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Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system
Physics Letters A 332 (2004) 244–249 www.elsevier.com/locate/pla
Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system Dong-chao Cheng, Cheng-pu Liu, Shang-qing Gong ∗ Key Laboratory for High Intensity Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, PR China Received 5 July 2004; received in revised form 24 September 2004; accepted 27 September 2004 Available online 6 October 2004 Communicated by B. Fricke
Abstract We investigate the optical bistability (OB) behavior of a nearly equispaced ladder-type three-level atomic system contained in a unidirectional ring cavity. We consider the three-level atoms under conditions that the atomic dipole moments are nonorthogonal so that the effect of spontaneously generated coherence (SGC) is important. We find that the SGC effect significantly affects the OB behavior of such system. Within certain parameter range, OB can be realized due to the SGC effect. The bistable hysteresis cycle becomes wider with the enhancement of SGC effect. Furthermore, such system is much sensitive to the relative phase between the coupling and probe fields. This property makes it possible to switch between bistability and multistability via adjusting the relative phase. 2004 Elsevier B.V. All rights reserved. PACS: 42.65.Pc; 42.50.Md; 42.50.Hz Keywords: Optical bistability; Optical multistability; Quantum interference; Spontaneously generated coherence; Three-level atom
1. Introduction Quantum coherence and interference in atomic systems can lead to many interesting optical phenomena such as lasing without population inversion, enhanced index of refraction without absorption, electromagnet* Corresponding author.
E-mail addresses: [email protected] (D.-c. Cheng), [email protected] (S.-q. Gong). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.09.052
ically induced transparency, and the cancellation of absorption etc. [1]. Optical bistability in two-level atomic systems has been extensively studied both experimentally and theoretically in the past years [2]. In recent years, optical bistability in systems with three-level atoms inside optical cavities has been theoretically studied [3] and experimentally demonstrated [4]. One major advantage of using three-level, instead of two-level, atoms as the nonlinear medium inside an optical cavity is
D.-c. Cheng et al. / Physics Letters A 332 (2004) 244–249
to make use of the atomic coherence induced in the three-level atomic systems, which can greatly modify the absorption, dispersion, and nonlinearity of the system [5]. Harshawaedhan and Agarwal [6] investigated the role of atomic coherence and interference in optical bistability. They manifested that these effects result in a considerable lowing of the threshold intensity. Gong et al. [7] investigated the bistable behavior of a ground-state doublet three-level atomic system via initial coherence of atoms, and showed that OB can be realized due to the initial coherence. Furthermore, the bistable hysteresis cycle becomes larger as the initial coherence increases. Gong et al. [8] also demonstrated that the phase fluctuations of the driven field can be used to control optical bistable response in Λ-type atoms. Antón et al. [9] studied the optical bistability behavior of a V -type three-level atomic system taking into account the possibility of quantum interference between the two decay channels from the two upper sublevels to the ground level. They found that OB can be achieved with a considerably lower threshold intensity and the cooperative parameter, and that the system is very sensitive to the detuning between the driven optical field and the optical transition, which controls the threshold and the width of the hysteresis loop. Very recently, Joshi et al. demonstrated the effect of spontaneously generated coherence on optical bistability in three-level V -type [10] and Λ-type [11] atomic system. They shown that the quantum interference reduces the bistability threshold, and that via adjusting the coupling field strength, one can alter the absorption and nonlinear optical properties of the atomic medium for the cavity field, and therefore, change the steady state behavior. It is now well known that spontaneously induced coherence can be created by the interference of spontaneous emission of either a single excited level to two closely lying atomic levels (Λ-type) [12] or two closely lying atomic levels to a common atomic level (V -type) [13]. In a ladder-type system, it can also be created in a nearly equispaced atomic levels case [14]. The closely lying, near-degenerate levels of a system have a coherence term due to interaction with the vacuum of the radiation field. The coherence can change the steady state response of the medium, and therefore, modify significantly the absorption or spontaneous emission spectra of a near-degnerate system.
