Physics Letters A 376 (2012) 1978–1985

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Physics Letters A

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Control of spontaneous emission from a microwave-field-driven four-level atomin an anisotropic photonic crystal

Duo Zhang a,b, Jiahua Li a,∗, Chunling Ding a, Xiaoxue Yang a

a Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of Chinab School of Electrical and Electronic Engineering, Wuhan Polytechnic University, Wuhan 430023, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 February 2012Received in revised form 17 April 2012Accepted 23 April 2012Available online 27 April 2012Communicated by R. Wu

Keywords:Photonic crystals (PCs)Spontaneous emissionMicrowave field

The spontaneous emission properties of a microwave-field-driven four-level atom embedded in aniso-tropic double-band photonic crystals (PCs) are investigated. We discuss the influences of the band-edgepositions, Rabi frequency and detuning of the microwave field on the emission spectrum. It is foundthat several interesting features such as spectral-line enhancement, spectral-line suppression, spectral-line overlap, and multi-peak structures can be observed in the spectra. The proposed scheme can beachieved by use of a microwave-coupled field into hyperfine levels in rubidium atom confined in a pho-tonic crystal. These theoretical investigations may provide more degrees of freedom to manipulate theatomic spontaneous emission.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Spontaneous emission control is an active research topic inquantum optics because of its potential applications in lasing with-out inversion (LWI) [1–3], high-precision spectroscopy and magne-tometry [4,5], quantum information processing [6,7], etc. The con-trol of spontaneous emission of a multi-level atomic system justby putting the atoms into reservoirs with various density of elec-tromagnetic modes, such as free space [8], optical cavities [9] andphotonic crystals (PCs) [10–14], can be achieved. PCs are a newtype of optical material with a periodic dielectric structure. Themost interesting feature of this material is the existence of a com-pletely photonic band gap (PBG), where a frequency range of theelectromagnetic waves are forbidden to propagate [15,16]. Thus theuse of PCs leads to many interesting optical effects, for instance,the localization of light [17], optical switching [18,19], electromag-netically induced transparency (EIT) [20,21], the enhancement ofquantum interference in spontaneous emission [22–24], suppres-sion and even complete cancellation of spontaneous emission [25,26], the dark lines in spontaneous emission spectra [27,28], andothers [29,30]. In the past few years, substantial efforts have beenmade to investigate the spontaneous emission properties of atomsin PCs. For example, Zhu and co-workers [31] investigated spon-taneous emission and Lamb shift in anisotropic three-dimensional(3D) PCs, they found that the properties of spontaneous emission

* Corresponding author. Tel.: +86 2787557477; fax: +86 2787557477.E-mail addresses: zhangduo10@gmail.com (D. Zhang), huajia_li@163.com (J. Li).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.04.046

are strongly dependent on the relative position of atomic transi-tion frequency to the band edge. Zhang et al. [32] studied quan-tum interference in spontaneous emission spectrum from a V-typethree-level and a double V-type four-level atom embedded in adouble-band PBG material, and compared the spontaneous emis-sion spectrum with the cases of single-band PBG reservoir. Yanget al. [33] calculated the spontaneous emission spectrum of a four-level atom which is coupled by three kinds of reservoirs, and dis-cussed the effect of the fine structure of atomic ground state levelson the spontaneous emission spectrum for the first time. More-over, the cancellation of the spontaneous emission in PBG materi-als in the presence of dipole–dipole interaction has been reportedby Singh [34]. The effect of atomic position on the spontaneousemission and optical spectra of a three-level atom embedded indouble-band PC has been reported in Ref. [35].

On the other hand, atomic spontaneous emission can be con-trolled by driving the atom with the externally applied field. Pas-palakis et al. [36] proposed a phase control scheme in a four-level atom driven by two lasers with the same frequency, wherethe phase difference of the two lasers was used to get partialcancellation, extreme linewidth narrowing and total cancellationin the spontaneous emission spectrum. Wu and his coworkers[37,38] studied the spontaneous emission properties of a ‘tripod’configuration four-level atom with two external driving fields,and showed some interesting phenomena such as fluorescencequenching, spectral-line narrowing, spectral-line enhancement andspectral-line elimination. They also pointed out that these phe-nomena could be observed in the experiment since the rigorouscondition of near-degenerate levels with non-orthogonal dipole

D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985 1979

Fig. 1. (a) Schematic diagram of a four-level atom in a double-band anisotropic PBG reservoir. The four-level atomic model consists of an upper level |4〉, and three lowerlevels |1〉, |2〉 and |3〉. The transition frequencies ω41 and ω42 lie near the lower and upper band edges of the PBG, respectively, while the transition |4〉 → |3〉 is coupled bythe vacuum-field mode in the free space. The transition between the two lower levels |1〉 and |2〉 is driven by a microwave field with a Rabi frequency Ω0. (b) Displays ofthe density of modes (DOMs) for the case of the double-band anisotropic PBG model. (c) A corresponding dressed-state description of the microwave field.

