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

Physics Letters A 376 (2012) 1978–1985

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Physics Letters A www.elsevier.com/locate/pla

Control of spontaneous emission from a microwave-field-driven four-level atom in an anisotropic photonic crystal Duo Zhang a,b , Jiahua Li a,∗ , Chunling Ding a , Xiaoxue Yang a a b

Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China 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

Article history: Received 27 February 2012 Received in revised form 17 April 2012 Accepted 23 April 2012 Available online 27 April 2012 Communicated by R. Wu Keywords: Photonic crystals (PCs) Spontaneous emission Microwave field

a b s t r a c t The spontaneous emission properties of a microwave-field-driven four-level atom embedded in anisotropic double-band photonic crystals (PCs) are investigated. We discuss the influences of the band-edge positions, Rabi frequency and detuning of the microwave field on the emission spectrum. It is found that several interesting features such as spectral-line enhancement, spectral-line suppression, spectralline overlap, and multi-peak structures can be observed in the spectra. The proposed scheme can be achieved by use of a microwave-coupled field into hyperfine levels in rubidium atom confined in a photonic crystal. These theoretical investigations may provide more degrees of freedom to manipulate the atomic spontaneous emission. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Spontaneous emission control is an active research topic in quantum optics because of its potential applications in lasing without inversion (LWI) [1–3], high-precision spectroscopy and magnetometry [4,5], quantum information processing [6,7], etc. The control of spontaneous emission of a multi-level atomic system just by putting the atoms into reservoirs with various density of electromagnetic modes, such as free space [8], optical cavities [9] and photonic crystals (PCs) [10–14], can be achieved. PCs are a new type of optical material with a periodic dielectric structure. The most interesting feature of this material is the existence of a completely photonic band gap (PBG), where a frequency range of the electromagnetic waves are forbidden to propagate [15,16]. Thus the use of PCs leads to many interesting optical effects, for instance, the localization of light [17], optical switching [18,19], electromagnetically induced transparency (EIT) [20,21], the enhancement of quantum interference in spontaneous emission [22–24], suppression and even complete cancellation of spontaneous emission [25, 26], the dark lines in spontaneous emission spectra [27,28], and others [29,30]. In the past few years, substantial efforts have been made to investigate the spontaneous emission properties of atoms in PCs. For example, Zhu and co-workers [31] investigated spontaneous emission and Lamb shift in anisotropic three-dimensional (3D) PCs, they found that the properties of spontaneous emission

*

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are strongly dependent on the relative position of atomic transition frequency to the band edge. Zhang et al. [32] studied quantum interference in spontaneous emission spectrum from a V-type three-level and a double V-type four-level atom embedded in a double-band PBG material, and compared the spontaneous emission spectrum with the cases of single-band PBG reservoir. Yang et al. [33] calculated the spontaneous emission spectrum of a fourlevel atom which is coupled by three kinds of reservoirs, and discussed the effect of the fine structure of atomic ground state levels on the spontaneous emission spectrum for the first time. Moreover, the cancellation of the spontaneous emission in PBG materials in the presence of dipole–dipole interaction has been reported by Singh [34]. The effect of atomic position on the spontaneous emission and optical spectra of a three-level atom embedded in double-band PC has been reported in Ref. [35]. On the other hand, atomic spontaneous emission can be controlled by driving the atom with the externally applied field. Paspalakis et al. [36] proposed a phase control scheme in a fourlevel atom driven by two lasers with the same frequency, where the phase difference of the two lasers was used to get partial cancellation, extreme linewidth narrowing and total cancellation in 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 fluorescence quenching, spectral-line narrowing, spectral-line enhancement and spectral-line elimination. They also pointed out that these phenomena could be observed in the experiment since the rigorous condition 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 lower levels |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 by the 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 of the 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 spectra of a five-level atomic system driven by two coherent laser fields and a radio-frequency/microwave field, and analyzed the reason in dressed-state picture. More recently, Jiang [40] studied the spontaneous emission spectrum of a three-level Λ-type atom coupled by a microwave field, and discussed the influence of PBG and Rabi frequency of the microwave field on the emission spectrum. Ding [41] investigated spontaneous emission of an RF-driven five-level atom embedded in a three-dimensional anisotropic double-band PC, they found that the PBG and the quantum interference effect induced by the RF-driven field have great influence on the spontaneous emission spectra. Motivated by this, we propose a four-level atomic system, in which two atomic transitions are coupled to the lower and upper band of a double-band PBG reservoir respectively and the other transition is coupled by the free space. In addition, the two lower levels coupled to PBG reservoir are driven by a microwave field. The spontaneous emission behavior of the atom– field interaction in such an anisotropic PC is studied in the present Letter. In the dressed-state picture, by using Laplace transform and final value theorem, we derive the analytic expression of spontaneous emission spectrum, and discuss the influence of the relative positions of the atomic transition frequencies from the band edges as well as Rabi frequency and detuning of the microwave field on atomic spontaneous emission properties in the free-space and PBG reservoirs. This Letter is organized as follows: In Section 2, the atomic model under consideration is presented, we deduce the analytical expression for describing the spontaneous emission spectra in the Markovian and non-Markovian reservoirs by use of the time-dependent Schrödinger equation. In Section 3, we analyze and discuss the spontaneous emission properties of the system under different reservoirs. Also, we put forward the possible experimental realization of our scheme with cold 87 Rb atoms and three-dimensional PCs. Finally, our conclusions are summarized in Section 4. 2. Theoretical model and basic formula We consider a five-level atom with one excited state |4, and three ground states |1, |2 and |3 as shown schematically in Fig. 1(a), where two lower levels |1 and |2 are coupled by the

