Abstract

The optical bistability (OB) and multistability (OM) in chiral molecules are investigated by placing the sample into a unidirectional ring cavity. Because of broken mirror symmetry of the effective potential, the chiral molecules have a cyclic three-level Δ-configuration structure, in which one- and two-photon transitions can coexist. We find that the OB is achievable in this system on exact one-, two- and three-photon resonance conditions but absent in the three-level Λ-type system under the two-photon resonance. Moreover, the OM and the switching between OB and OM are also realized by choosing parameters properly. Interestingly, the left- and right-handed chiral molecules exhibit different bistable and multistable behaviors. It is shown that the threshold intensity of OB is strongly dependent on the percentage of the two enantiomers in the mixture. This provides an effective approach to probe molecular chirality and to determine enantiomer excess, which may find potential application in organic chemistry, pharmacology, biochemistry, etc..

© 2016 Optical Society of America

1. Introduction

Quantum interference and atomic coherence effects play a crucial role in the most important quantum optics phenomena such as coherent population trapping (CPT) [1], electromagnetically induced transparency (EIT) [2–7], optical bistability (OB) and multistability (OM), and so on. In past years, there has been continuous interest in OB and OM because of their important applications in optical transistors, memory elements and all optical switches. A large number of schemes including natural atomic systems, quantum well structures and quantum dot molecules were proposed to realize OB and OM both theoretically and experimentally [8–30]. The earlier experimental observations of OB were reported in a two-level alkali atomic system [8–10]. In comparison with two-level system, the OB and OM showed different behaviors in three level systems since the atomic coherence effects can greatly modify the absorption, dispersion and nonlinearity of the systems [11, 12]. Harshawardhan and Agarwal proposed a scheme to demonstrate OB using field-induced atomic coherence and quantum interference effects [13], wherein the threshold intensity was rather small. Gong et al. showed that the OB was strongly influenced by the initial atomic coherence in a ground-state doublet three-level system [14]. In addition, by applying trichromatic fields to drive atomic system, the OM as well as the switching between OB and OM can be effectively controlled through modifying the relative intensities and phases of the fields [15]. The experimental realization of OB and OM in the three-level system has been studied extensively as well [16–18]. On the other hand, several groups have taken into account the spontaneously generated coherence or vacuum induced coherence effects on OB and OM behaviors in the three-level V-type, Λ-type and ladder-type system [19–22]. It is found that the linear and nonlinear response of the probe field are heavily dependent on the quantum interference, resulting in the occurrence of bistable and multistable behaviors. However, we note that the quantum interference arises from the coupling of two transitions with parallel moments to the same vacuum mode [23]. The parallel dipole moments are difficult to find in realistic system because it requires the same quantum numbers J and mJ.

To overcome this difficult, one usually employs a DC field or a microwave field to couple to the near-degenerate states. In this way, the quantum interference effects are simulated equivalently in the dressed-state representation. Henceforth, the closed three-level atomic structure could be formed by using a microwave field to couple with the near-degenerate states in Λ-type or V-type system. The OB can be effectively controlled by modifying the amplitude and the relative phase of the microwave fields in the three-level closed systems [24, 25]. In addition to the above mentioned schemes, the multilevel atomic systems, the quantum well structures and the quantum dot molecules have also been investigated widely to obtain OB and OM based on coherence effects [26–30]. Not only that, the Kerr nonlinearity, which is closely related to the OB/OM behaviors, has also been paid special attention in past and recent years [31–34]. Physically, the OB and OM are very sensitive to the coherence-controlled absorption-dispersion relation and the nonlinear interactions of lasers with different types of systems. Taking the three-level Λ-type system as an example, if the system is tuned to be away from two-photon resonance slightly, the OB or OM are obtainable because of the absorption and large nonlinearity experienced by the probe [17, 18]. Under the exact two-photon resonance, however, the system would evolve into dark-state and the linear absorption and the higher order nonlinear effects would vanish, leading to the absence of OB and OM.

Recently, an interesting cyclic structure in chiral molecules [35, 36] has been paid special attention, in which the coherence-related effects such as CPT and EIT were reported [37–39]. Because of broken mirror symmetry of the effective potential, the chiral molecules have a three-level Δ-configuration structure with three simultaneous transitions between the lowest three eigenstates states, thereby rendering the one- and two-photon processes to coexist [35, 36]. This feature makes the optical properties of the cyclic structure being different from other three-level systems. In general, the chiral molecules are divided into left- and right-handed chiral molecules (called “enantiomers”), which are coexistent because of the fundamental broken symmetries in nature [40, 41]. It is worthwhile to note that only one enantiomeric form is useful in biology, while the other one could be harmful or fatal. Consequently, how to distinguish the left- and right-hand chiral molecules is of great interest in organic chemistry, pharmacology, biochemistry, etc.. Král et al. [35] proposed a method to discriminate the left- and right-handed chiral molecules through a “cyclic population transfer” process. Li et al. [42] and Jia et al. [43] showed respectively that left- and right-handed chiral molecules can evolve into different final states by a purely dynamic transfer process or using two-step coherent pulses regardless of decay. Furthermore, several optical methods were suggested to probe effectively molecule chirality based on optical nonlinear processes [44–46]. Subsequently, a different approach to probe the chirality is provided by measuring their linear absorption spectra [47]. These results display that the absorption, dispersion and nonlinear effects in Δ-type molecule system have novel features. This motivates us to study the OB and OM behaviors in this special system. Our numerical results show that the OB and OM, as expected, are attained via quantum coherence effect in the cyclic structure under proper conditions. More interestingly, the OB and OM behaviors of the left- and right-handed molecules are different from each other. For a sample consisting of the two enantiomers, the threshold intensity varies monotonically as the ratio of the mixture changes. In this regard, via studying the bistable behaviors of the mixture, we provide a novel approach to identify molecular chirality and to determine enantiomer excess. The remainder of this paper is organized as follows. In section 2, the model is described and the master equations of the chiral molecules are derived. In section 3, the numerical results of OB and OM are given and the corresponding mechanisms are analyzed briefly. Section 4 ends with conclusion.

