Spontaneous decay of an excited atom placed near a rectangular plate

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by  Tuan Anh Nguyen

Spontaneous decay of an excited atom placed near a rectangular plate
Tuan Anh Nguyen and Ho Trung Dung
Institute of Physics, Academy of Sciences and Technology,
1 Mac Dinh Chi Street, District 1, Ho Chi Minh city, Vietnam
(Dated: Jan. 19, 2007)
UsingtheBornexpansionoftheGreentensor,weconsiderthespontaneousdecayrateofanexcited
atomplacedinthevicinityofarectangularplate. Wediscussthelimitationsofthecommonlyused
simplifying assumption that the plate extends to infinity in the lateral directions and examine the
effectsoftheatomicdipolemomentorientation, atomicposition, andplateboundaryandthickness
on the atomic decay rate. In particular, it is shown that in the boundary region, the spontaneous
decay rate can bestrongly modified.
7 PACSnumbers: 42.60.Da,32.80.-t,42.50.Nn,42.50.Pq
0
0
2 The ability to control the spontaneous decay process the Green tensor by writing it in terms of a Born se-
holds the key to powerful applications in micro- and ries and restrict ourselves to leading-order terms. The
n
nano-opticaldevices. Effectivecontrolcanbeachievedby boundary conditions enter the theory only via the inte-
a
J tailoring the environment surrounding the emitters. In gral limits. This approach is universal in the sense that
theoreticalanalysisofsurroundingenvironmentofdiffer- it works for an arbitrary geometry of the surrounding
9
1 entgeometries,themostinterestingonesbeingoftheres- media, and can be used to evaluate any characteristics
onatortype, the boundaryconditionsaretypicallytaken of the matter-electromagnetic field interaction express-

1 into account only in directions in which the electromag- ible in terms of the Green tensor. In this paper, we are
v netic field is confined or affected the most. For example, concerned mostly with the spontaneous decay rate of an
7
in a planar configuration, only the boundary conditions excited atom placed near a rectangular plate. Our aim
3
inthenormaldirectionaretakenintoaccountwhilethose is twofold: first, we compare our results with those for
1
inthe lateraldirectionsareneglected(see,e.g.,Ref. [1]). an infinitely extended plate in order to establish in a
1
0 Inacylindricalconfigurationthatextendstoinfinity,the quantitativewaytheconditionsunderwhichtheapprox-
7 reverse is true [2]. Under appropriate conditions, these imation of an infinitely extended plate is valid; second,
0 approximationsaregenerallyvalid. However,asthesizes we examine the effects brought about by the presence of
h/ ofdevicesdecreaseandfallinthemicro-andnano-meter the boundaries in the lateral directions.
p ranges as in the current trend of miniaturization, it is The (classical)Greentensor ofanarbitrarydispersing
- clearly of great importance to keep track of the effects
t and absorbing body satisfies the equation
n of all boundaries. One way to calculate the spontaneous
a decay rate in an arbitrary geometry is to directly solve
u theMaxwellequationsusingthefinitedifferencetimedo- HˆG(r,r′,ω)=δ(r−r′)I, (1)
q
v: mnuaminermiceatlhcoodm[p3u].taTtihoins,misentohtodw,itwhhoiucthwreealikenseesnsteisr.elIytroen- Hˆ(r)≡∇× 1 ∇×−ω2 ε(r,ω), (2)
µ(r,ω) c2
i quires that the whole computationaldomain be gridded,
X
leading to very large computational domains in cases of
ar extendedgeometries,orincaseswherethefieldvaluesat (I-unit tensor) together with the boundary condition
some distance are required. All curved surfaces must be at infinity, where ε(r,ω) [µ(r,ω)] is the frequency-
modelledbyastair-stepapproximation,whichcanintro- and space-dependent complex permittivity (permeabil-
duceerrorsintheresults. Additionally,thediscretization ity) which obeys the Kramers-Kronigrelations.
intimemaybeasourceoferrorsinthelongitudinalfield
Decomposing the permittivity and permeability as
[3].