moments was not required. In the following research, Li et al.[39] discussed the features of the spontaneous emission spectraof a five-level atomic system driven by two coherent laser fieldsand a radio-frequency/microwave field, and analyzed the reason indressed-state picture. More recently, Jiang [40] studied the spon-taneous emission spectrum of a three-level Λ-type atom coupledby a microwave field, and discussed the influence of PBG and Rabifrequency of the microwave field on the emission spectrum. Ding[41] investigated spontaneous emission of an RF-driven five-levelatom embedded in a three-dimensional anisotropic double-bandPC, they found that the PBG and the quantum interference effectinduced by the RF-driven field have great influence on the sponta-neous emission spectra. Motivated by this, we propose a four-levelatomic system, in which two atomic transitions are coupled to thelower and upper band of a double-band PBG reservoir respectivelyand the other transition is coupled by the free space. In addi-tion, the two lower levels coupled to PBG reservoir are driven by amicrowave field. The spontaneous emission behavior of the atom–field interaction in such an anisotropic PC is studied in the presentLetter. In the dressed-state picture, by using Laplace transform andfinal value theorem, we derive the analytic expression of sponta-neous emission spectrum, and discuss the influence of the relativepositions of the atomic transition frequencies from the band edgesas well as Rabi frequency and detuning of the microwave field onatomic spontaneous emission properties in the free-space and PBGreservoirs.

This Letter is organized as follows: In Section 2, the atomicmodel under consideration is presented, we deduce the analyt-ical expression for describing the spontaneous emission spectrain the Markovian and non-Markovian reservoirs by use of thetime-dependent Schrödinger equation. In Section 3, we analyzeand discuss the spontaneous emission properties of the systemunder different reservoirs. Also, we put forward the possible ex-perimental realization of our scheme with cold 87Rb atoms andthree-dimensional PCs. Finally, our conclusions are summarized inSection 4.

2. Theoretical model and basic formula

We consider a five-level atom with one excited state |4〉, andthree ground states |1〉, |2〉 and |3〉 as shown schematically inFig. 1(a), where two lower levels |1〉 and |2〉 are coupled by the

corresponding electric dipoles to a common excited level |4〉. Weassume that the transition |2〉 ↔ |1〉 is electric dipole forbiddentransition while magnetic dipole allowed, which is coupled by anexternal microwave field with a carrier frequency ω0 and a Rabifrequency Ω0 (the nature of this field is determined by the levelstructure, i.e., the related two levels may be coupled by a two-photon transition, or infrared, radio frequency, microwave, etc.,depending upon the structure of the specific atomic levels givenbelow in Section 3). The frequency spacing between the two lowerstates (|1〉 and |2〉) is ω21. The density of mode (DOM) for thedouble-band anisotropic PBG model with an upper band, a lowerband, and a band gap is shown in Fig. 1(b). The transitions fromthe upper level |4〉 to the two lower levels |1〉 and |2〉 can be cou-pled by the double-band anisotropic PBG reservoir (this will be re-ferred to as the non-Markovian reservoir). The transition |4〉 → |1〉is considered to be coupled with the upper band of the PBG reser-voir, and the transition |4〉 → |2〉 is taken to be coupled with thelower band of the PBG reservoir, while the transition |4〉 → |3〉 isconsidered to be coupled with the free-space reservoir (this willbe referred to as the Markovian reservoir). The atom is assumed tobe initially in level |4〉 via optical pumping [12]. In the interactionpicture with the rotating wave approximation (RWA), the Hamilto-nian for the system can be written as (taking h = 1) [42–44]

H = H A + H B(t), (1)

H A = δ0|2〉〈2| + Ω0(|2〉〈1| + |1〉〈2|), (2)

H B(t) =∑λ

gλe−iδλt |4〉〈3|aλ +∑

k

g41k e−iδk1t |4〉〈1|ak

+∑

k

g42k e−iδk2t |4〉〈2|ak + H.c., (3)

where H.c. means Hermitian conjugation, gλ is the coupling con-stant between the atomic transition |4〉 ↔ |3〉 and the mode λ ofthe vacuum radiation field. g4 j

k ( j = 1,2) is the coupling constantbetween the atomic transition |4〉 ↔ | j〉 and the mode k of thePBG radiation field. δ0 = ω21 − ω0 refers to the frequency detun-ing between the microwave-driven field and the atomic transition|2〉 ↔ |1〉. δλ = ωλ − ω43 stands for the detuning of the frequencyof the vacuum radiation mode ωλ from the transition frequencyω43, and δkj = ωk − ω4 j ( j = 1,2) represents the detuning of thePBG radiation mode frequency ωk from the transition frequency

1980 D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985

ω4 j . Here k and λ denote the momentum vectors of the emitted

photons. aq and a†q (q = k, λ) are the annihilation and creation op-

erators for reservoir modes with frequency ωq .From the dressed-state theory, due to the interaction of the

atom with the microwave field, the levels |1〉 and |2〉 can be re-placed by dressed levels |α〉 and |β〉, respectively. The dressedstates are defined by the eigenvalue equations H A |α〉 = λα |α〉 and

