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The influence on atomic decay

Eur.Phys.J.D41,403–411(2007) DOI:10.1140/epjd/e2006-00228-6T HE E UROPEAN

P HYSICAL J OURNAL D

The in?uence on atomic decay by inserting a LHM layer

into an ordinary one dimensional multi-layer structure

J.P.Xu1,2,a,Y.P.Yang1,N.H.Liu2,3,and S.Y.Zhu2,4

1Department of Physics,Tongji University,Shanghai200092,P.R.China

2Department of Physics,Hong Kong Baptist University,Kowloon Tong,Hong Kong

3Department of Physics,Nanchang University,Nanchang330047,P.R.China

4Key Laboratory of Quantum Communication and Quantum Computation,University of Science and Technology of China, Hefei230026,P.R.China

Received27February2006/Received in?nal form5June2006

Published online13October2006–c EDP Sciences,Societ`a Italiana di Fisica,Springer-Verlag2006

Abstract.Inserting left-handed material(LHM)layers into a one dimensional structure can in?uence the

spontaneous emission(SpE)of a two-level atom.This has been investigated,starting from the simplest

case of a three-layer system,where we?nd the re?ected?eld(atom can“see”)passing through LHM

layer is stronger than that through the corresponding normal layer.Indeed the induced decay is more

strongly in?uenced by re?ected?eld passing through LHM layer.Based on this and after further analysis

of re?ectivity,we?nd that,a quarter photonic crystal(PC)composed of alternately LHM and RHM can

inhibit the atomic spontaneous emission more intensely compared to an ordinary PC.

PACS.42.25.Bs Wave propagation,transmission and absorption–42.50.Lc Quantum?uctuations,quan-

tum noise,and quantum jumps–42.70.Qs Photonic bandgap materials

1Introduction

Recently a new type of material called left-handed mate-rials(LHM)[1]has attracted considerable attention.In 1968,Veselago[1]?rst suggested the concept of LHM, which refers to a material processing a negative refractive index with the permittivity and the permeability being negative simultaneously.The wave vector is anti-parallel to the direction of energy?ow,and the electric?eld,mag-netic?eld and the wave vector form a left-handed triplet. Some unusual phenomena,such as reverse Doppler shift, reverse Cerenkov radiation,negative refraction,reverse light pressure et al.are expected in LHM.Experimentally, LHM had been realized in the microwave band[2–4].As the result,its most potential application of the LHM,the perfect lens which can focus both propagation waves and the evanescent waves,has been predicted[5].In addition, Li et al.pointed out that a stack composed of alternating LHM layers and normal dielectric(RHM)layers can give rise to a new type of band gap with novel properties[6].

In1946,Purcell[7]pointed out that the atomic spon-taneous decay rate can be enhanced by putting the atom in a cavity.Since then,how to inhibit or enhance atomic SpE has became an interesting subject.Most research has focused on the inhibition of the atomic SpE,because it a e-mail:xx jj pp1980@https://www.sodocs.net/doc/628005268.html, prolongs the atomic coherent time and is useful in quan-tum computation and quantum information applications. The e?ective way to realize this is to put the atom into a micro-cavity or a PC[8–10],because the density of states in these environments can be controlled easily by vary-ing the dimension and other parameters of the structure. If we introduce LHM layers into a multi-layer structure, what are the in?uences of the LHM layers on atomic de-cay comparing with the in?uence of the RHM layer.Here we de?ne the PC containing alternate the LHM and the RHM layers as LHM-RHM PC,and the PC containing only the RHM layers as ordinary PC.After analyzing the re?ected?eld emitted by the atom from di?erent inter-faces in detail,we?nd that LHM layers have a stronger in?uence on the atomic decay.

This paper is organized as follows:in Section2,we in-troduce the model and the interaction Hamiltonian.The di?erent in?uences of LHM from RHM in a three-layer structure on the atomic decay have been analyzed in Sec-tion3.The case in a PC is presented in Section4.Finally we give the conclusion in Section5.