Here we employ an approach that, in a sense, com-
ε(r,ω)=ε¯(r,ω)+χ (r,ω), µ(r,ω)=µ¯(r,ω)+χ (r,ω),
bines analytical and numerical calculations, thereby sig- ε µ
(3)
nificantlyreducingthe numericalcomputationworkload.
andassumingthatthesolutionG¯(r,r′,ω)totheequation
This approach relies on first writing the atomic decay
rate in terms of the Green tensor characterizing the sur- Hˆ¯(r)G¯(r,r′,ω)=δ(r−r′)I,whereHˆ¯ isdefinedasinEq.
rounding media [4, 5]. Although this formula holds for (2) with ε¯instead of ε and µ¯ instead of µ, is known, the
arbitrary boundary conditions, exact analytical evalua- Green tensor G can be written as
tion of the Green tensors for realistic, finite-size systems
can be very cumbersome or even prohibitive. Following G(r,r′,ω)=G¯(r,r′,ω)+G′(r,r′,ω). (4)
[6]and[7],wheretheatom-bodyvanderWaalsforceand
the local-field correction, respectively, have been consid-
ered, we circumvent the task of an exact calculation of Substituting Eq. (4) into Eq. (1) and using the identity
2
(µ¯+χµ)−1 =µ¯−1 ∞l=0(χµ/µ¯)l, it can be found that z
P
Hˆ(r)G′(r,r′,ω)=Hˆχ(r)G¯(r,r′,ω)≡G˜(r,r′,ω) (5) r
1 ∞ χl(r,ω) ω2 A
Hˆχ(r)≡−∇× µ¯(r,ω) µ¯µl(r,ω)∇×+c2χε(r,ω), dx dz
l=1
X
(6) y
i.e., G′ satisfies the same differential equation as the
one governing the electric field, with the current be-
d
ing equal to G˜. Hence it can be written as a convo- x y
lution of this current with the kernel G: G′(r,r′,ω) =
d3sG(r,s,ω)G˜(s,r′,ω).SubstitutionofG′inthisform FIG. 1: A dipole emitter in the vicinity of a rectangular
in Eq. (4) and iteration lead to the desired Born series plate.
R
∞
G(r,r′,ω)=G¯(r,r′,ω)+ G (r,r′,ω), (7) Green tensor
k
Xk=1 k2
Gk(r,r′,ω)= k d3sj G1(r,r,ω)= 16π2 Z d3sHˆχ1(s)
! ×[a2I +(b2−2ab)uˆ⊗uˆ]e2iq (12)
j=1Z
Y
×G¯(r,s1,ω)G˜(s1,s2,ω)···G˜(sk,r′,ω). (8) [u≡r−s; q = ku; a=a(q); b=b(q); Hˆ (r) ≡ −∇×
χ1
χ (r,ω)∇×+ω2χ (r,ω) – linear part of Hˆ , Eq. (6)].
This formalexpansionforthe Greentensoris validfor µ c2 ε χ
Higher-orderterms can easily be obtained by repeatedly
an arbitrary geometry, permittivity, and permeability of
using Eq. (10) in Eq. (8).
the macroscopic bodies. Obviously, one of the situations
Our system consists of an excited two-level atom sur-
in which the Born series is particularly useful is when
rounded by macroscopic media, which can be absorbing
χ is a perturbation to ε¯and χ is a perturbation to µ¯,
ε µ and dispersing. In the electric-dipole and rotating-wave
thereby one makes only a small error cutting off higher-
approximations,the atomic decay rate reads as [4, 5]
order terms. For such weakly dielectric and magnetic
bodies, it is natural to choose 2k2
Γ= A d ImG(r ,r ,ω )d , (13)
~ε A A A A A
ε¯(r,ω)=µ¯(r,ω)=1 (9) 0
whered andω arethe atomicdipoleandshiftedtran-
A A
with χλ(r,ω)=χλR(r,ω)+iχλI(r,ω), |χλ(r,ω)| ≪ 1 sition frequency, kA = ωA/c, and G is the Green tensor
(λ = ε, µ) [cf. Eqs. (3)]. This implies we restrict our- describing the surrounding media.
selves to frequencies far from a medium resonance. In accordance with the linear Born expansion, Eqs.
