## 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 *ω _{c}* driving the transition |1⎤ ↔ |3⎤ and a probe field of angular frequency

*ω*driving the transition |2⎤ ↔ |3⎤ create the three-level ARG core [49, 50]. A CW signal field of angular frequency

_{p}*ω*drives the transition |2⎤ ↔ |4⎤. 2Ω

_{s}_{c}, 2Ω

_{p}, and 2Ω

_{s}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

$$+{\Omega}_{s}\mid 4\u3009\u30082\mid +H.c.],$$

where Δ_{3} = *ω _{c}* -(

*E*

_{3}-

*E*

_{1})/

*h*̄, and Δ

_{4}=

*ω*-

_{c}*ω*+

_{p}*ω*- (

_{s}*E*

_{4}-

*E*

_{1})/

*h*̄ are the one- and three-photon detunings (the two-photon detuning is set to be zero). The equations of motion for atomic response are

with the normalization condition Σ_{j=1}
^{4} |*A _{j}*|

^{2}= 1 , where

*A*(

_{j}*j*= 1 to 4) is the probability amplitude of the bare atomic state |

*j*〉 (with eigenenergy

*E*),

_{j}*d*= Δ

_{j}_{j}+

*iγ*(

_{j}*j*= 2 to 4, Δ

_{2}= 0) with

*γ*being the atomic decay rate of the state |

_{j}*j*〉.

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

where *κ* = *N _{a}*

*ω*|

_{p}**P**

_{23}|

^{2}/ (2

*ε*

_{0}

*ch*̄) with

*N*being the atomic density and

_{a}**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. |Δ_{3}| 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 ${A}_{1}^{\left(0\right)}=1/\sqrt{1+{\mid {\Omega}_{c}/{d}_{3}\mid}^{2}},$
${A}_{3}^{\left(0\right)}=-{\Omega}_{c}/\left({d}_{3}\sqrt{1+{\mid {\Omega}_{c}/{d}_{3}\mid}^{2}}\right),$
*A*
_{2}
^{(0)} = *A*
_{4}
^{(0)} = 0, and Ω_{p}
^{(0)} = 0.

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

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 + *ℬ*(*ω*
^{3}) with *K _{j}* =

*∂*(

^{j}K*ω*)/

*∂ω*)|

^{j}_{ω=0}(

*j*= 0,1,2,…). If the probe field at

*z*= 0 is a Gaussian pulse, i.e. Ω

_{p}(0,

*t*) = Ω

_{p}(0,0)exp[-

*t*

^{2}/(2

*τ*

_{0}

^{2})], we then readily obtain

where *b*
_{1}(*z*) = 1 + *z*Im(*K*
_{2})/*τ*
_{0}
^{2} and *b*
_{2}(*z*) = *z*Re(*K*
_{2})/*τ*
_{0}
^{2}. The expansion coefficients under the conditions |*d*
_{3}|≫| Ω_{c}| 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

$${K}_{2}=4\kappa \frac{{\mid {\Omega}_{c}\mid}^{2}{d}_{4}^{*}}{{\mid {\Omega}_{s}\mid}^{4}{\mid {d}_{3}\mid}^{2}}+2\kappa \frac{{\mid {\Omega}_{c}\mid}^{2}{d}_{4}^{*}}{{\mid {\Omega}_{s}\mid}^{6}{\mid {d}_{3}\mid}^{2}},$$

where *ϕ*=*κ*Δ_{4}|Ω_{c}|^{2}/(|Ω_{s}|^{2}|*d*
_{3}|^{2}) gives a phase shift per unit length and *α*=-2*κγ*
_{4}|Ω_{c}|^{2}/(|Ω_{s}|^{2}|*d*
_{3}|^{2}) leads to a (intensity) gain to the probe field. The probe field group velocity *V _{g}* =

