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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3509–3518
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Control of spontaneous emission from a micro-wave driven atomic system

Chun Liang Wang, Zhi Hui Kang, Si Cong Tian, and Jin Hui Wu  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 3509-3518 (2012)
http://dx.doi.org/10.1364/OE.20.003509


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Abstract

We demonstrate coherent control of spontaneous emission from an indirectly coupled transition in a microwave driven four-level atomic system. The transition of concern is not directly coupled by any laser fields, while the ground state is coupled to another ground state by a microwave field. We show that the coupling of the microwave field produces interesting features such as double narrow lines in the emission spectrum. The heights, widths and positions of the emission peaks can be controlled by modifying the Rabi frequency and detuning of the microwave field. We discuss the spectra in the dressed states basis.

© 2012 OSA

1. Introduction

Spontaneous emission is a fundamental phenomenon originating from the interaction between radiation and matter. It has been one of the most attractive topics in the field of quantum optics for the last decades. We can draw a lot of information of quantum systems by focusing on spontaneous emission. For example, Das and Agarwal showed that the photon-photon correlations of the radiation can be viewed as a probe of vacuum-induced coherence effects in a four-level system [1

1. S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A 77(3), 033850 (2008). [CrossRef]

]. We can also study many basic conceptions of quantum optics through the observation of spontaneous emission. In Ref [2

2. M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006). [CrossRef] [PubMed]

], the authors investigated the emission from a J=1/2 to J=1/2 transition that is driven by a monochromatic laser field. They showed that the spectrum and intensity of the fluorescence reflects the quantum interference enforced by the principle of time-energy complementarity. In Ref [3

3. V. V. Temnov and U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17(7), 5774–5782 (2009). [CrossRef] [PubMed]

], Temnov and Woggon found giant photon bunching in the cooperative spontaneous emission. In spontaneous emission from a continuously driven atomic ensemble, Norris et al. observed ground-state quantum beats [4

4. D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010). [CrossRef] [PubMed]

].

Apart from the interest of fundamentals, modification and control of spontaneous emission can also find applications in many fields such as lasing without inversion [5

5. S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62(9), 1033–1036 (1989). [CrossRef] [PubMed]

8

8. G. A. Koganov, B. Shif, and R. Shuker, “Field-driven super/subradiant lasing without inversion in three-level ladder scheme,” Opt. Lett. 36(15), 2779–2781 (2011). [CrossRef] [PubMed]

], transparent high index materials [9

9. A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996). [CrossRef] [PubMed]

], high-precision spectroscopy and magnetometry [10

10. O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011). [CrossRef] [PubMed]

,11

11. M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000). [CrossRef]

], spatial localization of atoms [12

12. R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28(1), 10–17 (2011). [CrossRef]

,13

13. C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011). [CrossRef]

], quantum information and computing [14

14. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000). [CrossRef] [PubMed]

16

16. P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010). [CrossRef] [PubMed]

], and so on. Based on the narrowing of spontaneous emission, the determination of atomic multipole moments by means of the detection of the fluorescence spectrum is anticipated to increase in accuracy by several orders of magnitude [10

10. O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011). [CrossRef] [PubMed]

]. In the study of spatial atom localization, Wan et al. and Ding et al. presented 2D atom localization schemes via controlled spontaneous emission from coherently driven four-level tripod [12

12. R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28(1), 10–17 (2011). [CrossRef]

] and five-level M-type [13

13. C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011). [CrossRef]

] atomic systems respectively. Entangled photons can also be generated through spontaneous emission. Grünwald and Vogel showed that resonance fluorescence from regular atomic systems may serve as a continuous source of non-Gaussian entangled radiation propagating in two different directions [16

16. P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010). [CrossRef] [PubMed]

]. Unwanted spontaneous emission prevents lasers from working at high frequencies, induces decoherence in a quantum system, set the ultimate precision of quantum measurements, therefore a great deal of work has been devoted to the elimination and suppression of this process [17

17. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996). [CrossRef] [PubMed]

21

21. W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A 79(1), 012310 (2009). [CrossRef]

].

