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Optik 122 (2011) 1262–1266

Contents lists available at ScienceDirect

Optik

journa l homepage: www.e lsev ier .de / i j leo

icrowave control of spontaneous emission of a driven four-level atom inhotonic crystals

ing Zhang, Xiudong Sun ∗, Xiangqian Jiangepartment of Physics, Harbin Institute of Technology, Harbin 150001, China

r t i c l e i n f o

rticle history:eceived 1 January 2010ccepted 2 August 2010

ACS:

a b s t r a c t

We study the coherent control of spontaneous emission of a double-driven four-level atom embeddedin photonic crystals. Combined effect of different relative locations between the upper band edge andthe two upper levels and the phase of microwave coupling field is discussed. It is shown that quantuminterference effect such as laser-induced dark line depends strongly on the phase of microwave field.

© 2010 Elsevier GmbH. All rights reserved.

2.50.Gy2.70.Qs2.80.−t

eywords:

hotonic crystalspontaneous emission spectrum

. Introduction

Atomic spontaneous emission is a basic problem in quantumlectrodynamics and it has attracted extensive attention in the lastew decades. In 1946, Purcell was the first to point out that sponta-eous emission depends not only on the level structure of atomicystem but also on nature of surrounding environment, especiallyn the density of states (DOS) of the radiation field [1]. In recentears, the DOS in the photonic crystal (PC) is shown to be signifi-antly different from that of vacuum [2–4]. Therefore, the study ofuantum and nonlinear optical phenomena for atoms embedded

n photonic crystals can lead to unusual effects such as localizationf light [5,6], photon-atom bound states [7], inhibition or enhance-ent of spontaneous emission [8,9], modified reservoir induced

ransparency [10].On the other hand, driving an atom with sufficiently strong res-

nant field can alter the radiative dynamic in a fundamental way,ven in ordinary vacuum. For instance, Zhu and co-workers havetudied quenching of spontaneous emission of an open V-type atom11,12]. Phase dependent effects in spontaneous emission spectraere investigated in a four-level atom driven by two lasers of the

ame frequency [13]. All of these phenomena can be attributed to

he effects of quantum interference, which is also referred to asGC (spontaneously generate coherence). However, the existencef SGC requires rigorous condition of near-degenerate levels and

∗ Corresponding author. Tel.: +86 451 86414129; fax: +86 451 86414129.E-mail address: dao1789@163.com (X. Sun).

030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved.oi:10.1016/j.ijleo.2010.08.012

nonorthogonal dipoles matrix elements, which are rarely met inreal atoms, so few experiments have been carried out to achievethese phenomena.

In a few papers, the combined effects of coherent control byexternal driving field and photon localization facilitated by PC onspontaneous emission from atoms embedded in photonic crys-tals have been discussed. For instance, the spontaneous radiationproperties of a driven three-level atom embedded in a three-dimensional anisotropic photonic crystal have been discussed [14].More recently, Yang et al. have investigated spontaneous emissionfrom a driven four-level atom embedded in a double-band PC [15],however these work have not considered the existence conditionof quantum interference.

In this paper, we introduce microwave generated coherence(MGC) in our scheme to avoid the problem of the dipole moments.The steady state behavior of a double-driven four-level atomembedded in photonic crystal is investigated. In our scheme, bothof the upper states coupled by the microwave field are allowedto decay spontaneously to the lower levels and the transition fre-quency between the two upper levels is assumed to be large, thusthere are no close-lying levels in our system. It is found that theproperties of the spontaneous emission are dependent strongly onthe relative position of the upper two levels from the band edge aswell as the phase of the microwave field.

2. Theoretical model and equations

We consider a four-level atom as shown in Fig. 1. It has twoupper levels |a1〉 and |a2〉 coupled by microwave field of frequency

B. Zhang et al. / Optik 122

Fig. 1. Schematic diagram of a coherently driven four-level atomic system: (a) intXi

ωwftttoaww

H

wamvcttss∣∣

w∣∣Mdf

i

i

i

i

E

i

he bare-state picture; (b) in the dressed-state picture.iudong Sun, Modified spontaneous emission spectra of a three-level V-type atom

n photonic crystals.

