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Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system
Physics Letters A 315 (2003) 203–207 www.elsevier.com/locate/pla
Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system Amitabh Joshi ∗ , Wenge Yang, Min Xiao Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA Received 4 June 2003; received in revised form 4 July 2003; accepted 9 July 2003 Communicated by P.R. Holland
Abstract Optical bistability is studied for a three-level atomic system in Λ-configuration contained in an optical ring cavity and the effects of spontaneously generated coherence in the presence of two arbitrary coherent fields are investigated. Condition for observing optical multistable behavior in this system has also been specified. 2003 Elsevier B.V. All rights reserved. PACS: 42.65.Pc; 42.65.Sf; 42.50.Gy
In recent years some new types of coherence produced due to decays of closely lying states have been discussed [1–5]. One of the main phenomena associated with these coherences is the modification of the line shapes of the spectral lines. In Λ-type three-level atomic system the spontaneously generated coherence (SGC) due to interaction with the vacuum bath of the radiation field has been discussed by Javanainen [4]. The important finding in the study of Ref. [4] is the disappearance of the dark state due to SGC. The SGC is very sensitive to the alignment of dipole moments of two transitions with respect to each other. The effects of SGC on electromagnetically induced transparency (EIT) [6] and coherent population trapping [7] phenomena were also examined [8]. Suppression of spontaneous emission [2] and subnatural linewidths [3] due
* Corresponding author.
E-mail address: [email protected] (A. Joshi). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)01046-6
to SGC were predicted in the three-level atomic systems. Optical bistability (OB) was extensively studied initially in the two-level system comprising of alkali atomic beams inside an optical resonator where a single mode beam circulating [9,10]. The experimental demonstration of OB in this system gave a great thrust to many potential applications such as optical transistors, memory elements, and all optical switches. This led to further investigation of OB in three-level atomic system inside an optical cavity both theoretically [11,12] as well as experimentally [13]. Multistable (multiple hysteresis) behaviors has also been observed in Fabry–Perot cavities filled with atoms having several degenerate or nearly degenerate sublevels in the ground state and driven by linearly polarized light [14]. Most of these experiments used magnetic fields and high pressure buffer gases, and relied on Zeeman coherence as an efficient mechanism for the observed behavior. More specifically, optical trista-
204
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
ρ˙11 = 2γ1 ρ22 − iΩP (ρ12 − ρ21 ), ρ˙22 = −2(γ1 + γ2 )ρ22 + iΩP (ρ12 − ρ21 ) + iΩC (ρ32 − ρ23 ), ρ˙33 = 2γ2 ρ22 − iΩC (ρ32 − ρ23 ), ρ˙21 = −(γ1 + γ2 − i∆P )ρ21 + iΩC ρ31 + iΩP (ρ11 − ρ22 ), ρ˙23 = −(γ1 + γ2 − i∆C )ρ23 + iΩP ρ13 − iΩC (ρ22 − ρ33 ), Fig. 1. Schematics of a three-level Λ-type atom.
bility in a three-level Λ-configuration was calculated under different conditions and also observed experimentally [15–18]. In this Letter, we deal with the role of SGC on OB in a three-level Λ-type atomic system contained in a ring optical cavity. The atomic system consists of two lower sublevels of same parity and a single excited level of different parity. The effect of quantum interference in spontaneous emission from the upper level to the lower two levels is included in the model. Such a model with quantum interference from spontaneous emission shows interference-assisted population inversion, trapping and probe gain at one sideband of the Autler–Townes spectrum [19]. However, the previous theoretical works [11,12] studying OB behavior in three-level atomic systems did not include such quantum interference in the decay channels or SGC in their models. We expect to see some novel changes in the usual OB brought out by the inclusion of this SGC in the three-level atomic system under consideration. The three-level atomic system considered here is shown in Fig. 1. It is a closed Λ-type configuration with one excited state |2 and two closely-lying lower states |1 and |3. The transition between |2 and |1 (with resonant frequency ω21 ) is mediated by the probe laser field EP (frequency ω1 ) while the transition |2 to |3 (with resonant frequency ω23 ) is driven by another laser field EC (frequency ω2 ) called coupling field in this work. The atomic dynamics of the system can be described by the Liouville equation for the density operator and the density matrix equations [4,8] with all decay terms included under rotating-wave approximation are
ρ˙13 = −i(∆P − ∆C )ρ13 − iΩC ρ12 √ + iΩP ρ23 + 2 γ1 γ2 ηρ22 .