corresponding electric dipoles to a common excited level |4. We assume that the transition |2 ↔ |1 is electric dipole forbidden transition while magnetic dipole allowed, which is coupled by an external microwave field with a carrier frequency ω0 and a Rabi frequency Ω0 (the nature of this field is determined by the level structure, i.e., the related two levels may be coupled by a twophoton transition, or infrared, radio frequency, microwave, etc., depending upon the structure of the specific atomic levels given below in Section 3). The frequency spacing between the two lower states (|1 and |2) is ω21 . The density of mode (DOM) for the double-band anisotropic PBG model with an upper band, a lower band, and a band gap is shown in Fig. 1(b). The transitions from the upper level |4 to the two lower levels |1 and |2 can be coupled by the double-band anisotropic PBG reservoir (this will be referred to as the non-Markovian reservoir). The transition |4 → |1 is considered to be coupled with the upper band of the PBG reservoir, and the transition |4 → |2 is taken to be coupled with the lower band of the PBG reservoir, while the transition |4 → |3 is considered to be coupled with the free-space reservoir (this will be referred to as the Markovian reservoir). The atom is assumed to be initially in level |4 via optical pumping [12]. In the interaction picture with the rotating wave approximation (RWA), the Hamiltonian for the system can be written as (taking h¯ = 1) [42–44]

ˆ = Hˆ A + Hˆ B (t ), H





ˆ A = δ0 |22| + Ω0 |21| + |12| , H ˆ B (t ) = H

 λ

+

g λ e −i δλ t |43|ˆaλ +





(1) (2)

gk41 e −i δk1 t |41|ˆak

k

gk42 e −i δk2 t |42|ˆak + H.c.,

(3)

k

where H.c. means Hermitian conjugation, g λ is the coupling constant between the atomic transition |4 ↔ |3 and the mode λ of 4j the vacuum radiation field. gk ( j = 1, 2) is the coupling constant between the atomic transition |4 ↔ | j  and the mode k of the PBG radiation field. δ0 = ω21 − ω0 refers to the frequency detuning between the microwave-driven field and the atomic transition |2 ↔ |1. δλ = ωλ − ω43 stands for the detuning of the frequency of the vacuum radiation mode ωλ from the transition frequency ω43 , and δkj = ωk − ω4 j ( j = 1, 2) represents the detuning of the PBG radiation mode frequency ωk from the transition frequency

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

t

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

−

† aˆ q

(q = k, λ) are the annihilation and creation opphotons. aˆ q and 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 replaced by dressed levels |α  and |β, respectively. The dressed ˆ A |α  = λα |α  and states are defined by the eigenvalue equations H 

