Abstract

We consider an active-Raman-gain scheme for realizing giant Kerr nonlinearity and superluminal optical solitons in a four-state atomic system with a gain doublet. We show that this scheme, which is fundamentally different from those based on electromagnetically induced transparency (EIT), is capable of working at room temperature and eliminating nearly all attenuation and distortion. We demonstrate that, due to the appearance of a gain spectrum hole induced by the quantum interference effect induced by a signal field, a significant enhancement of Kerr nonlin-earity of probe field can be realized effectively, which can be more than ten times larger than that arrived by the EIT-based scheme with the same energy-level configuration. Based on these important features, we obtain a giant cross-phase modulation effect and hence a stable long-distance propagation of optical solitons, which have superluminal propagating velocity and very low generating power.

©2010 Optical Society of America

1. Introduction

Kerr nonlinearity is the dispersive part of third-order susceptibility in optical media and is essential for most nonlinear optical processes [1]. Recent researches have shown that Kerr nonlinearity can be applied to realize quantum nondemolition measurement, quantum state teleportation, quantum logic gates, and nonlinear control of light [2, 3, 4, 5, 6]. In addition, Kerr nonlinearity can be used to balance dispersion and/or diffraction effects and thus form distortion-free optical wave packets, alias optical solitons [7]. However, up to now, Kerr non-linearity and optical solitons are usually produced in passive optical media such as glass-based optical fibers, in which far-off resonance excitation schemes are generally employed in order to avoid unmanageable optical attenuation and distortion. As a result, the nonlinear effect in such passive optical media is very weak, and hence a very long propagation distance or a very high light intensity is required for accumulating enough nonlinear phase shifts and forming optical solitons.

In recent years, much attention has been paid to the phenomena of electromagnetically induced transparency (EIT) in highly resonant optical media [8]. By means of the quantum interference effect induced by a coupling laser field, the absorption of a probe laser field tuned to a strong one-photon resonance can be largely suppressed. The wave propagation in EIT systems exhibits both significant reduction of group velocity and enhancement of Kerr nonlinearity. Based on these characters, the possibility of ultra-slow optical solitons in EIT systems has been studied in details [9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20]. However, weakly driven EIT-based schemes still have some inherent drawbacks, one of which is that the probe field usually experiences significant attenuation and distortion at room temperature. On the other hand, the wave propagation in resonant optical media with an active Raman gain (ARG) core have also attracted considerable attention both theoretically and experimentally [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Contrary to the EIT-based scheme which is absorptive in nature, the central idea of the ARG scheme is that the signal field operates in the stimulated Raman emission mode, and hence can eliminate signal attenuation and realize a stable superluminal propagation of the signal field even at room temperature.

We should mention that except for atomic systems, subluminal and superluminal lights has also been obtained in many solid-state systems, such as photorefractive crystals [40], optical fibers with stimulated Brillouin scattering [41, 42], crystals of molecular magnets[43], liquid crystal devices [44], photonic crystals [45], and coupled-resonator optical waveguides[46], etc.

In this work we investigate an ARG system to realize giant Kerr nonlinearity and superluminal solitons in a four-state atomic system with a gain doublet. Such system was proposed firstly by Agarwal and Dasgupta [34] for a linear propagation of a probe field. Here we explore the nonlinear propagation effect of this system and show that a giant cross Kerr nonlinearity, which can be more than ten times larger than that obtained by the EIT-based scheme with the same energy-level configuration, can be realized effectively. Furthermore, by means of the giant Kerr nonlinearity we predict the formation and propagation of shape-preserving nonlinear probe pulses in the system. We demonstrate also that in such system stable optical solitons with superluminal propagating velocity and very low generating power can be obtained. Our results may raise the possibility of rapidly responding nonlinear phase switching, phase gates, and information processing and transmission at low light intensity.

We note that some recent studies have considered possible superluminal optical solitons in three-state ARG systems [47, 48, 49, 50]. In Refs. [47, 48], in order to neglect decay rates of atomic levels the authors used a strong and short-pulse condition, and hence the solutions obtained are only valid in coherent transient regime; Meanwhile, Refs. [49, 50], without using coherent transient condition, showed that weak nonlinear superluminal optical solitons are possible in a three-level ARG medium. However, there exists a significant gain during soliton propagation because the gain spectrum in the three-level ARG system has no minimum near resonance. Nevertheless, the four-state ARG scheme suggested here has better characters than that of the three-state ARG scheme due to the existence of doublet structure in its gain spectrum. We will demonstrate that for the present four-level ARG system not only a rapid increase of probe field intensity appeared in the three-level ARG system can be avoided but also a giant enhancement of Kerr nonlinearity and hence a more stable superluminal optical soliton can be arrived when the system works near the minimum of the gain-spectrum hole.

The paper is arranged as follows. In the next section, we present theoretical model under study and discuss its solution in linear regime. In Sec. III, we calculate the nonlinear optical susceptibilities of probe field and study the giant cross Kerr nonlinear effect and hence large nonlinear phase shift contributed by a signal field. In Sec. IV, we make an asymptotic analysis on the Maxwell-SchrÖdinger equations and derive a nonlinear envelope equation governing the spatial-temporal evolution of the probe field. Based on this nonlinear envelope equation, we provide superluminal optical soliton solutions with their central carrier-wave frequency near the minimum of the gain doublet. We then investigate the stability and discuss the interaction property of two superluminal optical solitons. Finally, the last section contains a discussion and summary of the main results of our work.

2. Model and linear property

We start with considering a life-time broadened four-state atomic system, shown in Fig. 1(a). A strong continuous-wave (CW) coupling field of angular frequency ωc driving the transition |1⎤ ↔ |3⎤ and a probe field of angular frequency ωp driving the transition |2⎤ ↔ |3⎤ create the three-level ARG core [49, 50]. A CW signal field of angular frequency ωs drives the transition |2⎤ ↔ |4⎤. 2Ωc, 2Ωp, and 2Ωs are the Rabi frequencies of the coupling, probe, and signal fields, respectively. The probe field has a pulse length τ 0 at the entrance of the medium. Under electric-dipole and rotating-wave approximations, we obtain the Hamiltonian in the interaction picture

 

Fig. 1. (Color online) (a): Energy-level configuration and excitation scheme of the four-state ARG system, which interacts with a strong CW coupling field of angular frequency ω c and Rabi frequency 2ΩC (driving the transition |1⎤ ↔ |3⎤), a weak, pulsed probe field of angular frequency ωp and Rabi frequency 2Ωp (driving the transition |2⎤ ↔ |3⎤), and a CW signal field of angular frequency 2ωs and Rabi frequency Ωs (driving the transition |2⎤ ↔ |4⎤). Δj (j = 3, 4) are the corresponding detunings. (b): The four-state EIT system with the same energy-level configuration.

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Ĥint/h¯=[Δ333+Δ444+Ωc31+Ωp32
+Ωs42+H.c.],

where Δ3 = ωc -(E 3 - E 1)/h̄, and Δ4 = ωc - ωp + ωs - (E 4 - E 1)/h̄ are the one- and three-photon detunings (the two-photon detuning is set to be zero). The equations of motion for atomic response are

(it+d2)A2+Ωp*A3+Ωs*A4=0,
(it+d3)A3+ΩcA1+ΩpA2=0,
(it+d4)A4+ΩsA2=0

with the normalization condition Σj=1 4 |Aj|2 = 1 , where Aj (j = 1 to 4) is the probability amplitude of the bare atomic state |j〉 (with eigenenergy Ej), dj = Δj + j (j = 2 to 4, Δ2 = 0) with γj being the atomic decay rate of the state |j〉.