2. Model and equations

It has been demonstrated that a pair of chiral molecules can be treated as a system with mirror-symmetric double potential [35, 36]. When the mirror symmetry of the effective potential is broken, the three transitions between the lowest three localized chiral eigenstates are permitted simultaneously, leading to a cyclic transition structure [35]. As depicted in Fig. 1, the Δ-configuration chiral molecules have three levels |j〉 with transition frequencies ωjk and decay rates γjk between levels |j〉 and |k〉, where ωj are eigenfrequenies of the levels and ωjk = ωjωk for j,k = 1,2,3,(j > k). Here |jL and |jR represent the left- and right-handed chiral molecules, respectively. Three coherent driven fields are applied to three transitions |3〉 |1〉, |3〉 |2〉 and |2〉 |1〉. In electronic-dipole and rotating wave approximation, the Hamiltonian of the system is given as (h¯=1) [47]

H=j=13ωjσjj12(j,k=1,j>k3Ωjkei(νjkt+ϕjk)σjk+H.c.),
where Ωjk = μjkEjk are assumed to be real Rabi frequencies of the applied coherent fields with the frequencies νjk and initial phases ϕjk. μjk represent the electric-dipole moments. σjj = |j〉 〈j| and σjk = |j〉 〈k|, jk are the projection operators and spin-flip operators. In the appropriate rotating frame, we rewrite the system Hamiltonian as
HI=(Δ21σ22+Δ31σ33)12(Ω32σ32+Ω31σ31+Ω21σ21ei(δt+Φ)+H.c.),
where Δjk = νjk −ωjk, δ = Δ3221−Δ31, Φ = ϕ32 +ϕ21 ϕ31. We consider the three-photon resonance case, i.e., δ = 0 and assume that the strong field Ω21 is a control field and the two weak fields Ω31, Ω32 coupling with the two transitions |3〉 |1,2〉 are probe fields. The master equation describing the dynamical evolution of the system can be written as
ρ˙=i[HI,ρ]+[j,k=1,j>k3γjkσjk]ρ,
where ℒ [c] = c•c − {cc,•}/2. According to the density-operator master equation and the total Hamiltonian HI, we have the equations of motion of the density matrix elements
ρ˙33=(γ31+γ32)ρ33+i2Ω31(ρ13ρ31)+i2Ω32(ρ23ρ32),ρ˙22=γ21ρ22+γ32ρ33+i2Ω21(ρ12eiΦρ21eiΦ)+i2Ω32(ρ32ρ23),ρ˙12=Γ12ρ12+i2Ω21eiΦ(ρ22ρ11)+i2Ω31ρ32i2Ω32ρ13,ρ˙13=Γ13ρ13+i2Ω31(ρ33ρ11)+i2Ω21eiΦρ23i2Ω32ρ12,ρ˙23=Γ23ρ23+i2Ω21eiΦρ13+i2Ω32(ρ33ρ22)i2Ω31ρ21,
together with ρjk=ρkj*, (j, k = 1,2,3, jk). The closure of the system is ρ11+ρ22+ρ33 = 1. The damping rates of the off-diagonal elements are Γ12=12γ21+iΔ21, Γ13=12(γ21+γ31)+iΔ31, Γ23=12(γ21+γ31+γ32)+iΔ32. Particularly, the Rabi frequencies of the left-handed molecules are labeled as“Ω” while that of the right-handed molecules are “−Ω”, resulting in a π-difference of total phase factor [35, 42, 47]. As a result, the matrix elements ρjk of the left- and right-handed molecules are replaced by ρjk(+) and ρjk() with the relation of ρjk()=ρjk(+)(Φ+π).

 figure: Fig. 1

Fig. 1 The energy level structure of the Δ-type chiral molecules: (a) left-handed chiral molecules; (b) right-handed chiral molecules. The enantiomers are driven by three optical fields, wherein the transition |2〉L(|2〉R) |1〉L(1〉R) is driven by a strong control field while the other two transitions are coupled by two probe fields.

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In order to study the bistable behavior of the chiral molecules, we place the sample consisting of a mixture of left- and right-handed molecules into a unidirectional ring cavity as shown in Fig. 2, in which the reflectivity(transmissivity) of mirrors M1 and M2 are R and T (R + T = 1), while the mirrors M3 and M4 are perfect reflectors (R = 1). Adopting a standard method, the total electromagnetic field can be written as

E=E31eiν31t+E32eiν32t+E21eiν21t+c.c.,
wherein E31 = E32 = Ep are the amplitude of the probe fields, which circulate in the ring cavity and E21 is the amplitude of the control field denoted by Ec that does not circulate in the ring cavity. The symbol c.c. means the complex conjugation. Applying a slowly varying envelope approximation, the dynamic response of the probe fields is described by Maxwell’s equation
Ept+cEpz=i2ε0[ν31P(ν31)+ν32P(ν32)],
where c and ε0 are the speed of light and permittivity of free space, respectively. P(νjk) are the slowly varying polarization in the transitions |3〉 |1〉 and |3〉 |2〉, respectively, and given by P(νjk)=Nμjk(η(+)ρjk(+)+η()ρjk()), where N is the total molecular density, η(±) are the percentage of the left- and right-handed molecules in the mixture and η(+) +η(−) = 1.

 figure: Fig. 2

Fig. 2 Unidirectional ring cavity with a chiral molecule sample of length L. EpI and EpT are the incident and transmitted fields, respectively. Ec is the amplitude of the control field.

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We consider the field in equation (6) in the steady case, i.e., ∂Ep/∂t = 0, the Maxwell equation is reduced to

Epz=iN2cε0[ν31μ31ρ˜31+ν32μ32ρ˜32],
with ρ˜31=η(+)ρ31(+)+η()ρ31(), ρ˜32=η(+)ρ32(+)+η()ρ32(). For a perfectly tuned cavity, in the steady state limit, the boundary conditions between the incident field EpI and the transmitted field EpI are given as [48]
Ep(L)=EpI/T,
Ep(0)=TEpI+REp(L),
where L is the length of the molecule sample. The R term, which describes the feedback mechanism, is essential to the occurrence of bistability. Once R = 0, the bistability will not be existent any more. Using the boundary conditions (8)(9), normalizing the fields by letting Y=(μ31+μ32)EpIT and X=(μ31+μ32)EpT2T, we can get an input-output relationship
Y=2Xiγ(C1ρ˜31+C2ρ˜32).

The cooperative parameters are defined as C1=NLν31μ3122ε0cTγ and C2=NLν32μ3222ε0cTγ with γ31 = γ32 = γ. It should be pointed out that the coherence terms ρ˜31 and ρ˜32 in equation (7) are vital for OB and OM to occur. By setting all time derivatives of Eq. (4) to zero, we can obtain the steady values of ρ31 and ρ32. Arising from Eq. (4) and Eq. (10), we can study the bistable and multistable behaviors in this system.

3. Numerical results and discussions

In the following numerical calculation, for simplicity, we assume that γ31 = γ32 = γ and C1 = C2 = C, Ω21 = Ωc, Ω31 = Ω21 = Ωp and the relative phase Φ = 0. The Rabi frequencies, the detunings and the decaying rates are always scaled in units of γ throughout the paper. Our numerical results demonstrate that the OB and OM are possible to realize in the chiral molecules under different conditions. The range of the hysteresis cycle, the threshold value and the switching between OB and OM curves are controllable by adjusting effectively the detunings, the amplitude of the control field, and the percentages of the two enantiomers appropriately.