0

t + t

+ 4Ωo2 ),

− t −

(4a)

|β = − cos θ|2 + sin θ|1,

(4b)  where sin θ = λα / λ2α + Ω02 , cos θ = Ω0 / λ2α + Ω02 . Under the

t +

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







bα (t )|α {1k } +

k



− 



bβ (t )|β{1k } ,

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

∂ b4 (t )  i = g λ e −i δλ t b3 (t ) ∂t λ    + gk41 e −i δk1 t cos θ bα (t ) + sin θ bβ (t ) +



gk42 e −i δk2

 t

 sin θ bα (t ) − cos θ bβ (t ) ,

∂ b3 (t ) = g λ∗ e i δλ t b4 (t ), ∂t ∂ bα (t ) i = λα bα (t ) + gk14 e i δk1 t cos θ b4 (t ) ∂t + gk24 e i δk2 t sin θ b4 (t ), i

∂ bβ (t ) = λβ bβ (t ) + gk14 e i δk1 t sin θ b4 (t ) ∂t + gk24 e i δk2 t cos θ b4 (t ).

(6a)

(6b)

(6c)

t

  

dt  b4 t

−

(6d)



 

2



dt b4 t cos θ kα 1 t − t

− 0





K 0jl t − t  =







gkl4 gk41 e −i (λ j +δk1 )(t −t ) ,

(8a)

k





gkl4 gk42 e −i (λ j +δk1 +ω0 )(t −t ) .

(8b)

k

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

 Γ  δ t − t ,



g λ2 e −i δλ (t −t ) =

(9)

2

λ

where Γ = 2π | g λ |2 D (ωλ ) is the spontaneous-decay rate from level |4 to level |3, and D (ωλ ) is the vacuum-mode density at frequency ωλ in the free space. We consider the case of a doubleband anisotropic effective mass model of the PBG reservoir as shown in Fig. 1(b), which has an upper band, a lower band, and a band gap. The dispersion relations near the photonic band edges are approximated by [46]



2



2

ωk = ω g2 + A g2 (k − k0 ) , |k| > |k0 |,





+

 



+

√



α41 α4l exp{i [(δ41g2 − λ j )(t − t  ) + π4 ]}

2 4π (t − t  )3

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

K 0jl t − t  = ηl2

     dt  b4 t  cos θ sin θ e i ω0 t kα 2 t − t 

(10)

where A gl ≈ ω gl /|k0 |2 (l = 1, 2). Under the effective-mass approximation, the delay Green’s functions (8a) and (8b) can be expressed as [46]

K jl t − t  = η1l

(t −t  )

0

t



λ

0

t

g λ2 e −i δλ



K jl t − t  =

ωk = ω g1 − A g1 (k − k0 ) , |k| < |k0 |,

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

∂ b4 (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



λ

i

    dt  b4 t  cos2 θ k0β 2 t − t  ,

0

(5)

k

λ

     dt  b4 t  sin θ cos θ e −i ω0 t k0β 1 t − t 

0

t

       Ψ (t ) = b4 (t )|4{0} + b3 (t )|3{1λ } +

    dt  b4 t  sin2 θ k0α 2 t − t 

0



λ

     dt  b4 t  sin θ cos θ e −i ω0 t k0α 1 t − t 

0

pressions of the dressed state are

|α  = sin θ|2 + cos θ|1,

     dt  b4 t  sin θ cos θ e i ω0 t kβ 2 t − t 

0

ˆ A |β = λβ |β, where λα = 1/2(δ0 + H λβ = 1/2(δ0 −  δ02 + 4Ωo2 ) are the corresponding eigenvalues. The explicit exδ02

    dt  b4 t  sin2 θ kβ 1 t − t 

√



4π (t − t  )3

,

(11a)

α4l α42 exp{i [(δ42g2 − λ j )(t − t  ) + π4 ]}

2 4π (t − t  )3

 exp{i [(δ42g1 − λ j )(t − t  ) − π4 ]}

4π (t − t  )3

,

(11b)