The wave equation for the probe field is given by the Maxwell equation under slow varying amplitude approximation

i(z+1ct)Ωp+κA3A2*=0,

where κ = Na ωp|P 23|2/ (2ε 0 ch̄) with Na being the atomic density and P 23 the electric dipole matrix element for the transition |2〉 ↔ |3〉.

We assume that atoms are initially populated in the state |1〉. Under the action of the coupling field a significant transfer of the population from ground state |1〉 to excited state |3〉 may occur. In order to suppress temperature-related Doppler effect, we assume that the one-photon detuning is much larger than the other parameters of the system, i. e. |Δ3| is much larger than the Rabi frequencies, Doppler broadened line widths, atomic coherence decay rates, and frequency shift induced by the coupling laser field. The homogeneous stationary state solution of Eqs. (2) and (3) is then given by A1(0)=1/1+Ωc/d32, A3(0)=Ωc/(d31+Ωc/d32), A 2 (0) = A 4 (0) = 0, and Ωp (0) = 0.

The general procedure for calculating various dispersion effects is to seek the solution of Eq. (2) and (3) in linear regime. Assuming Ωp = F exp , A 2 * and A 4 * are proportional to exp() with θ = K(ω)z - ωt and F being the envelope of the probe field, we easily obtain linear dispersion relation

K(ω)=ωc+κ(ωd4*)Ωc2[(ωd2*)(ωd4*)Ωs2](d32+Ω22).

In most operation conditions K(ω) can be Taylor expanded around the center frequency of the probe field (ω = 0), i. e. K(ω) = K 0 + K 1 ω + K 2 ω 2/2 + (ω 3) with Kj = jK(ω)/∂ωj)|ω=0 (j = 0,1,2,…). If the probe field at z = 0 is a Gaussian pulse, i.e. Ωp(0,t) = Ωp(0,0)exp[-t 2/(2τ 0 2)], we then readily obtain

Ωp(z,t)=Ωp(0,0)eiK0zb1(z)ib2(z)exp[(K1tz)22[b1(z)ib2(z)]τ02],

where b 1(z) = 1 + zIm(K 2)/τ 0 2 and b 2(z) = zRe(K 2)/τ 0 2. The expansion coefficients under the conditions |d 3|≫| Ωc| and γ 2 ≃ 0 (if |2∪ is chosen as a hyperfine ground state then γ 2 is usually 3 or 4 orders smaller than the other parameters) are

K0=ϕ+iα2,K1=1cκΩc2Ωs2d32+κΩc2d4*Ωs4d32,
K2=4κΩc2d4*Ωs4d32+2κΩc2d4*Ωs6d32,

where ϕ=κΔ4c|2/(|Ωs|2|d 3|2) gives a phase shift per unit length and α=-2κγ 4c|2/(|Ωs|2|d 3|2) leads to a (intensity) gain to the probe field. The probe field group velocity Vg = c/ng = l/Re(K 1) whereas Im(K 1) contributes to probe wave gain. The group index ng = 1 - (|Ωs|2 + Δ4 2 - γ4 2)|Ωc|2/(|Ωs|4|d 3|2). It is noteworthy that one gets 0 < Vg < c when |Ωs|2 + Δ4 2 < γ 4 2 (corresponding to a sub luminal propagation), while Vg > c or Vg < 0 when |Ωs|2 + Δ4 2 < γ 4 2 (corresponding to a superluminal propagation [51]). K 2 in Eq. (6) represents group-velocity dispersion which leads to usually probe field deformation due to dispersion and additional gain.

In Fig. 2(a) and 2(b) we have plotted the curves of Re[K(ω)] characterizing the refractive spectrum and -Im[K(ω)] characterizing the gain spectrum as functions of ω with different values of Ωs. The parameters we used are related to a typical warm alkali atom gas, e.g. 87Rb atom gas, where κ = 1.0 × 109 cm-1 s-1, 2γ 2 = 300 Hz, 2γ 3 = 500 MHz, and 2γ 4 = 30 MHz [39]. The other parameters are taken as Δ3 = -2.0 × 109 s-1, Δ4 = 0, and Ωc = 2.0 × 107 s-1. From panel (b), we see that for small signal-field Rabi frequency (e. g. Ωs = 1.0 × 107 s-1) the gain profile of the probe field displays a maximum near resonance (i. e. ω) = 0). However, a gain minimum (i. e. a hole in line center of the gain profile) appears for large Ωs and hence a gain doublet (or called gain spectrum hole) is created [34]. The positions of two gain maxima on the two sides of the gain spectrum hole locate at ω = ω max = ±(1/2)(4|Ωs|2 - 2γ 4 2)1/2, which appear for Ωs exceeding its critical value Ωscr = γ 4/√2 ≃ 1.1 × 107 s-1. It is easy to get the depth of the gain spectrum hole, i. e.

DARG=ImK(ω=ωmax)ImK(ω=0)
=4κΩs2Ωc2γ4(4Ωs2γ42)d32κγ4Ωc2Ωs2d32.
 

Fig. 2. (Color online) (a) The curve of Re[K(ω)] characterizing refractive index of the probe field as a function of ω. (b) The curve of -Im[K(ω)] characterizing the appearance of the gain doublet of the probe field for increasing signal fiels. (c) The curve of Vg/c as a function of Δ4. The solid, dashed, and dot-dashed lines correspond to Ωs = 4.0 × 107 s-1, 2.0 × 107 s-1, and 1.0 × 107 s-1, respectively. The inset shows the detail of the curve of Vg/c with Ωs = 1.0 × 107 s-1. The other parameters are given in the text.

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To suppress the Raman gain in the line center, a deep gain spectrum hole is needed, which requires

Ωs22κγ4Ωc2/d32.

Condition (8) is the equivalent of EIT for a large absorption spectrum hole (i. e. EIT transparency window) [8]. In this case the width of the gain spectrum hole is given by

ΓARG=24Ωs22γ42+2γ44Ωs2γ42.

Consequently, we can implement an effective control on the width and depth of the gain spectrum hole by manipulating the intensity of the signal field. The appearance of the gain-doublet structure is very useful for a shaping-preserving wave propagation. Specifically, we can use a particular parameters to make the system work near the minimum of the gain spectrum hole, and hence a rapid growth of the probe-field intensity can be avoided. Moreover, from Fig. 2(a) we see that corresponding to the appearance of the gain spectrum hole an anomalous dispersion occurs, thus one can generate a superluminal propagation of the probe field in the system, which will be discussed in Sec. IV below.

We stress that the physical reason of the appearance of the gain doublet is due to the quantum interference effect induced by the signal field. This can be understood as follows. From the linear solution obtained above one obtains

Ωp*A3(0)+Ωs*A4=ωiγ2κA3*(0)(K*ωc)F*exp(iθ*),

which is vanishing near the center frequency of the probe field (i. e. ω = 0) and for |2〉 being chosen as a hyperfine ground state (i. e. γ 2 ≃ 0). As a result, from Eq. (2a) we have A 2 ≈ 0, i. e. the population of the state |2〉 is always very small. It is such restraint of the Raman gain that leads to the formation of the gain spectrum hole of the probe field.