2Model and interaction Hamiltonian

The1D LHM-RHM PC without dispersion and dissipa-tion is shown in Figure1.A two-level atom(transition

404The European Physical Journal

Fig.1.Sketch of the LHM-RHM1DPC structure.

frequencyω0,position r a(0,0,z a)and dipole moment p)is

placed in the middle layer(layer0).In order to have a com-

parison between the LHM and the RHM,absolute values

of the permittivity and the permeability for the RHM lay-

ers and for the LHM layers are set be equal.The center of

the structure is the origin of the z-coordinate.Each layer

is isotropic and in the x-y plane.Though most LHM are

anisotropic in experiments[2,3],however isotropic LHM

have been realized recently[4].Other parameters of the

structure are shown in Figure1.Note that the middle

layer is set to be vacuum in order to be consistent with

the real-cavity model[11–13].

Now we introduce the quantization of the three dimen-

sional electromagnetic?eld in the presence of the1DPC.

The positive frequency part of the electric?eld operator

for our system can be written as

E(+)(r,t)=

K+,λU(K+,λ,r)?e+ξm K a K

+λ

e?iνK t

K?,λU(K?,λ,r)?e?ξm K a K

?λ

e?iνK t(1)

whereνK is the frequency of K±andξm

K =

( νK/2|εm|V)1/2with m determined by the position r andεm be the permittivity of the m th layer.U(K+,r), U(K?,r)are the modi?ed mode function in the presence of the1D structure.According to the quantization scheme in reference[14],the mode functions together with the ?eld unit vectors(?e+,?e?)is expressed as the following piecewise functions,which

U(K+,λ,r)?e+=

?????????????????????[e i K m+·r?e m(K+,λ)

+Rλ

e i K m?·r+2iK z z?N?1?e m(K?,λ)],z

tλ

e iK z z?N?1[e i K m+·r?iK mz z m?1?e m(K+,λ)

+rλ

e i K m?·r+iK mz(z m?1+2d m)?e m(K?,λ)]/Dλm,

z m?1≤z

Tλ

e i K m+·r+iK z(z?N?1?z N)?e m(K+,λ),z≥z N

(2)

and

U(K?,λ,r)?e?=

???

[e i K m?·r?e m(K?,λ)+RλL e i K m+·r?2iK z z N?e m(K+,λ)],

z≥z N

tλ

e?iK z z N[e i K m?·r+iK mz z m?e m(K?,λ)

+rλ

e i K m+·r+iK mz(2d m?z m)?e m(K+,λ)]/Dλm,

z m?1≤z

Tλ

e i K m?·r+iK z(z?N?1?z N)?e m(K?,λ),z

(3)

where

K m±=(K x,K y,±K mz)

=K(sinθcosφ,sinθsinφ,±n m cosθm).(4)

Here the angleθm is determined by the angleθin vacuum

according to Snell’s law

sinθ=n m sinθm.(5)

The superscriptλ=TE,TM indicates two transverse

polarization directions,whose unit vectors are de?ned as

?e m(K±,TE)=(sinφ,?cosφ,0),

?e m(K±,TM)=(cosθm cosφ,cosθm sinφ,?sinθm).

(6)

In the expressions of the mode functions(2)and(3),

tλ

L/Rm

denotes the transmission coe?cient through the

left/right part of the m th layer(z m?1

R/Lm

denotes the re?ective coe?cient on the right/left interface

of the m th layer.Tλ

R/L

denotes the total transmission co-

e?cient of the entire structure coming from the right/left

interface of the region of zz N.Rλ

R/L

de-

notes the total re?ective coe?cient on the right/left inter-

face of the region of zz N.Dλm originates

from the multi-re?ection e?ect in the m th layer,

Dλm=1?rλLm rλRm e2iK mz d m.(7)

So the interaction Hamiltonian in the interaction picture is

V I(t)=

K+,λ

[gλK

(r a)σ+a K

+λ

e i(ω0?νK)t+H.C.]

K?,λ

[gλK

(r a)σ+a K

?λ

e i(ω0?νK)t+H.C.](8)

where gλK

(r a)are the atom-?eld coupling coe?cients,

gλK

(r a)=?