H A |β〉 = λβ |β〉, where λα = 1/2(δ0 +√

δ20 + 4Ω2

o ), λβ = 1/2(δ0 −√δ2

0 + 4Ω2o ) are the corresponding eigenvalues. The explicit ex-

pressions of the dressed state are

|α〉 = sin θ |2〉 + cos θ |1〉, (4a)

|β〉 = − cos θ |2〉 + sin θ |1〉, (4b)

where sin θ = λα/

√λ2α + Ω2

0 , cos θ = Ω0/

√λ2α + Ω2

0 . Under the

dressed-state description, the wave function of the system is inthe form

∣∣Ψ (t)⟩ = b4(t)|4〉∣∣{0}⟩ + ∑

λ

b3(t)|3〉∣∣{1λ}⟩

+∑

k

bα(t)|α〉∣∣{1k}⟩ + ∑

k

bβ(t)|β〉∣∣{1k}⟩, (5)

where b j(t) ( j = 3,4,α,β) stands for the time-dependent prob-ability amplitude of the atomic state, the initial value of whichdepends upon the initial quantum state of the atom being pre-pared. |{0}〉 denotes the vacuum of the radiation field, |{1k}〉 and|{1λ}〉 indicate that there is one photon of wave vector k or λ inthe photon reservoir mode. From Eqs. (1)–(3) and (5) we can de-rive the equations of motion for the expansion amplitudes:

i∂b4(t)

∂t=

∑λ

gλe−iδλtb3(t)

+∑λ

g41k e−iδk1t(cos θbα(t) + sin θbβ(t)

)

+∑λ

g42k e−iδk2t(sin θbα(t) − cos θbβ(t)

), (6a)

i∂b3(t)

∂t= g∗

λeiδλtb4(t), (6b)

i∂bα(t)

∂t= λαbα(t) + g14

k eiδk1t cos θb4(t)

+ g24k eiδk2t sin θb4(t), (6c)

i∂bβ(t)

∂t= λβbβ(t) + g14

k eiδk1t sin θb4(t)

+ g24k eiδk2t cos θb4(t). (6d)

We proceed by performing a formal time integration ofEqs. (6b)–(6d) and substitute the results into Eq. (6a), then theintegro-differential equations can be obtained

∂b4(t)

∂t= −

t∫0

dt′ b4(t′)∑

λ

g2λe−iδλ(t−t′)

−t∫

0

dt′ b4(t′) cos2 θkα1

(t − t′)

−t∫

dt′ b4(t′) cos θ sin θeiω0t′kα2

(t − t′)

0

−t∫

0

dt′ b4(t′) sin2 θkβ1

(t − t′)

+t∫

0

dt′ b4(t′) sin θ cos θeiω0t′kβ2

(t − t′)

−t∫

0

dt′ b4(t′) sin θ cos θe−iω0t′k0

α1

(t − t′)

−t∫

0

dt′ b4(t′) sin2 θk0

α2

(t − t′)

+t∫

0

dt′ b4(t′) sin θ cos θe−iω0t′k0

β1

(t − t′)

−t∫

0

dt′ b4(t′) cos2 θk0

β2

(t − t′), (7)

where K jl(t − t′) and K 0jl(t − t′) ( j = α,β , l = 1,2) are the delay

Green’s functions with the definitions

K jl(t − t′) =

∑k

gl4k g41

k e−i(λ j+δk1)(t−t′), (8a)

K 0jl

(t − t′) =

∑k

gl4k g42

k e−i(λ j+δk1+ω0)(t−t′). (8b)

Because the transition |4〉 → |3〉 is assumed to be far away fromthe PBG edges so that the density of states near ω43 is variedslowly, we can apply the usual Weisskopf–Wigner approximation[45] in the first term on the right-hand side of Eq. (7) and obtain

∑λ

g2λe−iδλ(t−t′) = Γ

2δ(t − t′), (9)

where Γ = 2π |gλ|2 D(ωλ) is the spontaneous-decay rate from level|4〉 to level |3〉, and D(ωλ) is the vacuum-mode density at fre-quency ωλ in the free space. We consider the case of a double-band anisotropic effective mass model of the PBG reservoir asshown in Fig. 1(b), which has an upper band, a lower band, anda band gap. The dispersion relations near the photonic band edgesare approximated by [46]

ωk = ωg1 − Ag1∣∣(�k − �k0)

∣∣2, |�k| < |�k0|,

ωk = ωg2 + Ag2∣∣(�k − �k0)

∣∣2, |�k| > |�k0|, (10)

where Agl ≈ ωgl/|�k0|2 (l = 1,2). Under the effective-mass approxi-mation, the delay Green’s functions (8a) and (8b) can be expressedas [46]