1 and phase �1, which decay to the lower level |c〉 via interactionith the vacuum field modes. Another coherent field with carrier

requency ω2, Rabi frequency ˝2 and phase �2 is used to pumphe atom into a superposition state of the two upper levels suchhat

∣∣ (t = 0)⟩

= (ei�2 |a1〉 + |a2〉)/√

2. At time t = 0, this atom startso interact with microwave field and we only consider the casef a square pulse of the microwave field. All the interactions aressumed to be resonant. In the interaction picture with the rotatingave approximation (RWA), the Hamiltonian for the system can beritten as

I = ˝1 |a1〉 〈a2| +˝2 |a2〉⟨b∣∣ +

∑k

g1ke−i(ωk−ω1c)t |a1〉

⟨b∣∣ ak

+∑k

g2ke−i(ωk−ω2c)t |a2〉

⟨b∣∣ ak +H.c., (1)

hereωic is the energy separation of the states∣∣ai⟩ and |c〉 (i = 1, 2),

k (a+k

) is the annihilation (creation) operator for the kth vacuumode with frequency ωk, k here represents both the momentum

ector and the polarization of the vacuum mode. g1k, g2k are theoupling constants between the kth vacuum mode and the atomicransitions |a1〉 → |c〉 and |a2〉 → |c〉, respectively, and are assumedo be real. For simplicity, h= 1 has been assumed. The atom-fieldtate vector at an arbitrary time t can be expressed in terms of baretate vector as

� (t)⟩

= a1(t)∣∣a1,0

⟩+ a2(t)

∣∣a2,0⟩

+ b(t)∣∣b,0

⟩

+∑k

Ck(t)∣∣c, {k}⟩ , (2)

here∣∣{0}

⟩represents the vacuum of electromagnetic field, and

{k}⟩

denotes that there is one photon in the kth vacuum mode.aking use of the time-dependent Schrödinger equation, the

ynamic equations of motion of the probability amplitudes can beound as

a1(t) =˝1a2(t) +∑k

g1ke−i(ωk−ω1c)tCk(t), (3a)

a2(t) =˝∗1a1(t) +˝2b(t) +

∑k

g2ke−i(ωk−ω2c)tCk(t), (3b)

b(t) =˝2a2(t), (3c)

ak(t) = g1ka1(t)ei(ωk−ω10)t + g2ka2(t)ei(ωk−ω20)t . (3d)

Formally integrating Eq. (3d), and substituting the result intoqs. (3a) and (3b) a pair of integro-differential equations is obtained

a1(t) = ˝1a2(t) − i∫ t

0

dt′∑k

∣∣g1k

∣∣2a1(t′)e−i(ωk−ω1c)(t−t′)

(2011) 1262–1266 1263

− i∫ t

0

dt′∑k

g1kg2ka2(t′)e−i(ωk−ω2c)(t−t′)eiω21t , (4a)

ia2(t) = ˝∗1a1(t) +˝2b(t) − i

∫ t

0

dt′∑k

∣∣g2k

∣∣2a2(t′)e−i(ωk−ω2c)(t−t′)

− i∫ t

0

dt′∑k

g2kg1ka1(t′)e−i(ωk−ω1c)(t−t′)e−iω21t . (4b)

Then, two kernels are introduced to simplify Eqs. (4)

Ki(t − t′) =∑k

∣∣gik∣∣2e−i(ωk−ωic)(t−t

′)

≈ c3/2i

∫dω�(ω)e−i(ωk−ωic)(t−t

′), (5)

where ci (i = 1, 2) denotes the coupling constants of the atom tothe reservoir. In the following, all parameters used in the calcula-tions are scaled by c, c1 = 1c and c2 = 0.9c are assumed, where c is anarbitrary constant. Substituting Eq. (5) into Eqs. (4), we get

a1(t) = −i˝1a2(t) −∫ t

0

dt′a1(t′)K1(t − t′),

a2(t) = −i˝∗1a1(t) − i˝2b(t) −

∫ t

0

dt′a2(t′)K2(t − t′). (6)

Performing the Laplace transform of (6) and solving the resultsdirectly, the Laplace transform ai(s) (i = 1, 2) for the amplitudes ai(t)can be found as

a1(s) =[s2 +

∣∣˝2

∣∣2 + sK2(s)]a1(0) − i[sa2(0) − i˝2b(0)]˝1

[s+ K1(s)][s2 +∣∣˝2

∣∣2 + sK1(s)] + s∣∣˝1

∣∣2, (7a)

a2(s) = [sa2(0) − i˝2b(0)][s+ K(s)] − i˝∗1sa1(0)

