We investigate the electromagnetically induced transparency (EIT) and nonlinear pulse propagation in a Λ-type three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We find that EIT effect can be greatly enhanced due to the reduction of optical-field mode volume contributed by waveguide geometry. Comparing with the atomic gases in free space, the EIT transparency window in the slot waveguide system can be much wider and deeper, and the Kerr nonlinearity of probe laser field can be much stronger. We also prove that using slot waveguide ultraslow optical solitons can be produced efficiently with extremely low generation power.
© 2013 OSA
Over the past two decades, quantum interference phenomena has attracted much attention due to their fundamental interest and promising applications for optical and quantum information processing. One of such phenomena is electromagnetically induced transparency (EIT), which can be used to substantially enhance the efficiency of nonlinear optical processes in addition to a large suppression of optical absorption . Another noticeable effect of EIT is drastic reduction of group velocity of optical pulses, which has important applications such as slow light [2, 3], quantum memory [4–6], quantum phase gates [7, 8], and slow light solitons [9, 10].
However, up to now most works on EIT and optical pulse propagations have been performed with atomic gases in bulk samples . Since optical pulses in such systems are unguided plane waves, interaction strength between optical pulses and quantum emitters is limited and thus EIT effect is weak. It is desirable to use optical waveguides where optical fields are guided and optical energy is concentrated in small spatial regions. Such structures can be used to not only for enhancing EIT effect, but also for raising the efficiency of nonlinear optical processes based on gases phase media such as atoms or molecules. In recent years, there have been several works on EIT and related studies in waveguide structures where atomic or molecular gases are filled in low-index regions, including hollow-core photonic crystal fibers [11–17], nanofibers , and so on.
Guiding light in low-refractive-index materials (e.g air) is thought to be prohibited in conventional waveguides based on total internal reflection. Usually, multiple dielectric layers  or hollow-core photonic crystal fibers [12–17] based on external reflections are adopted. However, in order to have high reflections, such structures have relatively large dimensions and are wavelength sensitive. In 2004, Almeida et al.  proposed a novel structure called slot waveguide, which consists of a nanometer-size slot filled with a low-index material and embedded in high-index materials. In such structure, light is also guided by total internal reflection but it can be tightly confined and hence largely enhanced in the slot region [20, 21]. Recently, There have been a large amount of research activities on guiding and confining light by using slot waveguides [22–32].
In this article, we propose a scheme to enhance quantum interference effect by using a Λtype three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We find that the EIT effect can be greatly enhanced due to reduction of optical-field mode volume contributed by waveguide geometry. Comparing with atomic gases in free space, the EIT transparency window in the slot-waveguide system is much wider and deeper, and Kerr nonlinearity of probe laser field is much stronger. We also demonstrate that using the slot waveguide ultraslow optical solitons can be produced more efficiently and their generation power is extremely low.
The rest of the article is organized as follows. Sec. 2 describes our theoretical model. Sec. 3 studies the linear propagation of probe field and analyzes its EIT characters. Sec. 4 discusses nonlinear pulse propagation in the slot waveguide system. Finally, the last section summarizes the main results obtained in this work.
The slot waveguide we adopt is similar to that suggested in Ref. , which is shown in the left part of Fig. 1. It consists of a very thin slot (with width 2a in z-direction) of low-index material (with index nS) embedded between two thick rectangular regions (with width b − a on both sides) of high-index material (with index nH), both surrounded by a low-index cladding (with index nC). Sizes in the x- and y-directions of each rectangular region are much larger than a and b. It has been shown  that such slot waveguide has the ability for concentrating electromagnetic (EM) field of transverse magnetic (TM) modes basically within the slot region, and hence is very attractive for enhancing radiation-matter interaction. The expressions of guided TM eigenmodes of eigenfrequencies ωm(k‖) are given in Appendix A. Here with kx and ky being respectively the wavenumbers in x- and y-directions. The physical reason of the confinement and enhancement of TM modes is quite simple. For interfaces with high-index contrast, Maxwell’s equations require a continuity of normal component of electric displacement vector, which gives that electric field in the slot region is (nH / nS)2 times higher than that in the high-index region.
Since kx, ky can take any continuous values, and m takes non-zero integers (i.e. m = 1, 2, 3,···), the guided TM eigenmodes propagate in the xy-plane but is confined in the slot region of the waveguide. For simplicity, we study the lowest-order (i.e. m = 1) TM guided mode, and assume kx = 0, ky = k without loss of generality. Then from the Appendix A we have k‖ = (0, k, 0), k̂‖ = ey, k‖ = k, andFig. 1). The concrete expressions of V1 and N1 (normalized constant) can be found in the Appendix A (i.e. Eq. (27)) for m = 1.
