Manipulation of Spontaneous Emission via Quantum Interference in an Elliptically Polarized Laser Field

Transcription

1 Commun. Theor. Phys. 59 (2013) Vol. 59, No. 5, May 15, 2013 Manipulation of Spontaneous Emission via Quantum Interference in an Elliptically Polarized Laser Field DING Chun-Ling (òë ), 1, LI Jia-Hua (Ó Ù), 2, YU Rong (ßÂ), 3 ZHANG Duo ( ), 2 and YANG Xiao-Xue ( ) 2 1 School of Physics and Electronics, Henan University, Kaifeng , China 2 Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan , China 3 School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan , China (Received January 4, 2013; revised manuscript received March 29, 2013) Abstract Manipulation of spontaneous emission from an atom confined in three kinds of modified reservoirs has been investigated by means of an elliptically polarized laser field. Some interesting phenomena such as the multi-peak structure, extreme spectral narrowing, and cancellation of spontaneous emission can be observed by adjusting controllable system parameters. Moreover, these phenomena depend on the constructive or destructive quantum interference between multiple decay channels and which can be changed appreciably by varying the phase difference between the two circularly polarized components of the probe field. These results demonstrate the importance of an elliptically polarized laser field in controlling the spontaneous emission and its potential applications in high-precision spectroscopy. PACS numbers: Gy, Qk, d Key words: spontaneous emission, elliptically polarized laser field, quantum interference 1 Introduction The control and modification of spontaneous emission [1 2] are of considerable importance in the area of atomic physics and quantum optics due to its many potential applications in high-precision spectroscopy and magnetometry, [3 6] lasing without inversion (LWI), [7 10] high-index transparent materials, [11 13] quantum information processing, [14 17] and so on. As is well known, quantum coherence and interference have become the basic mechanism for manipulating the spontaneous emission and optical spectra. The behavior of spontaneous emission relies not only on the energy-level structure of an atom, but on the characteristics of the surrounding environment, especially on the density of states (DOSs) of the radiation field. [18 20] Most existing work concentrates on the modification of the quantum interference between the spontaneous decay channels for the atom in free space, [21 31] which includes amplitude and phase control of spontaneous emission, [21 22] spectral line elimination and spontaneous emission cancellation via quantum interference, [23] the influence of quantum interference on the spontaneous emission from an excited atom, [24] the spectral narrowing and a black dark line induced by quantum interference, [25] etc. Moreover, the spontaneous emission cancellation has been proved experimentally in Ref. [32], and it is shown that the experimental results confirmed the theoretical prediction. Also, the laser control of collective spontaneous emission from a pair of two coupled radiators in vacuum has been reported in Ref. [33], it was specifically shown that how to trap population in and how to release it from a collective twoparticle system. In addition, it has been demonstrated that the planar dielectric microcavities, [34 36] quantum well laser diodes, [37 38] left-handed materials, [39 40] and photonic nanowires [41 42] have the ability to control the spontaneous emission under certain conditions. Photonic crystals (PCs), [43 45] also known as photonic band-gap (PBG) materials, are artificial periodic dielectric structures which can be used to control the propagation of electromagnetic waves. This concept originated from the pioneering work of Yablonovitch [46] and John, [47] they discussed the inhibition of spontaneous emission in solid-state systems and the strong localization of photons in certain disordered dielectric superlattices, respectively. To our knowledge, this material has become a powerful tool for the manipulation of spontaneous emis- Supported by the National Natural Science Foundation of China under Grant Nos and , by the Doctoral Foundation of the Ministry of Education of China under Grant No , and by the National Basic Research Program of China under Grant No. 2012CB clding2006@126.com huajia li@163.com c 2013 Chinese Physical Society and IOP Publishing Ltd