We can alter the emission of atoms by placing them in different environments such as cavities [22

22. M. L. Terraciano, R. O. Knell, D. L. Freimund, L. A. Orozco, J. P. Clemens, and P. R. Rice, “Enhanced spontaneous emission into the mode of a cavity QED system,” Opt. Lett. 32(8), 982–984 (2007). [CrossRef] [PubMed]

,23

23. X. D. Zeng, M. Z. Yu, D. W. Wang, J. P. Xu, and Y. P. Yang, “Spontaneous emission spectrum of a V-type three-level atom in a Fabry-Perot cavity containing left-handed materials,” J. Opt. Soc. Am. B 28(9), 2253–2259 (2011). [CrossRef]

], photonic crystals [24

24. X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011). [CrossRef]

], and nanostructures [25

25. S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011). [CrossRef]

]. For an atom in the free space, quantum coherence plays an important role in controlling spontaneous emission. In the presence of Spontaneously Generated Coherence (SGC), Zhu and Scully showed spectral line elimination and spontaneous emission cancellation in a four level system [17

17. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996). [CrossRef] [PubMed]

]. In a similar system, Paspalakis and Knight showed the possibility of controlling the emission spectrum with the relative phase of the driving fields [26

26. E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998). [CrossRef]

]. Interesting features of spontaneous emission, such as narrowing [27

27. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996). [CrossRef] [PubMed]

], squeezing [28

28. I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005). [CrossRef]

] and splitting [29

29. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008). [CrossRef]

], have been extensively studied in a variety of systems where SGC exists. We can also use quantum coherence generated by coherent fields to effectively modify the properties of spontaneous emission. Laser fields [30

30. L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990). [CrossRef] [PubMed]

33

33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005). [CrossRef]

] and microwave fields [34

34. F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000). [CrossRef]

36

36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008). [CrossRef]

], for example, have been used to couple different energy levels of atoms. It has been predicted [30

30. L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990). [CrossRef] [PubMed]

] and experimental demonstrated [32

32. D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991). [CrossRef] [PubMed]

] that laser fields may induce multi-peaks and sub-natural width peak in resonance fluorescence of three-level systems. In four-level systems, Wu et al. [33

33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005). [CrossRef]

] showed that coherence generated by a laser field could produce quenching and narrowing of the emission spectra. In a five-level system, Li et al. [35

35. J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006). [CrossRef]

] demonstrated similar features and showed the possibility of simulating SGC by coupling two ground states with a microwave field. In a four-level Y system where the upper two levels are coupled by a microwave field, the coupling of the microwave field is responsible for the phase dependent spectra [36

36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008). [CrossRef]

].

Coupling close-lying levels with microwave fields has been shown to be an efficient way of controlling spontaneous emission [34

34. F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000). [CrossRef]

36

36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008). [CrossRef]

]. It may produce enriched phenomena and provide flexibility for the control of spontaneous emission. However, most of the present researches on spontaneous emission deal with emission to such ground levels that is either unperturbed [17

17. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996). [CrossRef] [PubMed]

, 26

26. E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998). [CrossRef]

, 33

33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005). [CrossRef]

36

36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008). [CrossRef]

] or directly coupled to the excited levels by laser fields [27

27. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996). [CrossRef] [PubMed]

32

32. D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991). [CrossRef] [PubMed]

]. Spontaneous emission from an indirectly driven transition, which means it is not directly driven by any laser fields while the ground state is coupled to another ground level by a microwave field, has been seldom mentioned. Inspired by this situation, we investigate the spontaneous emission from a laser coupled excited state to a microwave coupled ground state in a four-level atomic system. We investigate the steady-state spectrum of the spontaneous emission, demonstrate phenomena such as double narrow lines of the spectrum, and show the possibility of controlling the spectrum with the Rabi frequency and detuning of the microwave field.

The paper is organized as follows. In Sec. 2, we introduce the model and the basic equations. In Sec. 3, we describe the numerical results and explain the corresponding features. Sec. 4 contains a summary of the results.

2. Schemes and equations

We consider a four-level system depicted in Fig. 1(a)
Fig. 1 The energy scheme under consideration. (a) In the bare-state basis. (b) In the dressed-state basis of the two laser fields and the microwave field. (c) In the dressed- state basis of the laser fieldω1.
. This set contains an excited state |3, and three ground states |1,|2, and |4. The transitions from |3 to |1,|2, and |4 are optical dipoles with decay rates γ31,γ32,and γ34, respectively. There are also nonradiative decaying rates (might owing to collision) between the three ground levels, denoted by γ12,γ24, and γ14. The transitions |3|1 and |3|2 are resonantly driven by two laser fields ω1 and ω2. The ground states |2 and |4 are coupled by a microwave field ω3. The detuning of the microwave field from the magnetic transition |2|4 is defined by Δ=ω42ω3. The Rabi frequencies of the two lasers and the microwave field are Ω1,Ω2, and Ω3, respectively.

