Control of spontaneous emission of an inverted Y-type atomic system coupled by three coherent fields
aa r X i v : . [ qu a n t - ph ] S e p Control of spontaneous emission of an inverted Y-type atomic system coupled bythree coherent fields
Jianbing Qi
Department of Physics and Astronomy, Penn State University,Berks Campus, Tulpehocken Road, P.O. Box 7009, Reading, PA 19610
We investigate the spontaneous emission from an inverted Y-type atomic system coupled bythree coherent fields. We use the Schr¨odinger equation to calculate the probability amplitudes ofthe wave function of the system and derive an analytical expression of the spontaneous emissionspectrum to trace the origin of the spectral features. Quantum interference effects, such as thespectral line narrowing, spectrum splitting and dark resonance are observed. The number of spectralcomponents, the spectral linewidth, and relative heights can be very different depending on thephysical parameters. A variety of spontaneous emission spectral features can be controlled by theamplitudes of the coupling fields and the preparation of the initial quantum state of the atom. Wepropose an ultracold atomic Rb system for experimental observation. PACS numbers: 42.50.Gy, 42.50.Ct, 32.80.Qk
I. INTRODUCTION
It is well-known that the spontaneous emission resultsfrom the inevitable interaction of the atomic system withthe quantized electromagnetic field vacuum [1]. Inter-ference effect in the spontaneous emission has becomean important topic and the control of the spontaneousemission has attracted intensive study in many atomicsystems in recent years [2, 3, 4]. The process of vacuumfields driving the excited atom to its decay target statecan be altered by coupling the excited state to other inter-nal atomic states or by modifying the vacuum states [5].The spontaneous emission of the atom will be influencedas either the excited state or the decay target state be-ing coupled to an internal state by a coherent field. It iswell known that the spontaneous emission spectrum of atwo-level atom driven by a strong resonant field is greatlymodified [6, 7].The early work of Agarwal has showed that the sponta-neous emission of a V-type three-level atom can be mod-ified through the quantum interference between the twodecay channels with a common ground level [8]. Thecompetition among the multiple decay path ways to acommon state can result in a complete destructive quan-tum interference as well as constructive quantum inter-ference [9]. It has been shown that the spontaneouslygenerated atomic coherence exists when two close-lyingexcited atomic levels are coupled by the same vacuummodes to a single lower level with non-orthogonal elec-tric dipole matrix elements between the upper pair andlower states [3, 10]. However, there are few atomic sys-tems that satisfy this condition, therefore few experi-ments have been reported [11, 12]. It is natural to thinkabout using coherent fields to couple the involved atomicstates to make the atomic system evolving in a control-lable way instead of seeking atomic systems with lim-ited operational parameters. Since the atomic states aredressed by the coherent fields, the dressed states will evolve complete differently in the vacuum. For exam-ple, the spectral linewidth which is associated with thedecay of one of the dressed states of the atoms dependscrucially on the relative strength of the coupling fieldsand the phase and can be extremely narrowed [3]. Under-standing the dynamics of the controlled system has manypotential applications, such as lasing without