Electromagnetically induced transparency and ultraslow optical solitons in a coherent atomic gas filled in a slot waveguide

doi:10.1364/OE.21.005149

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Electromagnetically induced transparency and ultraslow optical solitons in a coherent atomic gas filled in a slot waveguide

Jin Xu and Guoxiang Huang »View Author Affiliations

Optics Express, Vol. 21, Issue 4, pp. 5149-5163 (2013)
http://dx.doi.org/10.1364/OE.21.005149

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▸ Article Outline
– Abstract
1. Introduction
2. Model
3. EIT characters
4. Kerr nonlinearity and ultraslow optical solitons
5. Conclusion
– References and links

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Abstract

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.

1. Introduction

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 [1

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

]. Another noticeable effect of EIT is drastic reduction of group velocity of optical pulses, which has important applications such as slow light [2

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature. 397, 594–598 (1999). [CrossRef]

, 3

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999). [CrossRef]

], quantum memory [4

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009). [CrossRef]

–6

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett. 108, 210501 (2012). [CrossRef] [PubMed]

], quantum phase gates [7

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]

, 8

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,” Phys. Rev. A 74, 012319 (2006). [CrossRef]

], and slow light solitons [9

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]

, 10

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]

However, up to now most works on EIT and optical pulse propagations have been performed with atomic gases in bulk samples [1

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

]. 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

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef] [PubMed]

–17

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A 81, 053825 (2010). [CrossRef]

], nanofibers [18

F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A. 79, 013818 (2009). [CrossRef]

], 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 [11

] or hollow-core photonic crystal fibers [12

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005). [CrossRef] [PubMed]

–17

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A 81, 053825 (2010). [CrossRef]

] 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. [19

V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

] 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

Q. Xu, V. R. Almeida, R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29, 1626–1628 (2004). [CrossRef] [PubMed]

, 21

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett. 89, 241114 (2006). [CrossRef]

]. Recently, There have been a large amount of research activities on guiding and confining light by using slot waveguides [22

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett. 94, 251111 (2009). [CrossRef]

–32

R. Guo, B. Wang, X. Wang, L. Wang, L. Jiang, and Z. Zhou, “Optical amplification in Er/Yb silicate slot waveguide,” Opt. Lett. 37, 1427–1429 (2012). [CrossRef] [PubMed]

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.

2. Model

The slot waveguide we adopt is similar to that suggested in Ref. [19

V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

], 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 n_S) embedded between two thick rectangular regions (with width b − a on both sides) of high-index material (with index n_H), both surrounded by a low-index cladding (with index n_C). Sizes in the x- and y-directions of each rectangular region are much larger than a and b. It has been shown [19

V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

] 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 ω_m(k_‖) are given in Appendix A. Here

k_{| |} = {(k_{x}^{2} + k_{y}^{2})}^{1 / 2}

with k_x and k_y 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 (n_H / n_S)² times higher than that in the high-index region.

Fig. 1 Left: Schematic of slot waveguide structure. The slot width (index n_S) is 2a, the width of the high-index silicon slabs (index n_H) is 2b − 2a. The index of the cladding material is n_C. Right: Level diagram of the three-level atomic system. Ground state |1〉 couples to the exited |2〉 and |3〉 with the control field Ω_c and the probe field Ω_p. Δ₂ and Δ₃ are the detunings of control and probe fields, respectively. Γ₃₁ (Γ₃₂) is the spontaneous emission decay rate from |3〉 to |1〉 (|2〉). Γ₁₂ and Γ₂₁ are incoherent population exchange rates. Below the level diagram is the coordinate system chosen for theoretical calculations. The slot region is |z| ≤ a, and the slabs are in the region a < |z| < b.

Since k_x, k_y 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 k_x = 0, k_y = k without loss of generality. Then from the Appendix A we have k_‖ = (0, k, 0), k̂_‖ = e_y, k_‖ = k, and

E^{TM} (r, t) = \sum_{k} \sqrt{\frac{\bar{h} ω_{1} (k)}{2 ε_{0} V_{1}}} u_{1, k} (z) a_{1} (k) e^{i [k y - ω_{1} (k) t]} + c . c .,

(1a)

u_{1, k} (z) = \frac{c}{2 \sqrt{N_{1}} ω_{1} (k) n^{2} (z)} [k H_{1, k} (z) e_{z} + i \frac{d H_{1, k} (z)}{d z} e_{y}] .

(1b)

Here

ω_{1} (k) = {(κ_{H 1}^{2} + k^{2})}^{1 / 2} c / n_{H} = {(k^{2} - γ_{S 1}^{2})}^{1 / 2} c / n_{S}

, V₁ is the mode volume, e_y (e_z) is the unit vector along the y (z) direction, u_1,_k(z) is the mode function satisfying

\int_{- \infty}^{\infty} d z {| u_{1, k} (z) |}^{2} = V_{1}

, ε(z) ≡ n²(z) is dielectric function with n(z) = n_S (|z| < a), n_H (a < |z| < b), and n_C (|z| > b) (see Fig. 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

E^{TM} (r, t) = \sum_{ω} ℰ_{ω} {(\frac{W_{1}}{V_{1}})}^{\frac{1}{2}} u_{1, ω} (z) e^{i [(ω) y - ω t)]} + c . c .

, with

u_{1, ω} (z) \equiv {c / [\sqrt{N_{1}} ω n^{2} (z)]} {[k H_{1, ω} (z) e_{z} + i [d H_{1, ω} (z) / d z] e_{y}}

. Here

k (ω) = {[n_{H}^{2} ω^{2} / c^{2} - κ_{H 1}^{2}]}^{1 / 2} = {[γ_{S 1}^{2} + n_{S}^{2} ω^{2} / c^{2}]}^{1 / 2}

and

ℰ_{ω} = \sqrt{\frac{\bar{h} ω}{2 ε_{0} W_{1}}} a_{1} (ω)

, 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. n_S = n_C = 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 ω_p = k_p/c and half Rabi frequency Ω_p (i.e. |1〉 → |3〉 transition) and a strong, continuous-wave control field of center angular frequency ω_c = k_c/c and half Rabi frequency Ω_p (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

E^{TM} (r, t) = \sum_{l = p, c} ℰ_{1} {(\frac{W_{1}}{V_{1}})}^{\frac{1}{2}} u_{1, l} (z) exp {i [k (ω_{l}) y - ω_{l} t]} + c . c ..