In Fig. 2(c) we have plotted the curves of Vg/c as functions of Δ4 with different values of Ωs. The inset shows the detail of the curve of Vg/c with Ωs = 1.0 × 107 s-1. From the figure we see that Vg with Ωs = 1.0 × 107 s-1 changes sign from the positive to negative at Δ4 ≃ (γ 4 2 - |Ωs|2)1/2 = 1.1 × 107 s-1. Therefore, it is possible to tune the group velocity from negative to positive values by simply decreasing Δ4 below 1.1 × 107 s-1. Vg with Ωs = 2.0 × 107 s-1 and 4.0 × 107 s-1 are always negative because γ 4 2 - |Ωs|2 < 0.

3. Giant Kerr nonlinearity and cross-phase modulation

The main topic of the present study is the nonlinear effect of the system. For this aim we must have a detailed understanding of Kerr nonlinearity, which is characterized by the real part of third order optical susceptibilities. The probe-field susceptibility is defined as

χp=Nap232ε0h¯A3A2*Ωpχp(1)+χpp(3)p2+χps(3)s2,

where we have introduced the linear susceptibility χp (1), the third order self- and cross-Kerr susceptibilities χpp (3) and χps (3), respectively. The explicit expressions of static susceptibilities can be obtained by solving Eq. (2) under a steady state approximation, which read

A2=d4ΩcΩp*DA1,
A3=ΩcΩsDA1,
A4=ΩcΩsΩp*DA1,

with Ds = |Ωs|2 - d 2 d 4 and D = |Ωs|2 d 3 + |Ωp|2 d 4 - d 2 d 3 d 4. The expression of A 1 can be obtained by using the condition Σj=1 4|Aj|2 = 1 and hence

A1=[1+Ωc2Ds2+(Ωs2+d42)Ωp2D2]1/2.

After substituting A 1, A 2 and A 3 into Eq. (11) and making Taylor expansion, we obtain

χp(1)=Nap232ε0h̄iΩc2γ2(Ωc2+d32),
χpp(3)=Nap234ε0h¯3i(2γ2γ3+Ωc2)Ωc2γ23(Ωc2+d32)2,
χps(3)=Nap232p242ε0h¯3d4Ωc2γ22(Ωc2+d32)d42,

From Eq. (14), we obtain the following conclusions: (i) Both χp (1) and χpp (3) are Pure imaginary due to the exact two-photon resonance. Contrary to EIT-based systems here the sign of Im[χp (1)] is always negative, corresponding to a linear gain. (ii) The sign of Im[χpp (3)] is positive, corresponding to a nonlinear absorption from the self-Kerr effect, which increases with the growing of the probe-field intensity. (iii) The sign of Im[χps (3)] is also positive corresponding to a nonlinear absorption from the cross-Kerr effect, which increases as the signal-field intensity increases. Since the signal field is much stronger than the probe field, the linear gain effect can be suppressed by the nonlinear absorption from the cross-Kerr effect contributed by the signal field. This means that the gain spectrum hole can still maintains even under nonlinear excitation condition. In the following analysis, we will mainly focus on the third order cross-Kerr susceptibility, which leads to a cross-phase modulation (XPM) in the system.

The phase shift acquired by the probe field propagating through the atomic medium can be controlled by the signal field intensity. For |d 3| ≫ |Ωc| the phase shift due to such XPM effect is given by

ϕXPM=ωpL2cRe[χps(3)]s2κLΔ4Ωc2γ22d32d42Ωs2,

with L being the length of the medium. It is easily to obtain that ϕ XPM/(αL) ≈ -Δ4/(2γ 4). We note that the condition of three-photon off-resonance, i. e. Δ4 ≠ 0, is crucial to the observation of a nonzero XPM phase shift. We can also obtain Eq. (15) through expanding ϕ around |Ωs|2 = 0.

We recall that in order to enhance the cross-Kerr nonlinearity and hence the XPM, several schemes have been proposed in previous studies. The most important one is the four-state atomic medium with an N-type EIT configuration (i. e. Fig. 1(b)). In this scheme, the cross-Kerr nonlinearity was enhanced using an EIT core disturbed by an additional signal field [52]. In order to give a specific comparison between the present four-state ARG system and the four-state EIT system, we have calculated the linear and third-order nonlinear susceptibilities of the four-state EIT system with the energy-level configuration shown in Fig. 1(b), which consist of a weak, pulsed probe field driving the transition |1〉 ↔ |3〉, a strong CW coupling field driving the transition |2〉 ↔ |3〉, and a CW signal field driving the transition |2〉 ↔ |4〉. In Table I, we list the reduced expressions of χp (1), χpp (3), χps (3), and ϕ XPM for the four-state ARG (Fig. 1 (a)) and four-state EIT (Fig. 1(b)) systems, respectively.

Tables Icon

Table 1. Expressions of linear and nonlinear optical susceptibilities and XPM phase shift for the four-state ARG system (Fig. 1(a)) and the four-state EIT system (Fig. 1(b)).

In order to demonstrate the advantages of our four-state ARG scheme, we choose a set of system parameters to calculate the numerical values of the quantities in Table I for a one centimeter long atomic medium. Parameters for the ARG system are taken as Δ4 = 2.0 × 108 s-1, Ωc = 1.0 × 106 s-1, and Ωs = 1.0 × 105 s-1 with the other ones being the same with those use in Fig. 2. Parameters for the EIT system are taken as γ 3γ 4 = 10 MHz (corresponding to the cold atom gas) and Δ3 = 0 with the other ones being the same with those of the ARG system. The calculating result is given in Table II. We see that the cross Kerr susceptibility and hence the XPM phase shift of the ARG system is more tanh 10 times larger than that of the EIT system. Moreover, the ARG scheme have the advantage of being immune against a temperature-related Doppler effect, no loss, and fast response time due to the superluminal feature.

 

Fig. 3. (Color online) (a): The curve of ϕXPM for the EIT system (solid line) and the curve of -10ϕXPM of the EIT system (dashed line), as functions of Ωs. (b): The curves of ϕXPM (solid line) and ϕ =Re[K 0] (dashed line) versus Ωs for ARG system. The parameters are given in the text.

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Tables Icon

Table 2. Values of linear and nonlinear optical susceptibilities and XPM phase shift for the four-state ARG scheme and the four-state EIT scheme. The parameters used have been given in the text

Shown in Fig. 3(a) are curves of ϕXPM and -10ϕXPM as functions of Ωs for the ARG and the EIT systems, respectively. We see that the phase shift contributed from the XPM effect in the ARG system is much larger than that in the EIT system for the given system parameters. In panel (b) of this figure we have plotted the curves of ϕXPM and ϕ =Re[K 0] versus Ωs for the ARG system. The difference between the XPM phase shift and total phase shift, i.e. ϕXPM - ϕ, is negligible only for a weak signal field. This is because ϕXPM can be obtained by expanding ϕ around |Ωs|2 = 0 and keeping only the first few terms. However, when the signal-field intensity increases higher-order terms must be taken into account.

4. Envelope equation and superluminal optical solitons

4.1. Superluminal optical solitons

We now turn to study the possibility of superluminal optical solitons in the system. In order to obtain a shape-preserving nonlinear pulse, a nonzero three-photon off-resonance (Δ4 ≠ 0) is crucial, which provides not only superluminal propagation but also necessary nonlinearity to balance the dispersion effect. We apply the standard method of multiple-scales [53] to derive the nonlinear envelope equation of the probe field by taking the expansion Aj = Σn=0 εn Aj (n)(j = 1 to 4) and Ωp = Σn=1 εnΩp (n) , where ε is a small parameter characterizing the amplitude of the probe field. To obtain a divergence-free expansion, all quantities on the right hand side of the expansion are considered as functions of the multi-scale variables zl = εlz (l = 0 to 2) and tl = εlt (l = 0, 1). Substituting above expansion into Eqs. (2)-(3), we obtain a series of equations for Al (j) and Ωp (j) (l = 1,2,3,4; j = 1,2,3,…), which can be solved order by order.

