1. Introduction

Light propagating in a periodic medium is associated with many interesting phenomena. A fundamental change introduced by this periodicity is a forbidden band gap in the transmission spectrum. Light can not transmit in this medium if its frequency falls into the band gap. However if the medium has some nonlinearity, in some cases light can transmit in it even though its frequency lies within the forbidden frequency band. An important example is the so-called Bragg solitons, which can be supported by a periodic medium with Kerr nonlinearity[1]. This kind of solitons was firstly studied by Chen and Mills[2], and later by many other researchers[3, 4, 5, 6, 7, 8]. Some experiments in nonlinear optical fibers were reported[9, 10]. In these experiments pulsed lasers with very high peak intensities were required due to the small value of nonlinear coefficient.

On the other hand, electromagnetically induced transparency(EIT) is another fascinating phenomenon[11, 12, 13]. An optical opaque medium is rendered to be transparent for a probe light by a coupling field in a small frequency window. One important property is that in the same spectral region where there is a high degree of transmission, the nonlinear response χ⁽³⁾ displays constructive interference, i.e., its value at resonance could be very large. This effect is also termed as giant Kerr effect[14]. Another attractive application is that one can make a controllable photonic band gap by properly arranging coupling lights[15, 16, 17, 18, 19, 21, 20].

In this paper we try to combine these two effects. We will show that through a proper geometric configuration one can make an EIT medium to have both a tunable photonic band gap and a large Kerr nonlinearity. We propose a scheme to generate Bragg solitons in this medium with relatively low light power. Furthermore, the parameters of the medium can be easily controlled through adjusting the intensities and the detunings of lasers. This scheme may provide an opportunity to study the dynamics of gap solitons.

2. The theory

2.1. Energy structure of the atom and geometric setup

Consider an ensemble of atoms with energy diagram shown in Fig. 1. A probe laser with frequency ω_p is near resonant with transitions ∣1〉 ↔ ∣3〉 and ∣2〉 ↔ ∣4〉 (Here we assume ω₃ - ω₁ ≃ ω₄ - ω₂). The corresponding Rabi frequencies are Ω_p and Ω_p′ respectively. A coupling field with frequency coc and Rabi frequency Ω_c is tuned to ∣2〉 ↔ ∣3〉 transition. And a controlling field with frequency ω_s is tuned to ∣2〉 ↔ ∣5〉 transition.

 

Fig. 1. (a) Energy diagram of the atom. (b) Geometric configuration of the lights. Coupling beam and probe beam are co-propagating. A standing wave is formed by a forward and a backward controlling fields E→_sf and E→_sb. A small angle ϕ between the standing wave and probe beam is chosen so that k_scosϕ = k_p is fulfilled.

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2.2. The susceptibility of the medium

The nonlinear coefficient of the system can be obtained by calculating the coupled amplitude equations in the perturbative regime[22]. The susceptibility at the probe frequency is,

where ∆₁ = ω_p - ω₃₁, ∆₂ = ω_c - ω₃₂, ∆₄ = ω_p - ω₄₂, ∆₅ = ω_s - ω₅₂, $\tilde{\Delta }={\Delta }_1+\frac{i}{2}{\Gamma }_3$ $\tilde{\delta }={\Delta }_1-{\Delta }_2+\frac{i}{2}{\Gamma }_2$ ${\tilde{\Delta }}_{42}={\Delta }_1-{\Delta }_2+{\Delta }_4+\frac{i}{2}{\Gamma }_4,$ , ${\tilde{\Delta }}_{52}={\Delta }_1-{\Delta }_2+{\Delta }_5+\frac{i}{2}{\Gamma }_5,$Γ₂,Γ₃, Γ₄and Γ₅ are relaxation rates for energy level 2, 3, 4 and 5 respectively. K ₀ = ρ ∣μ₁₃∣²/h̅ε₀, ρ is the density of the atoms. Here we assume that (Γ₃, Γ₄, Γ₅, ∣Ω_c∣)≫(∣Ω_p∣, ∣Ω_p′∣). The susceptibility has three components χ⁽¹⁾ (ω_p), χ⁽³⁾ (ω_p;ω_s-ω_s, ω_p) and χ⁽³⁾ (ω_p, ω_p, -ω_p, ω_p). By defining χ_a = χ⁽¹⁾(ω_p)+χ⁽³⁾(ω_p;ω_s,-ω_s,ω_p)∣E_s∣² and noting ∣Ω_p′,∣² ≪ ∣Ω_c∣², (∆, δ)≪(∆₄₂,∆₅₂) we have,