(1)
Here, the atomic detunings are defined as ∆P = ω21 − ω1 , ∆C = ω23 − ω2 , respectively, and the Rabi frequencies for the probe and coupling fields are ΩP = d 12 .E P /h¯ and ΩC = d 32 .E C /h¯ , respectively. The transition dipole moments associated with the two transitions, e.g., d 12 and d 32 , can be nonorthogonal. We define a parameter η=
d 12.d 32 ≡ cos(θ ) |d 12 |.|d 32|
which is a measure of orthogonality of the dipole moments, i.e., η = 0 for θ = π/2. Physically, the √ term η γ1 γ2 accounts for the spontaneous emission induced quantum interference effect due to coupling between emission processes in the channels |2 → |1 and |2 → |3. In a recent experiment the ability of controlling η has been experimentally demonstrated [5] in sodium dimers by considering the superposition of singlet and triplet states due to spin-orbit coupling. However, a conflicting result was obtained in another experiment of similar kind [20]. It seems there are practical difficulties to find a perfect atomic system where dipole moments are parallel so that quantum interference due to SGC is maximum [21]. Recently, some novel methods are discussed in the literature [22–24] generating the quantum interference effects in more effective manners such as use of microwave field in a three-level system [22], multilevel schemes [23], or use of anisotropy of the electromagnetic vacuum [24]. The Rabi frequencies in our model are also functions of the angle θ , however, for the sake of convenience in comparison with different values of θ , the Rabi frequencies are kept unchanged by suitably adjusting the field strengths in this work.
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
tions in this configuration are √ EPT (t) = T EP (L, t), √ EP (0, t) = T EPI (t) + REP (L, t − t).
Fig. 2. Schematic diagram of a unidirectional ring cavity having four mirrors (M1–M4) and an atomic vapor cell of length L. The mirrors M3 and M4 are perfectly reflecting mirrors (R = 1 for each). The incident and the transmitted fields are represented by EPI and EPT , respectively, and the coupling field EC is noncirculating in the cavity.
We can study OB in this system by considering N such Λ-type atoms contained in a unidirectional optical ring resonator depicted in Fig. 2. The intensity reflection and transmission coefficients of mirrors M1 and M2 are R and T , respectively, such that R + T = 1. Further we assume that both the mirrors M3 and M4 are perfect reflectors. This is one of the standard model for studying OB in the literature [9]. The total electromagnetic field seen by N homogeneously-broadened atoms contained in a cell of length L is E = E P exp(−iω1 t) + E C exp(−iω2 t) + c.c.
(2)
The probe field E P is circulating in the ring cavity but the coupling field E C does not. So, the Maxwell’s equation, under slowly-varying envelope approximation is appropriate to describe the dynamics of the probe field in the optical cavity, ∂EP ∂EP +c = 2πiω1 d12 P (ω1 ), ∂t ∂z
(3)
where P (ω1 ) is the induced polarization in the transition |1 → |2 and is given by P (ω1 ) = Nd12 ρ12 . The probe field EPI enters the cavity through mirror M1, propagates in the cavity to interact with the atomic sample of length L, then circulates in the cavity, and partially transmits out from the mirror M2 as EPT . The probe field at the start of the atomic sample is EP (0) and propagates to the end of the atomic sample to be EP (L, t) in a single pass. The field boundary condi-
205
(4)
Time taken for light to travel from M2 to M1 is t and we keep cavity detuning to be zero in this study. The bistable behavior is due to the nonlinearity of the atomic medium and resonator feedback mechanism. If we set R = 0 in Eq. (4) above, we do not get any bistability. We do not consider the circulation of the coupling beam in the optical cavity and hence allow this beam to enter in the cavity via a polarizing beam splitter such that it copropagates with the probe beam to eliminate the first order Doppler effect for the three-level atoms in the vapor cell [25] as described experimentally in Ref. [13]. The size of the coupling beam is larger in comparison to the probe beam in the vapor cell so that there is a good overlap. It is rather difficult to have a closed form expressions for OB for this problem due to the complexity of the dynamic equations (1)–(3). Also, it is cumbersome to put P (ω1 ) in a