[s+ K1(s)][s2 +∣∣˝2

∣∣2 + sK2(s)] + s∣∣˝1

∣∣2, (7b)

here Ki(s) and K ′i(s) (i = 1, 2) are Laplace transforms of the kernels

defined in Eq. (5). The long-time spontaneous emission spectrum

in this V-configuration can be given by S(ık) = �(ωk)∣∣ak(t → ∞)

∣∣2,

here ık =ωk −ωc. For the sake of convenience, the following changeof variables A1(t) = a1(t)e−iω12t, A2(t) = a2(t), Ck(t) = ck(t)e−i(ωk−ω2c)t

was carried out in Eq. (3c).Taking Laplace transform of the result, we have

(s+ iık2)Ck(s) = −ig∗

1kA1(s) − ig∗2kA2(s), (8)

where ık2 =ωk −ω2c. Using Eq. (8) and the final value theoremexpressed by as s → iω (for the case where the Laplace transformF(s) has one pole at s = iω, and all other pole in the left complex splace) the steady state amplitude Ck(t → ∞ ) reads

Ck(t → ∞) = exp(−iık2t) × lims→−iık2

[−ig∗1kA1(s) − ig∗

2kA2(s)], (9)

because of A1(s) = a1(a+ iω21), we get

ck(t → ∞) = lims→iık2

[−ig∗1ka1(s+ iω12) − ig∗

2ka2(s)], (10)

note that the oscillating factor exp( − iık1) is now cancelled out.Considering the case of a single-band isotropic model of the

photonic crystal, the dispersion relation near the edge is approxi-mated byωk = ωc +ωc(k − k0)2/k2

0 [6,16], under which the densityof modes reads�(ω) = (1/�)(1/

√ω −ωc)�(ω −ωc),where� is the

Heaviside step function and ωc is the frequency of upper band gap

1 ik 122 (2011) 1262–1266

egem

�

hmb

K

K

K

K

hff

3

dtizt

prnafipFıuqtsbbbttisedvgaitsi

Fig. 2. Spontaneous emission spectra from a double-driven four-level atom embed-ded in photonics crystal versus detuning ık =ωk −ωc for different relative locationsof the upper levels from the upper band edge. (a) ı1 = 6, ı2 = 4; (b) ı1 = 1, ı2 = − 1; (c)ı1 = 0, ı2 = − 2; (d) ı1 = − 2, ı2 = − 4.

264 B. Zhang et al. / Opt

dge. Since the mode density tends to be infinite and presents sin-ularity near the gap edge, we use smoothing parameter ε [17] toliminate this singularity, and get a modified form of density ofodes, expressed as

(ω) = 1�

√ω −ωc

ε+ω −ωc �(ω −ωc), (11)

ere ε= 0.01 has been set. Substituting Eq. (11) into Eq. (5) andaking Laplace transform then, the memory kernel functions can

e obtained as

˜1(s) = c3/21

i√ε+

√is+ ı1

,

˜2(s) = c23/2

i√ε+

√is+ ı2

,

˜1(s+ iω12) = c13/2

i√ε+

√is+ ı2

,

˜ ′2(s+ iω21) = c23/2

i√ε+

√is+ ı2 −ω12

, (12)

ere ı1 =ω1c −ωc and ı2 =ω2c −ωc in the following section standor the relative positions of the upper levels

∣∣1⟩and

∣∣2⟩to the

orbidden gap edge respectively.

. The spontaneous spectrum

The spontaneous emission spectra were plotted as a function ofetuning frequency ık =ωk −ωc for different upper band-gap loca-ions in the following figures. For there is no emission of radiationn the photonic band gap (ωk −ωc < 0), and the emission goes toero at the photonic band edge (ωk −ωc = 0) as a consequence ofhe absence of electromagnetic modes at ωc.