In the leading order, O(ε), we have Ωp (1) = F exp() with θ = K(ω)z 0 - ωt 0 and F being a yet to be determined envelope function depending on the slow variables t 1 and zj (j = 1, 2).

The solution for Aj (1) (j = 1 to 4) in this order are given in the Appendix.

In the second order, O(ε 2), a divergence-free solution requires i[∂F /∂z 1 + (1/Vg)∂F/∂t 1] = 0 with Vg = 1/K 1 being the complex group velocity of the envelope F. The solution for Aj (1) in this order have been also given in the Appendix.

In the third order O(ε 3), the solvability condition yields the nonlinear SchrÖdinger (NLS) equation

iFz2K222Fτ2Weᾱz2FF2=0,

with α = ε 2 ᾱ and

W=κ(ωd4*)(ωd2*)(ωd4*)Ωs2(A3*(0)B3(2)+A3(0)B3*(2)),

being the coefficient contributed by the Kerr nonlinearity. The expression of B 3 (2) has been given in the Appendix.

Eq. (16) has complex coefficients and hence it is generally not integrable. However, the dominant part of these coefficients is real under realistic system parameters (see a practical numerical example given below) and hence it is possible to obtain shape-preserving, localized nonlinear solutions that propagate for a rather long distance without significant deformation. Combining the envelope equations of the second and the third orders we obtain the dimensionless equation

ius+2uσ2+2uu2=iLDLAu,

where u = (Ωp/U 0)exp(-iK˜0 z), s = z/(2LD), and σ = (t - z/Vg)/τ 0. LD = -τ 0 2/K˜2 is characteristic dispersion length, LA = -1/α is characteristic absorption length, and U 0 = (1/τ 0)(K˜2/W˜)1/2 is typical Rabi frequency of the probe field. Here the tilde denotes the real part of the coefficients.

When LDLA Eq. (17) reduces to the standard NLS equation i∂u/∂s+∂2 u/∂σ 2 + 2u|u|2 = 0. A simple soliton solution reads u = sech σ exp(is), or in terms of field

Ωp=1τ0K˜2W˜sech[1τ0(tzVg)]eiK˜0z+iz2LD,
 

Fig. 4. (Color online) (a): The wave shape of |Ωp/U 0|2 versus t/τ 0 and z. The initial condition is given by Eq. (18) with Ωs = 2.0 × 107 s-1 and τ 0 = 3.5 × 10-6 s. (b): The wave shape of |Ωp/U 0|2 versus t/τ 0 and z. The initial condition is given by Eq. (18) with Ωs = 1.0 × 107 s-1 and τ 0 = 1.0 × 10-5 s. The other parameters are given in the text. (c): The curves of η(z) = [A(z)/W(z)]/[A(0)/W(0)] - 1 versus the propagation length z with Ωs = 1.0 × 107 s-1 and 2.0 × 107 s-1.

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which describes a fundamental bright soliton travelling with propagating velocity Vg.

In Fig. 4(a), we show the wave shape of |Ωp/U 0|2 versus t/τ 0 and z. The result is obtained by numerical simulation directly from Eq. (2) and Eq. (3) without any approximation. The initial condition is given by Eq. (18) with the parameters Δ4 = -1.0 × 109 s-1, Ωs = 2.0 × 107 s-1, and τ 0 = 3.5 × 10-6 s. The other parameters are taken the same as those used in Fig. 2. We obtain K 0 = -(0.25 + i0.004) cm-1, K 1 = -(6.15 + i0.18) × 10-7 cm-1s (α = -7.6 Ö 10-3 cm-1), K 2 = -(3.08 + i0.14) × 10-12 cm-1s2, and W = -(7.58 + i0.12) × 10-16 cm-1s2. The characteristic lengths read LD = 4.0 cm, LA = 264 cm (LDLA) and U 0 = 1.8 × 107 s-1. With these parameters we obtain the group velocity

Vg=5.4×105c.

Hence the soliton obtained travels with a superluminal propagating velocity. We see that the superluminal optical soliton is fairly robust except for a small radiation appearing on its two wings due to higher-order dispersion and nonlinear effects that have not been included in Eq. (17). In Fig. 4 (b), we repeat the simulation by taking Ωs = 1.0 × 107 s-1 and τ 0 = 1.0 × 10-5s. We obtain LD = 0.5 cm, LA =61 cm (LDLA), Vg = -3.4 × 10-6 c, and U 0 = 1.0 × 107 s-1. We see the evolution distance for stable soliton propagation in panel (a) is much longer than that in panel (b). This is because of relatively shorter pulse length in the case of panel (b) and hence larger dispersion for the soliton.

In order to estimate the degree of shape variation of the propagating superluminal soliton, in Fig. 4(c) we have plotted the curves of η(z) = [A(z)/W(z)]/[A(0)/W(0)] - 1 versus the propagation distance z with Ωs = 1.0 × 107 s-1 and 2.0 × 107 s-1, respectively. Here, A(z) and W(z) are the maximum amplitude and half maximum full width of the soliton. From the figure, we see that the increase of η is about 7% if Ωs = 1.0 × 107 s-1 and only 1% if Ωs = 2.0 × 107 s-1 after the soliton propagating for 2.0 cm. Thus, the superluminal optical soliton can preserve its shape rather well after propagating a long distance when we make the system work in the gain spectrum hole by using a relative strong signal field.

 

Fig. 5. (Color online) The wave shape of |Ωp/U 0|2 versus t/τ 0 and z with the initial condition given by Ωp/U 0 = 0.8sech(σ - 1.0)exp( 1) + 1.2sech(σ + 3.0)exp( 2). Panel (a) and (b) show the collision for Δθθ 2 - θ 1 =0 and π/2. The lower panel show the corresponding contour maps. Both solitons in each panel have resumed their original shapes after their collision.

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The input power for generating the superluminal optical soliton can be estimated by calculating Poynting’s vector. The average flux of energy over carrier-wave period is P̄ = p̄max sech2 [(t - z/g)/τ 0], with the peak power P̄max = 2ε 0 cnp S 0|p|2 max = 2ε 0 cnp S 0(h̄/D 0)2 K˜2/(W˜τ 0 2). Here, np = 1 + cK/ωp is the refractive index and S 0 is the cross-section area of the probe field. Using the parameters in Fig. 4(a) and taking S 0 = 1.0 × 10-4 cm2, we obtain P̄max =1.4 μW. Thus, a superluminal optical soliton in the present system can be generated at very low input power.

4.2. Interaction between two superluminal optical solitons

Shape recovery after soliton collision is an interesting yet important feature of solitons. In Fig. 5 we demonstrate this feature in our system where an interaction between two bright superluminal solitons is shown by numerical means. The initial condition is chosen as

Ωp=0.8U0sech(σ1.0)eiθ1+1.2U0sech(σ+3.0)eiθ2,

where θj (j = 1, 2) is the initial phase of the jth soliton. The result for initial phase difference Δθ = θ 2 - θ 1=0 is shown in panel (a). We see that both solitons have resumed their original shapes after the collision, indicating that superluminal solitons created in our system are stable during the collision. We see also that the interaction between two solitons is repulsive. The physical reason for the repulsion is that the light intensity in the central region of the collision is decreased by the overlap of the two solitons, which leads to an decrease of the refractive index and hence ejects more light from the central region. In addition, a phase shift (i.e. position shift) is observed clearly after the collision. In panel (b) we show the result for Δθ = π/2. In this case the interaction is attractive. This can be understood as the result that the light intensity in the central region of the collision is increased by the overlap of the two solitons, which leads to an increase of the refractive index and hence attracts more light to the central region. Again the two superluminal optical solitons remain stably during their collision.