(2)

The Hamiltonian of the system in interaction picture reads

\hat{ℋ} = - \bar{h} \sum_{j}^{3} Δ_{j} | j 〉 〈 j | - \bar{h} [ζ_{p}^{*} (z) Ω_{p}^{*} | 1 〉 〈 3 | + ζ_{c}^{*} (z) Ω_{c}^{*} | 2 〉 〈 3 | + h . c .],

(3)

where Δ₃ ≡ ω_p − (ω₃ − ω₁) and Δ₂ ≡ ω_p − ω_c − (ω₂ − ω₁) are respectively the one- and two-photon detunings,

ζ_{p} (z) \equiv {(\frac{W_{1}}{V_{1}})}^{\frac{1}{2}} e_{31} \cdot u_{1, p} (z)

and

ζ_{c} (z) \equiv {(\frac{W_{1}}{V_{1}})}^{\frac{1}{2}} e_{32} \cdot u_{1, c} (z)

are respectively the mode functions of the probe and control fields, Ω_p ≡ p₃₁ℰ_p/h̄ and Ω_c ≡ p₃₂ℰ_c/h̄ are respectively their corresponding half Rabi frequencies, with e_jl the unit vector of the electric dipole matrix element p_jl, i.e. p_jl = e_jlp_jl.

Taking Doppler effect into account, the equation of motion of σ_jl, i.e. the density matrix elements in the interaction picture, are given by

i \frac{\partial}{\partial t} σ_{11} + i Γ_{21} σ_{11} - i Γ_{12} σ_{22} - i Γ_{13} σ_{33} + ζ_{p}^{*} (z) Ω_{p}^{*} σ_{31} - ζ_{p} (z) Ω_{p} σ_{31}^{*} = 0,

(4a)

i \frac{\partial}{\partial t} σ_{22} - i Γ_{21} σ_{11} + i Γ_{12} σ_{22} - i Γ_{23} σ_{33} + ζ_{c}^{*} (z) Ω_{c}^{*} σ_{32} - ζ_{c} (z) Ω_{c} σ_{32}^{*} = 0,

(4b)

\begin{matrix} i \frac{\partial}{\partial t} σ_{33} + i (Γ_{13} + Γ_{23}) σ_{33} - ζ_{p}^{*} (z) Ω_{p}^{*} σ_{31} + ζ_{p} (z) Ω_{p} σ_{31}^{*} \\ - ζ_{c}^{*} (z) Ω_{c}^{*} σ_{32} + ζ_{c} (z) Ω_{c} σ_{32}^{*} = 0, \end{matrix}

(4c)

(i \frac{\partial}{\partial t} + d_{21}) σ_{21} - ζ_{p} (z) Ω_{p} σ_{32}^{*} + ζ_{c}^{*} (z) Ω_{c}^{*} σ_{31} = 0,

(4d)

(i \frac{\partial}{\partial t} + d_{31}) σ_{31} - ζ_{p} (z) Ω_{p} (σ_{33} - σ_{11}) + ζ_{c} (z) Ω_{c} σ_{21} = 0,

(4e)

(i \frac{\partial}{\partial t} + d_{32}) σ_{32} - ζ_{c} (z) Ω_{c} (σ_{33} - σ_{22}) + ζ_{p} (z) Ω_{p} σ_{21}^{*} = 0,

(4f)

where d₂₁ = −(k_p − k_c)v + Δ₂ + iγ₂₁, d₃₁ = −k_pv + Δ₃ + iγ₃₁ and d₃₂ = −k_cv + (Δ₃ − Δ₂) + iγ₃₂ (with v the atom velocity), Γ_j = ∑_j>i Γ_ij and

γ_{i j} = (Γ_{i} + Γ_{j}) / 2 + γ_{i j}^{col}

are population and coherence decay rates, respectively. Here Γ_ij represents the spontaneous emission decay rate from state |j〉 to state |i〉,

γ_{i j}^{col}

are dephasing rates, and Γ₂₁ is the incoherent population exchange from state |1〉 to state |2〉. Note that the atom-photon interactions in waveguide geometries may introduce some undesirable effects, such as the atomic collisions with waveguide walls and the adhesion to the walls [17

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A 81, 053825 (2010). [CrossRef]

]. We assume these interfacial effects can be weakened by using similar experimental techniques as did in Refs. [13

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett. 97, 023603 (2006). [CrossRef] [PubMed]

, 16

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009). [CrossRef] [PubMed]

, 17

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A 81, 053825 (2010). [CrossRef]

], like coating the inner walls with some materials (i.e. paraffin) or using light-induced atomic desorption. Note that the controlling field Rabi frequency Ω_c 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 reads

P = 𝒩_{a} \int_{- \infty}^{\infty} d v f (v) [p_{13} σ_{31} e^{i (k_{p} y - ω_{p} t)} + p_{23} σ_{32} e^{i (k_{c} y - ω_{c} t)} + c . c .],

(5)

where 𝒩_a is atomic concentration,

f (v) = 1 / (\sqrt{π} v_{T}) exp [- {(v / v_{T})}^{2}]

is Maxwell velocity distribution,

v_{T} = \sqrt{2 k_{B} T / M}

is the most probable speed at temperature T, with M the atomic mass and k_B the Boltzmann constant. Because the integration over the Maxwell distribution is not easy to analyze, as did in Refs. [33

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B 76, 33 (2003). [CrossRef]

, 34

L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A 82, 023809 (2010). [CrossRef]

] we replace it by the modified Lorentzian velocity distribution

f (v) = v_{T} / [\sqrt{π} (v^{2} + v_{T}^{2})]