2 604 Communications in Theoretical Physics Vol. 59 sion and optical properties. Furthermore, remarkable achievements [48 62] have been obtained in such a periodic dielectric environment over the past few years, including simultaneous inhibition and redistribution of spontaneous emission in PCs, [48] controlling spontaneous emission in PBG materials doped with nanoparticles, [49 51] quantum interference effects in spontaneous emission from an atom embedded in a PBG structure, [52 54] and so on. However, in these proposed schemes, the atom embedded in a PBG reservoir which is driven by external coherent laser fields, only the case of linear polarization of the laser field was considered. On the contrary, if the atomic system coupled to a PBG reservoir is driven by an elliptically polarized laser field rather than a linearly polarized field, what will be the resulting emission spectrum by adjusting the phase difference between the two circularly polarized components of the laser field? Although extensive work has been done in this area, to the best of our knowledge, for the control of spontaneous emission in such a multilevel atomic system embedded in different modified reservoirs under the action of an elliptically polarized laser field via switching on and off an external magnetic field, it has rarely been studied theoretically and experimentally. On the other hand, it has been reported earlier [63] that the influence of the dark-state phase manipulation on the multitransparency windows is fully described by the use of an elliptically polarized laser field. More recently, Strelkov et al. [64] have studied the high-order harmonic generation (HHG) by an atom in an elliptically polarized laser field, and they discussed the effect of the driving field ellipticity on the HHG efficiency, harmonic ellipticity, and rotation angle under certain conditions. Additionally, the schemes [65 66] for implementing dual-channel all-optical switching are proposed by means of the switching on/off of two orthogonally polarized beams at different frequencies in the absence and presence of the signal beam and the external magnetic field. Motivated by the above-mentioned studies, here we put forward an alternative scheme to realize the control and modification of the spontaneous emission from an atom embedded in three kinds of modified reservoirs under the combined effects of an elliptically polarized field and an external magnetic field. Of particular interest is the introduction of a phase difference between the two circularly polarized components of the probe laser field. In fact, the phase difference stems from the rotation of a quarterwave plate (QWP), and simultaneously the QWP can be applied to control the polarization of the probe field. As a result, the phase difference can be used as a control parameter to investigate the behavior of spontaneous emission in our proposed scheme, which can also be verified by the discussions below. Another motivation for the present study is the application of an external magnetic field, due to the magnetic field is more easily available and more effective control compared with an extra laser field, and this is another important aspect to be considered in this paper. Naturally, the combination of an elliptically polarized laser field and an external magnetic field offers us further flexibility to manipulate the light-matter interactions, including the manipulation of spontaneous emission as discussed in the present paper. By adjusting these system parameters, the interesting results can be obtained as follows. The emission spectrum exhibits a four-peak structure with two normal width sideband lines and two ultranarrow central spectral lines in the presence of an external magnetic field. In contrast, when the external magnetic field is switched off, only three peaks remain in the spectrum, and the position of the ultranarrow central line is changed with the variation of the system parameters. Additionally, when the atom is initially prepared in an excited state, the polarization azimuth rotation and ellipticity only affect the central spectral lines and have no effect on the sideband lines. However, if the atom is initially prepared in a coherent superposition state of two ground levels, the two sideband lines are gradually suppressed with the increase in ellipticity due to the destructive quantum interference. Furthermore, in such a case, the extremely enhanced ultranarrow spectral lines can be clearly seen in the middle of the spectrum. In particular, these investigations may find potential applications in the design of optical elements and devices, as well as high-precision spectroscopy. This paper is organized as follows. In Sec. 2, we describe the atomic model under consideration and derive the analytical expression for the emission spectrum by employing the probability amplitude approach. In Sec. 3, we discuss the control and modification of the spontaneous emission by altering the system parameters under