245
Menon and Agarwal [15] have shown that the probe absorption of a Λ-type atom is crucially dependent on the relative phase between the drive and probe laser fields. Gong et al. [16] have shown that the steady state population behavior in a laser driven V -type system crucially depends on the parameter p, which determines the strength of the interference in spontaneous emission and on the relative phase of the two coherent driving fields. These early work has shown that when the system under consideration behaves as a generalized closed loop system, the behavior of the system crucially depends on the relative phase between the transition paths if the effects of spontaneous emission interference are accounted for [16]. It should be pointed out that the existence of the SGC effect depends on the nonorthogonality of dipole matrix elements [12]. The nonorthogonality can be obtained from the mixing of the levels arising from internal fields or external microwave fields [17]. Very recently, Ma et al. [18] investigated the effects of SGC on population inversion in a ladder-type threelevel atom with equispaced levels, subject to the condition that the atomic dipole moments are nonorthogonal. They found that the population inversion can be greatly enhanced on one of the optical transitions due to the spontaneous emission induced coherence. Furthermore, such coherence may also lead to population inversion on both of the optical transitions. In this Letter, we use the equispaced three-level ladder-type atom model to investigate the steady state OB behavior, by taking into account the effect of SGC. We find that the effect of SGC has considerable influence over the OB behavior of such atomic system. Within some parameter range, there is no OB in the absence of the SGC effect; but in the present of the SGC effect, OB occurs. The bistable hysteresis loop becomes wider due to the enhancement of SGC effect. Furthermore, such system is much sensitive to the relative phase between the coupling and probe fields. One can switch between bistability and multistability by means of varying the relative phase. This article is organized as follows: in Section 2, we present the ladder-type atomic model and establish the corresponding Maxwell–Bloch equations. In Section 3, we provide a numerical analysis. In Section 4, we offer some conclusions.
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2. Model and equations The atomic system of interest here is depicted in Fig. 1. We consider a ladder-type system with nearly equispaced levels, the transition |2 → |3 is driven by a coupling field of frequency ωc with Rabi frequency Ω2 (= µ 23 · Ec ), while the transition |1 → |2 is driven by a probe field of frequency ωp with Rabi frequency Ω1 (= µ 12 · Ep ). The level |2 (|3) spontaneously decays to the level |1 (|2) at the rate γ1 (γ2 ). 1 = ω21 − ωp and 2 = ω32 − ωc denote the frequency detuning of the coupling laser and the probe laser, respectively. The density matrix equation for such atomic system under the dipole interaction and the rotating wave approximation is [18]: ρ˙33 = −2γ2 ρ33 − iΩ2 ρ32 + iΩ2∗ ρ23 ,
(1a)
ρ˙11 = 2γ1 ρ22 + iΩ1 ρ21 − iΩ1∗ ρ12 , ρ˙32 = − iΩ2∗ (ρ33 − ρ22 )
(1b)
+ (i2 − γ1 − γ2 )ρ32 − iΩ1 ρ31 , ρ˙31 = iΩ2∗ ρ21
(1c)
− iΩ1∗ ρ32
+ (i1 + i2 − γ2 )ρ31 ,
(1d)
ρ˙21 = iΩ2 ρ31 − iΩ1∗ (ρ22
− ρ11 ) √ + (i1 − γ1 )ρ21 + 2p γ1 γ2 ρ32 , ρj∗i
(1e)
constrained by ρij = and ρ11 + ρ22 + ρ33 = 1. Here, in the case of nearly equispaced levels, the inclusion of two driving fields of different frequencies would lead to the optical Bloch equations with addi√ tional term (2p γ1 γ2 ρ32 ), which presents the effect of SGC. Note that, in the nonequispaced levels case, such a term of time dependence which includes an oscillation at the frequency |ωp − ωc | can be neglected. Because h|ω ¯ p − ωc | has been assumed to be much larger than either the interaction energies hΩ ¯ 1 and h¯ Ω2 or h¯ γ1 and h¯ γ2 , such terms are rapidly oscillating and will averaged out [14,18]. Therefore, in nonequispaced level situation, the SGC effect may be ignored. However, for the case of nearly equispaced levels, i.e., ωp − ωc ≈ 0, the SGC effect must be taken into account. The parameter p denotes the alignment of the two matrix elements which is nonparallel as well as nonorthogonal, and is defined as 23 /|µ 12 · µ 23 | = cos(θ ), where θ is the anp=µ 12 · µ gle between the two induced dipole moments µ 12 and µ 23 (as shown in Fig. 2). It is obvious that if the two matrix elements µ 12 , µ 23 are orthogonal then p = 0
Fig. 1. Three-level ladder-type atom interacting with two coherent field ωp (the probe laser field) and ωc (the coupling laser field). γ1 and γ2 are the decay rates. 1 and 2 are detunings of the coherent field.