where K ₁ = ρ∣μ₁₃∣² ∣μ₂₄∣² /ε₀ h̅ Eq. (2,3) governs the response of the medium to the incident probe field. From a practical point of view, the absorption of the medium to the probe light must be kept small. Note when ∣Ω_c∣² > Γ₂Γ₃, there is a transparent window near the two-photon resonance whose width is given by $\Delta {\omega }_{\mathrm{trans}}\approx \frac{{\mid {\Omega }_c\mid }^2}{{\Gamma }_3^2\sqrt{\mathrm{\rho \sigma L}}}$, where σ = 3λ²/2π is the absorption cross section of an atom and L is length of the sample. If we make a proper choice and let the frequency of the probe field be inside this window, the probe field will experience a negligible absorption while the medium still maintains a large Kerr nonlinearity. Fig. 2 shows the real and imaginary parts of χ_a and χ⁽³⁾. Itcan be clearly seen that absorptions are quite small near the two-photon resonance, while Re(χ⁽³⁾) has a relative large value.

 

Fig. 2. Real and imaginary parts of χ_a and χ⁽³⁾ versus ∆₁/γ_a (see the text for parameters). Near the two-photon resonance the absorption is small while the Kerr nonlinearity is large.

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2.3. The photonic band gap

In this section we will concentrate on engineering a medium, which has a periodic refraction index and a Kerr nonlinearity. Consider a geometric configuration shown in Fig. 1(b). E→ _p and E→ _c are co-propagating probe and coupling fields. A standing wave is formed by E→ _sf and E→ _sb, so Ω_s is spatially modulated and has the form ∣Ω_s(r→)∣² = Ω² ₁ cos²(k→_s∙r→), where k→_s is wave vector of the controlling field. A proper angle between E→ _p and E→ _s is chosen so that k_scosϕ = k_B ≈ k_p is fulfilled. This angle is small when the wavelengths of probe field and controlling field are close. Under this assumption we can work with a simplified 1D model. We choose the direction of wave propagation as z axis. Substitute Ω_s(z) into (2) and noting n ² = 1 + χ, we have

where ${\bar{\chi }}_a=\frac{K_0}{2}\frac{-4\tilde{\delta }{\tilde{\Delta }}_{52}+\frac{{\Omega }_1^2}{2}}{4\tilde{\delta }\tilde{\Delta }{\tilde{\Delta }}_{52}-{\tilde{\Delta }}_{52}{\mid {\Omega }_c\mid }^2},$, $\mathrm{\delta \chi }=\frac{K_0}{2}\frac{{\Omega }_1^2}{4\tilde{\delta }\tilde{\Delta }{\tilde{\Delta }}_{52}-{\tilde{\Delta }}_{52}{\mid {\Omega }_c\mid }^2}$. Thus we obtain a Bragg grating with a Kerr nonlinearity. The modulation depth of χ_a is controlled by Ω₁. The amplitude of χ⁽³⁾ (ω_p; (χ_p,- χ_p, χ_p) can be controlled by Ω_c and ∆₄₂. As a result of Floquet-Bloch theorem the periodic refraction index should produce a band gap in the transmission spectrum known as the photonic band gap. However there are two important differences between this Bragg grating and the traditional grating formed in an optical fiber. The first one is that the refraction index and modulation depth is strongly frequency dependent. The second one is the effect of absorption should be taken into account. Generally speaking, the absorption can make the edge of the band gap blur or even vanish[23, 24]. We can see this effect more clearly from the numerical result below. Transfer matrix method is used to get to the reflection coefficient and the dispersion relation [25].

