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Coherent control of spontaneous emission and radiation property in an anisotropic PBG
Optics Communications 283 (2010) 3714–3720
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Coherent control of spontaneous emission and radiation property in an anisotropic PBG Li Jiang, Shan Du, Ren-Gang Wan, Jun Kou, Han Zhang, Han-Zhuang Zhang ⁎ College of Physics, Jilin University, Changchun 130023, PR China
a r t i c l e
i n f o
Article history: Received 11 March 2010 Received in revised form 7 May 2010 Accepted 7 May 2010
a b s t r a c t The spontaneous emission and radiation properties of a three-level atom driven by a phase-sensitive laser field in the anisotropic photonic crystal have been investigated. Due to the modified density in PC and the dynamical coherence induced by the coupling field, the population becomes oscillated and a fractionalized steady-state inversion appears in the two upper levels under certain initial conditions. And the spontaneous spectra are influenced not only by the photonic band edge but also by the phase and the detuning of the coherent field. In addition, the phase dependent quantum interference leads to fluorescence quenching and enhancement in the spectra. Meanwhile, the radiation fields are greatly dependent on the intensity of the coherent field and the relative position of the resonant frequency to the band edge. Different kinds of fields within the regions we plotted are clearly calculated. When the resonant frequency moves from deep in the band gap to the band, localized field, diffusion field and propagating field regularly vary and the energy of them is exchanged to each other. © 2010 Published by Elsevier B.V.
1. Introduction Spontaneous emission of light results from the electromagnetic field's vacuum fluctuations and it is decided by the local density of the modes [1–3]. If the reservoir changes, such as near the reflecting surface or inside a microwave cavity, the spontaneous emission from the atom or molecule can be greatly modified and many novel phenomena will appear. Photonic crystal, as a new type material, is first discovered by S John [4] and E Yablonovitch [5] in the 1980s. This artificial periodic material has special density of the states which can greatly affect the atomic action [6]. The electromagnetic wave propagation in it within a range of frequencies is classically forbidden [7]. In a previous work, when a single atom is placed near the band edge where strong localization of light exists, many interesting phenomena have been studied, such as the photon-atom dressed state [8], the suppression and even complete cancellation or enhancement of the spontaneous emission [9–13], a fractionalized steady-state inversion [14], the occurrence of dark lines in spontaneous emission [15,16], electromagnetically induced transparency [17–19] and so on. Researchers also pay their attentions to the influence of the external fields on the atom in the PBG, which is so different from the case in the vacuum. If one of the two levels of the atom is coupled to a third level by a laser field with a specific phase relation to the pump laser pulse, the spontaneous emission may alter associated with the Aulter-
⁎ Corresponding author. E-mail address: [email protected] (H.-Z. Zhang). 0030-4018/$ – see front matter © 2010 Published by Elsevier B.V. doi:10.1016/j.optcom.2010.05.014
Townes doublets of the upper levels. The relative work has been done in the vacuum [20] and in the isotropic photonic crystal [21] from the theoretical point of view. Meanwhile the EIT phenomenon in the realistic atom driven by a 1064-nm laser beam is investigated by Harris [22]. In this paper, we consider a driven three-level atom embedded in anisotropic photonic crystal. The properties of the spontaneous emission and the radiation fields are analyzed here. Theoretical study is first presented and we find that the interference effect can lead to population trapping and fluorescence quenching with the suitable parameters. With the numerical calculation, the radiation field emitted from the atom is dependent on the intensity of the coherent field and the detuning of the resonant frequency from the band edge. The detailed expressions of the radiation field including localized, diffusion and propagating field are clearly given in the following parts. As a result, this phase-sensitive atomic system can be implemented for the optical memory device. 2. Basic theory The model considered in this paper is a three-level atom which consists of two upper levels |2〉,|3〉 and the ground level |1〉. The two upper levels are driven by a coherent laser field with the complex phase in the form of Rabi frequency Ω0 = |Ω0|eiθ, and the transition frequency between level |2〉 and ground level |1〉 is assumed to be near an anisotropic photonic band edge as shown in Fig. 1. In this model, the spontaneous emission of |3〉 → |2〉 and |3〉 → |1〉 are ignored just as referred in [21].
L. Jiang et al. / Optics Communications 283 (2010) 3714–3720
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where the functions are defined as follows:
f ðxÞ = C2 ð0Þ−
⁎
Ω0 C3 ð0Þ s−iΔ0
P ðxÞ = −x + iðω21c −Δ0 Þ
Fig. 1. A driven three-level atom (a) in the bare state picture; (b) in the dressed state picture.