inversion,coherent population trapping(CPT) [13], electromagnet-ically induced transparency(EIT) [14], and fluorescencequenching [15].A variety of atomic coherence and quantum interfer-ence phenomena have been discovered in many atomicand molecular systems based on two or three energystate models [16, 17, 18, 19], such as coherent popula-tion trapping (CPT) [20, 21], electromagnetic inducedtransparency (EIT) [22, 23, 24, 25] ultraslow propa-gation of light [26, 27], and Autler-Townes splitting[28, 29, 30]. Multilevel atomic and molecular systemsoffer many possibilities for the investigation of coherenceeffects and quantum control of the interactions amongthe quantum participants. The multilevel quantum sys-tems provide rich coupling schemes and thus degrees offreedom of controlling parameters. However, the cost ofrich coupling configurations will introduce more complex-ity and difficulty in the experiments and in the theoret-ical analysis, as well. The experimental realization of atheoretical model is very important to test the under-standing of the model. Particularly, the Doppler effectis a severe limit in the observation of many coherenceeffects [25, 31]. Recently, EIT in ultracold atomic gases,and Autler-Townes splitting effect in ultracold moleculeformation and detection have been reported [32, 33, 34].The optical information can be coherently controlled withmatter wave dynamics in Bose Einstein condensates [35].The development of ultracold physics makes the observa-tion of some subtle coherence effects possible in Dopplerfree environments otherwise difficult or even impossibleat high temperatures.In this paper, we study the spontaneous emission ofa coherently driven inverted-Y type atom as illustratedschematically in Fig. 1. This scheme can be applied tothe atomic Rb system and the corresponding energylevels can be chosen as shown in the parentheses of Fig.1. We propose an ultracold atomic Rb to be used toobserve the phenomena discussed in the following experi-mentally in which the Doppler effect is negligible. The ul-tracold atomic sample can be obtained using a magneto-optical trap(MOT). Similar schemes have been used forthe study of Autler-Townes effect in a sodium dimmer[36] and the suppression of two-photon absorption [37].The primary interest of this work is to investigate thecontrollability of the spectral features of the spontaneousemission of the atom by the coupling fields and other pa-rameters of the system. We use the wave function ap-proach in this paper to obtain an explicit expression forthe spontaneous emission spectrum. We find that thespectral features depend upon the amplitude of the cou-pling fields and the initial quantum state of the atom be-ing prepared. Quantum interference, such as the spectrallineshape narrowing, and fluorescence quenching is ob-served. We show that the spontaneous emission spectralfeatures can be controlled by the amplitude and detun-ing of three coupling lasers. We provide a numerical andqualitative analysis to trace the origin of the spectral fea-tures, which are attributed to the quantum interferencedue to the existence of competitive pathways generatedby the coherent couplings.The paper is organized as follows. In section II, wepresent the description of the model and the derivationof an analytical expression of the spontaneous emissionspectrum for the proposed atomic system. We discuss thespectral features and the corresponding numerical simu-lation in section III, and a summary of our results andsome conclusions are given in section IV. II. DESCRIPTION OF THE MODEL ANDEQUATIONS OF MOTION