ξ0

/Dλ0)e i K+·r a+iK(z?N?1+d0/2)cosθ

×[p·?e0(K+,λ)+p·?e0(K?,λ)rλR0e?iK(2z a?d0)cosθ],

(9)

gλK

(r a)=?

ξ0

/Dλ0)e i K?·r a+iK(d0/2?z N)cosθ

×[p·?e0(K?,λ)+p·?e0(K+,λ)rλL0e iK(2z a+d0)cosθ].

(10)

J.P.Xu et al.:The in?uence on atomic decay by inserting a LHM layer 405

Γ||Γ0=38n 0μ0π/2 0

dθsin θ t T E L 0D T E 0 2 1+r T E R 0e in 0k 0(d 0?2z a )cos θ 2+ t T M L 0D T M 0

2cos 2θ 1+r T M R 0e in 0k 0(d 0?2z a )cos θ 2+ t T E R 0D T E 0 2 1+r T E L 0e in 0k 0(d 0+2z a )cos θ 2+cos 2θ t T M R 0D T M 0

2 1+r T M L 0e in 0k 0(d 0+2z a )cos θ 2 ,(17a)Γ⊥Γ0=34n 0μ0π/2 0

dθsin 3θ t T M L 0D T M 0 2 1?r T M R 0e in 0k 0(d 0?2z a )cos θ 2+ t T M R 0D T M 0

2 1?r T M L 0e in 0k 0(d 0+2z a )cos θ 2 (17b)

The state vector of the system is |ψI (t ) =C a (t )|a,0 +

K +,λ

C b K +λ(t ) b,1K +λ

K ?,λ

C b K ?λ(t ) b,1K ?λ

.(11)

The state |a,0 refers to the atom in the excited state with no photon,and b,1K ±λ refers to the atom in the ground state with one photon in the mode of (K ±,λ).We assume initially,C a (0)=1and C b K ±λ(0)=0.Solving the Schr¨o dinger equation in the interaction picture

?t |ψI (t ) =?i

V I |ψI (t ) (12)

with a standard deduction,the equation of atomic upper level probability amplitude can be obtained [15]

˙C

a (t )=??

K +,λ

|g λ

K +

(r a )|2+

K ?,λ

|g λ

K ?

(r a )|2?

?×

t 0

dt e i (ω0?νK )(t ?t

)C a (t ).(13)

Now transforming the summation over K +and K ?into

an integral,which gives

K ±

→

V (2π)3

d K ±=

V (2π)3

∞

dK π/2 0

dθ2π 0

dφK 2sin θ.

(14)

Equation (13)becomes ˙C

a (t )=?V (2π)3

dt C a (t )

∞

dKK 2e ic (k 0?K )(t ?t

)

2π

dφπ/2 0

dθsin θT M

λ=T E

[|g λK +(r a )|2+|g λ

K ?

(r a )|2],(15)where k 0=ω0/c and K =νK /c .

With the Weissikopf-Wigner approximation [15](t 1/ω0),equation (15)can be reduced to

˙C

a (t )=?Γ2

C a (t )(16)

where Γis the steady decay rate and depends on the

atomic polarization and position.If the atomic dipole is along the x -axis,p =p (1,0,0),we have Γ=Γ||

see equation (17a)above

where Γ0is the decay rate in the free space (the vacuum).If the atomic dipole is perpendicular to the interfaces of the layers,i.e.p =p (0,0,1),we have Γ=Γ⊥

see equation (17b)above.

It is well-known that atomic spontaneous emission is re-lated to its environment.In current case,the environment is represented by the re?ection and transmission coe?-cients.

3Three-layer case

In order to see the di?erence on the atomic decay in the ordinary PC and in the LHM-RHM PC,we consider the symmetric three-layer structure (0|A |0|A |0)?rst.Here “|”indicates the interface between two materials.The “0”refers to the vacuum layers,and the “A”refers to the lay-ers with thickness d A and refractive index n A .The atom is at the center of the middle layer z a =0and the thickness of it is d 0.

