1. Introduction

Optical solitons, i.e. special types of optical wavepackets formed by the balance between nonlinearity and dispersion (and/or diffraction), have attracted much attention for decades due to their important applications for optical information processing and transmission [1–3]. However, up yo now most optical solitons are produced in far-off resonant optical media such as glass-based optical fibers. The nonlinear effect in such media is very weak and hence to form an optical soliton a very high-intensity is needed.

In recent years, interest has focused on wave propagation in resonant optical media via electromagnetically induced transparency (EIT) [4]. Propagation of a weak probe optical field in such media displays many new features due to the quantum interference effect induced by a strong control optical field, including large suppression of optical absorption, significant reduction of group velocity, and giant enhancement of Kerr nonlinearity [4]. Based on these features, it has been shown that it is possible to produce weak-light ultraslow optical solitons in various EIT media [5–8]. However, there are many drawbacks for the optical solitons obtained in EIT-based schemes, including pulse spreading at room temperature and very long response time due to ultraslow propagation, as indicated in Refs. [9, 10].

Contrary to the EIT-based scheme which is absorptive in nature, active Raman gain (ARG) scheme, which is based on a gain-assisted configuration, is able to support a superluminal propagation for a probe field. In recent years, parallel to the study on EIT, works based on ARG schemes have also attracted considerable attention both theoretically and experimentally [9–29]. Especially, Agarwal and Dasgupta investigate a linear propagation of probe field in a four-level N-type atomic system and showed that a gain doublet appears due to a quantum interference induced by an additional control field [30]. Recently, we have studied nonlinear effect of an ARG-based four-level Λ-type atomic system and demonstrated that it is possible to obtain giant Kerr nonlinearity in such system [31]. It is natural to consider the possibility of producing a superluminal optical soliton in ARG-based atomic ensemble. It is just this topic that will be addressed in this work. We shall show that, based on the ARG scheme and working on the gain minimum of the gain doublet, we can obtain a stable propagation of superluminal optical solitons. However, due to the resonant character of the system, the property of the superluminal optical solitons is very sensitive to probe pulse length. In general, solitons in such system must be described by a modified nonlinear Schrödinger (NLS) equation, which has high-order correction terms contributed from effects of linear and differential absorption, nonlinear dispersion, and delay response of nonlinear refractive index of the system. We shall show that superluminal optical solitons obtained in such system have many interesting features.

The paper is arranged as follows. In the next section, we present theoretical model and discuss its solution in linear regime. In Sec III, we make an asymptotic analysis on the Maxwell-Schrödinger equations and derive the modified NLS equation governing the evolution of the envelope of probe field. In sec IV, we discuss various soliton solutions of the modified NLS equation and make a numerical simulation to test their stability. Finally, in the last section we give a discussion and summary of main results obtained in our work.

2. Model and solution in linear regime

The model under study is shown in Fig. 1, where a lifetime-broadened four-state atom with bare energy states |j〉 (j = 1, 2, 3, 4) interacts with three laser fields. The weak, pulsed probe filed of angular frequency ω_p (half Rabi frequency Ω_p) couples to the transition |3〉 ↔ |4〉, the strong, continuous-wave control field of angular frequency ω_c (half Rabi frequency Ω_c) couples to the transition |1〉 ↔ |4〉, and the strong, continuous-wave control field of angular frequency ω_b (half Rabi frequency Ω_b) couples to the transition |2〉 ↔ |3〉. States |1〉, |3〉, |4〉, and half Rabi frequencies Ω_p and Ω_c consists of an ARG core [9, 10]. States |1〉, |2〉 and |3〉 can be chosen as hyperfine ground states. The transition |3〉 ↔ |2〉 is generally electric-dipole forbidden, and hence the control field of half Rabi frequency Ω_b is a microwave field in our system.

 

Fig. 1 (Color online) Energy-level diagram and excitation scheme of Λ-type four state atomic system with active Raman gain. |j〉 (j = 1, 2, 3, 4) are bare atomic states. Δ₄ is one-photon detuning. Ω_p is the half Rabi frequency of weak, pulsed probe field. Ω_c and Ω_b are half Rabi frequencies of two strong, continuous control fields.