We consider an inverted Y-type atomic system cou-pled by three lasers as shown in Fig. 1. Laser L1 andL2 couple the ground state level | i and | i to a com-mon excited state level | i , respectively, which forms awidely used Λ three-level system. In addition, a thirdlaser L3 couples the intermediate excited state level | i to a higher excited state level | i . Except for the decayof the excited state | i back to ground state levels | i and | i , and the upper state | i to the intermediate state | i , respectively, we assume that the intermediate level | i decays also to another ground state | g i and the upperexcited state | i decays to an auxiliary intermediate level | e i . Both transitions are assumed to be coupled by thevacuum modes in free space. If the separation betweenlevel | i and level | g i is much different from that of level L2L1 L3 |3> (|5D ,F=2,m F =0>)|2> (|5P ,F ’ =1,m ’F =0>)|1> (|5S ,F ’’ =1,m ’’F =-1>) |4> (|5S ,F ’’ =1,m ’’F =+1>)|g> (|5S ,F ’’ =2>)|e> (|5P ,F ’ =3>) FIG. 1: The energy level diagram and the coupling scheme.The proposed corresponding atomic Rb levels are shown inthe parentheses. | i from level | e i , we can assume that the vacuum modescoupling between | i and | g i is totally different from thatbetween | i and | e i . This is true in our scheme since twotransition frequencies are very different from each other.The intermediate excited level | i and the upper excitedlevel | i have opposite symmetry, and Level | e i has thesame symmetry as level | i . The interactions between thecoupling fields and the vacuum modes are neglected here.Under the electric dipole and rotating-wave approxima-tion the Hamiltonian of the system in the Schr¨odinger’spicture can be written as: H = H + H sint , (1)where H is H = X i =1 ~ ω i | i ih i | + ~ ω e | e ih e | + ~ ω g | g ih g | + X k ~ ω k b † k b k + X q ~ ω q b † q b q , (2)and H sint is the interaction Hamiltonian in the Shr¨odingerpicture, which is H sint = − ~ e − iν t | ih | + Ω e − iν t | ih | +Ω e − iν t | ih | ) + ~ ( X k g k b k | g ih | + X q g q b q | e ih | ) + h.c.. (3)The ~ ω i ( i = 1 , , . . . ) is the energy of the state | i i , ν i ( i = 1 , ,
3) is the laser frequency with the correspondingRabi frequency defined as Ω = µ · E ~ , Ω = µ · E ~ ,and Ω = µ · E ~ . µ ij is the electric dipole transition mo-ment of | i i ↔ | j i transition and E i is the field amplitudeof the corresponding coupling laser. b † k ( b † q ) and b k ( b q ) arethe photon creation and annihilation operators, and theindex k(q) stands for the kth ( qth ) field mode with fre-quency ω k ( ω q ). The g k ( g q ) stands for the vacuum cou-pling constant between the kth ( qth ) vacuum mode andthe atomic transitions | i ↔ | g i ( | i ↔ | e i ). The summa-tion over k(q) runs over modes near the correspondingatomic transition. For simplicity of the calculation butwithout loss of the generality, we take the energy of level | i as the reference, and let ω = 0. The Hamiltonianin the interaction representation is obtained through thefollowing transformation: H I = e iH t/ ~ H sint e − iH t/ ~ , (4)which is can be explicitly written as, H I ( t ) = − ~ e − iδ t | ih | + Ω e − iδ t | ih | +Ω e − iδ t | ih | ) + ~ ( X k g k e − iδ k t b k | ih g | + X q g q e − iδ q t b q | ih e | ) + h.c., (5)where δ = ν − ω , δ = ν − ω , and δ = ν − ω arethe frequency detunings of laser L1, L2, and L3, respec-tively; δ k = ω k − ω g and δ q = ω q − ω e are the frequencydetunings of the spontaneous emission with respect tothe transition, | i → | e i , and | i → | g i , respectively.The state vector of the system at any time t can be ex-panded in terms of bare-state eigenvectors of the systemas | Ψ( t ) i = [ a ( t ) | i + a ( t ) | i + a ( t ) | i + a ( t ) | i ] |{ }i +Σ k a g,k ( t ) | g i| k i + Σ q a e,q ( t ) | e i| q i , (6)where | i i ( i =1,2...4) is the i th unperturbed stationarystate of the atom, |{ }i represents for the absence ofphotons in all vacuum modes of the field, and | k i ( | q i )denotes that there is one photon in the kth ( qth ) vacuummode. The a i ( t ) ′ s , a g,k , and a e,q are the probability am-plitudes for the atomic state | i i , | g i , and | e i , respectively.The initial values of the corresponding probability ampli-tudes of the state vector depend upon the initial prepara-tion of the atom. We assume that a g,k (0) = a e,q (0) = 0,and a i (0) ′ s are arbitrary, apart from the normalizationrequirement, P i =1 | a i (0) | = 1. Then the Schr¨odingerequation in the interaction picture is ∂ | Ψ( t ) i ∂t = − i ~ H I | Ψ( t ) i . (7)The equations of motion for the probability amplitudesof the wave function are readily obtained by substitut-ing Eq. (5) and Eq. (6) into Eq. (7) and applying theWeisskopf-Wigner theory.˙ a ( t ) = i Ω ∗ e iδ t a ( t ) (8a) ˙ a ( t ) = i Ω e − iδ t a + i Ω e − iδ t a ( t )+ i Ω ∗ e iδ t a ( t ) − γ a ( t ) (8b)˙ a ( t ) = i Ω e − iδ t a ( t ) − γ a ( t ) (8c)˙ a ( t ) = i Ω ∗ e iδ t a ( t ) (8d)˙ a g,k ( t ) = − ig k, g e iδ k t a ( t ) (8e)˙ a e,q ( t ) = − ig q, e e iδ q t a ( t ) , (8f)where the γ = 2 π | g k ( q ) ( ω k ( q ) ) | D ( ω k ( q ) ) is the sponta-neous decay rate from level | i ( | i ) to level | g i ( | e i ), andthe D( ω k ( q ) ) is the vacuum mode density at frequency ω k ( q ) in the free space. Using following transformation, C ( t ) = a ( t ) (9a) C ( t ) = a ( t ) e iδ t (9b) C ( t ) = a ( t ) e i ( δ + δ ) t (9c) C ( t ) = a ( t ) e i ( δ − δ ) t , (9d)we have six coupled first order differential equations:˙ C ( t ) = i Ω ∗ C ( t ) (10a)˙ C ( t ) = iδ C ( t ) − γ C ( t ) + i Ω C ( t )+ i Ω ∗ C ( t ) + i Ω C ( t ) (10b)˙ C ( t ) = i ( δ + δ ) C ( t ) − γ C ( t ) + i Ω C ( t ) (10c)˙ C ( t ) = i ( δ − δ ) C ( t ) + i Ω ∗ C ( t ) (10d)˙ a g,k ( t ) = − ig k, g e i ( δ k − δ ) t C ( t ) (10e)˙ a e,q ( t ) = − ig q, e e i ( δ q − δ − δ ) t C ( t ) . (10f)Using Laplace transformations ˜ C j ( s ) = L ( C j ( t )) = R ∞ e − st C j ( t ) dt for equations (10a-d), and integratingequations 10e and 10f, we obtain the following six equa-tions: s ˜ C ( s ) − C (0) = i Ω ∗ C ( s ) (11a) s ˜ C ( s ) − C (0) = i ( δ + i γ C ( s ) + i Ω C ( s )+ i Ω ∗ C ( s ) + i Ω C ( s ) (11b) s ˜ C ( s ) − C (0) = i ( δ + δ ) ˜ C ( s ) − γ C ( s ) + i Ω C ( s ) (11c) s ˜ C ( s ) − C (0) = i ( δ − δ ) ˜ C ( s ) + i Ω ∗ C ( s ) (11d)˜ a g,k ( s ) = − ig k, g ˜ C ( s ) s + i ( δ k − δ ) (11e)˜ a e,q ( s ) = − ig q, e ˜ C ( s ) s + i ( δ q − δ − δ ) , (11f)where the C j (0) ′ s are the corresponding initial conditionsgiven by a j (0), which indicate how the atom is initiallyprepared. From equation 11a-d we obtain following re-sults for ˜ C ( s ) and ˜ C ( s ),˜ C ( s ) = a (0) + i Ω122 a (0) s − i Ω ∗ a (0) s − i ( δ + δ )+ γ + i Ω242 a (0) s − i ( δ − δ ) s + γ − iδ + | Ω122 | s + | Ω242 | s − i ( δ − δ ) + | Ω232 | s − i ( δ + δ )+ γ (12a)and ˜ C ( s ) = a (0) + i Ω ˜ C ( s ) s − i ( δ + δ ) + γ (12b)The spontaneous emission spectrum is proportional tothe Fourier transformation of the field correlation func-tion. h E − ( r , t + τ ) · E + ( r , t ) i t →∞ = h Ψ I ( t ) | E − ( r , t + τ ) · E + ( r , t ) | Ψ I ( t ) i t →∞ . (13)It can be shown that the spontaneous emission spectrumis S ( ω k ) = γ | a g,k ( t →∞ ) | π | g k ( ω k ) | for the transition of | i → | g i ,and S ( ω q ) = γ | a e,q ( t →∞ ) | π | g q ( ω q ) | , for the transition of | i →| e i , respectively. Using the final value theorem [38] andEq. 