K jl(t − t′) = η1l

√α41α4l

2

(exp{i[(δ41g2 − λ j)(t − t′) + π

4 ]}√4π(t − t′)3

+ exp{i[(δ41g1 − λ j)(t − t′) − π4 ]}√

4π(t − t′)3

), (11a)

K 0jl

(t − t′) = ηl2

√α4lα42

2

(exp{i[(δ42g2 − λ j)(t − t′) + π

4 ]}√4π(t − t′)3

+ exp{i[(δ42g1 − λ j)(t − t′) − π4 ]}√ ′ 3

), (11b)

4π(t − t )

D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985 1981

where η1l and ηl2 indicate the quantum interference in atomictransitions coupled to the modified reservoir. For simplicity butwithout loss of generality, we assume η1l = ηl2 = 1. Here,α4l (l = 1,2) is the modified coupling constant. δ41g2 = ω41 − ωg2,δ41g1 = ω41 −ωg1, δ42g2 = ω42 −ωg2, and δ42g1 = ω42 −ωg1 repre-sent the detuning of the atomic transition frequencies ω41 and ω42from the upper and lower band-edge frequencies ωg2 and ωg1.

Taking the Laplace transformations [45] b4(s) = ∫ ∞0 e−stb4(t)dt

for Eq. (7), and s is the time Laplace transform variable, we havethe result

sb4(s) − b4(0)

= −Γ

2b4(s) − cos2 θkα1(s)b4(s) − cos θ sin θkα2(s)b4(s − iω0)

− sin2 θkβ1(s)b4(s) + cos θ sin θkβ2(s)b4(s − iω0)

− cos θ sin θk0α1(s)b4(s + iω0) − sin2 θk0

α2(s)b4(s)

+ cos θ sin θk0β1(s)b4(s + iω0) − cos2 θk0

β2(s)b4(s), (12)

where k jl(s) (k0jl(s)) ( j = α,β; l = 1,2) is the Laplace transform of

k jl(t − t′) (k0jl(t − t′)) ( j = α,β; l = 1,2). For the anisotropic band

edges in the effective-mass approximation, by taking the Laplacetransform of Eqs. (11a) and (11b) we have

K jl(s) =√

α41α4l

2(

√is + δ41g2 − λ j − i

√is + δ41g1 − λ j ), (13a)

K 0jl(s) =

√α4lα42

2(

√is + δ42g2 − λ j − i

√is + δ42g1 − λ j ), (13b)

and we used the iterative method once in Eq. (12) to obtain

b4(s) = 1 + M + N

s + Γ2 + sin2 θ [kβ1(s) + k0

α2(s)] + cos2 θ [kα1(s) + k0β2(s)]

× b4(0), (14)

where

M = sin θ cos θ[kβ2(s) − kα2(s)

]b0

4(s − iω0),

N = sin θ cos θ[k0β1(s) − k0

α1(s)]b0

4(s + iω0)

and

b04(s) = 1

s + Γ2 + sin2 θ [kβ1(s) + k0

α2(s)] + cos2 θ [kα1(s) + k0β2(s)] .

First of all, we derive the long-time spontaneous emission spec-tra of the transition |4〉 ↔ |3〉 within the Markovian reservoir,namely, S(δλ) ∝ |b3(t → ∞)|2 with the detuning frequency δλ =ωλ − ω43. Performing the final-value theorem and Laplace trans-form for Eq. (6b), we have

S(δλ) = D(ωλ)∣∣b3(t → ∞)

∣∣2 = Γ

2π

∣∣b4(s = −iδλ)∣∣2

. (15)

Then, we derive the long-time spontaneous emission spectrain the non-Markovian reservoir. Using the Laplace transform andfinal-value theorem for Eqs. (6c) and (6d) we can obtain

bα(t → ∞) = −ig14k cos θb4(−iδk1 − iλα)

− ig24k sin θb4(−iδk2 − iλα), (16a)

bβ(t → ∞) = −ig14k sin θb4(−iδk1 − iλβ)

+ ig24k cos θb4(−iδk2 − iλβ). (16b)

The spontaneous emission spectrum S(ωk) is given by

S(ωk) = ρ(ωk)(∣∣bα(t → ∞)

∣∣2 + ∣∣bβ(t → ∞)∣∣2)

, (17)

where ρ(ωk) is the photonic DOM of the radiation field. For theanisotropic PBG reservoir, the photonic DOM can be derived as [46]

ρ(ωk) = 1

π

[√ωk − ωg2Θ(ωk − ωg2)

+ √ωg1 − ωkΘ(ωg1 − ωk)

], (18)

with the Heaviside step function Θ .