The Green tensor G¯ corresponding to ε¯(r,ω) = (7), (10), and (12) yield, for a purely electric material,
µ¯(r,ω)=1 is the vacuum Green tensor
Γk(⊥) 3k3
=1+ AIm d3sχ (s,ω )
G¯(r,r′,ω)=−δ(u)I + k (aI −buˆ⊗uˆ)eiq, (10) Γ0 8π (Z ε A
3k2 4π
1 (x−x )2
× a2+(b2−2ab) A e2iq (14)
1 i 1 1 3i 3
u2 (z−z )2
a≡a(q)= + − , b≡b(q)= + − (cid:20) A (cid:21) )
q q2 q3 q q2 q3
(11) [Γ =k3d2/(3π~ε )-free-spacedecayrate,s=(x,y,z),
0 A A 0
u = |s − r |, q = k u, a = a(q), b = b(q)] for
A A
(k= ω/c; u≡r−r′; uˆ =u/u, and q≡ ku). x-(z-)oriented dipole moments. Equations (14) are our
A description of quantities such as the emission pat- mainworkingequations. Justlike the Bornexpansionof
tern or the interatomic van der Waals forces requires the Green tensor,they hold for an arbitrarygeometryof
the knowledge of the Green tensor of different positions, the surrounding environment.
while adescriptionofquantities suchasthe spontaneous Next let us be specific about the shape of the macro-
decay rate of an excited atom or the atom-body van der scopicbodies. We considerarectangularplate ofdimen-
Waals forces requires the knowledge of the Green tensor sions d , d , and d and choose a Cartesian coordinate
x y z
of equal positions. Substituting Eq. (10) in Eq. (8) and system such that its origin is located at the center of
assuming that the position r lies outside the region oc- one surface of the plate, as sketched in Fig. 1. Then
cupiedbythemacroscopicbodies,wederiveforthefirst- Γk representsthespontaneousdecayrateofadipole mo-
order term in the Born expansion of the equal-position mentparalleltoaplatesurfaceandΓ⊥ –thatofadipole
3
1.04 χχεεRR==00..51 (a) 1.01 (a)
0
||ΓΓ/0 1 ||ΓΓ/ 1
0.99
0.96
0 0.5 1 1.5 0 0.5 1 1.5
1.4
(b) 1.06 (b)
0
⊥ΓΓ/ 1.2 χεR=0.5 ⊥Γ/0 1.03
χεR=0.1 Γ
1 1
0 0.5 1 1.5 0 0.5 1 1.5
z /λ z /λ
A A A A
FIG. 2: Atom-surface distance dependence of the normal- FIG. 3: The same as in Fig. 2 but for different sizes of
izedspontaneousdecayrateofanexcitedatom positioned at the rectangular plate: dx = dy = 3λA (dashed line), 0.4λA
(0,0,zA)nearaninfiniteplanarplate(solidline)andarectan- (dotted line), and 0.2λA (dash-dotted line). In the last case,
gular plate (dx =dy =10λA, dashed line) of equal thickness theplateisactually acube. Thecurvesforaninfiniteplanar
dz = 0.2λA and equal χε = χεR +i10−8. Case (a) is for a plate are shown by solid line and χε=0.1+i10−8.
x-oriented dipole moment, while case (b) is for a z-oriented
dipole moment.