We first consider the bistable behaviors of left- and right-handed molecules separately. In Fig. 3, the input-output curves for (a) left-handed molecules (η(+) = 1, η(−) = 0) and (b) right-handed molecules (η(+) = 0, η(−) = 1) are plotted under the two-photon resonance conditions, i.e., Δ31 = Δ32 = Δ. The probe absorption Im(ρ31 +ρ32) are numerically plotted in Fig. 3(c). The parameters are chosen as γ21 = γ, Ωc = 4γ, C = 100, Δ21 = 0. From these figures, it can be seen that the OB is obtainable in the two enantiomers when the two-photon resonance conditions are satisfied. In particular, under the exact one-, two- and three-photon resonance conditions, i.e., Δ = 0, the bistable curves of the left- and right-handed molecules are the same since the absorption is equal to each other at this time. As the detunings change from 0 to γ and 2γ, the threshold of the left-handed molecules (Fig. 3(a)) increases while that of the right-handed molecules (Fig. 3(b)) decreases. This can be understood from Fig. 3(c), wherein the absorption of the left-handed molecules first rises to a maximal value and then falls in the region of Δ > 0. For the righted-handed chiral molecules, the absorption curve is symmetrical about Δ = 0 with the left-handed chiral molecules. If we choose Δ < 0, the variation process of the threshold value for the two enantiomers would be reversed. As a matter of fact, when the control field Ωc is much stronger than the two probe fields, the analytical expressions of the linear probe response ρ31(1) and ρ32(1) can be solved by using the standard perturbation theory. Due to Ωc≫Ωp, by taking Δ21 = 0, we have ρ22s=0,, ρ22s=Ω2122Ω212+γ212, ρ11s=Ω212+γ2122Ω212+γ212. Then the linear probe response is derived from Eq. (4) as

ρ31(1)=2iΩpΓ32ρ11s+ΩpΩcρ22seiΦ4Γ31Γ32+Ωc2,ρ32(1)=2iΩpΓ31ρ22s+ΩpΩcρ11seiΦ4Γ31Γ32+Ωc2,
where Γ31=Γ13*,Γ32=Γ23*. In terms of the above expression, we can find that the probe absorption Im(ρ31(1)+ρ32(1)) is obviously phase dependent. At Φ = 0, it corresponds to the left-handed molecules while Φ = π corresponds to the right-handed molecules owing to the internal π-difference of phase for the two types of molecules [35, 42, 47]. Notably, the analytical result is approximately consistent with the numerical results plotted in Fig. 3(c) and Fig. 5(c, d). However, it is not applicable to the results shown in Fig. 4 because the perturbation theory is not valid at this time. Specially, the probe absorption under the exact resonance and non-exact resonance can be obtained from Eq. (11) as follows.
  1. For the exact resonant cases, i.e., Δ21 = Δ32 = Δ31 = 0, we have
    Im(ρ31(1)+ρ32(1))=Ωp[(γ31+γ32+γ21)ρ11s+(γ31+γ32)ρ22sΩc(ρ22sρ11s)sinΦ](γ31+γ32+γ21)(γ31+γ21)Ωc2
  2. For the non-exact resonant cases, i.e., Δ21 = 0,Δ32 = Δ31 = Δ, we have
    Im(ρ31(1)+ρ32(1))=ΩpAD+BA2+B2,
    with
    A=(γ31+γ32+γ21)(γ31+γ21)+Ωc24Δ2,B=2Δ(2γ31+γ32+γ21),=2Δ+ΩccosΦ,D=(γ31+γ32+γ21)ρ11s+(γ31+γ32)ρ11sΩc(ρ22sρ11s)sinΦ.
    From Eq. (12), it can be seen clearly that the probe absorption is identical for Φ = 0 and Φ = π, resulting in the same OB behaviors for left- and right-handed molecules. However, for the cases of Δ ≠ 0, the probe absorption for Φ = 0 and Φ = π is distinct from each other, which can be straightway deduced from Eq. (13) and Eq. (14).

 figure: Fig. 3

Fig. 3 Input-output field curves by choosing different detunings Δ under the two-photon resonance conditions for (a) left-handed chiral molecules, i.e., η(+) = 1,η(−) = 0; (b) right-handed chiral molecules, i.e., η(+) = 0,η(−) = 1. (c) Probe absorption Im(ρ31 +ρ32) of the two enantiomers as a function of the detuning Δ. The Other parameters are chosen as Ωc = 4γ,C = 100, γ21 = γ, Δ21 = 0.

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 figure: Fig. 4

Fig. 4 The probe absorption Im(ρ31 +ρ32) of left-handed molecules by choosing different decay rates γ21. The other parameters are chosen as Ωc = 0,C = 100,Δ31 = Δ32 = Δ,Δ21 = 0.

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The physical interpretation of the above results can be attributed to dark-state resonance effect. As is well known, in the three-level Λ-type atomic system, the light fields, if they satisfy two-photon resonance conditions, would drive the system into a dark state with vanishing absorption, leading to the absence of OB. In the present system, nevertheless, since the transition |2〉 |1〉 is permitted, the system would deviate away from dark state slightly although the two-photon resonance conditions are satisfied. The probe absorption is no longer equal to zero and then the bistable behaviors emerge. To verify that, the damping effects γ21 on the probe absorption under one- and two-photon resonance conditions are plotted in Fig. 4 by taking Ωc = 0. At γ21 = 0, corresponding to the three-level Λ-type system, the absorption is zero and the OB would be absence at this time. Once the transition channel between the two ground states exists, i.e., γ21 ≠ 0, the OB occurs and is enhanced as γ21 increases. In fact, the cyclic structure is equivalent approximately to a combination of three-level Λ-type and Ladder-type system under certain conditions [50], which can be attributed to the appearance of OB under exact resonance conditions.

In Fig. 5, the input-output curves for (a) left- and (b) right-handed chiral molecules are plotted by changing the intensity of the control field. The other parameters are the same as those in Fig. 3 except for Δ = 2γ. The probe absorption is also plotted in Fig. 5(c, d). With increasing Ωc from γ (solid line) to 5γ (dashed line), 10γ (dotted line), the threshold intensity and the range of the hysteresis cycle in Fig. 5(a) first increase and then decrease but in Fig. 5(b) they decrease always. From Fig. 5(c), it is easy to see that the absorption of the left-handed chiral molecules at Δ = 2γ is first enhanced and then reduced. Differently, shown in Fig. 5(d), the absorption of the right-handed molecules at Δ = 2γ decreases remarkably, resulting in the reduction of threshold intensity and the scope of hysteresis cycle.

 figure: Fig. 5

Fig. 5 (a,b) Input-output field curves by choosing different intensities of control field for left- and right-handed chiral molecules, respectively. (c, d) Probe absorption Im(ρ31 +ρ32) of the two enantiomers as a function of the detuning Δ by changing the strength of the control field. The other parameters are chosen as C = 100, γ21 = γ, Δ21 = 0.