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

where η1l and ηl2 indicate the quantum interference in atomic transitions coupled to the modified reservoir. For simplicity but without loss of generality, we assume η1l = ηl2 = 1. Here, α4l (l = 1, 2) is the modified coupling constant. δ41g2 = ω41 − ω g2 , δ41g1 = ω41 − ω g1 , δ42g2 = ω42 − ω g2 , and δ42g1 = ω42 − ω g1 represent the detuning of the atomic transition frequencies ω41 and ω42 from the upper and lower band-edge frequencies ω g2 and ω g1 . ∞ Taking the Laplace transformations [45] b4 (s) = 0 e −st b4 (t ) dt for Eq. (7), and s is the time Laplace transform variable, we have the result

Γ 2

b4 (s) − cos2 θ kα 1 (s)b4 (s) − cos θ sin θ kα 2 (s)b4 (s − i ω0 )

− sin2 θ kβ 1 (s)b4 (s) + cos θ sin θ kβ 2 (s)b4 (s − i ω0 ) − cos θ sin θ k0α 1 (s)b4 (s + i ω0 ) − sin2 θ k0α 2 (s)b4 (s) + cos θ sin θ k0β 1 (s)b4 (s + i ω0 ) − cos2 θ k0β 2 (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 Laplace transform of Eqs. (11a) and (11b) we have

K jl (s) = K 0jl (s) =

√

α41 α4l 

√

α4l α42 

 ( is + δ41g2 − λ j − i is + δ41g1 − λ j ), (13a)

2

 ( is + δ42g2 − λ j − i is + δ42g1 − λ j ), (13b)

2

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

b 4 (s) =

1+M +N s + Γ2 + sin2 θ [kβ 1 (s) + k0α 2 (s)] + cos2 θ [kα 1 (s) + k0β 2 (s)]

× b4 (0),

(14)

where





M = sin θ cos θ kβ 2 (s) − kα 2 (s) b04 (s − i ω0 ), N = sin θ cos θ k0β 1 (s) − k0α 1 (s) b04 (s + i ω0 ) and

b04 (s) =

1 . s + Γ2 + sin2 θ [kβ 1 (s) + k0α 2 (s)] + cos2 θ [kα 1 (s) + k0β 2 (s)]

First of all, we derive the long-time spontaneous emission spectra 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 transform for Eq. (6b), we have



2

S (δλ ) = D (ωλ )b3 (t → ∞) =

2 Γ  b4 (s = −i δλ ) .

2π

(15)

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

bα (t → ∞) = −igk14 cos θ b4 (−i δk1 − i λα )

− igk24 sin θ b4 (−i δk2 − i λα ), bβ (t → ∞) =





 

2 2 S (ωk ) = ρ (ωk ) bα (t → ∞) + bβ (t → ∞) ,

(17)

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

ρ (ωk ) =

1

ω − ω g2 Θ(ωk − ω g2 ) π k  + ω g1 − ωk Θ(ω g1 − ωk ) ,

(18)

with the Heaviside step function Θ . 3. Numerical results and analysis

sb4 (s) − b4 (0)

=−



1981

−igk14 sin θ b4 (−i δk1

(16a)

− i λβ )

+ igk24 cos θ b4 (−i δk2

− i λβ ).

The spontaneous emission spectrum S (ωk ) is given by

(16b)