simple analytic form in steady state for the three-level atomic system in comparison to its two-level counterpart. Therefore, we solve the density matrix equations (1) numerically and integrate the Maxwell’s equation (3) in the steady-state limit over the length of the sample together with the boundary conditions (4) to get the results for OB under various parametric conditions. It should be noticed that in the limiting condition of ΩC → 0, this system reduces to the ordinary two-level atomic system. In the following we discuss the numerical results for OB in the three-level Λ-system with SGC included. In Fig. 3 we plot the input–output field characteristics for different SGC (defined by the parameter η) with other parameters γ2 /γ1 = 1, ΩC /γ1 = 5, ∆C /γ1 = 1.0, ∆P /γ1 = 2.0, and C = 100. The definition of the cooperativity parameter C for atoms in a ring cavity is similar to what we have for the twolevel atom case [9], e.g., C = αL/2T where αL is single-pass absorption by atomic medium. Curves A, B, C, and D are for η = 0 (no quantum interference), η = 0.5, η = 0.8, and η = 0.95 (large quantum interference), respectively. Clearly, the quantum interference reduces the bistabilty threshold (the point where transition to upper branch takes place, see curves A to C), which can be easily explained by reduction
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Fig. 3. The input–output field characteristics of the optical cavity field for different values of quantum interference. The parametric conditions are γ2 /γ1 = 1, ΩC /γ1 = 5, ∆C /γ1 = 1, ∆P /γ1 = 2, and C = 100. Curves A, B, C and D are for η = 0.0, 0.5, 0.8, and 0.95, respectively.
Fig. 4. The input–output field characteristics of the optical cavity field for different values of coupling field strengths. The parametric conditions are γ2 /γ1 = 1, ∆C /γ1 = 1, ∆P /γ1 = 2, η = 0.95, and C = 100. Curves A, B, C and D are for ΩC /γ1 = 5, 4, 3, and 2.5, respectively.
in effective saturation intensity since quantum interference suppresses the radiative decay rate from |2 to |1. In curve D, we observe optical multistability (OM) in this system. The three-level atomic system has an advantage over the two-level one because of the additional controllability offered by the coupling field strength and its frequency detuning. By adjusting these parameters along with SGC, one can alter the absorption and nonlinear optical properties of the atomic medium for the cavity field and, therefore, change the
steady-state behaviors. Also, for the two-level atoms the atomic polarization responsible for OB is a ratio of polynomials of first order in ΩP (in the numerator) and second order in ΩP (in the denominator). However, the order of these polynomials can go higher (order 5 in numerator and order 6 in denominator) for three-level atoms depending on the relative strengths of various parameters associated with the three-level atoms [11,26]. The observed OM has certainly the roots in this complicated form of polarization P (ω1 ) in terms of the probe field amplitude ΩP . We have confirmed it in Fig. 4, where we see the sensitivity of observed OM on the coupling field strength. We keep the parameters in Fig. 4 as γ2 /γ1 = 1, ∆C /γ1 = 1.0, ∆P /γ1 = 2.0, η = 0.95, and C = 100. Curves A, B, C, and D are for ΩC /γ1 = 5, 4, 3, and 2.5, respectively. Clearly, with reducing the coupling field strength we find disappearance of the OM as well increase in the threshold of the OB. Note that, in the earlier works [11] it has been shown that the OB in a Λ-type threelevel atomic system can be controlled using either a coherent control field or with the initial coherence of the lower two levels. Here we have not used these mechanism but rather used an alternative mechanism of SGC to control the threshold of OB. Similar studies have been recently carried out in the collection of three-level atomic system in a V configuration [27]. In summary, we have demonstrated the controllability of atomic OB by using the theoretical model of three-level atoms in Λ-configuration inside an optical ring cavity. The controlling parameters is the SGC in the decay channels whose tunability has been experimentally demonstrated. The possibilities of obtaining OM is also discussed, by SGC in the system.
Acknowledgements We acknowledge the funding supports from the National Science Foundation and the Office of Naval Research.