and no interference from spontaneous decay occurs. Using the restriction that each of the linearly polarized lasers should only couple one of the optical transitions, we can find that the Rabi frequencies are connected to 2= = Ω 1 − p the parameter p by the relation Ω 1 10 Ω10 sin(θ ) and Ω2 = Ω20 1 − p2 = Ω20 sin(θ ), with Ω10 = |µ 12 ||E p | and Ω20 = |µ 23 ||E c |. Note that, in nonequispaced level situation, Rabi frequencies may be treated as real parameters. However, for the case of nearly equispaced levels, the SGC effect must be taken into account, moreover, this system becomes quite sensitive to the relative phase between the probe and the coupling fields. Thus Rabi frequencies have to be treated as complex parameters. For simplicity in calculation, we rewrite the Rabi fre−iϕ p = G10 1 − p2 e−iϕp = quencies as Ω1 ≡ G1 e −iϕ 2 p |µ 12 ||Ep | 1 − p e and Ω2 ≡ G2 e−iϕc = G20 1 − p2 e−iϕc = |µ 23 ||Ec | 1 − p2 e−iϕc . Redefining the atomic variables in Eq. (1) as σii ≡ ρii , σ21 ≡ ρ21 e−iϕp , σ32 ≡ ρ32 e−iϕc , σ31 ≡ ρ31 e−i(ϕp +ϕc ) , we can obtain equations for the redefined density matrix elements σij . The equations are identical to Eq. (1) except that p is replaced by peiϕ , ϕ = ϕc − ϕp , Ω1 is replaced by G1 , and Ω2 is replaced by G2 . In the following, we treat G1 and G2 as real parameters. Now, we put the ensemble of N homogeneously broadened ladder-type atoms in a unidirectional ring cavity (see Fig. 3). For simplicity, we assume that mirror 3 and 4 have 100% reflectivity, and the intensity reflection and transmission coefficient of mirrors 1 and 2 are R and T (with R + T = 1), respectively. The total electromagnetic field is E = Ep e−iωp t + Ec e−iωc t + c.c.
(2)
D.-c. Cheng et al. / Physics Letters A 332 (2004) 244–249
mitted field EpT [2]: √ Ep (L) = EpT / T , √ Ep (0) = T EpI + REp (L),
Fig. 2. The arrangement of field polarization required for a single field driving one transition if dipoles are nonorthogonal.
247
(6a) (6b)
where L is the length of the atomic sample, and the second term on the right-hand side of Eq. (6b) describes a feedback mechanism due to the mirror, which is essential to give rise to bistability, namely, there will be no bistability if R = 0. In the mean-field limit [19], using the boundary conditions Eqs. (6), √and normalizing the √ fields by letting y = µ12 EpI /h¯ T , x = µ12 EpT /h¯ T , we can get input–output relationship: y = x + Cγ1 Im(σ21 ) − iCγ1 Re(σ21 ),
(7)
= LNωp µ212 /2h¯ cε0 T γ1
Fig. 3. Unidirectional ring cavity with an atomic sample of length L, EpI and EpT are the incident and the transmitted field, respectively, and Ec is the control field which is noncirculating in the cavity.
The Maxwell’s equation, under slowly varying envelope approximation is [9] ∂Ep ωp ∂Ep +c =i P (ωp ), ∂t ∂z 2ε0
(3)
where P (ωp ) is the slowly oscillating term of the induced polarization in the transition |1 ↔ |2 and is given by P (ωp ) = Nµ12 σ21 ,
(4)
where N the density of the atoms. We consider the field equation (3) in the steadystate case. Setting the time derivative in Eq. (3) equal to zero for the steady state, we can obtain the field amplitude as follows: Nωp µ12 σ21 ∂Ep =i . ∂z 2cε0
(5)
For a perfectly tuned cavity, in the steady state limit, the boundary conditions impose the following conditions between the incident field EpI and the trans-
is cooperation parawhere C meter. The second and the third terms on the righthand side of Eq. (7) are vital for bistability to happen. We set the time derivatives σ˙ ij (i = 1, 2, 3) in the redefined density matrix equation equal to zero for the steady state, solve the corresponding density matrix equation together with the field equation Eq. (7), then we can obtain the steady-state solutions. In numerical calculations, we choose the parameters to be dimensionless units by scaling with γ and let γ = 1. 3. Numerical results Setting 1 = 2 = 0, γ1 = γ2 = 5γ , ϕ = 0, G20 = 50γ , and C = 600, we plot the input–output curves for different values of parameter p, as shown in Fig. 4. It is easy to see from Fig. 4 that in the case that p = 0, there is no OB; however, in the case of large values of p, such as p = 0.9, OB appears. Thus, the presence of the SGC effect is responsible for the occurrence of OB. From Fig. 4, we can also find that the bistable hysteresis loop for p = 0.99 is much wider than the one for p = 0.9. That is to say, in the two-photon resonance case, the bistable hysteresis loop becomes wider as the value of p increases. To gain an insight into the physical origin, we investigate the absorption of the probe field. Fig. 5 presents three absorption curves of the probe laser for p = 0 (dotted), p = 0.9 (dashed), and p = 0.99 (solid). From this figure, we can see that at the line center the absorption is greatly enhanced with the increase of the strength of SGC. As a result, OB appears in the presence of the SGC effect, and the
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Fig. 4. Plots of transmitted light versus incident light for different values of p. The dotted curve is for p = 0; the dashed curve for p = 0.9, the solid curves for p = 0.99. The other parameters are 1 = 2 = 0, γ1 = γ2 = 5γ , ϕ = 0, G20 = 50γ , and C = 600.
