Spontaneous decay dynamics in atomically doped carbon nanotubes

0.68 MB·English

by  I. V. Bondarev

Spontaneous decay dynamics in atomically doped carbon nanotubes
I. V. Bondarev∗ and Ph. Lambin
Facult´es Universitaires Notre-Dame de la Paix, 61 rue de Bruxelles, 5000 Namur, BELGIUM
We report a strictly non-exponential spontaneous decay dynamics of an excited two-level atom
4 placed inside or at different distances outside a carbon nanotube (CN). This is the result of strong
0 non-Markovianmemoryeffectsarisingfromtherapidvariationofthephotonicdensityofstateswith
0 frequency near the CN. The system exhibits vacuum-field Rabi oscillations, a principal signature
2 of strong atom-vacuum-field coupling, when the atom is close enough to the nanotube surface and
n theatomic transition frequencyis in thevicinityof theresonance of thephotonicdensityof states.
a Caused by decreasing the atom-field coupling strength, the non-exponential decay dynamics gives
J placetotheexponentialoneiftheatommovesawayfromtheCNsurface. Thus,atom-fieldcoupling
9 and the character of the spontaneous decay dynamics, respectively, may be controlled by changing
1 the distance between the atom and CN surface by means of a proper preparation of atomically
doped CNs. This opens routesfor new challenging nanophotonics applications of atomically doped

] CN systems as various sources of coherent light emitted bydopant atoms.
l
l
a PACSnumbers: 61.46.+w,73.22.-f,73.63.Fg,78.67.Ch
h
-
s
e I. INTRODUCTION electronic, mechanical, electromechanical, chemical and
m scanning probe devices and materials for macroscopic
composites [16]. Important is that their intrinsic prop-
. Ithaslongbeenrecognizedthatthespontaneousemis-
t
a sion rate of an excited atom is not an immutable prop- erties may be substantially modified in a controllable
m way by doping with extrinsic impurity atoms, molecules
erty, but that it can be modified by the atomic envi-
and compounds [17]. This opens routes for new chal-
- ronment. Generally called the Purcell effect [1], the phe-
d lenging nanophotonics applications of atomically doped
nomenon is qualitatively explained by the fact that the
n CN systems as various sources of coherent light emitted
localenvironmentmodifies the strengthanddistribution
o
by dopant atoms. Recent successful experiments on en-
c of the vacuum electromagnetic modes with which the
capsulation of single atoms into single-wall carbon nan-
[ atom can interact, resulting indirectly in the alteration
otubes [18] and their intercalation into single-wall CN
of atomic spontaneous emission properties.
1 bundles [17, 19] stimulate an analysis of atomic sponta-
v The Purcelleffect took on specialsignificance recently
neous emission in such systems as a first step towards
2 in view of rapid progress in physics of nanostructures.
their nanophotonics applications.
3 Here, the control of spontaneous emission has been pre-
3 Typically, there may be two qualitatively different
dicted to have a lot of useful applications, ranging from
1 regimes of interaction of an atomic excited state with
the improvement of existing devices (lasers, light emit-
0 avacuumelectromagneticfieldinthevicinity ofthe CN.
ting diodes) to such nontrivial functions as the emis-
4
They are the weak coupling regime and the strong cou-
0 sion of nonclassical states of light [2]. In particular, the
pling regime [20]. The former is characterized by the
/ enhancement of the spontaneous emission rate can be
t monotonous exponential decay dynamics of the upper
a the first step towards the realization of a thresholdless
m laser [3] or a single photon source [4]. The possibility to atomic state with the decay rate altered compared with
the free-space value. The latter is, in contrast, strictly
- controlatomic spontaneousemissionwas showntheoret-
d icallyformicrocavitiesandmicrospheres[5,6,7],optical non-exponential and is characterized by reversible Rabi
n oscillationswheretheenergyoftheinitiallyexcitedatom
fibers [8], photonic crystals [9], semiconductor quantum
o isperiodicallyexchangedbetweentheatomandthefield.
dots[10]. Recenttechnologicalprogressinthefabrication
c
In the present paper, we develop the quantum theory of
: oflow-dimensionalnanostructureshasenabledtheexper-
v the spontaneous decay of an excited two-levelatom near
imental investigation of spontaneous emission for micro-
i aCNandderivetheevolutionequationoftheupperstate
X cavities [11], photonic crystals [12], quantum dots [13].
ofthesystem. Bysolvingitnumerically,wedemonstrate
r Carbon nanotubes (CNs) are graphene sheets rolled-
a thestrictlynon-exponentialspontaneousdecaydynamics
up into cylinders of approximately one nanometer in di-
inthecasewheretheatomiscloseenoughtotheCNsur-
ameters. Extensive work carried out worldwide in re-
face. In certain cases, the system exhibits vacuum-field
cent years has revealed the intriguing physical proper-
Rabi oscillations — a result already detected for quasi-
ties of these novel molecular scale wires [14, 15]. Nan-
2D excitonic and intersubband electronic transitions in
otubes have been shown to be useful for miniaturized
semiconductor quantum microcavities [21, 22] and never
reported for atomically doped CNs so far.