References [1] D.A. Cardimona, M.G. Raymer, C.R. Stroud Jr., J. Phys. B 15 (1982) 55. [2] A. Imamoglu, Phys. Rev. A 40 (1989) 2835;
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
[3] [4] [5] [6] [7] [8] [9]
[10]
[11]
[12]
[13]
M.O. Scully, S.Y. Zhu, A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813; S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A 52 (1995) 710; G.C. Hegerfeldt, M.B. Plenio, Phys. Rev. A 46 (1992) 373. P. Zhou, S. Swain, Phys. Rev. Lett. 77 (1996) 3995; P. Zhou, S. Swain, Phys. Rev. Lett. 78 (1997) 832. J. Javanainen, Europhys. Lett. 17 (1992) 407. H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 (1996) 1032. S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033; S.E. Harris, J.J. Macklin, Phys. Rev. A 40 (1989) 4135. E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, Vol. 35, North-Holland, Amsterdam, 1996. S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) 4014. See, for example, a review by L.A. Ligiato, Theory of optical bistability, in: E. Wolf (Ed.), Progress in Optics, Vol. 21, NorthHolland, Amsterdam, 1984, p. 71, and references therein including of; H.J. Gibbs, S.L. McCall, T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. A.T. Rosenberger, L.A. Orozco, H.J. Kimble, Phys. Rev. A 28 (1983) 2569; L.A. Orozco, et al., Phys. Rev. A 39 (1989) 1235. W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812; S. Gong, S. Du, Z. Xu, Phys. Lett. A 226 (1997) 293. Some earlier interesting work on the three-level atomic bistability by D.F. Walls, P. Zoller, Opt. Commun. 34 (1980) 260; D.F. Walls, P. Zoller, M.L. Steyn-Ross, IEEE J. Quantum Electron. QE-17 (1981) 380. H. Wang, D.J. Goorskey, M. Xiao, Phys. Rev. A 65 (2002) 011801(R); A. Joshi, A. Brown, H. Wang, M. Xiao, Phys. Rev. A 67 (2003) 041801(R).
207
[14] G. Giusfredi, P. Salieri, S. Cecchi, F.T. Arecchi, Opt. Commun. 54 (1985) 39; E. Giacobino, Opt. Commun. 56 (1985) 249; M.W. Hamilton, et al., Opt. Commun. 48 (1983) 190; F. Mitschke, R. Deserno, W. Lange, J. Mlynek, Phys. Rev. A 33 (1986) 3219; J. Nalik, W. Lange, F. Mitschke, Appl. Phys. B 49 (1989) 191; J. Mlynek, F. Mitschke, R. Deserno, W. Lange, Phys. Rev. A 29 (1984) 1297. [15] M. Kitano, T. Yabuzaki, T. Ogawa, Phys. Rev. Lett. 46 (1981) 926. [16] S. Cecchi, G. Giusfredi, E. Petriella, P. Salieri, Phys. Rev. Lett. 49 (1982) 1928. [17] C.M. Savage, H.J. Carmichael, D.F. Walls, Opt. Commun. 42 (1982) 211. [18] F.T. Arecchi, J. Kurmann, A. Politi, Opt. Commun. 44 (1983) 421. [19] P. Zhou, Phys. Rev. A 63 (2001) 023810. [20] L. Li, et al., Phys. Rev. Lett. 84 (2000) 4016. [21] P. Zhou, S. Swain, L. You, Phys. Rev. A 63 (2001) 033818. [22] M.O. Scully, et al., Phys. Rev. Lett. 62 (1989) 2813; M. Fleischhauer, et al., Opt. Commun. 94 (1992) 599; S.Y. Zhu, et al., Phys. Rev. A 52 (1995) 4791; E. Paspalakis, et al., Phys. Rev. A 58 (1998) 4868. [23] A.K. Patnaik, G.S. Agarwal, Phys. Rev. A 59 (1999) 3015. [24] G.S. Agarwal, Phys. Rev. Lett. 84 (2000) 5500. [25] J. Gea-Banacloche, Y. Li, S. Jin, M. Xiao, Phys. Rev. A 51 (1995) 576. [26] U. Rathe, M. Fleischhauer, S.Y. Zhu, T.W. Hansch, M.O. Scully, Phys. Rev. A 47 (1993) 4994. [27] A. Joshi, et al., Phys. Rev. A (2003), in press.
Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system Amitabh Joshi ∗ , Wenge Yang, Min Xiao Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA Received 4 June 2003; received in revised form 4 July 2003; accepted 9 July 2003 Communicated by P.R. Holland
Abstract Optical bistability is studied for a three-level atomic system in Λ-configuration contained in an optical ring cavity and the effects of spontaneously generated coherence in the presence of two arbitrary coherent fields are investigated. Condition for observing optical multistable behavior in this system has also been specified. 2003 Elsevier B.V. All rights reserved. PACS: 42.65.Pc; 42.65.Sf; 42.50.Gy
In recent years some new types of coherence produced due to decays of closely lying states have been discussed [1–5]. One of the main phenomena associated with these coherences is the modification of the line shapes of the spectral lines. In Λ-type three-level atomic system the spontaneously generated coherence (SGC) due to interaction with the vacuum bath of the radiation field has been discussed by Javanainen [4]. The important finding in the study of Ref. [4] is the disappearance of the dark state due to SGC. The SGC is very sensitive to the alignment of dipole moments of two transitions with respect to each other. The effects of SGC on electromagnetically induced transparency (EIT) [6] and coherent population trapping [7] phenomena were also examined [8]. Suppression of spontaneous emission [2] and subnatural linewidths [3] due
* Corresponding author.
E-mail address: [email protected] (A. Joshi). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)01046-6
to SGC were predicted in the three-level atomic systems. Optical bistability (OB) was extensively studied initially in the two-level system comprising of alkali atomic beams inside an optical resonator where a single mode beam circulating [9,10]. The experimental demonstration of OB in this system gave a great thrust to many potential applications such as optical transistors, memory elements, and all optical switches. This led to further investigation of OB in three-level atomic system inside an optical cavity both theoretically [11,12] as well as experimentally [13]. Multistable (multiple hysteresis) behaviors has also been observed in Fabry–Perot cavities filled with atoms having several degenerate or nearly degenerate sublevels in the ground state and driven by linearly polarized light [14]. Most of these experiments used magnetic fields and high pressure buffer gases, and relied on Zeeman coherence as an efficient mechanism for the observed behavior. More specifically, optical trista-
204
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
ρ˙11 = 2γ1 ρ22 − iΩP (ρ12 − ρ21 ), ρ˙22 = −2(γ1 + γ2 )ρ22 + iΩP (ρ12 − ρ21 ) + iΩC (ρ32 − ρ23 ), ρ˙33 = 2γ2 ρ22 − iΩC (ρ32 − ρ23 ), ρ˙21 = −(γ1 + γ2 − i∆P )ρ21 + iΩC ρ31 + iΩP (ρ11 − ρ22 ), ρ˙23 = −(γ1 + γ2 − i∆C )ρ23 + iΩP ρ13 − iΩC (ρ22 − ρ33 ), Fig. 1. Schematics of a three-level Λ-type atom.
bility in a three-level Λ-configuration was calculated under different conditions and also observed experimentally [15–18]. In this Letter, we deal with the role of SGC on OB in a three-level Λ-type atomic system contained in a ring optical cavity. The atomic system consists of two lower sublevels of same parity and a single excited level of different parity. The effect of quantum interference in spontaneous emission from the upper level to the lower two levels is included in the model. Such a model with quantum interference from spontaneous emission shows interference-assisted population inversion, trapping and probe gain at one sideband of the Autler–Townes spectrum [19]. However, the previous theoretical works [11,12] studying OB behavior in three-level atomic systems did not include such quantum interference in the decay channels or SGC in their models. We expect to see some novel changes in the usual OB brought out by the inclusion of this SGC in the three-level atomic system under consideration. The three-level atomic system considered here is shown in Fig. 1. It is a closed Λ-type configuration with one excited state |2 and two closely-lying lower states |1 and |3. The transition between |2 and |1 (with resonant frequency ω21 ) is mediated by the probe laser field EP (frequency ω1 ) while the transition |2 to |3 (with resonant frequency ω23 ) is driven by another laser field EC (frequency ω2 ) called coupling field in this work. The atomic dynamics of the system can be described by the Liouville equation for the density operator and the density matrix equations [4,8] with all decay terms included under rotating-wave approximation are
ρ˙13 = −i(∆P − ∆C )ρ13 − iΩC ρ12 √ + iΩP ρ23 + 2 γ1 γ2 ηρ22 .
