Bistable behaviors of weak probe light via coherent and incoherent fields. H. Jafarzadeh, E. Ahmadi Sangachin and Seyyed Hossein Asadpour

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1 Page 1 of Bistable behaviors of weak robe light via coherent and incoherent fields H. Jafarzadeh, E. Ahmadi Sangachin and Seyyed Hossein Asadour Sama Technical and Vocational Training College, Islamic Azad University, Tabriz Branch, Tabriz, Iran Tel/Fax: S.Hosein.Asadour@gmail.com Abstract In this aer, the bistable behaviors of weak robe light in a unidirectional ring cavity with four-level atomic system are discussed. We find that in the resence of sontaneously generated coherence (SGC), the medium comletely hase deendent and intensity threshold of otical bistability can be controlled. The effect of incoherent uming field on otical bistability in the resence and absence of SGC is also analyzed. Moreover, the grou index behavior of weak robe light is investigated. We show that intensity threshold of otical bistability can be controlled by light roagation switching from slow to fast by different arameters of atomic medium. Keywords: otical bistability, otical multistability, incoherent uming field, sontaneously generated coherence, relative hase PACS Number: k; 4.5.-; 4.65.An Introduction By using a laser field interacting with an atomic medium, one can use a light to control another field roagation in the atomic medium, such as electromagnetically induced transarency (EIT) [1], lasing without inversion 1

2 Page of (LWI) [], electron localization [3], four-wave mixing [4], otical bistability (OB) and otical multistability (OM) [5-7], and so on [8-1]. On the basis of a threelevel configuration, by introducing a forth level and a third field a system of four level atoms interacting with these fields can be created. This kind of system has many interesting henomena such as electromagnetically induced absortion (EIA), a very narrow robe light line width, and robe light gain [11]. Some of the features have been demonstrated in cold and hot atom-light interacting systems, for examle, dark state interference, absortion and nonlinear sideband generation [1, 13]. Recently, OB in various atomic systems has been significantly investigated both theoretically and exerimentally [14, 15]. The studies show that one can control the bistable threshold intensity and the hysteresis loo via different aroaches such as the squeezed state field [16, 17], the quantum interference [18-1], and so on []. Li [3], has demonstrated the OB behavior based on intersubband transitions in a semiconductor quantum well system. Zhang et al [4] roosed a four-level triod configuration for controlling OB which driven by a single ellitically olarized light. Wang and Xu [5], investigated a hybrid absortive-disersive otical bistability and multistability behavior in a three-level V-tye system. They found that the intensity and frequency detuning of the couling field as well as the intensity of the microwave field can affect the OM behavior dramatically. There are also some new works about quantum coherence and interference henomenon in a semiconductor and solid state media [6-3]. For examle, in our recent work, we discussed the effect of Er 3+ ion concentration on transient and steady state roerties of Er 3+ : YAG crystal [6]. Or in another interesting work by Hamedi [7], he investigated the ultraslow ulse roagation through for quantum dot nanostructure by electron tunneling.

3 Page 3 of Recently, sontaneously generated coherence (SGC) has stimulated interest of researchers due to its otential alication in quantum and nonlinear otics henomena [31-33]. For examle, the effects of SGC on Kerr nonlinearity have been discussed in our recent work [34]. In another study by Sahrai, the effect of SGC on transient behaviors of the disersion and the absortion has also been discussed [35]. It is shown that the effect of SGC can dramatically be adjusted the resonse of an atomic medium to a weak robe light and alter the behavior of the disersion and absortion rofile [36]. The effect of SGC can be created by the interference of sontaneous emission of either a single excited level to two closely lying atomic levels [37] or two closely lying atomic levels to a common atomic level [38]. In a ladder tye system, it can be created in the nearly equisaced atomic level case [39]. Moreover, the diole moments are not orthogonal, which is necessary for the existence of the SGC effect. This makes the controlling of SGC in real exeriment works not easy. In order to observe SGC in atomic systems, a few methods have been roosed to simulate this effect. Agarwal suggested working in situations where the vacuum of the electromagnetic field is anisotroic [4]. Ficek and Swain simulated SGC with the couling of a DC field and laser field [41]. SGC has also been studied in dressed states icture of a microwave field [4]. In this aer, we theoretically studied the otical bistability and grou velocity of the robe field in a four-level cold 87 Rb atomic system. We find that in the resence of SGC the medium hase deendent and intensity threshold of OB and grou velocity of robe field can be controlled. Moreover, the effect of incoherent uming field on OB and grou velocity is also discussed. Here, the four-level atomic system is mainly based on Refs [43, 44], however, our results are comletely different from those aers. Firstly, we are mainly interested in 3