Therestofthe paperisarrangedasfollows. SectionII
∗OnleavefromtheInstituteforNuclearProblemsattheBelarusian presents a theoretical model we use to derive the evolu-
StateUniversity,BobruiskayaStr.11,220050 Minsk,BELARUS tion equation of the upper state population probability
2
result, their solutions cannot be expanded in power or-
thogonalmodes andthe concept ofmodes itself becomes
more subtle. We, therefore, use an alternative quantiza-
tion scheme developed in Refs. [5, 6], where the Fourier-
images of electric and magnetic fields are considered as
quantum mechanical observables of corresponding elec-
tric and magnetic field operators.The latter ones satisfy
theFourier-domainoperatorMaxwellequationsmodified
bythepresenceofaso-calledoperatornoisecurrentden-
sity Jˆ(r,ω) written in terms of a 3D vector bosonic field
operator ˆf(r,ω) and a medium dielectric tensor ǫ(r,ω)
(supposed to be diagonal) as
ω
Jˆ(r,ω)= ¯hImǫ (r,ω)fˆ(r,ω), i=1,2,3. (1)
i 2π ii i
FIG. 1: (Color online) The geometry of theproblem.
p
This operator is responsible for correct commutation re-
lationsofthe electricandmagneticfieldoperatorsinthe
amplitude ofthe composedquantumsystem”atwo-level presence of medium-induced absorbtion. In this formal-
atom interacting with a quantized CN-modified vacuum ism, the electric and magnetic field operators are ex-
radiation field”. In describing atom–field interaction we pressed in terms of a continuum set of the 3D vector
followtheoriginallineofRefs.[5,6],adoptingtheirelec- bosonic fields ˆf(r,ω) by means of the convolution over r
tromagnetic field quantization scheme for a particular ofthe current(1)withtheclassical electromagneticfield
case of the field near an infinitely long achiral single- Green tensor of the system. The bosonic field operators
wall CN. The carbon nanotube is considered as an in- ˆf†(r,ω)andˆf(r,ω)createandannihilatesingle-quantum
finitely thin anisotropically conducting cylinder. Its sur-
electromagnetic medium excitations. They are defined
faceconductivityisrepresentedintermsoftheπ-electron
by their commutation relations and play the role of the
dispersion law obtained in the tight-binding approxima-
fundamental dynamical variables in terms of which the
tionwithallowancemadeforazimuthalelectronmomen-
Hamiltonian of the composed system ”electromagnetic
tum quantization and axial electron momentum relax-
field + dissipative medium” is written in a standardsec-
ation [23]. Only the axial conductivity is taken into ac-
ondly quantized form as
count and the azimuthal one, being strongly suppressed
by transverse depolarization fields [24, 25, 26, 27], is ne- ∞
glected. In Section III, the time dynamics of the upper Hˆ = dr dω¯hωˆf†(r,ω) ˆf(r,ω). (2)
·
state population probability is analyzed qualitatively in Z Z0
terms of two different approximations admitting analyt-
Consider a two-level atom positioned at an arbitrary
ical solutions of the evolution equation derived in Sec-
point r near the CN. Since the problem has a cylin-
tion II. They are the Markovian approximation and the A
dric symmetry, it is convenient to assign the orthonor-
single-resonanceapproximationofthedensityofphotonic
mal cylindric basis e ,e ,e in such a way that r =
states in the vicinity of the CN. Section IV presents and r ϕ z A
r e = r ,0,0 an{d e is di}rected along the nanotube
discussesthe resultsofthe numericalsolutionofthe evo- A r A z
{ }
axis (see Figure 1). Let the atom interact with a quan-
lution equation for various particular cases where the
tized electromagnetic field via an electric dipole transi-
atom is placed in the center and near the wall inside,
tionoffrequencyω .Theatomicdipolemomentmaybe
and at different distances outside achiral CNs of differ- A
assumedtobedirectedalongtheCNaxis,d=d e .The
ent radii. A summary and conclusions of the work are z z
contribution of the transverse dipole moment orienta-
given in Section V.
tionissuppressedbecauseofstrongdepolarizationofthe
transversefield in anisolatedCN [24, 25, 26, 27]. Strong
transverse depolarization along with transverse electron
II. THE MODEL
momentum quantization allow one to neglect the az-
imuthal current and radial polarizability [23], in which
The quantum theory of the spontaneous decay of ex- case the dielectric tensor components ǫ and ǫ are
rr ϕϕ
cited atomic states in the presence of the CN involves identically equal to unit. The component ǫ is caused
zz
an electromagnetic field quantization procedure. Such a by the CN longitudinal polarizability and is responsible
procedure faces difficulties similar to those in quantum fortheaxialsurfacecurrentparalleltothee vector.This
z
optics of 3D Kramers-Kronigdielectric media where the current may be represented in terms of the 1D bosonic
canonical quantization scheme commonly used does not field operators by analogy with Eq. (1). Indeed, taking
work since, because of absorption, corresponding opera- into account the dimensionality conservation in passing
tor Maxwell equations become non-Hermitian [28]. As a from bulk to a monolayer in Eq. (2) and using a simple
3
Drude relation [24] [and Hˆ =(ik)−1 Eˆ accordingly],where G(r,R,ω) is
∇×
the Green tensor of the classical electromagnetic field in
σ (R,ω)= iωǫzz(R,ω)−1, (3) the vicinity of the CN. Its components satisfy the equa-
zz
− 4πSρT tion
pwohienrteoRf =th{eRCcnN,φs,uZrf}acise,thReradiisust-hveecrtaodrioufsaonfatrhbeitrCaNry, ∇×∇×−k2 zαGαz(r,R,ω)=δ(r−R), (10)
cn α=r,ϕ,z
σ (R,ω) is the CN surface axial conductivity per unit X (cid:0) (cid:1)
zz
length,Sistheareaofasinglenanotube,ρ isthetubule togetherwiththeradiationconditionsatinfinityandthe
T
density in a bundle, one immediately has from Eq. (1) boundary conditions on the CN surface. The Hamilto-
nian (5) along with Eqs. (6)–(10) and (4) is the modifi-
¯hωReσ (R,ω) cation of the Jaynes–Cummings model [28] for an atom
Jˆ(R,ω)= zz fˆ(R,ω)e (4)
z in the vicinity of a solitary CN. The classical electro-
π
r
magnetic field Green tensor of this system is derived in
withfˆ(R,ω)being the 1Dbosonicfieldoperatordefined Appendix A.
