Control of spontaneous emission in a five-level system

Journal of Russian Laser Research, Volume 29, Number 3, 2008

CONTROL OF SPONTANEOUS EMISSION

IN A FIVE-LEVEL SYSTEM

Rizwan Ahmed and Tasnim Azim

Department of Physics, Quaid-i-Azam UniversityIslamabad, Pakistan

email: [email protected]

Abstract

We study the control of spontaneous emission in a five-level atomic system driven by four fields. Weshow that with the variation of dynamical variables, namely, Rabi frequencies and carrier phases ofdriven fields, a wide variety of the spectral behavior can be obtained, including extreme spectralnarrowing. Our system can be found easily in real atoms.

Keywords: control of spontaneous emission, five-level atomic system, Rabi frequencies, spectral nar-rowing, quantum coherence.

1. Introduction

The modification and control of spontaneous emission have been an active research topic and widelydiscussed [1–11]. The control of spontaneous emission can be potentially useful to achieve high frequencylasing, since the spontaneous-emission rate is typically proportional to the frequency cubed w3 andcreating the inversion on the high-frequency transition, with allowed spontaneous emission, is thereforeproblematic. The basic mechanisms for controlling spontaneous emission [12] in free space are quantumcoherence and interference, which have led to a great number of novel effects such as lasing withoutinversion [13–17], electromagnetically induced transparency [18–22], enhanced nonlinear processes [23–25], correlated spontaneous-emission laser [26], etc. Recently, atomic coherence and quantum interferenceinduced by driving fields have become the basic phenomena for efficient control of spontaneous emission[27]. Paspalakis and Knight use the phase difference of two lasers with equal frequencies for the control ofspontaneous emission in a four-level atom [5]. In their calculations, parallel dipoles for the two transitionsare assumed. However, orthogonal dipoles for two transitions with small energy separation are easilyfound in nature. Ghafoor et al. studied the spontaneous emission in a four-level atomic system drivenby three fields and showed that, by controlling phase and amplitude of driving fields, a variety of thespectral behavior can be obtained [3]. Jin-Hui Wu et al. studied the spontaneous-emission properties ofcoherently driven four-level atom and showed a few interesting phenomena such as spectral-line narrowing,fluorescence quenching, spectral-line enhancement, and spectral-line elimination [28]. The relative phasedifference of two bichromatic fields of equal frequency differences for the coherent control of spontaneousemission of a three-level atom and the phase dependence of fluorescence spectra have been studied byXiao-xia et al.; they show that such effects as selective and total cancellation of fluorescence decay areobtained simply by varying the phase difference [29]. Recently the spontaneous-emission spectra in a

Manuscript submitted by the authors in English first on 14 September 2007 and in final form on 29 January 2008.

1071-2836/08/2903-0227 c©2008 Springer Science+Business Media, Inc. 227

Journal of Russian Laser Research Volume 29, Number 3, 2008

Fig. 1. The system under consideration. The ground state |a〉 is coupled to the excited state |b〉 with the Rabifrequency Ωab. The excited level |b〉 is coupled with |c〉 and |d〉 via Rabi frequencies Ωcb and Ωdb, respectively, theexcited state |c〉 is also coupled with |d〉 via the Rabi frequency Ωcd. The excited states |b〉, |c〉, and |d〉 decay solelyto the common state |e〉.

coherently driven five-level atomic system by means of a radiofrequency (rf) field driving a hyperfinestructure within the ground state has been studied and it is shown that spontaneous emission quenchingcan be realized by modulating the frequency and intensity of the rf-driving field [30].

In this paper, we present a scheme using a five-level atom, which consists of three upper levels andtwo lower metastable states. In our system, we can control spontaneous emission by amplitude andphase of the driving fields. The proposed scheme requires four driving fields but it is more convenient inthe experimental realization. We assume that the transition frequencies between the upper three levelsare large in comparison with their decay rates Γ1,Γ2, and Γ3. This approximation allows us to neglectthe quantum interference terms proportional to

√Γ1Γ2,

√Γ1Γ3, and

√Γ3Γ2 in the equations of motion

for the probability amplitudes [3, 31]. Our system is therefore independent of the alignment of dipolemoments. The main purpose of applying microwave field is to create quantum coherence in the system.The quantum coherence generated by the microwave field plays a fundamental role in line narrowing.