different conditions. Finally, our conclusions are summarized in Sec Proposed Model and Basic Formula We consider a five-level atomic system which is comprised of two upper states 3 and 4, and three lower states 0, 1, and 2, as depicted in Fig. 1. The atomic transition between 3 and 4 (transition frequency ω 43 ) is driven by a linearly polarized laser field with carrier frequency ω c and Rabi frequency 2Ω c. We assume that the transition from the upper state 3 to the lower state 0 is coupled by the modified reservoir modes (ω k ), which can be a double-band isotropic PBG reservoir as shown in Fig. 1(a), or a double-band anisotropic PBG reservoir as shown in Fig. 1(b), or a free vacuum reservoir, respectively. The lower states 1 and 2 are the degenerate Zeeman sublevels, which correspond to the magnetic quantum numbers m F = +1 and m F = 1, respectively,

3 No. 5 Communications in Theoretical Physics 605 and belong to the ground-state hyperfine level F = 1. Moreover, the degeneracy of the ground-state sublevels is removed by a longitudinal magnetic field B, where the amount B represents the Zeeman shift. The excited state 3 is the m F = 0 Zeeman sublevel and also belongs to the hyperfine level. Therefore, the transitions from the upper state 3 to the lower states 1 and 2 are the allowed electric dipole transitions. An elliptically polarized laser field with carrier frequency ω p is applied to couple the two transitions 3 1 and 3 2, simultaneously. Generally speaking, the elliptically polarized light can be viewed as a combination of the left-hand circularly (LHC) and right-hand circularly (RHC) polarized components. As we know, the QWP can be used to control the polarization of the probe field. An initial vertically polarized laser field with intensity I p and electric field amplitude E 0 = 2I p /(ε 0 c) (ε 0 is the permittivity of free space, c is the speed of light) becomes elliptically polarized when passing through the QWP which has been rotated by an angle θ. Elliptically polarized light can be expanded in terms of the circularly polarized basis as E p = E p + ˆσ+ + Ep ˆσ, in which E p + = (E 0/ 2)(cos θ + sin θ)e iθ and Ep = (E 0 / 2)(cosθ sin θ)e iθ, besides, ˆσ and ˆσ + are respectively the unit vectors of the LHC and RHC polarized basis, and the phase difference of ϕ = 2θ between the two circularly polarized components is introduced by the QWP. Consequently, Ep = E p + when θ = 0 or π/2, which means that the probe field is linearly polarized. And Ep = 0 (E+ p = 0) when θ = π/4 (3π/4), that is, the probe field is RHC (LHC) polarized. The phase parameter plays an important role in controlling the dynamics of spontaneous emission. Fig. 1 The schematic diagram of a five-level atom embedded in a double-band PBG material. ρ(ω k ) denotes the DOSs of the PBG modes. ω g1 and ω g2 are the lower and upper frequencies of the band gap, respectively. The atomic transition from the excited state 3 to the lower state 0 is coupled by the modified reservoir modes ω k. 2Ω p (2Ω + p ) is the Rabi frequency of the LHC (RHC) polarized component of the probe field with carrier frequency ω p coupling the transition from the excited state 3 to the ground state 1 ( 2 ). The transition 3 4 is driven by a linearly polarized laser field with Rabi frequency 2Ω c. p and c are the frequency detunings of the corresponding transitions. In the presence of a uniform magnetic field, the degeneracy among the ground levels 1 and 2 is lifted, in which B represents the Zeeman shift. The right panel of Fig. 1(a) shows the DOSs for the case of the double-band isotropic PBG model. The right panel of Fig. 1(b) shows the DOSs for the case of the double-band anisotropic PBG model. The Hamiltonian describing the dynamics of this system in the interaction picture can be represented as H I / = 2 B ( p + B ) ( p + B + c ) 4 4 [ + Ω p Ω + p Ω c ] g k e i( p+ B δk)t 3 0 ˆb k + H.c., (1) k where the rotating-wave approximation (RWA) has been used. Above the symbol H.c. denotes the Hermitian conjugate. We have defined the detunings of the atomic transition frequencies and the corresponding laser frequencies as p = ω 31 ω p B = ω 32 ω p + B and c = ω 43 ω c. δ k = ω 30 ω k stands for the detuning of the resonance transition 3 0 and the k-th reservoir mode. The quantities Ω p, Ω + p, and Ω c represent one-half Rabi frequencies for the relevant laser fields, i.e., Ω p = µ 13Ep /(2 ) = Ω p(cos θ sin θ)e iθ, Ω + p = µ 23E p + /(2 ) = Ω p(cosθ + sinθ)e iθ (here Ω p = µ 0 E 0 /(2 2 ) and µ 13 = µ 23 = µ 0 ), and Ω c = µ 34 E c /(2 ), with µ mn (m, n = 1 4) being the dipole moment for the transition between the states m and n. Here ˆb k (ˆb k ) denotes the annihilation (creation) operator for the k-th mode of the reservoir with frequency ω k. g k describes the frequency-dependent coupling constant between the atomic transition 3 0 and the k-th mode of the modified reservoir, which can be given by g k = ω 30µ 30 ( 2ε 0 ω k V ) 1/2 ek µ 30,