In the interaction picture, the coherent part of the Hamiltonian in the rotating wave approximation and electric dipole approximation reads

HI=Δ|44|[Ω1|13|+Ω2|23|+Ω3|24|+H.c.].
(1)

The master equation of motion for the density operator in an arbitrary multilevel atomic system is given by
ρt=1i[H,ρ]+Lρ,
(2)
where Lρ represents the decay part in the system. By expanding Eq. (2), we can arrive at the following density matrix equations of motion.

ρ˙22=iΩ2(ρ32ρ23)+iΩ3(ρ42ρ24)+γ32ρ33+γ12ρ11+γ24ρ44(γ12+γ24)ρ22ρ˙33=iΩ1(ρ13ρ31)+iΩ2(ρ23ρ32)(γ31+γ32+γ34)ρ33ρ˙44=iΩ3(ρ24ρ42)+γ34ρ33+γ14ρ11+γ24ρ22(γ12+γ14)ρ44ρ˙12=[γ12(γ14+γ24)/2]ρ12+iΩ1ρ32iΩ2ρ13iΩ3ρ14ρ˙13=[(γ12+γ14+γ31+γ32+γ34)/2]ρ13+iΩ1(ρ33ρ11)iΩ2ρ12ρ˙14=[γ14(γ12+γ24)+iΔ]ρ14iΩ1ρ34iΩ3ρ12ρ˙23=[(γ12+γ24+γ31+γ32+γ34)/2]ρ23+iΩ2(ρ33ρ22)+iΩ3ρ43ρ˙24=[γ24(γ12+γ14)/2+iΔ]ρ24+iΩ3(ρ44ρ22)+iΩ2ρ34ρ˙34=[(γ31+γ32+γ34+γ14+γ24)/2+iΔ]ρ34+iΩ1ρ14+iΩ2ρ24iΩ3ρ32.
(3)

Closure of this atomic system requires ρ11+ρ22+ρ33+ρ44=1, and ρij=ρji*.

We proceed to calculate the steady-state emission spectra with the common method [27

27. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996). [CrossRef] [PubMed]

, 29

29. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008). [CrossRef]

31

31. A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991). [CrossRef] [PubMed]

]. As is well known, the fluorescence emission spectrum is proportional to the Fourier transform of the steady-state correlation function limtE()(r,τ+t)E(+)(r,t) [37

37. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997), Chap. 10.

], whereE(±)(r,t) are the positive and negative frequency parts of the radiation field in the far zone, which consists of a free-field operator, and a source-field operator that is proportional to the atomic polarization operator. Therefore the incoherent fluorescence spectrum can be expressed in terms of the atomic correlation function,

S(ω)=Re0limt<ΔD+(τ+t)ΔD(t)>eiωtdτ.
(4)

Re denotes the real part and ΔD±(t)=D±(t)<D±()> represents the deviation of the dipole polarization operator D±(t) from its mean steady-state value, and
D+(t)=μ34a4a3,D(t)=[D+(t)],
(5)
where μ34 is the dipole moment of the atomic transition from |3 to |4.

In order to calculate the emission spectrum, we write the equations of motion [see Eq. (3)] in the form
ddtΨ=LΨ+I
(6)
where Ψ=(ρ12,ρ13,ρ14,ρ21,ρ22,ρ23,ρ24,ρ31,ρ32,ρ33,ρ34,ρ41,ρ42,ρ43,ρ44)T, andL is a (15×15) matrix. The elements of L and I can be found explicitly from Eq. (3).