4 Page 4 of studying the otical bistability and grou velocity of the robe field via coherent and incoherent fields in the resence of SGC. Secondly, we also show the effect of relative hase of alied fields on the OB and grou index of the robe field. To the best of our knowledge, so far no theoretical or exerimental work has been done to study stability switching via grou index switching in such four-level atomic system. In Ref [45], the authors discussed the effect of laser intensity and SGC on otical multistability and Kerr nonlinearity without introducing the exlicit deendence of the otical bistability and grou velocity on controlling arameters such as relative hase of the alied fields and incoherent uming rates. However, we have discussed the effect of controlling arameters, such as incoherent uming rate and relative hase of the alied fields as well as quantum interference induced by sontaneous emission on the OB behaviors and grou velocity of the weak robe light. A very imortant advantage of our scheme is that we rovide a realistic cold 87 Rb atomic system for realizing the OB/OM and grou index switching of the robe field, which may make our scheme much more convenient in exerimental realization Models and Equations We consider a closed four-level Y-tye system as shown in Fig. 1(a). The ground state and the next higher level 1 are couled by a coherent driving field (with amlitudee c ). The uermost excited levels and 3 are nondegenerate. The frequency sacing ( ω s in Fig. 1(a)) between these two closely lying levels is comarable to the natural linewidths of transitions from these two levels to the lower level 1. In the absence of two such closely saced levels in an atom, one can initially reare these levels by emloying a strong coherent field into the atom [8].In the resent model the levels and 3 are coherently connected to the 4 ω c

5 Page 5 of lower level 1 by alying the robe field ω (with amlitudee ). For the urose of incoherent uming, two broadband olarization fields (named as ΛandΛ 3 ) can be alied between energy level 1 and uer levels and 3. If µ 1 and µ 31 are the diole moments of the resective transitions 1 and 1 3, the olarizations of the incoherent fields can be arranged in such a way that r r r r µ. ε = µ. ε =. Consequently, each of the two incoherent fields acts only on 1 3 and 31 one transition so that one can avoid the interference induced by the incoherent um fields in stimulated emissions. ω1, ω 31, and ω1 are resonant frequencies which associates with the corresonding transitions 1, 3 1 and 1. The sontaneous emission rates from the levels 1, and 3 are reresented by γ1, γ and γ 3 resectively. To incororate the decay-interference effect in the resent r r model we considerµ 1. µ 31. The diole moment of transition 1 is taken asµ 1. The couling of the atom with the robe field and the driving field are denoted by the Rabi-frequencies µ E Ω = = h j1 1 j ( j,3) 5 µ 1Ec andω s = h, resectively. For two nearly degenerate levels and 3 the resective transition moments can be considered to be nearly equal i.e. µ 1 µ 31= µ. So we can assume, Ω1 Ω 13 =Ω. In our model, the system is sensitive to the hases of the robe and couling fields, therefore, we should treat the Rabi frequencies as comlex arameters: i Ω = Ω and eϕ i c c c eϕ Ω = Ω, where ϕ and ϕ c are the hases of the robe and couling fields, resectively. Here, the state of the system is not ure and it cannot be reresented by a single-state vector. It can be described by stating that the system has certain robabilities W 1, W, and of being in the ure states. In the case of incomlete rearations, it is therefore necessary to use a statistical descrition in the same sense as in classical statistical mechanics [46]. Therefore,

