+

1

r5A

( )

PSfrag repla ements ρA− 1

ρ8A+

1

ρ8A− 1

ρ7A+

1

ρ5A

(d)

PSfrag repla ements

zA

− 1

z6A+

1

z6A− 1

z5A+

1

z3A

(e)

PSfrag repla ements

zA

− 1

z5A+

1

z5A− 1

z4A+

1

z2A

(f)

PSfrag repla ements

z− 1

z4+

1

z4− 1

z3+1

z

Table 1

Asymptoti power laws for the for es between (a) two atoms, (b) an atom and a

small sphere, ( ) an atom and a thin ring, (d) an atom an a thin plate, (e) an atom

and a half spa e and (f) for the for e per unit area between two half spa es. In the

table heading, p stands for a polarizable obje t and m for a magnetizable one. The

signs + and − denote repulsive and attra tive for es, respe tively.

for es between polarizable/polarizable and polarizable/magnetizable obje ts

follow the same power laws, while in the non-retarded limit the for es between

polarizable and magnetizable obje ts are weaker than those between polariz-

able obje ts by two powers in the obje t separation. This an be understood

by regarding the for es as being due to the ele tromagneti eld reated by

the rst obje t intera ting with the se ond. While the ele tri and magneti

far elds reated by an os illating ele tri dipole display the same distan e de-

penden e, the ele tri near eld (whi h an intera t with a se ond polarizable

obje t) is stronger than the magneti near eld (whi h intera ts with a se ond

magnetizable obje t) by one power in the obje t separation (giving rise to a

dieren e of two powers in se ond-order perturbation theory).

60

4.2 Ex ited atoms

In a rst attempt, dispersion for es on atoms in ex ited energy eigenstates

an also be derived from perturbative energy shifts. A straightforward gener-

alization of the al ulation outlined in Se . 4.1.1 to an atom prepared in an

arbitrary energy eigenstate |m〉 yields the potential [465,491,492

Um(rA) = Uorm (rA) + U r

m(rA) (225)

where

Uorm (rA) =

~µ0

2π

∫ ∞

0dξ ξ2Tr

[αm(iξ)·G (1)(rA, rA, iξ)

](226)

and

U rm(rA) = −µ0

∑

k

Θ(ωmk)ω2mkdmk ·ReG (1)(rA, rA, ωmk)·dkm (227)

[Θ(z), unit step fun tion are the o-resonant and resonant ontributions to

the potential, respe tively, and

αm(ω) = limǫ→0

1

~

∑

k

[dmkdkm

ωkm − ω − iǫ+

dkmdmk

ωkm + ω + iǫ

](228)

is the atomi polarizability tensor. Equation (225) [together with Eqs. (226)

and (227) obviously redu es to the ground-state result (186) in the spe ial

ase m=0. Note that the resonant ontribution vanishes in the ground state;

it is only present for an ex ited atom that an undergo real transitions. Equa-

tions (225)(227) whi h an also be obtained by means of linear-response

theory [412,413, have been used to al ulate the potential of an ex ited atom

near a perfe tly ondu ting [412,413, diele tri [413, birefringent diele tri

half spa e [428 and an ele tri ylinder [425.

The above mentioned approa hes are time-independent and essentially per-

turbative and inspe tion of Eqs. (225)(228) reveals that the appli ation of

(stati ) perturbative methods to ex ited atoms is problemati in several re-

spe ts. Firstly, the potential is determined by quantities that are attributed to

the unperturbed atomi transitions whi h do not take into a ount the ee t

of line broadening, whereas in pra ti e nite line widths are observed whi h

are known to strongly ae t resonant transitions. Se ondly, the potential and

hen e also the for e remains onstant in time; this is not very realisti for

ex ited atoms whi h undergo spontaneous de ay with the allowed (dipole-)

transitions being the same as those entering the potential. And thirdly, per-

turbation theory does not apply to the ase of strong atomeld oupling.

These problems an be over ome by a dynami al approa h to the al ulation

of for es a ting on ex ited atoms.

61

4.2.1 Dynami al approa h

Instead of deriving the dispersion for e from an energy shift by some means or

other, we return to the origin of the for e by starting from the Lorentz for e

a ting on an atom and al ulating its expe tation value for a given initial

state. In parti ular, when the ele tromagneti eld is initially in its ground

state, then this expression yields the sought dispersion for e whi h is genuinely

time-dependent for atoms initially prepared in ex ited states.

Summing the physi al momenta mα˙rα of the parti les onstituting the atom

[as given by Eq. (114), one obtains for the atom as a whole

mA˙rA = pA +

∫d3r Pat(r)×B(r). (229)

Hen e, the enter-of-mass motion is governed by the Newton equation

mA¨rA = FL (230)

where, a ording to Eq. (229), the Lorentz for e is given by

FL =i

~

[H, pA

]+

d

dt

∫d3r Pat(r)×B(r) (231)

with Pat(r) from Eq. (79). The rst term in Eq. (231) an be further evaluated

by re alling Eq. (107) and using the ommutation relations (76). By making

use of the identity ∇APat(r)= −∇Pat(r) [re all Eq. (79), one an show that

i

~

[1

2ε0

∫d3r P2

at(r), pA

]=

1

2ε0

∫d3r∇P2

at(r) = 0 (232)

and by re alling Eq. (106) and using the denitions (79) and (80), one derives

i

~

[∑

α

1

2mα

pα +

∫d3r Ξα(r)×B(r)

2

, pA

]

= ∇A

∫d3r

[Mat(r) + Pat(r)× ˙rA

]·B(r)

. (233)

Equations (232) and (233) then imply that Eq. (231) an be written as

FL =∇A

∫d3r Pat(r)·E(r) +

∫d3r

[Mat(r) + Pat(r)× ˙rA

]·B(r)

+d

dt

∫d3r Pat(r)×B(r). (234)

It should be mentioned that by using Eqs. (81)(83) together with the Maxwell

equations (95) and (97), this equation an be given in the equivalent form

FL =∫

d3r[ρat(r)E(r) + jat(r)×B(r)

](235)

62

whi h orresponds to the Eq. (120) used in Se . 3 as a starting point for

al ulating dispersion for es on bodies.