2 Q ðxÞ = ð−x + iðω21c −Δ0 ÞÞð−x + iω21c Þ + jΩ0 j
iβ jΩ0 j ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + GðxÞ = x− pffiffiffiffiffiffi ; Im xj β x−iΔ0 ωc + −ix−ω21c ð1Þ N ω21 −ωc ; Re xj β = 0 3=2
Performing the rotating wave and the dipole approximation, the Hamiltonian of the system in the interaction picture can be written as −iδ t † iδ t † † HI = iℏ∑kλ gkλ e k bkλ a2 a1 −e k bkλ a1 a2
ð1Þ
−iΔ t † ⁎ iΔ t † + iℏ Ω0 e 0 a3 a2 −Ω0 e 0 a2 a3
Here b†kλ and bkλ are the creation and annihilation operators for the kth vacuum mode with the frequency ωk, and the frequency-dependent coupling constant between the resonant transitionffi and the mode {kλ} of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi radiation field is gkλ = ω21 d21 = ℏ ℏ = ð2ε0 ωk V0 Þ eˆkλ ⋅ uˆ d , in which d21 and ûd are the magnitude and unit vector of the atomic dipole moment, V0 is the quantized volume, and êkλ is polarization unit vector. The detuning parameters are defined as δk = ωk − ω21 and Δ0 = ω0 − ω32. The state vector of the system at time t can be written as
jΨðt Þ N = C2 ðt Þ j2; f0g N + C3 ðt Þj3; f0g N + ∑kλ C1k ðt Þj1; fkλg N ð2Þ Substituting Eqs. (1) and (2) into the Schrödinger equation, we can obtain the first-order differential equations for the amplitudes C˙ 1k ðt Þ = −gkλ eiδk t C2 ðt Þ ð3Þ
⁎ iΔ t −iδ t C˙ 2 ðt Þ = −Ω0 e 0 C3 ðt Þ + ∑kλ gkλ e k C1k ðt Þ
Then with the Laplace transformation method, the amplitude C2(t) is changed as
3=2 2 iβ jΩ0 j ð2Þ ; Im xj β F ðxÞ = x− pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + x−iΔ ωc −i ix + ω21c 0 ð2Þ b ω21 −ωc ; Re xj β b0
(2) Here x(1) are the imaginary and complex roots of the equation j , xj G(x) = 0 and F(x) = 0. From the expression derived above, we can see that time amplitude is greatly dependent on the properties and numbers of the roots, which are directly due to the intensity of the coherent field and the relative position of the resonant frequency to the band edge. With the help of numerical calculation, there are at most two roots and at least one root in regions plotted according to the properties of roots, as shown in Fig. 2. There are two type roots in (1) it: the imagery root x(1) with Im(x(1) j j β) N ω21 − ωc, Re(xj β) = 0 and (2) (2) the complex root x(2) with Im(x β) b ω − ω , Re(x β) b 0. In turns, 21 c j j j there are two pure imaginary roots x(1) in the region I, one x(1) in the j j region II, one x(1) and one complex root x(2) in the region III, one x(2) in j j j the region IV, and two x(2) in the region V. By this work, it is helpful to j analyze the spontaneous spectra and the compositions of the emitted fields.
In this section, we concentrate our attention on the emitted fields from the process of the transition |2〉 → |1〉. The radiation field amplitude at a particular space point r is written as → E r; t = ∑ kλ
" #" #−1 Ω⁎ C ð0Þ jΩ0 j 2 C˜ 2 ðsÞ = C2 ð0Þ− 0 3 s+Γ+ s−iΔ0 s−iΔ0 2 gkλ s + iδk
ð6Þ
3. The emitted field
−iΔ t C˙ 3 ðt Þ = Ω0 e 0 C2 ðt Þ
in which Γ = ∑kλ
2
= pffiffiffiffiffiffi
−iβ3 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −is−ω21c
ωc +
ð4Þ
is the Laplace form of
the delay function. The inverse transform of Eq. (4) yields ð1Þ ð2Þ f xj f xj ð1Þ ð2Þ xj βt x βt C2 ðt Þ = ∑j e + ∑j e j ð1Þ ð2Þ ′ ′ G xj F xj ð5Þ h i pffiffiffiffi ⁎ eiω21c t ∞ ixðωc −ixÞP ðxÞ C2 ð0ÞP ðxÞ−Ω0 C3 ð0Þ −xβt ∫0 e + dx pffiffiffiffiffiffi π Q ðxÞðωc −ixÞ−iP ðxÞ ωc 2 + ixP 2 ðxÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffi → ℏωk −i ωt−→ k⋅r e C1k ðt Þ eˆkλ 20 V0
ð7Þ
Based on Eqs. (5) and (7), we can derive the amplitude of the emitted field as → → → E r; t = El r; t + Ep r; t + ∑ Edj xj + Epj xj + Ed
ð8Þ
j
which means that the total radiation fields are composed of three parts: localized field El(→ r ,t), propagating field Ep(→ r ,t) and diffusion field Ed. With numerical calculation, we find that the characteristic of the emitted fields is related to the number and properties of the roots which obviously have two forms: the imagery root with Im(x(1) j β) N ω21 − ωc (2) and the complex root x(2) with Im(x(2) j j β) b ω21 − ωc, Re(xj β) b 0.
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Fig. 2. Regions of the root distribution with ωc = 100β, Δ0 = 0.1β.