2. Model and Equations

Consider a five-level atom that consists of three upper levels |b〉, |c〉, and |d〉 and two lower metastablestates |a〉 and |e〉 (see Fig. 1). In our system, five-level atom interacts with four driving fields. Thesefields resonantly couple the transitions

|b〉 ←→ |a〉 , |b〉 ←→ |c〉 , |c〉 ←→ |d〉 , |d〉 ←→ |b〉

with Rabi frequencies Ωab, Ωbc, Ωcd, and Ωbd, respectively. The levels |d〉, |c〉, and |b〉 decay to a lowerlevel |e〉 via interaction with the vacuum field modes. The spontaneous-emission-decay rates from theupper three levels |b〉, |c〉, and |d〉 to the ground level |e〉 are labeled by Γ1, Γ2, and Γ3, respectively.

Under the rotating-wave and electric-dipole approximations, the interaction Hamiltonian for the sys-

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Volume 29, Number 3, 2008 Journal of Russian Laser Research

tem composed of atom, driving fields, and vacuum modes reads

H(t) = [Ωab |a〉〈b|+ Ωbc |b〉〈c|+ Ωcd |c〉〈d|+ Ωbd |b〉〈d|] + ∑

k

(g(1)k ei(wbc−v)t |c〉〈e|

+g(2)k ei(wde−v)t |d〉〈e|+ g

(3)k ei(wbe−v)t |b〉〈e|) + H.c., (1)

where bk (b†k) is the annihilation (creation) operator for the kth mode with frequency v = ck, and g(j)k

(j = 1, 2, 3) is the coupling constant between the kth mode and atomic dipoles between the three upperlevels |b〉, |c〉, and |d〉 and the lower level |e〉. For the sake of simplicity, we assume the coupling constantsto be real. The state vector for our system at time t, whose evolution obeys the well-known Schrodingerequation, is written as follows:

|Ψ(t)〉 = [A(t) |a〉+ B(t) |b〉+ C(t) |c〉+ D(t) |d〉] |0〉+∑

k

Ek(t) |e, 1k〉 , (2)

where |0〉 denotes the vacuum of the radiation field and |1k〉 means that there is one photon in thekth vacuum mode. Here we use the wave-function approach and assume that the atom is excited to asuperposition of states |a〉, |b〉, |c〉, and |d〉. We apply the Weisskopf–Wigner theory [32] and obtain thefollowing dynamical equations for atomic probability amplitudes in the interaction picture:

A(t) = −iΩabB(t), (3)

B(t) = −iΩabA(t)− iΩbcC(t)− iΩbdD(t)− Γ3

2B(t)− C(t)

p3

√Γ1Γ3

2eiwbct −D(t)

p2

√Γ2Γ3

2eiwbdt,

(4)

C(t) = −iΩcbB(t)− iΩcdD(t)− Γ1

2C(t)−D(t)

p1

√Γ1Γ2

2e−iwdct −B(t)

p3

√Γ1Γ3

2e−iwbct, (5)

D(t) = −iΩdcC(t)− iΩdbB(t)− Γ2

2D(t)− C(t)

p1

√Γ1Γ2

2eiwdct −B(t)

p2

√Γ2Γ3

2e−iwbdt, (6)

E(t) = −ig(1)k C(t)eiδ1t − ig

(2)k D(t)eiδ2t − ig

(3)k B(t)eiδ3t, (7)

where δ1 = v − wce, δ2 = v − wde, δ3 = v − wbe and δ1 = δ2 + wdc, δ2 = δ3 + wbc, δ3 = δ2 − wbd.Furthermore, the alignment of matrix elements of the dipole moments is given by

p1 =〈c| r |e〉 · 〈d| r |e〉|〈c| r |e〉| |〈d| r |e〉|

, p2 =〈b| r |e〉 · 〈d| r |e〉|〈b| r |e〉| |〈d| r |e〉|

, p3 =〈c| r |e〉 · 〈b| r |e〉|〈c| r |e〉| |〈b| r |e〉|

.