30

From Eq. (234), the Lorentz for e in long-wavelength approximation an be

obtained by performing a leading-order expansion in the relative parti le o-

ordinates rα, resulting in

FL = ∇A

[d·E(rA) + m·B(rA) + d× ˙rA ·B(rA)

]+

d

dt

[d×B(rA)

], (236)

[re all Eqs. (87) and (88) where

d

dt

[d×B(rA)

]=

i

~

[H, d×B(rA)

]=

˙d×B(rA) + d× ˙

B(r)∣∣∣r=rA

+ 12d×

[(˙rA ·∇A

)B(rA) + B(rA)

(←−∇A · ˙rA

)]. (237)

Dis arding all terms proportional to

˙rA (whi h are of the order v/c and thus

negligible for nonrelativisti enter-of-mass motion), as well as the ontribution

from the magneti intera tions, Eq. (236) redu es to

FL =

∇

[d·E(r)

]+

d

dt

[d×B(r)

]

r=rA

, (238)

whose expe tation value

F = 〈FL〉 (239)

quite generally provides a basis for al ulating ele tromagneti for es on non-

magneti atoms, in luding dispersion for es. Needless to say that Eq. (238) is

valid regardless of the state the atom and the body-assisted eld are prepared

in.

At this point we re all that, a ording to Eqs. (44) and (47), the ele tri

and the magneti indu tion elds are expressed in terms of the dynami al

variables fλ(r, ω) and f†λ(r, ω). It is not di ult to prove that in ele tri -dipole

approximation, fλ(r, ω) obeys the Heisenberg equation of motion

˙fλ(r, ω) =

i

~

[H, fλ(r, ω)

]= −iωfλ(r, ω) +

i

~d·G ∗

λ(rA, r, ω) (240)

[re all the Hamiltonian (109) together with Eqs. (110), (111) and (119), whose

formal solution reads

fλ(r, ω, t) = fλfree(r, ω, t) + fλsource(r, ω, t) (241)

30Note that the eld reated by the atom only gives rise to internal for es, so that

one may equivalently write Eq. (235) with the total elds E(r) and B(r) [Eq. (93)instead of E(r) and B(r).

63

where

fλfree(r, ω, t) = e−iω(t−t0)fλ(r, ω) (242)

and

fλsource(r, ω, t) =i

~

∫ t

t0dτ e−iω(t−τ)d(τ)·G ∗

λ[rA(τ), r, ω] (243)

(t0, initial time), respe tively, determine the free-eld parts Efree(r, ω, t) andBfree(r, ω, t) and the sour e-eld parts Esource(r, ω, t) and Bsource(r, ω, t) of theele tri and the indu tion eld in the ω domain. Substitution of Eqs. (241)

(243) together with Eqs. (44) and (47) into Eq. (239) together with Eq. (238)

and use of Eq. (45) leads to the following expression for the mean for e

[465,492:

F(t) = Ffree(t) + Fsource(t) (244)

with

Ffree(t) =∫ ∞

0dω

∇

⟨d(t)·Efree(r, ω, t)

⟩

+d

dt

[⟨d(t)×Bfree(r, ω, t)

⟩]

r=rA(t)

+ C.c. (245)

and

Fsource(t) = Felsource(t) + Fmag

source(t) (246)

where the omponents

Felsource(t) =

iµ0

π

∫ ∞

0dω ω2

∫ t

t0dτ e−iω(t−τ)

×∇

⟨d(t)·ImG [r, rA(τ), ω]·d(τ)

⟩

r=rA(t)

+ C.c. (247)

and

Fmagsource(t) =

µ0

π

∫ ∞

0dω ω

d

dt

∫ t

t0dτ e−iω(t−τ)

×⟨d(t)×

(∇×ImG [r, rA(τ), ω]

)·d(τ)

⟩

r=rA(t)

+ C.c. (248)

are related to the sour e-eld parts of the ele tri and the indu tion eld,

respe tively.

While Eq. (244) together with Eqs. (245)(248) gives the for e on a (non-

magneti ) atom subje t to an arbitrary ele tromagneti eld, the pure disper-

sion for e an be obtained by onsidering the ase where the (body-assisted)

ele tromagneti eld is initially prepared in the ground state |0〉 so that

〈0| · · · Efree(r, ω, t)|0〉 = 〈0| · · · Bfree(r, ω, t)|0〉 = 0 (249)

64

[re all Eq. (63) whi h implies that Ffree(t)=0. Hen e, Eq. (244) simply redu es

to

F(t) = Fsource(t) (250)

in this ase. In parti ular, for hosen atomi position, rA may be regarded as

a time-independent -number parameter [rA(t) 7→ rA, so that the expe tation

values to be taken in Eqs. (247) and (248) only refer to the internal state

of the atom. It should be pointed out that the on ept is not restri ted to

the al ulation of the mean for e but an be extended to higher-order for e

moments (for a dis ussion of for e u tuations, see also Ref. [294).

4.2.2 Weak atomeld oupling

The remaining task now onsists in the determination of the dipoledipole

orrelation fun tion

⟨d(t)d(τ)

⟩=∑

m,n

∑

m′,n′

dmndm′n′

⟨Amn(t)Am′n′(τ)

⟩(251)

in Eqs. (247) and (248) [Amn = |m〉〈n|, re all Eq. (111). To that end, the

problem of the internal atomi dynami s must be solved. Let rst onsider

the ase of weak atomeld oupling where the Markov approximation an

by used to onsiderably simplify the problem. Under the assumption that the

relevant atomi transition frequen ies are well separated from one another so

that diagonal and o-diagonal density matrix elements evolve independently,

appli ation of the quantum-regression theorem (see, e.g., Ref. [450) yields the

familiar result

⟨Amn(t)Am′n′(τ)

⟩= δnm′

⟨Amn′(τ)

⟩eiωmn(rA)−[Γm(rA)+Γn(rA)]/2(t−τ)

(252)

(t≥ τ , m 6= n). Here,

ωmn(rA) = ωmn + δωm(rA)− δωn(rA) (253)

are the atomi transition frequen ies in luding the position-dependent energy-

level shifts

31

δωm(rA) =∑

k

δωkm(rA), (254)

δωkm(rA) =

µ0

π~P∫ ∞

0dω ω2 dkm ·ImG

(1)(rA, rA, ω)·dmk

ωmk(rA)− ω(255)

31The Lamb shifts observed in free spa e are thought of as being already in luded

in the frequen ies ωmn.

