atomic spontaneous decay rate enhancement near a …cophen04/talks/bondarev.pdf · 2004-07-12 ·...

NEARNEAR--FIELD ELECTRODYNAMICS FIELD ELECTRODYNAMICS OF ATOMICALLY DOPED OF ATOMICALLY DOPED

CARBON NANOTUBESCARBON NANOTUBES

Igor Bondarev

The Institute for Nuclear ProblemsThe Belarusian State University

Minsk, BELARUS

E-mail: [email protected]

Collaborators:

Prof. Philippe Lambin

Laboratoire de Physique du SolideFacultes Universitaires Notre-Dame de la Paix

Namur, BELGIUM

Acknowledgements:

Belgian Office for Scientific, Technical and Cultural Affairs

OUTLINEOUTLINE

Basic physical properties of single-walled carbon nanotubes (CNs)

Motivation

Near-field electrodynamics. The model

Atomic spontaneous decay dynamics near a carbon nanotube

van der Waals energy of an atom near a carbon nanotube

Conclusions

BASIC PHYSICAL PROPERTIES BASIC PHYSICAL PROPERTIES OF SINGLEOF SINGLE--WALLED CNsWALLED CNs

(1) (1) classification of singleclassification of single--walled CNswalled CNs

Graphene single sheet

a1

a2

ma1 + na2

x

y

300

Single-walled carbon nanotube of (m,n) type

BASIC PHYSICAL PROPERTIES BASIC PHYSICAL PROPERTIES OF SINGLEOF SINGLE--WALLED CNsWALLED CNs

(2) (2) Brillouin zone structure of achiral CNsBrillouin zone structure of achiral CNs

0,0 0,5 1,0 1,5 2,0 2,5

-10-8-6-4-202468

10

Con

duct

ivity

nor

mal

ized

x

Zigzag (9,0) Re σzz Im σzz

Typical frequency dependence of the conductivity

00

, 2.7 eV2

x ω γγ

= =

pφ

pzpz

pφ

Zigzag: (m,0)m = 3q – metallic (q =1,2,3,…)

Armchair: (m,m)metallic for all m

,

1,2, ,cn

spR

s m

φ =

= …

MOTIVATION MOTIVATION Experimental observationsExperimental observations

TEM image of a sample partially filled by Cs atoms. The inset sets off the encapsulated atoms by dotting with black circles [after J.-H.Jeong et al. PRB68,075410(2003)]

Recent experiments on single atom encapsulation into single-wall carbon nanotubes and atom intercalation into single-wall carbon nanotube bundles: J.-H.Jeong et al. PRB68,075410(2003)

H.Shimoda et al. PRL88,015502 (2002)L.Duclaux Carbon40,1751 (2002).

Nanotube/atomic physical properties are substantially modified ina controllable way when doping nanotubes with impurity atoms.

New challenging nanophotonics applications of atomically doped carbon nanotube systems

as various sources of coherent light emitted by dopant atoms

NEARNEAR--FIELD ELECTRODYNAMICS. FIELD ELECTRODYNAMICS. THE MODELTHE MODEL

(1) (1) the geometry of the problemthe geometry of the problem

I.V.Bondarev & Ph.Lambin, Phys. Lett. A, to be published I.V.Bondarev & Ph.Lambin, PRB69,235xxx(2004)I.V.Bondarev & Ph.Lambin, cond-mat/0401332I.V.Bondarev & Ph.Lambin, cond-mat/0404211 [submitted to SSC]I.V.Bondarev et al., Carbon42,997(2004)I.V.Bondarev et al., PRB68,073310(2003)I.V.Bondarev et al., PRL89,115504(2002)

( , , )cnR Zφ=R

THE MODELTHE MODEL(2)(2) HamiltonianHamiltonian

Field quantization in the presence of dispersing and absorbing bField quantization in the presence of dispersing and absorbing bodies:odies:L.KnL.Knööll, S.Scheel, Dll, S.Scheel, D--G.Welsch, in: G.Welsch, in: Coherence and Statistics of Coherence and Statistics of

Photons and Atoms, Photons and Atoms, New York,Wiley, 2001New York,Wiley, 2001

1 ||

0 0ˆ ˆ ˆˆ( ) ( ) ( , ) . ., ( ) ( , ) . .d i h c d h cω ω ω ϕ ω ω

∞ ∞− ⊥= + − = +∫ ∫A r E r r E r∇

(||) (||)ˆ ˆ( , ) ( ) ( , ), { , , }d r zω ω ϕ⊥ ⊥′ ′ ′= − ⋅ =∫E r r r r E r rδ

(1) (2)ˆ ˆ ˆ ˆ ˆF A AF AFH H H H H= + + +Hamiltonian:

0ˆ ˆˆ ( , ) ( , )FH d d f fω ω ω ω

∞ += ∫ ∫ R R RField:2

,

ˆ 1ˆ2 2 | |

i jA

i i ji i j

q qH

m= +

−∑ ∑pr r

Atom:

(1) ˆˆ ˆˆ ˆ( ) ( )iAF i A A A

i i

qHm c

ϕ= − ⋅ + ⋅∑ p A r d r∇Interaction I:(electric dipole)

2(2) 2

2ˆˆ ( )

2i

AF Ai i

qHm c

=∑ A rInteraction II:(diamagnetic)

ˆ ˆ ˆ ˆ( , ) ( , ), ( , ) 4, ) ˆ ( ,( )c

i ic cω ωω ω πω ω ω× = × = − +E r H r H r Ir rE∇ ∇

ˆ ˆ( , ) ( ) ( , ) ˆ ( , , , )2 ( )cncnR zd r Rω δ ω δϕ ω= − = −∫I r R r R J R J

ˆˆ ( , ) Re ( , ) / ( , { ,, }) ,cz nzz cnR f R Zω ω σ ω π ω φ==J R R e R

Maxwell equations with an external source (CN)

12

ˆ( , , ) (4ˆ ˆ ˆ( , ) , ( /, ))i d i cc

ωπωω ωω −= ⋅ = ×∫ G rE Rr R JR H E∇

is the classical field Green tensor( , , )ωG r R

THE MODELTHE MODEL(3) (3) twotwo--level approximationlevel approximation

2

2 0

2( ) 1 ( , , )( )A A A A

A

d d gω ω ω ω ωω

∞ ⊥⎛ ⎞= −⎜ ⎟

⎝ ⎠∫ ∫r R r R

Renormalized atomic frequency (due to diamagnetic interaction)

( ) ||( , , ) ( , , ) ( , , )A A AA

g g gωω ω ωω

± ⊥= ±r R r R r R

(||( )2

)|| 4( , , ) R ( , , )e ( )AA zz Ag i d

cGα αβ

ωω π ω σ ω ω⊥⊥ = −R r Rr

Electric dipole coupling (electric dipole interaction)

0 00 2

8( ) Im ( ), Im ( , , )6A Ad d G G

c cα β αβ αβ αβπ ωω ω ω δ

πΓ = =r r

(2

2(||) 2 ||)0 ( )( , , )2

( , )A AAd g ω ω ξ ωω

π ω⊥⊥ Γ ⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ rR r R

(||) (||)( (||)||)

0

Im ( , , )( , ) 1 ,

m ( )(

I)A

AA

A

GGαβ

αβ

ξω

ξ ωω

ω⊥⊥ ⊥

⊥ = = + rr r

r

Local photonic DOS

( ) ( )

0ˆˆ ˆ( , , ) ( , , ) ( , ) . .A Ad d g g f h cω ω σ ω σ ω

∞ + + −⎡ ⎤− +⎣ ⎦∫ ∫ R r R r R R

0ˆ ˆˆ ( , ) ( , )H d d f fω ω ω ω

∞ += +∫ ∫ R R R

ˆ2 z

Aω σ +

Local photonic DOS for an atom Local photonic DOS for an atom near a carbon nanotubenear a carbon nanotube

I.V.Bondarev & Ph.Lambin, I.V.Bondarev & Ph.Lambin, PRBPRB6969,235xxx(2004) [cond,235xxx(2004) [cond--mat/0401332]mat/0401332]

4 2 2

3 4

:

( ) ( ) ( )3( , ) 1 Im2 1 ( ) ( ) ( )

A cn

p cn p AcnA C

p cn p cn p cn

r R

v I vR K vrRr dhk R v I vR K vR

β ωξ ω

π β ω

∞

=−∞

>

= ++∑ ∫

Rcn

2 2 ; ;

( , )( ) 4 zz cn

v h k kc

Ri

ω

σ ωβ ω πω

= − =

= kk−

h

:A cn

p p

r R

I K

<

↔ in the numerator of the integrand

Rcn

Example of local photonic DOS for an atom Example of local photonic DOS for an atom near a carbon nanotubenear a carbon nanotube


rA= 2.5Rcn ;

rA= 1.5Rcn ;

0 1 2 3100

101

102

103

104

105

106

107

108

(9,0)

rA= 3.5Rcn

ξ(x)

x

Local photonic DOS for an atom outside the (9,0) CN at different distances from the wall