3. Numerical results and analysis

In this section, we will discuss some properties about the spon-taneous emission spectra via a few numerical calculations based onEqs. (15) and (17) within the Markovian and non-Markovian reser-voirs, respectively. All the parameters used in the following discus-sion are in units of γ , and γ is the decay rate for the transitionfrom the upper level |4〉 to the lower level |3〉, i.e., Γ = γ , whichshould be in order of MHz for rubidium atom. And we assume theatom is initially prepared in level |4〉 [12], i.e., b4(0) = 1. In thefollowing discussions, we will use some of the detuning parame-ters, in which δk is the frequency detuning of the radiation fieldfrom the middle of the lower levels, i.e., δk = ωk − 0.5(ω41 + ω42).ωk − ωg2 = δk + 0.5δ41g2 + 0.5δ42g1 − 0.5δg2g1 and ωg1 − ωk =−δk − 0.5δ41g2 − 0.5δ42g1 − 0.5δg2g1 are the frequency detuningsof the radiation field from the upper and lower band edges, thesymbol δg2g1 = ωg2 − ωg1 is the width of the band gap.

3.1. The spontaneous emission spectra in the non-Markovian reservoir

We begin with a study of the spontaneous emission spectra as-sociated with the transitions |4〉 → |1〉 and |4〉 → |2〉 within thenon-Markovian reservoir, and the level |1〉 is coupled to |2〉 bya microwave field. In the following, we will discuss the influenceof the band-edge positions, as well as the intensity and detuningof the microwave field on the spontaneous emission spectra. Forthe case that both of the transitions coupled to the same modi-fied reservoir from the microwave-field-coupled three-level Λ-typeatom in PCs has been discussed in Ref. [40], and the case that onetransition is coupled to the lower band and the other is coupled tothe upper band in a double-band anisotropic PBG reservoir fromthe RF-driven five-level atom is also reported in Ref. [41]. However,the situation is quite different if one atomic transition is coupledto the lower band, the other is coupled to the upper band in adouble-band anisotropic PBG reservoir, and the corresponding twolower levels are coupled by a microwave field as well.

First of all, we show the spontaneous emission spectra S(δk)

versus the detuning δk with different relative positions of the tran-sition frequencies from the band edges in Fig. 2 when the detuningand intensity of the microwave-driven field is fixed, i.e., δ0 = 0 andΩ0 = 2. From Fig. 2 we can see that the spectral profile is fairlysensitive to the detunings of the atomic transition frequencies fromthe band edges. We can observe that the emission spectrum showstwo peaks at both sides of δk = 0 when one transition frequencyis within the upper band (δ41g2 = 3) and the other is within thelower band (δ42g1 = −3) as shown in Fig. 2(a). However, for thecase that the transition frequency ω42 moves into the band gap(δ42g1 = 0.3) and the transition frequency ω41 is within the up-per band (δ41g2 = 3), one of two peaks on the left-hand side issuppressed as can be seen from Fig. 2(b). When the transitionfrequency ω41 moves into the band gap (δ41g2 = −0.3) and thetransition frequency ω42 is within the lower band (δ42g1 = −3),one of two peaks on the right-hand side disappears [see Fig. 2(c)].It is more interesting that when both transition frequencies ofthe atom are inside the band gap (δ41g2 = −0.3, δ42g1 = 0.3),the emission spectrum exhibits only one peak at both sides of

1982 D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985

Fig. 2. The spontaneous emission spectra S(δk) (in arbitrary units) as a function of detuning δk for both transitions occur at different location of the gap for g14 = 1, g24 = 1,δ0 = 0, Ω0 = 2, α41 = 0.5, α42 = 0.5, and Γ = 1, respectively. (a) (δ41g2, δ42g1, δg2g1) = (3,−3,3); (b) (δ41g2, δ42g1, δg2g1) = (3,0.3,3); (c) (δ41g2, δ42g1, δg2g1) = (−0.3,−3,3);(d) (δ41g2, δ42g1, δg2g1) = (−0.3,0.3,3).