65

( f. also Refs. [493495) whi h are due to the intera tion of the atom with

the body-assisted ele tromagneti eld, and similarly,

Γm(rA) =∑

k

Γkm(rA), (256)

Γkm(rA) =

2µ0

~Θ[ωmk(rA)]ω

2mk(rA)dkm ·ImG [rA, rA, ωmk(rA)]·dmk (257)

are the position-dependent level widths. Note that the position-dependent

energy shifts ~δωm(rA) as given by Eq. (254) together with Eq. (255) redu e

to those obtained by leading-order perturbation theory, Eqs. (225)(227), if

the frequen y shifts in the denominator on the r.h.s. of Eq. (255) are ignored.

Substituting Eqs. (251) and (252) into Eqs. (246)(248), one an then show

that the for e on an atom that is initially prepared in an arbitrary state an

be represented in the form [465,491,492

F(t) =∑

m,n

σnm(t)Fmn(rA) (258)

where the atomi density matrix elements σnm(t) = 〈Amn(t)〉 solve the intra-atomi master equation together with the respe tive initial ondition, and we

have

Fmn(rA) = Fel,ormn (rA) + Fel,r

mn(rA) + Fmag,ormn (rA) + Fmag,r

mn (rA) (259)

with the various ele tri /magneti , o-resonant/resonant for e omponents

being given as follows:

Fel,ormn (rA) = −

~µ0

2π

∫ ∞

0dξ ξ2

×(∇Tr

[αmn(rA, iξ) +αmn(rA,−iξ)]·G (1)(rA, r, iξ)

)r=rA

, (260)

Fel,rmn(rA) = µ0

∑

k

Θ(ωnk)Ω2mnk(rA)

×∇dmk ·G (1)[r, rA,Ωmnk(rA)]·dkn

r=rA

+ C.c., (261)

Fmag,ormn (rA) =

~µ0

2π

∫ ∞

0dξ ξ2Tr

[ωmn(rA)

iξαT

mn(rA, iξ)

− ωmn(rA)

iξαT

mn(rA,−iξ)]×[∇×G (1)(r, rA, iξ)

]

r=rA

, (262)

66

Fmag,rmn (rA) = µ0

∑

k

Θ(ωnk)ωmn(rA)Ωmnk(rA)

×(dmk×

∇×G (1)[r, rA,Ωmnk(rA)]·dkn

)r=rA

+ C.c. (263)

Here, Ωmnk(rA) and αmn(rA, ω), respe tively, are the omplex atomi transi-

tion frequen ies and the generalized polarizability tensor:

Ωmnk(rA) = ωnk(rA) + i[Γm(rA) + Γk(rA)]/2, (264)

αmn(rA, ω) =1

~

∑

k

[dmkdkn

−Ωmnk(rA)− ω+

dkndmk

−Ω∗nmk(rA) + ω

]. (265)

Equations (258)(263) show that the for e omponents Fmn(rA) (m 6=n) asso- iated with (non-vanishing) o-diagonal elements of the atomi density matrix

ontain ontributions arising from the intera tion of the atom with both the

ele tri and the magneti eld, where the magneti for e omponents display

a ve tor stru ture whi h is entirely dierent from that of the ele tri ones.

Sin e under the assumptions made, diagonal and o-diagonal density matrix

elements are not oupled to ea h other so that

σnm(t) = eiωmn(rA)−[Γm(rA)+Γn(rA)]/2(t−t0)σnm(t0) (266)

(m 6= n), for e omponents asso iated with o-diagonal density-matrix ele-

ments an only be observed if the atom is initially prepared in an at least par-

tially oherent superposition of energy eigenstates. A ordingly, if the atom is

initially prepared in an in oherent superposition of energy eigenstates, then

only for e omponents Fmm(rA) whi h are asso iated with diagonal density-

matrix elements and whi h are ele tri al by their nature,

Fmm(rA) = Fel,ormm (rA) + Fel,r

mm(rA) ≡ Formm(rA) + Fr

mm(rA), (267)

are observed, with the density matrix elements obeying the balan e equations

σmm(t) = −Γm(rA)σmm(t) +∑

k

Γmk (rA)σkk(t). (268)

In parti ular, when the level shifts and broadenings are negle ted, then the

for e omponents Formm(rA) and Fr

mm(rA) as follow from Eqs. (260) and (261)

obviously redu e to those that are obtained from the perturbative poten-

tial (225)(227) by means of Eq. (183). Note that the gradient in Eqs. (260)

and (261) a ts only on the Green tensor and not on the additional position-

dependent quantities, so that this result annot be derived from a potential in

the usual way. Sin e the for e omponents asso iated with ex ited-state den-

sity matrix elements are transient, they are only observable on time s ales of

the order of magnitude of the respe tive de ay times Γ−1m (rA) whi h are known

to sensitively depend on the atomi position [450. Needless to say that the

67

-3

0

3

1.12 1.13 1.14

PSfrag repla ements

ω10/ωTe

Fr 11(z

A)3

2πε 0

λ4 T

e/(

3[d

2 01+

(d01·e

z)2

]) 3 × 109

0

−3 × 109

1.12 1.13 1.14

Fig. 6. The resonant omponent of the for e on a two-level atom in the upper state

pla ed in front of a diele tri half spa e, Eq. (271), is shown as a fun tion of the

unperturbed transition frequen y ω10 (solid line) (ωPe/ωTe = 0.75, γe/ωTe = 0.01,ω2Te[d

201 + (d01 ·ez)2]/3π~ε0c3 = 10−7

, zA/λTe = 0.0075, λTe = 2πc/ωTe). For om-

parison, both the perturbative result as obtained from Eq. (227) (dashed lines) and

the separate ee ts of level shifting (dotted lines) and level broadening (dash-dotted

lines), are shown.

for e F(t) that a ts on an initially ex ited atom approa hes the ground-state

for e F00(rA) after su iently long times, limt→∞〈F(t)〉= F00(rA).