*c*/

*n*= l/Re(

_{g}*K*

_{1}) whereas Im(

*K*

_{1}) contributes to probe wave gain. The group index

*n*= 1 -

_{g}*cκ*(|Ω

_{s}|

^{2}+ Δ

_{4}

^{2}- γ

_{4}

^{2})|Ω

_{c}|

^{2}/(|Ω

_{s}|

^{4}|

*d*

_{3}|

^{2}). It is noteworthy that one gets 0 <

*V*<

_{g}*c*when |Ω

_{s}|

^{2}+ Δ

_{4}

^{2}<

*γ*

_{4}

^{2}(corresponding to a sub luminal propagation), while

*V*>

_{g}*c*or

*V*< 0 when |Ω

_{g}_{s}|

^{2}+ Δ

_{4}

^{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 Ω_{s}. The parameters we used are related to a typical warm alkali atom gas, e.g. ^{87}Rb atom gas, where *κ* = 1.0 × 10^{9} cm^{-1} s^{-1}, 2*γ*
_{2} = 300 Hz, 2*γ*
_{3} = 500 MHz, and 2*γ*
_{4} = 30 MHz [39]. The other parameters are taken as Δ3 = -2.0 × 10^{9} s^{-1}, Δ_{4} = 0, and Ω_{c} = 2.0 × 10^{7} s^{-1}. From panel (b), we see that for small signal-field Rabi frequency (e. g. Ω_{s} = 1.0 × 10^{7} 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 Ω_{s} 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|Ω_{s}|^{2} - 2*γ*
_{4}
^{2})^{1/2}, which appear for Ω_{s} exceeding its critical value Ω* _{s}^{cr}* =

*γ*

_{4}/√2 ≃ 1.1 × 10

^{7}s

^{-1}. It is easy to get the depth of the gain spectrum hole, i. e.

$$=\frac{4\kappa {\mid {\Omega}_{s}\mid}^{2}{\mid {\Omega}_{c}\mid}^{2}}{{\gamma}_{4}\left(4{\mid {\Omega}_{s}\mid}^{2}-{\gamma}_{4}^{2}\right){\mid {d}_{3}\mid}^{2}}-\frac{\kappa {\gamma}_{4}{\mid {\Omega}_{c}\mid}^{2}}{{\mid {\Omega}_{s}\mid}^{2}{\mid {d}_{3}\mid}^{2}}.$$

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

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

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

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 *V _{g}*/

*c*as functions of Δ

_{4}with different values of Ω

_{s}. The inset shows the detail of the curve of

*V*/

_{g}*c*with Ω

_{s}= 1.0 × 10

^{7}s

^{-1}. From the figure we see that

*V*with Ω

_{g}_{s}= 1.0 × 10

^{7}s

^{-1}changes sign from the positive to negative at Δ

_{4}≃ (

*γ*

_{4}

^{2}- |Ω

_{s}|

^{2})

^{1/2}= 1.1 × 10

^{7}s

^{-1}. Therefore, it is possible to tune the group velocity from negative to positive values by simply decreasing Δ

_{4}below 1.1 × 10

^{7}s

^{-1}.

*V*with Ω

_{g}_{s}= 2.0 × 10

^{7}s

^{-1}and 4.0 × 10

^{7}s

^{-1}are always negative because

*γ*

_{4}

^{2}- |Ω

_{s}|

^{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

where we have introduced the linear susceptibility *χ _{p}*

^{(1)}, the third order self- and cross-Kerr susceptibilities

*χ*

_{pp}^{(3)}and

*χ*

_{ps}^{(3)}, respectively. The explicit expressions of static susceptibilities can be obtained by solving Eq. (2) under a steady state approximation, which read

with *D _{s}* = |Ω

_{s}|

^{2}-

*d*

_{2}

*d*

_{4}and

*D*= |Ω

_{s}|

^{2}

*d*

_{3}+ |Ω

_{p}|

^{2}

*d*

_{4}-

*d*

_{2}

*d*

_{3}

*d*

_{4}. The expression of

*A*

_{1}can be obtained by using the condition Σ

_{j=1}

^{4}|

*A*|

_{j}^{2}= 1 and hence

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

From Eq. (14), we obtain the following conclusions: (i) Both *χ _{p}*

^{(1)}and

*χ*

_{pp}^{(3)}are Pure imaginary due to the exact two-photon resonance. Contrary to EIT-based systems here the sign of Im[

*χ*

_{p}^{(1)}] is always negative, corresponding to a linear gain. (ii) The sign of Im[

*χ*

_{pp}^{(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[

*χ*

_{ps}^{(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}| ≫ |Ω_{c}| the phase shift due to such XPM effect is given by

with *L* being the length of the medium. It is easily to obtain that *ϕ*
_{XPM}/(*αL*) ≈ -Δ_{4}/(2*γ*
_{4}). We note that the condition of three-photon off-resonance, i. e. Δ_{4} ≠ 0, is crucial to the observation of a nonzero XPM phase shift. We can also obtain Eq. (15) through expanding *ϕ* around |Ω_{s}|^{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 *χ _{p}*

^{(1)},

*χ*

_{pp}^{(3)},

*χ*

_{ps}^{(3)}, and

*ϕ*

_{XPM}for the four-state ARG (Fig. 1 (a)) and four-state EIT (Fig. 1(b)) systems, respectively.