If the matrix elements are orthogonal, there is no interference between the decay paths and p1 = p2 =p3 = 0. If the matrix elements are parallel, there is maximum interference and p1 = p2 = p3 = 1.

In this paper, we assume that the transition frequencies between the upper levels are large in com-parison with their decay rates Γ1, Γ2, and Γ3. Due to this, we can neglect the quantum interferenceterms proportional to

√Γ1Γ2,

√Γ1Γ3, and

√Γ3Γ2 in the equations of motion for the probability ampli-

tudes. However, under the approximation wbc, wdc, wdb Γ1,2,3 [3, 31], we can solve Eqs. (4)–(6) for thesteady-state expression for the probability amplitude Ek(t → ∞). The spontaneous-emission spectrumis the Fourier transform of 〈E−(t + T )E+(t)〉t−→∞ and can be written as

S(δK) =Γm |Ek(t −→∞)|2

2π∣∣gm

k

∣∣2 (m = 1, 2, 3)

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for the system under consideration. We employ the Laplace-transform method and final-value theoremto obtain

Ek(t −→∞) = −ig(1)k

F (δ)N1(δ)

− ig(2)k

G(δ)N2(δK)

− ig(3)k

H(δ)N3(δ)

, (8)

where we have used the following relationships:

F (δ) = −i(x1 − iΓ1/2)B(0)[ΩdbΩcd − x3 x2]− C(0)[|Ωdb|2 − x3 x2] + D(0)[Ωcd x3 + ΩcbΩbd]−iΩba−A(0)[Ωcb x2 + ΩdbΩcd] + Ωba[C(0) x2 + D(0)Ωcd], (9)

G(δ) = −i(y1 − iΓ1/2)D(0)[|Ωcb|2 − y1 y3]− C(0)[Ωdc y3 + ΩbcΩdb]−B(0)[Ωbd y1 + ΩcbΩdc]−iΩba−A(0)[Ωbdy1 + ΩcbΩcd] + C(0)ΩabΩdc + D(0)Ωaby1, (10)

H(δ) = −i(z1 − iΓ1/2)B(0)[|Ωdc|2 − z1 z2]− C(0)[Ωbc z2 + ΩdbΩdc]

−D(0)[ΩbcΩcd + Ωbd z1] − iΩbaA(0)[|Ωdc|2 − z1 z2] (11)

N1 = −i(x1 − iΓ1/2)−ix3 [|Ωdc|2 − x2x1]− iΩbc[Ωcbx2 + ΩdbΩcd]− iΩbd[ΩcbΩdc

−Ωdbx1]+ |Ωba|2 [|Ωdc|2 − x2x1], (12)

N2 = −i(y1 − iΓ1/2)−i y3 [|Ωdc|2 − y2 y1]− iΩbc[Ωcby2 + ΩdbΩcd]− iΩbd[ΩcbΩdc

−Ωdb y1]+ |Ωba|2 [|Ωdc|2 − y2 y1], (13)

N3 = −i(z1 − iΓ1/2)−iz3 [|Ωdc|2 − z2 z1]− iΩbc[Ωcbz2 + ΩdbΩcd]− iΩbd[ΩcbΩdc

−Ωdb z1]+ |Ωba|2 [|Ωdc|2 − z2 z1]. (14)

In the above equations, we put

δ = δ2, δ = δ1 − wbd − wdc, δ = δ3 − wbd,

and

x1 = δ + wbd + wdc +iΓ1

2, x2 = δ + wbd + wdc +

iΓ2

2, x3 = δ + wbd + wdc +

iΓ3

2,

y1 = δ + wbd +iΓ1

2, y2 = δ + wbd +

iΓ2

2, y3 = δ + wdc +

iΓ3

2,

z1 = δ +iΓ1

2, z2 = δ +

iΓ2

2, z3 = δ +

iΓ3

2.