5. Discussion and Summary

It is possible to give an experimental demonstration of the giant Kerr nonlinearity and superlu-minal optical solitons predicted above by using a 87Ru atomic gas. At room-temperature, atoms suffer Doppler effect due to their significant thermal velocity, which will generally shallow the depth and narrow the width of the gain spectrum hole shown in Fig. 2(b). However, this difficulty can be overcome largely by using following scheme. When an atom has thermal velocity V, the radiating frequencies ωj (j = c,p,s) should be replaced by ωj - k j · V = ωj - kjZVz (we assume the coupling, probe, and signal fields propagate along the z direction). Thus in Eq. (1) the one-photon and three photon detunings should be replaced by Δ3 → Δ3 (Vz) = ωc - (E 3 - E 1)/h̄ - kcz Vz and Δ4 → Δ4(VZ) = (ωc - ωp + ωs) - (E 4-E 1)h̄ + (kp - kc - ks)zVz, respectively. Then the second term on the left hand side of Eq. (3) should be averaged over Vz for a given thermal velocity (e.g. Maxwell) distribution function f(Vz). To suppress the Doppler effect of high-lying levels at room temperature, state |4〉 should be chosen as a member of the ground state manifold (notice that in our parameters, γ 4 < γ 3), and therefore Ωs should be a circularly polarized microwave field. Notice that, since kp and kc are much larger than ks, the Doppler effect in the three-photon detuning Δ4(Vz) for copropagating case (kpkc > 0) is much smaller than that for the counterpropagating case (kpkc < 0). Therefore, one can take all light fields to propagate in the same (i.e. copropagating) direction to suppress the Doppler effect appearing in the three-photon detuning Δ4(VZ). In addition, the Doppler effect appearing in one-photon detuning Δ3 (Vz) can be efficiently avoided by taking a large ωc - (E 3 - E 1)/h̄, and hence the influence by kczVz is largely suppressed.

Notice that in the present work the dependence of the optical field on the spatial transverse coordinates is not taken into account because the light considered is assumed to be guided in the z direction. If the probe field is not guided, one needs to add diffraction terms (in the transverse x - y plane) into the model. In this way, Eq. (16) will be replaced in a multidimensional nonlinear SchrÖdinger equation. Although solutions for the spatiotemporal solitons (bullets) in such model can be found, the solutions are usually unstable in a pure Kerr medium because of the spatiotemporal spread or collapse. For securing the stability, one has to introduce different types of saturable Kerr nonlinearity and/or higher-order dispersion effects, which exist in the present system. Thus it is possible to observe superluminal light bullets in the system we study, but it is a topic beyond the scope of the present work.

In conclusion, in this work we have investigated an ARG scheme for realizing giant Kerr nonlinearity and superluminal optical solitons in a four-state atomic system with a gain doublet. We have shown that this scheme, which is fundamentally different from the EIT-based four-state atomic system, is capable of working at room temperature and eliminating nearly all attenuation and distortion. Because of the quantum interference contributed by a signal field, a gain spectrum hole is opened and a significant enhancement of the cross Kerr nonlinear effect can be obtained, which is more than ten times larger than that arrived by the EIT-based scheme under the same energy-level configuration. Based on these important features, we have obtained a giant cross-phase modulation effect and demonstrated that stable optical solitons with superluminal propagating velocity and very low generating power can be produced in the system. The results presented in this work may have potential applications in rapidly responding optical information processing and transmission.

A. Appendix: First and second order solutions of Aj (j = 1,2,3,4)

The solution of Aj in the first order reads

A1(1)=A3(1)=0,
A2(1)=[(K*ω/c)F*κA3*(0)]eiθ*,
A4(1)=B4(1)F*eiθ*

with B 4 (1) = [Ωs * A 3 *(0)]/κ] [(ω - d 2) (K * - ω/c) - κ|A 3 (0)|2].

The solution of Aj at the second order is given by

Aj(2)=Bj(2)F2eᾱz2,(j=1,3),
A2(2)=iκA3*(0)(1c1Vg*)F*t1eiθ*,
A4(2)=iκA3*(0)Ωs*[K*ωc+(ω+d2)(1Vg*1c)]
×F*t1eiθ*,

with

B1(2)=12A1(0)(1+Ωc/d32)[Kω/cκd3*+K*ω/cκd3
Kω/c2κ2A3(0)2B4(1)2],
B3(2)=1d3(K*ω/cκA3*(0)+ΩcB1(2)).

Acknowledgments

C. H. was supportedby the FCT under Grant No. SFRH/BPD/36385/2007.G. H. was supported by the NSF-China under Grant Nos. 10674060 and 10874043, and by the Key Development Program for Basic Research of China under Grant No. 2005CB724508 and 2006CB921104.

References and links

1. R. W. Boyd, Nonlinear Optics, 2cd edition (Academic, San Diego, 2003).

2. J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997). [CrossRef]  

3. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995). [CrossRef]   [PubMed]  

4. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003). [CrossRef]   [PubMed]  

5. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006). [CrossRef]  

6. S. E. Harris and Y. Yamamoto, “Photon Switching by Quantum Interference,” Phys. Rev. Lett. 81, 3611(1998). [CrossRef]  

7. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

8. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein. [CrossRef]  

9. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef]   [PubMed]  

10. Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064 (2004). [CrossRef]   [PubMed]  

11. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005). [CrossRef]  

12. Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005). [CrossRef]  

13. G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006). [CrossRef]  

14. L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005). [CrossRef]  

15. C. Hang, G. Huang, and L. Deng, “Generalized nonlinear SchrÖdinger equation and ultraslow optical solitons in a cold four-state atomic system,” Phys. Rev. E 73, 036607 (2006). [CrossRef]  

16. C. Hang, G. Huang, and L. Deng, “Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system,” Phys. Rev. E 74, 046601 (2006). [CrossRef]  

17. Y. Wu and X. Yang, “Giant Kerr nonlinearity and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007). [CrossRef]  

18. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008). [CrossRef]  

19. W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008). [CrossRef]  

20. Chao Hang, V. V. Konotop, and Guoxiang Huang, “Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field,” 79, 033826 (2009).

21. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993). [CrossRef]  

22. A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994). [CrossRef]   [PubMed]  

23. L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000). [CrossRef]  

24. A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001). [CrossRef]  

25. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001). [CrossRef]   [PubMed]  

26. A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003). [CrossRef]  

27. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003). [CrossRef]   [PubMed]  

28. M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003). [CrossRef]  

29. M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003). [CrossRef]  

30. R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003). [CrossRef]  

31. K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003). [CrossRef]  

32. L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003). [CrossRef]  

33. E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004). [CrossRef]  

34. G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]  

35. A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006). [CrossRef]  

36. M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006). [CrossRef]  

37. J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnet-ically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006). [CrossRef]   [PubMed]  

38. K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007). [CrossRef]  

39. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef]   [PubMed]  

40. E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003). [CrossRef]   [PubMed]  

41. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005). [CrossRef]   [PubMed]  

42. L. Thévenaz “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]  

43. Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Applied Physics Letters , 91094104 (2007). [CrossRef]  

44. S. Residori, U. Bortolozzo, and J. P. Huignard, “Slow and Fast Light in Liquid Crystal Light Valves,” Phys. Rev. Lett. 100, 203603 (2008). [CrossRef]   [PubMed]  

45. T. Baba, “Slow light in photonic crystals,” Nat Photon. 2, 465–473 (2008). [CrossRef]  

46. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33, 2389–2391 (2008). [CrossRef]   [PubMed]  

47. B. D. Clader, Q-Han Park, and J. H. Eberly, “Fast light in fully coherent gain media,” Opt. Lett. 31, 2921 (2006). [CrossRef]   [PubMed]  

48. G. S. Agarwal and T. N. Dey, “Fast light solitons in resonant media,” Phys. Rev. A 75, 043806 (2007). [CrossRef]  

49. G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008). [CrossRef]  

50. H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008). [CrossRef]  

51. The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol. 43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

52. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936 (1996). [CrossRef]   [PubMed]  

53. A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

References

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  1. R. W. Boyd, Nonlinear Optics, 2cd edition (Academic, San Diego, 2003).
  2. J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
    [Crossref]
  3. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
    [Crossref] [PubMed]
  4. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
    [Crossref] [PubMed]
  5. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
    [Crossref]
  6. S. E. Harris and Y. Yamamoto, “Photon Switching by Quantum Interference,” Phys. Rev. Lett. 81, 3611(1998).
    [Crossref]
  7. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
  8. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
    [Crossref]
  9. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
    [Crossref] [PubMed]
  10. Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064 (2004).
    [Crossref] [PubMed]
  11. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
    [Crossref]
  12. Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
    [Crossref]
  13. G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
    [Crossref]
  14. L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005).
    [Crossref]
  15. C. Hang, G. Huang, and L. Deng, “Generalized nonlinear SchrÖdinger equation and ultraslow optical solitons in a cold four-state atomic system,” Phys. Rev. E 73, 036607 (2006).
    [Crossref]
  16. C. Hang, G. Huang, and L. Deng, “Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system,” Phys. Rev. E 74, 046601 (2006).
    [Crossref]
  17. Y. Wu and X. Yang, “Giant Kerr nonlinearity and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
    [Crossref]
  18. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
    [Crossref]
  19. W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008).
    [Crossref]
  20. Chao Hang, V. V. Konotop, and Guoxiang Huang, “Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field,”  79, 033826 (2009).
  21. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
    [Crossref]
  22. A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
    [Crossref] [PubMed]
  23. L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
    [Crossref]
  24. A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
    [Crossref]
  25. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
    [Crossref] [PubMed]
  26. A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
    [Crossref]
  27. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
    [Crossref] [PubMed]
  28. M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
    [Crossref]
  29. M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
    [Crossref]
  30. R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
    [Crossref]
  31. K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
    [Crossref]
  32. L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
    [Crossref]
  33. E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
    [Crossref]
  34. G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
    [Crossref]
  35. A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
    [Crossref]
  36. M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
    [Crossref]
  37. J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnet-ically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006).
    [Crossref] [PubMed]
  38. K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
    [Crossref]
  39. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
    [Crossref] [PubMed]
  40. E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003).
    [Crossref] [PubMed]
  41. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
    [Crossref] [PubMed]
  42. L. Thévenaz “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
    [Crossref]
  43. Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Applied Physics Letters,  91094104 (2007).
    [Crossref]
  44. S. Residori, U. Bortolozzo, and J. P. Huignard, “Slow and Fast Light in Liquid Crystal Light Valves,” Phys. Rev. Lett. 100, 203603 (2008).
    [Crossref] [PubMed]
  45. T. Baba, “Slow light in photonic crystals,” Nat Photon. 2, 465–473 (2008).
    [Crossref]
  46. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33, 2389–2391 (2008).
    [Crossref] [PubMed]
  47. B. D. Clader, Q-Han Park, and J. H. Eberly, “Fast light in fully coherent gain media,” Opt. Lett. 31, 2921 (2006).
    [Crossref] [PubMed]
  48. G. S. Agarwal and T. N. Dey, “Fast light solitons in resonant media,” Phys. Rev. A 75, 043806 (2007).
    [Crossref]
  49. G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
    [Crossref]
  50. H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
    [Crossref]
  51. The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).
  52. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936 (1996).
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  53. A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

2009 (1)

Chao Hang, V. V. Konotop, and Guoxiang Huang, “Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field,”  79, 033826 (2009).

2008 (8)

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008).
[Crossref]

S. Residori, U. Bortolozzo, and J. P. Huignard, “Slow and Fast Light in Liquid Crystal Light Valves,” Phys. Rev. Lett. 100, 203603 (2008).
[Crossref] [PubMed]

T. Baba, “Slow light in photonic crystals,” Nat Photon. 2, 465–473 (2008).
[Crossref]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33, 2389–2391 (2008).
[Crossref] [PubMed]

L. Thévenaz “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[Crossref]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
[Crossref]

2007 (5)

Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Applied Physics Letters,  91094104 (2007).
[Crossref]

G. S. Agarwal and T. N. Dey, “Fast light solitons in resonant media,” Phys. Rev. A 75, 043806 (2007).
[Crossref]

Y. Wu and X. Yang, “Giant Kerr nonlinearity and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[Crossref]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

2006 (8)

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnet-ically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006).
[Crossref] [PubMed]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Generalized nonlinear SchrÖdinger equation and ultraslow optical solitons in a cold four-state atomic system,” Phys. Rev. E 73, 036607 (2006).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system,” Phys. Rev. E 74, 046601 (2006).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
[Crossref]

B. D. Clader, Q-Han Park, and J. H. Eberly, “Fast light in fully coherent gain media,” Opt. Lett. 31, 2921 (2006).
[Crossref] [PubMed]

2005 (5)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

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

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

2004 (4)

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064 (2004).
[Crossref] [PubMed]

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

2003 (9)

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[Crossref] [PubMed]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

2002 (1)

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

2001 (2)

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

2000 (1)

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

1998 (1)

S. E. Harris and Y. Yamamoto, “Photon Switching by Quantum Interference,” Phys. Rev. Lett. 81, 3611(1998).
[Crossref]

1997 (1)

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

1996 (1)

1995 (1)

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
[Crossref] [PubMed]

1994 (1)

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

1993 (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and T. N. Dey, “Fast light solitons in resonant media,” Phys. Rev. A 75, 043806 (2007).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

Akulshin, A. M.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Artoni, M.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat Photon. 2, 465–473 (2008).
[Crossref]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

Bortolozzo, U.

S. Residori, U. Bortolozzo, and J. P. Huignard, “Slow and Fast Light in Liquid Crystal Light Valves,” Phys. Rev. Lett. 100, 203603 (2008).
[Crossref] [PubMed]

Boyd, R. W.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

R. W. Boyd, Nonlinear Optics, 2cd edition (Academic, San Diego, 2003).

Cataliotti, F.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Chiao, R. Y.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
[Crossref]

Cimmino, A.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Clader, B. D.

Dasgupta, S.

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

Deng, L.

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
[Crossref]

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Generalized nonlinear SchrÖdinger equation and ultraslow optical solitons in a cold four-state atomic system,” Phys. Rev. E 73, 036607 (2006).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system,” Phys. Rev. E 74, 046601 (2006).
[Crossref]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Dey, T. N.