Fig. 5. Enhancement of the absorption of probe field with the increase of the value of p for G10 = 10γ . The dotted, dashed, and solid curve correspond to p = 0, p = 0.9, and p = 0.99, respectively. The other parameters are the same as Fig. 4.
bistable hysteresis loop becomes wider as the strength of SGC effect increases. Similar results can also be obtained in the offresonance case for appropriate parameters. In Fig. 6, we plot the bistable curves in the case that 1 = 2γ , 2 = 3γ , and G20 = 30γ . The dotted, the dashed and the solid curves correspond to p = 0, p = 0.9, and p = 0.99, respectively. Clearly, the bistable hysteresis loop still widens as the effect of SGC strengthens. As mentioned in the introduction, atomic systems with SGC effect are usually sensitive to the relative phase between the applied fields. Such effect will
Fig. 6. Widening of the bistable hysteresis loop produced by the SGC effect for 1 = 2γ , 2 = 3γ , G20 = 30γ . The dotted curve is for p = 0, the dashed curve for p = 0.9, and the solid curve for p = 0.99. The other parameters are the same as Fig. 4.
without doubt be reflected in the OB behavior of such system. We find that one can transit between bistability and multistability by adjusting the relative phase ϕ. Fig. 7 displays bistability curve for ϕ = π . Once altering the value of ϕ to π/4, the system exhibits multistable behavior, as shown in Fig. 8. Note that, in Fig. 8, we only depicted the part |y| < 1400 in order to show the multistability behavior more clearly, which leads to the appearance of discontinuity in the figure. In fact, the curve in continuous. From Figs. 7 and 8, we can see the dramatic influence of the relative phase between the probe and coupling fields, which is the result of the strong influence of the relative phase on the absorption and the dispersion property of atomic medium when the SGC effect is considered. Thus the relative phase can be used as a very effective approach to control the transition between the bistable and the multistable behavior. Joshi and Xiao [20] have experimentally achieved the transition between bistability and multistability by varying the frequency detuning of the coupling laser beam or the atomic number density. 4. Conclusions In summary, by using the model of ladder-type three-level atoms with equispaced levels inside unidirectional ring cavity, we have demonstrated that the spontaneously generated coherence can significantly affect the OB behavior of such system. Within a certain parameter range, for p = 0, there is no optical
D.-c. Cheng et al. / Physics Letters A 332 (2004) 244–249
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the National Science Foundation of Shanghai (Grant No. 03ZR14102).
References
Fig. 7. Bistability for ϕ = π . The other parameters are 1 = 1γ , 2 = 2γ , γ1 = γ2 = 5γ , p = 0.99, G20 = 50γ , and C = 600.
Fig. 8. Multistability for ϕ = π/4. The other parameters are the same as Fig. 7.
bistability; for p = 0, bistability exists. The enhancement of SGC effect makes the bistable hysteresis loop become wider. Moreover, the system is much sensitive to the relative phase between the coupling and probe fields when the effect of SGC is accounted for. Via adjusting the relative phase, one can switch between bistability and multistability.
Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 10234030) and
[1] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge Univ. Press, Cambridge, UK, 1997, Chapter 7, p. 220; E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, vol. XXXV, Elsevier, Amsterdam, 1996, p. 257. [2] L.A. Lugiato, in: E. Wolf (Ed.), Progress in Optics, vol. XXI, North-Holland, Amsterdam, 1984, p. 71, and references therein. [3] D.F. Walls, P. Zoller, Opt. Commun. 34 (1980) 260. [4] H. Wang, D.J. Goorskey, M. Xiao, Phys. Rev. A 65 (2001) 011801(R); H. Wang, D.J. Goorskey, M. Xiao, Phys. Rev. A 65 (2002) 051802(R); A. Joshi, A. Brown, H. Wang, M. Xiao, Phys. Rev. A 67 (2003) 041801(R). [5] A. Brown, A. Joshi, M. Xiao, Appl. Phys. Lett. 83 (2003) 1301. [6] W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. [7] S.-q. Gong, S.-d. Du, Z.-z. Xu, Phys. Lett. A (1997) 293. [8] S.-q. Gong, S.-d. Du, Z.-z. Xu, S.-h. Pan, Phys. Lett. A 222 (1996) 237. [9] M.A. Antón, O.G. Calderón, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) 91. [10] A. Joshi, W. Yang, M. Xiao, Phys. Rev. A 68 (2003) 015806. [11] A. Joshi, W. Yang, M. Xiao, Phys. Lett. A 315 (2003) 203. [12] J. Javanainen, Europhys. Lett. 17 (1992) 407. [13] P. Zhou, S. Swain, Phys. Rev. Lett. 77 (1996) 3995. [14] R.M. Whitley, C.R. Stroud Jr., Phys. Rev. A 14 (1976) 1498; Z. Ficek, B.J. Dalton, P.L. Knight, Phys. Rev. A 51 (1995) 4062. [15] S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) 4014. [16] S.-q. Gong, E. Paspalakis, P.L. Knight, J. Mod. Opt. 45 (1998) 2433. [17] H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 (1996) 1032; H. Schmidt, A. Imamoglu, Opt. Commun. 131 (1996) 333. [18] H.-m. Ma, S.-q. Gong, C.-p. Liu, Z.-r. Sun, Z.-z. Xu, Opt. Commun. 223 (2003) 97. [19] R. Bonifacio, L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 505; R. Bonifacio, L.A. Lugiato, Phys. Rev. A 18 (1978) 1129; P. Meystre, Opt. Commun. 26 (1978) 277. [20] A. Joshi, M. Xiao, Phys. Rev. Lett. 91 (2003) 143904.
Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system Dong-chao Cheng, Cheng-pu Liu, Shang-qing Gong ∗ Key Laboratory for High Intensity Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, PR China Received 5 July 2004; received in revised form 24 September 2004; accepted 27 September 2004 Available online 6 October 2004 Communicated by B. Fricke
Abstract We investigate the optical bistability (OB) behavior of a nearly equispaced ladder-type three-level atomic system contained in a unidirectional ring cavity. We consider the three-level atoms under conditions that the atomic dipole moments are nonorthogonal so that the effect of spontaneously generated coherence (SGC) is important. We find that the SGC effect significantly affects the OB behavior of such system. Within certain parameter range, OB can be realized due to the SGC effect. The bistable hysteresis cycle becomes wider with the enhancement of SGC effect. Furthermore, such system is much sensitive to the relative phase between the coupling and probe fields. This property makes it possible to switch between bistability and multistability via adjusting the relative phase. 2004 Elsevier B.V. All rights reserved. PACS: 42.65.Pc; 42.50.Md; 42.50.Hz Keywords: Optical bistability; Optical multistability; Quantum interference; Spontaneously generated coherence; Three-level atom
1. Introduction Quantum coherence and interference in atomic systems can lead to many interesting optical phenomena such as lasing without population inversion, enhanced index of refraction without absorption, electromagnet* Corresponding author.
E-mail addresses: [email protected] (D.-c. Cheng), [email protected] (S.-q. Gong). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.09.052
ically induced transparency, and the cancellation of absorption etc. [1]. Optical bistability in two-level atomic systems has been extensively studied both experimentally and theoretically in the past years [2]. In recent years, optical bistability in systems with three-level atoms inside optical cavities has been theoretically studied [3] and experimentally demonstrated [4]. One major advantage of using three-level, instead of two-level, atoms as the nonlinear medium inside an optical cavity is
D.-c. Cheng et al. / Physics Letters A 332 (2004) 244–249
to make use of the atomic coherence induced in the three-level atomic systems, which can greatly modify the absorption, dispersion, and nonlinearity of the system [5]. Harshawaedhan and Agarwal [6] investigated the role of atomic coherence and interference in optical bistability. They manifested that these effects result in a considerable lowing of the threshold intensity. Gong et al. [7] investigated the bistable behavior of a ground-state doublet three-level atomic system via initial coherence of atoms, and showed that OB can be realized due to the initial coherence. Furthermore, the bistable hysteresis cycle becomes larger as the initial coherence increases. Gong et al. [8] also demonstrated that the phase fluctuations of the driven field can be used to control optical bistable response in Λ-type atoms. Antón et al. [9] studied the optical bistability behavior of a V -type three-level atomic system taking into account the possibility of quantum interference between the two decay channels from the two upper sublevels to the ground level. They found that OB can be achieved with a considerably lower threshold intensity and the cooperative parameter, and that the system is very sensitive to the detuning between the driven optical field and the optical transition, which controls the threshold and the width of the hysteresis loop. Very recently, Joshi et al. demonstrated the effect of spontaneously generated coherence on optical bistability in three-level V -type [10] and Λ-type [11] atomic system. They shown that the quantum interference reduces the bistability threshold, and that via adjusting the coupling field strength, one can alter the absorption and nonlinear optical properties of the atomic medium for the cavity field, and therefore, change the steady state behavior. It is now well known that spontaneously induced coherence can be created by the interference of spontaneous emission of either a single excited level to two closely lying atomic levels (Λ-type) [12] or two closely lying atomic levels to a common atomic level (V -type) [13]. In a ladder-type system, it can also be created in a nearly equispaced atomic levels case [14]. The closely lying, near-degenerate levels of a system have a coherence term due to interaction with the vacuum of the radiation field. The coherence can change the steady state response of the medium, and therefore, modify significantly the absorption or spontaneous emission spectra of a near-degnerate system.
