equations (1)±(5) reduce to a set of algebraic equations where, after we split them into real and .... the laser and displays a typical Autler±Townes splitting.
journal of modern optics, 1998, vol. 45, no. 12, 2433±2442
Letter E ects of spontaneous emission interference on population inversions of a V-type atom SHANG-QING GONG{, EMMANUEL PASPALAKIS and PETER L. KNIGHT Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, UK (Received 18 June 1998) Abstract. The steady-state population behaviour in a laser driven V-type system has been analysed. The e ects of spontaneous emission-induced coherence and of the relative phase of the two coherent driving ®elds are considered in detail. A large and unexpected population inversion is found on one of the optical transitions due to these coherent e ects.
There has been a growing interest in targeted population transfer in quantum systems over the past decade. The most robust technique for achieving e cient population transfer is stimulated-Raman adiabatic passage [1]. In its basic approach, this method uses two lasers in a counterintuitive order and can transfer population with up to 100% e ciency between the two lower levels of a L type scheme [2]. There are also techniques where chirped lasers (lasers with time-dependent frequency) are used to transfer e ciently population [3]. This latter method applies favourably to ladder-type atoms [4]. Both of the above techniques, however, fail in transferring population between the two upper levels of a V-type con®guration, where complicated laser combinations need to be applied [5]. As has been shown extensively in the past decade, atomic coherence and quantum interference can lead to many interesting optical phenomena such as lasing without inversion [6], electromagnetically-induced transparency [7], enhancement of the index of refraction [8] and spontaneous emission modi®cation [9] in multilevel systems. Using a V-type con®guration which was driven by two laser ®elds, Meduri et al. [10] showed that an unexpected population inversion can be achieved in one of the optical transitions due to atomic coherence e ects. In the range of parameters they studied, the maximum inversion found was of the order of 10%. In an earlier article Whitley and Stroud [11] also predicted unexpected population inversion in a ladder-type atom. Also quite recently, Malakyan and Unanyan [12] studied the possibility of large inversions in a ladder-type system using Fano interference. We have recently shown that spontaneous emissioninduced coherence can lead to large modi®cations of the dispersion and absorption { Permanent address: Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China. 0950 ± 0340/98 $12´00 # 1998 Taylor & Francis Ltd.
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Figure 1. The system under consideration. The ground state j1i is coupled to the excited state j2i with the drive laser !a and also to state j3i with the probe laser !b . The dashed lines indicate the spontaneous emission process from the closely spaced excited levels to the ground state.
properties of a closed V-type atom driven by two lasers [13]. The purpose of this article is to study the e ects of this kind of interference on the steady state population inversion of such an atom. The atomic model used is similar to that studied by Meduri et al. [10], but the e ects of quantum interference from spontaneous emission [9], which arises if the two upper levels are close, are included in our model. We ®nd that the interference from spontaneous emission greatly enhances the population inversion in one of the optical transitions. Such population inversions are associated with substantial radiation ampli®cation [13]. We furthermore show that the steady state population inversion is very sensitive to the relative phase between the two driving lasers. Thus, the relative phase can be used to control coherently the population inversion. The e ects of spontaneous emission-induced coherence on the steady state population dynamics of a driven L -type atom has been studied by Javanainen [14]. Also, Lee et al. [15] studied the e ects of this interference on both the transient and steady state behaviour of a four level atom, and Zhou and Swain [16] studied the e ects of probe absorption in a V-type system driven by a single weak laser and associated the gain with population inversion. The atomic system, considered here, is displayed in ®gure 1. We study a closed V-type con®guration with two close lying excited states j2i and j3i and a single ground state j1i. The optical transition j1i $ j2i is driven by a strong laser pulse (drive laser) with frequency !a and Rabi frequency O ˆ l12 a . Also, the optical transition j1i $ j3i is driven by a weaker laser pulse (probe laser) with frequency !b and Rabi frequency E ˆ l13 b . Both upper levels are allowed to decay spontaneously to the ground level with decay rates 2®2 , 2®3 , respectively. The analysis of this closed dissipative system is best described by the density matrix approach [17]. We begin with the Liouville equation for the motion of the density matrix and employ the generalized reservoir version of the Weisskopf± Wigner theory of spontaneous emission [17] to obtain the following set of di erential equations (in a rotating frame)