11e-f, we obtain an analytical expression for the spontaneous emission of the intermediate excited level | i S ( ω k ) = γ | a g,k ( t → ∞ ) | π | g k, g ( ω k ) | = γ | ˜ C ( s = − i ( δ k − δ )) | π , (14a)and similarly, the expression for the upper level | i S ( ω q ) = γ | a e,q ( t → ∞ ) | π | g q, e ( ω q ) | = γ | ˜ C ( s = − i ( δ q − δ − δ )) | π . (14b)From equation 12a-b, the explicit form of the | ˜ C ( s = − i ( δ k − δ )) | is | ˜ C ( δ k ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (0) − Ω122 a (0) δ k − δ − Ω ∗ a (0) δ k + δ + i γ − Ω242 a (0) δ k − δ γ − iδ k − | Ω122 | i ( δ k − δ ) − | Ω242 | i ( δ k − δ ) − | Ω232 | i ( δ k + δ ) − γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (15a)and similarly for | ˜ C ( s = − i ( δ q − δ − δ )) | , | ˜ C ( δ q ) | = | a (0)( γ + iδ q ) + i Ω ( γ + iδ q ) ˜ C ( s = − i ( δ q − δ − δ )) | δ q + γ , (15b)with ˜ C ( s = − i ( δ q − δ − δ )) = a (0) − Ω122 a (0) δ q − δ − δ + i Ω ∗ a (0)( γ + iδ q ) δ q + γ − Ω242 a (0) δ q − δ − δ γ − i ( δ q − δ ) + i | Ω122 | δ q − δ − δ + i | Ω242 | δ q − δ − δ + | Ω232 | ( γ + iδ q ) δ q + γ . (15c) III. DISCUSSION AND NUMERICAL RESULTS
From Eqs. (15a-c) we can see that the sponta-neous emission depends upon the initial probabilityamplitudes( a i (0)) of the atom, or the initial quantumstate of the atom being prepared, the Rabi frequency andthe frequency detuning of the lasers. Even the analyticalspectrum expression is succinct, however it is still verycomplicated and difficult to identify the physical origin ofthe spectral features. For this reason, we limit our discus-sion to the resonant coupling situation, that is, all threelasers are on resonance with respect to the correspondingtransitions. We hope that the equations can be simpli-fied enough but not prevent us from understanding theessential physics of the system and analyzing the effectsof each physical parameter to the spectral features. Byinspecting Eq. (15c), if the atom is initially not preparedin level | i or at any superposition state involving level | i , for the resonant coupling we can have the similardiscussion for level | i as for level | i . Therefore we fo-cus our discussion to the spontaneous emission spectrum S ( ω k ) of the intermediate level | i in this paper. In Fig. 1, if there is no laser L3 the system will be awidely studied Λ coupling scheme( | i − | i − | i ), while ifwithout the coupling laser L2 the system ( | i − | i − | i )will be a so called cascade scheme. Both schemes havebeen discussed in references [39, 40, 41]. The inverted Y-type scheme might be regarded as an extension of the Λscheme, however it shows that introducing the couplingbetween the excited state | i and an upper level | i notonly changes the dynamics of the system significantly butalso brings additional options for controlling spontaneousemission of level | i .For the resonant coupling case, δ = δ = δ = 0, Eq.(15a) reads˜ C ( s ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (0) + i Ω122 a (0)+ i Ω242 a (0) − iδ k + i Ω ∗ a (0) − iδ k + γ − iδ k + γ + | Ω122 | + | Ω242 | − iδ k + | Ω232 | − iδ k + γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (16)Substituting Eq. (16) into Eq. (14a), the spontaneousemission spectrum S ( ω k ) can be written as S ( ω k ) = γ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( Ω a (0) + Ω a (0) − a (0) δ k )( δ k + i γ ) − Ω ∗ a (0) δ k ( − iδ k + Λ )( − iδ k + Λ )( − iδ k + Λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)where Λ , Λ , and Λ are the roots of the following cubicequation. s + s ( γ + γ s ( | Ω | + | Ω | + | Ω | + γ γ γ | Ω | + | Ω | = y + + y − − γ + γ , (19a)Λ = − ( y + + y − )2 − γ + γ i √