The motion of the electric field is controlled by Maxwell equation, which under the slowly varying envelope approximation reduces to

i (\frac{\partial}{\partial y} + \frac{1}{c} \frac{〈 n^{2} (z) 〉}{n_{eff}} \frac{\partial}{\partial t}) Ω_{p} + \frac{c}{2 ω_{p} n_{eff}} \frac{\partial^{2} Ω_{p}}{\partial x^{2}} + κ_{13} \int_{- \infty}^{\infty} d v f 〈 v 〉 〈 σ_{31} (v, z) 〉 = 0,

(6)

where κ₁₃ = Nω_p|p₁₃|²/(2ε₀h̄cn_eff) and n_eff = ck(ω_p)/ω_p is the effective refraction index. The quantity

〈 Q (z) 〉 \equiv \int_{- \infty}^{\infty} d z ζ^{*} (z) Q (z) / \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2}

for any function of Q(z).

3. EIT characters

3.1. Base state

When the probe field is absent (i.e. Ω_p = 0), the Maxwell-Bloch (MB) Eqs. (4), (6) have the steady-state solution

σ_{11}^{(0)} = \frac{Γ_{12} (Γ_{13} + Γ_{23}) X_{1} + (Γ_{12} + Γ_{13}) {| ζ (z) Ω_{c} |}^{2}}{X_{2}},

(7a)

σ_{22}^{(0)} = \frac{Γ_{21} (Γ_{13} + Γ_{23}) X_{1} + Γ_{21} {| ζ (z) Ω_{c} |}^{2}}{X_{2}},

(7b)

σ_{33}^{(0)} = \frac{Γ_{21} {| ζ (z) Ω_{c} |}^{2}}{X_{2}},

(7c)

σ_{32}^{(0)} = - \frac{ζ (z) Ω_{c}}{d_{32}} \frac{Γ_{21} (Γ_{13} + Γ_{23}) X_{1}}{X_{2}},

(7d)

and

σ_{21}^{(0)} = σ_{31}^{(0)} = 0

, where

X_{1} = {{[k_{c} v - (Δ_{3} - Δ_{2})]}^{2} + γ_{32}^{2}} / 2 γ_{32}

and X₂ = (Γ₁₂ + Γ₂₁)(Γ₁₃ + Γ₂₃)X₁ + (Γ₁₂ + 2Γ₂₁ + Γ₁₃)|ζ(z)Ω_c|². Note that we have taken ζ_c(z) ≈ ζ_p(z) ≡ ζ(z) because ω_p ≈ ω_c.

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 σ_jl = 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

σ_{j j} = σ_{j j}^{(0)}

(j = 1, 2, 3),

σ_{32} = σ_{32}^{(0)}

, and assuming Ω_p and σ_j₁ are small quantities proportional to exp[i(K(ω)y − ωt)], we obtain the linear dispersion relation

\begin{array}{l} K (ω) = & \frac{1}{\int_{- \infty}^{\infty} d z {| ζ (z) |}^{2}} \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2} {\frac{ω}{c} \frac{n^{2} (z)}{n_{eff}} \\ + κ_{13} \int_{- \infty}^{\infty} d v f (v) \frac{ζ (z) Ω_{c} σ_{32}^{* (0)} + (ω + d_{21}) (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1)}{{| ζ (z) Ω_{c} |}^{2} - (ω + d_{21}) (ω + d_{31})}} . \end{array}

(8)

Different from the case in free space [34

L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A 82, 023809 (2010). [CrossRef]

], 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 [34

L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A 82, 023809 (2010). [CrossRef]

]. Taking k_pv as a complex number, we find two poles in the lower half complex plane, given by k_pv = −ik_pv_T and

k_{p} v = - i {γ_{32}^{2} + 2 γ_{32} (Γ_{12} + 2 Γ_{21} + Γ_{13}) {| ζ (z) Ω_{c} |}^{2} / [(Γ_{12} + Γ_{21}) Γ_{3}]}^{1 / 2} \equiv - i η

. For calculating the integration, we take a contour consisting of the lower half complex plane and real axis. The use of residue theorem gives

K (ω) = \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2} (\frac{ω}{c} \frac{n^{2} (z)}{n_{eff}} + 𝒦_{1} (z) + 𝒦_{2} (z)) / \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2},

(9)

where 𝒦₁ is the contribution from the first pole point k_pv = −ik_pv_T :

\begin{array}{l} 𝒦_{1} (z) = & \sqrt{π} κ_{13} {{| ζ (z) Ω_{c} |}^{2} Γ_{21} Γ_{3} (- i Δ ω_{D} - i γ_{32}) \\ + (ω + i γ_{21}) [Γ_{12} Γ_{3} (- Δ ω_{D}^{2} + γ_{32}^{2}) + 2 γ_{32} (Γ_{12} - Γ_{21} + Γ_{13}) {| ζ (z) Ω_{c} |}^{2}]} \\ / {(- Δ ω_{D}^{2} + η^{2}) (Γ_{12} + Γ_{21}) Γ_{3} [{| ζ (z) Ω_{c} |}^{2} - (ω + i γ_{21}) (ω + i Δ ω_{D} + i γ_{31})]}, \end{array}

(10)

and 𝒦₂ is the contribution from the second pole point k_pv = −iη

\begin{array}{l} 𝒦_{2} (z) = & \sqrt{π} κ_{13} Δ ω_{D} {{| ζ (z) Ω_{c} |}^{2} Γ_{21} Γ_{3} (- i η - i γ_{32}) \\ + (ω + i γ_{21}) [Γ_{12} Γ_{3} (- η^{2} + γ_{32}^{2}) + 2 γ_{32} (Γ_{12} - Γ_{21} + Γ_{13}) {| ζ (z) Ω_{c} |}^{2}]} \\ / {η (Δ ω_{D}^{2} - η^{2}) (Γ_{12} + Γ_{21}) Γ_{3} [{| ζ (z) Ω_{c} |}^{2} - (ω + i γ_{21}) (ω + i η + i γ_{31})]} . \end{array}