In this section, we will discuss some properties about the spontaneous emission spectra via a few numerical calculations based on Eqs. (15) and (17) within the Markovian and non-Markovian reservoirs, respectively. All the parameters used in the following discussion are in units of γ , and γ is the decay rate for the transition from the upper level |4 to the lower level |3, i.e., Γ = γ , which should be in order of MHz for rubidium atom. And we assume the atom is initially prepared in level |4 [12], i.e., b4 (0) = 1. In the following discussions, we will use some of the detuning parameters, in which δk is the frequency detuning of the radiation field from the middle of the lower levels, i.e., δk = ωk − 0.5(ω41 + ω42 ). ωk − ω g2 = δk + 0.5δ41g2 + 0.5δ42g1 − 0.5δ g2g1 and ω g1 − ωk = −δk − 0.5δ41g2 − 0.5δ42g1 − 0.5δ g2g1 are the frequency detunings of the radiation field from the upper and lower band edges, the symbol δ g2g1 = ω g2 − ω g1 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 associated with the transitions |4 → |1 and |4 → |2 within the non-Markovian reservoir, and the level |1 is coupled to |2 by a microwave field. In the following, we will discuss the influence of the band-edge positions, as well as the intensity and detuning of the microwave field on the spontaneous emission spectra. For the case that both of the transitions coupled to the same modified reservoir from the microwave-field-coupled three-level Λ-type atom in PCs has been discussed in Ref. [40], and the case that one transition is coupled to the lower band and the other is coupled to the upper band in a double-band anisotropic PBG reservoir from the RF-driven five-level atom is also reported in Ref. [41]. However, the situation is quite different if one atomic transition is coupled to the lower band, the other is coupled to the upper band in a double-band anisotropic PBG reservoir, and the corresponding two lower 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 transition frequencies from the band edges in Fig. 2 when the detuning and intensity of the microwave-driven field is fixed, i.e., δ0 = 0 and Ω0 = 2. From Fig. 2 we can see that the spectral profile is fairly sensitive to the detunings of the atomic transition frequencies from the band edges. We can observe that the emission spectrum shows two peaks at both sides of δk = 0 when one transition frequency is within the upper band (δ41g2 = 3) and the other is within the lower band (δ42g1 = −3) as shown in Fig. 2(a). However, for the case that the transition frequency ω42 moves into the band gap (δ42g1 = 0.3) and the transition frequency ω41 is within the upper band (δ41g2 = 3), one of two peaks on the left-hand side is suppressed as can be seen from Fig. 2(b). When the transition frequency ω41 moves into the band gap (δ41g2 = −0.3) and the transition 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 of the atom are inside the band gap (δ41g2 = −0.3, δ42g1 = 0.3), the emission spectrum exhibits only one peak at both sides of

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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 g 14 = 1, g 24 = 1, δ0 = 0, Ω0 = 2, α41 = 0.5, α42 = 0.5, and Γ = 1, respectively. (a) (δ41g2 , δ42g1 , δ g2g1 ) = (3, −3, 3); (b) (δ41g2 , δ42g1 , δ g2g1 ) = (3, 0.3, 3); (c) (δ41g2 , δ42g1 , δ g2g1 ) = (−0.3, −3, 3); (d) (δ41g2 , δ42g1 , δ g2g1 ) = (−0.3, 0.3, 3).

δk = 0 [see Fig. 2(d)]. These interesting phenomena can be quantitatively explained by using the dressed-state picture. According to the dressed-state theory, the levels |1 and |2 are replaced by two dressed levels |α  and |β under the action of the microwave field, which are the linear superpositions of levels |1 and |2. Therefore, both transitions |4 → |1 and |4 → |2 consist of two transitions, respectively. From Eqs. (16a), (16b) and (17), we can see that the spectral structure of |bα |2 consists of two lines centered near δ41g2 − |Ω0 | and δ42g1 − |Ω0 |, while that of |bβ |2 consists of two lines centered near δ41g2 + |Ω0 | and δ42g1 + |Ω0 |, respectively. And the lower peak at the right side of δk = 0 and the 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 peak at the right side of δk = 0 and the lower peak on the left side of 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 get δ41g2 − |Ω0 | = 1 > 0, δ42g1 − |Ω0 | = −5 < 0, δ41g2 + |Ω0 | = 5 > 0 and δ42g1 + |Ω0 | = −1 < 0. These indicate that all the transition frequencies associated with |4 → |α  and |4 → |β fall outside the band gap. Therefore the emission spectrum shows the superposition of four Lorentzian shapes as shown in Fig. 2(a). When one transition frequency is inside the band gap and the other is outside the band gap (δ41g2 = 3 and δ42g1 = 0.3), the transition frequency of |4 → |β associated with the spectral line centered near δ42g1 + |Ω0 | = 2.3 > 0 falls within the band gap and the other three transition frequencies remain outside the band gap. Consequently, the lower peak on the left side of δk = 0 is inhibited and the emission spectrum has a three-peak structure as shown in Fig. 2(b). In the case of δ41g2 = −0.3 and δ42g1 = −3, the transition frequency of |4 → |α  associated with the spectral line centered near δ41g2 − |Ω0 | = −2.3 < 0 moves deeply into the band gap and the other three transition frequencies are outside the band gap, so there are only one peak at the right side of δk = 0 as can be seen from Fig. 2(c). From Fig. 2(d), we can find that when both transition 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  and the other with transition |4 → |β fall into the band gap and the other two transitions still outside the band gap, hence the two lower peaks move into band gap at the both sides of δk = 0 and the emission spectrum shows a two-peak structure.