32 ( y + − y − ) , (19b)Λ = − ( y + + y − )2 − γ + γ i √
32 ( y − − y + ) , (19c) where y ± = s − q ± r ( q + ( p , (20)and p = | Ω | + | Ω | + | Ω | + γ γ − ( γ + γ ) , (21) q = − γ + γ | Ω | + | Ω | + | Ω | + γ γ γ | Ω | + | Ω | γ + γ . (22)Further inspecting the structure of Eq. (17) we canrewrite the denominator of Eq. (17) as following. S ( ω k ) ∝ (cid:12)(cid:12)(cid:12)(cid:12) δ k + i Λ )( δ k + i Λ )( δ k + i Λ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) β δ k + i Λ + β δ k + i Λ + β δ k + i Λ (cid:12)(cid:12)(cid:12)(cid:12) , (23)where the coefficient β i can be determined by some sim-ple algebraic calculations. β i = (Λ k − Λ j ) ǫ kji Λ (Λ − Λ ) + Λ (Λ − Λ ) + Λ (Λ − Λ ) (24) ǫ kji is the permutation symbol and i=1,2,3. Eq. (23)shows that the spontaneous emission spectrum S ( ω k ) isa square of the sum of three complex quantities, thereforethe interference effect is inherent. Of course, the interfer-ence can be constructive or destructive depending uponthe physical parameters which we will discuss in detailsin the following. (A) Dark Line and Dark States We assume that both | i and | i have the same decayrate in our following discussions. By inspecting the nu-merator of Eq. (17), if the atom is initially prepared inthe excited state | i ( a (0) = 1) or | i ( a (0) = 1) orin a superposition state of | i and | i ( ψ (0) = a (0) | i + a (0) | i ), but not in a dark state, we find that the spec-trum of S ( ω k ) has a complete dark line( S ( ω k ) = 0) dueto a destructive interference at the resonant frequency δ k = 0. The spectrum always has a dark line at the reso-nance frequency and two components as long as the atomis initially prepared in the excited states as shown in Fig.2(a)-(c). This result is similar to the earlier work for acascade three level system by Zhu et al. [42]. We notethat there can be a dark state, in which the atom willbe completely decoupled from the interaction of laser 3and stay at the dark state. As we can see in Eq. (17)that if the atom is initially not prepared in the excitedstates, a (0) = a (0) = 0, but in a superposition oftwo ground states: | ψ (0) i = a (0) | i + a (0) | i , and theRabi frequencies of laser L1 and L2 are chosen such that Ω a (0) + Ω a (0) = 0, then the atom stays in a darkstate and the spontaneous emission of the atom will becompletely suppressed. One such example is illustratedin Fig. 2(d). (B) Spectral Line Splitting and Narrowing From an experiment perspective, the atom is typicallyprepared in a single ground state at start, such as in level | i (that is, a (0) = 1) or in level | i (that is, a (0) = 1). - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H a L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H b L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H c L - - ∆ k (cid:144) Γ- - S H Ω k LH a r b . un i t s L H d L FIG. 2: (Color online) The spontaneous emission spectra S ( ω k ) of the intermediate state | i . The parameters for thecalculations are Ω = Ω = Ω = 0 . γ , δ = δ = δ = 0, γ = γ = γ , and γ = 6 . a (0) = a (0) = a (0) =0, a (0) = 1, (b) a (0) = a (0) = a (0) = 0, a (0) = 1and (c) a (0) = a (0) = 0, a (0) = √ . , a (0) = √ . a (0) = − a (0) = √ .