In the following we consider the special LHM and RHM for layer A,the absolute refractive index equals to 1(εA =1/μA and εA =1).For the atomic dipole along the x -axis,p =p (1,0,0),the steady decay rate can be decomposed as follows under the condition of t 1/ω0

406The European Physical Journal D

Γ≈Γ0

(a)

+32Γ0Re 2 m =1

(?r 0)m e imk 0d 0 2

imk 0d 0+2(mk 0d 0)2?2i (mk 0d 0)3

(b)+32Γ0Re 2 m =1

(t 20r 0)m e im (2n A k 0d A +k 0d 0)

2im (2n A k 0d A +kd 0)+2m 2(2n A k 0d A +kd 0)2?2im 3(2n A k 0d A +k 0d 0)3 (c)+32Γ0Re 2 m =1

(?2t 20r 20)m e im (2n A k 0d A +2k 0d 0)

2i (2n A k 0d A +2k 0d 0)+2(2n A k 0d A +2k 0d 0)2?2i (2n A k 0d A +2k 0d 0)3 (d)

+...

(18)

Fig.2.The path of each term in equation (18).

(see the Appendix)

see equation (18)above

where r 0is the Fresnel re?ectivity incident from layer A to vacuum for the TE wave,t 20is expressed in equation (A.8)(detailed deductions are shown in the Appendix).Here the Lamb shift is neglected.Each term in equation (18)rep-resents the contribution of the re?ected ?elds back to the atom with the di?erent paths.The terms (b),(c)and (d)are shown in Figure 2.Term (a)describes the free decay which is immune to the surroundings;term (b)describes the contribution of the ?eld re?ected at the nearest inter-faces;term (c)represents the contribution re?ected at the outside interfaces;term (d)the contribution re?ected from one nearest interface and its opposing outer interface.From equation (18),the induced decay rate (terms (b),(c)and (d))caused by the re?ected ?eld is mainly in-versely proportional to the phase shift.Furthermore,x ?1,x ?2and x ?3in each term correspond to the dipole ra-diation ?eld,the induced ?eld and the electrostatic ?eld respectively.In the LHM,the phase shift decreases with wave propagation.On the other hand,the phase shift is always increasing in RHM.Finally the re?ected ?eld passing through LHM layer is stronger in amplitude than that through RHM layers,which leads to a more intense change in the decay rate.As the re?ectivity and transmit-ted coe?cients (r 0,t 20)at the interface are the same for both the LHM and RHM cases,from equation (18),the only di?erence between LHM and RHM is the phase shift (mn A k 0d A +lk 0d 0,m and l are arbitrary integrals).The e?ect of the RHM or the LHM layer A on SpE can be summarized as follows.

Firstly,terms (b),(c)and (d)will be small enough that they can be omitted as d 0tends to in?nity,and consequently only term (a)will contribute to the decay,which is just as in free space,Γ≈Γ0.Secondly,term (b)is identical for the LHM and the RHM,because it don’t include the contribution of the re?ected ?eld pass-ing through layer A.Thirdly,term (c)is smaller than (b)for RHM case,because the denominator of (c)is larger than (b).However,for LHM layer,i.e.n A <0,the de-nominator |m (2n A k 0d A +n 0k 0d 0)|can be smaller than |mk 0d 0|,in which case the contribution of term (c)will be larger than that of term (b).Consequently,the re?ected ?eld through the LHM layer is stronger than that through the RHM layer,which leads to stronger e?ect on the decay rate.Finally,terms (c)and (d)can be neglected when d A is large enough,and so the di?erence between LHM and RHM will disappear.That means layer 0is connected to two half-in?nite layers.

There also exist many other contributions of the re-?ected ?eld along other paths.They must be much weaker than terms (b),(c)and (d)for the RHM case (due to |r 0|<1and a larger phase shift).However,for the LHM case,they may be equal to or even larger than terms (b),(c)and (d)due to the phase compensation e?ect of the LHM layer.We neglect them here in order to be concision.Now we perform the numerical calculations according to equation (17a)to con?rm the deductions made https://www.sodocs.net/doc/628005268.html,yers A with thickness d A can be LHM (εA =?0.25and μA =?4)or RHM (εA =+0.25and μA =+4).From equation (18),we know that the thickness of layer A plays the dominant role in distinguishing between the e?ect of LHM and RHM.So we plot the SpE rate as function of d A according to equation (17a)in Figure 3with ?xed d 0.From Figure 3,the normalized decay rate oscillates with d A .The oscillation is caused by the periodically vari-ation of the re?ectivity with increasing d A .In other words,the density of states changes periodically with d A .