on the CN surface. The total Hamiltonian of the system When the atom is initially in the upper state and the
under consideration is then written in terms of fˆ†(R,ω) field subsystem is in vacuum, the time-dependent wave
and fˆ(R,ω) operators as function of the whole system can be written as
ˆ = dR ∞dω¯hωfˆ†(R,ω)fˆ(R,ω)+ 1¯hω σˆ |ψ(t)i=Cu(t)e−i(ωA/2)t|ui|{0}i
A z
H 2
Z Z0 ∞
−[σˆ†Eˆz(+)(rA)dz +h.c.]. (5) +Z drZ0 dωCl(r,ω,t)e−i(ω−ωA/2)t|li|{1(r,ω)}i, (11)
Here, the three terms representthe electromagneticfield where 0 is the vacuum state of the field subsystem,
modified by the presence of the CN, the two-level atom 1(r,ω|{) }iisitsexcitedstatewherethefieldisinasingle-
and their interaction (within the framework of electric |q{uantum}Fiockstate,C andC arethepopulationproba-
u l
dipole, and rotating wave approximations [5]), respec- bilityamplitudesoftheupperstateandlowerstateofthe
tively. The Pauli operators σˆz= u u l l , σˆ= l u, whole system, respectively. In view of Eqs. (6), (9), (4)
σˆ†= u l describethetwo-levela|toihmi|c−su|bishy|stemw| ihher|e and the integral relationship
| ih |
u and l aretheupperandloweratomicstates,respec-
|tiviely.T|hieoperatorsEˆ(±)(rA)representtheelectricfield ImGαβ(r,r′,ω)=
the atom interacts with. For an arbitrary r= r,ϕ,z ,
{ }
they are defined as follows 4π
k dRReσ (R,ω)G (r,R,ω)G∗ (r′,R,ω)
c zz αz βz
Eˆ(r)=Eˆ(+)(r)+Eˆ(−)(r), Z
(which is nothing but a particular case of the gen-
∞ eral 3D Green tensor integral relationship rigorously
Eˆ(+)(r)= Eˆ(r,ω)dω, Eˆ(−)(r)=[Eˆ(+)(r)]†. (6)
proven in Ref. [6] with Eq. (3) taken into account), the
Z0 Schr¨odingerequationwiththeHamiltonian(5)andwave
Here,Eˆ(r,ω)satisfiesthe Fourier-domainMaxwellequa- function (11) yields the following evolution law for the
tions population probability amplitude of the upper state of
the system
Eˆ(r,ω)=ikHˆ(r,ω),
∇× τ
C (τ)=1+ K(τ τ′)C (τ′)dτ′, (12)
u u
4π −
Hˆ(r,ω)= ikEˆ(r,ω)+ ˆI(r,ω), (7) Z0
∇× − c
whereHˆ(r,ω)standsforthemagneticfieldoperator[de- K(τ τ′)= ¯hΓ0(xA) ∞dxx3ξ(x)e−i(x−xA)(τ−τ′)−1.
fined by analogy with Eq. (6)], k =ω/c, and − 4πx3γ i(x x )
A 0 Z0 − A
(13)
ˆI(r,ω)= dRδ(r R)Jˆ(R,ω)=2Jˆ(R ,ϕ,z,ω)δ(r R ) Here,
cn cn
− −
Z
(8) ¯hω 2γ t
0
x= and τ = (14)
istheexterioroperatorcurrentdensity[withJˆ(R,ω)de- 2γ ¯h
0
finedby Eq.(4)]associatedwiththe presenceofthe CN.
are the dimensionless frequency and time, respectively,
FromEqs.(7)and(8)inviewofEq.(4),itfollowsthat
withγ =2.7eVbeingthe carbonnearestneighborhop-
0
4π ping integral [29] appearing in the CN surface axial con-
Eˆ(r,ω)=i k dRG(r,R,ω) Jˆ(R,ω) (9)
c · ductivity in Eq. (4), Γ0 is the rate of the exponential
Z
4
atomic spontaneous decay in vacuum obtained from the III. QUALITATIVE ANALYSIS
general expression of the form [30, 31]
In this Section we will qualitatively analyze the time
8πd2 2γ x 2 dynamicsoftheupperstatepopulationprobabilityC (τ)
Γ(x)= z 0 ImG (r ,r ,x) (15) u
¯hc2 ¯h zz A A in terms of two different approximations admitting an-
(cid:18) (cid:19)
alytical solutions of the evolution problem (12), (13).
by substituting They are the Markovian approximation and the single-
resonance approximation of the relative density ξ(x) of
1 2γ x photonic states in the vicinity of the CN.