E ects of spontaneous emission interference on population inversion
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1=2 »_ 22 ˆ ¡2®2 »22 ¡ p…®2 ®3 † …»23 exp …¡i¯¿† ‡ »32 exp …i¯¿†† ‡ iO…»12 ¡ »21 †;
…1 †
1=2 »_ 33 ˆ ¡2®3 »33 ¡ p…®2 ®3 † …»23 exp …¡i¯¿† ‡ »32 exp …i¯¿†† ‡ iE …»13 ¡ »31 †;
…2 †
1=2 »_ 23 ˆ ¡‰®2 ‡ ®3 ‡ i…D3 ¡ D2 †Š»23 ¡ p…®2 ®3 † exp …i¯¿†…»22 ‡ »33 †
‡ iO»13 ¡ iE»21 ;
…3 †
1=2 »_ 21 ˆ ¡…®2 ¡ iD2 †»21 ¡ p…®2 ®3 † exp …i¯¿†»31 ‡ iO…»11 ¡ »22 † ¡ iE»23 ;
…4 †
1=2 »_ 31 ˆ ¡…®3 ¡ iD3 †»31 ¡ p…®2 ®3 † exp …¡i¯¿†»21 ‡ iE…»11 ¡ »33 † ¡ iO»32 ;
…5 †
where »11 ‡ »22 ‡ »33 ˆ 1 and »nm ˆ »mn . Also, the assumption D2 ¡ D3 !3 ¡ !2 ² !32 (or !a !b ) has been imposed in the derivation of equations (1)±(5). The radiative shifts have been included in the detunings D2 , D3 . We immediately note that the equations considered here are quite di erent from those studied by Meduri et al. [10]. The main di erence between our equations and 1=2 those of [10] are the terms p…®2 ®3 † which appear in equations (1)±(5). These terms are the e ect of quantum interference of spontaneous emission from the two upper levels. We should note that if the above assumption does not hold, then these terms will average out and no interference from spontaneous emission is possible. An intuitive explanation of these terms are that they represent the e ect of virtual emission of a photon in the transition j2i ! j1i and virtual absorption of the same photon in the transition j1i ! j3i (or vice versa). The parameter p denotes the alignment of the two matrix elements and is de®ned as p ˆ l21 · l13 =jl21 jjl13 j ˆ cos ³. It is obvious that if the two matrix elements l12 , l31 are orthogonal then p ˆ 0 and no interference from spontaneous decay occurs. Using now the restriction that each of the linearly polarized lasers should only couple one of the optical transitions (as shown in ®gure 1) we arrive at the following relations between the Rabi frequencies and the parameter p: 1=2 1=2 O ˆ O0 …1 ¡ p2 † ˆ O0 sin ³ and E ˆ E0 …1 ¡ p2 † ˆ E0 sin ³, with O0 ˆ jl12 jj a j and E0 ˆ jl13 jj b j. Furthermore, we note that the p dependent terms are always accompanied by a phase dependent term exp … i¯¿† where ¯¿ ˆ ¿a ¡ ¿b denotes the relative phase between the two laser ®elds. The phase dependence in this system occurs only if p 6ˆ 0. This is a consequence of the fact that this system behaves as a generalized closed loop system only in the case when p 6ˆ 0. Closed loop systems possess the property of phase dependence, in that the behaviour of the system crucially depends on the relative phase between the transition paths [18±20]. Several successful experiments have been performed where the phase dependent population dynamics of closed loop systems have been observed [21, 22]. Also, Korsunsky et al. have recently observed phase dependent electromagnetically-induced transparency using a double L con®guration [23]. Recently, Martinez et al. [24] showed that the spontaneous emission spectrum of a L -type atom is crucially dependent on the phase of a microwave ®eld used for coupling the two lower levels. Menon and Agarwal [25] have also recently shown that the probe absorption of a L -type atom is also dependent on the relative phase between the drive and probe lasers. Both of these phase-dependent e ects occur only if the e ects of spontaneous emission interference are accounted for. Also, quite recently, two of the present authors
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have studied the e ects of phase dependence on the spontaneous emission spectra in three [26] and four level systems [27] and show such e ects as extreme spectral narrowing, selective and total cancellation of ¯uorescence decay can occur. In order to make clear to the reader why this system is a generalized closed loop system, we should note that, in the case when p 6ˆ 0, then the two upper states can be coupled via two di erent transition routes. The ®rst transition route is a laser!a !b induced one, i.e. a Raman-type transition where j2i ! j1i ! j3i and the phase of this transition route is ¯¿. The second transition route is the one explained above, i.e. the Raman coupling due to the spontaneous emission interference. This is a laser independent transition route and has zero phase. So, the relative phase between the two possible transition routes is ¯¿ and the system is expected to show changes in its behaviour by varying the relative phase ¯¿. In order to study the steady state behaviour of the system we set »_ nm ˆ 0 so that equations (1)±(5) reduce to a set of algebraic equations where, after we split them into real and imaginary parts, we obtain an 8 8 matrix equation. This equation can be easily treated in all orders using the symbolic computation package Mathematica . Here, we are interested in the steady state population di erence W ² »~33 ¡ »~11 (where the tilde denotes the steady state value) in the j1i $ j3i optical transition. This population di erence can be expressed, using equations (1)±(5), as 2E = … † ‡ O = … † ¡ 2®2 ‡ ®3 ‰