4 606 Communications in Theoretical Physics Vol. 59 where e k is the polarization unit vector of the radiation modes, µ 30 means the atomic dipole moment unit vector for the transition 3 0, and V is the sample volume. The dynamics of this system can be described by utilizing the probability amplitude equations. Then the wave function of our considered system at time t can be expressed in terms of the state vectors as Ψ(t) = [a 1 (t) 1 + a 2 (t) 2 + a 3 (t) 3 + a 4 (t) 4 ] {0} + a k (t) 0 {1 k }, (2) k where the probability amplitude a j (t) (j = 1 4) denotes the state of an atom at time t and {0} signifies the vacuum state of the modified reservoir, a k (t) gives the probability amplitude to find the atom in the lower state 0 with one emitted photon appearing in the k-th reservoir mode. Substituting the interaction Hamiltonian (Eq. (1)) and the wave function of the atom (Eq. (2)) into the timedependent Schrödinger equation i Ψ(t) / t = H I Ψ(t), we arrive at the following set of coupled differential equations a 1 (t) = i(ω p t a 3 (t), (3a) a 2 (t) = 2i B a 2 (t) i(ω + p ) a 3 (t), t (3b) a 3 (t) = i( p + B )a 3 (t) iω p a 1 (t) iω + p a 2 (t) iω t ca 4 (t) i g k e i( p+ B δk)t a k (t), k (3c) a 4 (t) = i( p + B + c )a 4 (t) iω c a 3 (t), t (3d) a k (t) = igk e i( p+ B δk)t a 3 (t). t (3e) We proceed by performing a formal time integration of Eq. (3e), and substitute the result into Eq. (3c). Thus, the integrodifferential equation can be readily obtained as a 3 (t) t = i( p + B )a 3 (t) iω p a 1(t) iω + p a 2(t) iω c a 4(t) where the delay Green s function can be written as t 0 e i( p+ B)(t t ) G(t t )a 3 (t )dt, (4) G(t t ) = g k 2 e i(ω k ω 30)(t t ). k According to the dispersion relations near the photonic band edges in the double-band isotropic and anisotropic models of the PBG materials, as well as the Markovian approximation in free space, the delay Green s function can be given in the following forms [19] G(t t ) = β3/ G(t t ) = α 30 2 ( exp{i[δ30g2 (t t ) π/4]} π(t t ) ( exp{i[δ30g2 (t t ) + π/4]} 4π(t t ) 3 + exp{i[δ 30g1(t t ) + π/4]} ), π(t t ) + exp{i[δ 30g1(t t ) π/4]} ), 4π(t t ) 3 G(t t ) = γ 30 2 δ(t t ). (5) for isotropic PBG, anisotropic PBG, and free vacuum reservoirs, respectively. The definitions of these parameters [19] are β 3/2 30 = 1 ω30 2 µ2 30 ω3/2 g2 2πε 0 3 c 3, α 30 = 1 ω30 2 µ2 30 ω1/2 g2 2πε 0 3 c 3, γ 30 = 2 ω30 3 µ2 30 πε 0 3 c 3, respectively, in which β 30 and α 2 30 indicate the resonant frequency splittings, [55] γ 30 is the effective decay rate of the transition from the upper state 3 to the lower state 0. β 30, α 2 30, and γ 30 possess the same dimension. The central idea of this paper is to derive an analytical expression for the spontaneous emission spectrum, so that we are able to investigate the dynamical properties of spontaneous emission. As is well known, the emission spectrum S(ω k ) of the atom is the Fourier transformation of the field-correlation function [25] E (t + τ)e + (t) t = Ψ(t) ˆb kˆb k e iωk(t+τ) e iω k t Ψ(t) t k,k = + e iωkτ dω k ρ(ω k ) 2 [a ke ( )a ke( )]dω, (6) e=1

5 No. 5 Communications in Theoretical Physics 607 here dω is the space angle component, ρ(ω k ) is the DOSs of the radiation modes, which can be deduced as follows [19] ρ(ω k ) 1 1 Θ(ω k ω g2 ) + Θ(ω g1 ω k ), ωk ω g2 ωg1 ω k ρ(ω k ) ω k ω g2 Θ(ω k ω g2 ) + ω g1 ω k Θ(ω g1 ω k ), ρ(ω k ) 1. for isotropic PBG, anisotropic PBG, and free vacuum reservoirs, respectively. Where Θ is the Heaviside step function. From Eqs. (2), (6), and (7), we can obtain the spontaneous emission spectrum S(ω k ) = ρ(ω k ) a k (t ) 2 = g k 2 ρ(ω k ) ã 3 [s i( p + B δ k )] 2, (8) here ã j (s) is the Laplace transform of a j (t). We solve this by employing the Laplace transformations ã j (s) = e st a 0 j (t)dt for Eqs. (3a), (3b), (3d), and (4) with respect to t, and thus we can get ã 3 (s) = a 3 (0) iω p s a 1(0) iω+ p s+2i B a 2 (0) iω c s+i( p+ B+ c) a 4(0) s + i( p + B ) + Ω p 2 s + Ω+ p 2 Ω s+2i B + c 2 s+i( + G[s p+ B+ c) + i( p + B )] (7), (9) where G(s) is the Laplace transform of G(t). By carrying out the Laplace transformations for Eqs. (5), we have the results ( G(s) = β3/2 30 i 1 ) +, (10a) 2 is + δ30g1 is + δ30g2 G(s) = α 30 ( ) i is + δ 30g1 + is + δ 30g2, (10b) 2 G(s) = γ (10c) Based on the above calculations and Eqs. (8), (9), and (10), the resulting emission spectrum can be expressed as a 3 (0) + S(δ k ) = g k 2 ρ(ω k ) Ω p p+ B δ k a 1 (0) + Ω + p p B δ k a 2 (0) 2 Ω c δ k + c a 4 (0) δ k + Ω p 2 