Next, we will discuss the influence of Rabi frequency Ω0 of the microwave field on the spontaneous emission spectra for the case that δ41g2 = 3 and δ42g1 = −3 when the atom is embedded in the anisotropic double-band PBG reservoir. In Fig. 3(a), we can observe a symmetric double-peak structure [see dotted line in 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 reduced, while the peak on the right is nearly the same height as the case of Ω0 = 0 [see the solid line in Fig. 3(a)]. When the Rabi frequency increases from 0.5 to 2, each spontaneous emission peak splits into two as can be seen from Figs. 3(b) and 3(c). These phenomena can be explained as follows. Under the condition that Rabi frequency of the microwave field increases from 0.5 to 2, we get δ41g2 − |Ω0 | > 0, δ42g1 − |Ω0 | < 0, δ41g2 + |Ω0 | > 0 and δ42g1 + |Ω0 | < 0. These suggest that all the transition frequencies associated with |4 → |α  and |4 → |β are outside the band gap, therefore the spontaneous emission spectra show a four-peak structure. However, for the case of Ω0 = 0.5, the difference between δ41g2 − |Ω0 | and δ41g2 + |Ω0 | or the difference between δ42g1 − |Ω0 | and δ42g1 − |Ω0 | is very small, and the corresponding peaks overlap each other. As Ω0 increasing to 1, the corresponding differences reduce and the distance between the left (right) peaks becomes wider, hence the spontaneous emission spectra show a four-peak structure gradually [see Fig. 3(b)]. With Ω0 increasing further to 2, the emission spectra display a clear four-peak structure as shown in Fig. 3(c). However, it is more interesting that when Ω0 increases to 4, the two peaks (δ42g1 + |Ω0 | > 0 and δ41g2 − |Ω0 | < 0) near the band gap will move deeply into the band gap, and the emission spectra display one peak on both sides of δk = 0, respectively. In order to further explore the effect of the detuning of the microwave field on the spontaneous emission spectra, we also plot S (δk ) versus the detuning δk under the condition that both transition frequencies are outside the band gap as shown in Fig. 4. It is clearly shown that, when the microwave field is tuned to the resonant interaction with the atomic transition |1 ↔ |2, an higher peak and a lower peak can be observed simultaneously on both sides of δk = 0 [see Fig. 4(a)]. As the detuning of microwave field increases to 1 or 2, the two peaks near band gap are suppressed gradually compared with the situation at resonance [see Figs. 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 g 14 = 1, g 24 = 1, δ0 = 0, α41 = 0.5, α42 = 0.5, Γ = 1, δ41g2 = 3, δ42g1 = −3, δ g2g1 = 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 g 14 = 1, g 24 = 1, Ω0 = 2, α41 = 0.5, α42 = 0.5, Γ = 1, δ41g2 = 3, δ42g1 = −3, δ g2g1 = 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 = 0 is enhanced obviously as shown in Fig. 4(d). The reason for the phenomena we have been discussed above can be explained as follows. We take the third peak for example, when δ0 increases from 0 to 2, the eigenvalue of the dressed state λα increases gradually and one component of bα , i.e., δ41g2 − λα decreases correspondingly. That is to say, the peak near the band gap on the right side of δk = 0 moves to the band edge slowly and the height of peak is suppressed obviously as shown in Figs. 4(a)–(c). However, when δ0 = −2, the eigenvalue of the dressed state λα decreases greatly and δ41g2 − λα increases accordingly, as a result, the peak moves far from the band edge and the third peak corresponds to the line centered near δ41g2 −λα , which is enhanced obviously [see Fig. 4(d)].