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Next, we show the OM curves and the switching between OB and OM under different conditions. In Fig. 6, we plot the input-output curves for (a) left- and (b) right-handed chiral molecules by choosing the probe detunings as Δ32 = Δ31 = 0 (solid line), 0.2γ (dashed line), 0.4γ (dotted line). The other parameters are chosen as γ21 = 0.1γ, Ωc = 4γ, C = 100, Δ21 = Δ31 Δ32. It is found that the OM behaviors arise in the two enantiomers but the variation trends are different from each other. In Fig. 6(a), when the asymmetrical detunings are changed from 0 to 0.2γ, the OB switches to OM. By further increasing the detunings to 0.4γ, the threshold of the first hysteresis circle falls while the second one rises as the detunings increase. In comparison with left-handed chiral molecules, both the threshold intensities of the right-handed molecules shown in Fig. 6(b) are enhanced during this process. In addition, as shown in Fig. 7, the input-output curves are plotted by changing the strength of the control field. We choose the different Rabi frequencies as Ωc = γ (solid line), 5γ (dashed line), 10γ (dotted line). The other parameters are the same as those in Fig. 6 except for Δ32 = Δ31 = 0.2γ. Evidently, in spite of the variation trends of the curves for the two enantiomers are similar to each other in this case, the threshold values of OM in the left-handed chiral molecules (Fig. 7(a)) are smaller than that in the right-handed ones (Fig. 7(b)).

 figure: Fig. 6

Fig. 6 Input-output field curves by choosing different detunings Δ32 = −Δ31 = 0 (solid line), 0.2γ (dashed line), 0.4γ (dotted line) for (a) left-handed chiral molecules; (b) right-handed chiral molecules. The other parameters are chosen as γ21 = 0.1γ, Ωc = 4γ, C = 100, Δ21 = Δ31 Δ32.

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 figure: Fig. 7

Fig. 7 Input-output field curves by choosing different intensities of control field: Ωc = γ (solid line), 5γ (dashed line), 10γ (dotted line) for (a) left- and (b) right-handed chiral molecules, respectively. The other parameters are the same as those in Fig. 6 except for Δ32 = Δ31 = 0.2γ.

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In brief, we can draw a conclusion from the above discussion that the bistable and multistable behaviors of the left- and right-handed chiral molecules are different from each other. Nevertheless, the two enantiomers are usually in coexistence because of the fundamental broken symmetries in nature [40, 41]. Henceforth, it is of realistic interest to study the sample consisting of a mixture of left- and right-handed (Δ-type) molecules. In Fig. (8), the bistable curves are plotted by taking different ratio of the two enantiomers. The parameters are the same as those in Fig. 5 except for Ωc = 4γ. It can be seen that the threshold intensity of the OB becomes smaller as the ratio of left-handed chiral molecules η(+) falls. To make it more clearly, we plot the evolution of threshold intensity as a function of η+ for two cases (a) Δ > 0 and (b) Δ < 0 in Fig. 9. Remarkably, the threshold is monotonically enhanced in the region of Δ > 0 but decreases in the region of Δ < 0. The threshold intensity and the percentage satisfy one-to-one relation, which can be utilized to identify molecular chirality and to determine enantiomer excess.

 figure: Fig. 8

Fig. 8 Input-output field curves by varying the ratios of the two enantiomers. The parameters are chosen as C = 100,Ωc = 4γ31 = Δ32 = 2γ21 = 0,γ21 = γ.

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 figure: Fig. 9

Fig. 9 The threshold intensity versus the percentages η+ of the left-handed chiral molecules for two cases (a) Δ > 0; (b) Δ < 0. The other parameters are the same as those in Fig. 8

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Before ending this section, we would like to emphasize the main differences between the present scheme and the previous works. Firstly, the three dipole transitions between the lowest three eigenstates of the chiral molecules are permitted simultaneously [35, 36, 42]. This leads to the fact that the one- and two-photon processes are coexistent. The bistable behavior can be obtained under exact one-, two- and three-photon resonance conditions but it would be absent in the three-level Λ-type system under two-photon resonance. The transition channel between the two ground states plays an important role in the generation of OB. Secondly, it is explored that the bistable and multistable behaviors of the left- and right-handed molecules are different from each other. The threshold intensity and scope of the hysteresis loop are strongly dependent on the percentage of the two “enantiomers” in the mixture, which provides a novel way to probe left- and right-handed chiral molecules. Ultimately, it is well known that a similar cyclic transition structure can be found in the superconducting flux quantum circuits [49, 50]. However, one of the differences between the two systems is that the transition frequencies for the chiral molecules are in the optical frequency region while those for the superconducting system are in the microwave frequency domain. Besides, in general, merely a single superconducting atom mediates into the interaction in the 1D open space. When a large number of fluxonium qubits are put together, there is the inductively coupling between those fluxonium qubits. This would be harmful for the generation of OB in experiment. What is noteworthy is that the optical response in molecule system is relatively weak. In order to solve this problem in experiment, one has proposed a method by confining the molecules into a hollow-core photonic crystal fiber (HC-PCF) to enhance the light-matter interactions [51]. To do so, the typically quantum coherence effects, such as EIT [37, 38] and four-wave mixing [52], are realized. In addition, it has been reported that the EIT related coherence effects could be achievable in laser cooled molecule ensemble [53]. These pioneering works demonstrate that the experimental investigation of OB and OM in chiral molecules are implementable.

4. Conclusion

In summary, it is shown that the optical bistable and multistable behaviors are obtained based on the quantum coherence in three-level Δ-type chiral molecules. Since three transitions between the lowest three eigenstates of the chiral molecules are permitted simultaneously, the one- and two-photon processes can coexist. This gives rise to the distinctive quantum coherence effect being different from natural three-level atomic systems. By applying two probe fields and a control field to the cyclic structure, the OB is generated even under the exact one-, two-, and three-photon resonance conditions for both left- and right-handed chiral molecules. Moreover, the OM patterns and the switching between OB and OM are also realized by choosing the parameters properly. More importantly, it has been found that the two enantiomers display different OB and OM behaviors under the same conditions. For a sample consisting of the two enantiomers, the threshold intensity and scope of hysteresis loop increase or decrease as the percentages change in the mixture. This provides a different method to probe molecular chirality and to determine enantiomer excess.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11574179), the Natural Science Foundation of Hubei Province (Grant No. 2014CFC1148), the Natural Science Foundation of Shanghai (Grant No. 15ZR1430600), the Singapore Ministry of Education (Grant No. WBS: R-710-006-104-112), and the government of China through CSC.

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14. S. Gong, S. Du, and Z. Xu, “Optical bistability via atomic coherence,” Phys. Lett. A 226, 293–297 (1997). [CrossRef]  

15. X. M. Hu and J. Wang, “Sideband control of optical bistability and multistability,” Phys. Lett. A 365, 253–257 (2007). [CrossRef]  

16. H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001). [CrossRef]  

17. A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003). [CrossRef]   [PubMed]  

18. A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003). [CrossRef]  

19. M. A. Antón and O. G. Calderón, “Optical bistability using quantum interference in V -type atoms,” J. Opt. B 4, 91–98 (2002). [CrossRef]  

20. A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003). [CrossRef]  

21. A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003). [CrossRef]  

22. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004). [CrossRef]  

23. D. A. Cardimona, M. G. Raymer, and C. R. Stroud Jr, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).

24. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006). [CrossRef]  

25. J. H. Li, “Coherent control of optical bistability in a microwave-driven V-type atomic system,” Physica D 228, 148–152 (2007). [CrossRef]  

26. J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006). [CrossRef]  

27. H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014). [CrossRef]  

28. J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007). [CrossRef]  

29. H. R. Hamedi and M. R. Mehmannavaz, “Behavior of optical bistability in multifold quantum dot molecules,” Laser Phys. 25, 025403 (2015). [CrossRef]  