δk = 0 [see Fig. 2(d)]. These interesting phenomena can be quan-titatively explained by using the dressed-state picture. Accordingto the dressed-state theory, the levels |1〉 and |2〉 are replacedby two dressed levels |α〉 and |β〉 under the action of the mi-crowave field, which are the linear superpositions of levels |1〉 and|2〉. Therefore, both transitions |4〉 → |1〉 and |4〉 → |2〉 consist oftwo transitions, respectively. From Eqs. (16a), (16b) and (17), wecan see that the spectral structure of |bα |2 consists of two linescentered near δ41g2 − |Ω0| and δ42g1 − |Ω0|, while that of |bβ |2consists of two lines centered near δ41g2 + |Ω0| and δ42g1 + |Ω0|,respectively. And the lower peak at the right side of δk = 0 andthe higher peak at the left side of δk = 0 associate with bα , i.e.,δ41g2 − |Ω0| and δ42g1 − |Ω0| in Fig. 2(a), while the higher peakat the right side of δk = 0 and the lower peak on the left sideof δk = 0 associate with bβ , i.e., δ41g2 + |Ω0| and δ42g1 + |Ω0|in Fig. 2(a). When Ω0 = 2, δ41g2 = 3 and δ42g1 = −3, we getδ41g2 − |Ω0| = 1 > 0, δ42g1 − |Ω0| = −5 < 0, δ41g2 + |Ω0| = 5 > 0and δ42g1 + |Ω0| = −1 < 0. These indicate that all the transitionfrequencies associated with |4〉 → |α〉 and |4〉 → |β〉 fall outsidethe band gap. Therefore the emission spectrum shows the super-position of four Lorentzian shapes as shown in Fig. 2(a). Whenone transition frequency is inside the band gap and the other isoutside the band gap (δ41g2 = 3 and δ42g1 = 0.3), the transitionfrequency of |4〉 → |β〉 associated with the spectral line centerednear δ42g1 +|Ω0| = 2.3 > 0 falls within the band gap and the otherthree transition frequencies remain outside the band gap. Con-sequently, the lower peak on the left side of δk = 0 is inhibitedand the emission spectrum has a three-peak structure as shown inFig. 2(b). In the case of δ41g2 = −0.3 and δ42g1 = −3, the transitionfrequency of |4〉 → |α〉 associated with the spectral line centerednear δ41g2 − |Ω0| = −2.3 < 0 moves deeply into the band gap andthe other three transition frequencies are outside the band gap, sothere are only one peak at the right side of δk = 0 as can be seenfrom Fig. 2(c). From Fig. 2(d), we can find that when both transi-tion frequencies are inside the band gap (δ41g2 = −0.3 and δ42g1 =−3), where δ41g2 − |Ω0| = −2.3 < 0 and δ42g1 + |Ω0| = 2.3 > 0,one transition frequency associated with transition |4〉 → |α〉 andthe other with transition |4〉 → |β〉 fall into the band gap and theother two transitions still outside the band gap, hence the twolower peaks move into band gap at the both sides of δk = 0 andthe emission spectrum shows a two-peak structure.

Next, we will discuss the influence of Rabi frequency Ω0 ofthe microwave field on the spontaneous emission spectra for thecase that δ41g2 = 3 and δ42g1 = −3 when the atom is embed-ded in the anisotropic double-band PBG reservoir. In Fig. 3(a), wecan observe a symmetric double-peak structure [see dotted linein Fig. 3(a)] in the spectra when there is no microwave field,i.e., Ω0 = 0. When Ω0 = 0.5, the peak on the left side is re-duced, while the peak on the right is nearly the same height asthe case of Ω0 = 0 [see the solid line in Fig. 3(a)]. When theRabi frequency increases from 0.5 to 2, each spontaneous emis-sion peak splits into two as can be seen from Figs. 3(b) and 3(c).These phenomena can be explained as follows. Under the condi-tion that Rabi frequency of the microwave field increases from 0.5to 2, we get δ41g2 − |Ω0| > 0, δ42g1 − |Ω0| < 0, δ41g2 + |Ω0| > 0and δ42g1 + |Ω0| < 0. These suggest that all the transition frequen-cies associated with |4〉 → |α〉 and |4〉 → |β〉 are outside the bandgap, therefore the spontaneous emission spectra show a four-peakstructure. However, for the case of Ω0 = 0.5, the difference be-tween δ41g2 − |Ω0| and δ41g2 + |Ω0| or the difference betweenδ42g1 − |Ω0| and δ42g1 − |Ω0| is very small, and the correspondingpeaks overlap each other. As Ω0 increasing to 1, the correspondingdifferences reduce and the distance between the left (right) peaksbecomes wider, hence the spontaneous emission spectra show afour-peak structure gradually [see Fig. 3(b)]. With Ω0 increasingfurther to 2, the emission spectra display a clear four-peak struc-ture as shown in Fig. 3(c). However, it is more interesting thatwhen Ω0 increases to 4, the two peaks (δ42g1 + |Ω0| > 0 andδ41g2 − |Ω0| < 0) near the band gap will move deeply into theband gap, and the emission spectra display one peak on both sidesof δk = 0, respectively.

In order to further explore the effect of the detuning of the mi-crowave field on the spontaneous emission spectra, we also plotS(δk) versus the detuning δk under the condition that both tran-sition frequencies are outside the band gap as shown in Fig. 4.It is clearly shown that, when the microwave field is tuned tothe resonant interaction with the atomic transition |1〉 ↔ |2〉, anhigher peak and a lower peak can be observed simultaneously onboth sides of δk = 0 [see Fig. 4(a)]. As the detuning of microwavefield increases to 1 or 2, the two peaks near band gap are sup-pressed gradually compared with the situation at resonance [seeFigs. 4(b) and 3(c)]. While in the case of δ0 = −2, the two peaks

D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985 1983

Fig. 3. The spontaneous emission spectra S(δk) (in arbitrary units) as a function of detuning δk for different Rabi frequencies Ω0 of the microwave field for g14 = 1, g24 = 1,δ0 = 0, α41 = 0.5, α42 = 0.5, Γ = 1, δ41g2 = 3, δ42g1 = −3, δg2g1 = 3. (a) Ω0 = 0 (dotted line), Ω0 = 0.5 (solid line); (b) Ω0 = 1; (c) Ω0 = 2; (d) Ω0 = 4.