3.2. The spontaneous emission spectra in the Markovian reservoir We now turn to what happens to the spontaneous emission spectra associated with the transitions |4 → |3 within the Markovian reservoir. In the following, we will discuss the influence of the band-edge positions, and the intensity of the microwave field on the spontaneous emission spectra. Firstly, we analyze the influence of both the lower and upper band-edge positions on the spontaneous emission spectra, considering the external microwave field resonates with the atomic transition |1 ↔ |2 and Rabi frequency Ω0 = 2. We plot the spontaneous emission spectra S (δλ ) versus the detuning δλ as shown in Fig. 5. For the case that one transition frequency ω41 is within the upper band (δ41g2 = 3) and the other ω42 is within the lower band (δ42g1 = −3) as the dashed line shown in Fig. 5,

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

Fig. 5. The spontaneous emission spectra S (δλ ) (in arbitrary units) for both transitions occur at different location of the gap for g 14 = 1, g 24 = 1, δ0 = 0, Ω0 = 2, α41 = 0.5, α42 = 0.5, and Γ = 1, respectively. Here, the spontaneous emission spectra of separate transitions are given within Markovian reservoir.

we can observe that the spontaneous emission spectra show a single-peak structure and the height of peak is small. When the transition 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 obviously [see the solid line in Fig. 5]. As the both transition frequencies ω41 and ω42 move into the band gap, i.e., δ41g2 = −0.3 and δ42g1 = 0.3, the spontaneous emission peak continues to enhance as the dotted line shown in Fig. 5. We can quantitatively explain the reason with spontaneous emission theory. From Fig. 1, we can see that there exist three emission channels from upper energy level |4 to lower energy levels |1, |2 and |3. When both transition frequencies ω41 and ω42 are outside the band gap (δ41g2 = 3 and δ42g1 = −3), the spontaneous emission in the PCs reservoir are strong, hence the free-space spontaneous emission is slightly suppressed. As one or both transition frequencies ω41 and ω42 move into PBG, we can find that the spontaneous emission from |4 to |1 and |2 are suppressed gradually. Consequently, the free-space spontaneous emission from |4 to |3 is greatly enhanced and the peak of emission spectra increases obviously as can be seen from the solid line and dotted line shown in Fig. 5. Finally, we investigate the influence of Rabi frequency of the microwave field on the free-space spontaneous emission spectra. As can be seen from Fig. 6, in the case of Ω0 = 1, the emission spectra display a single-peak structure, and the center of the peak is near δλ = 0 [see dotted line in Fig. 6]. For the case of Ω0 = 2, spontaneous emission spectra is slightly enhanced as the solid line shown in Fig. 6. While, under the condition of Ω0 = 4, it is very interesting that the free-space spontaneous emission is obviously enhanced and a new lower peak emerges on the right side of the zero detuning of δλ as can be seen from the dashed line in Fig. 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 accordingly and the constructive quantum interference between the decay channels of |4 to |α  and |4 to |β is suppressed. Therefore, the decay from |4 to |α  and |β reduces and the emission from |4 to |3 enhances accordingly. The right-hand lower peak [see the insets of Fig. 6] can be explained by quantum interference of the emission channels.

Fig. 6. The spontaneous emission spectra S (δλ ) (in arbitrary units) for different Rabi frequencies Ω0 of the microwave field for g 14 = 1, g 24 = 1, δ0 = 0, α41 = 0.5, α42 = 0.5, Γ = 1, δ41g2 = 3, δ42g1 = −3, and δ g2g1 = 3, respectively. Here, the spontaneous emission spectra of separate transitions are given within Markovian reservoir.