(1) If there is a pure imagery root x(1) with b(1) j = ibj j N ω21 − ωc, the emitted field E(r ,t) can be derived as
→ → Eð r; tÞ = Ep ð r; tÞ + Epd + Ed
→ → E r; t = El r; t + Eld + Ed → E0 r =
El = E0
iki r ω21 d21 ∑i e 0 8π2 0 ri
→ u21 −
ð2Þ
Ep = E0
→→ → ! k k ⋅ d21
ð1Þ
G
′
ð1Þ ðxj Þ
∞ ×∫−∞
−iðωc t−r 2 = ð4AtÞÞ +
⋅e
3πi
ð9Þ
! 2
−Atρ ρe 4 + r = ð2AtÞ e dρ
!2
ð1Þ xj β + iðωc −ω21 Þ + iA ρe 4 + r = ð2AtÞ
Ed =
πi 2 ∞ E ðrÞ −iω t + ir = ð4AtÞ− 4 ∫−∞ − 0 e c π
ð2Þ ðxj Þ
ð2Þ
ve = −aj
3πi 4
3πi
AF
′
ð2Þ
ð2Þ
−iðω21 −bj Þðt−r = vp Þ + aj ðt−ve Þ
⋅e
Θ
! 3πi pffiffiffiffiffi −ρ2 ρe 4 + r = 2 At e dρ 3iπ pffiffiffiffiffi ð1Þ −xj βt + i ρe 4 + r =2 At
ð10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ ð2Þ vp = ðω21 −bj βÞ A = Im ω21 −ωc + ixj β;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi × −r + 2 At ðRe−ImÞ ωc −ω21 + bð1Þ β f ðxj Þ
πf ðxj Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ð2Þ ×ð−r + 2 At ðRe+ImÞ ωc −ω21 −ixj βÞ
k2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ πf xj iðb β−ω21 Þt−r ðωc −ω21 + bð1Þ βÞ = A ⋅e Θ ð1Þ ′ AG xj
Edl = E0
Meantime, if there is a complex root x(2) = a(2) + ib(2) with j j j (2) b(2) b ω − ω , a b 0, the emitted field can be written as j 21 c j
!2
pffiffiffiffi ⁎ ∞ ixðωc −ixÞPðxÞ½C2 ð0ÞPðxÞ−Ω0 C3 ð0Þ ⋅∫0 dx pffiffiffiffiffiffi 2 ½Q ðxÞðωc −ixÞ−iPðxÞ ωc Þ + ixP 2 ðxÞ
These formulas indicate that the frequency of the emitted field El is b(1) j − ω21 which is less than ωc. We call it localized field within the forbidden band, which represents that the spontaneous photons can't propagate far away from the atom within the range of localized length qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ l = A = ðωc −ω21 + bj Þ and the field's amplitude drops exponentially along the distance from the atom. This conclusion is consistent with the population and spontaneous spectra which are discussed in the next section. In addition, the last two parts of the emitted field are diffusion fields that show power-law decay in the radiation process, which are coming from the integral calculation.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ A = Re ω21 −ωc + ixj β
The frequency of this field is b(2) j − ω21 N ωc within the traveling band. The first term of the field is the propagating field that can transmit out of the photonic crystal in the form of a traveling pulse ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ ð2Þ with the energy velocity ve = −aj A = Re ω21 −ωc + ixj β and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ ð2Þ the phase velocity vp = ðω21 −bj βÞ A = Im ω21 −ωc + ixj β. Owing to the minus real part of the root, the propagating field decays along with time. The last two parts are also the diffusion fields. Attribute to the regions plotted in Fig. 2, we know that the main parts of the emitted fields can be totally written as 8 E + E + E + Edl2 + Ed > > > El1 + El2 + Edl1 > < l1 dl1 d → E r; t = El1 + Ep1 + Edl1 + Edp1 + Ed > > E + Edp1 + Ed > > : p1 Ep1 + Ep2 + Edp1 + Edp2 + Ed
ðRegion IÞ ðRegion IIÞ ðRegion IIIÞ ðRegion IVÞ ðRegion VÞ
So far, the definite expressions of the emitted fields are clear. When the resonant frequency is deep in the photonic band gap just in region I or II, most of the emitted fields are localized fields. Compared with the localized field, diffusion fields are very small. It is to say that the emitted photons are localized around the atom. But if we move the resonant frequency into the traveling band, the emitted photon can propagate out. Both of the propagating field and diffusion field make effects on the traveling process. During this process, the energy of the propagating field is partly transferred into the diffusion field due to the influence of the photonic crystal.
L. Jiang et al. / Optics Communications 283 (2010) 3714–3720
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Fig. 3. The population of the two upper levels P2, P3 and total population P (dash solid, dash-dot line respectively) as a function of the scaled time βt for different resonant transition frequencies with (a) ω21 = 99.5β, (b) ω21 = 100.4β, and (c) ω21 = 101β. The other parameters are ωc = 100β, C2(0) = 1, C3(0) = 0, Δ0 = 0.1β, Ωc = 0.5βexp(iθ) where θ is any value.
4. The time-evolution properties and spontaneous emission 4.1. Time-evolution of the population The phase and detuning of the coherent field, and the atomic initial states play an important role in the exchanging process between the photon and the atom in an anisotropic photonic crystal. The population in every level and spontaneous emission spectra are used to analyze the influences. The time-evolution of the population in the upper level is obtained by Pi(t) = |Ci(t)|2, (i = 2, 3). For the complicated factors, there are many new interesting phenomena depicted in the figures. First, as shown in Fig. 3, if the initial population is completely in one of the upper levels, whatever the phase of external field is, it can't make any effect on the time-evolution of the population. And what's more, the oscillation behaviors are found in P2 and P3 due to the interference
between the dressed state and quasi-dressed state that comes from the single-valued branch-cut contribution at the band edge frequency. When the resonant frequency ω21 moves from the inside to the outside of the band edge, the amplitudes of the time-evolution decay much faster. While ω21 = 99.5β is deep in the forbidden band gap in region I in Fig. 2, there are two imagery roots. So no decay in the amplitudes appears in Fig. 3(a), which represents that photons in the status of emitting and reabsorbing to the atom can't propagate out. But when ω21 = 100.4β (in region III) with one imagery root and one complex root or ω21 = 101β (in region V) with two complex roots, time decay comes out. The difference between these two regions is that the population in the upper levels will eventually stay invariable in the former region, but total population will decay to zero in the latter. This is because when the resonant frequency is beside the band edge, due to the coherent field, the upper level forms two dressed states with one in the band gap and the other in the band. Then part of
Fig. 4. The population of the two upper levels P2, P3 and total population P (dash solid, dash-dot line respectively) as a function of the scaled time βt for different phases of the coherent field with (a) θ = 0, (b) θ = π/2, (c) θ = π, and (d) θ = 3π/2. The other parameters are ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Δ0 = 0.1β, |Ωc| = 0.5β, ω21 = 100.4β.
