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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 15 — Jul. 19, 2010
  • pp: 15591–15596
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Controlled light-pulse propagation via dynamically induced double photonic band gaps

Ren-Gang Wan, Jun Kou, Shang-Qi Kuang, Li Jiang, and Jin-Yue Gao  »View Author Affiliations


Optics Express, Vol. 18, Issue 15, pp. 15591-15596 (2010)
http://dx.doi.org/10.1364/OE.18.015591


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Abstract

We analyze the optical response of a standing-wave driven four-level atomic system with double dark resonances. Fully developed double photonic band gaps arise as a result of periodically modulated refractive index within the two electromagnetically induced transparency widows. We anticipate that the dynamically induced band gaps can be used to coherently control the propagation of light-pulses with different center frequencies and may have applications in all-optical switching and routing for quantum information networks.

© 2010 OSA

1. Introduction

The phenomenon of electromagnetically induced transparency (EIT) [1

1. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36 (1997). [CrossRef]

,2

2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

], which based on the laser induced atomic coherence, plays an important role in the interaction between light and matter and has found numerous applications in light propagation control [3

3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

,4

4. 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(26), 5229–5232 (1999). [CrossRef]

], light storage [5

5. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84(22), 5094–5097 (2000). [CrossRef] [PubMed]

], enhancement of nonlinearity at low light levels [6

6. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82(23), 4611–4614 (1999). [CrossRef]

], etc. The usual EIT-scheme is three-level Λ system driven by a strong traveling-wave (TW) coupling field. Yet, when the coupling field has a standing-wave (SW) pattern, the optical response of the probe is modulated periodically in the space. This has been explored to obtain photonic band gap (PBG) [7

7. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006). [CrossRef] [PubMed]

9

9. J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77(11), 113106 (2008). [CrossRef]

], to generate stationary light pulse [10

10. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003). [CrossRef] [PubMed]

], to enhance optical nonlinearity [11

11. A. André, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94(6), 063902 (2005). [CrossRef] [PubMed]

], to realize deterministic quantum logic [12

12. I. Friedler, G. Kurizki, and D. Petrosyan, “Deterministic quantum logic with photons via optically induced photonic bandgaps,” Phys. Rev. A 71(2), 023803 (2005). [CrossRef]

], and to devise optical cavity [13

13. J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103(13), 133601 (2009). [CrossRef] [PubMed]

], etc. Most recently, multiple PBGs are obtained in some systems, such as photonic-crystal fibers [14

14. P. S. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]

] and multi-standing-wave driven atoms [15

15. C. L. Cui, J. H. Wu, J. W. Gao, Y. Zhang, and N. Ba, “Double photonic bandgaps dynamically induced in a tripod system of cold atoms,” Opt. Express 18(5), 4538–4546 (2010). [CrossRef] [PubMed]

].

It is well know that basis of EIT is the existence of dark resonance. As the single dark state is coherently coupled to another level by a control field, double dark resonances arise and the atom exhibits two EIT windows [16

16. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

,17

17. Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64(5), 053806 (2001). [CrossRef]

]. The interferences between the dark resonances allow us to well manipulate the optical responses of atoms [18

18. S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003). [CrossRef]

20

20. C. P. Liu and S. Q. Gong, “D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006). [CrossRef]

]. Inspired by these studies, we here demonstrate a coherent control of light-pulse propagation via induced double PBGs in a SW driven double-dark-resonance system. Due to the spatial modulation of probe refractive index, a pair of PBGs open up in the two EIT regions and can be tuned dynamically. The double-dark-resonance scheme has more degree of freedom in manipulating light-pulse than the single-dark-resonance scheme with only one PBG.

2. Theoretical model

Consider the four-level system in cold 87Rb atoms shown in Fig. 1
Fig. 1 Four-level atomic system with double dark resonances.
. The levels |1, |2, |3 and |4may correspond to the hyperfine states |5P3/2,F=2,mF=1, |5S1/2,F=2,mF=0, |5S1/2,F=2,mF=2 and |5S1/2,F=1,mF=0. A strong SW coupling field with space-dependent Rabi frequency Ωc(z) couples levels |2and |3, a weak probe field Ωp couples levels |1 and |2, and a microwave control field Ωd drives levels |3 and |4. In real experiments, the coupling and probe fields are circularly polarized with right and left helicities, and propagate along the z axis defined by a static magnetic field, while the control field is linearly polarized with polarization direction close to the z axis.