30. M. R. Mehmannavaz and H. Sattari, “A quintuple quantum dot system for electrical and optical control of multi/bistability in a telecommunication window,” Laser Phys. Lett. 12, 025201 (2015). [CrossRef]  

31. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001). [CrossRef]   [PubMed]  

32. T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015). [CrossRef]  

33. H. R. Hamedi and G. Juzeliūnas, “Phase-sensitive Kerr nonlinearity for closed-loop quantum systems,” Phys. Rev. A 91, 053823 (2015). [CrossRef]  

34. C. Hang and G. X. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010). [CrossRef]   [PubMed]  

35. P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001). [CrossRef]  

36. P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003). [CrossRef]   [PubMed]  

37. J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999). [CrossRef]  

38. J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002). [CrossRef]   [PubMed]  

39. P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009). [CrossRef]  

40. R. G. Wooley, “Quantum theory and molecular structure,” Adv. Phys. 25, 27–52 (1976). [CrossRef]  

41. D. C. Walker, ed., Origins of Optical Activity in Nature (Elsevier, 1979).

42. Y. Li and C. Bruder, “Dynamic method to distinguish between left- and right-handed chiral molecules,” Phys. Rev. A 77, 015403 (2008). [CrossRef]  

43. W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010). [CrossRef]  

44. P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003). [CrossRef]  

45. M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000). [CrossRef]   [PubMed]  

46. N. Ji and Y. R. Shen, “A novel spectroscopic probe for molecular chirality,” Chirality 18, 146–158 (2006). [CrossRef]   [PubMed]  

47. W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011). [CrossRef]  

48. L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984). [CrossRef]  

49. Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005). [CrossRef]   [PubMed]  

50. H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014). [CrossRef]  

51. F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef]   [PubMed]  

52. S. O. Konorov, A. B. Fedotov, and A. M. Zheltikov, “Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,” Opt. Lett. 28, 1448–1450 (2003). [CrossRef]   [PubMed]  

53. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009). [CrossRef]   [PubMed]  

References

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  16. H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001).
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  18. A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
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  19. M. A. Antón and O. G. Calderón, “Optical bistability using quantum interference in V -type atoms,” J. Opt. B 4, 91–98 (2002).
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  20. A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003).
    [Crossref]
  21. A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003).
    [Crossref]
  22. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004).
    [Crossref]
  23. D. A. Cardimona, M. G. Raymer, and C. R. Stroud, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).
  24. D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006).
    [Crossref]
  25. J. H. Li, “Coherent control of optical bistability in a microwave-driven V-type atomic system,” Physica D 228, 148–152 (2007).
    [Crossref]
  26. J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
    [Crossref]
  27. H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
    [Crossref]
  28. J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).
    [Crossref]
  29. H. R. Hamedi and M. R. Mehmannavaz, “Behavior of optical bistability in multifold quantum dot molecules,” Laser Phys. 25, 025403 (2015).
    [Crossref]
  30. M. R. Mehmannavaz and H. Sattari, “A quintuple quantum dot system for electrical and optical control of multi/bistability in a telecommunication window,” Laser Phys. Lett. 12, 025201 (2015).
    [Crossref]
  31. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
    [Crossref] [PubMed]
  32. T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
    [Crossref]
  33. H. R. Hamedi and G. Juzeliūnas, “Phase-sensitive Kerr nonlinearity for closed-loop quantum systems,” Phys. Rev. A 91, 053823 (2015).
    [Crossref]
  34. C. Hang and G. X. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010).
    [Crossref] [PubMed]
  35. P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
    [Crossref]
  36. P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
    [Crossref] [PubMed]
  37. J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
    [Crossref]
  38. J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
    [Crossref] [PubMed]
  39. P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
    [Crossref]
  40. R. G. Wooley, “Quantum theory and molecular structure,” Adv. Phys. 25, 27–52 (1976).
    [Crossref]
  41. D. C. Walker, ed., Origins of Optical Activity in Nature (Elsevier, 1979).
  42. Y. Li and C. Bruder, “Dynamic method to distinguish between left- and right-handed chiral molecules,” Phys. Rev. A 77, 015403 (2008).
    [Crossref]
  43. W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010).
    [Crossref]
  44. P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
    [Crossref]
  45. M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
    [Crossref] [PubMed]
  46. N. Ji and Y. R. Shen, “A novel spectroscopic probe for molecular chirality,” Chirality 18, 146–158 (2006).
    [Crossref] [PubMed]
  47. W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
    [Crossref]
  48. L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984).
    [Crossref]
  49. Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
    [Crossref] [PubMed]
  50. H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
    [Crossref]
  51. F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
    [Crossref] [PubMed]
  52. S. O. Konorov, A. B. Fedotov, and A. M. Zheltikov, “Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,” Opt. Lett. 28, 1448–1450 (2003).
    [Crossref] [PubMed]
  53. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
    [Crossref] [PubMed]

2015 (4)

H. R. Hamedi and M. R. Mehmannavaz, “Behavior of optical bistability in multifold quantum dot molecules,” Laser Phys. 25, 025403 (2015).
[Crossref]

M. R. Mehmannavaz and H. Sattari, “A quintuple quantum dot system for electrical and optical control of multi/bistability in a telecommunication window,” Laser Phys. Lett. 12, 025201 (2015).
[Crossref]

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

H. R. Hamedi and G. Juzeliūnas, “Phase-sensitive Kerr nonlinearity for closed-loop quantum systems,” Phys. Rev. A 91, 053823 (2015).
[Crossref]

2014 (2)

H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
[Crossref]

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

2011 (1)

W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
[Crossref]

2010 (2)

2009 (2)

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

2008 (1)

Y. Li and C. Bruder, “Dynamic method to distinguish between left- and right-handed chiral molecules,” Phys. Rev. A 77, 015403 (2008).
[Crossref]

2007 (3)

J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).
[Crossref]

J. H. Li, “Coherent control of optical bistability in a microwave-driven V-type atomic system,” Physica D 228, 148–152 (2007).
[Crossref]

X. M. Hu and J. Wang, “Sideband control of optical bistability and multistability,” Phys. Lett. A 365, 253–257 (2007).
[Crossref]

2006 (3)

J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

N. Ji and Y. R. Shen, “A novel spectroscopic probe for molecular chirality,” Chirality 18, 146–158 (2006).
[Crossref] [PubMed]

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006).
[Crossref]

2005 (3)

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

2004 (1)

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004).
[Crossref]

2003 (8)

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[Crossref] [PubMed]

A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
[Crossref]

A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003).
[Crossref]

A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003).
[Crossref]

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[Crossref]

S. O. Konorov, A. B. Fedotov, and A. M. Zheltikov, “Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,” Opt. Lett. 28, 1448–1450 (2003).
[Crossref] [PubMed]

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref] [PubMed]

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

2002 (3)

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

M. A. Antón and O. G. Calderón, “Optical bistability using quantum interference in V -type atoms,” J. Opt. B 4, 91–98 (2002).
[Crossref]

2001 (4)

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[Crossref] [PubMed]

P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
[Crossref]

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
[Crossref]

H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001).
[Crossref]

2000 (1)

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

1999 (1)

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

1998 (1)

J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998).
[Crossref]

1997 (2)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[Crossref]

S. Gong, S. Du, and Z. Xu, “Optical bistability via atomic coherence,” Phys. Lett. A 226, 293–297 (1997).
[Crossref]

1996 (2)

W. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref] [PubMed]

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[Crossref]

1989 (1)

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

1983 (1)

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[Crossref]

1982 (1)

D. A. Cardimona, M. G. Raymer, and C. R. Stroud, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).

1981 (1)

N. Tsukada and T. Nakayama, “Optical bistability from interference between one- and two-photon transition processes,” Phys. Rev. A 25, 947–955 (1981).
[Crossref]

1980 (1)

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

1976 (2)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

R. G. Wooley, “Quantum theory and molecular structure,” Adv. Phys. 25, 27–52 (1976).
[Crossref]

Agarwal, G. S.

W. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref] [PubMed]

Antón, M. A.

M. A. Antón and O. G. Calderón, “Optical bistability using quantum interference in V -type atoms,” J. Opt. B 4, 91–98 (2002).
[Crossref]

Antonopoulos, G.

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

Arimondo, E.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[Crossref]

Asquini, M. L.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

Bajcsy, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Balic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Beckwitt, K.

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

Belkin, M. A.

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

Benabid, F.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

Bird, D. M.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

Brambilla, M.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

Brown, A.

A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
[Crossref]

Bruder, C.

Y. Li and C. Bruder, “Dynamic method to distinguish between left- and right-handed chiral molecules,” Phys. Rev. A 77, 015403 (2008).
[Crossref]

Buckingham, A. D.

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

Calderón, O. G.

M. A. Antón and O. G. Calderón, “Optical bistability using quantum interference in V -type atoms,” J. Opt. B 4, 91–98 (2002).
[Crossref]

Cardimona, D. A.

D. A. Cardimona, M. G. Raymer, and C. R. Stroud, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).

Cheng, D. C.

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006).
[Crossref]

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004).
[Crossref]

Christensen, T.

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

Cohen, D.

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref] [PubMed]

Couny, F.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

Du, S.

S. Gong, S. Du, and Z. Xu, “Optical bistability via atomic coherence,” Phys. Lett. A 226, 293–297 (1997).
[Crossref]

Ernst, K. H.

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

Fedotov, A. B.

Field, R. W.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

Fischer, P.

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Gong, S.

S. Gong, S. Du, and Z. Xu, “Optical bistability via atomic coherence,” Phys. Lett. A 226, 293–297 (1997).
[Crossref]

Gong, S. Q.

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006).
[Crossref]

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004).
[Crossref]

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[Crossref] [PubMed]

Goorskey, D. J.

H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001).
[Crossref]

Hafezi, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Hamedi, H. R.

H. R. Hamedi and M. R. Mehmannavaz, “Behavior of optical bistability in multifold quantum dot molecules,” Laser Phys. 25, 025403 (2015).
[Crossref]

H. R. Hamedi and G. Juzeliūnas, “Phase-sensitive Kerr nonlinearity for closed-loop quantum systems,” Phys. Rev. A 91, 053823 (2015).
[Crossref]

H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
[Crossref]

Hang, C.

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[Crossref]

Harshawardhan, W.

W. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref] [PubMed]

Hofferberth, S.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Hu, X. M.

X. M. Hu and J. Wang, “Sideband control of optical bistability and multistability,” Phys. Lett. A 365, 253–257 (2007).
[Crossref]

Huang, G. X.

Huang, Q. J.

J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

Ian, H.

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

Il’ichev, E.

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
[Crossref]

Jauho, A.

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

Ji, N.

N. Ji and Y. R. Shen, “A novel spectroscopic probe for molecular chirality,” Chirality 18, 146–158 (2006).
[Crossref] [PubMed]

Jia, W. Z.

W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
[Crossref]

W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010).
[Crossref]

Joshi, A.

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[Crossref] [PubMed]

A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003).
[Crossref]

A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
[Crossref]

A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003).
[Crossref]

Juzeliunas, G.

H. R. Hamedi and G. Juzeliūnas, “Phase-sensitive Kerr nonlinearity for closed-loop quantum systems,” Phys. Rev. A 91, 053823 (2015).
[Crossref]

Khaledi-Nasab, A.

H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
[Crossref]

Kimble, H. J.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[Crossref]

Kirova, T.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

Knight, J. C.

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

Konorov, S. O.

Král, P.

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref] [PubMed]

P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
[Crossref]

Kulakov, T. A.

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

Lazarov, G.

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

Lazoudis, A.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

Li, J. H.

J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).
[Crossref]

J. H. Li, “Coherent control of optical bistability in a microwave-driven V-type atomic system,” Physica D 228, 148–152 (2007).
[Crossref]

J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

Li, L.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

Li, Y.

Y. Li and C. Bruder, “Dynamic method to distinguish between left- and right-handed chiral molecules,” Phys. Rev. A 77, 015403 (2008).
[Crossref]

Light, P. S.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

Liu, C. P.

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006).
[Crossref]

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004).
[Crossref]

Liu, Y.-x.

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

Lü, X. Y.

J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

Lugiato, L. A.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984).
[Crossref]

Lukin, M. D.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
[Crossref]

Luo, J.M.

J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

Lyyra, A. M.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

Magnes, J.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998).
[Crossref]

McCall, S. L.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Mehmannavaz, M. R.

H. R. Hamedi and M. R. Mehmannavaz, “Behavior of optical bistability in multifold quantum dot molecules,” Laser Phys. 25, 025403 (2015).
[Crossref]

M. R. Mehmannavaz and H. Sattari, “A quintuple quantum dot system for electrical and optical control of multi/bistability in a telecommunication window,” Laser Phys. Lett. 12, 025201 (2015).
[Crossref]

Mortensen, N. A.

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

Nakayama, T.

N. Tsukada and T. Nakayama, “Optical bistability from interference between one- and two-photon transition processes,” Phys. Rev. A 25, 947–955 (1981).
[Crossref]

Narducci, L. M.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

Nori, F.

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

Orozco, L. A.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[Crossref]

Pearce, G. J.

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

Peyronel, T.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Qi, J.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

Raheli, A.

H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
[Crossref]

Raymer, M. G.

D. A. Cardimona, M. G. Raymer, and C. R. Stroud, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).

Rosenberger, A. T.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[Crossref]

Russell, J.

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

Sahrai, M.

H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
[Crossref]

Saldana, J.

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[Crossref]

Sattari, H.

M. R. Mehmannavaz and H. Sattari, “A quintuple quantum dot system for electrical and optical control of multi/bistability in a telecommunication window,” Laser Phys. Lett. 12, 025201 (2015).
[Crossref]

Shapiro, M.

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref] [PubMed]

P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
[Crossref]

Shen, Y. R.

N. Ji and Y. R. Shen, “A novel spectroscopic probe for molecular chirality,” Chirality 18, 146–158 (2006).
[Crossref] [PubMed]

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

Spano, F. C.

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

St, P.

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

Stroud, C. R.

D. A. Cardimona, M. G. Raymer, and C. R. Stroud, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).

Sun, C.

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

Sun, H. C.

