Enhanced van der Waals Enhanced van der Waals interaction at interfacesinteraction at interfaces

Marin-Slobodan TomašRuđer Bošković Institute, Zagreb,

Croatia

Introduction Consider an excited (e) atom A and a ground-state (g) atom B in

an inhomogeneous system, e.g., near a body.

A

B

Assuming two-level isotropic atoms the vacuum forceon the atom A is obtained from the potential

.

3

||2)(

),();,();,(Tr

)](Re[3

||)(),(

),();,(ReTr3

||)()(

),,()(),(

22

22

4

42nres

sc2

22nres

BB

B

B

BegB

g

AAABABA

ABg

AAeg

AABAAB

AAAAA

Aeg

AAAA

BAABAABAA

i

cUU

cUU

UUU

d

rrGrrG

drrr

rrGd

rr

rrrrr

Wylie and Sipe (1985) [G. S. Agarwal (1974)]

Y. Sherkunov (2007)

Assuming two-level isotropic atoms the vacuum forceon the atom A is obtained from the potential

.3

||2)(

),();,();,(Tr)](Re[3

||

);,();,(Tr)()(2

),(

),();,(ReTr3

||

);,(Tr)(2

)(

),,()(),(

22

22

4

42

0

44

sc2

22

sc

0

22

BB

B

B

BegB

g

AAABABAABg

AAeg

ABBABg

AeBAAB

AAAAA

Aeg

AAAeAA

BAABAABAA

i

c

iiiidc

U

c

iidc

U

UUU

d

rrGrrGd

rrGrrGrr

rrGd

rrGr

rrrrr

Wylie and Sipe (1985)[G. S. Agarwal (1974)]

Y. Sherkunov (2007)

Now, near a surface (resonant) mode frequency

.~)( 122sc iSG

);,();,();,( ,sc,0, rrGrrGrrG

Accordingly

.

)(

)](Re[~)()]([Re~)(

,)(

~)(Re~)(

2222

2scres

2222

22scres

AAS

ABg

AABgAAB

AAS

ASAAA

U

U

G

G

R. R. Chance et al. (1975) H. Failache et al. (1999)

SystemdzzId )ˆˆ)((' r d

Green function

.ˆ,ˆ;)()(

)()()(

,)ˆˆ()3(

)(3

)();,(

||,

||

5

2

5

2

2

2

nr

zRrrRzRrrR

zzIIRRIRRrrG

ZZr

R

Rr

R

Rc

BABAm

m

BA

For zA+zB the Green function reads

Note that .)(2

)()];,([Tr

32

2scnr

AAA z

rc

rrG

Resonant two-atom potential

.|)(|)(3

3

)](Re[1

,)()(

)(

|)(|

)0(||2),(

6

62

4

22224||

2222

222

62

2res

0

R

Rr

R

RRZZR

R

Rrg

gR

U

U

AA

BAAB

ABB

A

Bg

Aeg

BAAB

AB

drr

enhancement factor{ normalization unit

Model calculation

,1)()(,1)(22

2

iT

Pm

.1)(

)(,

1)(

1)(

1)0(

1)0(

,1)(

1)()(

22

2

22

22

TP

S

S

S

ir

Assuming atoms in the vacuum near a Lorentz type medium

we have [Barton (1997)]

It is immediately seen that at resonance A= S

.4

1)(3

22

24

R

zzg BAS

S

Since for insulators /S ~ 10-2 and for (noble) metals /S ~ 10-3, this implies that under circumstances of the resonant coupling of the atom A to the surface polariton mode at nearby surface the van der Waals (two-atom) potential can be enhanced with respect to its free-space value by several orders of magnitudes.

Estimate

Results for the sapphire surface

SBSBA Rz 310,9.0,1.0

||

gS)=299.2gS)=113.7

[ε(∞)=2.71, ε(0)=6.57, S=1.54x1014 s-1, S]

B=0.9 S

B=1.1 S

||

Double resonance

B=S

Conclusion- The (generalized) Sherkunov formula implies a strong enhancement of the van der Waals (atom*-atom) interaction near an interface under the circumstances of the resonant coupling of the excited atom with the surface polariton mode of the system.

It is very likely that the same holds for the retarded atom*-atom interaction where, owing to the dispersionof the surface polariton mode, the atom*-surface resonance can be obtained much easier. This promotes the van der Waals interaction as another example of the surface enhanced phenomena.

and a guess