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 Δ_{4} = 2.0 × 10^{8} s^{-1}, Ω_{c} = 1.0 × 10^{6} s^{-1}, and Ω_{s} = 1.0 × 10^{5} s^{-1} with the other ones being the same with those use in Fig. 2. Parameters for the EIT system are taken as *γ*
_{3} ≈ *γ*
_{4} = 10 MHz (corresponding to the cold atom gas) and Δ_{3} = 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.

Shown in Fig. 3(a) are curves of *ϕ _{XPM}* and -10

*ϕ*as functions of Ω

_{XPM}_{s}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

*ϕ*and

_{XPM}*ϕ*=Re[

*K*

_{0}] versus Ω

_{s}for the ARG system. The difference between the XPM phase shift and total phase shift, i.e.

*ϕ*-

_{XPM}*ϕ*, is negligible only for a weak signal field. This is because

*ϕ*can be obtained by expanding

_{XPM}*ϕ*around |Ω

_{s}|

^{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 (Δ_{4} ≠ 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 *A _{j}* = Σ

_{n=0}

^{∞}

*ε*

^{n}*A*

_{j}^{(n)}(

*j*= 1 to 4) and Ω

_{p}= Σ

_{n=1}

^{∞}

*ε*Ω

^{n}

_{p}^{(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

*z*=

_{l}*ε*(

^{l}z*l*= 0 to 2) and

*t*=

_{l}*ε*(

^{l}t*l*= 0, 1). Substituting above expansion into Eqs. (2)-(3), we obtain a series of equations for

*A*

_{l}^{(j)}and Ω

_{p}

^{(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 Ω_{p}
^{(1)} = *F* exp(*iθ*) 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 *z _{j}* (

*j*= 1, 2).

The solution for *A _{j}*

^{(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/*V _{g}*)

*∂F*/

*∂t*

_{1}] = 0 with

*V*= 1/

_{g}*K*

_{1}being the complex group velocity of the envelope

*F*. The solution for

*A*

_{j}^{(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

with *α* = *ε*
^{2}
*α*̄ and

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

where *u* = (Ω_{p}/*U*
_{0})exp(-*iK*˜_{0}
*z*), *s* = *z*/(2*L _{D}*), and σ = (

*t*-

*z*/

*V*)/

_{g}*τ*

_{0}.

*L*= -

_{D}*τ*

_{0}

^{2}/

*K*˜

_{2}is characteristic dispersion length,

*L*= -1/

_{A}*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 *L _{D}* ≪

*L*Eq. (17) reduces to the standard NLS equation

_{A}*i∂u*/

*∂s*+∂

^{2}

*u*/

*∂σ*

^{2}+ 2

*u*|

*u*|

^{2}= 0. A simple soliton solution reads

*u*= sech σ exp(

*is*), or in terms of field

which describes a fundamental bright soliton travelling with propagating velocity *V _{g}*.

In Fig. 4(a), we show the wave shape of |Ω_{p}/*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 Δ_{4} = -1.0 × 10^{9} s^{-1}, Ω_{s} = 2.0 × 10^{7} 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 + *i*0.004) cm^{-1}, *K*
_{1} = -(6.15 + *i*0.18) × 10^{-7} cm^{-1}s (*α* = -7.6 Ö 10^{-3} cm^{-1}), *K*
_{2} = -(3.08 + *i*0.14) × 10^{-12} cm^{-1}s^{2}, and *W* = -(7.58 + *i*0.12) × 10^{-16} cm^{-1}s^{2}. The characteristic lengths read *L _{D}* = 4.0 cm,

*L*= 264 cm (

_{A}*L*≪

_{D}*L*) and

_{A}*U*

_{0}= 1.8 × 10

^{7}s

^{-1}. With these parameters we obtain the group velocity

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 Ω_{s} = 1.0 × 10^{7} s^{-1} and *τ*
_{0} = 1.0 × 10^{-5}s. We obtain *L _{D}* = 0.5 cm,

*L*=61 cm (

_{A}*L*≪

_{D}*L*),

_{A}*V*= -3.4 × 10

_{g}^{-6}

*c*, and

*U*

_{0}= 1.0 × 10

^{7}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 Ω_{s} = 1.0 × 10^{7} s^{-1} and 2.0 × 10^{7} 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 Ω_{s} = 1.0 × 10^{7} s^{-1} and only 1% if Ω_{s} = 2.0 × 10^{7} 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.