6 Page 6 of the dynamical evolution of the system under sontaneous daming can be reresented by the density matrix equation of motion. The rate of change of density oerator can be written: ρ ( ρ ) ( ρ = reversible + ) son. daming + ( ρ ) inco. uming (1) t t t t Where the reversible art reresents the interaction between coherent fields and the medium can be given as: ρ i ( ) reversible = [ H, ρ]. () t h Under diole-aroximation, the erturbation Hamiltonian H can be exresses by excluding the counter rotating terms in the interaction icture as follows: H = h Ω e + HC + Ω e i 1t i 13t [( 1 1. ) ( ) i ct + ( Ω e 1 ) + HC. )] (3) c Where the detunings of the fields from their resective resonances are written as 1 = + ωs and 13 = ωs for = ω ωm, and s = ωs ω1, resectively. The 1 average atomic transition frequency, ωm = ( ω1+ ω31 ). Using Weisskof-Wigner aroximation in the generalized reservoir theory [], we can derive the exression for the rate of change of density oerator relating to the sontaneous relaxation as given below, ρ γ γ ρ ρ ρ ρ 1 j ( ) son. daming = [{1 1, } 1 1 ] [{ j j, } 1 j j 1 ] t j=,3 η [({ 3, ρ} 1 ρ 3 1 ) + ({ 3, ρ} 1 3 ρ 1 )] (4) 6

7 Page 7 of The terms with η = γ γ 3 in Eq. (4) reresent the SGC effect resulted from the cross-couling between two decay aths 3 1 and 1. The arameter is r r r r defined as = µ 13. µ 1 ( µ 1 µ 13 = cosθ where θ is the angle between diole matrix elements r µ 1 andµ 13. For the rate of change of density oerator corresonding to the incoherent uming rocess we consider the case where the henomenon of uming will oulate a small number of atoms in the levels and 3 from level 1 i.e. the mode of uming is unidirectional. Thus, we can reresent ρ Λj ( ) = [{1 1, ρ} j 1 ρ 1 j ]. (5) inco. uming t j= 1, The density matrix equation of motion in the rotating wave aroximation and electric diole aroximation can be written as: & ρ = γ ρ + iω ( ρ ρ ) 1 11 c 1 1 & ρ = ( γ +Λ +Λ ) ρ + γ ρ + γ ρ + iω ( ρ ρ ) c 1 1 iφ + iω ( ρ ρ ) + iω ( ρ ρ ) + η( ρ e + ρ e ) iφ η iφ iφ & ρ =Λ1ρ11 γ ρ+ iω ( ρ1 ρ1) ( ρ3e + ρ3e ) η iφ iφ & ρ33 =Λρ11 γ 3ρ33 iω ( ρ31 ρ13 ) ( ρ3e + ρ3e ) γ1+λ +Λ1 & ρ1= ( + i c ) ρ1+ iωc( ρ11 ρ ) iω ( ρ+ ρ3) 7

8 Page 8 of γ η iφ & ρ = ( + i( c+ + ωs )) ρ+ iωcρ1 iωρ1 ρ3e γ 3 η iφ & ρ3 = ( + i( c+ ωs )) ρ3+ iωcρ13 iωρ1 ρe γ1+ γ +Λ +Λ1 & ρ1 = ( + i( + ωs )) ρ1 + iω ( ρ ρ11) + iωcρ η iφ + iωρ3 ρ13e γ1+ γ 3+Λ +Λ1 & ρ13 = ( + i( ωs )) ρ13+ iω ( ρ33 ρ11) + iωcρ3 η iφ + iωρ3 ρ1e γ + γ 3 η iφ & ρ3 = ( iωs ) ρ3+ iωρ13 iωρ1 ( ρ+ ρ33) e (6) φ ϕ ϕ Where, = c. The density-matrix element ρjkfollows the symmetryrelationshi, ρjk ρ j= 3 = kj. For closeness of the system we have = 1. 8 ρ j= jj Now, we consider a medium of length L comosed of the above described atomic structure immersed in unidirectional ring cavity (Fig. 1(c) based Ref [14-5]. The robe laser E enters the cavity through mirror M 1, roagates in the cavity and interacts with the atomic medium of length L, and artially transmits out of the mirror M. Under slowly varying envelo aroximation; the dynamic resonse of the robe field is governed by Maxwell s equations, E E iω + c = P( ω ). (7) t z ε P( ω ) is induced olarization in the transitions 1 and 1 3 it is given by: P( ω ) = Nµ ( ρ + ρ ), (8) 1 31