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the population will be trapped in the dressed state which is in the band gap. However, when the initial population distributes in both of the upper levels, the phase plays an important role in the amplitude of the population's exchange between the two upper levels just as shown in Fig. 4. When we set the resonant frequency beside the band edge and change the phase from 0 to 3π/2 every other π/2, we can see that the population exchange between the two upper levels is present estimated periodic oscillation per 2π corresponding to the detail process: maximum exchange and fractional population trapping when θ = 0 in Fig. 4(a), small exchange and almost entire population trapping when θ = π/2 in Fig. 4(b), the opposite population exchange different from the case when θ = 0 and fractional population trapping when θ = π in Fig. 4(c), few exchange and non-population trapping in upper levels when θ = 3π/2 in Fig. 4(d). 4.2. Spontaneous emission From the formula S(ωk) = D(ωk) ⋅ |C1k(∞)|2 and Eq. (4), we can obtain the spectra as
Sðδk Þ =
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωk −ωc ⋅Θðωk −ωc Þ
j
C2 ð0Þðδk + Δ0 Þ−iΩ⁎0 C3 ð0Þ 3=2
iβ ðδk +Δ0 Þ −iδk ðδk + Δ0 Þ−qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωc + ωc −ω21 −δk
+ i jΩ0 j
2
j
2
ð11Þ As shown in Fig. 5, when the resonant frequency moves towards the photonic band edge, the spontaneous emission peak changes as follows: the long "tail" line deep in the band gap, one peak slightly outside the band edge, two peaks far from the band edge in Fig. 5(a),
(b),(c). This phenomenon is very easy to be understood by the influence of the photonic crystal and the coherent field. When the external field is added to the upper level |2〉, this level is split into two dressed states, which refers to the Autler-Townes splitting [23] as shown in Fig. 1(b) When the upper level is deep in the band gap, it means two dressed states are also in the band gap and there will be no spontaneous emission from the atom as shown in Fig. 5(a). Then as this level moves to the band edge or far away from the band edge, one or two of the dressed states is in the band, so spontaneous emission peak will appear in the spectrum such as Fig. 5(b),(c). However, after considering the phase of external field, the spontaneous spectra are also influenced by its values just as shown in Fig. 5(c),(d) and Fig. 6(a). Directly, we can deduce that the larger phase value makes the left peak increase and the right peak decrease. More importantly, a fluorescence quenching happens on the condition in Fig. 5(a). Destructive interference leads to this result and we can calculate the quenching point which can be derived from the expression C2(0)(δk + Δ0) − iΩ0⁎C3(0) = 0. If we change the parameter of the external field's detuning and keep other parameters unchanged, the destructive interference changes into the constructive interference and the right peak appears again. Meanwhile, when we set the resonant frequency beside the band edge and change the phase of the driven field from 0 to 3π/2 as shown in Fig. 5(b) and Fig. 7, the spontaneous peaks change in period of 2π and the phenomenon of the fluorescence quenching and tail-like spontaneous spectra with the phase of π/2 in Fig. 7(a), the same spontaneous emission line with the phase of 0 and π in Fig. 5(b) and Fig. 7(b), the maximum spontaneous peak with the phase of 3π/2 in Fig. 7(c) are observed. In substance, when the phase is π/2, destructive interference between two channels of the dressed states as shown in Fig. 1(b) leads to fluorescence quenching, no spontaneous emission and population trapping in the
Fig. 5. The spontaneous emission spectrum (in arbitrary units) as the function of the detuning δk. (a) ω21 = 99.5β, θ = 0, (b) ω21 = 100.4β, θ = 0, (c) ω21 = 101β, θ = 0, and (d) ω21 = 101β, θ = π/4. The other parameters are chosen as ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Δ0 = 0.1β, δk = ωk − ω21, |Ωc| = 0.5β.
L. Jiang et al. / Optics Communications 283 (2010) 3714–3720
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Fig. 6. The spontaneous emission spectrum (in arbitrary units) as the function of the detuning δk. (a) Δ0 = 0.1β, (b) Δ0 = 0. The other parameters are chosen as ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Ωc = 0.5βexp(iπ/2), ω21 = 101β.
Fig. 7. The spontaneous emission spectrum (in arbitrary units) as the function of the detuning δk. (a) θ = π/2, (b) θ = π, (c) θ = 3π/2. The other parameters are chosen as ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Δ0 = 0.1β, |Ωc| = 0.5β, ω21 = 100.4β.
upper levels in Fig. 4(b). In the same way, when the phase is 3π/2, constructive interference between the two channels makes the spontaneous emitting maximum and no population trapping in Fig. 4(d). So we can make the conclusion that under the coherent field's phase condition, the interference effects are much stronger than the band edge localization effects on the atomic system's spontaneous emission.