Here, we define the relative phase between the pump and cou-ling fields as ı� =�1 −�2, which will show later plays a crucialole in the steady state behavior of the system. Also for conve-ience, we set arbitrarily the pump field’s phase to zero, so thebove phase difference simply reduces to the phase of the couplingeld, i.e. ˝1 =

∣∣˝1

∣∣ eiı� and. ˝2 =∣∣˝2

∣∣. The atom is initially pre-ared in a symmetrical superposition of the two upper levels. Inig. 2 we show the spontaneous emission spectra in the case of� = 0 and

∣∣˝1

∣∣ =∣∣˝2

∣∣ = for different relative positions of thepper two levels from the photonic band edge. When both fre-uencies of the two transitions are outside the gap (ı1 > 0, ı2 > 0),he long-time behavior of radiation is similar to that in the freepace, which can be evaluated in dressed states representation ofoth of the two fields. As shown in Fig. 1b, the system turns out toe two sets of three dressed states decaying to level |c〉, the spaceetween the dressed states is determined by the Rabi frequency ofhe two fields, so the spontaneous emission spectrum is expectedo have six peaks, but the spectra-lines can be eliminated or fusento each other under certain conditions. As one of the atomic tran-ition frequencies moves into the gap (ı1 > 0, ı2 < 0), the inhibitingffect of the gap turns strong and there is a deep valley (but not aark line) between two peaks in the spectrum due to the uncon-entional density of states of radiation field near the photonic bandap. In Fig. 2c, spectrum under the situation that one transition ist the edge of band gap and the other is in the gap (ı1 = −2, ı2 = 0)

s plotted. The spectrum presents two peaks and a dark line andhe peak near the gap becomes weak. For the case that both tran-ition frequencies are inside the gap (ı1 = −2, ı2 = −4), as is plottedn Fig. 2d, the spontaneous emission is strongly inhibited in both

Xiudong Sun, Modified spontaneous emission spectra of a three-level V-type atomin photonic crystals.

B. Zhang et al. / Optik 122

Fig. 3. Spontaneous emission spectra from a double-driven four-level atom embed-ded in photonics crystal versus detuning ık =ωk −ωc as one transition frequency isat the band edge (ı1 = 0) and the other is inside the photonic band gap. (a) ı� = 0, (b)ı� =�/2, (c) ı� =�, (d) ı� = 3�/2.Xiudong Sun, Modified spontaneous emission spectra of a three-level V-type atomin photonic crystals.

(2011) 1262–1266 1265

initially superposition states. So, the shape of the spectral lines isnon-Lorentzian and the amounts of the radiation are very small.

In a laser-driven atomic system, although embedded in photoniccrystal, the shape of the spontaneous emission spectrum dependsstrongly on the dynamical variables of the atomic system, such asRabi frequency, phases and frequency of driving fields. The vari-ation of the spontaneous emission spectra with the phase valuesof microwave field is shown in a periodicity of 2� in Fig. 3, underthe condition that one transition frequency is at the band edge andthe other is inside the photonic band gap (ı1 = −2, ı2 = 0). The influ-ence of the phase of the microwave field on the spectral line shapeis now obvious. In case of ı� = 0, as shown in Fig. 3a, the left sidepeak is much lower than the right side one and there is a dark linebetween the two peaks. However, as ı� increases to �/2 the darkline disappears in the spectrum. With continue increase of ı� to�, the shape of spectra changes greatly, the left side peak growshigher than the right peak and a deep valley presents in the spec-trum. If ı� is varied to 3�/2 (see Fig. 3d), the characteristic of thespectrum is quite similar to that in Fig. 2b, except for some quantita-tive differences in peak heights. From these spectra, it is also notedthat, using the modified density of modes, the basic features of thespectra are still present with only some quantitative modificationscompared to those making use of the usual density of mode directlydeduced from the isotropic model of the photonic crystal, however,the singularity near the photonic band edge has been eliminated.

4. Conclusion

In summary, we have theoretically investigated the steady statebehavior of spontaneous emission of a double-driven four-levelatom embedded in the photonic crystal. The results clearly showthat the property of the spontaneous emission spectra can beaffected by the relative position of upper band gap edge to theupper levels as well as the phase of microwave field. As the atomictransitions frequency is near the photonic band edge, though thespontaneous emission is partly depressed, dark lines appear due tothe microwave generated coherence. Furthermore, quantum inter-ference property between the decay channels depends strongly onthe phase of microwave field, with increase of the phase, the darkline disappears. Furthermore, since close-lying levels with orthog-onal dipoles for two transitions are easily found in real atoms, ournumerical results for atomic spontaneous emission may be observ-able in realistic experiments.

Acknowledgments

This work was supported by the National Natural Science Foun-dation of China (Grant No: 10904025 and 10674037), the MOST ofChina (973 project No. 2007CB307001), and program of excellentTeam in Harbin Institute of Technology.

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[

[

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