9 Page 9 of where N is the atomic number density of the medium. Substituting Eq. (8) into Eq. (7), one can obtain the field amlitude relation for the steady state as follow: E Nω µ = i ( ρ 1+ ρ 31), (9) z cε For a erfectly tuned cavity, the boundary conditions in the steady-state limit between the incident field T E and transmitted field E are: I T E E ( L) =, (1) T I E () = TE + RE ( L), (11) P Where L the length of the samle, R is the feedback mechanism and it is resonsible for the bistable behavior and T, is the transmittance coefficients of the mirror M. Therefore; one does not exect any bistability for R = in Eq. (11). According to the mean field limit and by using of the boundary conditions the steady state behavior of transmitted field is given by: y= x ic ( ρ + ρ ) (1) where 1 31 y µ E T T = h and x= µ E h T are the normalized inut and outut field, I resectively. The arameter 9 = ω µ h ε ct is the cooeratively arameter in a C N L ring cavity. Transmitted field deends on the incident robe field and the coherence terms ρ1+ ρ31via Eq. 1. Results and discussion In the following, we solve the density matrix equations numerically in the steadystate condition and use equation (1) to obtain the results for OB under assumtion of secific arameters. In the following numerical calculations, all the arameters used are scaled byγ, which should be of the order of MHz for rubidium atoms.

10 Page 1 of Next, we show that OB can be realized in the resent atomic system, and discuss the influence of the corresonding system arameters on the behaviors of OB as illustrated in figs -5. Fig (a) shows the deendence of the OB on the frequency sacing between two uer levels; we find that the threshold and width of the bistable region can be maniulated by the frequency sacing between uer levels. It is clearly shown that with increasing ω from.5γ to3.5γ, the threshold of OB increase. The reason for the above results can be interreted as follows. The changed frequency sacing between two uer levels will modify the absortion and the Kerr nonlinearity of the atomic medium, which make the intensity threshold of OB increases. In this case, the cavity field more difficult to reach saturation.fig. (b) deicts the effect of frequency sacing between two uer levels on the grou velocity of weak robe field. In fact, the resonse of the medium to the alied fields is determined by the suscetibilityχ, which defines as: N. µ χ = ( ρ31+ ρ1), (13) ε E ' '' Where N is the atomic density number in the medium, and χ = χ + iχ. The sloe of disersion, with resect to the coule and robe fields detuning has a major role in the determination of the grou velocity. We introduce the grou indexn g where the grou velocity v g of the robe field in given by χ ( ν ) vg = c 1+ πχ ( ν ) + πν ( ) (14) ν 1 c =, v Equation (14) imlies that for a negligible real art ofχ, the grou velocity can be significantly reduced via a stee disersion. Moreover, a strong negative disersion can lead to an increase in the grou velocity and even a negative grou velocity. g

11 Page 11 of From fig. (b), we find that by enhancing arameter ωs from.5γ to3.5γ, the grou velocity can be switched from subluminal to suerluminal or vice versa at some frequency detuning of robe field. Therefore, suerluminal or subluminal light roagation can be obtained by tuning the frequency of robe light and frequency sacing between uer levels. In this case, we also find that by switching the grou velocity from subluminal to suerluminal the intensity threshold of otical bistability is also reduced. This, henomena is suitable for otical switching which is essential for the next generation of all-otical system. The effect of SGC on OB and grou velocity of weak robe light is shown in Fig. 3(a, b). As deicted in fig. 3(a), the intensity threshold of OB can be controlled by changing the value of arameter. However, for= 1, the OB converts to OM. Therefore, switching from OB to OM can be obtained by changing the arameter. The effect of SGC on grou velocity of robe light is also discussed in art (b) of Fig. 3. It is found that for any values of arameter, the light roagates in subluminal case. The effect of incoherent uming field on OB, in the resence and absence of SGC is dislayed in Fig. 4 (a, b). From fig. 4(a), we find that by increasing the incoherent uming rate, the intensity threshold of OB decreases. Physically, in the resence of SGC and by increasing the rate of incoherent uming field the absortion and Kerr nonlinearity of the medium will be modified which makes the cavity field easier to reach saturation. However, in the absence of SGC (fig. 4(b)), the results are in contrast with revious results. It can be seen that, initially, by increasing the incoherent uming rate, the OB threshold decreases and then enhances. Here, forλ =Λ 1= γ, the OB threshold has a minimum value and forλ =Λ 1= 3γ, has maximum value. However, the intensity thresholds of OB are less than obtained results in art (a) of Fig. 4. At the end, the effect of relative hase between alied fields in the resence of SGC is shown in Fig. 5. It can be seen that in the resence 11