For convenience, we take k as a function of ω1. Replacing ω1 by ω, the electric-field expression of the lowest-order TM guided mode reads , with . Here and , with W1 being the mode volume without the slot (its expression is given in the Appendix B (i.e. Eq. (30)) for m = 1), which is taken to be a reference mode volume for the discussions in the following.
Our aim is to investigate the resonant interaction between the TM-mode of EM field and quantum emitters that are embedded in the slot of the waveguide. For simplicity, we assume the media in the slot and the cladding regions are air (i.e. nS = nC = 1), and the quantum emitters are gaseous atoms with a Λ-type three-level configuration (see the right part of Fig. 1), which is filled in the slot region. Atomic energy levels are two ground states |1〉 and |2〉 and one exited state |3〉, which couple with a weak, pulsed probe field with center angular frequency ωp = kp/c and half Rabi frequency Ωp (i.e. |1〉 → |3〉 transition) and a strong, continuous-wave control field of center angular frequency ωc = kc/c and half Rabi frequency Ωp (i.e. |2〉 → |3〉 transition). Γ31 (Γ32) is the spontaneous emission decay rate from |3〉 to |1〉 (|2〉). Γ21 (Γ12) is the incoherent population exchange rate from |1〉 to |2〉 (|2〉 to |1〉), introduced to reflect the transient relaxation process of the atoms entering and leaving interaction region. We assume that both the probe and control fields belong to the lowest-order TM guided mode given in Eq. (1), which has the form
The Hamiltonian of the system in interaction picture reads
Taking Doppler effect into account, the equation of motion of σjl, i.e. the density matrix elements in the interaction picture, are given by17]. We assume these interfacial effects can be weakened by using similar experimental techniques as did in Refs. [13, 16, 17], like coating the inner walls with some materials (i.e. paraffin) or using light-induced atomic desorption. Note that the controlling field Rabi frequency Ωc in our system is strong enough, so that the change of the decay rate due to the confinement of atoms plays no significant role.
The electric polarization intensity of the system reads33, 34] we replace it by the modified Lorentzian velocity distribution .
The motion of the electric field is controlled by Maxwell equation, which under the slowly varying envelope approximation reduces to
3. EIT characters
3.1. Base state
From (7) we see that, due to the incoherent population exchange, there are population occupation in all three levels. However, the population in states |2〉 and |3〉 are small because generally Γ21 and Γ12 are small. When Γ21 = Γ12 = 0, one has σ11 = 1 and σjl = 0 (j, l ≠ 1).
3.2. EIT characters
We now study the solution of linear excitations of the system, which can be done by linearizing the MB Eqs. (4) and (6) around the base state (7). Taking (j = 1, 2, 3), , and assuming Ωp and σj1 are small quantities proportional to exp[i(K(ω)y − ωt)], we obtain the linear dispersion relation
Different from the case in free space , here we must calculate two-fold integration. The first one is the second term in the brace, which is a statistical average on atomic velocity v. Such integration can be calculated by the use of residue theorem . Taking kpv as a complex number, we find two poles in the lower half complex plane, given by kpv = −ikpvT and . For calculating the integration, we take a contour consisting of the lower half complex plane and real axis. The use of residue theorem givesEq. (9) is a spatial average due to the EM field confinement by the waveguide geometry, which must be done numerically.
The expression of the imaginary part of K(ω) at ω = 0, i.e. Im(K0), is given byEq. (9) to Eq. (12) we obtain the following conclusions:
- The linear dispersion relation K(ω) depends strongly on the slot width 2a due to the factor (W1/V1)1/2 appeared in the mode function ζ(z). Shown in Fig. 2(a) is the probe-field absorption spectrum Im(K) as a function of ω for different slot width. The red solid, black dashed and blue dashed-dotted lines are for 2a = 50, 30 and 10 nm, respectively. We see that: (i) For three different slot widths, an EIT transparency window (i.e. the dip near ω = 0) is opened. (ii) The width of the EIT transparency window becomes larger as the slot width 2a decreases, which means that quantum interference effect is enhanced when the slot width decreases. The physical reason of the enhancement of EIT effect is due to the reduction of EM-field mode volume, which results in (W1/V1)1/2 ≫ 1 and hence the giant enhancement of the interaction between light and atoms.