In order to illustrate the ee t of the body-indu ed level shifting and broaden-

ing on the for e, let us onsider a two-level atom whi h is situated at distan e

zA very lose to a diele tri half spa e. By means of the respe tive Green ten-

sor (App. B), it turns out that in the non-retarded limit the shift and width

of the transition frequen y are determined by

δω(zA) = δω1(zA)−δω0(zA) = −d201 + (d01 ·ez)232π~ε0z3A

|ε[ω10 + δω(zA)]|2 − 1

|ε[ω10 + δω(zA)] + 1|2 (269)

and

Γ(zA) = Γ1(zA) =d201 + (d01 ·ez)28π~ε0z3A

Im ε[ω10 + δω(zA)]

|ε[ω10 + δω(zA)] + 1|2 , (270)

respe tively, where the transition-dipole matrix element has been assumed

to be real and the (small) o-resonant ontribution to the frequen y shift

has been omitted. Note that due to the appearan e of the frequen y shift

on the r.h.s. of Eq. (269), this equation determines the shift only impli itly.

68

The dominant ontribution to the for e on the atom in the upper state is the

resonant one, i.e., F11(rA)≃Fr11(rA). Substituting the half-spa e Green tensor

(App. B) into Eqs. (261) and (267), one an show that [Fr11(rA) =F r

11(zA)ez[465,491,492

F r11(zA) = −

3[d201 + (d01 ·ez)2]32π~ε0z

4A

|ε[Ω110(zA)]|2 − 1

|ε[Ω110(zA)] + 1|2 (271)

where, a ording to Eq. (264),

Ω110(zA) = ω10(zA) + iΓ(zA)/2 = ω10 + δω(zA) + iΓ(zA)/2. (272)

In parti ular, for a medium whose permittivity is of DrudeLorentz type,

Eq. (217) leads to (γe,Γ≪ωTe)

ε[Ω110(zA)] = 1 +ω2Pe

ω2Te − ω2

10(zA)− i[Γ(zA) + γe]ω10(zA), (273)

showing that the absorption parameter of the half-spa e medium, γe, is re-pla ed with the total absorption parameter, i.e., the sum of γe and the spon-

taneous-de ay onstant Γ(zA) of the atom. Figure 6 displays the resonant

omponent of the for e on a two-level atom in the upper state pla ed near a

(single-resonan e) diele tri half spa e as a fun tion of the unperturbed tran-

sition frequen y ω10. It is seen that in the vi inity of the (surfa e-plasmon)

frequen y ωS=√ω2Te + ω2

Pe/2, an enhan ed for e is observed whi h is attra -

tive (repulsive) for red (blue) detuned atomi transition frequen ies ω10<ωS

(ω10>ωS)a result already known from perturbation theory [413. However,

it is also seen that due to body-indu ed level shifting and broadening the ab-

solute value of the for e an be noti eably redu ed. Interestingly, the positions

of the extrema of the for e remain nearly un hanged, be ause level shifting

and broadening give rise to ompeting ee ts that almost an el.

The al ulation of the o-resonant omponent of the for e, Eq. (267) together

with Eq. (260), leads to [For11(rA)= F or

11(zA)ez

F or11(zA) =

3[d201 + (d01 ·ez)2]32π2~ε0z4A

∫ ∞

0dξ

ε(iξ)− 1

ε(iξ) + 1

× ω10(zA)

ω210(zA) + [ξ + Γ(zA)/2]2

ω210(zA) + ξ2 + Γ2(zA)/4

ω210(zA) + [ξ − Γ(zA)/2]2

. (274)

Equation (274) reveals that the o-resonant omponent of the for e is only

weakly inuen ed by the level broadening [the leading-order dependen e be-

ing O(Γ2) whi h is in agreement with the physi al requirement that the vir-

tual emission and absorption pro esses governing the o-resonant omponent

should be only weakly ae ted by de ay-indu ed broadening. Formally, the

69

0.0172

0.0173

0.0174

1.12 1.13 1.14

0

3 10

6 10

PSfrag repla ements

ω10/ωTe

For

11(z

A)3

2πε 0

λ4 T

e/(

3[d

2 01+

(d01·e

z)2

])

1.74 × 107

1.73 × 107

1.72 × 107

1.12 1.13 1.14

1.12 1.13 1.14

60

30

0

Fig. 7. The o-resonant omponent of the for e on a two-level atom in the upper

state pla ed in front of a diele tri half spa e, Eq. (274), is shown as a fun tion of

the unperturbed transition frequen y (solid line), the parameters being the same as

in Fig. 6. For omparison, the perturbative result is also shown (dashed lines). The

inset displays the dieren e between the for e with and without onsideration of

level broadening (solid lines). In addition, the same dieren e is displayed when the

level shifts are ignored (dashed lines).

absen e of a linear-order term O(Γ) is due to the fa t that the atomi polariz-

ability (265) enters the o-resonant for e omponents (260) only in the om-

bination [αmm(rA, iξ)+αmm(rA,−iξ)], whi h ould not have been anti ipated

from the perturbative result (226) [where in fa t, αm(rA, iξ) and αm(rA,−iξ) oin ide, re all Eq. (228). The ee ts of level shifting and broadening are il-

lustrated in Fig. 7. Compared to the perturbative result, the frequen y shift

has the ee t of raising or lowering the for e for ω10<ωS or ω10>ωS, respe -

tively, whereas the ee t of broadening is not visible in the urves. Only by

plotting the dieren e between the results with and without broadening, a

slight redu tion of the for e be omes visible in the vi inity of ωS where Γ is

largest. Sin e this behavior is generally typi al of o-resonant omponents, the

perturbative result may be regarded as a good approximation for the for es

on ground-state atoms where no resonant omponents are present.