1.02 Γ⊥/Γ
momentnormaltothesamesurface. Sincenofurtheran- 0 1.0006
alyticalcalculationinEqs. (14)seemspossible,weresort 1
to numerical computation. 1.01 0.9994 Γ||/Γ0
0 6 12
In Fig. 2 we present the spontaneous decay rate in
1
accordance with the linear Born expansion (14), as a Γ||/Γ
function of the atom-surface distance for two different 0
0 3 6
values of the permittivity. The same quantity but for d /λ
z A
anatomplacednearaninfinitely extendedplanarslabis
plotted using the Green tensor given in Ref. [8]. It can
FIG. 4: Plate-thickness dependence of the spontaneous de-
be seen that when the lateral dimensions of the rectan-
cayrateofanexcitedatomlocatednearaninfinitelyextended
gular plate are sufficiently large and the absolute value
ooff tχhe=pe0rm.1it+tivi1it0y−8isisnuffithceiefingtluyrec)l,ostehetotwonoere(tshueltscaasle- p10laλnAa,rdpalsahteed(lsionleid)olifneeq)uaanldχεa=re0c.t1an+giu1l0a−r8p.laTthee(adtxom=icdypo=-
ε sitionisfixedat(0,0,0.2λA)inthemainfigure,and(0,0,5λA)
most coincide for both dipole moment orientations. As in the inset.
χ increases, the agreement worsens but is still quite
εR
good at χ = 0.5. Further numerical calculations indi-
εR
<
catethatintherangeof|χε(ωA)|∼0.5,thespontaneous and compare the resulting spontaneous decay rates with
decay rate can be well approximated by the linear Born that for an infinite slab. To be on the conservative side,
expansion. InFig. 2,theatomhasbeenmovedalongthe the permittivity is set equal to ε(ω ) = 1.1+i10−8 – a
A
z-axiswithxA =yA =0. Whentheatomismovedalong value which is very close to one (cf. Fig. 2). For lengths
otherlinesparalleltothez-axisbutnearertotheborder, of the lateral sides comparable to the atomic transition
the zA-dependence of the normalized spontaneous decay wavelength,the infinite slabapproximationstarts to dif-
rates behaves in a similar way as in Fig. 2 but with its fer noticeably from the linear Born expansion (see the
value being generally closer to one. Having determined figure, case of d = d = 3λ ), which in this situation
x y A
the range of permittivities where the linear Born expan- is regarded as a good and nondegrading approximation.
sion provides a good approximation to the spontaneous As the lateral sizes of the plate decrease further and be-
decayrate,wemoveonnowtoinvestigatewhentheplate come smaller than λ , the infinite-slab approximation
A
can be regarded as an infinite slab. fails completely (see cases of d =d =0.4λ and0.2λ
x y A A
In Fig. 3, we gradually reduce the lateral sizes of the in the figure). In other words, while a rectangular plate
rectangular plate while keeping its thickness constant, with lateral sizes much larger than the atomic transi-
4
1.6 method, one obtains from Eq. (14)
Γ⊥/Γ
1.4 0 Γk 3k3 dz
≃1+ AIm χ dza2(q )e2iqz
ε z
Γ 2π
1.2 0 " Z0
1 Γ||/Γ0 × d2x dx d2y dyeikqA2z(x2+y2) (15)
Z0 Z0 #
0.8
4 5 6 [q =k (z+z )]. Theintegralsoverxandy inEq. (15)
x /λ z A A
A A containFresnelintegrals. In the limit of aninfinite plate
d , d →∞, Eq. (15) becomes
x y
FIG. 5: Effects of the presence of a boundary in the x-
direction on the spontaneous decay rate of an excited atom Γk 3k dz
located near a rectangular plate of permittivity ε(ωA) = ≃1+ AIm χε dza2(qz)qze2iqz(1+i)2 .