5, and a (0) = a (0) = 0. Of course, S ( ω k ) should have a similar spectral lineshapefor both cases. This can readily be checked in Eq. (17).Without the loss of generality, we assume the atom isinitially prepared in level | i with a (0) = 1. SubstitutingEq. (19a-c) into Eq. (17) we obtain S ( ω k ) = γ π | Ω | ( δ k + γ )[ δ k + Γ ][( δ k − δ λ ) + Γ ][( δ k + δ λ ) + Γ ] , (25)where δ λ = Im Λ = − Im Λ = √
32 ( y + − y − ) , (26a)Γ = Λ = y + + y − − γ + γ , (26b)Γ = Re Λ = Re Λ = − ( y + + y − )2 − γ + γ . (26c)Eq. (25) indicates that the spectrum of S ( ω k ) has threepeaks, one at resonance frequency δ k = 0 with a spec-tral width of 2 | Γ | , and two symmetric sidebands at δ k = ± δ λ with a spectral width of 2 | Γ | , respectively.The linewidth of the two sidebands is always larger thanthat of the central component. Both the linewidth andthe position of the sidebands depend upon the Rabi fre-quencies of lasers. However, the spectral features can besignificantly different for various combinations of Rabifrequencies of three lasers as we will see in the followingnumerical calculations. We scale the decay rates, Rabifrequencies and the frequency of the spontaneous emis-sion by the decay rate of level | i in our calculations.There is no surprise that the spectrum shows just asingle resonance peak when the Rabi frequency of thelasers is smaller than the decay rate of level | i (as shownin Fig. 3(a)). As the Rabi frequencies of lasers increaseto the decay rate of level | i the spectrum starts to dis-play some broadening structures (Fig. 3(b)). To findthe role of each coupling laser, for a given pump laserL1, we keep the Rabi frequency of the upper couplinglaser L3, and only increase the Rabi frequency of laserL2, then the spectrum displays a double-peak structure.The spectrum is similar to a Λ coupling scheme as long asthe Rabi frequency of L2 is much larger than that of L3,but with a little bump at resonance frequency ( δ k = 0) asshown in Fig. 3(c). If we keep the Rabi frequency of laserL2 but increase the Rabi frequency of the upper couplinglaser L3 the central component emerges(Fig. 3(d)). Aswe further increase the strength of Laser L3, the cen-tral component is enhanced and its linewidth decreases,while two sidebands are suppressed and separates morewith respect to the central component as shown in Fig.3(e). When we keep the Rabi frequency of the uppercoupling laser L3 as the same as in Fig. 3(e) but in-crease the Rabi frequency of laser L2 the sidebands will - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H a L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H b L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H c L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H d L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H e L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H f L FIG. 3: (Color online) The spontaneous emission spectra S ( ω k ) of level | i for the atom is initially prepared in oneof the ground states | i , a (0) = 1. The parameters forthe calculations are δ = δ = δ = 0, γ = γ = γ , γ = 6 . = 0 . γ . (a) Ω = Ω = 0 . γ .(b) Ω = Ω = 1 . γ . (c) Ω = 2 . γ , Ω = 1 . γ . (d)Ω = 2 . γ , Ω = 2 . γ . (e) Ω = 2 . γ , Ω = 4 . γ . (f)Ω = 3 . γ , Ω = 4 . γ . be enhanced and the central component is suppressedand the linewidth is broadened as shown in Fig. 3(f).This is quite remarkable, because the spontaneous emis-sion spectral features of level | i| can easily be controlledby the combination of the Rabi frequencies of two cou-pling lasers L2 and L3 through the quantum interfer-ence effects. The desired frequency component can beenhanced or suppressed, narrowed or broadened. In fact,the linewidth of the central component can be subnat-ural by adjusting the Rabi Frequencies of laser L2 andlaser L3 to have y + + y − ≈ γ + γ according to Eq. (26b).From the experiment point of view, controlling the laserintensity therefore the Rabi frequency of the laser can beeasily achieved. (C) Effects of the Decay Rate of the Upper Level Above analysis shows that the central component ofthe spectrum is due to the upper coupling laser L3. Toillustrate this we plot two spectra to compare the spec-tral features for the Λ coupling scheme( | i - | i - | i , with-out L3) with the current inverted