(1)
Here, the atomic detunings are defined as ∆P = ω21 − ω1 , ∆C = ω23 − ω2 , respectively, and the Rabi frequencies for the probe and coupling fields are ΩP = d 12 .E P /h¯ and ΩC = d 32 .E C /h¯ , respectively. The transition dipole moments associated with the two transitions, e.g., d 12 and d 32 , can be nonorthogonal. We define a parameter η=
d 12.d 32 ≡ cos(θ ) |d 12 |.|d 32|
which is a measure of orthogonality of the dipole moments, i.e., η = 0 for θ = π/2. Physically, the √ term η γ1 γ2 accounts for the spontaneous emission induced quantum interference effect due to coupling between emission processes in the channels |2 → |1 and |2 → |3. In a recent experiment the ability of controlling η has been experimentally demonstrated [5] in sodium dimers by considering the superposition of singlet and triplet states due to spin-orbit coupling. However, a conflicting result was obtained in another experiment of similar kind [20]. It seems there are practical difficulties to find a perfect atomic system where dipole moments are parallel so that quantum interference due to SGC is maximum [21]. Recently, some novel methods are discussed in the literature [22–24] generating the quantum interference effects in more effective manners such as use of microwave field in a three-level system [22], multilevel schemes [23], or use of anisotropy of the electromagnetic vacuum [24]. The Rabi frequencies in our model are also functions of the angle θ , however, for the sake of convenience in comparison with different values of θ , the Rabi frequencies are kept unchanged by suitably adjusting the field strengths in this work.
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
tions in this configuration are √ EPT (t) = T EP (L, t), √ EP (0, t) = T EPI (t) + REP (L, t − t).
Fig. 2. Schematic diagram of a unidirectional ring cavity having four mirrors (M1–M4) and an atomic vapor cell of length L. The mirrors M3 and M4 are perfectly reflecting mirrors (R = 1 for each). The incident and the transmitted fields are represented by EPI and EPT , respectively, and the coupling field EC is noncirculating in the cavity.
We can study OB in this system by considering N such Λ-type atoms contained in a unidirectional optical ring resonator depicted in Fig. 2. The intensity reflection and transmission coefficients of mirrors M1 and M2 are R and T , respectively, such that R + T = 1. Further we assume that both the mirrors M3 and M4 are perfect reflectors. This is one of the standard model for studying OB in the literature [9]. The total electromagnetic field seen by N homogeneously-broadened atoms contained in a cell of length L is E = E P exp(−iω1 t) + E C exp(−iω2 t) + c.c.
(2)
The probe field E P is circulating in the ring cavity but the coupling field E C does not. So, the Maxwell’s equation, under slowly-varying envelope approximation is appropriate to describe the dynamics of the probe field in the optical cavity, ∂EP ∂EP +c = 2πiω1 d12 P (ω1 ), ∂t ∂z
(3)
where P (ω1 ) is the induced polarization in the transition |1 → |2 and is given by P (ω1 ) = Nd12 ρ12 . The probe field EPI enters the cavity through mirror M1, propagates in the cavity to interact with the atomic sample of length L, then circulates in the cavity, and partially transmits out from the mirror M2 as EPT . The probe field at the start of the atomic sample is EP (0) and propagates to the end of the atomic sample to be EP (L, t) in a single pass. The field boundary condi-
205
(4)
Time taken for light to travel from M2 to M1 is t and we keep cavity detuning to be zero in this study. The bistable behavior is due to the nonlinearity of the atomic medium and resonator feedback mechanism. If we set R = 0 in Eq. (4) above, we do not get any bistability. We do not consider the circulation of the coupling beam in the optical cavity and hence allow this beam to enter in the cavity via a polarizing beam splitter such that it copropagates with the probe beam to eliminate the first order Doppler effect for the three-level atoms in the vapor cell [25] as described experimentally in Ref. [13]. The size of the coupling beam is larger in comparison to the probe beam in the vapor cell so that there is a good overlap. It is rather difficult to have a closed form expressions for OB for this problem due to the complexity of the dynamic equations (1)–(3). Also, it is cumbersome to put P (ω1 ) in a