- The minimum of the absorption, i.e. Im(K0), depends not only on Ωc but also on the slot width. Fig. 2(b) shows the profile of Im(K0) as a function of Rabi frequency |Ωc| for different slot width 2a, where the red solid, black dashed and blue dashed-dotted lines are for 2a = 50,30 and 10 nm, respectively. One sees that: (i) For a given Ωc, Im(K0) for smaller slot width is obviously much smaller than that for larger slot width. As the slot width decreases, the EIT transparency window can be not only widened but also deepened dramatically. (ii) The EIT transparency window with a smaller slot width can be obtained with a much smaller Ωc than that with a larger slot width, which means that the confinement provided by the waveguide geometry can be used to get an EIT more easily than that without the confinement. Note that when plotting Fig. 2 we have used a practical example with the D1 line transition of 87Rb atoms, by selecting |5S1/2, F = 1〉, |5S1/2, F = 2〉, and |5P1/2, F = 1〉 as the atomic states |1〉, |2〉, and |3〉, respectively. The system parameters used are κ13 = 1.0 × 109 cm−1s−1, Ωc = 1.0 × 108 s−1, Γ31 = Γ32 = 1.0 × 107 s−1, and Γ21 = 10−4Γ31.
- The incoherent population exchange (i.e. nonzero Γ21) plays no significant role on the probe-field absorption when Ωc is very large. However, it has non-negligible influence when Ωc is not too large, reflected in the second term (i.e. the term related to Γ21) in the bracket of Eq. (12) which contributes a obvious reduction to the absorption of the probe field. In fact, by the incoherent population exchange the atoms undergo an active Raman gain process from |2〉 → |3〉 → |1〉. Shown in Fig. 3(a) is Im(K) as function of frequency ω for Ωc = 1.0 × 106 s−1 with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively. We see that the absorption for Γ21 = Γ12 is much smaller than that for Γ21 = 0. So a incoherent population exchange can be used to widen and deepen the EIT transparency window. Fig. 3(b) shows the profile of Im(K0) as a function of |Ωc| with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively. We see that for large |Ωc|, there is no obvious difference in Im(K0) for different Γ21; but for small |Ωc| the effect caused by Γ21 can be observed clearly.
4. Kerr nonlinearity and ultraslow optical solitons
4.1. Kerr nonlinearity of the system
The Kerr effect can be enhanced due to the confinement effect induced by the waveguide geometry. When the slot width 2a decreases, the confinement of the light field increases because the factor (W1/V1) in the expression |ζ(z)|2 increases. Shown in Fig. 4 is the real part of the third-order susceptibility, i.e. , as a function of detunning Δ3 for different slot width 2a, where the red solid, black dashed and blue dashed-dotted lines are for 2a = 50,25 and 5 nm, respectively. Parameters are the same as those used in Fig. 2 with a small air (εS = 1) slot embedded between the silicon (εH = 14) slabs. We see that the Kerr effect for large confinement (2a = 5 nm) is much larger than that for small confinement (2a = 50 nm). Hence the effect of self-phase modulation becomes stronger as the slot size 2a goes smaller, which indicates that in the slot waveguide the efficiency of producing ultraslow optical solitons may be higher than that in free space.
4.2. Asymptotic expansion and nonlinear envelope equation
We now turn to study possible optical solitons in the system, which is especially interesting for the present slot waveguide geometry because the light power density in such system is increased and diffraction is suppressed in the confined direction, and thus optical solitons are easy to produce than in free space [9, 10]. Such study is also of practical interest in optical information processing and transmission in quantum hybrid systems when shape-preserving probe pulses with low light power are needed.
To this end, we employ the method of multiple scales to solve the MB equations for nonlinear propagation problems developed in Ref. . Taking the asymptotic expansion , , where is the base state solution given by Eq. (7) and ε is a dimensionless 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 yβ = εβy (β = 0, 1, 2) and tβ = εβt (β = 0,1). Substituting the expansion into the MB Eqs. (4) and (6), we obtain a series of equations for and , which can be solved order by order.
At the first order (l = 1), we obtain the linear solutionEq. (8).
At the second order (l = 2), the condition of the solution in this order is divergence-free requires i[∂F/∂y1 + (1/vg)∂F/∂t1] = 0, where vg = ∂K/∂ω is the group velocity of the envelope function F. The explicit expressions of the second-order solution are omitted here for saving space.
At the third order (l = 3), we obtain the closed equation for F:Appendix C.
Returning to the original variables, Eq. (16) becomesEquation (18) is of the form of nonlinear Schrödinger (NLS) equation, but has complex coefficients and hence is not integrable generally. If a nonlinear localized pulse is produced, it may be highly unstable during propagation. However, if a realistic set of system parameters under some conditions can be found so that the imaginary part of the coefficients can be much smaller than their real part, it is possible to obtain a shape-preserving soliton solution that can propagate for a rather long distance without significant distortion. In fact, such parameter set can indeed be found near the EIT transparency window (see below), so the imaginary parts of the coefficients are very small. In this way Eq. (18) can be written into the dimensionless form Eq. (19), d0 = LD/LA and d1 = LD/Ldiff are two dimensionless coefficients, with LA = 1/2α the characteristic absorption length and the characteristic diffraction length, respectively. Under the condition d0, d1 ≪ 1, Eq. (19) reduces to an integrable NLS equation, which allows multi-soliton solutions. A single soliton solution reads Equation (21) describes a bright soliton traveling with velocity ṽg = Re(vg).