ImGv (r ,r ,x)= 0 (16)
zz A A 6πc ¯h
for the vacuum imaginary Green tensor [32]. The func- 1. Markovian approximation
tionξ(x)istherelativedensity(withrespecttovacuum)
of photonic states near the CN given for rA>Rcn (see In the case where the Markovianapproximationis ap-
Appendix B) by plicable,or,inotherwords,whentheatom-fieldcoupling
strength is weak enough for atomic motion memory ef-
Γ(x) 3 ∞ fects to be insignificant, so that they may be neglected,
ξ(x)= =1+ Im (17)
the time-dependent factor in the kernel (13) may be re-
Γ (x) π
0
p=X−∞ placed by its long-time approximation
dys(R ,x)v(y)4I2[v(y)u(R )x]K2[v(y)u(r )x] e−i(x−xA)(τ−τ′) 1 1
cn p cn p A , − πδ(x xA)+i
ZC1+s(Rcn,x)v(y)2Ip[v(y)u(Rcn)x]Kp[v(y)u(Rcn)x] i(x−xA) →− − Px−xA
( denotes a principalvalue). Then, inview ofEq.(17),
where Ip and Kp are the modified cylindric Bessel func- tPhe kernel becomes
tions, v(y)= y2 1, u(r)=2γ r/¯hc, and s(R ,x)=
0 cn
2diimαue(nRsiconn)lxeσsszpzc(oRncdn−u,cxt)ivwitiythanσdzzα==2eπ2¯h/¯hσczz=/e12/1b3e7inrgeptrhee- K(τ −τ′)=−¯hΓ4(γx0A) +i∆(xA)
senting the fine-structure constant. The integrationcon- with
tour C goes along the real axis of the complex plane
¯hΓ (x ) ∞ ξ(x)
and envelopes the branch points y= 1 of the function ∆(x )= 0 A dxx3 ,
± A 4γ πx3 P x x
v(y)intheintegrandfrombelowandfromabove,respec- 0 A Z0 − A
tively. For rA<Rcn, Eq. (17) is modified by the simple and Eq. (12) yields
replacement r R in the Bessel function arguments
A cn
inthenumerato↔roftheintegrand.Notethedivergenceof ¯hΓ(xA)
C (τ)=exp +i∆(x ) τ (19)
u A
ξ(x) at r =R , i. e. when the atom is located righton − 4γ
A cn (cid:26)(cid:20) 0 (cid:21) (cid:27)
theCNsurface.ThepointisthattheCNdielectrictensor
— the exponential decay dynamics of the [shifted by
longitudinalcomponentǫ [which, accordingto Eq.(3),
zz ∆(x )]upperatomiclevelwiththerateΓ(x ).Thiscase
is responsible for the surface axial conductivity σ in A A
zz was analyzed in Ref. [34].
Eq.(17)]isobtainedasaresultofastandardprocedureof
physical averaging a local electromagnetic field over the
twospatialdirectionsinthegrapheneplane[33].Suchav-
2. Single-resonance approximation of the relative density
eragingdoesnotassumeextrinsicatomsonthegraphene
of photonic states
surface. To take them into consideration the averaging
procedure must be modified. Thus, the applicability do-
Another approximation that admits an analytical so-
main of our model is restricted by the condition
lution of the evolution problem (12), (13) is a single-
resonance approximation. Suppose that at x = x the
r R >a, (18) r
| A− cn| photonic density of states ξ(x) has a sharp peak of half-
width-at-half-maximum δx . For all x in the vicinity
wherea=1.42˚A isagrapheneinteratomicdistance[29]. r
of x , the shape of ξ(x) may then be roughly approxi-
r
Eq. (12) is a well-known Volterra integral equation of
mated by the Lorentzian function of the form
the second kind. In our case, it describes the sponta-
neousdecaydynamicsoftheexcitedtwo-levelatominthe ξ(x) ξ(xr)δx2r .
vicinity of the CN. All the CN parameters that are rele- ≈ (x x )2+δx2
− r r
vantforthespontaneousdecayarecontainedinthe rela-
The kernel (13) is then easily calculated analytically to
tive density of photonic states (17) appearing in the ker-
give
nel(13).Thedensityofphotonicstatesis,inturn,deter-
minedbytheimaginaryclassicalGreentensoroftheCN- ¯hΓ(x )δx
K(τ τ′) r r
modifiedelectromagneticfieldviaΓ(x)givenbyEq.(15). − ≈ 2γ 2
0
5
exp[ i(x iδx x )(τ τ′) 1]
− r− r − A − − , τ >τ′.
× i(x iδx x )
r r A
− −
SubstitutingthisintoEq.(12)andmakingthedifferenti-
ationofbothsidesoftheresultingequationovertime,fol-
lowedbythechangeoftheintegrationorderandonemore
time differentiation, one straightforwardly arrives at a
secondorderordinaryhomogeneousdifferentialequation
C¨ (τ)+i(x iδx x )C˙ (τ)+(Ω/2)2C (τ)=0,
u r r A u u
− −
where Ω = 2δx ¯hΓ(x )/2γ , with the solution given
r r 0
for x x by
A r
≈ p
1 δx τ
C (τ) 1+ r exp δx δx2 Ω2
u ≈2 δx2r −Ω2! h−(cid:16) r−p r − (cid:17)2i
p
1 δx τ
+ 1 r exp δx + δx2 Ω2 .
2 − δx2r −Ω2! h−(cid:16) r p r − (cid:17)2i(20)
p
This solution is approximately valid for those atomic
transition frequencies x which are located in the vicin-
A
ity of the photonic density-of-states resonances what-
ever the atom-field coupling strength is. In particular,
if (Ω/δx )2 1, Eq. (20) yields the exponential de-
r
≪
cay of the upper atomic state population probability
C (τ)2 withthe rateh¯Γ(x )/2γ infullagreementwith
u r 0
| |
Eq. (19) obtained within the Markovian approximation
for weak atom-field coupling. In the opposite case, when
FIG.2: (Coloronline)Relativedensityofphotonicstates(a)
(Ω/δx )2 1, one has
r ≫ andupper-levelspontaneousdecaydynamics(b)fortheatom
inthecenterofdifferentCNs.Theatomictransitionfrequency
Ωτ
C (τ)2 e−δxrτ cos2 , is indicated bythe dashed line in Fig. 2 (a).