5. Conclusion We have investigated the properties of radiation field and spontaneous emission from a coherent three-level atom in an anisotropic crystal. As the relative position of the upper level to the band edge or the intensity of the coherent field increase, components of the radiation fields regularly vary and the detailed expressions are clearly given. The variety of the emitted fields can be used to design the optical multi-channels switch. With the help of Fourier transform, on the viewpoint of the time-evolution properties and spontaneous spectrum, we analyze the effects of the photonic crystal and the phase-sensitive coherent field on the emitted modes in different regions. The results analyzed here are consistent with the distribution of emitted fields. Meanwhile, quantum interference plays an im-
portant role in the process and leads to population oscillation and fluorescence quenching with suitable parameters. As a result, this atomic system with phase-sensitive may be implemented for the optical memory device. Acknowledgement This work was supported by the National Natural Science Foundation of China (NSFC) (grants 10974071, 10774060, J0730311). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Coherent control of spontaneous emission and radiation property in an anisotropic PBG Li Jiang, Shan Du, Ren-Gang Wan, Jun Kou, Han Zhang, Han-Zhuang Zhang ⁎ College of Physics, Jilin University, Changchun 130023, PR China
a r t i c l e
i n f o
Article history: Received 11 March 2010 Received in revised form 7 May 2010 Accepted 7 May 2010
a b s t r a c t The spontaneous emission and radiation properties of a three-level atom driven by a phase-sensitive laser field in the anisotropic photonic crystal have been investigated. Due to the modified density in PC and the dynamical coherence induced by the coupling field, the population becomes oscillated and a fractionalized steady-state inversion appears in the two upper levels under certain initial conditions. And the spontaneous spectra are influenced not only by the photonic band edge but also by the phase and the detuning of the coherent field. In addition, the phase dependent quantum interference leads to fluorescence quenching and enhancement in the spectra. Meanwhile, the radiation fields are greatly dependent on the intensity of the coherent field and the relative position of the resonant frequency to the band edge. Different kinds of fields within the regions we plotted are clearly calculated. When the resonant frequency moves from deep in the band gap to the band, localized field, diffusion field and propagating field regularly vary and the energy of them is exchanged to each other. © 2010 Published by Elsevier B.V.
1. Introduction Spontaneous emission of light results from the electromagnetic field's vacuum fluctuations and it is decided by the local density of the modes [1–3]. If the reservoir changes, such as near the reflecting surface or inside a microwave cavity, the spontaneous emission from the atom or molecule can be greatly modified and many novel phenomena will appear. Photonic crystal, as a new type material, is first discovered by S John [4] and E Yablonovitch [5] in the 1980s. This artificial periodic material has special density of the states which can greatly affect the atomic action [6]. The electromagnetic wave propagation in it within a range of frequencies is classically forbidden [7]. In a previous work, when a single atom is placed near the band edge where strong localization of light exists, many interesting phenomena have been studied, such as the photon-atom dressed state [8], the suppression and even complete cancellation or enhancement of the spontaneous emission [9–13], a fractionalized steady-state inversion [14], the occurrence of dark lines in spontaneous emission [15,16], electromagnetically induced transparency [17–19] and so on. Researchers also pay their attentions to the influence of the external fields on the atom in the PBG, which is so different from the case in the vacuum. If one of the two levels of the atom is coupled to a third level by a laser field with a specific phase relation to the pump laser pulse, the spontaneous emission may alter associated with the Aulter-
⁎ Corresponding author. E-mail address: [email protected] (H.-Z. Zhang). 0030-4018/$ – see front matter © 2010 Published by Elsevier B.V. doi:10.1016/j.optcom.2010.05.014
Townes doublets of the upper levels. The relative work has been done in the vacuum [20] and in the isotropic photonic crystal [21] from the theoretical point of view. Meanwhile the EIT phenomenon in the realistic atom driven by a 1064-nm laser beam is investigated by Harris [22]. In this paper, we consider a driven three-level atom embedded in anisotropic photonic crystal. The properties of the spontaneous emission and the radiation fields are analyzed here. Theoretical study is first presented and we find that the interference effect can lead to population trapping and fluorescence quenching with the suitable parameters. With the numerical calculation, the radiation field emitted from the atom is dependent on the intensity of the coherent field and the detuning of the resonant frequency from the band edge. The detailed expressions of the radiation field including localized, diffusion and propagating field are clearly given in the following parts. As a result, this phase-sensitive atomic system can be implemented for the optical memory device. 2. Basic theory The model considered in this paper is a three-level atom which consists of two upper levels |2〉,|3〉 and the ground level |1〉. The two upper levels are driven by a coherent laser field with the complex phase in the form of Rabi frequency Ω0 = |Ω0|eiθ, and the transition frequency between level |2〉 and ground level |1〉 is assumed to be near an anisotropic photonic band edge as shown in Fig. 1. In this model, the spontaneous emission of |3〉 → |2〉 and |3〉 → |1〉 are ignored just as referred in [21].
L. Jiang et al. / Optics Communications 283 (2010) 3714–3720
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where the functions are defined as follows:
f ðxÞ = C2 ð0Þ−
⁎
Ω0 C3 ð0Þ s−iΔ0
P ðxÞ = −x + iðω21c −Δ0 Þ
Fig. 1. A driven three-level atom (a) in the bare state picture; (b) in the dressed state picture.