Because the intensity of the re?ected ?eld passing through LHM layer back to atom is stronger than that passing through RHM,the superposition of the re?ected ?eld and the emitted ?eld in LHM case can have stronger constructive or deconstructive interference.In Figure 3,

J.P.Xu et al.:The in?uence on atomic decay by inserting a LHM layer

407

Fig.3.The normalized decay rate as function of d A .Solid line for the LHM layer A,dashed line for the RHM layer

Fig.4.The decay rate as function of d A ,(a)for d 0=0.75λ0,(b)for d 0=0.5λ0,(c)for d 0=0.25λ0.Solid line for the LHM layer A,dashed line for the RHM layer A.

the stronger deconstruction (deeper inhibition)is pre-sented.The di?erence between the RHM and the LHM layers will disappear as d A 3λ0.

A similar result will be obtained if the index of layer A |n A |=1.For example,we plot the decay rate versus d A in Figure 4for layer A of LHM (εA =?8and μA =?2)or of RHM (εA =+8and μA =+2).The change of the decay rate (the di?erence between the decay rate and the free space decay rate)for LHM (solid line)is larger than that for RHM (dashed line),see Figure 4.The amplitude of the change of SpE rate for LHM increases linearly with d A ,while decreases for RHM.

There is always a focal point of the re?ected ?eld to the left of the LHM with n A =?1according to Snell’s law,see Figure 5a.The position of the focus will leave away from the interface with increasing d A .The intensity of the re?ected ?eld is strongest at the focus.So the change of decay rate is largest when d A ≈d 0,as shown by the solid line in Figure 3.

However,when n A =?1,there is no clear focal point of re?ected ?eld,as shown in Figure 5b.Due to the non-perfect focus of re?ected ?eld in the case of Figure 4,

the

Fig.5.(a)A perfect focusing of the re?ected ?eld as n A =?1,(b)non-perfect focusing of the re?ected ?eld as LHM’s index n A =?1(we omit other re?ected paths including transmission to the right and multi-re?ection).

tendency to increase for LHM is longer than in Figure 3.The decay rate in Figure 4will have the same value for the LHM and the RHM at a much larger d A compared with the case of Figure 3.

4Photonic crystals

The method giving equation (18)only ?ts for the simplest case.For more a complicated case,such as in a PC with an arbitrary refractive index,the decay rate cannot be de-composed into the formation in equation (18),because the integral over an angle is impossible to resolve analytically.However,equation (18)provides us with a clear physical picture to distinguish the di?erent in?uences of the LHM layer from the RHM layer on the atomic decay.We can predict that,for an arbitrary structure,if it contains the LHM layer (but not all),the re?ected ?eld passing through the LHM layer must be larger than that passing through the RHM case.Furthermore,the LHM-RHM PC will have the stronger in?uence on the inhibition or enhancement of the SpE than the ordinary PC.

It is known that the LHM-RHM PC has a near omni-directional gap for the TE mode,while the ordinary PC hasn’t [16].In the following we will consider the quar-ter wavelength PC,the optical length of each layer being a quarter of one wavelength,because the quarter wave-length PC has a high re?ectivity (or gap around centre frequency)in the normal direction.All layer Bs are the vacuum (εB =1.0,μB =1.0)and d B =d 0=λ0/4.For comparison,layer As can be LHM or RHM with the same absolute permittivity and permeability (εA =±2,μA =±0.5,|n A |d A =λ0/4).The total structure has the form (0|A |(|B |A |)40(|A |B |)4|A |0).The re?ectivity as func-tion of frequency and incident angle are plotted in Fig-ure 6.