Fig. 4. The spontaneous emission spectra S(δk) (in arbitrary units) as a function of detuning δk for different detunings δ0 of the microwave field for g14 = 1, g24 = 1, Ω0 = 2,α41 = 0.5, α42 = 0.5, Γ = 1, δ41g2 = 3, δ42g1 = −3, δg2g1 = 3. (a) δ0 = 0; (b) δ0 = 1; (c) δ0 = 2; (d) δ0 = −2.

far from band gap is inhibited greatly, and the middle two peaks,especially the one near the band gap on the right side of δk = 0is enhanced obviously as shown in Fig. 4(d). The reason for thephenomena we have been discussed above can be explained as fol-lows. We take the third peak for example, when δ0 increases from0 to 2, the eigenvalue of the dressed state λα increases graduallyand one component of bα , i.e., δ41g2 − λα decreases correspond-ingly. That is to say, the peak near the band gap on the rightside of δk = 0 moves to the band edge slowly and the height ofpeak is suppressed obviously as shown in Figs. 4(a)–(c). However,when δ0 = −2, the eigenvalue of the dressed state λα decreasesgreatly and δ41g2 − λα increases accordingly, as a result, the peakmoves far from the band edge and the third peak corresponds tothe line centered near δ41g2 −λα , which is enhanced obviously [seeFig. 4(d)].

3.2. The spontaneous emission spectra in the Markovian reservoir

We now turn to what happens to the spontaneous emissionspectra associated with the transitions |4〉 → |3〉 within the Marko-vian reservoir. In the following, we will discuss the influence of theband-edge positions, and the intensity of the microwave field onthe spontaneous emission spectra.

Firstly, we analyze the influence of both the lower and upperband-edge positions on the spontaneous emission spectra, con-sidering the external microwave field resonates with the atomictransition |1〉 ↔ |2〉 and Rabi frequency Ω0 = 2. We plot thespontaneous emission spectra S(δλ) versus the detuning δλ asshown in Fig. 5. For the case that one transition frequency ω41is within the upper band (δ41g2 = 3) and the other ω42 is withinthe lower band (δ42g1 = −3) as the dashed line shown in Fig. 5,

1984 D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985

Fig. 5. The spontaneous emission spectra S(δλ) (in arbitrary units) for both tran-sitions occur at different location of the gap for g14 = 1, g24 = 1, δ0 = 0, Ω0 = 2,α41 = 0.5, α42 = 0.5, and Γ = 1, respectively. Here, the spontaneous emission spec-tra of separate transitions are given within Markovian reservoir.

we can observe that the spontaneous emission spectra show asingle-peak structure and the height of peak is small. When thetransition frequency ω41 is within the upper band (δ41g2 = 3)and the other transition frequency ω42 moves into the band gap(δ42g1 = 0.3), the height of the emission peak increases obvi-ously [see the solid line in Fig. 5]. As the both transition fre-quencies ω41 and ω42 move into the band gap, i.e., δ41g2 = −0.3and δ42g1 = 0.3, the spontaneous emission peak continues to en-hance as the dotted line shown in Fig. 5. We can quantitativelyexplain the reason with spontaneous emission theory. From Fig. 1,we can see that there exist three emission channels from upperenergy level |4〉 to lower energy levels |1〉, |2〉 and |3〉. Whenboth transition frequencies ω41 and ω42 are outside the bandgap (δ41g2 = 3 and δ42g1 = −3), the spontaneous emission in thePCs reservoir are strong, hence the free-space spontaneous emis-sion is slightly suppressed. As one or both transition frequenciesω41 and ω42 move into PBG, we can find that the spontaneousemission from |4〉 to |1〉 and |2〉 are suppressed gradually. Con-sequently, the free-space spontaneous emission from |4〉 to |3〉 isgreatly enhanced and the peak of emission spectra increases obvi-ously as can be seen from the solid line and dotted line shown inFig. 5.

Finally, we investigate the influence of Rabi frequency of themicrowave field on the free-space spontaneous emission spectra.As can be seen from Fig. 6, in the case of Ω0 = 1, the emissionspectra display a single-peak structure, and the center of the peakis near δλ = 0 [see dotted line in Fig. 6]. For the case of Ω0 = 2,spontaneous emission spectra is slightly enhanced as the solid lineshown in Fig. 6. While, under the condition of Ω0 = 4, it is veryinteresting that the free-space spontaneous emission is obviouslyenhanced and a new lower peak emerges on the right side ofthe zero detuning of δλ as can be seen from the dashed line inFig. 6. We can use a dressed-state theory to explain these results.The bare states |1〉 and |2〉 can be substituted with dressed states|α〉 and |β〉. As Rabi frequency of the microwave field increases,the separation of two dressed states |α〉 and |β〉 increases ac-cordingly and the constructive quantum interference between thedecay channels of |4〉 to |α〉 and |4〉 to |β〉 is suppressed. There-fore, the decay from |4〉 to |α〉 and |β〉 reduces and the emissionfrom |4〉 to |3〉 enhances accordingly. The right-hand lower peak[see the insets of Fig. 6] can be explained by quantum interferenceof the emission channels.