245
Menon and Agarwal [15] have shown that the probe absorption of a Λ-type atom is crucially dependent on the relative phase between the drive and probe laser fields. Gong et al. [16] have shown that the steady state population behavior in a laser driven V -type system crucially depends on the parameter p, which determines the strength of the interference in spontaneous emission and on the relative phase of the two coherent driving fields. These early work has shown that when the system under consideration behaves as a generalized closed loop system, the behavior of the system crucially depends on the relative phase between the transition paths if the effects of spontaneous emission interference are accounted for [16]. It should be pointed out that the existence of the SGC effect depends on the nonorthogonality of dipole matrix elements [12]. The nonorthogonality can be obtained from the mixing of the levels arising from internal fields or external microwave fields [17]. Very recently, Ma et al. [18] investigated the effects of SGC on population inversion in a ladder-type threelevel atom with equispaced levels, subject to the condition that the atomic dipole moments are nonorthogonal. They found that the population inversion can be greatly enhanced on one of the optical transitions due to the spontaneous emission induced coherence. Furthermore, such coherence may also lead to population inversion on both of the optical transitions. In this Letter, we use the equispaced three-level ladder-type atom model to investigate the steady state OB behavior, by taking into account the effect of SGC. We find that the effect of SGC has considerable influence over the OB behavior of such atomic system. Within some parameter range, there is no OB in the absence of the SGC effect; but in the present of the SGC effect, OB occurs. The bistable hysteresis loop becomes wider due to the enhancement of SGC effect. Furthermore, such system is much sensitive to the relative phase between the coupling and probe fields. One can switch between bistability and multistability by means of varying the relative phase. This article is organized as follows: in Section 2, we present the ladder-type atomic model and establish the corresponding Maxwell–Bloch equations. In Section 3, we provide a numerical analysis. In Section 4, we offer some conclusions.
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2. Model and equations The atomic system of interest here is depicted in Fig. 1. We consider a ladder-type system with nearly equispaced levels, the transition |2 → |3 is driven by a coupling field of frequency ωc with Rabi frequency Ω2 (= µ 23 · Ec ), while the transition |1 → |2 is driven by a probe field of frequency ωp with Rabi frequency Ω1 (= µ 12 · Ep ). The level |2 (|3) spontaneously decays to the level |1 (|2) at the rate γ1 (γ2 ). 1 = ω21 − ωp and 2 = ω32 − ωc denote the frequency detuning of the coupling laser and the probe laser, respectively. The density matrix equation for such atomic system under the dipole interaction and the rotating wave approximation is [18]: ρ˙33 = −2γ2 ρ33 − iΩ2 ρ32 + iΩ2∗ ρ23 ,
(1a)
ρ˙11 = 2γ1 ρ22 + iΩ1 ρ21 − iΩ1∗ ρ12 , ρ˙32 = − iΩ2∗ (ρ33 − ρ22 )
(1b)
+ (i2 − γ1 − γ2 )ρ32 − iΩ1 ρ31 , ρ˙31 = iΩ2∗ ρ21
(1c)
− iΩ1∗ ρ32
+ (i1 + i2 − γ2 )ρ31 ,
(1d)
ρ˙21 = iΩ2 ρ31 − iΩ1∗ (ρ22
− ρ11 ) √ + (i1 − γ1 )ρ21 + 2p γ1 γ2 ρ32 , ρj∗i
(1e)
constrained by ρij = and ρ11 + ρ22 + ρ33 = 1. Here, in the case of nearly equispaced levels, the inclusion of two driving fields of different frequencies would lead to the optical Bloch equations with addi√ tional term (2p γ1 γ2 ρ32 ), which presents the effect of SGC. Note that, in the nonequispaced levels case, such a term of time dependence which includes an oscillation at the frequency |ωp − ωc | can be neglected. Because h|ω ¯ p − ωc | has been assumed to be much larger than either the interaction energies hΩ ¯ 1 and h¯ Ω2 or h¯ γ1 and h¯ γ2 , such terms are rapidly oscillating and will averaged out [14,18]. Therefore, in nonequispaced level situation, the SGC effect may be ignored. However, for the case of nearly equispaced levels, i.e., ωp − ωc ≈ 0, the SGC effect must be taken into account. The parameter p denotes the alignment of the two matrix elements which is nonparallel as well as nonorthogonal, and is defined as 23 /|µ 12 · µ 23 | = cos(θ ), where θ is the anp=µ 12 · µ gle between the two induced dipole moments µ 12 and µ 23 (as shown in Fig. 2). It is obvious that if the two matrix elements µ 12 , µ 23 are orthogonal then p = 0
Fig. 1. Three-level ladder-type atom interacting with two coherent field ωp (the probe laser field) and ωc (the coupling laser field). γ1 and γ2 are the decay rates. 1 and 2 are detunings of the coherent field.