In Figure 6,the white region implies nearly complete re?ection.The lighter the color,the higher the re?https://www.sodocs.net/doc/628005268.html,paring with Figures 6a and 6b,we ?nd that the LHM-RHM PC has a much wider gap than the ordinary PC not only in frequency but also in incident angle.Near 2ω0,there is a resonant tunneling line in both cases.The di?erence between Figures 6a and 6b can be explained by the energy band theory used in reference [6],and also can

408The European Physical Journal

Fig.6.The re?ective index for TE waves versus frequency and incident angle for(a)RHM PC and(b)for LHM-RHM PC(TE and TM have no di?erence

here).

Fig.7.The SpE rate as a function ofωwhen the atom lies

between two symmetrical PCs((AB)4A).

be understood by the propagation analysis in the previous

section.All the e?ects originate from the phase compen-

sation e?ect of the LHM.It should be pointed out that,

in Figure6b,there is no omni-directional gap because the

absolute refractive indexes are the same for each layer.

The SpE rates of the atom with p=p(1,0,0)as function

of frequency are drawn in Figure7.

In Figure7,the solid line and the dashed line are the

SpE rates for the LHM-RHM PC and the ordinary PC,

respectively.The SpE in the LHM-RHM PC is inhibited

much more profoundly than in the ordinary PC in the fre-

quency region near the gap.This is easy to understand

by comparing Figure6a with Figure6b.High re?ection

means strong localization and fewer channels to propa-

gate out.The LHM-RHM PC has a much wider gap than

the ordinary PC both in frequency and in incident angle,

which leads to a lower density of states within it.

A similar result is also obtained in other quarter wave-

length PC(|n A|d A=n B d B=d0=λ0/4)with the

ab-

Fig.8.The SpE rate as a function ofωwhen the atom lies

between two symmetrical PCs((AB)4A).

solute value of the index not equal to1.Consider the

LHM-RHM PC and the ordinate PC withεA=±2,

μA=±1for the layer A and the vacuum for the layer B

(εB=1.0,μB=1.0).

The SpE rate in the LHM-RHM PC is inhibited much

stronger than in the ordinate PC at the frequency region

near the gap,which is similar to Figure7.

It should be noted that there is an apparent di?erence

between the atomic decay in the F-P cavity and in the

1DPC.Though the1DPC is a natural extension of the F-

P cavity(by replacing the wall of the cavity by the1DPC),

the atomic decay is much more complicated in the1DPC

than in the F-P cavity due to the multiple re?ection in

each layer and the dependence of the re?ectivity in the

1DPC on incident angles(the re?ectivity at the wall of

a cavity is taken to be a constant),which leads to less

inhibition in the PC than in the F-P cavity,see dashed

line in Figure7and8.From Figure6b,we can?nd that

the re?ectivity for the LHM-RHM PC is insensitive to the

J.P.Xu et al.:The in?uence on atomic decay by inserting a LHM layer409 angleθat the frequency region near the gap,similar to

the wall of the cavity,and the atomic steady decay rate is

always inhibited more strongly in the LHM-RHM PC(see

Figs.7and8),which is similar to that in the cavity[14].

5Conclusion

The spontaneous emission of the two-level atom located

in a multi-layer structure containing LHM has been stud-

ied,and the results are compared with that in ordinary

structure.In a three-layer structure,we?nd that the SpE

rate oscillates with d A for both the RHM layer A and the

LHM layer A.The change of the SpE rate for the LHM

case is much larger than that for RHM when d A is small.

Such a di?erence can be explained from the viewpoint of

the re?ected?eld which feed back to the atom.The re-

?ected?eld passing through the LHM layer is stronger

than that passing through the RHM layer.Due to the su-

perposition(interference)of the re?ected?elds from all

paths and the emitted?eld,the structure containing the

LHM has a stronger e?ect on the atomic decay than the

ordinary structure.Such a conclusion can be extended to

the case of the quarter wavelength https://www.sodocs.net/doc/628005268.html,pared with

the case in the ordinary PC,the LHM-RHM PCs have a

higher re?ectivity which results in a strong suppression of

the SpE under the appropriate conditions.

From our analysis,we?nd a new way to enhance or in-

hibit the atomic spontaneous decay without changing the

dimension of the structure and it can be easily expanded

to multi-atom interference.