Fig. 6. The spontaneous emission spectra S(δλ) (in arbitrary units) for different Rabifrequencies Ω0 of the microwave field for g14 = 1, g24 = 1, δ0 = 0, α41 = 0.5, α42 =0.5, Γ = 1, δ41g2 = 3, δ42g1 = −3, and δg2g1 = 3, respectively. Here, the spontaneousemission spectra of separate transitions are given within Markovian reservoir.

Before ending this section, let us briefly discuss the possibleexperimental realization of our proposed scheme by means ofalkali-metal atoms, microwave field, and double-band anisotropicPCs. For example, we consider D2 line of the cold 87Rb atoms(nuclear spin I = 3/2) as a possible candidate [47]. The desig-nated states can be chosen as follows: |1〉 = |5S1/2, F = 1,mF = 1〉,|2〉 = |5S1/2, F = 2,mF = 1〉, |3〉 = |5S1/2, F = 2,mF = 2〉 and |4〉 =|5P1/2, F = 2,mF = 1〉, respectively. In this case, a microwave fieldcouples the allowed magnetic dipole transition between |1〉 =|5S1/2, F = 1,mF = 1〉 and |2〉 = |5S1/2, F = 2,mF = 1〉 with thefrequency spacing ω21 = 8.6 GHz, while the transitions from theexcited level |4〉 = |5P1/2, F = 2,mF = 1〉 to the lower levels |1〉 =|5S1/2, F = 1,mF = 1〉 and |2〉 = |5S1/2, F = 2,mF = 1〉 can be cou-pled by the lower and upper bands of the double-band anisotropicPBG reservoir, respectively. At the same time, the transition fromthe upper level |4〉 = |5P1/2, F = 2,mF = 1〉 to the lower level|3〉 = |5S1/2, F = 2,mF = 2〉 can be coupled by the vacuum-fieldmode in the free space.

4. Conclusion

In summary, we have investigated the properties of sponta-neous emission from a four-level atom embedded in a double-bandanisotropic PBG material. It is clearly shown that the behavior ofthe spontaneous emission is strongly dependent on the relativeposition between the transition frequency and the PBG edge, thedetunings and Rabi frequencies of the microwave field. On the onehand, for the spontaneous emission spectra in the non-Markovianreservoir, i.e., the transitions from the common upper level |4〉 tothe two lower levels |1〉 and |2〉 interact with the upper and lowerbands of the PBG reservoir, respectively. By adjusting the systemparameters, we can obtain some interesting phenomena: (i) Whentransition frequency ω41 is within the upper band and the otherω42 is within the lower band, the spectral line is a four-peakstructure. While the transition frequency ω42 (ω41) moves into theband gap and the other transition frequency is outside the bandgap, one of two peaks on the left(right)-hand side is suppressed.When both transition frequencies of the atom are inside the bandgap, the emission spectrum exhibits only a double-peak structure.(ii) The emission spectrum shows a double-peak structure in theabsence of the microwave field. As Rabi frequency increases grad-ually, each spontaneous emission peak splits into two peaks and

D. Zhang et al. / Physics Letters A 376 (2012) 1978–1985 1985

the separation between the two peaks becomes wider. (iii) Thenumber and height of the spontaneous emission peaks can becontrolled effectively by regulating the detuning of the microwavefield. On the other hand, for the spontaneous emission spectra inthe Markovian reservoir, the spectra exhibit a single-peak structurewhether two transition frequencies ω41 and ω42 are in or out-side the band gap. While when two transition frequencies ω41 orω42 move from energy band to band gap, the spontaneous emis-sion from |4〉 to |3〉 increases obviously. Similarly, the spontaneousemission from |4〉 to |3〉 increases with the increasing of Rabi fre-quency of microwave field, but when Rabi frequency Ω0 increasesto a certain value, there will be a new lower peak at the right sideof the main peak due to quantum interference channels.

These results show that the spectral behaviors are very sensi-tive to the positions of the band edges, Rabi frequency and thedetuning of the microwave field, so we can control the sponta-neous emission by adjusting these system parameters under re-alistic experimental conditions. Finally, we would like to mentionthat the application of the PCs offers us further flexibility to ma-nipulate the light–matter interactions, including the manipulationof spontaneous emission here, since the PBG materials can be usedto better control the atomic spontaneous emission. Moreover, theseinvestigations open up possibilities for various applications of PCsusing spontaneous emission properties.

Acknowledgements

The research is supported in part by the National NaturalScience Foundation of China (Grant Nos. 10975054, 91021011and 11004069), by the Doctoral Foundation of the Ministryof Education of China under Grant No. 20100142120081 andby National Basic Research Program of China under ContractNo. 2012CB922103. We would like to thank Professor Ying Wufor helpful discussion and his encouragement.

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