(11)

The integration on z in Eq. (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 by

\begin{matrix} Im (K_{0}) = \frac{1}{\int_{- \infty}^{\infty} d z {| ζ (z) |}^{2}} \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2} \int_{- \infty}^{\infty} d v f (v) \frac{κ_{13} γ_{21}}{{| ζ (z) Ω_{c} |}^{2} + γ_{21} γ_{31}} \\ \times (1 - \frac{Γ_{21} Γ_{3} X_{1} + 3 Γ_{21} {| ζ (z) Ω_{c} |}^{2}}{(Γ_{21} + Γ_{12}) Γ_{3} X_{1} + (2 Γ_{21} + Γ_{12} + Γ_{13}) {| ζ (z) Ω_{c} |}^{2}}) . \end{matrix}

(12)

From Eq. (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 Ω_c but also on the slot width. Fig. 2(b) shows the profile of Im(K₀) as a function of Rabi frequency |Ω_c| 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 Ω_c, 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 Ω_c 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 |5S_1/2, F = 1〉, |5S_1/2, F = 2〉, and |5P_1/2, F = 1〉 as the atomic states |1〉, |2〉, and |3〉, respectively. The system parameters used are κ₁₃ = 1.0 × 10⁹ cm⁻¹s⁻¹, Ω_c = 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 Ω_c is very large. However, it has non-negligible influence when Ω_c 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 Ω_c = 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 |Ω_c| 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 |Ω_c|, there is no obvious difference in Im(K₀) for different Γ₂₁; but for small |Ω_c| the effect caused by Γ₂₁ can be observed clearly.

Fig. 2 (a): Im(K) as a function of frequency ω for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30 and 10 nm, respectively. (b): Im(K₀) as a function of |Ω_c| for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30, 10 nm, respectively.

Fig. 3 (a): Im(K) as a function of ω. (b): Im(K₀) as a function of |Ω_c| with Γ₂₁ = 0 (red solid line), Γ₂₁ = 0.5Γ₁₂ (black dashed line) and Γ₂₁ = Γ₁₂ (blue dashed-dotted line), respectively.

4. Kerr nonlinearity and ultraslow optical solitons

4.1. Kerr nonlinearity of the system

From the MB Eqs. (4) and (6), we obtain the probe field susceptibility

χ_{p} = \int_{- \infty}^{\infty} d v f (v) \frac{𝒩_{a} {| p_{13} |}^{2}}{ε_{0} \bar{h}} \frac{〈 σ_{31} (v, z) 〉}{Ω_{p}} \approx χ_{p}^{(1)} + χ_{p p}^{(3)} {| ℰ_{p} |}^{2},

(13)

where ℰ_p = h̄Ω_p/p₃₁,

χ_{p}^{(1)}

and

χ_{p p}^{(3)}

are respectively first-order (linear) and third-order (Kerr) susceptibilities, with the expressions given by

\begin{array}{l} χ_{p}^{(1)} & = & \frac{𝒩_{a} {| p_{13} |}^{2}}{ε_{0} \bar{h}} \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2} \int_{- \infty}^{\infty} d v f (v) \\ \times \frac{d_{21} d_{32}^{*} (σ_{33}^{(0)} - σ_{11}^{(0)}) - {| ζ (z) Ω_{c} |}^{2} (σ_{33}^{(0)} - σ_{22}^{(0)})}{d_{32}^{*} (d_{21} d_{31} - {| ζ (z) Ω_{c} |}^{2})} / \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2}, \end{array}

(14a)

\begin{array}{l} χ_{p p}^{(3)} & = & \frac{𝒩_{a} {| p_{13} |}^{4}}{ε_{0} {\bar{h}}^{3}} \frac{1}{\int_{- \infty}^{\infty} d z {| ζ (z) |}^{2}} \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2} \int_{- \infty}^{\infty} d v f (v) \\ \times {| ζ (z) |}^{2} [i d_{32} ζ (z) Ω_{c} Z_{1} Z_{4} - (2 d_{21} {| d_{32} |}^{2} - d_{32} {| ζ (z) Ω_{c} |}^{2}) Z_{2} \\ - (d_{21} {| d_{32} |}^{2} - 2 d_{32} {| ζ (z) Ω_{c} |}^{2}) Z_{3}] / [i Z_{1} {| d_{32} |}^{2} ({| ζ (z) Ω_{c} |}^{2} - d_{21} d_{31})], \end{array}

(14b)

where Z₁ = (Γ₁₂ + Γ₂₁)(Γ₁₃ + Γ₂₃)|d₃₂|² + 2γ₃₂(Γ₁₂ + 2Γ₂₁ + Γ₁₃)|ζ(z)Ω_c|²,

Z_{2} = [(Γ_{12} + Γ_{23}) {| d_{32} |}^{2} + 4 γ_{32} {| ζ (z) Ω_{c} |}^{2}] (Z_{5} - c . c .) + (Γ_{12} - Γ_{13}) [(d_{32}^{*} ζ^{*} (z) Ω_{c}^{*} Z_{4}^{*}) - c . c .]

Z_{3} = (Γ_{21} + Γ_{13}) [(d_{32}^{*} ζ^{*} (z) Ω_{c}^{*} Z_{4}^{*}) - c . c .] - [(- Γ_{21} + Γ_{23}) {| d_{32} |}^{2} + 2 γ_{32} {| ζ (z) Ω_{c} |}^{2}] (Z_{5} - c . c .)

Z_{4} = [- d_{31} σ_{32}^{* (0)} - ζ^{*} (z) Ω_{c}^{*} (σ_{11}^{(0)} - σ_{33}^{(0)})] / ({| ζ (z) Ω_{c} |}^{2} - d_{21} d_{31})

and

Z_{5} = [ζ (z) Ω_{c} σ_{32}^{(0)} + d_{21}^{*} (σ_{11}^{(0)} - σ_{33}^{(0)})] / ({| ζ (z) Ω_{c} |}^{2} - d_{21}^{*} d_{31}^{*})

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.