00

, 2.7 eV2

x ω γγ

= =

Rcn

(I)(I)SPONTANEOUS DECAY DYNAMICSSPONTANEOUS DECAY DYNAMICS


u

Aω2

Aω

2Aω

− l

0

{ }0u

{ }1( , )l ωRω

{ }0l

0A cnr R− →

( / 2)

( / 2)

0

( ) {0}

( , , ) {1( , )

( )

}

A

A

i t

i

u

tl

t e u

d d C t e

C t

l

ω

ω ω

ψ

ω ω ω

−

∞− −

=

+ ∫ ∫ R R R

Wave function of the system

0

( )( ) 2( )2

0 0

( ) 1 ( ) ( )

1 1( ) ( , , )( )

A

t

u u

i t t

AA

C t K t t C t dt

eK t t d d gi

ω ω

ω ωω ω

∞ ∞′− − −+

′ ′ ′= + −

−′− =−

∫

∫ ∫ R r R

Evolution equation of the upper state

,||~ ( , )Aξ ω⊥ r

00

( ,( )( ) 1 )A A A Ad ω ξ ωω ω ωπ ω

⊥∞ Γ⎛ ⎞= −⎜ ⎟⎝ ⎠∫r r

SPONTANEOUS DECAY DYNAMICSSPONTANEOUS DECAY DYNAMICSSolution of evolution equationSolution of evolution equation

I.V.Bondarev and Ph.Lambin, I.V.Bondarev and Ph.Lambin, PRBPRB6969,235xxx(2004) [cond,235xxx(2004) [cond--mat/0401332]mat/0401332]

0

( )2 2

0

0(

( )( ) ( ) ( )4

( ) ( ))

AxA

A u

A AA

xK i x C e

x xx

τγτ τ τ

ξγ

Γ−Γ′− ≈ − + ∆ → =

Γ = Γ

0

( )( )30

3 0

( ) 1 ( ) ( )

( ) ( ) 1( )2 ( )

A

u u

i x xA

A A

x

C K C d

x eK dx xx i x x

τ

τ τ

τ τ

ξ

τ τ τ

τ τπ

′− − −∞

′ ′= + −

Γ −′− = −−

∫

∫

0 00 0

0 0

( ) 2( ) , , , 2.7 eV2 2

xx x tω γτ γγ γ

ΓΓ = = = =

Markovian approximation( )( ) 1 1( )( )

Ai x x

AA A

e x x iPi x x x x

τ τ

πδ′− − − −

→ − − +− −

NUMERICALLY !!!

SPONTANEOUS DECAY DYNAMICSSPONTANEOUS DECAY DYNAMICSNumerical resultsNumerical resultsI.V.Bondarev & Ph.Lambin, I.V.Bondarev & Ph.Lambin,

PRBPRB6969,235xxx(2004) [cond,235xxx(2004) [cond--mat/0401332]mat/0401332]

rA= 2.5Rcn ;

rA= 1.5Rcn ;

0 1 2 3100

101

102

103

104

105

106

107

108

(9,0)

rA= 3.5Rcn

ξ(x)

x

Local photonic DOS for an atom outside the (9,0) CN at different distances from the wall

00

, 2.7 eV2

x ω γγ

= =

0.33

Rcn



1 2 rA= 2.5Rcn ;

2

rA= 1.5Rcn ;

1

0 10 20 30 40 50 60 70 800,0

0,2

0,4

0,6

0,8

1,02

1

(9,0), x = 0.33

rA= 3.5Rcn

1 - exact, 2 - exp. decay |Cu(τ)

|2

τ

00

2 , 2.7 eVtγτ γ= =

Spontaneous decay dynamics for an atom outside the (9,0) CN at different distances from the wall Rcn



Atom outside the (9,0) CN at different distances from the wall

0.52

( )xξ⊥

00

, 2.7 eV2

x ω γγ

= =

(a) local photonic DOS

0 1 2 3100

101

102

103

104

105

106

107

108 (9,0)

rA= 1.5Rcn ; rA= 2.5Rcn ; rA= 3.5Rcn

x

0 10 20 30 40 50 60 70 800,0

0,2

0,4

0,6

0,8

1,0

(a) (9,0), x = 0.52 rA= 1.5Rcn ; rA= 2.5Rcn ;

rA= 3.5Rcn

1 - exact, 2 - exp. decay 1

11

2

|Cu(τ)

|2

τ

02 tγτ =

(b) upper-level spontaneous decay dynamics compared with exponential decay (Markovian approximation)