In our system, we do not need to apply any trapping conditions, since there is no physical condition forthem to arise; we control spontaneous emission without trapping the atom. To analyze the spontaneous-emission spectrum, we assume the atom to be initially prepared in the state |a〉, so that A(0) = 1 andB(0) = C(0) = D(0) = 0. According to basic mathematical rules, we can predict that N1(δ), N2(δ), andN3(δ) have four roots each, and we name them λi (i = 1, . . . , 12). After partially fractioning, Eq. (8)becomes

Ek(t→∞) =X1

(δ − λ1)+

X2

(δ − λ2)+

X3

(δ − λ3)+

X4

(δ − λ4)+

X5

(δ − λ5)+

X6

(δ − λ6)

+X7

(δ − λ7)+

X8

(δ − λ8)+

X9

(δ − λ9)+

X10

(δ − λ10)+

X11

(δ − λ11)+

X12

(δ − λ12).

(15)

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As we know, Xi and λi have real and imaginary parts (where i = 1, . . . , 12), and we can put Xi = αi + iβi

and λi = ηi + iχi, where

X1 = α1 + iβ1 =iΩab[ΩdbΩcde

iφ + Ωbc(λ1 + wbd + wdc + iΓ2/2)](λ1 − λ2)(λ1 − λ3)(λ1 − λ4)

, (16)



, (17)



, (18)



, (19)

X5 = α5 + iβ5 =iΩab[ΩbcΩcde

iφ + Ωbd(λ5 + wbd + iΓ1/2)](λ5 − λ6)(λ5 − λ7)(λ5 − λ8)

, (20)



, (21)



, (22)



, (23)

X9 = α9 + iβ9 =−iΩab[|Ωcd|2 − (λ9 + iΓ1/2)(λ9 + iΓ2/2)]

(λ9 − λ6)(λ9 − λ7)(λ9 − λ8), (24)


(λ10 − λ9)(λ10 − λ11)(λ10 − λ12), (25)


(λ11 − λ9)(λ11 − λ10)(λ11 − λ12), (26)


(λ12 − λ9)(λ12 − λ10)(λ12 − λ11). (27)

The spontaneous-emission spectrum S(δ) is proportional to |Ek(t→∞)|2; thus, apart from a propor-tionality constant, the spontaneous-emission spectrum is given by the following expression:

S(δ) = Γ1

∣∣∣∣ α1 + iβ1

δ − (η1 + iχ1)+

α2 + iβ2

δ − (η2 + iχ2)+

α3 + iβ3

δ − (η3 + iχ3)+

α4 + iβ4

δ − (η4 + iχ4)

∣∣∣∣2+Γ2

∣∣∣∣ α5 + iβ5

δ − (η5 + iχ5)+

α6 + iβ6

δ − (η6 + iχ6)+

α7 + iβ7

δ − (η7 + iχ7). +

α8 + iβ8

δ − (η8 + iχ8)

∣∣∣∣2+Γ3

∣∣∣∣ α9 + iβ9

δ − (η9 + iχ9)+

α10 + iβ10

δ − (η10 + iχ10)+

α11 + iβ11

δ − (η11 + iχ11). +

α12 + iβ12

δ − (η12 + iχ12)

∣∣∣∣2 .

(28)

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3. Results and Discussion

In this section, we discuss some numerical results and their plots. To analyze the spontaneous-emission spectrum, we assume the atom to be initially prepared in the state |a〉, so that A(0) = 1and B(0) = C(0) = D(0) = 0. The coupling constants g

(1)k , g

(2)k , and g

(3)k are chosen in such a way

that g(1)k = g

(2)k = g

(3)k = 1. Further, to take into account the effect of the microwave-field phase

on the spontaneous emission, we replace Ωdc by |Ωdc| e−iφ. We assume Ωba, Ωcb, and Ωdb to be real,i.e., Ωba = Ωab = |Ωba|, Ωbc = Ωcb = |Ωcb|, and Ωdb = Ωbd = |Ωdb|. In the following discussion,Γ1 = Γ2 = Γ3 = Γ, wbc = wdc = 5Γ, wbd = 10 Γ, and Rabi frequencies Ωba, Ωcb, Ωdc, and Ωdb are givenin units of [Γ].