Re (χ_{p p}^{(3)})

, 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 (ε_S = 1) slot embedded between the silicon (ε_H = 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.

Fig. 4 Third-order susceptibility

Re (χ_{p p}^{(3)})

as a function of detunning Δ₃ for different slot width 2a. The red solid, black dashed and blue dashed-dotted lines are for 2a = 50,25 and 5 nm, respectively.

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

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]

, 10

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]

]. 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. [10

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]

]. Taking the asymptotic expansion

σ_{i j} = \sum_{l = 0} ε^{(l)} σ_{i j}^{(l)}

Ω_{p} = \sum_{l = 1} ε^{(l)} Ω_{p}^{(l)}

, where

σ_{i j}^{(0)}

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 y_β = ε^βy (β = 0, 1, 2) and t_β = ε^βt (β = 0,1). Substituting the expansion into the MB Eqs. (4) and (6), we obtain a series of equations for

σ_{i j}^{(l)}

and

Ω_{p}^{(l)}

, which can be solved order by order.

At the first order (l = 1), we obtain the linear solution

Ω_{p}^{(1)} = {Fe}^{i θ},

(15a)

σ_{31}^{(1)} = \frac{(ω + d_{21}) (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1) + ζ (z) Ω_{c} σ_{32}^{* (0)}}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}) (ω + d_{31})} ζ (z) {Fe}^{i θ},

(15b)

σ_{21}^{(1)} = - \frac{(ω + d_{31}) σ_{32}^{* (0)} + ζ_{c}^{*} (z) Ω_{c}^{*} (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1)}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}) (ω + d_{31})} ζ (z) {Fe}^{i θ},

(15c)

with other

σ_{i j}^{(1)} = 0

. Here θ = K(ω)y₀ − ωt₀, with F being a yet to be determined envelope function depending on the slow variables (t₁, y₁, y₂) and K(ω) being the linear dispersion relation given by Eq. (8).

At the second order (l = 2), the condition of the solution in this order is divergence-free requires i[∂F/∂y₁ + (1/v_g)∂F/∂t₁] = 0, where v_g = ∂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:

i \frac{\partial F}{\partial y_{2}} + \frac{c}{2 ω_{p} n_{eff}} \frac{\partial^{2} F}{\partial x_{1}^{2}} - \frac{K_{2}}{2} \frac{\partial^{2} F}{\partial t_{1}^{2}} - W {| F |}^{2} {Fe}^{- 2 \bar{α} y_{2}} = 0,

(16)

where K₂ ≡ ∂²K/∂ω², and

W = - κ_{13} \int_{- \infty}^{\infty} d z {| ζ (z) |}^{4} \int_{- \infty}^{\infty} d v f (v) \frac{ζ (z) Ω_{c} a_{32}^{* (2)} + (ω + d_{21}) (2 a_{11}^{(2)} + a_{22}^{(2)})}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}) (ω + d_{31})} / \int_{- \infty}^{\infty} d z {| ζ (z) |}^{2},

(17)

and α = Im(K) = ε²ᾱ. The explicit expressions of

a_{11}^{(2)}

a_{22}^{(2)}

a_{32}^{(2)}

have been given in Appendix C.

Returning to the original variables, Eq. (16) becomes

i (\frac{\partial}{\partial y} + α) U + \frac{c}{2 ω_{p} n_{eff}} \frac{\partial^{2} U}{\partial x^{2}} - \frac{K_{2}}{2} \frac{\partial^{2} U}{\partial τ^{2}} - W {| U |}^{2} U = 0,

(18)

where τ = t − y/v_g and U = εFe⁻^αy. Equation (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

i \frac{\partial u}{\partial s} + \frac{\partial^{2} u}{\partial σ^{2}} + 2 u {| u |}^{2} = i d_{0} u + d_{1} \frac{\partial^{2} u}{\partial ξ^{2}},

(19)

where s = −z/(2L_D), σ = τ/τ₀, ξ = x/R_⊥ and u = U/U₀.

L_{D} = τ_{0}^{2} / {\tilde{K}}_{2}

is the characteristic dispersion length, R_⊥ is the beam radius in x-direction and

U_{0} = 1 / τ_{0} \sqrt{{\tilde{K}}_{2} / \tilde{W}}

is typical Rabi frequency of the probe field, with K̃₂ and W̃ denoting respectively the real parts of K₂ and W. In Eq. (19), d₀ = L_D/L_A and d₁ = L_D/L_diff are two dimensionless coefficients, with L_A = 1/2α the characteristic absorption length and

L_{diff} = ω_{p} n_{eff} R_{x}^{2}

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

u = 2 β sech [2 β (σ - σ_{0} + 4 δ_{s})] exp [- 2 i δ σ - 4 i (δ^{2} - β^{2}) s - i ϕ_{0}],

(20)

where β, δ, σ₀ and ϕ₀ are real free parameters that determine the amplitude (also width), propagating velocity, initial position, and initial phase of the soliton, respectively. Taking β = 1/2, δ = σ₀ = ϕ₀ = 0, we have u = 2β sechexp(is); or in terms of Rabi frequency

Ω_{p} = \frac{1}{τ_{0}} \sqrt{\frac{{\tilde{K}}_{2}}{\tilde{W}}} sech [\frac{1}{τ_{0}} (t - \frac{y}{{\tilde{v}}_{g}})] exp [i {\tilde{K}}_{0} y + i \frac{y}{2 L_{D}}],

(21)

with K̃₀ = Re(K₀). Equation (21) describes a bright soliton traveling with velocity ṽ_g = Re(v_g).

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, R_x = 0.05 cm we have the characteristic lengths L_D = 1.4 cm, L_A = 38.4 cm and L_diff = 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 V_g = 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 = (P̄_max/S₀)sech²[(t − z/Ṽ_g)/τ₀] with the peak power

{\bar{P}}_{\max} = 2 ε_{0} c n_{p} S_{0} {(h / p_{31})}^{2} {\tilde{K}}_{2} / (\tilde{W} τ_{0}^{2})

. Here, n_p = n_neff + cK̃₀/ω_p is the refractive index and S₀ is the cross section area of the probe beam. With the values of coefficients given above, we obtain P̄_max = 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 |Ω_p/U₀|² as a function of z/L_D and t/τ₀. The initial condition of the simulation is given by Ω_p(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 Ω_p(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.