simple analytic form in steady state for the three-level atomic system in comparison to its two-level counterpart. Therefore, we solve the density matrix equations (1) numerically and integrate the Maxwell’s equation (3) in the steady-state limit over the length of the sample together with the boundary conditions (4) to get the results for OB under various parametric conditions. It should be noticed that in the limiting condition of ΩC → 0, this system reduces to the ordinary two-level atomic system. In the following we discuss the numerical results for OB in the three-level Λ-system with SGC included. In Fig. 3 we plot the input–output field characteristics for different SGC (defined by the parameter η) with other parameters γ2 /γ1 = 1, ΩC /γ1 = 5, ∆C /γ1 = 1.0, ∆P /γ1 = 2.0, and C = 100. The definition of the cooperativity parameter C for atoms in a ring cavity is similar to what we have for the twolevel atom case [9], e.g., C = αL/2T where αL is single-pass absorption by atomic medium. Curves A, B, C, and D are for η = 0 (no quantum interference), η = 0.5, η = 0.8, and η = 0.95 (large quantum interference), respectively. Clearly, the quantum interference reduces the bistabilty threshold (the point where transition to upper branch takes place, see curves A to C), which can be easily explained by reduction
206
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
Fig. 3. The input–output field characteristics of the optical cavity field for different values of quantum interference. The parametric conditions are γ2 /γ1 = 1, ΩC /γ1 = 5, ∆C /γ1 = 1, ∆P /γ1 = 2, and C = 100. Curves A, B, C and D are for η = 0.0, 0.5, 0.8, and 0.95, respectively.
Fig. 4. The input–output field characteristics of the optical cavity field for different values of coupling field strengths. The parametric conditions are γ2 /γ1 = 1, ∆C /γ1 = 1, ∆P /γ1 = 2, η = 0.95, and C = 100. Curves A, B, C and D are for ΩC /γ1 = 5, 4, 3, and 2.5, respectively.
in effective saturation intensity since quantum interference suppresses the radiative decay rate from |2 to |1. In curve D, we observe optical multistability (OM) in this system. The three-level atomic system has an advantage over the two-level one because of the additional controllability offered by the coupling field strength and its frequency detuning. By adjusting these parameters along with SGC, one can alter the absorption and nonlinear optical properties of the atomic medium for the cavity field and, therefore, change the
steady-state behaviors. Also, for the two-level atoms the atomic polarization responsible for OB is a ratio of polynomials of first order in ΩP (in the numerator) and second order in ΩP (in the denominator). However, the order of these polynomials can go higher (order 5 in numerator and order 6 in denominator) for three-level atoms depending on the relative strengths of various parameters associated with the three-level atoms [11,26]. The observed OM has certainly the roots in this complicated form of polarization P (ω1 ) in terms of the probe field amplitude ΩP . We have confirmed it in Fig. 4, where we see the sensitivity of observed OM on the coupling field strength. We keep the parameters in Fig. 4 as γ2 /γ1 = 1, ∆C /γ1 = 1.0, ∆P /γ1 = 2.0, η = 0.95, and C = 100. Curves A, B, C, and D are for ΩC /γ1 = 5, 4, 3, and 2.5, respectively. Clearly, with reducing the coupling field strength we find disappearance of the OM as well increase in the threshold of the OB. Note that, in the earlier works [11] it has been shown that the OB in a Λ-type threelevel atomic system can be controlled using either a coherent control field or with the initial coherence of the lower two levels. Here we have not used these mechanism but rather used an alternative mechanism of SGC to control the threshold of OB. Similar studies have been recently carried out in the collection of three-level atomic system in a V configuration [27]. In summary, we have demonstrated the controllability of atomic OB by using the theoretical model of three-level atoms in Λ-configuration inside an optical ring cavity. The controlling parameters is the SGC in the decay channels whose tunability has been experimentally demonstrated. The possibilities of obtaining OM is also discussed, by SGC in the system.
Acknowledgements We acknowledge the funding supports from the National Science Foundation and the Office of Naval Research.