u
| | ≈ 2
(cid:18) (cid:19)
andthedecayoftheupperatomicstatepopulationprob-
centerandnearthewallinside,andatdifferentdistances
abilityproceedsviadampedRabioscillations.Thisisthe
outsideachiralCNsofdifferentradii.Therelativedensity
principalsignatureofstrongatom-fieldcouplingwhichis
ofphotonicstatesξ(x)inEq.(13)wascomputedaccord-
beyondthe Markovianapproximation. ExpressingΓ(x )
r
ing to Eq. (17). The CN surface axial conductivity σ
in Ω in terms of ξ(x ) by means of Eq. (17) and using zz
r
the approximationΓ (x) α32γ x/¯h valid for hydrogen- appearing in Eq. (17) was calculated in the relaxation-
0 ≈ 0 time approximationwith the relaxationtime 3 10−12 s;
like atoms [35], one may conveniently rewrite the strong ×
thespatialdispersionofσ wasneglected[23].Thefree-
atom-field coupling condition in the form zz
space spontaneous decay rate was approximated by the
2α3x ξ(xr) 1, (21) expression Γ0(x)≈α32γ0x/¯h [35].
r δx ≫ Figure2(a)presentsξ(x)fortheatominthecenterof
r
the (5,5), (10,10) and (23,0) CNs. It is seen to decrease
from which it follows that the strong coupling regime is with increasingthe CN radius,representing the decrease
fostered by high and narrow resonances in the relative of the atom-field coupling strength as the atom moves
density of photonic states. away from the CN surface [34]. To calculate C (τ)2 in
u
| |
thisparticularcase,wehavefixedx =0.45(indicatedby
A
the vertical dashed line), firstly, because this transition
IV. NUMERICAL RESULTS AND DISCUSSION islocatedwithinthe visiblelightrange0.305<x<0.574,
secondly, because this is the approximate peak position
To get beyondthe Markovianand single-peakapprox- of ξ(x) for all the three CNs. The functions C (τ)2 cal-
u
| |
imations, we have solved Eqs. (12) and (13) numeri- culated are shown in Figure 2 (b) in comparison with
cally. The exact time evolution of the upper state pop- those obtained in the Markovian approximation yield-
ulation probability C (τ)2 was obtained for the atom ingtheexponentialdecay.Theactualspontaneousdecay
u
| |
placed[inawaythatEq.(18)wasalwayssatisfied]inthe dynamics is clearly seen to be non-exponential. For the
6
cific transition frequencies x (dashed lines) for which
A
the functions C (t)2 presentedinFigure3 (b)werecal-
u
| |
culated.Rabi oscillationsare clearlyseen to occur in the
vicinity of the highest peak (x 0.22) of the photonic
A
≈
density of states. Importantis that they persistfor large
enough detuning values x 0.21 0.25. For x =0.30,
A A
≈ ÷
the density of photonic states has a dip, and the decay
dynamics exhibits no Rabi oscillations, being strongly
non-exponential nevertheless. For x =0.45, the inten-
A
sity of the peak of the photonic density of states is not
large enough and the peak is too broad, so that strong
atom-fieldcouplingcondition(21)isnotsatisfiedandthe
decay dynamics is close to the exponential one.
Figure 4 (a) shows the density of photonic states for
the atom outside the (9,0) CN at the different distances
from its surface. The vertical dashed lines indicate the
atomic transitions for which the functions C (τ)2 in
u
| |
Figures 4 (b), (c), and (d) were calculated. The tran-
sitions x = 0.33 and 0.58 are the positions of sharp
A
peaks (at least for the shortest atom-surface distance),
while x = 0.52 is the position of a dip of the func-
A
tion ξ(x). Very clear underdamped Rabi oscillations are
seen for the shortest atom-surface distance at x =0.33
A
[Figure4(b)], indicatingstrongatom-fieldcouplingwith
strong non-Markovity. For x =0.58 [Figure 4(d)], the
A
value of ξ(0.58) is not large enoughfor strong atom-field
coupling to occur, so that Eq. (21) is not fulfilled. As a
consequence, the decay dynamics, being strongly non-
exponential in general, exhibits no clear Rabi oscilla-
tions.Forx =0.52[Figure4(c)],thoughξ(0.52)iscom-
FIG.3: (Coloronline)(a)Fragmentoftherelativedensityof A
parativelysmall,the spontaneousdecay dynamicsis still
photonicstatesfortheatominsidethe(10,10)CNatdistance
of 3 ˚A from the wall (the situation observed experimentally non-exponential, approaching the exponential one only
when the atom is far enough from the CN surface.
for Cs in Ref. [18]). (b) The upper-level spontaneous decay
dynamicsfordifferentatomictransitionfrequencies[indicated Thereasonfornon-exponentialspontaneousdecaydy-
by thedashed lines in Fig. 3 (a)] in this particular case. namicsinallthecasesconsideredissimilartothattaking
place in photonic crystals [9]. When the atom is closed
enough to the CN surface, an absolute value of the rela-
tive density of photonic states is large and its frequency
smallradius(5,5)CN,Rabioscillationsareobserved,in-
variation in the neighborhood of a specific atomic tran-
dicating a strong atom-field coupling regime related to
strong non-Markovian memory effects. Eq. (21) is satis- sition frequency essentially influences the time behavior
of the kernel (13) of evolution equation (12). Physically,
fied in this case. With increasing the CN radius, as the
this means that the correlation time of the electromag-
valueofξ(0.45)decreases,Eq.(21)ceasestobevalidand
netic vacuum is not negligible on the time scale of the
thedecaydynamicsapproachestheexponentialonewith
evolution of the atomic system, so that atomic motion
the decay rate enhanced by several orders of magnitude
memoryeffectsareimportantandtheMarkovianapprox-
compared with that in free space. Note that, though the
imation in the kernel (13) is inapplicable.