Rcn



Atom outside the (9,0) CN at different distances from the wall

0.58

( )xξ⊥

00

, 2.7 eV2

x ω γγ

= =


0 1 2 3100

101

102

103

104

105

106

107

108 (9,0)

rA= 1.5Rcn ; rA= 2.5Rcn ; rA= 3.5Rcn

x

02 tγτ =


0 10 20 30 40 50 60 70 800,0

0,2

0,4

0,6

0,8

1,0

(b) (9,0), x = 0.58 rA= 1.5Rcn ; rA= 2.5Rcn ;

rA= 3.5Rcn

1 - exact, 2 - exp. decay

1

2

1

2 1

2

τ

|Cu(τ)

|2

Rcn



Atom in the center of different CNs:

0.45

02 tγτ =

0 10 20 30 40 50 60 70 800,0

0,2

0,4

0,6

0,8

1,0

τ

|Cu(τ)

|2

2 1

2 1

rA= 0 , x = 0.45 (5,5); (10,10); (23,0)

1 - exact, 2 - exp. decay

12


( )xξ⊥

00

, 2.7 eV2

x ω γγ

= =


0 1 2 3100

101

102

103

104

105

106

107

x

(5,5); (10,10); (23,0) Rcn



Atom inside the (10,10) CNs at 3A from the wall [PRB68,075410]:

0.21

02 tγτ =


( )xξ⊥

00

, 2.7 eV2

x ω γγ

= =


0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1103

104

105

106

107

(10,10); rA= 0.56Rcn

x

0 10 20 30 40 50 60 70 800,0

0,2

0,4

0,6

0,8

1,0

12

(10,10); rA= 0.56Rcn; x = 0.21;

1 - exact; 2 - exp.decay

τ

|Cu(τ)

|2

x = 0.23;

2

1

x = 0.30; x = 0.25;

12

x = 0.45;

0.230.250.30 0.45

Rcn

How to observe experimentally?How to observe experimentally?

2Aω

2Aω

−

0

{ }0u

{ }1( , )l ωR

{ }0l×

Weak coupling

Optical absorbance profile in the neighborhoodof the atomic transition energy

ω~ Aω

Rabiω

ω~ Aω

2Aω

2Aω

−

0

{ } { }( )1 0 1( , )2

u l ω− R

{ } { }( )1 0 1( , )2

u l ω+ R

{ }0l

Strong coupling

Rabiω

Optical absorption experiments

SPONTANEOUS DECAY DYNAMICSSPONTANEOUS DECAY DYNAMICSCONCLUSIONSCONCLUSIONS

(I.1) Spontaneous decay dynamics of an excited atom near a carbon nanotube is in general NON-EXPONENTIAL – because of the strong non-Markovian memory effects arising from the rapid variationof the photonic density of states with frequency near the nanotube

(I.2) Spontaneous decay may proceed via damped Rabi oscillations –a principal signature of STRONG atom-vacuum-field coupling in the vicinity of the nanotube

(I.3) Atom-field coupling and spontaneous decay dynamics may be controlled experimentally by changing the distance between an atom and a nanotube surface – by means of a proper preparation of atomically doped carbon nanotube systems

This opens routes for possible new challenging applicationsof atomically doped carbon nanotube systems in:

• cavity QED• nanophotonics

• quantum information processing

(II)(II)GROUNDGROUND--STATE van der WAALS ENERGYSTATE van der WAALS ENERGY

I.V.Bondarev & Ph.Lambin, I.V.Bondarev & Ph.Lambin, condcond--mat/0404211 [submitted to SSC]mat/0404211 [submitted to SSC]

u

Aω2

Aω

2Aω

− l

0

{ }0u{ }1( , )u ωR

ω

{ }0l

0A cnr R− →

0

{0} ( , ) {1( , )}l uC l d d C uψ ω ω ω∞

= + ∫ ∫ R R R

Wave function of the system

2( )

0 0

( , , )

2

2(

2

)

AA

A

Avw AE

dE d gE

E

ω ω ωωω

ω

∞ ∞−= − −

+ −

= − +

∫ ∫

r

R r R

van der Waals energy,||~ ( , )Aξ ω⊥ r

00

( ,( )( ) 1 )A A A Ad ω ξ ωω ω ωπ ω

⊥∞ Γ⎛ ⎞= −⎜ ⎟⎝ ⎠∫r r

GROUNDGROUND--STATE van der WAALS ENERGYSTATE van der WAALS ENERGYIntegral equationIntegral equation

I.V.Bondarev and Ph.Lambin, I.V.Bondarev and Ph.Lambin, condcond--mat/0404211mat/0404211 [submitted to SSC][submitted to SSC]

020

2||

20

20

( )( ) ( , )2

( , ) ( , )( )

2 ( )