Y-type scheme in Fig.4. One may intuitively (but mistakenly) think that thesidebands are two Rabi splitting (Autler-Townes) com-ponents due to laser 2 or laser 3. Though there are ATsplitting when laser L2 and L3 are strong, the spectrumillustrated here is not an AT splitting of the excitation - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L FIG. 4: (Color online) A comparison of the spontaneous emis-sion of level | i with a Lambda scheme. The solid line isfor the inverted Y-type scheme with Ω = 4 . γ , γ = γ and the dashing lines are for the Λ scheme( | i − | i − | i ).other parameters for the calculations are δ = δ = δ = 0, γ = γ = 6 . MHz , a (0) = 1 , a (0) = a (0) = a (0) = 0,Ω = 0 . γ , and Ω = 3 . γ . spectrum which has been reported in our recent exper-iment [36], where the fluorescence of the excited stateswas detected by scanning the probe laser L1 while thelaser L2 and laser L3 were tuned at the correspondingresonance transitions. It is easy to check this by inspect-ing the separation of the two sidebands of the spectrum.For example in Fig. 4a, the two sidebands are not sepa-rated by the corresponding Rabi frequencies(Ω or Ω ),but by 2 | δ λ | = | √ ( y + − y − ) | according to Eq. (26a). Byinspecting the spectral linewidth of the central compo-nent with respect to the decay rate of the upper level | i γ , we find that the linewidth decreases as γ decreasesfor a given Rabi frequency of laser L3 as we show twoexamples in Fig. 5(a)-(b). The origin of this result isnot so obvious by checking Eq. (26b), but if we thinkabout the extreme case: when the upper state is not de-caying the coupling between level | i to level | i wouldbe equivalent to a situation that the laser L3 coupling | i to another ground state level | i . This can be checked inEq. (16) by setting γ = 0 as shown as the dashing linesin Fig. 5(b). This result can be useful for measuring thedecay rate of the upper state. For example, in the caseof direct fluorescence detection of | i → | e i transitionbeing not convenient, one can detect the fluorescence ofa convenient transition | i → | g i and then calculate thecorresponding γ by fitting the experimental spectrumusing Eq. (26b). IV. SUMMARY AND CONCLUSION
We have investigated the spontaneous emission for aninverted Y-type atom driven by three coherent fields. Awave function approach is used to derive an analytical ex-pression of the spontaneous emission spectrum. We show - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H a L - - ∆ k (cid:144) Γ S H Ω k LH a r b . un i t s L H b L FIG. 5: (Color online) The dependence of the spontaneousemission spectrum S ( ω k ) upon the decay rate of the upperstate | i . (a) γ = 0 . γ (b) γ = 0 . γ for the solid line, γ =0for the dashing lines. Other parameters for the calculationsare δ = δ = δ = 0; γ = γ and γ = 6 . a (0) = 1and a (0) = a (0) = a (0) = 0, Ω = 0 . γ , Ω = 2 . γ andΩ = 4 . γ . that quantum interference leads to the spectral line nar-rowing, spectrum splitting and dark fluorescence. Theorigin of the spectral characteristics can be explained bythe analytical expression with the corresponding physicalparameters. We find that the number of spectral compo-nents, the spectral linewidth, and relative heights can becontrolled by the amplitudes of the coupling fields andthe preparation of the initial quantum state of the atom.The numerical results have been presented based on thetheoretical model and the role of each parameters is ex-amined. The limitations of this approach are that it maynot be able to treat the situation that there is an inco-herent pumping among the levels, such as repopulationterms. In these cases a density matrix approach with useof the quantum regression theorem has to be used as inreferences [15, 42]. The results obtained in this paperdo not include the motion of the atom, therefore onlyvalid for Doppler free environments. For a realistic ex-perimental observation and to eliminate Doppler effect,we propose an ultracold atomic system, such as a Rb for experimental observation since Doppler effect can benegligible in an ultracold system. ACKNOWLEDGEMENTS
This work is supported by the RDG grant from PennState University. [1] Marlan O. Scully and M. Suhail Zubairy,
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