 

Fig. 3. (a) Reflectivity of the medium(see text for parameters). A band gap appears near the two-photon resonance. Solid lines are results when absorption is included while dash lines are results when there is no absorption. (b) Dispersion relation (with absorption). d is the period of the Bragg grating. (c) Dispersion relation (without absorption). The number of periodic structure is 4000 corresponding to a propagation distance (size of the cold atomic cloud) of 3mm.

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Fig. 3 is the calculated reflectivity of the sample and the dispersion relation. We can see there is a band gap inside the electromagnetically induced transparent window. The width of the band gap is very narrow because of the frequency dependence of $\overline{\chi }$ _a and δχ. This is not a surprising result. The modulation of refraction index comes from the interaction between atoms and light fields. When the probe light is far off resonant the interaction strength is very weak, therefore the modulation of refraction index should disappear. In other words, strong refraction index modulation exists only in the vicinity of ∆₁ =0, where two-photon detuning is small. Consequently, the band gap is considerably narrowed. Note the edges of the gap are blurred due to the effect of small absorption. As a comparison, we also give the reflectivity and dispersion relation when absorption is not taken into account. From the discussion above we can conclude that a relatively strong coupling field is needed for the photonic band gap to survive.

2.4. Propagation of lights inside the medium

Now we will consider the propagation of light whose frequency is near or falls into the band gap. The propagation of probe field is governed by Maxwell equations,

Decompose E_p into a forward wave and a backward wave, i.e.,

where A ₊(z,t) and E _-(z,t) are the envelopes of forward and backward waves respectively. k_p = n ₀ω_p/c is the wave vector. Substitute Eq. (7) into Eq. (6). After applying slowly varying envelopes approximation and expanding χ_a, δχ and χ⁽³⁾ around ω_p, we have[26],

where, ∆_k = k_p - k_B, $v_g={\left(\frac{\bar{n}}{c}+\frac{{\omega }_p}{2\mathrm{nc}}\frac{\partial \bar{\chi }}{\partial \omega }\right)}^{-1}$, $\kappa =\frac{\mathrm{\delta \chi }}{4\mathrm{nc}}{\omega }_p$, $\gamma =\frac{{\omega }_p}{2\mathrm{nc}}{\chi }^{\left(3\right)}$. We also used $\mathrm{\delta \chi }\gg \frac{\partial \bar{\chi }}{\partial \omega }\gg \frac{\partial \mathrm{\delta \chi }}{\partial \omega },$ ${\chi }^{\left(3\right)}\gg \frac{\partial {\chi }^{\left(3\right)}}{\partial \omega }$ in deriving Eq.(8). v_g is group velocity of the light pulse inside the medium without the light induced grating. k is coupling strength between the front propagating wave A ₊ and back propagating wave A _-. γ represents the self phase modulation (SPM) and cross phase modulation (XPM) due to Kerr nonlinearity. These coupled mode equations have been studied extensively. These equations are related to the well known massive Thirring model of quantum field theory. Although it is non-integrable when the SPM term is non-zero, shape-persevering solitary waves can still be obtained[5]. The solution is,

where

This solution represents a two-parameter family of Bragg solitons. The parameter v is in the range - 1 < v < 1, while the parameter ψ can be chosen anywhere in the range 0 < ψ < π. The velocity of the Bragg soliton is given by V_G = Vv_g.

The scenario here is quite different from that in nonlinear fiber gratings. In fibers the group velocity v_g is always comparable with c. But in our case there is a strong normal dispersion near the two-photon resonance beside the dispersion induced by the grating, so v_g can be considerably less than c. The ultra slow group velocity will cause the width of the band gap to decrease sharply because the band gap width ∆ω is given by 2 ∣v_gκ∣. When there is no Kerr nonlinearity, the group velocity and the group velocity dispersion (GVD) are given by V_g = v_g√1 - κ²/δ² _ωand ${\beta }_2=\frac{\frac{\mathrm{sgn}\left({\delta }_{\omega }\right){\kappa }^2}{v_g^2}}{{\left({\delta }_{\omega }^2-{\kappa }^2\right)}^{\frac{3}{2}}}$ where δ_ω = v ^-1 _g (ω - ω_p). Here the GVD is enhanced due to the small v_g.