distance from the atom to the CN surface is larger for
the (23,0) CN than for the (10,10) CN, the deviation of
the actual decay dynamics from the exponential one is
larger for the (23,0) CN. This is an obvious consequence V. CONCLUSIONS
ofthe influence ofa smallneighboringpeak in the (23,0)
CN photonic density of states [Figure 2 (a)]. Theeffectswepredictwillyieldanadditionalstructure
In Ref. [18], formation of Cs-encapsulating single-wall in optical absorbance/reflectance spectra (see, e.g., [26,
CNs was reported. In a particular case of the (10,10) 27]) of atomically doped CNs in the vicinity of the en-
CN, the stable Cs atom/ion position was observed to ergyofanatomictransition. Weaknon-Markovityofthe
be at distance of 3 ˚A from the wall. We have simulated spontaneous decay (non-exponential decay with no Rabi
the spontaneous decay dynamics for a number of typical oscillations) will cause an asymmetry of an optical spec-
atomic transition frequencies for this case. Figure 3 (a) tralline-shape(similartoexcitonopticalabsorbtionline-
shows the density of photonic states and the five spe- shape in quantum dots [36]). Strong non-Markovity of
7
FIG. 4: (Color online) (a) Relativedensity of photonic states for theatom located at differentdistances outsidethe (9,0) CN.
(b, c, d) Upper-levelspontaneous decay dynamics for thethree atomic transition frequencies [indicated by the dashed lines in
Fig. 4 (a)] at different atom-nanotube-surface distances.
the spontaneous decay (non-exponential decay with fast nature of strong atom-vacuum-fieldcoupling, — a result
Rabi oscillations) originates from strong atom-vacuum- alreadydetectedforquasi-2Dexcitonicandintersubband
field coupling with the upper state of the system split- electronic transitions in semiconductor quantum micro-
ted into two ”dressed” states. This will yield a two- cavities [21, 22] and never reportedfor atomically doped
component structure of optical spectra similar to that CNs so far. This is the result of strong non-Markovian
observed for excitonic and intersubband electronic tran- memoryeffectsarisingfromtherapidfrequencyvariation
sitions in semiconductor quantum microcavities [21, 22]. of the photonic density of states near the nanotube. The
Summarizing, we have developed the quantum theory non-exponential decay dynamics gives place to the ex-
of the spontaneous decay of an excited two-level atom ponential one if the atom moves away from the CN sur-
near a carbon nanotube. In describing atom-field inter- face.Thus,the atom-vacuum-fieldcouplingstrengthand
action,we followedanelectromagneticfieldquantization thecharacterofthespontaneousdecaydynamics,respec-
scheme developed for dispersing and absorbing media in tively, may be controlled by changing the distance be-
Refs. [5, 6]. This quantization formalism was adopted tween the atom and CN surface by means of a proper
by us for a particular case of an atom near an infinitely preparationof atomically doped CNs. This opens routes
longsingle-wallCN.Wederivedtheevolutionequationof for new challenging nanophotonics applications of atom-
the upper state of the system and, by solving it numer- ically doped CN systems as various sources of coherent
ically, demonstrated a strictly non-exponential sponta- light emitted by dopant atoms.
neousdecaydynamicsinthecasewheretheatomisclose Finally, we would like to emphasize a general char-
enoughtotheCNsurface.Incertaincases,namelywhen acter of the results we obtained. We have shown that
the atom is close enough to the nanotube surface and similarto semiconductormicrocavities[22]andphotonic
the atomic transition frequency is in the vicinity of the band-gap materials [9], carbon nanotubes may quali-
resonance of the photonic density of states, the system tatively change the character of atom-electromagnetic-
exhibits vacuum-field Rabi oscillations, a principal sig- field interaction, yielding strong atom-field coupling —
8
an important phenomenon necessary, e.g., for quantum (ǫ is the totally antisymmetricunit tensorofrank3),
αβγ
information processing [37, 38, 39]. The present pa- they can be derived from the classical electromagnetic
per dealt with the simplest manifestation of this gen- field boundary conditions of the form
eral phenomenon — vacuum-field Rabi oscillations in
the atomic spontaneous decay dynamics near a single- Eϕ|r=Rcn+0−Eϕ|r=Rcn−0 =0, (A6)
wall carbon nanotube. However, similar manifestations
of strong atom-field coupling may occur in many other E E =0, (A7)
atom-electromagnetic-field interaction processes in the z|r=Rcn+0− z|r=Rcn−0
presence of CNs, such as, e.g., dipole-dipole interac- 4π
tion between atoms by means of a vacuum photon ex- Hϕ|r=Rcn+0−Hϕ|r=Rcn−0 = c σzz(Rcn,ω)Ez|r=Rcn,
change [40], or cascade spontaneous transitions in three- (A8)
level atomic systems [41].
H H =0 (A9)
z|r=Rcn+0− z|r=Rcn−0
(spatial dispersion neglected) obtained in Ref. [23].
Acknowledgments
Letr >R (the atomoutsidethe CN) tobe specific.