Fig. 5 (a) The three-dimensional plot of the wave shape |Ω_p/U₀|² as a function of z/L_D and t/τ₀. The solution is numerically obtained from Eq. (14) with full complex coefficients taken into account. The values of parameters are given in the text. (b): The interaction between two identical bright solitons.

5. Conclusion

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.

Appendix

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 = E^TE +E^TM and H = H^TE +H^TM. 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 as

H_{m, k_{| |}}^{TM} (r, t) = ({\hat{k}}_{| |} \times e_{z}) H_{m, k_{| |}} (z) e^{i (k_{| |} \cdot r - ω_{m} t)} + c . c .,

(22a)

E_{m, k_{| |}}^{TM} (r, t) = \frac{i}{ω_{m} (k_{| |}) ε_{0} ε (z)} [- i k_{| |} H_{m, k_{| |}} (z) e_{z} + \frac{d H_{m, k_{| |}} (z)}{d z} {\hat{k}}_{| |}] e^{i (k_{| |} \cdot r - ω_{m} t)} + c . c .,

(22b)

where k_‖ = (k_x, k_y, 0) (k_x, k_y are arbitrary real numbers) is the wavevector in the xy plane, k̂_‖ = k_‖/|k_‖| and e_z are respectively the unit vectors in the k_‖- and z-directions, ω_n(k_‖) is the eigenfrequency with m = 1, 2, 3···, and ε(z) ≡ n²(z) is dielectric function with n(z) (refractive index) taking the value n_S for |z| < a, n_H for a < |z| < b, and n_C for |z| > b (see Fig. 1).

The function H_{m,k_‖}(z) in Eq. (22) satisfies the equation

\frac{d^{2}}{d z^{2}} H_{m, k_{| |}} (z) + [{(\frac{ω}{c})}^{2} n^{2} (z) - k_{| |}^{2}] H_{m, k_{| |}} (z) = 0,

(23)

with the boundary conditions H_{m,k_‖}(z), dH_{m,k_‖}(z)/dz being continuous at the interfaces z = ±a, ±b. For guided modes, an additional condition _{m,k_‖}(z) → 0 for z → ±∞ is also required. Then one obtains [19

V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

]

H_{m, k_{| |}}^{TM} (z) = {\begin{matrix} cosh (γ_{S m} z), & | z | < a \\ C_{m} cos [κ_{H m} (| z | - a)] + D_{m} sin [κ_{H m} (| z | - a)], & a < | z | < b \\ E_{m} exp [- γ_{S m} (| z | - b)], & | z | > b \end{matrix}

(24)

where C_m = cosh(γ_Sma),

D_{m} = [n_{H}^{2} γ_{S m} / (n_{S}^{2} κ_{H m})] sinh (γ_{S m} a)

E_{m} = {cosh (γ_{S m} a) cos [κ_{H m} (b - a)] + [n_{H}^{2} γ_{S m} / (n_{S}^{2} κ_{H m})] sinh (γ_{S} a) sin [κ_{H m} (b - a)]}

, with

κ_{H m} = {[n_{H}^{2} ω_{m}^{2} (k_{| |}) / c^{2} - k_{| |}^{2}]}^{1 / 2}

and

γ_{S m} = {[k_{| |}^{2} - n_{S}^{2} ω_{m}^{2} (k_{| |}) / c^{2}]}^{1 / 2}

. The eigenvalue ω_m(k_‖) is determined by the equation

tan [κ_{H m} (b - a) - arctan (\frac{γ_{S m} n_{H}^{2}}{κ_{H m} n_{S}^{2}})] = \frac{γ_{S n} n_{H}^{2}}{κ_{H m} n_{S}^{2}} tanh (γ_{S m} a) .

(25)

In fact, the eigenfrequency ω_m depends only on

k_{| |} = {(k_{x}^{2} + k_{y}^{2})}^{1 / 2}

, i.e. ω_m = ω_m(k_‖). Obviously, the guided eigenmodes given above propagate in the xy-plane but confined basically in the slot region.

Using the formula

U = \frac{1}{2} \int d^{3} r (ε E^{2} + μ H^{2})

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

E^{TM} (r, t) = \sum_{k_{x}, k_{y}} \sum_{m = 1}^{\infty} \sqrt{\frac{\bar{h} ω_{m}}{2 ε_{0} V_{m}}} u_{m, k_{| |}} (z) {\hat{a}}_{m} (k) e^{i (k_{| |} \cdot r - ω_{m} t)} + h . c .,

(26a)

H^{TM} (r, t) = \sum_{k_{x}, k_{y}} \sum_{m = 1}^{\infty} \sqrt{\frac{\bar{h} ω_{m}}{2 ε_{0} V_{m}}} ({\hat{k}}_{| |} \times e_{z}) \frac{ε_{0} c}{\sqrt{N_{m}}} H_{m, k_{| |}} (z) {\hat{a}}_{m} (k) e^{i (k_{| |} \cdot r - ω_{m} t)} + h . c .,

(26b)

where

u_{m, k_{| |}} (z) = {c / [\sqrt{N_{m}} ω_{m} n^{2} (z)]} [e_{z} k_{| |} H_{m, k_{| |}} (z) + i {\hat{k}}_{| |} d H_{m, k_{| |}} (z) / d z]

is the mode function with

\int_{- \infty}^{\infty} d z {| u_{m, k_{| |}} (z) |}^{2} = V_{m}

, â_m(k) and

{\hat{a}}_{m}^{†} (k)

are creation and annihilation operators of TM photons. In our present study, we assume that the photon numbers in both control and probe fields are much larger than one, so â_m(k) and