Avw A A

A AAA

vw A

xx dx xx

xx xxxx dx

x x

ε ξπ

ξ ξ

π ε

∞ ⊥

⊥

∞

Γ≈ −

⎛ ⎞+ ⎜ ⎟Γ ⎝ ⎠−

∫

∫

r r

r r

r

Strong coupling approximation0

20

2||

20

200

20

( )( ) ( , )2

( , ) ( , )( )

2 ( )1 ( , )

Avw A A

A AAA

A A

xx dx xx

xx xxxx dx

x xx x dx xx

ε ξπ

ξ ξ

πξ

∞ ⊥

⊥

∞

∞⊥

Γ≈ −

⎛ ⎞+ ⎜ ⎟Γ ⎝ ⎠

⎛ ⎞Γ+ −⎜ ⎟

⎝ ⎠

∫

∫∫

r r

r r

r

Weak coupling approximation

( ) 0vw Aε ≈r 020

( ) ( , ) 1Axdx x

xξ

∞ ⊥Γ≈∫ r

020

2

20

20

||

020

( )( )2

( )

( , )

( , ) ( , )

( ,(1 ))2 ) (

A

A

Avw A

AA

A vw

A

A A

xx dxx

xxxx d

x

xx

x xd

x

x xx xx

επ

ξ

ε

ξ

π

ξ

ξ

⊥

∞

⊥

⊥

∞

∞

Γ=

⎛ ⎞+ ⎜ ⎟Γ ⎝ ⎠−

⎛ ⎞Γ+ − −⎜ ⎟

⎝ ⎠

∫

∫∫

r

r

r

r

r

r

00 0

0 0 0

( ) ( )( ) , ( ) , , 2.7 eV2 2 2vw A

vw AE xx x ωε γ

γ γ γΓ

= Γ = = =rr

GROUNDGROUND--STATE van der WAALS ENERGY STATE van der WAALS ENERGY Numerical resultsNumerical results


strong coupling approx. exact solution

Atom outside the (9,0) CN: exact solution for the vdW energy compared with

weak/strong coupling approximation.

00

, 2.7 eV2

AAx ω γ

γ= =

1,4 1,6 1,8 2,0 2,2 2,4

-0,4

-0,3

-0,2

-0,1

0,0

(9,0), xA= 0.33 weak coupling approx.

E vw /2

γ 0

rA /Rcn

GROUNDGROUND--STATE van der WAALS ENERGY STATE van der WAALS ENERGY Numerical resultsNumerical results


1,4 1,6 1,8 2,0 2,2 2,4

-0,4

-0,3

-0,2

-0,1

0,0

(b) (9,0) exact solution strong coupling approx. weak coupling approx.

1 - xA= 0.33; 2 - xA= 0.52; 3 - xA= 0.581

1

1

2

2

3

3

3

E vw /2

γ 0

rA /Rcn

2

Atom outside the (9,0) CN: exact solution for the vdW energy compared

with weak/strong coupling approximation for different atomic transition frequencies

00

, 2.7 eV2

AAx ω γ

γ= =

GROUNDGROUND--STATE van der WAALS ENERGY STATE van der WAALS ENERGY Numerical resultsNumerical resultsI.V.Bondarev & Ph.Lambin, I.V.Bondarev & Ph.Lambin,

condcond--mat/0404211 [submitted to SSC]mat/0404211 [submitted to SSC]

Atom in/outside the (10,10) CN: van der Waals energy in/outside the nanotube

for a few atomic transition frequencies

0,0 0,2 0,4 0,6 1,4 1,6 1,8

-0,3

-0,2

-0,1

0,0

rA /Rcn

E vw /2

γ 0

(10,10) exact solution

xA= 0.22; xA= 0.30; xA= 0.45

00

, 2.7 eV2

AAx ω γ

γ= =

GROUNDGROUND--STATE van der WAALS ENERGYSTATE van der WAALS ENERGYCONCLUSIONSCONCLUSIONS

(II.1) Due to STRONG atom-vacuum field coupling in a close vicinity of the nanotube surface, one has to be careful when calculating the van der Waals energy of an atom near a carbon nanotube

(II.2) An integral equation for the van der Waals energy has been derived which is valid for both strong and weak atom-vacuum-field coupling

(II.3) The inapplicability has been demonstrated of weak-coupling-based van der Waals interaction models in a close vicinity of the nanotube surface

(II.4) Encapsulation of doped atoms into the nanotube has been shown to be energetically more favorable than their outside adsorption by the nanotube surface

atomic spontaneous decay rate enhancement near a …cophen04/talks/bondarev.pdf · 2004-07-12 ·...

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