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 *P*̄ = *p*̄_{max} sech^{2} [(*t* - *z*/*V˜ _{g}*)/

*τ*

_{0}], with the peak power

*P*̄

_{max}= 2

*ε*

_{0}

*cn*

_{p}*S*

_{0}|

*ℰ*|

_{p}^{2}

_{max}= 2

*ε*

_{0}

*cn*

_{p}*S*

_{0}(

*h*̄/

*D*

_{0})

^{2}

*K*˜

_{2}/(

*W*˜

*τ*

_{0}

^{2}). Here,

*n*= 1 +

_{p}*cK*/

*ω*is the refractive index and

_{p}*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}cm

^{2}, we obtain

*P*̄

_{max}=1.4

*μ*W. 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

where *θ _{j}* (

*j*= 1, 2) is the initial phase of the

*j*th soliton. The result for initial phase difference Δ

*θ*=

*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 Δ

*θ*=

*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 ^{87}Ru 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 *ω _{j}* (

*j*=

*c*,

*p*,

*s*) should be replaced by

*ω*-

_{j}**k**

_{j}·

**V**=

*ω*-

_{j}*k*(we assume the coupling, probe, and signal fields propagate along the

_{jZ}V_{z}*z*direction). Thus in Eq. (1) the one-photon and three photon detunings should be replaced by Δ

_{3}→ Δ

_{3}(

*V*) =

_{z}*ω*- (

_{c}*E*

_{3}-

*E*

_{1})/

*h*̄ -

*k*

_{cz}*V*and Δ

_{z}_{4}→ Δ

_{4}(

*V*) = (

_{Z}*ω*-

_{c}*ω*+

_{p}*ω*) - (

_{s}*E*

_{4}-

*E*

_{1})

*h*̄ + (

*k*-

_{p}*k*-

_{c}*k*)

_{s}*, respectively. Then the second term on the left hand side of Eq. (3) should be averaged over*

_{z}V_{z}*V*for a given thermal velocity (e.g. Maxwell) distribution function

_{z}*f*(

*V*). 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,

_{z}*γ*

_{4}<

*γ*

_{3}), and therefore Ω

_{s}should be a circularly polarized microwave field. Notice that, since

*k*and

_{p}*k*are much larger than

_{c}*k*, the Doppler effect in the three-photon detuning Δ

_{s}_{4}(

*V*) for copropagating case (

_{z}*k*> 0) is much smaller than that for the counterpropagating case (

_{p}k_{c}*k*< 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 Δ

_{p}k_{c}_{4}(

*V*). In addition, the Doppler effect appearing in one-photon detuning Δ

_{Z}_{3}(

*V*) can be efficiently avoided by taking a large

_{z}*ω*- (

_{c}*E*

_{3}-

*E*

_{1})/

*h*̄, and hence the influence by

*k*is largely suppressed.

_{cz}V_{z}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 *A*_{j} (*j* = 1,2,3,4)

_{j}

The solution of *A _{j}* in the first order reads

with *B*
_{4}
^{(1)} = [Ω_{s}
^{*}
*A*
_{3}
^{*(0)}]/*κ*] [(*ω* - *d*
_{2}) (*K*
^{*} - *ω*/*c*) - *κ*|*A*
_{3}
^{(0)}|^{2}].

The solution of *A _{j}* at the second order is given by

$$\times \frac{\partial {F}^{*}}{\partial {t}_{1}}{e}^{-i{\theta}^{*}},$$

with

$$-\frac{{\mid K-\omega /c\mid}^{2}}{{\kappa}^{2}{\mid {A}_{3}^{\left(0\right)}\mid}^{2}}-{\mid {B}_{4}^{\left(1\right)}\mid}^{2}],$$

## 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.

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