Following the common procedure [27

27. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996). [CrossRef] [PubMed]

, 29

29. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008). [CrossRef]

31

31. A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991). [CrossRef] [PubMed]

], we can obtain the steady spectrum of the spontaneous emission from the excited state |3 to the ground state |4 by means of the quantum regression theorem [38

38. M. Lax, “Quantum noise. XI. multitime correspondence between quantum and classical Stochastic Processes,” Phys. Rev. 172(2), 350–361 (1968). [CrossRef]

,39

39. S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A 14(10), 2577–2580 (1981). [CrossRef]

]. The result is
S(δk)=Re{M14,12ρ¯31+M14,13ρ¯32+M14,14ρ¯33+M14,15ρ¯34+lN14,lIlρ¯34},
(7)
where
Mi,j=[(zL)1|z=iδk]i,j,Ni,j=[L1(zL)1|z=iδk]i,j.
(8)
ρ¯ijis the steady-state population (i=j) and atomic coherence (ij), which can be obtained by setting ρ˙ij=0 and solving numerically Eq. (6). δk is the detuing between the fluorescence and the transition |3|4.

We assume that the decay rates from |3 to |1,|2, and |4 are γ12=γ14=γ24=γ. The Rabi frequencies of the two laser fields are fixed at Ω1=Ω2=γ. We tune the Rabi frequency Ω3 and the detuning Δ of the microwave field, and calculate the corresponding emission spectrum.

3. Results and discussions

First of all, we assume that the microwave field is resonant with the corresponding transition and concentrate on controlling the emission properties with the Rabi frequency of the microwave field. The numerical results at different values of Ω3 are shown in Fig. 2
Fig. 2 Numerical results of the emission spectrum with different Rabi frequency of the microwave field. The parameters used are (a)Ω3=0,(b)Ω3=0.05γ,(c)Ω3=0.1γ,(d)Ω3=0.2γ,(e)Ω3=1.5γ,(f)Ω3=2γ.
. When Ω3=0, which means no microwave field is applied, the fluorescence spectrum is doubly peaked with normal width restricted by the spontaneous decay rate γ [see Fig. 2(a)]. Then we apply a weak microwave field to the system by setting Ω3=0.05γ. Two narrow lines can be observed in the middle of the spontaneous emission spectrum [see Fig. 2(b)]. The heights of theses narrow lines exceed the two broad peaks when we increase the Rabi frequency of the microwave field to Ω3=0.1γ [see Fig. 2(c)]. It is clear that the narrow lines are more pronounced when we increase the value of the Rabi frequency to Ω3=0.2γ[see Fig. 3(d)
Fig. 3 Properties as a function of the Rabi frequency Ω3, (a) The eigen energies given by Eq. (2), (b) Steady-state population, (c) decay rates of |b and |c.
]. We can also see the distance between the two narrow lines grows larger with an increase of Ω3.

If the increase of Ω3 continues, narrow lines become broadened [see Fig. 2(e)]. Meanwhile, an emission peak emerges at the frequency of ω=ω24[see Fig. 2(e)]. Moreover, the central peak can be greatly enhanced with a larger Ω3 [see Fig. 2(f)]. We can also observe more peaks at both sides of the central peak.

The observed spectrum can be analyzed in the dressed state basis. Under the resonant coupling of the two laser fields and the microwave field, we obtain the dressed levels by finding the eigenvectors of the coherent part of the interaction Hamiltonian [see Eq. (1)]. The eigen energies are

λa=(Ω12+Ω22+Ω32)+(Ω12+Ω22+Ω32)24Ω12Ω322λb=(Ω12+Ω22+Ω32)(Ω12+Ω22+Ω32)24Ω12Ω322λc=(Ω12+Ω22+Ω32)(Ω12+Ω22+Ω32)24Ω12Ω322λd=(Ω12+Ω22+Ω32)+(Ω12+Ω22+Ω32)24Ω12Ω322.
(9)

The corresponding dressed levels can be expressed as
|a=x1|1+x2|2+x3|3+x4|4|b=x1|1x2|2+x3|3+x4|4|c=x1|1x2|2+x3|3+x4|4|d=x1|1+x2|2+x3|3+x4|4
(10)
where
x1=Ω1(λi2Ω32)Ω2Ω3λi1+(λiΩ3)2+(λi2Ω32Ω2Ω3)2+(Ω1(λi2Ω32)Ω2Ω3λi)2x2=λiΩ31+(λiΩ3)2+(λi2Ω32Ω2Ω3)2+(Ω1(λi2Ω32)Ω2Ω3λi)2x3=λi2Ω32Ω2Ω31+(λiΩ3)2+(λi2Ω32Ω2Ω3)2+(Ω1(λi2Ω32)Ω2Ω3λi)2x4=11+(λiΩ3)2+(λi2Ω32Ω2Ω3)2+(Ω1(λi2Ω32)Ω3Ω2λi)2,
(11)
with i=a,b,c,d.