To deal with the optical properties of such a SW driven system, we start with its steady response. By solving the density-matrix equations under weak field approximation, the linear susceptibility χp and the refractive index np experienced by the probe are derived as
χp(Δp,z)=3π𝓍γ21(Γ31Γ41Ωd2)Γ21(Γ31Γ41Ωd2)Γ41Ωc2(z),np(Δp,z)=1+χp(Δp,z),
(1)
where 𝓍=N0(λp/2π)3, N0 is the atomic density. Γ21=Δpiγ21, Γ3=ΔpΔciγ31, and Γ4=ΔpΔcΔdiγ41, with γij being the dephasing rate of the respective atomic coherence ρij, and Δp=ω21ωp, Δc=ω23ωc, and Δp=ω21ωp are the detunings of the three fields from the corresponding transitions.

Unlike in typical double-dark-resonance system where a TW coupling field is used, the coupling here has a SW pattern which generated from the retroreflection on a mirror of reflectivityRm. Then the resulting squared coupling Rabi frequency varies periodically along z as
Ωc2(z)=Ω02[(1+Rm)2cos2(kcz)+(1Rm)2sin2(kcz)],
(2)
where kc is the wave vector of the coupling field. It is clear that Ωc2(z) has a spatial periodicity of a=λc/2, which may be changed into a=λc/[2cos(θ/2)] via slightly misaligning the forward and backward coupling fields by an angle θ. Clearly, the complex refractive index np(Δp,z) also varies periodically in the z direction. With np(Δp,z) in hand, we further obtain the 2×2 unimodular transfer matrix M(Δp) describing the probe propagation through a single period of length a. The translational invariance of the periodic medium requires that the Bloch condition must be imposed on the photonic eigenstates, i.e
(E+(x+a)E(x+a))=M(Δp)(E+(x)E(x))=(eiκaE+(x)eiκaE(x)),
(3)
where E+ and E are the amplitudes of the forward and backward electric field of the probe. κ=κ+iκis the complex Bloch wave vector, which represents the photonic band gap structure, and can be obtained from solutions of the equation e2iκaTr[M(Δp)]eiκa+1=0 [7

7. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006). [CrossRef] [PubMed]

]. For a sample of length l=Na with N being the number of the SW periods, the total transfer matrix can be expressed as MN=MN, and the reflection and transmission amplitudes for the probe are given by
r(ΔP)=MN(12)(ΔP)MN(22)(ΔP)=M12sin(Nκa)M22sin(Nκa)sin[(N1)κa], t(ΔP)=1MN(22)(ΔP)=sin(κa)M22sin(Nκa)sin[(N1)κa],
(4)
with MN(ij) being the matrix elements of MN. Then the reflectivity R(ΔP)=|r(ΔP)|2 and the transmissivity T(ΔP)=|t(ΔP)|2 can be calculated easily.

By utilizing Eq. (4) and the Fourier transform method [9

9. J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77(11), 113106 (2008). [CrossRef]

], we can further study the propagation dynamics of an incident probe pulse. Here, we assume that the input probe has Gaussian profiles in the time and frequency domains as
EIt(t)=E0te(tt0)2/δt2, EIf(Δp)=E0fe(ΔpΔp0)2/δp2,
(5)
where E0f=πδtE0t, δp=2/δt, t0 and δt (Δp0 and δp) are the center and width of the input probe pulse in the time (frequency) domain. Then we obtain the reflected and the transmitted Fourier components from ERf(Δp)=EIf(Δp)r(Δp) and ETf(Δp)=EIf(Δp)t(Δp), so that the reflected and transmitted probe pulse in the time domain can be derived via inverse Fourier transform, given by

ERt(Δp)=ERf(Δp)ei(ΔpΔp0)td(Δp),ETt(Δp)=ETf(Δp)ei(ΔpΔp0)td(Δp).
(6)

3. Numerical results

As pointed out by Lukin et al. [16

16. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

], the coherent interaction between the single dark state and the control field Ωd leads to the emergence of double dark resonances at the frequencies Δ±=(2Δc+Δd±Δd2+4Ωd2)/2, which indicate two distinct EIT windows for the probe. In the present scheme, as a result of the SW coupling field, the probe absorption and dispersion are modified along z axis with the same periodicity as the SW. Therefore, the probe field propagates as in a one-dimensional multilayer periodic structure which has two transparent regions determined by the double EIT. Consequently, double PBGs are expected to occur at Brillouin zone boundary π/a.