p+ B δ k + Ω+ p 2 p B δ k Ωc 2 δ k + c i G(s iδ k ). (11) where equation (11) is the main result of the present study. Specifically, we display the two different cases as follows: Case 1 Under the conditions a 3 (0) = 1 and a 1 (0) = a 2 (0) = a 4 (0) = 0, so that the atomic spontaneous emission spectrum S(δ k ) in Eq. (11) can be explicitly reduced into the form S(δ k ) = g k 2 1 ρ(ω k ) δ k + Ωp 2 (1 sin 2θ) p+ B δ k + Ωp 2 (1+sin 2θ) p B δ k Ωc 2 δ k + c i G(s iδ k ). (12) 2 Case 2 Under the conditions a 1 (0) = a 2 (0) = 1/ 2 and a 3 (0) = a 4 (0) = 0, we have the result Ω p(cos θ sin θ)e iθ 2 Ωp(cos θ+sin θ)eiθ S(δ k ) = g k 2 ρ(ω k ) 2( p+ + B δ k ) 2( p B δ k ) δ k + Ωp 2 (1 sin 2θ) p+ B δ k + Ωp 2 (1+sin 2θ) p B δ k Ωc 2 δ k + c i G(s. (13) iδ k ) Equations (12) and (13) show that the spontaneous emission spectrum can be modified, on the one hand, by putting atoms into different environments, i.e., the DOSs ρ(ω k ) of the radiation field, on the other hand, by using external coherent fields to couple atoms, i.e., the detuning p and the intensity Ω p of the polarized probe laser field, the switching on and off of the external magnetic field B ( B B), as well as the detuning c and the intensity Ω c of the control field. As a consequence, we can control and manipulate the spontaneous emission by adjusting these system parameters under certain conditions. Furthermore, from the above expressions (12) and (13) we can find that the spontaneous emission spectrum S(δ k ) is strongly dependent on the laser-polarization-dependent phase θ and also is a periodic function of the relative phase θ with a period π. Alternatively, it can be easily seen from the expressions (12) and (13) that the relations S(δ k ; θ) = S(δ k ; π/2 θ) = S(δ k ; 3π/2 θ) hold for cases 1 and 2. 3 Results and Discussion Our main focus in this section is to illustrate the dynamical properties of the spontaneous emission with the numerical simulations based on Eqs. (12) and (13). All

6 608 Communications in Theoretical Physics Vol. 59 parameters used in this paper are scaled by β 30, α 2 30, and γ 30 for isotropic PBG, anisotropic PBG, and free vacuum modes, respectively, which should be in the order of MHz for rubidium atoms. Fig. 2 The spontaneous emission spectrum S(δ k ) (in arbitrary units) as a function of the detuning δ k. (a) (a ) θ = 0, π/2; (b) (b ) θ = π/6; (c) (c ) θ = π/4. The other parameters used are Ω c = 4, Ω p = 3, c = 0.3, p = 1.2, B = 0.5, δ 30g2 = 1.0, and δ g2g1 = 0.4. The atom is initially prepared in the excited state 3, i.e., a 3(0) = 1. (a) (c) correspond to the double-band isotropic PBG reservoir and β 30 = 1; (a ) (c ) show the double-band anisotropic PBG reservoir and α 2 30 = 1; (a ) (c ) denote the free vacuum reservoir and γ 30 = 1. All parameters in this paper are in units of β 30, α 2 30, and γ 30, for isotropic PBG, anisotropic PBG, and free vacuum modes, respectively. To comprehend clearly the effect of the phase difference between the two circularly polarized components of the probe field on the spontaneous emission, we first consider the case where the atom is initially prepared in the upper state 3, i.e., a 3 (0) = 1 corresponding to the above case 1. The spontaneous emission spectrum S(δ k ) as a function of detuning δ k is plotted in Fig. 2 for the cases of isotropic PBG (Figs. 2(a) 2(c)), anisotropic PBG (Figs. 2(a ) 2(c )), and free vacuum (Figs. 2(a ) 2(c )) reservoirs, respectively. We can observe some unusual phenomena in three kinds of reservoirs by modulating the phase difference between the LHC and RHC polarized elements of the probe laser field. It can be seen that the emission spectrum exhibits a three-peak structure with two normal width sideband lines and an ultranarrow central line under the action of a linearly polarized laser field (i.e., θ = 0, π/2) and a circularly polarized laser field (i.e., θ = π/4) for the cases of isotropic and anisotropic PBG reservoirs (see Figs. 2(a), 2(a ), 2(c), and 2(c )). From Figs. 2(a) and 2(c), we see that two enhanced sharp peaks are distributed on both sides and a small ultranarrow spectral line at line center in the case of isotropic PBG reservoir. In contrast, for the case of anisotropic PBG reservoir, an enhanced ultranarrow line appears in the middle of the spectrum and two relatively wide peaks are located at both sides of the central line (see Figs. 2(a ) and 2(c )). Interestingly, when θ = π/6, that is, the probe field is elliptically polarized, an extra ultranarrow spectral line appears in the spectrum and hence the line shape changes from a three-peak structure into a four-peak structure in both cases of isotropic and anisotropic PBG reservoirs (see Figs.2(b) and 2(b )). However, the variation of the spectral lines is different when the atomic transition 3 0 is coupled by a free vacuum reservoir. More precisely, in the presence of a linearly polarized field (i.e., θ = 0, π/2) and an elliptically polarized field (e.g., θ = π/6), the emission spectrum displays a four-peak configuration with two ultranarrow spectral lines occurring at line center and two narrow peaks are symmetrically located at both sides (see Figs. 2(a ) and 2(b )). While, the ultranarrow line which