{\hat{a}}_{m}^{†} (k)

are taken as dimensionless numbers with â_m(k) = a_n(k) and

{\hat{a}}_{m}^{†} (k) = a_{m}^{*} (k)

The quantity V_m appeared in Eq. (26) is the (effective) mode volume given by

\begin{array}{l} V_{m} = & S {\frac{1}{2 γ_{S m}} [sinh (2 γ_{S m} a) + 2 E_{m}^{2}] + \frac{a}{2} \\ + (C_{m}^{2} + D_{m}^{2}) (b - a) + \frac{1}{2 κ_{H m}} (C_{m}^{2} - D_{m}^{2}) sin [2 κ_{H m} (b - a)]} / N_{m}, \end{array}

(27a)

N_{m} = ω_{m}^{2} / {(c P)}^{2},

(27b)

where S is the transverse area of the waveguide in the xy-plane, and P =

(k_{| |}^{2} + γ_{S}^{2}) [sinh (2 γ_{S m} a) + 2 E_{m}^{2}] / (2 γ_{S} n_{S}^{4})

n_{H}^{2} ω_{m}^{2} (C_{m}^{2} + D_{m}^{2}) (b - a) / c^{2}

C_{m} D_{m} ω_{m}^{2} cos [2 κ_{H} (b - a)] / (4 c^{2} κ_{H m} n_{H}^{2})

ω_{m}^{2} (C_{m}^{2} - D_{m}^{2}) sinh [2 κ_{H} (b - a)] / (4 c^{2} κ_{H m} n_{H}^{2})

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 n_H for |z| < b, and n_C for |z| > b, we have

H_{m, k_{| |}}^{TM} (z) = {\begin{matrix} sin ϕ_{m} exp (\frac{z + b}{2 b} ψ_{m} cos ϕ_{m}), & z < - b \\ cos (ψ_{m} \frac{z}{b} sin ϕ_{m}), & - b < z < b \\ sin ϕ_{m} exp (\frac{b - z}{2 b} ψ_{m} cos ϕ_{m}), & z > b \end{matrix}

(28)

where

ϕ_{m} = {(n_{H}^{2} ω_{m}^{2} (k_{| |}) / c^{2} - k_{| |}^{2})}^{1 / 2} L

and

ψ_{m} = (2 b ω_{m} / c) {(n_{H}^{2} - n_{C}^{2})}^{1 / 2}

. The eigenvalue ω_m(k_‖) is determined by the equation ψ_m sinϕ_m = mπ − 2ϕ_m (m = 1, 2, 3, ···). The second-quantization form of the EM field reads

E^{TM} (r, t) = \sum_{k_{x}, k_{y}} \sum_{m = 1}^{\infty} \sqrt{\frac{\bar{h} ω_{m}}{2 ε_{0} W_{m}}} u_{m, k_{| |}} (z) {\hat{a}}_{n} (k) e^{i (k_{| |} \cdot r - ω_{m} t)} + h . c .,

(29a)

H^{TM} (r, t) = \sum_{k_{x}, k_{y}} \sum_{m = 1}^{\infty} \sqrt{\frac{\bar{h} ω_{m}}{2 ε_{0} W_{m}}} ({\hat{k}}_{| |} \times e_{z}) \frac{ε_{0} c}{\sqrt{M_{m}}} H_{m, k_{| |}} (z) {\hat{a}}_{m} (k) e^{i (k_{| |} \cdot r - ω_{m} t)} + h . c .,

(29b)

where

u_{m, k_{| |}} (z) = {c / [\sqrt{M_{m}} ω_{m} n^{2} (z)]} [e_{z} k_{| |} H_{m, k_{| |}} (z) + i {\hat{k}}_{| |} d H_{m, k_{| |}} (z) / d z]

is the mode function with

\int_{- \infty}^{\infty} {| u_{m, k_{| |}} (z) |}^{2} d z = W_{m} \cdot {\hat{a}}_{m} (k)

and

{\hat{a}}_{m}^{†} (k)

are creation and annihilation operators of TM photons, W_m is the mode volume given by

\begin{array}{l} W_{m} = & S {b [2 {sin}^{2} ϕ_{m} + ψ_{m} cos ϕ_{m}] / (ψ_{m} cos ϕ_{m}) \\ + b sin (2 ψ_{m} sin ϕ_{m}) / (2 ψ_{m} sin ϕ_{m})} / M_{m}, \end{array}

(30a)

M_{m} = ω_{m}^{2} / {(c G)}^{2},

(30b)

with

G = [2 k_{| |}^{2} b {sin}^{2} ϕ_{m}] / (ψ_{m} cos ϕ_{m}) + {sin}^{2} ϕ_{m} [ψ_{m} cos ϕ_{m} + ψ_{m}^{2} / 2 + ψ_{m}^{2} sin (2 ψ_{m} sin ϕ_{m})] / (2 b)

C. Expressions of $a_{i j}^{(2)}$ appearing in Eq. (17)

\begin{array}{l} a_{11}^{(2)} & = & - i {[i (Γ_{12} + Γ_{23}) - 2 {| ζ^{2} (z) Ω_{c} |}^{2} (\frac{1}{d_{32}} - \frac{1}{d_{32}^{*}})] \\ \cdot [\frac{(ω + d_{21}^{*}) (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1) + ζ^{*} (z) Ω_{c}^{*} σ_{32}^{(0)}}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}^{*}) (ω + d_{31}^{*})} - c . c .] \\ + i (Γ_{12} - Γ_{13}) [- \frac{ζ^{*} (z) Ω_{c}^{*}}{d_{32}} \frac{(ω + d_{31}^{*}) σ_{32}^{(0)} + ζ (z) Ω_{c} (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1)}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}^{*}) (ω + d_{31}^{*})} + c . c .]} \\ / [i (Γ_{12} + Γ_{21}) (Γ_{13} + Γ_{23}) - (2 Γ_{21} + Γ_{12} + Γ_{13}) {| ζ^{2} (z) Ω_{c} |}^{2} (\frac{1}{d_{32}} - \frac{1}{d_{32}^{*}})], \end{array}