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

Thanopulos, I.

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref] [PubMed]

Tsukada, N.

N. Tsukada and T. Nakayama, “Optical bistability from interference between one- and two-photon transition processes,” Phys. Rev. A 25, 947–955 (1981).
[Crossref]

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Vuletic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Walls, D. F.

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

Wang, H.

A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
[Crossref]

H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001).
[Crossref]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[Crossref] [PubMed]

Wang, J.

X. M. Hu and J. Wang, “Sideband control of optical bistability and multistability,” Phys. Lett. A 365, 253–257 (2007).
[Crossref]

Wang, X.

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

Wei, L.

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

Wei, L. F.

W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
[Crossref]

W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010).
[Crossref]

Wiersma, D. S.

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

Wise, F. W.

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

Wooley, R. G.

R. G. Wooley, “Quantum theory and molecular structure,” Adv. Phys. 25, 27–52 (1976).
[Crossref]

Wu, Y.

Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[Crossref]

Wubs, M.

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

Xiao, M.

A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
[Crossref]

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[Crossref] [PubMed]

A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003).
[Crossref]

A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003).
[Crossref]

H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001).
[Crossref]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[Crossref] [PubMed]

Xu, Z.

S. Gong, S. Du, and Z. Xu, “Optical bistability via atomic coherence,” Phys. Lett. A 226, 293–297 (1997).
[Crossref]

Yan, L.

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

Yan, W.

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

Yang, W.

A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003).
[Crossref]

A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003).
[Crossref]

Yang, X. X.

Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

You, J.

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

You, J. Q.

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

Zheltikov, A. M.

Zhu, Y. F.

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[Crossref]

Zibrov, A. S.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Zoller, P.

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

Adv. Phys. (1)

R. G. Wooley, “Quantum theory and molecular structure,” Adv. Phys. 25, 27–52 (1976).
[Crossref]

Appl. Phys. Lett. (1)

P. S. Light, F. Benabid, G. J. Pearce, F. Couny, and D. M. Bird, “Electromagnetically induced transparency in acetylene molecules with counterpropagating beams in V and Λ schemes,” Appl. Phys. Lett. 94, 141103 (2009).
[Crossref]

Chirality (1)

N. Ji and Y. R. Shen, “A novel spectroscopic probe for molecular chirality,” Chirality 18, 146–158 (2006).
[Crossref] [PubMed]

J. Mod. Opt. (1)

J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998).
[Crossref]

J. Opt. B (1)

M. A. Antón and O. G. Calderón, “Optical bistability using quantum interference in V -type atoms,” J. Opt. B 4, 91–98 (2002).
[Crossref]

J. Phys. (1)

D. A. Cardimona, M. G. Raymer, and C. R. Stroud, “Steady-state quantum interference in resonance fluorescence,” J. Phys. 15, 55–64 (1982).

J. Phys. B (1)

W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010).
[Crossref]

Laser Phys. (1)

H. R. Hamedi and M. R. Mehmannavaz, “Behavior of optical bistability in multifold quantum dot molecules,” Laser Phys. 25, 025403 (2015).
[Crossref]

Laser Phys. Lett. (1)

M. R. Mehmannavaz and H. Sattari, “A quintuple quantum dot system for electrical and optical control of multi/bistability in a telecommunication window,” Laser Phys. Lett. 12, 025201 (2015).
[Crossref]

Nature (London) (1)

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
[Crossref]

Opt. Commun. (3)

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability via amplitude and phase control of a microwave field,” Opt. Commun. 263, 111–115 (2006).
[Crossref]

H. R. Hamedi, A. Khaledi-Nasab, A. Raheli, and M. Sahrai, “Coherent control of optical bistability and multistability via double dark resonances (DDRs),” Opt. Commun. 312, 117–122 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (4)

A. Joshi, W. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203–207 (2003).
[Crossref]

D. C. Cheng, C. P. Liu, and S. Q. Gong, “Optical bistability and multistability via the effect of spontaneously generated coherence in a three-level ladder-type atomic system,” Phys. Lett. A 332, 244–249 (2004).
[Crossref]

S. Gong, S. Du, and Z. Xu, “Optical bistability via atomic coherence,” Phys. Lett. A 226, 293–297 (1997).
[Crossref]

X. M. Hu and J. Wang, “Sideband control of optical bistability and multistability,” Phys. Lett. A 365, 253–257 (2007).
[Crossref]

Phys. Rev. A (14)

H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2001).
[Crossref]

N. Tsukada and T. Nakayama, “Optical bistability from interference between one- and two-photon transition processes,” Phys. Rev. A 25, 947–955 (1981).
[Crossref]

W. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref] [PubMed]

A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68, 015806 (2003).
[Crossref]

Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[Crossref]

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with two-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[Crossref]

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, L. A. Lugiato, M. L. Asquini, M. Brambilla, and L. M. Narducci, “Single-mode instability in optical bistability,” Phys. Rev. A 39, 1235–1252 (1989).
[Crossref]

A. Joshi, A. Brown, H. Wang, and M. Xiao, “Controlling optical bistability in a three-level atomic system,” Phys. Rev. A 67, 041801 (2003).
[Crossref]

J. H. Li, X. Y. Lü, J.M. Luo, and Q. J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

H. R. Hamedi and G. Juzeliūnas, “Phase-sensitive Kerr nonlinearity for closed-loop quantum systems,” Phys. Rev. A 91, 053823 (2015).
[Crossref]

H. C. Sun, Y.-x. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89, 063822 (2014).
[Crossref]

W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
[Crossref]

Y. Li and C. Bruder, “Dynamic method to distinguish between left- and right-handed chiral molecules,” Phys. Rev. A 77, 015403 (2008).
[Crossref]

Phys. Rev. B (2)

T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75, 155329 (2007).
[Crossref]

Phys. Rev. Lett. (11)

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[Crossref] [PubMed]

P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
[Crossref]

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref] [PubMed]

J. Qi, G. Lazarov, X. Wang, L. Li, L. M. Narducci, A. M. Lyyra, and F. C. Spano, “Autler-Townes splitting in molecular lithium: prospects for all-optical alignment of nonpolar molecules,” Phys. Rev. Lett. 83, 288–291 (1999).
[Crossref]

J. Qi, F. C. Spano, T. Kirova, A. Lazoudis, J. Magnes, L. Li, L. M. Narducci, R. W. Field, and A. M. Lyyra, “Measurement of transition dipole moments in lithium dimers using electromagnetically induced transparency,” Phys. Rev. Lett. 88, 173003 (2002).
[Crossref] [PubMed]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[Crossref] [PubMed]

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, “New electro-optic effect: sumfrequency generation from optically active liquids in the presence of a dc electric field,” Phys. Rev. Lett. 91, 173901 (2003).
[Crossref]

M. A. Belkin, T. A. Kulakov, K. H. Ernst, L. Yan, and Y. R. Shen, “Sum-frequency vibrational spectroscopy on chiral liquids: a novel technique to probe molecular chirality,” Phys. Rev. Lett. 85, 4474–4477 (2000).
[Crossref] [PubMed]

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009).
[Crossref] [PubMed]