G. S. Agarwal and T. N. Dey, “Fast light solitons in resonant media,” Phys. Rev. A 75, 043806 (2007).
[Crossref]

Dogariu, A.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

Eberly, J. H.

Ferrari, C.

Fleischhauer, M.

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

Gaeta, A. L.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Gauthier, D. J.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

Ghulghazaryan, R. G.

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

Grangier, Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

Grelu, Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

Hagley, E. W.

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Hang, C.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Generalized nonlinear SchrÖdinger equation and ultraslow optical solitons in a cold four-state atomic system,” Phys. Rev. E 73, 036607 (2006).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system,” Phys. Rev. E 74, 046601 (2006).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
[Crossref]

Hang, Chao

Chao Hang, V. V. Konotop, and Guoxiang Huang, “Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field,”  79, 033826 (2009).

Hannaford, P.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Harris, S. E.

S. E. Harris and Y. Yamamoto, “Photon Switching by Quantum Interference,” Phys. Rev. Lett. 81, 3611(1998).
[Crossref]

Hernandez, G.

Hood, C. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
[Crossref] [PubMed]

Hou, J.-M.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008).
[Crossref]

Huang, G.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system,” Phys. Rev. E 74, 046601 (2006).
[Crossref]

C. Hang, G. Huang, and L. Deng, “Generalized nonlinear SchrÖdinger equation and ultraslow optical solitons in a cold four-state atomic system,” Phys. Rev. E 73, 036607 (2006).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
[Crossref]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Huang, Guoxiang

Chao Hang, V. V. Konotop, and Guoxiang Huang, “Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field,”  79, 033826 (2009).

Huignard, J. P.

S. Residori, U. Bortolozzo, and J. P. Huignard, “Slow and Fast Light in Liquid Crystal Light Valves,” Phys. Rev. Lett. 100, 203603 (2008).
[Crossref] [PubMed]

Imamoglu, A.

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

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936 (1996).
[Crossref] [PubMed]

Janowicz, M.

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

Jeffery, A.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

Jiang, K.

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

Jiang, K. J.

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

Kawahawa, T.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

Kim, J. B.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Kim, K.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Kim, S. K.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Kimble, H. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
[Crossref] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

Konotop, V. V.

Chao Hang, V. V. Konotop, and Guoxiang Huang, “Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field,”  79, 033826 (2009).

Kuzmich, A.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

Lange, W.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
[Crossref] [PubMed]

Lee, C.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Lee, R.-K.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008).
[Crossref]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

Lezama, A.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

Li, H.

H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
[Crossref]

Li, Y.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
[Crossref]

Lin, Q.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Liu, N.-H.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Ma, L.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
[Crossref]

Mabuchi, H.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
[Crossref] [PubMed]

Malakyan, Y. P.

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

Marangos, J. P.

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

Martinelli, M.

Melloni, A.

Mikhailov, E. E.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Milonni, P. W.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

Moon, H. S.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Morichetti, F.

Mostowski, J.

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

Neifield, M. A.

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

Novikova, I.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Odoulov, S.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[Crossref] [PubMed]

Okawachi, Y.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Opat, G. I.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Ottaviani, C.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Park, Q-Han

Payne, M. G.

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Podivilov, E.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[Crossref] [PubMed]

Pogariu, P.

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

Poizat, J.-Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

Residori, S.

S. Residori, U. Bortolozzo, and J. P. Huignard, “Slow and Fast Light in Liquid Crystal Light Valves,” Phys. Rev. Lett. 100, 203603 (2008).
[Crossref] [PubMed]

Roch, J.-F.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

Sautenkov, V. A.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Schmidt, H.

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Sharping, J. E.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Shumelyuk, A.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[Crossref] [PubMed]

Sidorov, A. I.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Sinatra, A.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

Stenner, M. D.

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

Sturman, B.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, “Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling,” Phys. Rev. Lett. 91, 083902 (2003).
[Crossref] [PubMed]

Thévenaz, L.

L. Thévenaz “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[Crossref]

Tombesi, P.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Turchette, Q. A.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of Conditional Phase Shifts for Quantum Logic,” Phys. Rev. Lett. 75, 4710 (1995).
[Crossref] [PubMed]

Vigneron, K.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78, 634 (1997).
[Crossref]

Vitali, D.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Wang, L. J.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

Wang, L.-G.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Welch, G. R.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Wu, Y.

Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Applied Physics Letters,  91094104 (2007).
[Crossref]

Y. Wu and X. Yang, “Giant Kerr nonlinearity and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[Crossref]

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064 (2004).
[Crossref] [PubMed]

Yamamoto, Y.

S. E. Harris and Y. Yamamoto, “Photon Switching by Quantum Interference,” Phys. Rev. Lett. 81, 3611(1998).
[Crossref]

Yang, W.-X.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008).
[Crossref]

Yang, X.

Y. Wu and X. Yang, “Giant Kerr nonlinearity and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[Crossref]

Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Applied Physics Letters,  91094104 (2007).
[Crossref]

Zhang, J.

Zhu, S.-Y.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Zhu, Y.

Zhu, Z.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

Y. Wu and X. Yang, “Giant Kerr nonlinearity and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[Crossref]

Applied Physics Letters (1)

Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Applied Physics Letters,  91094104 (2007).
[Crossref]

Nat Photon. (1)

T. Baba, “Slow light in photonic crystals,” Nat Photon. 2, 465–473 (2008).
[Crossref]

Nat. Photonics (1)

L. Thévenaz “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[Crossref]

Nature (London) (2)

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

Opt. Lett. (5)

Phy. Rev. A (1)

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phy. Rev. A 74, 012319 (2006).
[Crossref]

Phys. Rev. A (17)

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008).
[Crossref]

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
[Crossref]

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

G. S. Agarwal and T. N. Dey, “Fast light solitons in resonant media,” Phys. Rev. A 75, 043806 (2007).
[Crossref]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

H. Li, C. Hang, G. Huang, and L. Deng, “High-order nonlinear SchrÖdinger equation and superluminal optical solitons in room-temperature active-Raman-gain media,” Phys. Rev. A 78, 023822 (2008).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Phys. Rev. E (7)

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

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[Crossref]

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The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, seeR. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol.  43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, “Controlling the Velocity of Light Pulses,” Science 326, 1074 (2009).

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Figures (5)