For simplicity, we write Eq. (10) in the following form:
|i=Cik|k,(i=a,b,c,d;k=1,2,3,4),
(12)
where Cik is a function of Ω3.

We can see that both the excited state |3 and the ground state |4 [see Fig. 1(a)] are split into four dressed levels [see Fig. 1(b)], which are |a,|b,|c, and |d for the bare-state level |3 while |a,|b,|c, and |d for the bare-state level |4. Therefore the spontaneous emission from the excited state |3 to the ground state |4 has 16 dipole transitions in the dressed-state representation. This result is similar to that in Ref [40

40. C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977). [CrossRef]

], where a two-level system has 4 dipole transitions in the dressed-state representation. Note that the dressed levels |i has the same expressions as the dressed levels |i with i=a,b,c,d, but |i and |i are different in energy by a constant value, i. e. by the energy difference between level |3 and |4. It is well known that positions, widths, and heights of emission peaks are determined by the energies, electronic dipole moments, and steady-state populations of dressed states. In order to interpret the numerical results, we investigate the properties of the dressed levels. We plot the eigen energies (λi,i=a,b,c,d) and the populations of the dressed levels (ρaa,ρbb,ρcc,ρdd) as functions of the Rabi frequency Ω3 [see Fig. 3]. We can see that the energies of the dressed levels |a and |d depend weakly on Ω3, and the splitting between the dressed levels |b and |c increases with a larger value of Ω3 [see Fig. 3(a)]. When Ω3 is relatively small (i. e. Ω3<2γ), all the four dressed levels are well populated [see Fig. 3(b)]. We can also see the influence of Ω3 on the populations of the dressed levels.

The decay rate of the transition between the dressed levels |i to |jis proportional to the squared dipole moments Rij=|i|P|j|2, where P=μ43|43| is the transition dipole moment operator between |3 and |4 in the bare state basis [see Fig. 1(a)]. Rij can be calculated with the expression:
Rij=|j|P|i|2=|Cj4|2μ432|Ci3|2,(i,j=a,b,c,d),
(13)
where Cij is the coefficient in Eq. (12).

The two broad side bands [see Fig. 2(b)2(d)] are associated to the decay of the four transitions |a|b,|a|c,|d|b, and |d|c. The splitting of the dressed levels |b and |c is smaller than the linewidths of the emission peaks, so we can see two broad bands instead of four. With a weak coupling of the microwave field (e. g. Ω3<0.2γ), the decay rates of the dressed levels |b and |c takes very low values (approximately 1/400 of the decay rates of |a and |d when Ω3=0.2γ). The slow decay of the transition |b|c and |c|b leads to the double narrow lines in the middle of the emission spectrum. The position of the emission peaks are determined by the eigen energies of the dressed levels [see Fig. 3(a)]. The dependence of λb and λc on Ω3 are responsible for the variation of the splitting between the two narrow lines. The positions of the two broad side bands are almost unchanged owing to the fact that λa and λd depend weakly on Ω3. The decay rate of the dressed levels |b and |c are sensitive to Ω3 [see Fig. 3(c)], so the heights of the two narrow lines are greatly enhanced with an increase of this Rabi frequency [see Fig. 2(b)2(d)]. Under the weak coupling of the microwave field, the decay of other transitions, such as |a|d and |d|a, are too weak to be observed.When Ω3 is comparable to the Rabi frequencies of the laser fields Ω1 and Ω2, the decay of the dressed levels |b and |c becomes much faster [see Fig. 3(c)], and results in the broadening of the corresponding emission [see the peaks marked with arrows in Fig. 2(e), 2(f)]. When we tune the value of Ω3, the populations and the decay rate of the dressed levels change accordingly. Then all the sixteen transitions between the dressed levels contribute to the emission spectrum, and results in more peaks in the spectrum [see Fig. 2(e), 2(f)]. Owing to the degeneracy of the transitions, we can obtain as many as nine peaks with sufficient Rabi frequencies of the driving fields (not shown here).