The spectrum contains three main parts due to the three upper bare states. We study the effects of thedynamical variables, namely, the amplitudes (or more precisely the Rabi frequencies) and carrier phasesof the driven fields on the spontaneous-emission spectrum. The variation of the phase φ associated withthe microwave field influences the spontaneous-emission spectrum efficiently. When the Rabi frequenciesΩba = 1, Ωcb = Ωdb = |Ωdc| = 2 and phase φ = 0, the plot of these values gives a triple-peak spectrum,with two peaks of equal heights and the third peak with less height in comparison with the others.Dominantly the triple-peak spectrum is due to the upper three bare states as shown in Fig. 2a. Ourspectrum is very sensitive to the phase φ associated with microfield. When φ changes from 0 to π/2,each of the three peaks splits into two; in this way, we get a six-peak spectrum. The heights of the twopeaks on the left are equal and the heights of the two peaks on the right are also equal, but overall thetwo peaks on the right have nearly one-fourth of the heights of the other peaks. The central two peakshave small difference in their heights (see Fig. 2b). With equal values of Rabi frequencies at φ = π,we again get a triple peak spectrum with two peaks of equal heights and a third peak on the right withsomewhat less height as shown in Fig. 2c. When φ further varies from π to 3π/2 (Fig. 2d), the newspectrum consists of six peaks, with two peaks on the right having the same height, two peaks on theleft also having equal heights, and two peaks in the middle having a small difference in their heights.Comparing Figs. 2d and 2b, we can easily see that the height of the left two peaks remains the same,while the height of the other four peaks decreases. We note that the peak height varies with φ; however,there is no appreciable change in the position of the spectral lines on the frequency axis. This behavioris in agreement with the coherently driven three-level atom of Martinez et al. [35].

The shape of the spontaneous-emission spectrum is strongly influenced by the variation of Rabifrequencies also. For instance, when the Rabi frequencies Ωba = Ωbd = 0.5 and |Ωdc| reduces to 0.1 andthe Rabi frequency Ωbc is set to 5, then the spectrum behavior with variation of phase φ shows extremenarrowing, and a single peak spectrum is observed for φ = 0 (see Fig. 3a). There are two other peakswith very small heights. These small peaks do have extreme narrowing, which reduces their effect evenfurther. At φ = π/2, the height of the central peak decreases, but the height of the small peak on theleft increases, and the heights of small peaks on the right also decrease, as shown in Fig. 3b. Studyingthe behavior of the spectrum for φ = π, we get an extremely narrow single peak with two small peaks,but this time the height of the central peak increases and the heights of the side peaks decrease. Withthe same set of Rabi frequencies, extreme narrowing is observed (see Fig. 3d) with negligible heights ofside peaks for φ = 3π/2. Obviously, the width and peak heights are strongly influenced by variation ofRabi frequencies, as well as the phase associated with microwave field. It is interesting to note that onappropriate values of Rabi frequencies and phase, the side peaks start to disappear (see Fig. 3d). Wealso note that the position of spectral lines is independent of the phase, while the heights of spectral lines

232


Fig. 2. Spontaneous emission spectra S(δ) (in units of[Γ−1]) for Ωba = Γ, Ωcb = Ωdb = |Ωdc| = 2Γ with φ = 0(a), 0.5π (b), π (c), and 1.5 π (d).

Fig. 3. Spontaneous emission spectra S(δ) (in units of[Γ−1]) for Ωba = Ωdb = 0.5 Γ, |Ωdc| = 0.1 Γ, and Ωbc =5 Γ with φ = 0 (a), 0.5π (b), π (c), and 1.5π (d).

233


Fig. 4. Spontaneous emission spectra S(δ) (in units of [Γ−1]) for φ = 0.6π with Ωba = Ωbc = |Ωdc| = Γ andΩdb = 0.1 Γ (a) and Ωba = Ωbc = Ωdb = 0.1 Γ and |Ωdc| = Γ (b).

depend on the phase. This behavior agrees with [3].