(31a)

\begin{array}{l} a_{22}^{(2)} & = & \frac{i}{Γ_{12} - Γ_{13}} [\frac{(ω + d_{21}^{*}) (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1) + ζ^{*} (z) Ω_{c}^{*} σ_{32}^{(0)}}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}^{*}) (ω + d_{31}^{*})} - c . c . \\ - i (Γ_{21} + Γ_{13}) a_{11}^{(2)}], \end{array}

(31b)

a_{32}^{(2)} = \frac{1}{d_{32}} [\frac{(ω + d_{31}^{*}) σ_{32}^{(0)} + ζ (z) Ω_{c} (2 σ_{11}^{(0)} + σ_{22}^{(0)} - 1)}{{| ζ^{2} (z) Ω_{c} |}^{2} - (ω + d_{21}^{*}) (ω + d_{31}^{*})} - ζ (z) Ω_{c} (a_{11}^{(2)} + 2 a_{22}^{(2)})] .

(31c)

Acknowledgments

The authors thanks L. Deng and M. Xiao for useful discussions. This work was supported by NSF-China under Grant numbers 10874043 and 11174080.

References and links

1.	M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]
2.	L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature. 397, 594–598 (1999). [CrossRef]
3.	M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999). [CrossRef]
4.	A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009). [CrossRef]
5.	H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics 5, 628–632 (2011). [CrossRef]
6.	H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett. 108, 210501 (2012). [CrossRef] [PubMed]
7.	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]
8.	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,” Phys. Rev. A 74, 012319 (2006). [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.	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]
11.	S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef] [PubMed]
12.	S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005). [CrossRef] [PubMed]
13.	S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett. 97, 023603 (2006). [CrossRef] [PubMed]
14.	P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett. 32, 1323–1325 (2007). [CrossRef] [PubMed]
15.	F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express 13, 5694–5703 (2005). [CrossRef] [PubMed]
16.	M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009). [CrossRef] [PubMed]
17.	A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A 81, 053825 (2010). [CrossRef]
18.	F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A. 79, 013818 (2009). [CrossRef]
19.	V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]
20.	Q. Xu, V. R. Almeida, R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29, 1626–1628 (2004). [CrossRef] [PubMed]
21.	M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett. 89, 241114 (2006). [CrossRef]
22.	K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett. 94, 251111 (2009). [CrossRef]
23.	T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16, 13809–13817 (2008). [CrossRef] [PubMed]
24.	Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express 18, 23844–23856 (2010). [CrossRef] [PubMed]
25.	M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express 17, 7295–7303 (2009). [CrossRef] [PubMed]
26.	H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express 20, 13189–13194 (2012) [CrossRef] [PubMed]
27.	C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]
28.	P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). [CrossRef] [PubMed]
29.	Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A. 80, 011810(R) (2009). [CrossRef]
30.	L. Zhang, Y. Yue, Y. X. Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18, 13187–13193 (2010). [CrossRef] [PubMed]
31.	Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express 18, 22061–22066 (2010). [CrossRef] [PubMed]
32.	R. Guo, B. Wang, X. Wang, L. Wang, L. Jiang, and Z. Zhou, “Optical amplification in Er/Yb silicate slot waveguide,” Opt. Lett. 37, 1427–1429 (2012). [CrossRef] [PubMed]
33.	H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B 76, 33 (2003). [CrossRef]
34.	L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A 82, 023809 (2010). [CrossRef]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(130.2790) Integrated optics : Guided waves
(270.5530) Quantum optics : Pulse propagation and temporal solitons

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: November 30, 2012
Revised Manuscript: January 13, 2013
Manuscript Accepted: January 15, 2013
Published: February 22, 2013

Citation
Jin Xu and Guoxiang Huang, "Electromagnetically induced transparency and ultraslow optical solitons in a coherent atomic gas filled in a slot waveguide," Opt. Express 21, 5149-5163 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-5149

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References

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005). [CrossRef]
L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature.397, 594–598 (1999). [CrossRef]
M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999). [CrossRef]
A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009). [CrossRef]
H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011). [CrossRef]
H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012). [CrossRef] [PubMed]
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]
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,” Phys. Rev. A74, 012319 (2006). [CrossRef]
Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett.93, 143904 (2004). [CrossRef] [PubMed]
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]
S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express9, 748–779 (2001). [CrossRef] [PubMed]
S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005). [CrossRef] [PubMed]
S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006). [CrossRef] [PubMed]
P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett.32, 1323–1325 (2007). [CrossRef] [PubMed]
F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express13, 5694–5703 (2005). [CrossRef] [PubMed]
M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009). [CrossRef] [PubMed]
A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010). [CrossRef]
F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009). [CrossRef]
V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett.29, 1209–1211 (2004). [CrossRef] [PubMed]
Q. Xu, V. R. Almeida, R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett.29, 1626–1628 (2004). [CrossRef] [PubMed]
M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006). [CrossRef]
K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009). [CrossRef]
T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express16, 13809–13817 (2008). [CrossRef] [PubMed]
Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express18, 23844–23856 (2010). [CrossRef] [PubMed]
M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express17, 7295–7303 (2009). [CrossRef] [PubMed]
H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express20, 13189–13194 (2012) [CrossRef] [PubMed]
C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express15, 5976–5990 (2007). [CrossRef] [PubMed]
P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express17, 9282–9287 (2009). [CrossRef] [PubMed]
Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009). [CrossRef]
L. Zhang, Y. Yue, Y. X. Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express18, 13187–13193 (2010). [CrossRef] [PubMed]
Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express18, 22061–22066 (2010). [CrossRef] [PubMed]
R. Guo, B. Wang, X. Wang, L. Wang, L. Jiang, and Z. Zhou, “Optical amplification in Er/Yb silicate slot waveguide,” Opt. Lett.37, 1427–1429 (2012). [CrossRef] [PubMed]
H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003). [CrossRef]
L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A82, 023809 (2010). [CrossRef]

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