3. Numerical estimation and discussion

In this section we will discuss under what conditions would we observe the optical solitary waves in a cold atomic clouds. The low power limit (γP ₀ ≪ κ) is of particular interest because in this limit the coupled-mode equations reduced to the nonlinear Schrodinger equation (NLS), where P ₀ is the peak power of the pulse propagating inside the grating. In this case, the Bragg soliton is actually reduced to the fundamental NLS solitons and is found to be stable. We take some typical values of ⁸⁷Rb in our estimation. μ₁₃ = 2.5 × 10^-29Cm, N = 10¹²cm^-3,Γ₂ = 0.01γ_a, Γ₃ = Γ₄ = Γ₅ = γ_a, ∆₁ = ∆₂ = 0, ∆₄ = 5γ_a, ∆₅ = 20γ_a, Ω_c = 10γ_a and Ω₁ = 10Ω_a where γ_a = 6MHz. The results are v_g ≈ 4200m/s, κ ≈ -2600m^-1, γ ≈ -0.60m/W In order to form a band gap, ∣κ∣L ≥ 2 is required. This can be trivially fulfilled since the typical size of the atomic cloud is about 1 ∼ 5mm corresponding to ∣κ∣L = 2.6 ∼ 13. The width of the band gap is about 0.6γ_a. Since the bandwidth of the input pulse should be much less than the width of the photonic band gap, we can also estimate the minimum width of the input pulse T_FWHN > 1/∆v = 0.29μs. The possibility of observing Bragg solitons also depends on the soliton order N_s and the soliton period z ₀[26]. These parameters are given by $N_s^2=\frac{{\left(3-v\right)}^2\gamma T_0^2{v}_g^2{\mathrm{\kappa v}}^2}{2{\left(1-v^2\right)}^{\frac{3}{2}}}{P}_0$ and $z_0=\frac{{\mathrm{\pi v}}^2{v}_g^2{T}_0^2\kappa }{2{\left(1-v^2\right)}^{\frac{3}{2}}},$, where T ₀ is related to the FWHM of the input pulse as T_FWHM ≃ 1.76T ₀. A Bragg soliton can form only if $N_s>\frac{1}{2},$, so the peak power P_in required to excite the fundamental Bragg soliton can be estimated through relation $P_{\mathrm{in}}=\frac{2{\left(1-v^2\right)}^{\frac{3}{2}}}{v\left(3-v^2\right)v_g^2{T}_0^2\mathrm{\kappa \gamma }}.$. Here we use P_in = P ₀ v. The parameter z ₀ sets the length scale over which optical soliton evolve and gives the minimal length of the grating. Note P_in and z ₀ depend only on v and T ₀ when the power and detuning of the coupling laser is fixed. Fig. 4 gives P_in(v) and T ₀(v) with different T ₀. The region where Bragg soliton is expected is given by P_in ≥ P_c/10 ≪ P_c and z ₀ < L, where P_c is the power of coupling laser. L is the length of the cold atomic sample. When T ₀ =2μs the workable region of v is 0.05 < ∣v∣ < 0.25. When T ₀ = 10μs, the workable region of v is 0.0005 < ∣v∣ < 0.05. So we can work in different ranges of parameter v by choosing different pulse widthes T ₀. The lower limit of T ₀ is set by the width of the band gap. One thing worth noting is that the input power P_in is very low. Compared with the experiments performed in nonlinear fiber gratings (typical peak intensity required is about 10 GW/cm²), the power requirement here is modest. This advantage is due to the combination of the EIT effect and the giant Kerr effect.

 

Fig. 4. P_in(v) and T ₀(v) with different T ₀.

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4. Conclusion

In conclusion we have discussed the possibility of producing Bragg solitons in a coherent medium. We have shown that by using a proper geometric configuration one can make an EIT medium to have both a tunable photonic band gap and a large Kerr nonlinearity. This scheme requires very low light power and provides a large controllability over the properties of the medium.