A cn
Then, g(r,r ,ω) may be represented in the form
We gratefully acknowledge numerous discussions with A
Dr. G.Ya.SlepyanandProf. I.D.Feranchuk. I.B.thanks
g (r,r ,ω)+g(+)(r,ω), r >R
the Belgian OSTC. The work was performed within the g(r,r ,ω)= 0 A cn
framework of the Belgian PAI-P5/01 project. A ( g(−)(r,ω), r <Rcn
(A10)
where g (r,r ,ω) is the point radiative atomic source
0 A
APPENDIX A: GREEN TENSOR OF function defined in Eq. (A3) and g(±)(r,ω) are un-
A SINGLE-WALL CARBON NANOTUBE knownnonsingularfunctionssatisfyingthehomogeneous
Helmholtz equation and the radiation conditions at in-
The form of the classical electromagnetic field Green finity. We seek them using integral decompositions over
tensor in Eqs. (9) and (10) depends on the presence the modifiedcylindric Besselfunctions I andK as fol-
p p
of external radiative sources (an atom in our case) and lows [42]
their position (inside or outside) with respect to a nan-
Foatsiusgiubgrene. tS1hi)necionerttshhuoecnhporraombwalealmycythlhianasdtraieczcybilasinsiddsirri{ececstrye,dmeϕma,leeoztnr}gy,(tswheeee g(±)(r,ω)=p=X∞−∞eipϕZC(cid:26)BApp((hh))KIpp(v(vrr))(cid:27)eihzdh (A11)
nanotube axis and rA =rAer = rA,0,0 . and
{ }
We use the representation
∞
1
1 g (r,r ,ω)= eipϕ
Gαz(r,rA,ω)= k2∇α∇z +δαz g(r,rA,ω), (A1) 0 A (2π)2 p=−∞
(cid:18) (cid:19) X
where α = r,ϕ,z and the function g(r,rA,ω) is the I (vr)K (vr )eihzdh, r r, (A12)
{ } p p A A
Green function of the Helmholtz equation. Substituting × ≥
ZC
Eq. (A1) into Eq. (10), one straightforwardly obtains
where A (h) and B (h) are unknown functions to be
p p
(∆+k2)g(r,rA,ω)= δ(r rA) (A2) found from the boundary conditions (A6)–(A9) in view
− −
of Eqs. (A1), (A4) and (A5), v = v(h,ω) = √h2 k2.
with a known solution −
TheintegrationcontourC goesalongthe realaxisofthe
g0(r,rA,ω)= 41π erik|r−rrA| (A3) cboemlopwleaxnpdlafrnoemanadboevnev,erloesppeesctthiveeblyr.anchpoints±k from
A
| − |
The boundary conditions (A6)–(A9) with Eqs. (A1),
satisfying the radiation condition at infinity (see, e.g.,
(A4) and (A5) taken into account yield the following
[35]). In our case, however, the functions G (r,r ,ω)
αz A two independent equations for the scalar Green func-
and g(r,r ,ω) are imposed one more set of boundary
A tion (A10)
conditions.Theyaretheboundaryconditionsonthesur-
face of the CN. Using simple relations (valid for r = rA g (r,r ,ω) +g(+)(r,ω) =g(−)(r,ω) ,
under the Coulomb-gauge condition) 6 0 A |r=Rcn |r=Rcn |r=Rcn
E (r,ω)=ikG (r,r ,ω) (A4)
α αz A ∂g(+)(r,ω) ∂g(−)(r,ω)
+
and ∂r |r=Rcn − ∂r |r=Rcn
Hα(r,ω)=−kiβ,γ=r,ϕ,zǫαβγ∇βEγ(r,ω) (A5) β(ω) ∂∂z22+k2 g(−)(r,ω)|r=Rcn=−∂g0(r∂,rrA,ω)|r=Rcn,
X (cid:18) (cid:19)
9
where β(ω) = 4πiσ (R ,ω)/ω. Substituting the inte- ratio [see Eqs. (14)–(17)]
zz cn
gral decompositions (A11) and (A12) into these equa-
tions, one obtains the set of two simultaneous algebraic
equations for the functions Ap(h) and Bp(h). The func- ImGzz(rA,rA,ω)
ξ(ω)= , (B1)
tion Ap(h) we need (we only need the Green function in ImGvzz(rA,rA,ω)
the region where the atom is located) is found by solv-
ing this set with the use of basic properties of cylindric
Bessel functions (see, e.g., [43]). In so doing, one has where
R β(ω)v2I2(vR )K (vr )
cn p cn p A
A (h)= .
p −(2π)2[1+β(ω)v2R I (vR )K (vR )] 1 ∂2
cn p cn p cn Gzz(rA,rA,ω)= k2∂z2 +1 g(r,rA,ω)|r=rA (B2)
These A (h), being substituted into Eq. (A11), yield (cid:18) (cid:19)
p
the function g(+)(r,ω) sought. The latter one, in view
of Eq. (A10), results in the scalar electromagnetic field according to Eq. (A1), g(r,r ,ω) is given by Eq. (A13)
A
Green function of the form
for r r > R and by the same equation with the
A cn
≥
symbol replacement I K in the numerator of the
R p p
g(r,rA,ω)=g0(r,rA,ω)− (2πcn)2 integrand for r≤rA<Rc↔n.
Substituting Eq. (A13) into Eq. (B2), making differ-
∞ eipϕ β(ω)v2Ip2(vRcn)Kp(vrA)Kp(vr) eihzdh, entiation and passing to the limit r rA, followed by
×p=−∞ ZC 1+β(ω)v2RcnIp(vRcn)Kp(vRcn) the substitution of the result into Eq→. (B1), one has for
X (A13) rA>Rcn
where r r > R . One may show in a similar way
A cn
that the fu≥nction g(r,r ,ω) for r r <R is obtained
A A cn
≤ 3R
fromEq.(A13)bymeansofasimplesymbolreplacement cn
ξ(ω)=1+ (B3)
I K in the numerator of the integrand. 2πk3
p p
K↔nowing g(r,r ,ω), one may easily compute the
A
components of the electromagnetic field Green tensor ∞ β(ω)v4I2(vR )K2(vr )
G (r,r ,ω) according to Eq. (A1). Im p cn p A dh.
αz A × 1+R β(ω)v2I (vR )K (vR )
p=−∞ZC cn p cn p cn
X
APPENDIX B: DENSITY OF PHOTONIC
STATES NEAR A SINGLE-WALL CARBON For r <R , Eq. (B3) is modified either by the symbol
A cn
NANOTUBE replacementI K or by the Besselfunction argument
p p
↔
replacementr R inthenumeratoroftheintegrand.
