Enhanced van der Waals interaction at interfaces Marin-Slobodan Tomaš Ruđer Bošković Institute,...
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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