Fig. 5. The time evolution of the population in three upper levelsof the system for atomic parameters Ωba = Ωdb = 0.5 Γ, |Ωdc| =0.1 Γ, and Ωbc = 5Γ with φ = 1.5π.

The expression of spontaneous emis-sion spectrum given in Eq. (28) consists ofthree parts. Each part corresponds to fourpeaks associated to the four dressed statesof which it is composed. In Eq. (28),we neglect the interference terms betweenthe three sets of dressed states correspond-ing to the three bare states due to a largeseparation between them. The spectrumtherefore consists, in general, of 12 peakslocated at δ = ηi; their heights are givenby (α2

i + β2i )/χ2

i (i = 1, . . . , 12). Whenwe set φ = 0.6π, Ωba = Ωbc = |Ωdc| = 1and reduce Ωbd to 0.1, the spectrum showstwelve-peak behavior (see Fig. 4a). Onthe same value of phase, we set |Ωdc| = 1and Ωbd = Ωba = Ωbc = 0.1; then the spec-trum gives again a narrow single peak (seeFig. 4b). This behavior of the spectrumshows that there is a variety of combinations of the values of Rabi frequencies and phase for which wecan get spectral narrowing. We should also note that, as Fig. 4 indicates, that this scheme is quite robustand large modification of the spontaneous emission can be achieved. The assumption that the decayrates from the upper three levels are nearly the same gives a narrow single-peak spectrum of the sponta-neous emission which agrees with [33]. If we turn off all the microwave fields, the spontaneous emissionspectra exhibit a symmetric double-peak structure with equal height and normal linewidth restricted byspontaneous-emission-decay rates, which is in good agreement with [30].

Using the same parameters of Fig. 3d, we plot the evolution of the population in the upper levels|b〉, |c〉, and |d〉 (see Fig. 5). It is clear that the population in the upper levels is oscillating at small

234


values of t, but with increase in time it becomes slowly decaying, which explains the line narrowing seenin Fig. 3. Such a behavior of the population evolution shows that the physical trapping conditions donot exist in our system. This spectral narrowing is associated with slow decay rates. The result agreeswith Zhou and Swain [15] in obtaining line width narrowing near the quenching condition in the contextof resonance fluorescence of a closed V-type atom.

According to the experimental point of view, our system can be found easily in real atoms (e.g., alkalimetals, etc.). All parameters can be set easily, for example, the Rabi frequency can be set according toour required value just by changing the applied-field intensity. After neglecting the interference termsin Eqs. (4)–(6), our system becomes independent of the alignments of dipole moments. Most of theschemes presented for the control of spontaneous emission, e.g., [4,5,7,8], requires closely-spaced exciteddoublets with parallel dipole moments. It is more restrictive in finding the proper atomic (or molecular)configuration, since it requires the spontaneous-emission matrix elements to be parallel and the quantumstates to be close in energy. But orthogonal dipoles are easily found in nature. Since our system isindependent of the alignments of dipole moments, there is no need to achieve the condition of exciteddoublet parallel dipole moments that are close in energy. The proposed scheme is equally valid for atomic(or molecular) configurations with excited state of orthogonal dipole moments as well as for parallel dipolemoments. Using Racah algebra [34] we can show that the interference effects are only observable whenthe upper two states have identical angular momenta J ′s but, generally, the two levels with identicalangular momentum have energy separation much larger than their decay widths as discussed in [31].However, to achieve this condition in real atoms for observing interference effects is not an easy job. Inour system, we assume that the upper three levels are not closely spaced, which makes it easy to find inreal atoms.

4. Conclusions

In summary, we study the control of spontaneous emission in a five-level atomic system driven byfour fields. We show that, in the absence of quantum interference due to the sharing of vacuum modesby three transitions, we can control spontaneous emission by generating quantum coherence with thehelp of an applied microwave field. The variation of dynamical variables, namely, Rabi frequencies andcarrier phases of driven fields, provides a wide variety of spectral behavior, including extreme spectralnarrowing. We obtain an extremely narrow single-peaked spectrum by the selection of certain values ofRabi frequencies; it proves to be very useful for the purpose of atom localization, since it reduces theuncertainty to obtain the atom within a single peak only. The number of peaks, ranging from 12 to 1, canbe obtained by varying the Rabi frequencies and the phases. The benefits obtained by the reduction ofpeaks are mentioned in [11]. Thus, an efficient control of the spontaneous-emission process in a tailoredmanner can be achieved.