The spontaneous emission is also sensitive to the detuning of the microwave field relative to the magnetic transition |4|2. We illustrate the effects of the detuning Δ on the emission spectrum in Fig. 4
Fig. 4 Numerical results of the emission spectrum with different detuning of the microwave field. The parameters used are (a) Δ=0.1, (b) Δ=0.2γ, (c) Δ=0.4γ.
. The Rabi frequency of the microwave field is fixed at Ω3=0.2γ.

By varying the detuning Δ, we get the results in Fig. 4(a)4(c). When the microwave field is slightly detuned, the spectrum becomes asymmetric [see Fig. 4(a)]. We can also observe a narrow line at the frequency of ω=ω34. When the detuning is increased to Δ=0.2γ, the central peak turns to be comparable with the other two narrow lines in height [see Fig. 4(b)]. The narrow peak in the center continues to grow higher with a greater detuning [see Fig. 4(c), where Δ=0.4γ]. There are also larger splitings between the three narrow peaks with an increase of the detuning.

With the detuning of the microwave field Δ, it is difficult to obtain the analytical expression of the dressed levels as Eqs. (9)(11). We give qualitative explanations for the results in Fig. 4. Under the coupling of the two laser fields and the microwave field, we get dressed levels similar to those in Fig. 1(b). Compared with the case of resonant coupling, the properties of the dressed levels, such as the positions, populations and dipole moments between them, are modified by the detuing of the microwave field Δ. The modified populations and decay rates of the dressed levels lead to the asymmetry and additional narrow peak in the emission spectra shown in Fig. 4. The larger is the detuing Δ, the greater is the modification. As a result, we see the variation of the spectrum from Fig. 4(a) to 4(c). Furthermore, the detuing Δ modifies the positions of the dressed levels and induces splitings between the degenerated transitions in Fig. 3(b). Therefore the spectrum is further affected. As a matter of fact, we can get as much as 13 peaks in the emission spectrum (no shown here) with proper chosen Rabi frequencies and detunings of the coupling fields.

The observed results can also be viewed as the combined effects of the microwave coupling and SGC between dressed levels. In the dressed-state representation of the laser field ω1, the system turns to be a four-level scheme with two close-lying excited levels [see Fig. 1(c)], where |+=(|1|3)/2 and |=(|1+|3)/2. The coupling of the microwave field together with SGC between |+ and | affect the decay from the two excited levels to the ground state |4. The combined effects of SGC and the coupling of the microwave field result in the interesting emission spectrum such as narrow peaks. We can certainly realize the modification and control of spontaneous emission by tuning the parameters of the microwave field.

As is well known, it is very difficult to realize SGC in real atoms owing to the rigorous requirements: there are at least two near degenerate levels subject to the condition that the dipole moments from them to another level are not orthogonal. The system presented here is more realistic as compared with the systems with SGC [17

17. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996). [CrossRef] [PubMed]

, 26

26. E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998). [CrossRef]

29

29. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008). [CrossRef]

] because no stringent condition is required.

Our system is a modified version of a tripod four-level atomic system [33

33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005). [CrossRef]

], where fluorescence quenching and spectral-line narrowing can be obtained by tuning the detunings of the laser fields. With the coupling of the microwave field, we observe enriched phenomena in this work, such as more narrow peaks in the spectrum. It has been shown that emission spectrum can be modified by detunings and phases of laser fields [33

33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005). [CrossRef]

, 35

35. J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006). [CrossRef]

,36

36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008). [CrossRef]

], while we demonstrate an alternative way for the coherent control of spontaneous emission with the Rabi frequency and detuning of the microwave field. Our work can be extended to other systems such as N-type and M-type atomic systems. We can anticipate more interesting phenomena and flexible control of spontaneous emission by including the coupling of microwave fields.

Finally, we would like to mention the limitation of the current scheme. We obtain the emission spectrum with numerical calculations, and resort to the dressed-state basis for the physics behind the phenomena. In more sophisticated systems, it might be difficult to give explicit expressions of dressed levels and we might need to solve the problem numerically.

4. Conclusion

We have investigated the steady-state spectrum of spontaneous emission from an indirectly driven transition in a four-level atomic system driven by a microwave field and two laser fields. We have obtained a few interesting features such as double narrow lines between two broad bands. We can control the heights, linewidths and splittings of the narrow peaks by tuning the Rabi frequency and detuning of the microwave field. In the mean time, the number and the relative heights of emission peaks can also be modified. The numerical results have been analyzed with the decay rates of the dressed levels.