70

4.2.3 Strong atomeld oupling

Strong atomeld oupling may o ur if an initially ex ited atom intera ts

resonantly with a sharply peaked (quasi-)mode of a body-assisted eld, as

it is observed in avity-like systems. In this ase, the atomeld dynami s

an no longer be des ribed within the Markov approximation. Typi ally, a

single atomi transition is in resonan e with su h a mode, so the (resonant

part of the) dispersion for e an be studied, to a good approximation, by

employing the two-level model in rotating-wave approximation with respe t to

the intera tion Hamiltonian (119) (see, e.g., Ref. [450). To be more spe i ,

let us onsider the intera tion of a two-level atom initially prepared in the

upper state |1〉 with the body-assisted ele tromagneti eld in the ground

state |0〉 and al ulate the ele tri part of the resonant omponent of the

for e a ting on the atom, i.e., in the S hrödinger pi ture,

F(t) ≃ 〈ψ(t)|∇

[d·E(r)

]r=rA

|ψ(t)〉, (275)

with the state ve tor |ψ(t)〉 [|ψ(t = t0)〉 = |0〉|1〉 being represented in the

form

|ψ(t)〉 = ψ1(t)|0〉|1〉

+∑

λ=e,m

∫d3r

∫ ∞

0dω

ψ0(ω, t)

~g(rA, ω)d01 ·G ∗

λ(rA, r, ω)·|1(rA, ω)〉|0〉. (276)

It is normalized to unity provided that

|ψ1(t)|2 +∫ ∞

0dω |ψ0(ω, t)|2 = 1 (277)

and

g2(rA, ω) =µ0

π~ω2d10 ·ImG(rA, rA, ω)·d01. (278)

Substituting Eq. (276) into the S hrödinger equation

i~∂

∂t|ψ(t)〉 = H|ψ(t)〉, (279)

with H being given a ording to Eq. (109) together with Eqs. (110), (111) and

(119), one obtains the following oupled dierential equations for ψ1(t) andψ0(ω, t):

ψ1(t) = −i

~E1ψ1(t) + i

∫ ∞

0dω g(rA, ω)ψ0(ω, t), (280)

ψ0(ω, t) = −i

~(E0 + ~ω)ψ0(ω, t) + ig(rA, ω)ψ1(t). (281)

Equation (281) together with the initial ondition ψ0(ω, t= t0)=0 an be for-

mally integrated in a straightforward way. Inserting the result into Eqs. (276)

71

and (280) then yields

|ψ(t)〉 = ψ1(t)|0〉|1〉+i

~

∑

λ=e,m

∫d3r

∫ ∞

0dω

∫ t

t0dτ e−i(E0/~+ω)(t−τ)ψ1(τ)

× d01 ·G∗λ(rA, r, ω)·|1(rA, ω)〉|0〉 (282)

and

ψ1(t) = −iE1

~ψ1(t)−

∫ ∞

0dω g2(rA, ω)

∫ t

t0dτ e−i(E0/~+ω)(t−τ)ψ1(τ), (283)

respe tively. Combining Eqs. (275) and (282) and making use of the integral

relation (45), one nds that

F(t) =iµ0

π

∫ ∞

0dω ω2

∇

[d10 ·ImG

(1)(r, rA, ω)·d01

]r=rA

×∫ t

t0dτ ψ∗

1(t)ψ1(τ)e−i(E0/~+ω)(t−τ) + C.c. (284)

Sin e so far nothing has been said about the strength of the atomeld ou-

pling, Eq. (284) gives the ele tri part of the resonant omponent of the dis-

persion for e on a two-level atom whi h is initially prepared in the upper state

for arbitrary oupling strengths. Let us now approximate that part of the ex-

itation spe trum of the body-assisted ele tromagneti eld whi h may give

rise to strong atomeld oupling in the resonant transition by a quasi-mode

(labeled by ν) of Lorentzian shape,

g2(rA, ω) = g2(rA, ων)γ2ν/4

(ω − ων)2 + γ2ν/4+ g′2(rA, ω) (285)

and assume that the ee t of the residual part of the eld whi h is des ribed

by the term g′2(rA, ω) is weakly oupled to the atom, so that it an be treated

in the Markov approximation. From Eq. (283) it then follows that [498

ψ1(t) = e[−iE1/~−iδω′1(rA)−Γ′

1(rA)/2](t−t0)φ1(t) (286)

where φ1(t) is the solution to the dierential equation

φ1(t) + i∆ω(rA) + [γν − Γ′1(rA)] /2 φ1(t) +

14Ω2

R(rA)φ1(t) = 0 (287)

together with the initial onditions φ1(t = t0) = 1, φ1(t = t0) = 0. Here,

∆ω(rA)=ων − ω′10(rA) and ΩR(rA) =

√2πγνg2(rA, ων), respe tively, are the

detuning and the va uum Rabi frequen y and

δω′1(rA) = δω1(rA) +

Ω2R(rA)

4

∆ω(rA)

[∆ω(rA)]2 + γ2ν/4(288)

72

and

Γ′1(rA) = Γ1(rA)−

Ω2R(rA)

4

γν[∆ω(rA)]2 + γ2ν/4

, (289)

respe tively, are the shift and width of the upper level asso iated with the

residual part of the eld where δω1(rA) and Γ1(rA), are dened a ording to

Eqs. (254)(257) with the shifted transition frequen y being given by

32

ω′10(rA) = ω10 + δω′

1(rA) (290)

in pla e of Eq. (253). Equation (287) an easily be solved to obtain

φ1(t) = c+(rA)eΩ+(rA)(t−t0) + c−(rA)e

Ω−(rA)(t−t0)(291)

where

c±(rA) =Ω∓(rA)

Ω∓(rA)− Ω±(rA)(292)

and

Ω±(rA) =− 12

i∆ω(rA)+[γν−Γ′

1(rA)]/2

∓ 12

√i∆ω(rA)+[γν−Γ′

1(rA)]/22 − Ω2

R(rA) . (293)

Combination of Eqs. (284), (286) and (291) then leads to the sought for e

[498:

F(t) =µ0

π

∫ ∞

0dω ω2s(rA, ω, t− t0)

∇

[d10 ·ImG

(1)(r, rA, ω)·d01

]

r=rA

+ C.c.