Download Full Size | PDF

We assume that two control fields are strong enough and thus are undepleted during probe-field propagation. In interaction picture, equations of motion for the atomic-state amplitudes A_j under electric dipole and rotating wave approximations and Maxwell equation on the probe-field half Rabi frequency Ω_p under the slow varying envelope approximation are given by

In order to achieve shape-preserving propagation of the probe field, we first examine linear dispersion property of the system. When the probe field is absent, the state of the system reads A₁ = a₁, A₂ = A₃ = 0, and A₄ = −Ω_ca₁/d₄, where $a_1=a_1=\left|d_4\right|/\sqrt{{\left|d_4\right|}^2+{\left|{\Omega }_c\right|}^2}$ is a constant. Such steady state can be obtained by using the condition ${\sum }_{j=1}^4{\left|A_j\right|}^2=1$. Note that this approximation can only be used when the detuning Δ₄ is very large or γ_jt ≪ 1, which result in the population in the energy levels |2〉 and |3〉 close to zero. For a very weak probe field Ω_p = F exp(iθ), here θ = Kz – ωt and F is a very small amplitude, one can easily get the linear dispersion of the system

Shown in Fig. 2 are −Im(K(ω)) (Fig. 2(a)) and Re(K(ω)) (Fig. 2(b)), representing the negative imaginary part and real part of K(ω), respectively. System parameters are those given by typical alkali atomic (⁸⁷Rb) gas, with [9] γ₂ = 1.5 × 10⁴s⁻¹, γ₃ = 5.5 × 10⁴s⁻¹ and γ₄ = 1.8 × 10⁸s⁻¹. Other parameters are taken as Ω_c = 1.5 × 10⁷s⁻¹, Δ₂ = Δ₃ = 0s⁻¹, Δ₄ = −1.0 × 10⁹s⁻¹, κ₃₄ = 1 × 10⁹cm⁻¹s⁻¹. From Fig. 2(a), we see that for small microwave control field (Ω_b = 1.0 × 10⁴ s⁻¹) the gain spectrum has only a single peak (dashed line). In this case, near the central frequency (i.e. ω = 0) the probe field is highly amplified and hence unstable during propagation; however, for large microwave control field (Ω_b = 1.0 × 10⁶ s⁻¹) the gain spectrum displays two peaks (solid line), i.e. a gain spectrum hole appears. In this case, within the spectrum hole between two gain peaks the probe field has a very small gain and thus can keep its shape during propagation if the dispersion of the system can be arrested by some other effect of the system. The appearance of such gain spectrum hole (gain doublet) comes from the quantum interference effect induced by the microwave control field [30, 31]. From Fig. 2(b) we see that for small microwave control field (dashed line) the slope of ReK(ω) near ω = 0 is positive, and thus the probe field in this case has a subluminal (but unstable) propagation. For large microwave control field (solid line) the slope of ReK(ω) corresponding to the region of the gain spectrum hole is negative, and hence in this region the probe field has a superluminal propagation.

 

Fig. 2 (Color online) (a) Negative imaginary part (gain) −Im(K) and (b) real part Re(K) as functions of ω. Solid and dashed lines correspond to Ω_b = 1.0 × 10⁴ s⁻¹ and Ω_b = 1.0 × 10⁶ s⁻¹, respectively.

Download Full Size | PDF

3. Asymptotic expansion and envelope equation

We are interested in a shape-preserving propagation of probe field. As shown above, in the ARG scheme the gain of the probe field can be largely suppressed by using a large microwave control field. However, the system still has large dispersion originated from its resonant feature. It is natural to use the nonlinearity to balance the dispersion effect of the system. In this section, we apply a weak nonlinear perturbation theory to search for stable soliton formation and propagation in the system. For this aim we first derive a nonlinear envelope equation describing the evolution of the probe field by using a standard multiscale approach [32]. We start by making the following asymptotic expansion:

To obtain a divergence-free expansion, all quantities on the right hand side of Eq. (3) are considered as function of multiscale variables z_l = ɛ^lz (l = 0,1, 2, 3), and t_l = ɛ^lt (l =0,1). Then Eqs. (1a)–(1d) become

To make the procedure of obtaining iterative solutions of Eqs. (4a)–(4d) more trackable, we express Eqs. (4a)–(4d) in the following forms:

3.1. First-order approximation

The case for l = 1 is just the linear problem solved in the last section. we thus have the linear dispersion relation (2). The first-order approximation solution reads

3.2. Second-order approximation

For l = 2 we can obtain M⁽²⁾, N⁽²⁾, Q⁽²⁾ and R⁽²⁾ by substituting the first-order solution Eqs. (7a)–(7c) into Eqs. (4a)–(4d), and also obtain the explicit expression of S⁽²⁾. Equation (5c) becomes

3.3. Third-order approximation

Similarly, for l = 3 we obtain

As indicated above, for pulse propagation in resonant systems, the dispersion and nonlinear effects are very sensitive to pulse length. For a shorter pulse, the third-order approximation is not enough and hence one must go to fourth-order. Using the solutions from the first- to third-orders we can solve the fourth-order approximation equations. After a detailed calculation we obtain the solvability condition