A cn
↔
The relative density of photonic states in an atomic In dimensionless variables (14), Eq. (B3) is rewritten to
spontaneous decay process near a CN is defined by the give Eq. (17).
[1] E.M. Purcell, Phys. Rev. 69, 681 (1946). [11] H. Schniepp and V. Sandoghdar, Phys. Rev. Lett. 89,
[2] J.D.RarityandC.Weisbuch,MicrocavitiesandPhotonic 257403 (2002).
Bandgaps: Physics and Applications, NATO ASI Series [12] E.P.Petrov,V.N.Bogomolov,I.I.Kalosha,andS.V.Ga-
(Kluwer, Dordrecht,1996), Vol. E324. ponenko, Phys.Rev. Lett.81, 77 (1998).
[3] M. Pelton and Y. Yamamoto, Phys. Rev. A 59, 2418 [13] B. Gayral, J.-M. G´erard, B. Sermage, A. Lemaˆitre, and
(1999). C. Dupuis,Appl.Phys. Lett. 78, 2828 (2001).
[4] J. Vuˇckovi´c, D. Fattal, C. Santori, G.S. Solomon, and [14] M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund,
Y.Yamomoto, Appl.Phys.Lett. 82, 3596 (2003). Science of Fullerens and Carbon Nanotubes (Academic
[5] H.T. Dung, L. Kn¨oll, and D.-G. Welsch, Phys. Rev. A Press, New York,1996).
62, 053804 (2000) [and refs. therein]. [15] H. Dai, Surf. Sci. 500, 218 (2002).
[6] L. Kn¨oll, S. Scheel, and D.-G. Welsch, in: Coherence [16] R.H.Baughman,A.A.Zakhidov,andW.A.deHeer,Sci-
andStatistics ofPhotons andAtoms,editedbyJ.Peˇrina ence 297, 787, 2002.
(Wiley,New York,2001). [17] L. Duclaux, Carbon 40, 1751 (2002).
[7] J.R. Buck and H.J. Kimble, Phys. Rev. A 67, 033806 [18] G.-H. Jeong, A.A. Farajian, R. Hatakeyama, T. Hirata
(2003). et al.,Phys.Rev. B 68, 075410 (2003).
[8] V.V.Klimov and M. Ducloy,e-print: physics/0206048. [19] H. Shimoda, B. Gao, X.P.Tang, A.Kleinhammes et al.,
[9] M.FlorescuandS.John,Phys.Rev.A64,033801(2001). Phys. Rev.Lett. 88, 015502 (2002).
[10] M. Sugawara, Phys. Rev.B 51, 10743 (1995). [20] L. Allen and J. H. Eberly, Optical Resonance and Two-
10
Level Atoms (Wiley, New York,1975). (Dover, New York,1975).
[21] C.Weisbuch,M.Nishioka,A.Ishikawa,andY.Arakawa, [33] L.HenrardandPh.Lambin,J.Phys.B29,5127(1996).
Phys.Rev.Lett. 69, 3314 (1992). [34] I.V. Bondarev, G.Ya.Slepyan, and S.A. Maksimenko,
[22] D. Dini, R. K¨ohler, A. Tredicucci, G. Biasiol, and Phys. Rev.Lett. 89, 115504 (2002).
L. Sorba, Phys.Rev.Lett. 90, 116401 (2003). [35] A.S. Davydov, Quantum Mechanics (NEO, Ann Arbor,
[23] G.Ya.Slepyan,S.A.Maksimenko,A.Lakhtakia,O.Yev- MI, 1967).
tushenko, and A.V. Gusakov, Phys. Rev. B 60, 17136 [36] I.V.Bondarev,S.A.Maksimenko,G.Ya.Slepyan,I.L.Kre-
(1999). stnikov, and A. Hoffmann, Phys. Rev. B 68, 073310
[24] S. Tasaki, K. Maekawa, and T. Yamabe, Phys. Rev. B (2003).
57, 9301 (1998). [37] J.M. Raimond, M. Brune, and S. Haroche, Rev. Mod.
[25] A.Jorio,A.G.SouzaFilho,V.W.Brar,A.K.Swanetal., Phys. 73, 565 (2001).
Phys.Rev.B 65, 121402R (2002). [38] A.Zrenner,E.Beham,S.Stufler,F.Findeis,M.Bichler,
[26] Z.M.Li,Z.K.Tang,H.J.Liu,N.Wangetal.,Phys.Rev. and G. Abstreiter, Nature418, 612 (2002).
Lett.87, 127401 (2001). [39] X. Li, Y. Wu, D. Steel, D. Gammon et al., Science 301,
[27] A.G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, 809 (2003).
Phys.Rev.Lett. 91, 046402 (2003). [40] G.S.AgarwalandS.DuttaGupta,Phys.Rev.A57,667
[28] W. Vogel, D.-G. Welsch and S. Wallentowitz, Quantum (1998).
Optics: an Introduction (Wiley-VCH,New York,2001). [41] B.J.DaltonandB.M.Garraway,Phys.Rev.A68,033809
[29] P.R.Wallace, Phys.Rev. 71, 622 (1947). (2003).
[30] S.M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. [42] J.D. Jackson, Classical Electrodynamics (Wiley, New
Lett.68, 3698 (1992). York, 1975).
[31] G.S. Agarwal, Phys. Rev.A 42, 1475 (1975). [43] Handbook of Mathematical Functions, eds. M. Abramo-
[32] A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, vitz and I.A. Stegun (Dover,New York,1972).
Methods of Quantum Field Theory in Statistical Physics