We now consider a practical example for the formation of the optical soliton given above. We choose 87Rb D1-line transition, with system parameters given by κ13 = 1.0 × 109 cm−1s−1, Δ2 = 2.5 × 105 s−1, Δ3 = 5.9 × 107 s−1 and slot width 2a = 10 nm. In this case, the coefficients in Eq. (21) are K2 = (1.59 + 0.14i) × 10−14 cm−1s−2 and W = (5.1 + 0.36i) × 10−15 cm−1s−2. We see that the imaginary parts of these coefficients are indeed much smaller than their corresponding real parts. The physical reason of so small imaginary parts is due to the quantum interference effect induced by the control field, by which the role of population and coherence decay rates for the propagation of the soliton is largely suppressed. When taking τ0 = 1.5 × 10−7 s, Rx = 0.05 cm we have the characteristic lengths LD = 1.4 cm, LA = 38.4 cm and Ldiff = 107 cm, which ensure the validity of neglecting absorption and diffraction of the probe pulse when propagating a distance not much larger than the dispersion length, i.e. d0 ≪ 1 and d0 ≪ 1 is satisfied. With these parameters we obtain the group velocity Vg = 1.6 × 10−5c. Consequently, the optical soliton obtained travels with an ultraslow propagating velocity in the system.
The input power for generating the ultraslow optical soliton may be estimated by calculating Poyntings vector. The average flux of energy over carrier-wave period is P̄/S0 = (P̄max/S0)sech2[(t − z/Ṽg)/τ0] with the peak power . Here, np = nneff + cK̃0/ωp is the refractive index and S0 is the cross section area of the probe beam. With the values of coefficients given above, we obtain P̄max = 1.19 μW. Thus, very low input power is needed for generating the ultraslow optical soliton in the slot waveguide system.
In order to make a further confirmation of the soliton solutions and check their stability, a numerical simulation is carried out. Shown in Fig. 5(a) is the three-dimensional plot for the wave shape |Ωp/U0|2 as a function of z/LD and t/τ0. The initial condition of the simulation is given by Ωp(0, σ) = U0sech(t/τ0). We see that the amplitude of the soliton undergoes only a slight decrease and its width undergoes slight increase due to the influence of the small imaginary parts of the coefficients. The properties of collision between two ultraslow optical solitons are also investigated numerically by taking Ωp(0, σ) = U0sech(t/τ0 − 5)+U0sech(t/τ0 + 5) as the initial condition without any approximation. As time goes on, they collide, pass through, and depart from each other, as shown in Fig. 5(b). The two solitons recover their initial waveforms after the collision.
We have investigated the EIT and nonlinear pulse propagation in a Λ-type three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We have found that the EIT effect can be largely enhanced due to reduction of optical-field mode volume contributed by the waveguide geometry. In comparison with the atomic gases in free space, the EIT transparency window in the slot waveguide system are much wider and deeper, and the Kerr nonlinearity of the probe laser field are much stronger. We have also proved that by using the slot waveguide ultraslow optical solitons via EIT can be produced efficiently with extremely low generation power. The present work opens an avenue to the study EIT-related quantum coherence in nano-sized systems and the results presented may have promising applications for optical information processing and transmission.
A. TM-modes of EM field and mode volume in the slot waveguide
For the slot waveguide, the EM field can be divided into transverse electric (TE) and transverse magnetic (TM) parts, i.e. E = ETE +ETM and H = HTE +HTM. The confinement and enhancement of EM field near the slot region is contributed by the TM part. By solving Maxwell’s equations in the absence of atoms, we can obtain the TM eigenmode solutions with the coordinate system chosen in Fig. 1 asFig. 1).
The function Hm,k‖(z) in Eq. (22) satisfies the equation19]
Using the formula with U being EM-field energy and ε and μ (= μ0) being respectively the permittivity and permeability, we obtain the second-quantization form of the TM part
The quantity Vm appeared in Eq. (26) is the (effective) mode volume given by
B. TM-modes and mode volume of EM field without the slot
For conventional slab waveguide (i.e. the waveguide shown in Fig. 1 but with the slot width 2a = 0), the TM-modes of the EM field have the same form of Eq. (22), but here n(z) takes the values nH for |z| < b, and nC for |z| > b, we have
C. Expressions of appearing in Eq. (17)
The authors thanks L. Deng and M. Xiao for useful discussions. This work was supported by NSF-China under Grant numbers 10874043 and 11174080.
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