Combining Eqs. (9), (12) and (17) we obtain an envelope equation with complex coefficients of the form (after returning to original variables)

4. Formation of superluminal optical solitons

Equation (18) derived in the preceding section is in fact a Ginzburg-Landau equation [33] with some high-order terms since its coefficients are complex. In general case, such equation does not allows soliton solutions. However, if a realistic set of system parameters can be found so that the imaginary part of these coefficients can be made much smaller than their corresponding real parts, it is possible to obtain a shape-preserving, localized probe pulse that can propagate for an extended distance without significant distortion. We show that this is indeed possible for the present ARG system. We take the parameters used in Ref. [9], which are the same as those in Fig. 2 Other parameters are chosen as Ω_c = 1.5 × 10⁷s⁻¹, Ω_b = 1.0 × 10⁶s⁻¹,Δ₂ = 0.1 × 10⁶s⁻¹,Δ₃ = 0.5 × 10⁶s⁻¹,Δ₄ = −1.0 × 10⁹s⁻¹,κ₃₄ = 1 × 10⁹cm⁻¹s⁻¹. With the above parameters, we obtain K₀ = −(2.16 + i0.14)cm⁻¹, K₁ = −(14.97 + i0.04) × 10⁻⁷cm⁻¹s, K₂ = −(21.35 + i0.01) × 10⁻¹³cm⁻¹s², K₃ = −(39.26 + i0.07) × 10⁻¹⁹cm⁻¹s³, W = (20.52 + i0.008) × 10⁻¹⁶cm⁻¹s², β₁ = (27.62 + i0.01) × 10⁻²²cm⁻¹s³, β₂ = −(2.39 + i0.1) × 10⁻²¹cm⁻¹s³. We see that the imaginary part of each coefficient of the high-order Ginzburg-Landau Eq. (17) is indeed much smaller than its real part. The physical reason of so small imaginary part is due to the quantum destructive interference effect induced by the control field Ω_b, by which the transition passage |2〉 ↔ |3〉 superposes on the transition passage |3〉 ↔ |4〉 destructively to make the population in the energy level |3〉 vanish, and hence suppress the gain of the probe field largely.

When neglecting the small imaginary part of the coefficients, the Ginzburg-Landau Eq. (18) is converted into the dimensionless high-order NLS equation

Notice that each term in Eq. (19) has clear physical meaning. The second and the third terms on the left hand side describe respectively the second-order dispersion and Kerr nonlinearity of the system. The terms from the first to the fourth ones in the square bracket on the right hand side describe linear absorption (proportional to g₀), nonlinear dispersion (proportional to g₁), delay response of nonlinear refractive index (proportional to g₂), and third-order dispersion (proportional to g₃), respectively. The last term describes differential absorption (proportional to g₄).

The solution property of Eq. (19) is determined by coefficients g_j (j = 0 – 4). Using the system parameters given above we have calculated values of g_j as functions of the pulse length τ₀ of the probe field, which is shown in Fig. 3. We see that g₃ and g₄ are not sensitive to τ₀, but g₀, g₁ and g₂ change rapidly as τ₀ varies. Based on the result of Fig. 3(a) and Fig. 3(b) we can divide the Eq. (19) into several regimes and hence obtain different soliton solutions, which are given as follows:

 

Fig. 3 (Color online) Values of coefficients g₀, g₄ (panel (a)), and g₁, g₂, g₃ (panel (b)) of Eq. (19) as function of the pulse length τ₀. Both panels (a) and (b) are plotted by the parameters given by γ₂ = 1.5 × 10⁴s⁻¹, γ₃ = 5. 5 × 10⁴s⁻¹, γ₄ = 1.8 × 10⁸s⁻¹, Ω_c = 1. 5 × 10⁷s⁻¹, Ω_b = 1.0 × 10⁶s⁻¹, Δ₂ = 0.1 × 10⁶s⁻¹, Δ₃ = 0.5 × 10⁶s⁻¹, Δ₄ = −1.0 × 10⁹s⁻¹, and κ₃₄ = 1 × 10⁹cm⁻¹s⁻¹.