(294)

with

s(rA, ω, t)

= |c+(rA)|2e[−Γ′

1(rA)+Ω∗+(rA)+Ω+(rA)]t − ei[ω

′10(rA)−ω]−Γ′

1(rA)/2+Ω∗+(rA)t

ω − ω′10(rA) + iΓ′

1(rA)/2− iΩ+(rA)

+ c∗+(rA)c−(rA)e[−Γ′

1(rA)+Ω∗+(rA)+Ω−(rA)]t − ei[ω

′10(rA)−ω]−Γ′

1(rA)/2+Ω∗+(rA)t

ω − ω′10(rA) + iΓ′

1(rA)/2− iΩ−(rA)

+ c∗−(rA)c+(rA)e[−Γ′

1(rA)+Ω∗−(rA)+Ω+(rA)]t − ei[ω

′10(rA)−ω]−Γ′

1(rA)/2+Ω∗−(rA)t

ω − ω′10(rA) + iΓ′

1(rA)/2− iΩ+(rA)

+ |c−(rA)|2e[−Γ′

1(rA)+Ω∗−(rA)+Ω−(rA)]t − ei[ω

′10(rA)−ω]−Γ′

1(rA)/2+Ω∗−(rA)t

ω − ω′10(rA) + iΓ′

1(rA)/2− iΩ−(rA). (295)

Let us rst make onta t with result obtained in the limit of weak atomeld

oupling where the rst term under the square root in Eq. (293) is mu h larger

32Note that ontrary to Eq. (253), the ground-state shift is absent here as a onse-

quen e of the rotating-wave approximation.

73

than the se ond one. This is the ase when for given transition dipole moment,

the spe trum of the eld in the resonan e region is su iently at,

γν ≫ 2ΩR(rA) (296)

or when the atomi transition is su iently far detuned from the eld reso-

nan e,

|∆ω(rA)| ≫ 2Ω2R(rA)/γν . (297)

By means of Taylor expansion it an then be shown that

Ω±(rA) ≃

−i∆ω(rA)− [γν − Γ′1(rA)]/2,

iΩ2R(rA)

4

∆ω(rA)

[∆ω(rA)]2 + γ2ν/4− Ω2

R(rA)

8

γν[∆ω(rA)]2 + γ2ν/4

,

(298)

so c+(rA)≃ 1, c−(rA)≃ 0 and Eq. (284) [together with Eqs. (286) and (291)

approximates to [498

F(t) = e−Γ1(rA)(t−t0)µ0

π

∫ ∞

0dω ω2

[∇d10 ·ImG

(1)(r, rA, ω)·d01

]r=rA

ω − ω10(rA)− iΓ1(rA)/2+ C.c.

≃ e−Γ1(rA)(t−t0)F1(rA) (299)

with

F1(rA) = µ0Ω210(rA)

∇d10 ·G (1)[r, rA,Ω10(rA)]·d01

r=rA

+ C.c. (300)

and

Ω10(rA) = ω10(rA) + iΓ1(rA)/2 (301)

whi h orresponds to the term σ11(t)Fel,r11 (rA) in Eqs. (258) and (259) with

Fel,r11 (rA) being given a ording to Eq. (261).

The strong- oupling limit is realized if the spe trum of the eld features a

su iently sharp peak and the atomi transition is near resonant with this

peak, su h that

γν ≤ 2ΩR(rA) and |∆ω(rA)| ≪ 2Ω2R(rA)/γν. (302)

In this ase, the square root in Eq. (293) be omes approximately real,

Ω±(rA) ≃ −12

i∆ω(rA) +

12[γν − Γ′

1(rA)]∓ 1

2iΩ(rA) (303)

where

Ω(rA) =√Ω2

R(rA) + [∆ω(rA)]2 − [γν − Γ′1(rA)]

2/4 (304)

74

so that the oe ients c±(rA) [Eq. (292) an be given in the form

c±(rA) =Ω(rA)∓∆ω(rA)± i[γν−Γ′

1(rA)]/2

2Ω(rA). (305)

Substituting Eqs. (303) and (305) into Eq. (294) [together with Eq. (295),

one nds that for real dipole matrix elements F(t) approximates to [498

F(t) = 2e−[γν+Γ′1(rA)](t−t0)/2 sin2[Ω(rA)(t− t0)/2]

× [∆ω(rA)]2 − [γν−Γ′

1(rA)]2/4

Ω2R(rA) + [∆ω(rA)]2 − [γν−Γ′

1(rA)]2/4

F1(rA) (306)

where F1(rA) is given a ording to Eq. (300) with

Ω′10(rA) = ω′

10(rA) + iΓ′1(rA)/2 (307)

in pla e of Eq. (301). Note that [γν +Γ′1(rA)]/2=γν/2 if |∆ω(rA)|≪ γν/2 and

[γν + Γ′1(rA)]/2=Γ1(rA)/2 if γν/2≪|∆ω(rA)|≪ 2Ω2

R(rA)/γν.

Comparing Eq. (306) with Eq. (299), we see that while the resonant ompo-

nent of the for e in the limit of weak atomeld oupling simply exponentially

de reases as a fun tion of time, Rabi os illations of the for e are typi ally

observed in the strong- oupling limitin agreement with the well-known fea-

tures of spontaneous emission in the two oupling regimes. As a onsequen e

of the appearan e of Rabi os illations, the magnitude of the for e hanges

periodi ally; for appropriate spatial stru ture of the resonantly intera ting

quasi-mode, the atom may be trapped with the trap being set by the atom

itself, f. also Refs. [499,500. Rabi os illations do not o ur if the system is

initially prepared in a dressed state; in this ase the (exponentially de aying)

for e is simply given by the gradient of the position-dependent part±~Ω(rA)/2[re all Eq. (304) of the respe tive dressed-state energy [501,502.