The phenomena predicted in this paper can be experimentally observed in cold atoms R87b. The magnetic sublevels of 5S1/2,F=2 can serve as the ground states |1 and |2. The excited state |3 and ground state |4 can be provided by5P1/2,F=2 and 5S1/2,F=1. We note that the populations in the ground state |4 can be pumped by the microwave field in our system, so that we do not need processes such as incoherent pumping to maintain the intensity of the fluorescence. Moreover, the spontaneous emission of interest can be spectrally separated from the scattering of the lasers. These features may facilitate the experimental observation.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (09QNJJ009), the National Natural Science Foundation of China (11174110), the Science and Technology Development program of Jilin Province (201101006), and the Basic Scientific Research Foundation of Jilin University.

References and links

1.

S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A 77(3), 033850 (2008). [CrossRef]

2.

M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006). [CrossRef] [PubMed]

3.

V. V. Temnov and U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17(7), 5774–5782 (2009). [CrossRef] [PubMed]

4.

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010). [CrossRef] [PubMed]

5.

S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62(9), 1033–1036 (1989). [CrossRef] [PubMed]

6.

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999). [CrossRef]

7.

S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008). [CrossRef] [PubMed]

8.

G. A. Koganov, B. Shif, and R. Shuker, “Field-driven super/subradiant lasing without inversion in three-level ladder scheme,” Opt. Lett. 36(15), 2779–2781 (2011). [CrossRef] [PubMed]

9.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996). [CrossRef] [PubMed]

10.

O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011). [CrossRef] [PubMed]

11.

M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000). [CrossRef]

12.

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28(1), 10–17 (2011). [CrossRef]

13.

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011). [CrossRef]

14.

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000). [CrossRef] [PubMed]

15.

A. Kalachev and S. Kröll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A 74(2), 023814 (2006). [CrossRef]

16.

P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010). [CrossRef] [PubMed]

17.

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996). [CrossRef] [PubMed]

18.

G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett. 86(19), 4271–4274 (2001). [CrossRef] [PubMed]

19.

E. Frishman and M. Shapiro, “Complete suppression of spontaneous decay of a manifold of states by infrequent interruptions,” Phys. Rev. Lett. 87(25), 253001 (2001). [CrossRef] [PubMed]

20.

J. Evers and C. H. Keitel, “Spontaneous-emission suppression on arbitrary atomic transitions,” Phys. Rev. Lett. 89(16), 163601 (2002). [CrossRef] [PubMed]

21.

W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A 79(1), 012310 (2009). [CrossRef]

22.

M. L. Terraciano, R. O. Knell, D. L. Freimund, L. A. Orozco, J. P. Clemens, and P. R. Rice, “Enhanced spontaneous emission into the mode of a cavity QED system,” Opt. Lett. 32(8), 982–984 (2007). [CrossRef] [PubMed]

23.

X. D. Zeng, M. Z. Yu, D. W. Wang, J. P. Xu, and Y. P. Yang, “Spontaneous emission spectrum of a V-type three-level atom in a Fabry-Perot cavity containing left-handed materials,” J. Opt. Soc. Am. B 28(9), 2253–2259 (2011). [CrossRef]

24.

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011). [CrossRef]

25.

S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011). [CrossRef]

26.

E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998). [CrossRef]

27.

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996). [CrossRef] [PubMed]

28.

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005). [CrossRef]

29.

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008). [CrossRef]

30.

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990). [CrossRef] [PubMed]

31.

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991). [CrossRef] [PubMed]

32.

D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991). [CrossRef] [PubMed]

33.

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005). [CrossRef]

34.

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000). [CrossRef]

35.

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006). [CrossRef]

36.

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008). [CrossRef]

37.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997), Chap. 10.

38.

M. Lax, “Quantum noise. XI. multitime correspondence between quantum and classical Stochastic Processes,” Phys. Rev. 172(2), 350–361 (1968). [CrossRef]

39.

S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A 14(10), 2577–2580 (1981). [CrossRef]

40.