Download Full Size | PDF

By considering small imaginary part of the coefficients and using the above data, we obtain U₀ = 1.8×10⁷s⁻¹, L_D = L_NL = 1.5cm, α = −0.14cm⁻¹. The propagating velocity of the soliton (21) for τ₀ = 3.5 × 10⁻⁶ s is

In Fig. 4 we have shown the propagation of the superluminal optical solitons and their interaction. Shown in Fig. 4(a) is the waveshape of |Ω_p/U₀|² versus time t and propagating z based on Eqs. (1a)–(1d). The initial condition (black solid line) is given by Eq. (23) with the parameters τ₀ = 1.5 × 10⁻⁶s, γ₂ = 1.5 × 10⁴s⁻¹, γ₃ = 5.5 × 10⁴s⁻¹, γ₄ = 1.8 × 10⁸s⁻¹, Ω_c = 1.5 × 10⁷s⁻¹, Ω_b = 1.0 × 10⁶s⁻¹, Δ₂ = 0.1 × 10⁶s⁻¹, Δ₃ = 0.5 × 10⁶s⁻¹, Δ₄ = −1.0 × 10⁹s⁻¹, and κ₃₄ = 1 × 10⁹cm⁻¹s⁻¹. The (red) dot-dashed line is the soliton profile at the propagation distance z = 2L_D. The (green) dotted line is the result for τ₀ = 3.5 × 10⁻⁶s, with the other parameters the same as above. The (blue) dashed line the soliton profile at the propagation distance z = 4L_D. From these results we see that the superluminal optical soliton is fairly stable when propagates to 2L_D. However, there exists a small radiation appearing on its right wing for τ₀ = 3.5 × 10⁻⁶s and for the soliton propagating to 4L_D. The physical reason for the appearance of such small radiation on the tail is due to that the approximation necessary to obtain the analytic solution (i.e. Eq. (23)) breaks down for a large pulse width for long propagation distance. We have also investigated two-soliton collisions based on the high-order NLS equation (19). The result is potted in Fig. 4(b)–4(d). Shown in Fig. 4(b) is the result of a collision between two solitons with the input condition u(0,σ) =1.0sech(σ – 3.0) exp(−i1.2σ + iπ)+1.0sech(σ +5.0) exp(i1.2σ). Figure. 4(c) reports the result of a collision between two solitons, which occurs for the initial condition u(0, σ) = 1.0sech(σ − 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ). We see the both solitons preserve their wave shapes after collisions. However, there exists many radiations when two large-amplitude solitons propagate to a long distance with the initial condition u(0, σ) = 1.5sech(σ − 3.0)exp(−i1.2σ)+1.0sech(σ+5.0) exp(i1.2σ), as shown in Fig. 4(d).

 

Fig. 4 (Color online) (a) The wave shape of |Ω_p/U₀|² versus time t and propagating distance z based on Eqs. (1a)–(1d). The initial condition (black solid line) is given by Eq. (23) with τ₀ = 1.5 × 10⁻⁶s. The (red) dot-dashed line is the soliton profile at the propagation distance z = 2L_D. The (green) dotted line is the result for τ₀ = 3.5 × 10⁻⁶s. The (blue) dashed line the soliton profile at the propagation distance z = 4L_D. (b) Collision between two solitons with the initial condition u(0,σ) = 1.0sech(σ – 3.0) exp(–i1.2σ + iπ) + 1.0sech(σ + 5.0) exp(i1.2σ). (c) Collision between two solitons with the initial condition u(0, σ) = 1.0sech(σ − 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ). (d) Collision between two solitons with larger amplitude amplitudes. The initial condition is u(0,σ) = 1.5sech(σ – 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ).

Download Full Size | PDF

The propagating velocity of the soliton (23) Ṽ_H is determined by $1/{\tilde{V}}_g^H=1/{\tilde{V}}_g-\beta {\tau }_0/\left(2L_D{g}_3\right)$. We obtain

It is easy to calculate the threshold of optical power density required to generate the super-luminal optical solitons given above. Note that the energy flux of the probe filed is given by Poynting’s vector i.e. P = ∫dS(E_p × H_p) · e_z, where e_z is the unit vector in the propagation direction. Using E_p = (E_p, 0, 0), H_p = (0,ɛ₀cn_pE_p, 0) (n_p = 1 + c Re(K/ω_p) is refractive index), and E_p = (h̄/|μ₃₄|) Ω_p exp[i(k_pz − ω_pt)] + c.c., we can get the expressions of P for different soliton solutions. Then the integration of P over a carrier-wave period gives the average energy flux of the probe field, i.e.

5. Conclusion

We have proposed a scheme for generating superluminal optical solitons in a four-level ARG atomic system with two control fields. By using a standard method of multiple-scales, we have derived a Ginzburg-Landau equation with high-order corrections contributed from linear and differential absorption, nonlinear dispersion, and delay response of nonlinear refractive index of the system. Based on the quantum interference effect induced by the control field, the imaginary part of the coefficients of the Ginzburg-Landau equation can be made to be much smaller than their real part, and hence the Ginzburg-Landau equation is reduced to a high-order nonlinear Schrödinger equation. We have made predictions of various optical solitons in different regimes of system parameters, and shown that the optical solitons obtained display many interesting features, including superluminal propagating velocity and very low generation power. Such optical solitons may have potential applications in optical information processing and engineering.