It is necessary to point out the reason why we have not

considered dispersion and dissipation.Firstly,the disper-

sion information will be lost with the Markov approxima-

tion and dispersion has no in?uence on the steady atomic

decay.Secondly,the main di?erence between the LHM and

the RHM is the opposite phase shift within them.How-

ever,the dissipation only provides a decay channel called

non-radiated decay to the atom[13,17,18],and the non-

radiated decay has no contribution to distinguish the dif-

ferent in?uences between the LHM and RHM.Introducing

dissipation in the model just weaken the in?uence between

the LHM and the RHM on the atomic decay,because only

the propagating radiated?eld can distinguish the LHM

from the RHM.When considering the non-Markov pro-

cess,the dispersion of the LHM has to be taken into ac-

count,which will be done in further research.

This work was supported in part by the National Natural Sci-

ence Fundation of China(No.10674103),RGC from HK Gov-

ernment,FRC from HKBU,the Shanghai Phosphor Tracing

Plan(No.04QMH1407)and the foundation of Shanghai Sci-

ence Committee.

Appendix:The evolution of C a(t)

in the three-layer structure for a special case

For the symmetric structure(rλ

L0=rλ

=rλ0)and

p=p(0,0,1),after inserting equations(9,10)into equa-tion(15),we get

˙C

a(t)≈?

8π

Γ0

π/2

dθsinθ

dt C a(t )

∞

?∞

dνK e i(ω0?νK)(t?t )

t T E

1+r T E

iKd0cosθ

D T E

8π

Γ0

π/2

dθsinθcos2θ

dt C a(t )

∞

?∞

dνK e i(ω0?νK)(t?t )

t T M

1+r T M

iKd0cosθ

D T M

(A.1)

whereΓ0=p2k3/(3ε0π )is the vacuum decay rate of the atom.In the above integration,K2is replaced by k20as it is not the exponential function.From equation(7),the last factor in the above equation can be rewritten as

|tλ0|2

1+rλ0e iKd0cosθ

Dλ0

=1+

∞

n=1

(rλ0e iKd0cosθ)n

+(rλ?0e?iKd0cosθ)n.(A.2) In the derivation of(A.2),

tλ0

=1?

rλ0

has been used due to lossless of all the layers.Inserting(A.2)into(A.1), we get

˙C

a(t)=

8π

Γ0

π/2

dθsinθ

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )

∞

m=1

(r T E

iKd0cosθ)m+(r T E?

?iKd0cosθ)m

I,II,III

8π

Γ0

π/2

dθsinθcos2θ

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )×

∞

m=1

(r T M

iKd0cosθ)m+(r T M?

?iKd0cosθ)m

IV,V,VI.

(A.3)

Next we will calculate the six integrations marked by I, II,III,IV,V and VI for t 1/ω0.The following well-known results(valid for t 1/ω0)are useful to perform

410The European Physical Journal D the next calculations

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )=πC a(t)(A.4a) t

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )e imKd0=

2πe imω0d0/c C a(t?md0/c)Θ(t?md0/c)

(A.4b) t

0dt C a(t )

∞

dνK e i(ω0?νK)(t?t )e?imKd0=0.(A.4c)

The most simpli?ed example is the cavity formed by two single layers which its refractive index is1or?1which we call it as special case.For example,εA=0.5,μA=2 then n A=1,orεA=?0.5,μA=?2then n A=?1.The advantage of such a case is that the re?ectivity between each interface is independent of the incidence angle.

The total re?ectivity of the layer can be expanded as

rλ0=rλout+tλin tλout rλin e2in A Kd A cosθ

∞

m=0

(rλ2in e2in A Kd A cosθ)m λ=TE or TM.(A.5)

According to Fresnel’s law,note that cosθA=cosθ0

r T E in =

n A?μA

n A+μA

,r T M

=r T E

r T E out =?r T E

,r T M

out

=?r T E

,(A.6)

where rλin is the re?ectivity incident from layer A to vacuum.rλout is the re?ectivity incident from vacuum to layer A

t T E in =

2μA

μA+n A

,t T M

=t T E

t T E out =

2n A

μA+n A

,t T M

out

=t T E

out

,(A.7)

where tλout is the transmission incident from layer A to vacuum.tλin is the transmission incident from vacuum to layer A.