Before ending this section, let us briefly discuss the possible experimental realization of our proposed scheme by means of alkali-metal atoms, microwave field, and double-band anisotropic PCs. For example, we consider D 2 line of the cold 87 Rb atoms (nuclear spin I = 3/2) as a possible candidate [47]. The designated states can be chosen as follows: |1 = |5S 1/2 , F = 1, m F = 1, |2 = |5S 1/2 , F = 2, m F = 1, |3 = |5S 1/2 , F = 2, m F = 2 and |4 = |5P 1/2 , F = 2, m F = 1, respectively. In this case, a microwave field couples the allowed magnetic dipole transition between |1 = |5S 1/2 , F = 1, m F = 1 and |2 = |5S 1/2 , F = 2, m F = 1 with the frequency spacing ω21 = 8.6 GHz, while the transitions from the excited level |4 = |5P 1/2 , F = 2, m F = 1 to the lower levels |1 = |5S 1/2 , F = 1, m F = 1 and |2 = |5S 1/2 , F = 2, m F = 1 can be coupled by the lower and upper bands of the double-band anisotropic PBG reservoir, respectively. At the same time, the transition from the upper level |4 = |5P 1/2 , F = 2, m F = 1 to the lower level |3 = |5S 1/2 , F = 2, m F = 2 can be coupled by the vacuum-field mode in the free space. 4. Conclusion In summary, we have investigated the properties of spontaneous emission from a four-level atom embedded in a double-band anisotropic PBG material. It is clearly shown that the behavior of the spontaneous emission is strongly dependent on the relative position between the transition frequency and the PBG edge, the detunings and Rabi frequencies of the microwave field. On the one hand, for the spontaneous emission spectra in the non-Markovian reservoir, i.e., the transitions from the common upper level |4 to the two lower levels |1 and |2 interact with the upper and lower bands of the PBG reservoir, respectively. By adjusting the system parameters, we can obtain some interesting phenomena: (i) When transition frequency ω41 is within the upper band and the other ω42 is within the lower band, the spectral line is a four-peak structure. While the transition frequency ω42 (ω41 ) moves into the band gap and the other transition frequency is outside the band gap, one of two peaks on the left(right)-hand side is suppressed. When both transition frequencies of the atom are inside the band gap, the emission spectrum exhibits only a double-peak structure. (ii) The emission spectrum shows a double-peak structure in the absence of the microwave field. As Rabi frequency increases gradually, each spontaneous emission peak splits into two peaks and

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

the separation between the two peaks becomes wider. (iii) The number and height of the spontaneous emission peaks can be controlled effectively by regulating the detuning of the microwave field. On the other hand, for the spontaneous emission spectra in the Markovian reservoir, the spectra exhibit a single-peak structure whether two transition frequencies ω41 and ω42 are in or outside the band gap. While when two transition frequencies ω41 or ω42 move from energy band to band gap, the spontaneous emission from |4 to |3 increases obviously. Similarly, the spontaneous emission from |4 to |3 increases with the increasing of Rabi frequency of microwave field, but when Rabi frequency Ω0 increases to a certain value, there will be a new lower peak at the right side of the main peak due to quantum interference channels. These results show that the spectral behaviors are very sensitive to the positions of the band edges, Rabi frequency and the detuning of the microwave field, so we can control the spontaneous emission by adjusting these system parameters under realistic experimental conditions. Finally, we would like to mention that the application of the PCs offers us further flexibility to manipulate the light–matter interactions, including the manipulation of spontaneous emission here, since the PBG materials can be used to better control the atomic spontaneous emission. Moreover, these investigations open up possibilities for various applications of PCs using spontaneous emission properties. Acknowledgements The research is supported in part by the National Natural Science Foundation of China (Grant Nos. 10975054, 91021011 and 11004069), by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081 and by National Basic Research Program of China under Contract No. 2012CB922103. We would like to thank Professor Ying Wu for helpful discussion and his encouragement. References [1] A.S. Zibrov, M.D. Lukin, D.E. Nikonov, L. Hollberg, M.O. Scully, V.L. Velichansky, H.G. Robinson, Phys. Rev. Lett. 75 (1995) 1499. [2] J.Y. Gao, C. Guo, X.Z. Guo, G.X. Jin, Q.W. Wang, J. Zhao, H.Z. Zhang, Y. Jiang, D.Z. Wang, D.M. Jiang, Opt. Commun. 93 (1992) 323.

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