5 Con luding remarks

Dispersion for es are a parti ular signature of the quantum nature of the inter-

a tion of matter with the ele tromagneti eld. As soon as the intera ting mat-

ter onsists of a large number of elementary atomi parti les, exa t mi ros opi

al ulations be ome very involved. Therefore most theoreti al approa hes to

dispersion for es make use of assumptions typi al of ma ros opi ele trody-

nami s by introdu ingsooner oder laterfamiliarma ros opi on epts su h

as boundary onditions at surfa es of dis ontinuity and/or onstitutive rela-

tions averaged over a su iently large number of the elementary onstituents

of the respe tive material obje ts. Ma ros opi ele trodynami s whose appli-

ability surprisingly ranges even to nano-stru tures, has the benet of being

75

universally valid, be ause it uses only very general physi al properties, without

the need of involved ab initio al ulations. Moreover, all the relevant quan-

tities used for hara terizing the material obje ts an easily be inferred from

measurements. This on ept does not only apply to lassi al ele trodynami s

but also to QED. Ma ros opi QED has been well elaborated for the ase

of lo ally responding media des ribed in terms of omplex-valued, position-

and frequen y-dependent permittivities and permeabilities. It an be extended

to arbitrary linear media, in luding spatially dispersing media, sin e the de-

s ription of the quantized eld in terms of urrent densities and the Green

tensor asso iated with the ma ros opi Maxwell equations is independent of

the parti ular medium des ription. When supplemented with standard atom

eld oupling terms, a powerful tool for studying medium-assisted quantum

ee ts in QED is obtained. Clearly, the appli ability of the theory is restri ted

to bodybody and bodyatom separations that are su iently large ompared

with the length s ale on whi h the atomisti stru ture of the bodies begins to

play a role.

In parti ular, the so established ma ros opi QED provides a unied approa h

to the various types of dispersion for esan approa h whi h in orporates the

benets of normal-mode and linear-response approa hes, while exa tly taking

into a ount real material properties. In parti ular, dispersion for es between

ele tri ally neutral, unpolarized and unmagnetized ground-state bodies sim-

ply ree t the for es whi h are due to the a tion of the u tuating body-

assisted ele tromagneti va uum on the u tuating harge and urrent den-

sities of the bodies. Sin e all the relevant hara teristi s of the bodies enter

the so obtained for e formulas via the Green tensor of the asso iated ma ro-

s opi Maxwell equations, they are valid for arbitrary (linear) bodies. Both

the Casimir stress and the Casimir for e density an thus be introdu ed in

a natural way. Moreover, by appropriate Born-series expansions of the Green

tensor, relations between dispersion for es on bodies and dispersion for es on

atoms an be established whi h learly demonstrate the ommon origin of all

these for es. In parti ular, the for e on an atom in the presen e of arbitrary

bodies as well as the for e between two atoms an be obtained as limiting

ases of the bodybody for e.

In this arti le we have restri ted our attention to ground-state bodies, i.e,

to bodies that at zero temperature intera t with the ele tromagneti eld

where the ee t of dispersion for es is purely quantum by nature. As outlined,

an extension of the entral results to in lude equilibrium systems at nite

temperatures an be obtained in a straightforward way by simply repla ing

the va uum averages by thermal averages. In this ontext it should be pointed

out that the entral assumption of linear-response theories a ording to whi h

the thermal average of the eld u tuations is related to the imaginary part

of the eld response fun tion, is expli itly fullled within the framework of

ma ros opi QED.

76

As we have seen, dispersion for es on ground-state atoms turn out to be lim-

iting ases of dispersion for es on ma ros opi bodies, so the orresponding

formulas an be obtained without expli itly addressing the underlying atom

eld intera tion. Of ourse, they an also be derived by expli itly solving the

quantum-me hani al problem of individual atoms intera ting with the ele tro-

magneti eld, with the presen e of ma ros opi bodies being again des ribed

within the framework of ma ros opi QED. For ground-state atoms where

only virtual transitions o ur, this leads to results that agree with the ones

obtained from the purely ma ros opi approa h, as expe ted. In fa t, expli -

itly addressing the atomeld intera tion is more exible, be ause it an also

be applied to non-equilibrium systems, su h as initially ex ited atoms where

also real transitions are involved in the atomeld intera tion. In this ase,

a dynami al des ription is in general preferred to be employed, leading to

time-dependent expressions for the for es, a ording to the temporal evolu-

tion of the atomi quantum state. In parti ular for weak atomeld oupling,

the for e on an initially ex ited atom is a sum of omponents whose tempo-

ral evolution follows that of the asso iated atomi density matrix elements

whi h is in turn governed by the familiar master equation of an atomi system

undergoing radiative damping. For strong atomeld oupling, damped Rabi

os illations may o ur whi h periodi ally hange the magnitude of the for e.

The dynami al approa h ould serve as a starting point for studying disper-

sion for es on bodies that are not in thermal equilibrium, by appropriately

modeling su h bodies as olle tions of ex ited atoms [281,282.

In luding linearly responding bodies in ma ros opi QED has the advantage

that from the very beginning of all al ulations the ee t of the bodies is taken

into a ount in a onsistent manner, without the need to spe ify the properties

of the bodies at an early stage. In this way very general results of broad appli-

ability an be obtained. This naturally applies not only to dispersion for es,

but also to other quantum phenomena of radiationmatter intera tion whi h

are strongly inuen ed by the presen e of ma ros opi bodiesphenomena

that may be subsumed under the term Casimir ee t in the broadest sense

of the word. Typi al examples are the enhan ement and inhibition of sponta-

neous emission, resonant energy transfer between atoms or mole ules and the

wide eld of avity-QED ee ts.

A knowledgements

We would like to a knowledge fruitful ollaboration with Ho Trung Dung, T.

Kampf, C. Raabe and H. Safari. Furthermore, we are grateful to L. Arntzen,

G. Barton, I. Bondarev, M. DeKieviet, A. Guzmán, A. Lambre ht, S. Linden,

E. Shamonina, Y. Sherkunov, L. Rizzuto, M. S. Toma² and C. Víllarreal for

stimulating dis ussions.

77

A Overview over s enarios

In order to provide for an overview over the various theoreti al works on

dispersion for es, referen es ontaining work on bodybody, atombody and

atomatom for es are given in separate tables. In the tables, the referen es are

stru tured a ording to the s enarios onsidered, regardless of the methods

used to address these s enarios.