1.5+i10−8 and dimensions dz =0.2λA and dx =dy =10λA. Γ0 16 " Z0 #
The atom is located at (xA,0, 0.01λA). (16)
As a function of z/λ , the integrand in Eq. (16) has a
A
tionwavelengthcanbemoreorlesstreatedasaninfinite period of 1. Since this period is z-independent, the re-
2
slab, care should be taken when the sizes are reduced sulting integral must exhibit oscillations with the same
to about or below a wavelength. This happens for both period, asconfirmedby Fig. 4, solidcurves. These oscil-
normal and parallel to the surface dipole moment ori- lations survivefor plates of finite lateralextensions (Fig.
entations. When the rectangular plate can be roughly 4, dashed curves). The beating clearly arises from the
regarded as an infinite slab, the infinite-slab curve and finite values of d and d . Note that as a function of z,
x y
the linear Born expansion curve agree better when the the inner integrands in Eq. (15) have a ‘period’ that is
atomisplacedclosertothesurface(seedashedandsolid z- dependent and that increases with increasing z.
curves in the figure). This can be explained by that the Nextweturntoelucidatingtheinfluenceofthebound-
closer to the surface the atom is situated, the more it is ariesinthex-andy-directionsonthespontaneousdecay
inclined to see the plate as infinite. rates. As can be seen from Fig. 5, where an edge is
Besides the dependence on the lateral sizes, whether present at x =5λ , the decay rates exhibit oscillations
A A
a rectangular plate can be treated as an infinitely ex- near the boundary with a particularly strong magnitude
tended one clearly depends on its thickness as well. It righton either side of it, and damping tails. The oscilla-
is intuitively obvious that even when plate lateral sizes tions are more pronouncedfor a dipole moment oriented
are much larger than the atomic transition wavelength, parallel to the (xy)-plane than for a z-oriented dipole
the plate cannot be treated as extending to infinity if its moment. Onecannoticethatwhenthe projectionofthe
thickness is comparable with the lateral sizes. In Fig. 4, atomic position on the xy-plane lies outside and suffi-
we compare the dz-dependence of the spontaneous de- ciently far from the boundaries, the spontaneous decay
cay rate for a rectangular plate with that for an infinite rate approaches one in free space, as it should.
slab. The agreementbetween the two curves, being very In summary, using the Born expansion of the Green
good for sufficiently thin plates, gradually worsens with tensor, we have consideredthe decay rate of an atom lo-
an increasing plate thickness. The disagreement is al- cated near a plate of rectangular shape. We have shown
readynoticeableatdz ∼λA –avalue whichis stillmuch thatarectangularplatecanbetreatedasextendingtoin-
smaller than the lateral sizes dx = dy = 10λA, and it finityonlywhenitslateralsizesaremuchlargerthanthe
sets in earlier for Γ⊥ than for Γk. The two calculations atomic transition wavelength, its thickness sufficiently
predictquite differentlarge-thicknesslimits. This means small, and the atom is located close enough to the plate
that in the case of thick plates, one must take into ac- surface. We have also shown that a boundary in the lat-
count the boundary conditions in the lateral directions eral directions can give rise to significant modifications
in order to obtain reliable results. ofthedecayrateineithersideofit. Thefirst-orderBorn
Account of the boundary conditions in the lateral di- expansion remains quite reliable even at a value of per-
rections also gives rise to some curious beating in the mittivity as high as 1.5. Inclusion of higher-order terms
d -dependence of the spontaneous decay rate, which is would allow one to investigate more dense media.
z
especially visible when the atom has a dipole moment H.T.D. thanks S. Y. Buhmann and D.-G. Welsch for
orientedparalleltothe surfaceandis situatedsomewhat enlighteningdiscussions. WearegratefultoJ.Weinerfor
away from the surface (see Fig. 4, inset). Let’s have a critical reading of the manuscript. This work has been
a closer look at, say, the decay rate for a dipole ori- supported by the Ho Chi Minh city National University
ented parallelto the surface. Using the stationary phase andtheNationalProgramforBasicResearchofVietnam.
5
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