and no interference from spontaneous decay occurs. Using the restriction that each of the linearly polarized lasers should only couple one of the optical transitions, we can find that the Rabi frequencies are connected to 2= = Ω 1 − p the parameter p by the relation Ω 1 10 Ω10 sin(θ ) and Ω2 = Ω20 1 − p2 = Ω20 sin(θ ), with Ω10 = |µ 12 ||E p | and Ω20 = |µ 23 ||E c |. Note that, in nonequispaced level situation, Rabi frequencies may be treated as real parameters. However, for the case of nearly equispaced levels, the SGC effect must be taken into account, moreover, this system becomes quite sensitive to the relative phase between the probe and the coupling fields. Thus Rabi frequencies have to be treated as complex parameters. For simplicity in calculation, we rewrite the Rabi fre−iϕ p = G10 1 − p2 e−iϕp = quencies as Ω1 ≡ G1 e −iϕ 2 p |µ 12 ||Ep | 1 − p e and Ω2 ≡ G2 e−iϕc = G20 1 − p2 e−iϕc = |µ 23 ||Ec | 1 − p2 e−iϕc . Redefining the atomic variables in Eq. (1) as σii ≡ ρii , σ21 ≡ ρ21 e−iϕp , σ32 ≡ ρ32 e−iϕc , σ31 ≡ ρ31 e−i(ϕp +ϕc ) , we can obtain equations for the redefined density matrix elements σij . The equations are identical to Eq. (1) except that p is replaced by peiϕ , ϕ = ϕc − ϕp , Ω1 is replaced by G1 , and Ω2 is replaced by G2 . In the following, we treat G1 and G2 as real parameters. Now, we put the ensemble of N homogeneously broadened ladder-type atoms in a unidirectional ring cavity (see Fig. 3). For simplicity, we assume that mirror 3 and 4 have 100% reflectivity, and the intensity reflection and transmission coefficient of mirrors 1 and 2 are R and T (with R + T = 1), respectively. The total electromagnetic field is E = Ep e−iωp t + Ec e−iωc t + c.c.
(2)
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mitted field EpT [2]: √ Ep (L) = EpT / T , √ Ep (0) = T EpI + REp (L),
Fig. 2. The arrangement of field polarization required for a single field driving one transition if dipoles are nonorthogonal.
247
(6a) (6b)
where L is the length of the atomic sample, and the second term on the right-hand side of Eq. (6b) describes a feedback mechanism due to the mirror, which is essential to give rise to bistability, namely, there will be no bistability if R = 0. In the mean-field limit [19], using the boundary conditions Eqs. (6), √and normalizing the √ fields by letting y = µ12 EpI /h¯ T , x = µ12 EpT /h¯ T , we can get input–output relationship: y = x + Cγ1 Im(σ21 ) − iCγ1 Re(σ21 ),
(7)
= LNωp µ212 /2h¯ cε0 T γ1
Fig. 3. Unidirectional ring cavity with an atomic sample of length L, EpI and EpT are the incident and the transmitted field, respectively, and Ec is the control field which is noncirculating in the cavity.