References [1] D.A. Cardimona, M.G. Raymer, C.R. Stroud Jr., J. Phys. B 15 (1982) 55. [2] A. Imamoglu, Phys. Rev. A 40 (1989) 2835;
A. Joshi et al. / Physics Letters A 315 (2003) 203–207
[3] [4] [5] [6] [7] [8] [9]
[10]
[11]
[12]
[13]
M.O. Scully, S.Y. Zhu, A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813; S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A 52 (1995) 710; G.C. Hegerfeldt, M.B. Plenio, Phys. Rev. A 46 (1992) 373. P. Zhou, S. Swain, Phys. Rev. Lett. 77 (1996) 3995; P. Zhou, S. Swain, Phys. Rev. Lett. 78 (1997) 832. J. Javanainen, Europhys. Lett. 17 (1992) 407. H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 (1996) 1032. S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033; S.E. Harris, J.J. Macklin, Phys. Rev. A 40 (1989) 4135. E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, Vol. 35, North-Holland, Amsterdam, 1996. S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) 4014. See, for example, a review by L.A. Ligiato, Theory of optical bistability, in: E. Wolf (Ed.), Progress in Optics, Vol. 21, NorthHolland, Amsterdam, 1984, p. 71, and references therein including of; H.J. Gibbs, S.L. McCall, T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. A.T. Rosenberger, L.A. Orozco, H.J. Kimble, Phys. Rev. A 28 (1983) 2569; L.A. Orozco, et al., Phys. Rev. A 39 (1989) 1235. W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812; S. Gong, S. Du, Z. Xu, Phys. Lett. A 226 (1997) 293. Some earlier interesting work on the three-level atomic bistability by D.F. Walls, P. Zoller, Opt. Commun. 34 (1980) 260; D.F. Walls, P. Zoller, M.L. Steyn-Ross, IEEE J. Quantum Electron. QE-17 (1981) 380. H. Wang, D.J. Goorskey, M. Xiao, Phys. Rev. A 65 (2002) 011801(R); A. Joshi, A. Brown, H. Wang, M. Xiao, Phys. Rev. A 67 (2003) 041801(R).
207
[14] G. Giusfredi, P. Salieri, S. Cecchi, F.T. Arecchi, Opt. Commun. 54 (1985) 39; E. Giacobino, Opt. Commun. 56 (1985) 249; M.W. Hamilton, et al., Opt. Commun. 48 (1983) 190; F. Mitschke, R. Deserno, W. Lange, J. Mlynek, Phys. Rev. A 33 (1986) 3219; J. Nalik, W. Lange, F. Mitschke, Appl. Phys. B 49 (1989) 191; J. Mlynek, F. Mitschke, R. Deserno, W. Lange, Phys. Rev. A 29 (1984) 1297. [15] M. Kitano, T. Yabuzaki, T. Ogawa, Phys. Rev. Lett. 46 (1981) 926. [16] S. Cecchi, G. Giusfredi, E. Petriella, P. Salieri, Phys. Rev. Lett. 49 (1982) 1928. [17] C.M. Savage, H.J. Carmichael, D.F. Walls, Opt. Commun. 42 (1982) 211. [18] F.T. Arecchi, J. Kurmann, A. Politi, Opt. Commun. 44 (1983) 421. [19] P. Zhou, Phys. Rev. A 63 (2001) 023810. [20] L. Li, et al., Phys. Rev. Lett. 84 (2000) 4016. [21] P. Zhou, S. Swain, L. You, Phys. Rev. A 63 (2001) 033818. [22] M.O. Scully, et al., Phys. Rev. Lett. 62 (1989) 2813; M. Fleischhauer, et al., Opt. Commun. 94 (1992) 599; S.Y. Zhu, et al., Phys. Rev. A 52 (1995) 4791; E. Paspalakis, et al., Phys. Rev. A 58 (1998) 4868. [23] A.K. Patnaik, G.S. Agarwal, Phys. Rev. A 59 (1999) 3015. [24] G.S. Agarwal, Phys. Rev. Lett. 84 (2000) 5500. [25] J. Gea-Banacloche, Y. Li, S. Jin, M. Xiao, Phys. Rev. A 51 (1995) 576. [26] U. Rathe, M. Fleischhauer, S.Y. Zhu, T.W. Hansch, M.O. Scully, Phys. Rev. A 47 (1993) 4994. [27] A. Joshi, et al., Phys. Rev. A (2003), in press.