7 No. 5 Communications in Theoretical Physics 609 is situated on the right side of δ k = 0 is completely vanished when the probe field is circularly polarized, leaving a three-peak structure with equal height as can be seen from Fig. 2(c ). Fig. 3 The spontaneous emission spectrum S(δ k ) (in arbitrary units) as a function of the detuning δ k. (a)-(a ) θ = 0, π/2; (b) θ = π/9, (b ) θ = 2π/9, (b ) θ = π/12; (c)-(c ) θ = π/4. The system parameters used are the same as Fig. 2 except that δ 30g2 = 0.1, δ g2g1 = 1.5 and the atom is initially prepared in an equal superposition of two ground states 1 and 2, i.e., a 1(0) = a 2(0) = 1/ 2. In order to further investigate the influence of the QWP rotation angle on the spontaneous emission spectrum S(δ k ), we plot in Fig. 3 the corresponding spectral response versus the detuning δ k by adjusting the phase parameter under the condition that the atom is initially prepared in an equal superposition of the lower states 1 and 2, i.e., a 1 (0) = a 2 (0) = 1/ 2 corresponding to the above case 2. In the case of isotropic PBG reservoir, we can observe a four-peak profile in the emission spectrum, and the two sideband peaks are gradually inhibited with an increasing in the rotation angle of the QWP (see Figs. 3(a), 3(b), and 3(c)). Moreover, the ultranarrow spectral line which is located at δ k = 0 is slightly enhanced as the ellipticity increases, the ultranarrow line which is situated on the right side of zero detuning is enhanced under the influence of an elliptically polarized probe field (Fig. 3(b)) and then almost completely disappeared in the presence of a circularly polarized field (Fig. 3(c)). For the case of anisotropic PBG reservoir, an asymmetric double-peak structure appears in the spectrum when the probe field is linearly polarized (i.e., θ = 0, π/2 in Fig. 3(a )) or circularly polarized (i.e., θ = π/4 in Fig. 3(c )). It can be seen from Fig. 3(b ) that the emission spectrum shows a three-peak pattern characteristic under the action of an elliptically polarized field. From Figs. 2(b ) and 3(b ), we obtain the result that there is a strong correlation between the polarization of the probe laser field and the ultranarrow spectral line which lies at the right side of zero detuning, that is, the ultranarrow line emerges only when an elliptically polarized laser field is applied. Additionally, the emission spectrum S(δ k ) is also plotted in Figs. 3(a ) 3(c ) for the case of free vacuum reservoir, the extremely enhanced ultranarrow spectral line can be observed by tuning the system parameters. Similar to those shown in Figs. 2(a ) 2(c ), the spectrum has a four-peak structure with two prominent ultranarrow lines under the influence of a linearly polarized field (i.e., θ = 0, π/2 in Fig. 3(a )) or an elliptically polarized field (e.g., θ = π/12 in Fig. 3(b )). With the increase of ellipticity, when the probe field becomes circularly polarized (i.e., θ = π/4 in Fig. 3(c )), the spectral profile changes from a fourpeak structure into a three-peak structure. The change of

8 610 Communications in Theoretical Physics Vol. 59 the ultranarrow spectral line is strongly dependent upon the phase difference between the LHC and RHC polarized components of the probe field. The specific results are as follows. When the relative phase θ is tuned from θ = 0, π/2 to π/12, the right ultranarrow line is rapidly enhanced whereas the left one is slightly reduced (see insets of Figs. 3(a ) and 3(b )). It should be noted that the left ultranarrow line is greatly enhanced and the right one is completely disappeared under the condition of θ = π/4. In a word, the ultranarrow spectral line which is located at the right side of δ k = 0 is greatly enhanced under the action of an elliptically polarized field and the two asymmetric peaks which are situated at the positive and negative sides of the detuning are gradually decreased with the increase of ellipticity. Fig. 4 The spontaneous emission spectrum S(δ k ) (in arbitrary units) as a function of the detuning δ k in a free vacuum reservoir. (a) θ = π/12; (b) θ = π/6; (c) θ = 2π/9; (a ) θ = π/12; (b ) θ = π/6; (c ) θ = 2π/9. The system parameters used are the same as Fig. 2 except that c = 0. We have investigated