We set

r0=r T E

in ,t20=t T E

t T E

out

<1(A.8)

and with the relationship between the di?erent kinds of re?ectivity and transmission in(A.6)and(A.7),we get

r T E

0=?r0+t 2

0r0e

2in A Kd A cosθ

∞

m=0

(r20e2in A Kd A cosθ)m

(A.9)

r T M

0=r T E

0.(A.10)Inserting(A.9)and(A.10)into(A.3),we can perform the six integrations in(A.3)

I+IV=?1

2π

Γ0

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )

Γ0

C a(t)(A.11)

II=?3

8π

Γ0

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )

∞

m=1

π/2

dθsinθ

r T E

iKd0cosθ

.(A.12)

For equation(A.9),we only retain the terms up to r20.So that

r T E

iKd0cosθ≈?r0e iKd0cosθ

+t20r0e2in A Kd A cosθe iKd0cosθ

(r T E

2e2iKd0cosθ≈r2

2iKd0cosθ

?2t20r20e2in A Kd A cosθe2iKd0cosθ

+t40r20e4in A Kd A cosθe2iKd0cosθ. Consequently,we have

∞

m=1

r T E

iKd0cosθ

≈

m=1

?r0e iKd0cosθ

m=1

t20r0e2in A Kd A cosθe iKd0cosθ

?2t20r20e2in A Kd A cosθe i2Kd0cosθ(A.13)

III=?3

8π

Γ0

dt C a(t )

∞

dνK e i(ω0?νK)(t?t )

∞

m=1

π/2

dθsinθ

r T E

?imKd0cosθ

.(A.14) Similarly the sum can be written as

∞

m=1

r T E

iKd0cosθ

≈

m=1

?r0e?iKd0cosθ

m=1

t20r0e?2in A Kd A cosθe?iKd0cosθ

?2t20r20e?2in A Kd A cosθe i2Kd0cosθ.(A.15)

J.P.Xu et al.:The in?uence on atomic decay by inserting a LHM layer411

˙C

a(t)≈?Γ0

C a(t)

4Γ0Re

m=1

(?r0)m e imk0d0

imk0d0

(mk0d0)2

i(mk0d0)3

C a(t)

4Γ0Re

m=1

(t20r0)m e im(2n A k0d A+k0d0)

im(2n A k0d A+k0d0)

m2(2n A k0d A+k0d0)2

im3(2n A k0d A+k0d0)3

C a(t)

4Γ0Re

m=1

(?2t20r20)m e im(2n A k0d A+2k0d0)

i(2n A k0d A+2k0d0)

(2n A k0d A+2k0d0)2

i(2n A k0d A+2k0d0)3

C a(t)

+...(A.18)Γ≈Γ0

Γ0Re

m=1

(?r0)m e imk0d0

imk0d0

(mk0d0)2

i(mk0d0)3

Γ0Re

m=1

(t20r0)m e im(2n A k0d A+k0d0)

im(2n A k0d A+k0d0)

m2(2n A k0d A+k0d0)2

im3(2n A k0d A+k0d0)3

Γ0Re

m=1

(?2t20r20)m e im(2n A k0d A+2k0d0)

i(2n A k0d A+2k0d0)

(2n A k0d A+2k0d0)2

i(2n A k0d A+2k0d0)3

+...(A.19) The result of IV and V can be obtained in a similar way.

With the help of(A.4)and the following two formulas,

π/2

0dθsinθe iF cosθ=

dxe iF x=

e iF0?1

(A.16)

π/2

0dθsinθcos2θe imKd0cosθ=e imKd0

imKd0

(mKd0)2

?21

i(mKd0)3

(A.17)

and note that C a(t?md n/c)≈C a(t)at t 1/ω0,we get the?nal equation for(A.3)without the Lamb shift

see equation(A.18)above

and the steady decay rate can be written as

see equation(A.19)above.

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黄自艺术歌曲钢琴伴奏及艺术成就

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The influence on atomic decay

黄自艺术歌曲钢琴伴奏及艺术成就

我国艺术歌曲钢琴伴奏-精

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