C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977). [CrossRef]

OCIS Codes
(270.1670) Quantum optics : Coherent optical effects
(270.4180) Quantum optics : Multiphoton processes

ToC Category:
Quantum Optics

History
Original Manuscript: November 28, 2011
Revised Manuscript: December 24, 2011
Manuscript Accepted: January 6, 2012
Published: January 30, 2012

Citation
Chun Liang Wang, Zhi Hui Kang, Si Cong Tian, and Jin Hui Wu, "Control of spontaneous emission from a micro-wave driven atomic system," Opt. Express 20, 3509-3518 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3509


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References

  1. S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A77(3), 033850 (2008). [CrossRef]
  2. M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett.96(10), 100403 (2006). [CrossRef] [PubMed]
  3. V. V. Temnov and U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express17(7), 5774–5782 (2009). [CrossRef] [PubMed]
  4. D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett.105(12), 123602 (2010). [CrossRef] [PubMed]
  5. S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett.62(9), 1033–1036 (1989). [CrossRef] [PubMed]
  6. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A61(1), 013807 (1999). [CrossRef]
  7. S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett.100(17), 173601 (2008). [CrossRef] [PubMed]
  8. G. A. Koganov, B. Shif, and R. Shuker, “Field-driven super/subradiant lasing without inversion in three-level ladder scheme,” Opt. Lett.36(15), 2779–2781 (2011). [CrossRef] [PubMed]
  9. A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett.76(21), 3935–3938 (1996). [CrossRef] [PubMed]
  10. O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett.106(3), 033001 (2011). [CrossRef] [PubMed]
  11. M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A62(1), 013808 (2000). [CrossRef]
  12. R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B28(1), 10–17 (2011). [CrossRef]
  13. C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A83(6), 063834 (2011). [CrossRef]
  14. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature404(6775), 247–255 (2000). [CrossRef] [PubMed]
  15. A. Kalachev and S. Kröll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A74(2), 023814 (2006). [CrossRef]
  16. P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett.104(23), 233602 (2010). [CrossRef] [PubMed]
  17. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett.76(3), 388–391 (1996). [CrossRef] [PubMed]
  18. G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett.86(19), 4271–4274 (2001). [CrossRef] [PubMed]
  19. E. Frishman and M. Shapiro, “Complete suppression of spontaneous decay of a manifold of states by infrequent interruptions,” Phys. Rev. Lett.87(25), 253001 (2001). [CrossRef] [PubMed]
  20. J. Evers and C. H. Keitel, “Spontaneous-emission suppression on arbitrary atomic transitions,” Phys. Rev. Lett.89(16), 163601 (2002). [CrossRef] [PubMed]
  21. W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A79(1), 012310 (2009). [CrossRef]
  22. M. L. Terraciano, R. O. Knell, D. L. Freimund, L. A. Orozco, J. P. Clemens, and P. R. Rice, “Enhanced spontaneous emission into the mode of a cavity QED system,” Opt. Lett.32(8), 982–984 (2007). [CrossRef] [PubMed]
  23. X. D. Zeng, M. Z. Yu, D. W. Wang, J. P. Xu, and Y. P. Yang, “Spontaneous emission spectrum of a V-type three-level atom in a Fabry-Perot cavity containing left-handed materials,” J. Opt. Soc. Am. B28(9), 2253–2259 (2011). [CrossRef]
  24. X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A83(5), 053823 (2011). [CrossRef]
  25. S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A83(5), 055805 (2011). [CrossRef]
  26. E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett.81(2), 293–296 (1998). [CrossRef]
  27. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett.77(19), 3995–3998 (1996). [CrossRef] [PubMed]
  28. I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A72(3), 033809 (2005). [CrossRef]
  29. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A77(3), 033820 (2008). [CrossRef]
  30. L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A42(3), 1630–1649 (1990). [CrossRef] [PubMed]
  31. A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A43(7), 3748–3763 (1991). [CrossRef] [PubMed]
  32. D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett.66(19), 2460–2463 (1991). [CrossRef] [PubMed]
  33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A72(2), 023802 (2005). [CrossRef]
  34. F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A62(1), 013811 (2000). [CrossRef]
  35. J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A74(3), 033816 (2006). [CrossRef]
  36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A77(5), 053806 (2008). [CrossRef]
  37. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997), Chap. 10.
  38. M. Lax, “Quantum noise. XI. multitime correspondence between quantum and classical Stochastic Processes,” Phys. Rev.172(2), 350–361 (1968). [CrossRef]
  39. S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A14(10), 2577–2580 (1981). [CrossRef]
  40. C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B10(3), 345–363 (1977). [CrossRef]

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