the influence of the phase difference between the two circularly polarized components of the probe field on the atomic spontaneous emission under different conditions. Now we study the effect of the degree of ellipticity and the direction of polarization of the probe field on the spontaneous emission behavior for the case of free vacuum reservoir. The emission spectrum is plotted in Fig. 4 as a function of detuning parameter (δ k ) via the numerical calculations based on Eq. (12) under the condition that the control field is tuned to the resonant with the corresponding atomic transition. Under the action of an elliptically polarized field, the spectrum shows a four-peak structure which has two ultranarrow spectral lines and two sideband peaks with equal height. From Fig. 4, we can see that the four-peak structure evolves into a three-peak line shape when the ellipticity is sufficiently large. In the presence of positive polarization angle, the right ultranarrow spectral line is gradually suppressed till disappeared as the ellipticity increases, while the left ultranarrow line is enhanced (Figs. 4(a) 4(c)). In the opposite direction, that is, when the polarization azimuth angle is negative, the two ultranarrow spectral lines are close to each other with the increase of ellipticity, furthermore, the two spectral lines is transformed into one ultranarrow line when the ellipticity is adjusted to a certain value (Figs. 4(a ) 4(c )). In the following, let us consider the influence of the sets of system parameters on the spontaneous emission spectrum by switching on and off the external magnetic field under the action of an elliptically polarized laser field. In Fig. 5, we present the results based on Eq. (12) for the case of isotropic PBG reservoir by adjusting the relative position of the atom from the band edges and the width of the forbidden gap, as well as the detuning of the control field from the corresponding atomic transition. With

9 No. 5 Communications in Theoretical Physics 611 an external magnetic field applied, the emission spectrum has a four-peak shape or a double-peak profile as shown in Figs. 5(a) and 5(c). When the control field is far detuned from the corresponding transition (e.g., c = 1.5), the resonant transition frequency lies within the upper band and keeps away from the band edge (δ 30g2 = 1.2), and the band gap is narrower (δ g2g1 = 0.2), the spectrum exhibits a four-peak structure (see Fig. 5(a)). While, the four-peak structure changes into a double-peak configuration as potted in Fig. 5(c) when the control field is near resonance (i.e., c = 0.1), and the transition frequency ω 30 lies close to the upper band edge (δ 30g2 = 0.1) and the band gap gets larger (δ g2g1 = 2). On the other hand, without an external magnetic field applied, the emission spectrum displays a triple-peak structure as can be seen from Figs. 5(b) and 5(d). When the transition frequency ω 30 lies near the band edge (δ 30g2 = 0.1) and the band gap is much wider (δ g2g1 = 2), the right ultranarrow spectral line is entirely suppressed, the left ultranarrow line is close to the zero detuning of δ k = 0 and becomes lower and wider, as shown in Fig. 5(b). Moreover, when the control field is tuned from far-resonance ( c = 1.5 in Fig. 5(a)) to resonance ( c = 0 in Fig. 5(d)), and the excited level 3 moves from the upper band (δ 30g2 = 1.2 in Fig. 5(a)) to the lower band (δ 30g2 = 1 and δ g2g1 = 0.8 in Fig. 5(d)) and near the band-gap edge, the left ultranarrow spectral line is completely vanished, and the right ultranarrow line shifts towards the left and is slightly enhanced (Fig. 5(d)). Fig. 5 The spontaneous emission spectrum S(δ k ) (in arbitrary units) as a function of the detuning δ k in a double-band isotropic PBG reservoir. (a) c = 1.5, B = 0.5, δ 30g2 = 1.2, and δ g2g1 = 0.2; (b) c = 1.5, B = 0, δ 30g2 = 0.1, and δ g2g1 = 2; (c) c = 0.1, B = 0.5, δ 30g2 = 0.1, and δ g2g1 = 2; (d) c = 0, B = 0, δ 30g2 = 1, and δ g2g1 = 0.8. The other parameters used are the same as Fig. 2 except that θ = π/12. Interesting results are also found in the case of doubleband anisotropic PBG reservoir. In Fig. 6 we depict the spontaneous emission spectrum S(δ k ) versus the detuning parameter δ k by numerical calculations of Eq. (13) for the cases with and without the external magnetic field. We find that there is a strong connection between the external magnetic field and the ultranarrow central spectral line. To examine the effect of the external magnetic field on the spectral profile, we have analyzed the case when other parameters keep fixed and only by turning on and off the magnetic field. Figs. 6(a) and 6(b) present the emission spectrum with and without the application