Y.-x. Liu, J. You, L. Wei, C. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[Crossref]

Physica D (1)

J. H. Li, “Coherent control of optical bistability in a microwave-driven V-type atomic system,” Physica D 228, 148–152 (2007).
[Crossref]

Prog. Opt. (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[Crossref]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Science (1)

F. Benabid, J. C. Knight, G. Antonopoulos, P. St, and J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
[Crossref] [PubMed]

Other (2)

L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1984).
[Crossref]

D. C. Walker, ed., Origins of Optical Activity in Nature (Elsevier, 1979).

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Figures (9)

Fig. 1
Fig. 1 The energy level structure of the Δ-type chiral molecules: (a) left-handed chiral molecules; (b) right-handed chiral molecules. The enantiomers are driven by three optical fields, wherein the transition |2〉L(|2〉R) |1〉L(1〉R) is driven by a strong control field while the other two transitions are coupled by two probe fields.
Fig. 2
Fig. 2 Unidirectional ring cavity with a chiral molecule sample of length L. E p I and E p T are the incident and transmitted fields, respectively. Ec is the amplitude of the control field.
Fig. 3
Fig. 3 Input-output field curves by choosing different detunings Δ under the two-photon resonance conditions for (a) left-handed chiral molecules, i.e., η(+) = 1,η(−) = 0; (b) right-handed chiral molecules, i.e., η(+) = 0,η(−) = 1. (c) Probe absorption Im(ρ31 +ρ32) of the two enantiomers as a function of the detuning Δ. The Other parameters are chosen as Ωc = 4γ,C = 100, γ21 = γ, Δ21 = 0.
Fig. 4
Fig. 4 The probe absorption Im(ρ31 +ρ32) of left-handed molecules by choosing different decay rates γ21. The other parameters are chosen as Ωc = 0,C = 100,Δ31 = Δ32 = Δ,Δ21 = 0.
Fig. 5
Fig. 5 (a,b) Input-output field curves by choosing different intensities of control field for left- and right-handed chiral molecules, respectively. (c, d) Probe absorption Im(ρ31 +ρ32) of the two enantiomers as a function of the detuning Δ by changing the strength of the control field. The other parameters are chosen as C = 100, γ21 = γ, Δ21 = 0.
Fig. 6
Fig. 6 Input-output field curves by choosing different detunings Δ32 = −Δ31 = 0 (solid line), 0.2γ (dashed line), 0.4γ (dotted line) for (a) left-handed chiral molecules; (b) right-handed chiral molecules. The other parameters are chosen as γ21 = 0.1γ, Ωc = 4γ, C = 100, Δ21 = Δ31 Δ32.
Fig. 7
Fig. 7 Input-output field curves by choosing different intensities of control field: Ωc = γ (solid line), 5γ (dashed line), 10γ (dotted line) for (a) left- and (b) right-handed chiral molecules, respectively. The other parameters are the same as those in Fig. 6 except for Δ32 = Δ31 = 0.2γ.
Fig. 8
Fig. 8 Input-output field curves by varying the ratios of the two enantiomers. The parameters are chosen as C = 100,Ωc = 4γ31 = Δ32 = 2γ21 = 0,γ21 = γ.
Fig. 9
Fig. 9 The threshold intensity versus the percentages η+ of the left-handed chiral molecules for two cases (a) Δ > 0; (b) Δ < 0. The other parameters are the same as those in Fig. 8

Equations (14)

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H = j = 1 3 ω j σ jj 1 2 ( j , k = 1 , j > k 3 Ω j k e i ( ν j k t + ϕ j k ) σ j k + H . c . ) ,
H I = ( Δ 21 σ 22 + Δ 31 σ 33 ) 1 2 ( Ω 32 σ 32 + Ω 31 σ 31 + Ω 21 σ 21 e i ( δ t + Φ ) + H . c . ) ,
ρ ˙ = i [ H I , ρ ] + [ j , k = 1 , j > k 3 γ j k σ j k ] ρ ,
ρ ˙ 33 = ( γ 31 + γ 32 ) ρ 33 + i 2 Ω 31 ( ρ 13 ρ 31 ) + i 2 Ω 32 ( ρ 23 ρ 32 ) , ρ ˙ 22 = γ 21 ρ 22 + γ 32 ρ 33 + i 2 Ω 21 ( ρ 12 e i Φ ρ 21 e i Φ ) + i 2 Ω 32 ( ρ 32 ρ 23 ) , ρ ˙ 12 = Γ 12 ρ 12 + i 2 Ω 21 e i Φ ( ρ 22 ρ 11 ) + i 2 Ω 31 ρ 32 i 2 Ω 32 ρ 13 , ρ ˙ 13 = Γ 13 ρ 13 + i 2 Ω 31 ( ρ 33 ρ 11 ) + i 2 Ω 21 e i Φ ρ 23 i 2 Ω 32 ρ 12 , ρ ˙ 23 = Γ 23 ρ 23 + i 2 Ω 21 e i Φ ρ 13 + i 2 Ω 32 ( ρ 33 ρ 22 ) i 2 Ω 31 ρ 21 ,
E = E 31 e i ν 31 t + E 32 e i ν 32 t + E 21 e i ν 21 t + c . c . ,
E p t + c E p z = i 2 ε 0 [ ν 31 P ( ν 31 ) + ν 32 P ( ν 32 ) ] ,
E p z = i N 2 c ε 0 [ ν 31 μ 31 ρ ˜ 31 + ν 32 μ 32 ρ ˜ 32 ] ,
E p ( L ) = E p I / T ,
E p ( 0 ) = T E p I + R E p ( L ) ,
Y = 2 X i γ ( C 1 ρ ˜ 31 + C 2 ρ ˜ 32 ) .
ρ 31 ( 1 ) = 2 i Ω p Γ 32 ρ 11 s + Ω p Ω c ρ 22 s e i Φ 4 Γ 31 Γ 32 + Ω c 2 , ρ 32 ( 1 ) = 2 i Ω p Γ 31 ρ 22 s + Ω p Ω c ρ 11 s e i Φ 4 Γ 31 Γ 32 + Ω c 2 ,
Im ( ρ 31 ( 1 ) + ρ 32 ( 1 ) ) = Ω p [ ( γ 31 + γ 32 + γ 21 ) ρ 11 s + ( γ 31 + γ 32 ) ρ 22 s Ω c ( ρ 22 s ρ 11 s ) sin Φ ] ( γ 31 + γ 32 + γ 21 ) ( γ 31 + γ 21 ) Ω c 2
Im ( ρ 31 ( 1 ) + ρ 32 ( 1 ) ) = Ω p A D + B A 2 + B 2 ,
A = ( γ 31 + γ 32 + γ 21 ) ( γ 31 + γ 21 ) + Ω c 2 4 Δ 2 , B = 2 Δ ( 2 γ 31 + γ 32 + γ 21 ) , = 2 Δ + Ω c cos Φ , D = ( γ 31 + γ 32 + γ 21 ) ρ 11 s + ( γ 31 + γ 32 ) ρ 11 s Ω c ( ρ 22 s ρ 11 s ) sin Φ .

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