12 Page 1 of (a) and absence (b) of incoherent uming fields, the OB converts to OM forφ = π. However, for other relative hases of alied fields, the behaviors of OB are comletely different with together. In this case, the incoherent uming field makes the intensity threshold of OB enhances and cavity field harder to reach saturation. Possible exerimental realization Before ending this aer, let us briefly discuss the ossible exerimental realization of our roosed scheme for the resent study, which are given as follows: We consider, for instance, the cold atoms 87 Rb (nuclear sin I=3/) on the 5S-5P- 5D transitions as a ossible candidate [47, 48], the detailed couling diagram is shown in Fig. 1(b). The designed states can be chosen as follows: = S1/ F = P1/ F 3/ 5,, 1 = 5, =, = 5 D, F = 1, and 3 = 5 D3/, F =, resectively. Moreover, we have found that [3] the decay rates from the excited state 1 to the ground state, from excited stated to excited state 1, and from excited state 3 to excited state 1, are γ1 5.3 MHz, γ =.67MHz, and γ 3 =.67 MHz, resectively. Conclusion In a summary, we have theoretically investigated the OB behaviors a four-level atomic system driven by a couling laser field and a robe field inside the unidirectional ring cavity. The results show that the OB behavior and grou velocity of weak robe light can be controlled by roer tuning of incoherent uming fields, SGC and relative hase of alied fields. We find that by changing the relative hase between alied fields the switching between otical bistability and otical multistability occurs. By controlling the threshold intensity and the hysteresis of otical multistability, we can build more efficient all-otical switches 1

13 Page 13 of and logic-gate devises for otical comuting and quantum information rocessing. Moreover, we find that by switching the grou velocity from subluminal to suerluminal or vice versa the intensity threshold of OB can be controlled. We rovide a clue for achieving an otical switch in which the roagation of a laser ulse (slow to fast or vice versa) can be controlled with another laser field. 13

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19 Page 19 of Figure Cations Fig. 1: (a) Schematic diagram of a four-level atomic system. (b) Level hyerfine structure and the laser-couling scheme for a four-level 87 Rb atomic system [6]. (c) Unidirectional ring cavity with atomic samle of length L. incident and transmittance resectively. 19 I EP and T EP are the Fig. : (a) Outut intensity versus inut intensity and (b) grou index versus robe field detuning for different values of frequency sacing. The selected arameters areω c = 3 γ, = c =, Λ =Λ 1=, P=, φ =, and 1 Fig. 3: C=. (a) Outut intensity versus inut intensity and (b) grou index versus robe field detuning for different values of frequency sacing. The selected arameters areω c = 3 γ, = c =, Λ =Λ 1=, ω =.5 γ, φ =, and 1 C=. Fig. 4: Outut intensity versus inut intensity in the resence (a) and absence of SGC (b) for different values of incoherent uming fields. The selected arameters are same as Fig. 3. Fig. 5: Outut intensity versus inut intensity in the resence (a) and absence of incoherent uming field (b) for different values of relative hase between alied fields. The arameter P=1 and other arameters are same as Fig. 3.

20 Page of I E 5D 3/ 76nm 5P 1/ 795nm 5S 1/ (c) (b) I E 44MHz 15MHz 816MHz 6.834GHz Λ 1E Λ c 3MHz F=3 F= F=1 F= F= F=1 F= F=1 T E 1 4 Fig. 1 3

21 Page 1 of Outut(Ω T ) Outut(Ω T ) a Inut(Ω I ) a Inut(Ω I ) ω=.5 ω=.5 ω=3.5 1 Grou Index Fig. Fig P=.5 P=-.5 P=1. P=-1. Grou Index b b

22 Page of Outut(Ω T ) O utu t(ω T ) a Inut(Ω I ) a Inut(Ω I ) Λ 1 =Λ =1 9 Λ 1 =Λ = Λ 1 =Λ =3 8 O utu t(ω T ) Fig Φ=π/6 Φ=π/4 Φ=π/ Φ=π Fig Outut(Ω T ) b Inut(Ω I ) b Inut(Ω I )