Material → Magneto-

Geometry ↓Perfe t ond. Ele tri

ele tri

[242,314316 [175,204,205,208,239241,281,282 [333,334

[322,385,386 [323,325,230232,236,321,326,329 [336,372

Half spa e [397,398 [362365,370,371,374,378,379,382 [393,485

+ half spa e [391

T[387,388,399,433,436438, [324

T[335

T

[318320

L[328,330332,359361,366369

T[337

T

[389

L, [348,390

Tq[339

T

Half spa e [445,446

+ plate [487,480

Half spa e [377,397,398 [231,436, [330

T

+ sphere

Half spa e [343,377

+ ylinder

Plate [327,373,381,396,435,483,484

+ plate [347

Tq

Plate [345

+ sphere

Sphere [393,397 [235,237,344

+ sphere

Table A.1

S hedular summary of referen es asso iated with theoreti al work on bodybody

for es. The supers ripts indi ate that nite temperature (T), lateral for es (L)

and/or torques (Tq) are in luded.

78

Material → Magneto-

Geometry ↓Perfe t ondu tor Ele tri

ele tri

[107,177,189191 [174,175,189,190,194,196199 [416,443

[245,192,247251 [201203,206209,211213,216 [491,492

[290,294,404,405 [217,219,220,252,305,306,405 [497, [309

M

[407, [287,295

E[407,414,415,417,419,433,441

Half spa e

[296,304,412

E[490,[195,218,281,282,304

E

[413

E, [269

M[307,308,413,418,428,439

E

[284,408

T[465,491,492

E, [406

M, [360

T

[361,409,410,442

T

Plate [444

T[491,497

[221,420 [221223,395,414,421,422,440 [488

Sphere

[496

[224226,414,423,424,440

Cylinder

[493495, [425

E, [444

T

[227,290,310,426 [310,426, [311

T[491,497

Planar

[291,297301

E

Cavity

[292

E,M, [293

T

Spher. av. [228,229,427,440, [312

E

Cyl. av. [424,440

Parab. av. [302,303

Table A.2

S hedular summary of referen es asso iated with theoreti al work on atombody

for es. Unless otherwise stated, nonmagneti ground-state atoms are onsidered.

The supers ripts indi ate that ex ited atoms (E), magneti atoms (M) and/or nite

temperature (T) are in luded.

79

Material → Magneto-

Geometry ↓Perfe t ondu tor Ele tri

ele tri

[107,164,168170,174,175,181183,243257,265,404,433,489,490

Free spa e [276282

E,[179,184

E,M, [178,180,258,266,268272,274,489

M

[273

M,T, [283286,360,361,408,447

T

[429 [467,488,503

Bulk medium

[446

M

Half Spa e [214,287289, [430

T[215,216,267,415,431 [467,489

Planar avity [411, [430

T[267, [432

T

Table A.3

S hedular summary of referen es asso iated with theoreti al work on atomatom

for es, possibly in the presen e of bodies. Unless otherwise stated, nonmagneti

ground-state atoms are onsidered. The supers ripts indi ate that ex ited atoms

(E), magneti atoms (M) and/or nite temperature (T) are in luded.

B Green tensors

The Green tensor in free spa e is given by [448

G free(r, r′, iξ) =

1

3

(c

ξ

)2

δ(ρ)I +c2e−ξρ/c

4πξ2ρ3

[a(ξρ/c)I − b(ξρ/c)eρeρ

](B.1)

(ρ= r− r′; ρ= |ρ|; eρ=ρ/ρ) where

a(x) = 1 + x+ x2, b(x) = 3 + 3x+ x2. (B.2)

The s attering Green tensor for the planar magneto-ele tri stru ture hara -

terized by Eqs. (143) and (144) is given by [482,504

G

(1)(r, r′, iξ) =∫

d2q eiq·(r−r′)G

(1)(q, z, z′, iξ) for 0 < z, z′ < d (B.3)

(q ⊥ ez) with

G

(1)(q, z, z′, iξ) =µ(iξ)

8π2b

∑

σ=s,p

rσ−rσ+e

−2bd

Dσ

[e+σ e

+σ e

−b(z−z′) + e−σ e−σ e

b(z−z′)]

+1

Dσ

[e+σ e

−σ rσ−e

−b(z+z′) + e−σ e+σ rσ+e

−2bdeb(z+z′)]. (B.4)

80

Here, b and Dσ are dened by Eqs. (148) and (149), respe tively,

e±s = eq×ez, e±p = −1

k(iqez ± beq) (B.5)

(eq =q/q, q= |q|) withk =

ξ

c

√ε(iξ)µ(iξ) (B.6)

are the polarization ve tors for s- and p-polarized waves propagating in the

positive (+) and negative (−) z-dire tions, and r±σ = r±σ (ξ, q) with σ = s, pdes ribe the ree tion of these waves at the right (+) and left (−) walls,

respe tively.

In parti ular, assume that both walls are multi-slab magneto-ele tri s on-

sisting of N± homogeneous layers of thi knesses dj± (j= 1, . . . , N±) with dN±

±

=∞, permittivity εj±(ω) and permeability µj±(ω). In this ase the ree tion

oe ients an be obtained from the re urren e relations [rσ±≡ r0σ±; d≡ d0±;ε(ω)≡ ε0±(ω); µ(ω)≡ µ0

±(ω)

rjs± =(µj+1

± bj± − µj±b

j+1± ) + (µj+1

± bj± + µj±b

j+1± ) e−2bj+1

±dj+1± rj+1

s±

(µj+1± bj± + µj

±bj+1± ) + (µj+1

± bj± − µj±b

j+1± ) e−2bj+1

±dj+1± rj+1

s±

, (B.7)

rjp± =(εj+1

± bj± − εj±bj+1± ) + (εj+1

± bj± + εj±bj+1± ) e−2bj+1

±dj+1± rj+1

p±

(εj+1± bj± + εj±b

j+1± ) + (εj+1

± bj± − εj±bj+1± ) e−2bj+1

±dj+1± rj+1

p±

(B.8)

(j=0, . . . , N±− 1) with rN±

σ± =0 where

bj± =

√ξ2

c2εj±(iξ)µ

j±(iξ) + q2 . (B.9)

For single, semi-innite slabs, Eqs. (B.7) and (B.8) redu e to the Fresnel o-

e ients

rs± =µ1±b− µb1±µ1±b+ µb1±

, rp± =ε1±b− εb1±ε1±b+ εb1±

. (B.10)

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