2 Q ðxÞ = ð−x + iðω21c −Δ0 ÞÞð−x + iω21c Þ + jΩ0 j
iβ jΩ0 j ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + GðxÞ = x− pffiffiffiffiffiffi ; Im xj β x−iΔ0 ωc + −ix−ω21c ð1Þ N ω21 −ωc ; Re xj β = 0 3=2
Performing the rotating wave and the dipole approximation, the Hamiltonian of the system in the interaction picture can be written as −iδ t † iδ t † † HI = iℏ∑kλ gkλ e k bkλ a2 a1 −e k bkλ a1 a2
ð1Þ
−iΔ t † ⁎ iΔ t † + iℏ Ω0 e 0 a3 a2 −Ω0 e 0 a2 a3
Here b†kλ and bkλ are the creation and annihilation operators for the kth vacuum mode with the frequency ωk, and the frequency-dependent coupling constant between the resonant transitionffi and the mode {kλ} of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi radiation field is gkλ = ω21 d21 = ℏ ℏ = ð2ε0 ωk V0 Þ eˆkλ ⋅ uˆ d , in which d21 and ûd are the magnitude and unit vector of the atomic dipole moment, V0 is the quantized volume, and êkλ is polarization unit vector. The detuning parameters are defined as δk = ωk − ω21 and Δ0 = ω0 − ω32. The state vector of the system at time t can be written as
jΨðt Þ N = C2 ðt Þ j2; f0g N + C3 ðt Þj3; f0g N + ∑kλ C1k ðt Þj1; fkλg N ð2Þ Substituting Eqs. (1) and (2) into the Schrödinger equation, we can obtain the first-order differential equations for the amplitudes C˙ 1k ðt Þ = −gkλ eiδk t C2 ðt Þ ð3Þ
⁎ iΔ t −iδ t C˙ 2 ðt Þ = −Ω0 e 0 C3 ðt Þ + ∑kλ gkλ e k C1k ðt Þ
Then with the Laplace transformation method, the amplitude C2(t) is changed as
3=2 2 iβ jΩ0 j ð2Þ ; Im xj β F ðxÞ = x− pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + x−iΔ ωc −i ix + ω21c 0 ð2Þ b ω21 −ωc ; Re xj β b0
(2) Here x(1) are the imaginary and complex roots of the equation j , xj G(x) = 0 and F(x) = 0. From the expression derived above, we can see that time amplitude is greatly dependent on the properties and numbers of the roots, which are directly due to the intensity of the coherent field and the relative position of the resonant frequency to the band edge. With the help of numerical calculation, there are at most two roots and at least one root in regions plotted according to the properties of roots, as shown in Fig. 2. There are two type roots in (1) it: the imagery root x(1) with Im(x(1) j j β) N ω21 − ωc, Re(xj β) = 0 and (2) (2) the complex root x(2) with Im(x β) b ω − ω , Re(x β) b 0. In turns, 21 c j j j there are two pure imaginary roots x(1) in the region I, one x(1) in the j j region II, one x(1) and one complex root x(2) in the region III, one x(2) in j j j the region IV, and two x(2) in the region V. By this work, it is helpful to j analyze the spontaneous spectra and the compositions of the emitted fields.
In this section, we concentrate our attention on the emitted fields from the process of the transition |2〉 → |1〉. The radiation field amplitude at a particular space point r is written as → E r; t = ∑ kλ
" #" #−1 Ω⁎ C ð0Þ jΩ0 j 2 C˜ 2 ðsÞ = C2 ð0Þ− 0 3 s+Γ+ s−iΔ0 s−iΔ0 2 gkλ s + iδk
ð6Þ
3. The emitted field
−iΔ t C˙ 3 ðt Þ = Ω0 e 0 C2 ðt Þ
in which Γ = ∑kλ
2
= pffiffiffiffiffiffi
−iβ3 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −is−ω21c
ωc +
ð4Þ
is the Laplace form of
the delay function. The inverse transform of Eq. (4) yields ð1Þ ð2Þ f xj f xj ð1Þ ð2Þ xj βt x βt C2 ðt Þ = ∑j e + ∑j e j ð1Þ ð2Þ ′ ′ G xj F xj ð5Þ h i pffiffiffiffi ⁎ eiω21c t ∞ ixðωc −ixÞP ðxÞ C2 ð0ÞP ðxÞ−Ω0 C3 ð0Þ −xβt ∫0 e + dx pffiffiffiffiffiffi π Q ðxÞðωc −ixÞ−iP ðxÞ ωc 2 + ixP 2 ðxÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffi → ℏωk −i ωt−→ k⋅r e C1k ðt Þ eˆkλ 20 V0
ð7Þ
Based on Eqs. (5) and (7), we can derive the amplitude of the emitted field as → → → E r; t = El r; t + Ep r; t + ∑ Edj xj + Epj xj + Ed
ð8Þ
j
which means that the total radiation fields are composed of three parts: localized field El(→ r ,t), propagating field Ep(→ r ,t) and diffusion field Ed. With numerical calculation, we find that the characteristic of the emitted fields is related to the number and properties of the roots which obviously have two forms: the imagery root with Im(x(1) j β) N ω21 − ωc (2) and the complex root x(2) with Im(x(2) j j β) b ω21 − ωc, Re(xj β) b 0.
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Fig. 2. Regions of the root distribution with ωc = 100β, Δ0 = 0.1β.