The Maxwell’s equation, under slowly varying envelope approximation is [9] ∂Ep ωp ∂Ep +c =i P (ωp ), ∂t ∂z 2ε0
(3)
where P (ωp ) is the slowly oscillating term of the induced polarization in the transition |1 ↔ |2 and is given by P (ωp ) = Nµ12 σ21 ,
(4)
where N the density of the atoms. We consider the field equation (3) in the steadystate case. Setting the time derivative in Eq. (3) equal to zero for the steady state, we can obtain the field amplitude as follows: Nωp µ12 σ21 ∂Ep =i . ∂z 2cε0
(5)
For a perfectly tuned cavity, in the steady state limit, the boundary conditions impose the following conditions between the incident field EpI and the trans-
is cooperation parawhere C meter. The second and the third terms on the righthand side of Eq. (7) are vital for bistability to happen. We set the time derivatives σ˙ ij (i = 1, 2, 3) in the redefined density matrix equation equal to zero for the steady state, solve the corresponding density matrix equation together with the field equation Eq. (7), then we can obtain the steady-state solutions. In numerical calculations, we choose the parameters to be dimensionless units by scaling with γ and let γ = 1. 3. Numerical results Setting 1 = 2 = 0, γ1 = γ2 = 5γ , ϕ = 0, G20 = 50γ , and C = 600, we plot the input–output curves for different values of parameter p, as shown in Fig. 4. It is easy to see from Fig. 4 that in the case that p = 0, there is no OB; however, in the case of large values of p, such as p = 0.9, OB appears. Thus, the presence of the SGC effect is responsible for the occurrence of OB. From Fig. 4, we can also find that the bistable hysteresis loop for p = 0.99 is much wider than the one for p = 0.9. That is to say, in the two-photon resonance case, the bistable hysteresis loop becomes wider as the value of p increases. To gain an insight into the physical origin, we investigate the absorption of the probe field. Fig. 5 presents three absorption curves of the probe laser for p = 0 (dotted), p = 0.9 (dashed), and p = 0.99 (solid). From this figure, we can see that at the line center the absorption is greatly enhanced with the increase of the strength of SGC. As a result, OB appears in the presence of the SGC effect, and the
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Fig. 4. Plots of transmitted light versus incident light for different values of p. The dotted curve is for p = 0; the dashed curve for p = 0.9, the solid curves for p = 0.99. The other parameters are 1 = 2 = 0, γ1 = γ2 = 5γ , ϕ = 0, G20 = 50γ , and C = 600.
Fig. 5. Enhancement of the absorption of probe field with the increase of the value of p for G10 = 10γ . The dotted, dashed, and solid curve correspond to p = 0, p = 0.9, and p = 0.99, respectively. The other parameters are the same as Fig. 4.
bistable hysteresis loop becomes wider as the strength of SGC effect increases. Similar results can also be obtained in the offresonance case for appropriate parameters. In Fig. 6, we plot the bistable curves in the case that 1 = 2γ , 2 = 3γ , and G20 = 30γ . The dotted, the dashed and the solid curves correspond to p = 0, p = 0.9, and p = 0.99, respectively. Clearly, the bistable hysteresis loop still widens as the effect of SGC strengthens. As mentioned in the introduction, atomic systems with SGC effect are usually sensitive to the relative phase between the applied fields. Such effect will
Fig. 6. Widening of the bistable hysteresis loop produced by the SGC effect for 1 = 2γ , 2 = 3γ , G20 = 30γ . The dotted curve is for p = 0, the dashed curve for p = 0.9, and the solid curve for p = 0.99. The other parameters are the same as Fig. 4.
without doubt be reflected in the OB behavior of such system. We find that one can transit between bistability and multistability by adjusting the relative phase ϕ. Fig. 7 displays bistability curve for ϕ = π . Once altering the value of ϕ to π/4, the system exhibits multistable behavior, as shown in Fig. 8. Note that, in Fig. 8, we only depicted the part |y| < 1400 in order to show the multistability behavior more clearly, which leads to the appearance of discontinuity in the figure. In fact, the curve in continuous. From Figs. 7 and 8, we can see the dramatic influence of the relative phase between the probe and coupling fields, which is the result of the strong influence of the relative phase on the absorption and the dispersion property of atomic medium when the SGC effect is considered. Thus the relative phase can be used as a very effective approach to control the transition between the bistable and the multistable behavior. Joshi and Xiao [20] have experimentally achieved the transition between bistability and multistability by varying the frequency detuning of the coupling laser beam or the atomic number density. 4. Conclusions In summary, by using the model of ladder-type three-level atoms with equispaced levels inside unidirectional ring cavity, we have demonstrated that the spontaneously generated coherence can significantly affect the OB behavior of such system. Within a certain parameter range, for p = 0, there is no optical
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the National Science Foundation of Shanghai (Grant No. 03ZR14102).
References
Fig. 7. Bistability for ϕ = π . The other parameters are 1 = 1γ , 2 = 2γ , γ1 = γ2 = 5γ , p = 0.99, G20 = 50γ , and C = 600.
Fig. 8. Multistability for ϕ = π/4. The other parameters are the same as Fig. 7.
bistability; for p = 0, bistability exists. The enhancement of SGC effect makes the bistable hysteresis loop become wider. Moreover, the system is much sensitive to the relative phase between the coupling and probe fields when the effect of SGC is accounted for. Via adjusting the relative phase, one can switch between bistability and multistability.
Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 10234030) and
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