Fig. 1.
Fig. 1. (Color online) (a): Energy-level configuration and excitation scheme of the four-state ARG system, which interacts with a strong CW coupling field of angular frequency ω c and Rabi frequency 2Ω C (driving the transition |1⎤ ↔ |3⎤), a weak, pulsed probe field of angular frequency ωp and Rabi frequency 2Ω p (driving the transition |2⎤ ↔ |3⎤), and a CW signal field of angular frequency 2ωs and Rabi frequency Ω s (driving the transition |2⎤ ↔ |4⎤). Δ j (j = 3, 4) are the corresponding detunings. (b): The four-state EIT system with the same energy-level configuration.
Fig. 2.
Fig. 2. (Color online) (a) The curve of Re[K(ω)] characterizing refractive index of the probe field as a function of ω. (b) The curve of -Im[K(ω)] characterizing the appearance of the gain doublet of the probe field for increasing signal fiels. (c) The curve of Vg /c as a function of Δ4. The solid, dashed, and dot-dashed lines correspond to Ω s = 4.0 × 107 s-1, 2.0 × 107 s-1, and 1.0 × 107 s-1, respectively. The inset shows the detail of the curve of Vg /c with Ω s = 1.0 × 107 s-1. The other parameters are given in the text.
Fig. 3.
Fig. 3. (Color online) (a): The curve of ϕXPM for the EIT system (solid line) and the curve of -10ϕXPM of the EIT system (dashed line), as functions of Ω s . (b): The curves of ϕXPM (solid line) and ϕ =Re[K 0] (dashed line) versus Ω s for ARG system. The parameters are given in the text.
Fig. 4.
Fig. 4. (Color online) (a): The wave shape of |Ω p /U 0|2 versus t/τ 0 and z. The initial condition is given by Eq. (18) with Ω s = 2.0 × 107 s-1 and τ 0 = 3.5 × 10-6 s. (b): The wave shape of |Ω p /U 0|2 versus t/τ 0 and z. The initial condition is given by Eq. (18) with Ω s = 1.0 × 107 s-1 and τ 0 = 1.0 × 10-5 s. The other parameters are given in the text. (c): The curves of η(z) = [A(z)/W(z)]/[A(0)/W(0)] - 1 versus the propagation length z with Ω s = 1.0 × 107 s-1 and 2.0 × 107 s-1.
Fig. 5.
Fig. 5. (Color online) The wave shape of |Ω p /U 0|2 versus t/τ 0 and z with the initial condition given by Ω p /U 0 = 0.8sech(σ - 1.0)exp( 1) + 1.2sech(σ + 3.0)exp( 2). Panel (a) and (b) show the collision for Δθθ 2 - θ 1 =0 and π/2. The lower panel show the corresponding contour maps. Both solitons in each panel have resumed their original shapes after their collision.

Tables (2)

Tables Icon

Table 1. Expressions of linear and nonlinear optical susceptibilities and XPM phase shift for the four-state ARG system (Fig. 1(a)) and the four-state EIT system (Fig. 1(b)).

Tables Icon

Table 2. Values of linear and nonlinear optical susceptibilities and XPM phase shift for the four-state ARG scheme and the four-state EIT scheme. The parameters used have been given in the text

Equations (40)

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H ̂ int / h ¯ = [ Δ 3 3 3 + Δ 4 4 4 + Ω c 3 1 + Ω p 3 2
+ Ω s 4 2 + H . c . ] ,
( i t + d 2 ) A 2 + Ω p * A 3 + Ω s * A 4 = 0 ,
( i t + d 3 ) A 3 + Ω c A 1 + Ω p A 2 = 0 ,
( i t + d 4 ) A 4 + Ω s A 2 = 0
i ( z + 1 c t ) Ω p + κ A 3 A 2 * = 0 ,
K ( ω ) = ω c + κ ( ω d 4 * ) Ω c 2 [ ( ω d 2 * ) ( ω d 4 * ) Ω s 2 ] ( d 3 2 + Ω 2 2 ) .
Ω p ( z , t ) = Ω p ( 0,0 ) e i K 0 z b 1 ( z ) i b 2 ( z ) exp [ ( K 1 t z ) 2 2 [ b 1 ( z ) i b 2 ( z ) ] τ 0 2 ] ,
K 0 = ϕ + i α 2 , K 1 = 1 c κ Ω c 2 Ω s 2 d 3 2 + κ Ω c 2 d 4 * Ω s 4 d 3 2 ,
K 2 = 4 κ Ω c 2 d 4 * Ω s 4 d 3 2 + 2 κ Ω c 2 d 4 * Ω s 6 d 3 2 ,
D ARG = Im K ( ω = ω max ) Im K ( ω = 0 )
= 4 κ Ω s 2 Ω c 2 γ 4 ( 4 Ω s 2 γ 4 2 ) d 3 2 κ γ 4 Ω c 2 Ω s 2 d 3 2 .
Ω s 2 2 κ γ 4 Ω c 2 / d 3 2 .
Γ ARG = 2 4 Ω s 2 2 γ 4 2 + 2 γ 4 4 Ω s 2 γ 4 2 .
Ω p * A 3 ( 0 ) + Ω s * A 4 = ω i γ 2 κ A 3 * ( 0 ) ( K * ω c ) F * exp ( i θ * ) ,
χ p = N a p 23 2 ε 0 h ¯ A 3 A 2 * Ω p χ p ( 1 ) + χ pp ( 3 ) p 2 + χ ps ( 3 ) s 2 ,
A 2 = d 4 Ω c Ω p * D A 1 ,
A 3 = Ω c Ω s D A 1 ,
A 4 = Ω c Ω s Ω p * D A 1 ,
A 1 = [ 1 + Ω c 2 D s 2 + ( Ω s 2 + d 4 2 ) Ω p 2 D 2 ] 1 / 2 .
χ p ( 1 ) = N a p 23 2 ε 0 h ̄ i Ω c 2 γ 2 ( Ω c 2 + d 3 2 ) ,
χ pp ( 3 ) = N a p 23 4 ε 0 h ¯ 3 i ( 2 γ 2 γ 3 + Ω c 2 ) Ω c 2 γ 2 3 ( Ω c 2 + d 3 2 ) 2 ,
χ ps ( 3 ) = N a p 23 2 p 24 2 ε 0 h ¯ 3 d 4 Ω c 2 γ 2 2 ( Ω c 2 + d 3 2 ) d 4 2 ,
ϕ XPM = ω p L 2 c Re [ χ ps ( 3 ) ] s 2 κL Δ 4 Ω c 2 γ 2 2 d 3 2 d 4 2 Ω s 2 ,
i F z 2 K 2 2 2 F τ 2 W e α ̄ z 2 F F 2 = 0 ,
W = κ ( ω d 4 * ) ( ω d 2 * ) ( ω d 4 * ) Ω s 2 ( A 3 * ( 0 ) B 3 ( 2 ) + A 3 ( 0 ) B 3 * ( 2 ) ) ,
i u s + 2 u σ 2 + 2 u u 2 = i L D L A u ,
Ω p = 1 τ 0 K ˜ 2 W ˜ sech [ 1 τ 0 ( t z V g ) ] e i K ˜ 0 z + i z 2 L D ,
V g = 5.4 × 10 5 c .
Ω p = 0.8 U 0 sech ( σ 1.0 ) e i θ 1 + 1.2 U 0 sech ( σ + 3.0 ) e i θ 2 ,
A 1 ( 1 ) = A 3 ( 1 ) = 0 ,
A 2 ( 1 ) = [ ( K * ω / c ) F * κ A 3 * ( 0 ) ] e i θ * ,
A 4 ( 1 ) = B 4 ( 1 ) F * e i θ *
A j ( 2 ) = B j ( 2 ) F 2 e α ̄ z 2 , ( j = 1,3 ) ,
A 2 ( 2 ) = i κ A 3 * ( 0 ) ( 1 c 1 V g * ) F * t 1 e i θ * ,
A 4 ( 2 ) = i κ A 3 * ( 0 ) Ω s * [ K * ω c + ( ω + d 2 ) ( 1 V g * 1 c ) ]
× F * t 1 e i θ * ,
B 1 ( 2 ) = 1 2 A 1 ( 0 ) ( 1 + Ω c / d 3 2 ) [ K ω / c κ d 3 * + K * ω / c κ d 3
K ω / c 2 κ 2 A 3 ( 0 ) 2 B 4 ( 1 ) 2 ] ,
B 3 ( 2 ) = 1 d 3 ( K * ω / c κ A 3 * ( 0 ) + Ω c B 1 ( 2 ) ) .

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