(1) If there is a pure imagery root x(1) with b(1) j = ibj j N ω21 − ωc, the emitted field E(r ,t) can be derived as
→ → Eð r; tÞ = Ep ð r; tÞ + Epd + Ed
→ → E r; t = El r; t + Eld + Ed → E0 r =
El = E0
iki r ω21 d21 ∑i e 0 8π2 0 ri
→ u21 −
ð2Þ
Ep = E0
→→ → ! k k ⋅ d21
ð1Þ
G
′
ð1Þ ðxj Þ
∞ ×∫−∞
−iðωc t−r 2 = ð4AtÞÞ +
⋅e
3πi
ð9Þ
! 2
−Atρ ρe 4 + r = ð2AtÞ e dρ
!2
ð1Þ xj β + iðωc −ω21 Þ + iA ρe 4 + r = ð2AtÞ
Ed =
πi 2 ∞ E ðrÞ −iω t + ir = ð4AtÞ− 4 ∫−∞ − 0 e c π
ð2Þ ðxj Þ
ð2Þ
ve = −aj
3πi 4
3πi
AF
′
ð2Þ
ð2Þ
−iðω21 −bj Þðt−r = vp Þ + aj ðt−ve Þ
⋅e
Θ
! 3πi pffiffiffiffiffi −ρ2 ρe 4 + r = 2 At e dρ 3iπ pffiffiffiffiffi ð1Þ −xj βt + i ρe 4 + r =2 At
ð10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ ð2Þ vp = ðω21 −bj βÞ A = Im ω21 −ωc + ixj β;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi × −r + 2 At ðRe−ImÞ ωc −ω21 + bð1Þ β f ðxj Þ
πf ðxj Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ð2Þ ×ð−r + 2 At ðRe+ImÞ ωc −ω21 −ixj βÞ
k2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ πf xj iðb β−ω21 Þt−r ðωc −ω21 + bð1Þ βÞ = A ⋅e Θ ð1Þ ′ AG xj
Edl = E0
Meantime, if there is a complex root x(2) = a(2) + ib(2) with j j j (2) b(2) b ω − ω , a b 0, the emitted field can be written as j 21 c j
!2
pffiffiffiffi ⁎ ∞ ixðωc −ixÞPðxÞ½C2 ð0ÞPðxÞ−Ω0 C3 ð0Þ ⋅∫0 dx pffiffiffiffiffiffi 2 ½Q ðxÞðωc −ixÞ−iPðxÞ ωc Þ + ixP 2 ðxÞ
These formulas indicate that the frequency of the emitted field El is b(1) j − ω21 which is less than ωc. We call it localized field within the forbidden band, which represents that the spontaneous photons can't propagate far away from the atom within the range of localized length qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ l = A = ðωc −ω21 + bj Þ and the field's amplitude drops exponentially along the distance from the atom. This conclusion is consistent with the population and spontaneous spectra which are discussed in the next section. In addition, the last two parts of the emitted field are diffusion fields that show power-law decay in the radiation process, which are coming from the integral calculation.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ A = Re ω21 −ωc + ixj β
The frequency of this field is b(2) j − ω21 N ωc within the traveling band. The first term of the field is the propagating field that can transmit out of the photonic crystal in the form of a traveling pulse ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ ð2Þ with the energy velocity ve = −aj A = Re ω21 −ωc + ixj β and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ ð2Þ the phase velocity vp = ðω21 −bj βÞ A = Im ω21 −ωc + ixj β. Owing to the minus real part of the root, the propagating field decays along with time. The last two parts are also the diffusion fields. Attribute to the regions plotted in Fig. 2, we know that the main parts of the emitted fields can be totally written as 8 E + E + E + Edl2 + Ed > > > El1 + El2 + Edl1 > < l1 dl1 d → E r; t = El1 + Ep1 + Edl1 + Edp1 + Ed > > E + Edp1 + Ed > > : p1 Ep1 + Ep2 + Edp1 + Edp2 + Ed
ðRegion IÞ ðRegion IIÞ ðRegion IIIÞ ðRegion IVÞ ðRegion VÞ
So far, the definite expressions of the emitted fields are clear. When the resonant frequency is deep in the photonic band gap just in region I or II, most of the emitted fields are localized fields. Compared with the localized field, diffusion fields are very small. It is to say that the emitted photons are localized around the atom. But if we move the resonant frequency into the traveling band, the emitted photon can propagate out. Both of the propagating field and diffusion field make effects on the traveling process. During this process, the energy of the propagating field is partly transferred into the diffusion field due to the influence of the photonic crystal.
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Fig. 3. The population of the two upper levels P2, P3 and total population P (dash solid, dash-dot line respectively) as a function of the scaled time βt for different resonant transition frequencies with (a) ω21 = 99.5β, (b) ω21 = 100.4β, and (c) ω21 = 101β. The other parameters are ωc = 100β, C2(0) = 1, C3(0) = 0, Δ0 = 0.1β, Ωc = 0.5βexp(iθ) where θ is any value.
4. The time-evolution properties and spontaneous emission 4.1. Time-evolution of the population The phase and detuning of the coherent field, and the atomic initial states play an important role in the exchanging process between the photon and the atom in an anisotropic photonic crystal. The population in every level and spontaneous emission spectra are used to analyze the influences. The time-evolution of the population in the upper level is obtained by Pi(t) = |Ci(t)|2, (i = 2, 3). For the complicated factors, there are many new interesting phenomena depicted in the figures. First, as shown in Fig. 3, if the initial population is completely in one of the upper levels, whatever the phase of external field is, it can't make any effect on the time-evolution of the population. And what's more, the oscillation behaviors are found in P2 and P3 due to the interference
between the dressed state and quasi-dressed state that comes from the single-valued branch-cut contribution at the band edge frequency. When the resonant frequency ω21 moves from the inside to the outside of the band edge, the amplitudes of the time-evolution decay much faster. While ω21 = 99.5β is deep in the forbidden band gap in region I in Fig. 2, there are two imagery roots. So no decay in the amplitudes appears in Fig. 3(a), which represents that photons in the status of emitting and reabsorbing to the atom can't propagate out. But when ω21 = 100.4β (in region III) with one imagery root and one complex root or ω21 = 101β (in region V) with two complex roots, time decay comes out. The difference between these two regions is that the population in the upper levels will eventually stay invariable in the former region, but total population will decay to zero in the latter. This is because when the resonant frequency is beside the band edge, due to the coherent field, the upper level forms two dressed states with one in the band gap and the other in the band. Then part of
Fig. 4. The population of the two upper levels P2, P3 and total population P (dash solid, dash-dot line respectively) as a function of the scaled time βt for different phases of the coherent field with (a) θ = 0, (b) θ = π/2, (c) θ = π, and (d) θ = 3π/2. The other parameters are ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Δ0 = 0.1β, |Ωc| = 0.5β, ω21 = 100.4β.