References

1. S. John and T. Quang, Phys. Rev. Lett., 78, 1888 (1997).2. P. Zhou and S. Swain, Phys. Rev. A, 55, 772 (1997).3. F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, Phys. Rev. A, 62, 013811 (2000).4. E. Paspalakis, C. H. Keitel, and P. L. Knight, Phys. Rev. A, 58, 4868 (1998).

235


5. E. Paspalakis and P. L. Knight, Phys. Rev. Lett., 81, 293 (1998).6. E. Paspalakis, N. J. Kylstra, and P. L. Knight, Phys. Rev. A, 61, 045802 (2000).7. H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, Phys. Rev. A, 55, 4454 (1997).8. S. Y. Zhu and M. O. Scully, Phys. Rev. Lett., 76, 388 (1996).9. P. Zhou and S. Swain, Phys. Rev. Lett., 78, 832 (1997).

10. Y. Wu and X. Yang, Phys. Rev. A, 70, 053818 (2004).11. F. Ghafoor, S. Qamar, and M. S. Zubairy, Phys. Rev. A, 65, 043819 (2002).12. S. Yuan and J. Y. Gao, Eur. Phys. J. D, 11, 267 (1999).13. G. C. Hegerfeldt and M. B. Plenio, Phys. Rev. A, 46, 373 (1992).14. S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett., 71, 3637 (1993).15. P. Zhou and S. Swain, Phys. Rev. Lett., 77, 3995 (1996).16. M. O. Scully, Phys. Rep., 219, 191 (1992).17. P. Mandel, Contemp. Phys., 34, 235 (1994).18. K. Hakuta, L. Marmet, and B. P. Stoicheff, Phys. Rev. Lett., 66, 596 (1991).19. J. E. Field, K. H. Hahn, and S. E. Harris, Phys. Rev. Lett., 67, 3062 (1991).20. S. E. Harris, Phys. Today, 50, 36 (1997).21. J. P. Marangos, J. Mod. Opt., 45, 471 (1998).22. M. D. Lukin and A. Imamoglu, Nature, 413, 273 (2001).23. Y. Wu, L. Wen, and Y. Zhu, Opt. Lett., 28, 631 (2003).24. X. H. Tu, J. Wang, K. J. Jiang, et al., Chinese Phys. Lett., 20, 1954 (2003).25. F. Xiao, R. M. Guo, S. Chen, et al., Chinese Phys. Lett., 20, 1257(2003).26. K. Zaheer and M. S. Zubairy, Phys. Rev. A, 38, 5227 (1988).27. E. Arimondo, in: E. Wolf (ed.), Progress in Optics, Elsevier, Amstredam (1996), p. 222.28. Jin-Hui Wu, Ai-Jun Li, Yue Ding, et al., Phys. Rev. A, 72, 023802 (2005).29. Li Xiao-Xia, Hu Xiang-Ming, Shi Wen-Xing, and Xu Qing, Chinese Phys. Lett., 23, 340 (2006).30. Li Jia-Hua, Liu Ji-Bing, Chen Ai-Xi, and Qi Chun-Chao, Phys. Rev. A, 74, 033816 (2006).31. A. Imamoglu, Phys. Rev. A, 40, 2835 (1989).32. V. Weisskopf and E. P. Wigner, Z. Phys., 54, 63 (1930).33. S. Y. Zhu, R. C. F. Chan, and C. P. Lee, Phys. Rev. A, 52, 710 (1995).34. R. D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, Berkeley

(1981), pp. 142–147; 400–402.35. M. A. G. Martinez, P. R. Herezfeld, C. Samuels, et al., Phys. Rev. A, 55, 4483 (1997).

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Control of spontaneous emission in a five-level system

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