of an external magnetic field, respectively. It is shown that the spectrum has two enhanced ultranarrow central lines (see inset in Fig. 6(a)) and two wider sideband peaks under the influence of an external magnetic field (Fig. 6(a)). On the contrary, when the external magnetic field is switched off, the right ultranarrow spectral line is completely vanished and the left ultranarrow line moves towards the right a little and is slightly inhibited (see inset in Fig. 6(b)). Specifically, for the case that the external magnetic field exists and the control field is tuned to the resonant interaction with the corresponding atomic transition (i.e., c = 0), the spectrum also exhibits a double-peak structure under the condition of δ 30g2 = 0.1 and δ g2g1 = 2, that is, the transition frequency ω 30 is in the vicinity of the upperband edge and the width of the band gap is large enough (Fig. 6(c)). One can see from Figs. 6(a) and 6(d) that the right ultranarrow spectral line becomes more pronounced and is close to the zero detuning (δ k = 0), whereas the

10 612 Communications in Theoretical Physics Vol. 59 left ultranarrow line is disappeared when the transition frequency ω 30 moves from the upper band to the lower band. As a result, any desired spectral shape can be observed by adjusting the combination of system parameters under certain conditions. Fig. 6 The spontaneous emission spectrum S(δ k ) (in arbitrary units) as a function of the detuning δ k in a double-band anisotropic PBG reservoir. (a) c = 1.5, B = 0.5, δ 30g2 = 1.2, and δ g2g1 = 0.2; (b) c = 1.5, B = 0, δ 30g2 = 1.2, and δ g2g1 = 0.2; (c) c = 0, B = 0.5, δ 30g2 = 0.1, and δ g2g1 = 2; (d) c = 0.2, B = 0, δ 30g2 = 1, and δ g2g1 = 0.6. The other parameters used are the same as Fig. 2 except that the atom is initially prepared in a superposition of two ground states 1 and 2, i.e., a 1(0) = a 2(0) = 1/ 2, and θ = π/12. From the numerical simulations mentioned above, we can conclude as follows. In the presence of an external magnetic field, the emission spectrum shows a four-peak structure which includes two normal width sideband lines and two ultranarrow central spectral lines. In the absence of the external magnetic field, a triple-peak spectral profile emerges in the spectrum, and the position of the central line varies with the change of system parameters. Moreover, when the atom is initially prepared in the excited state, the sign of the rotation angle and the ellipticity only influence on the central spectral line and have no effect on the sideband lines. Under the condition that the atom is initially prepared in a superposition state of two ground levels, the height of two sideband lines is slowly decreased with increasing ellipticity. The origin of these interesting phenomena discussed above can be explained in terms of a dressed-state picture. On the one hand, the two ground states 1 and 2 are the degenerate sublevels without the external magnetic field, consequently, the transitions 1 ( 2 ) 3 4 together with the control and probe fields are regarded as a coupled atom+field system, thus the bare-state level 3 is split into three dressed-state sublevels (not shown here). There are three decay channels from the dressed-state sublevels to the lower level 0, which correspond to three spectral lines in the emission spectrum. On the other hand, the ground-state degeneracy would be removed when the external magnetic field was applied. Under the action of the LHC and RHC polarized components of the probe field as well as the control field, the excited level 3 is divided into four dressedstate sublevels (not shown here). Accordingly, there exist four spontaneous decay channels in the dressed-state picture, thus leading to a four-peak structure emerges in the spectrum. Furthermore, phase-induced constructive quantum interference between the four transition pathways results in spectral-line enhancement and narrowing. While, the phase-induced destructive interference between the multiple decay channels leads to spectral-line suppression, spontaneous emission cancellation, and the multipeak structure. Before ending this section, let us now discuss why the results are different when the atom is located in different environments. This is due to the fact that the spontaneous emission of an atom is dependent on the structure of atomic energy levels and the characteristics of the surrounding environment, particularly, on the DOSs of the radiation field, and these have been demonstrated in Refs. [19 20]. The significant difference between the isotropic and anisotropic PBG reservoirs is that there exist two singularities in the DOS of the isotropic modes, and

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