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the population will be trapped in the dressed state which is in the band gap. However, when the initial population distributes in both of the upper levels, the phase plays an important role in the amplitude of the population's exchange between the two upper levels just as shown in Fig. 4. When we set the resonant frequency beside the band edge and change the phase from 0 to 3π/2 every other π/2, we can see that the population exchange between the two upper levels is present estimated periodic oscillation per 2π corresponding to the detail process: maximum exchange and fractional population trapping when θ = 0 in Fig. 4(a), small exchange and almost entire population trapping when θ = π/2 in Fig. 4(b), the opposite population exchange different from the case when θ = 0 and fractional population trapping when θ = π in Fig. 4(c), few exchange and non-population trapping in upper levels when θ = 3π/2 in Fig. 4(d). 4.2. Spontaneous emission From the formula S(ωk) = D(ωk) ⋅ |C1k(∞)|2 and Eq. (4), we can obtain the spectra as
Sðδk Þ =
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωk −ωc ⋅Θðωk −ωc Þ
j
C2 ð0Þðδk + Δ0 Þ−iΩ⁎0 C3 ð0Þ 3=2
iβ ðδk +Δ0 Þ −iδk ðδk + Δ0 Þ−qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωc + ωc −ω21 −δk
+ i jΩ0 j
2
j
2
ð11Þ As shown in Fig. 5, when the resonant frequency moves towards the photonic band edge, the spontaneous emission peak changes as follows: the long "tail" line deep in the band gap, one peak slightly outside the band edge, two peaks far from the band edge in Fig. 5(a),
(b),(c). This phenomenon is very easy to be understood by the influence of the photonic crystal and the coherent field. When the external field is added to the upper level |2〉, this level is split into two dressed states, which refers to the Autler-Townes splitting [23] as shown in Fig. 1(b) When the upper level is deep in the band gap, it means two dressed states are also in the band gap and there will be no spontaneous emission from the atom as shown in Fig. 5(a). Then as this level moves to the band edge or far away from the band edge, one or two of the dressed states is in the band, so spontaneous emission peak will appear in the spectrum such as Fig. 5(b),(c). However, after considering the phase of external field, the spontaneous spectra are also influenced by its values just as shown in Fig. 5(c),(d) and Fig. 6(a). Directly, we can deduce that the larger phase value makes the left peak increase and the right peak decrease. More importantly, a fluorescence quenching happens on the condition in Fig. 5(a). Destructive interference leads to this result and we can calculate the quenching point which can be derived from the expression C2(0)(δk + Δ0) − iΩ0⁎C3(0) = 0. If we change the parameter of the external field's detuning and keep other parameters unchanged, the destructive interference changes into the constructive interference and the right peak appears again. Meanwhile, when we set the resonant frequency beside the band edge and change the phase of the driven field from 0 to 3π/2 as shown in Fig. 5(b) and Fig. 7, the spontaneous peaks change in period of 2π and the phenomenon of the fluorescence quenching and tail-like spontaneous spectra with the phase of π/2 in Fig. 7(a), the same spontaneous emission line with the phase of 0 and π in Fig. 5(b) and Fig. 7(b), the maximum spontaneous peak with the phase of 3π/2 in Fig. 7(c) are observed. In substance, when the phase is π/2, destructive interference between two channels of the dressed states as shown in Fig. 1(b) leads to fluorescence quenching, no spontaneous emission and population trapping in the
Fig. 5. The spontaneous emission spectrum (in arbitrary units) as the function of the detuning δk. (a) ω21 = 99.5β, θ = 0, (b) ω21 = 100.4β, θ = 0, (c) ω21 = 101β, θ = 0, and (d) ω21 = 101β, θ = π/4. The other parameters are chosen as ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Δ0 = 0.1β, δk = ωk − ω21, |Ωc| = 0.5β.
L. Jiang et al. / Optics Communications 283 (2010) 3714–3720
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Fig. 6. The spontaneous emission spectrum (in arbitrary units) as the function of the detuning δk. (a) Δ0 = 0.1β, (b) Δ0 = 0. The other parameters are chosen as ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Ωc = 0.5βexp(iπ/2), ω21 = 101β.
Fig. 7. The spontaneous emission spectrum (in arbitrary units) as the function of the detuning δk. (a) θ = π/2, (b) θ = π, (c) θ = 3π/2. The other parameters are chosen as ωc = 100β, C2(0) = sin(π/4), C3(0) = cos(π/4), Δ0 = 0.1β, |Ωc| = 0.5β, ω21 = 100.4β.
upper levels in Fig. 4(b). In the same way, when the phase is 3π/2, constructive interference between the two channels makes the spontaneous emitting maximum and no population trapping in Fig. 4(d). So we can make the conclusion that under the coherent field's phase condition, the interference effects are much stronger than the band edge localization effects on the atomic system's spontaneous emission.
5. Conclusion We have investigated the properties of radiation field and spontaneous emission from a coherent three-level atom in an anisotropic crystal. As the relative position of the upper level to the band edge or the intensity of the coherent field increase, components of the radiation fields regularly vary and the detailed expressions are clearly given. The variety of the emitted fields can be used to design the optical multi-channels switch. With the help of Fourier transform, on the viewpoint of the time-evolution properties and spontaneous spectrum, we analyze the effects of the photonic crystal and the phase-sensitive coherent field on the emitted modes in different regions. The results analyzed here are consistent with the distribution of emitted fields. Meanwhile, quantum interference plays an im-
portant role in the process and leads to population oscillation and fluorescence quenching with suitable parameters. As a result, this atomic system with phase-sensitive may be implemented for the optical memory device. Acknowledgement This work was supported by the National Natural Science Foundation of China (NSFC) (grants 10974071, 10774060, J0730311). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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