Figure 2
Fig. 2 Band gap structure near the first Brillouin zone boundary in terms of Bloch wave vector, and the probe reflectivity and transmissivity spectra for a 1.0 cm long atomic sample. Parameters are N0=1.0×1012cm3, η=0.12, θ=20mrad, γ21=6MHz, γ31=γ41=1KHz, Ω0=40MHz, Ωd=3MHz, Δc=0, and Δd=1MHz.
shows the PBGs structure in terms of the Bloch wave vector and the reflection and transmission spectra. First of all, it is worth mentioning that an imperfect SW coupling with unequal forward and backward components is applied so that double EIT are established everywhere, even at nodes of the SW. Therefore, the probe experiences no absorption and the band gaps can be well developed. We here define a parameter η=(1Rm)/(1+Rm) to describe the degree of imperfectness of the SW. As can be seen in Fig. 2, a pair of PBGs open up in the frequency ranges where κ=π/a and κ0. Within the gaps, κ0 corresponds to reflection rather than absorption and the probe reflectivity is nearly 95%. On the left of the band edges, there are two narrow regions with high transmissivity (95% at maximum) for the probe, and the positions of the two left band edges are determined by the dark resonances which occur at Δp=Δ±. Therefore, we can tune the gap positions by changing Δ±. Figure 3
Fig. 3 (Color online) Tuning of the band-gap reflectivity for different control intensities with Ωd=2MHz (black-solid), Ωd=3MHz (red-dashed), and Ωd=4MHz (blue-dotted). Other parameters are the same as in Fig. 2.
displays the variation of the band-gap structure for different control Rabi frequencies. Moreover, the detunings Δc and Δd, and the coupling Rabi frequency Ω0 can also be applied to tuning the positions and widths of the gaps.

The coherently induced double PBGs allow us control the light-pulse propagation all-optically. Consider an incident Gaussian pulse of which the amplitude of electric field can be written as EIt(t)=E0te(tt0)2/δt2 and the central probe detuning is denoted by Δp0, Fig. 4
Fig. 4 (Color online) Pulse dynamics of an incident (black-solid) probe impinging upon a long sample with the reflected (red-dashed) and transmitted (blue-dotted) parts shown. The frequency widths and centers of the incident pulsed are δt=20μs, t0=0 and (a) Δp0=2.6MHz, (b) Δp0=2.4MHz, (c) Δp0=2.0MHz, (d) Δp0=3.5MHz, (e) Δp0=3.65MHz, (f) Δp0=4.0MHz. Other parameters are the same as in Fig. 2.
shows the reflected part at z=0 (red-dashed) and transmitted part at z=l (blue-dotted). It is obviously that the dynamics of light-pulse propagation are sensitive to the central detuningΔp0 of the incident probe. If most of the carrier frequencies of the probe are inside the band gaps (Δp0=2.0MHz and Δp0=4.0MHz), the pulse is reflected with little attenuation or distortion except a short time delay [Fig. 4(c) and Fig. 4(f)]. As the center frequencies of the probe move near the band edge (Δp0=2.4MHz and Δp0=3.65MHz), i.e. only partial carrier frequencies fall into the PBGs, the reflected part is increasingly suppressed and distorted, while a stronger transmitted part is observed [Fig. 4(b) and Fig. 4(e)]. Both parts are broadened due to the decrease of carrier-frequency components, and the latter experiences slow-light propagation as a result of strong EIT dispersion. If most of the carrier frequencies lie inside the transmission regions (Δp0=2.6MHz and Δp0=3.5MHz), the probe pulse transmits subluminally through the atoms with a large time delay [Fig. 4(a) and Fig. 4(d)]. The loss and broadening of the pulse is owing to the narrow transmission and group-velocity dispersion.

The double PBGs scheme is more appealing than the single PBG scheme in real applications, because it can synchronously manipulate two weak light signals with distinct frequencies. For two pulses, by turning on the SW (double PBGs) or the TW (double EIT windows) coupling, we can either let them be reflected or be transmitted with little attenuation and deformation. This can be applied to all-optical switching or routing with two channels.

4. Conclusion

In conclusion, the optical response of a double-dark-resonance atomic system driven by a SW is investigated. The results show that fully developed double PBGs open up in the two EIT windows as a result of the periodically modulated refractive index, and the band-gap structure can be coherently controlled via tuning the positions and widths of the dark resonances. Moreover, the dynamically induced double PBGs provide an avenue to manipulate the light-pulse propagation. Such a double stop-bands mechanism can easily be exploited to implement a double-channel all-optical switching or routing for simultaneous information processing of two weak light-pulses with different central frequencies. We expect that it may have applications in optical network and quantum information processing.

Acknowledgements

The authors would like to thank the financial supports from NSFC (Grant No. 10774059), and the National Basic Research Program (Grant No. 2006CB921103) of P. R. China.

References and links

1.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36 (1997). [CrossRef]

2.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

3.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

4.

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(26), 5229–5232 (1999). [CrossRef]

5.

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84(22), 5094–5097 (2000). [CrossRef] [PubMed]

6.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82(23), 4611–4614 (1999). [CrossRef]

7.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006). [CrossRef] [PubMed]

8.

Q. Y. He, J. H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73(5), 053813 (2006). [CrossRef]

9.

J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77(11), 113106 (2008). [CrossRef]

10.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003). [CrossRef] [PubMed]

11.

A. André, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94(6), 063902 (2005). [CrossRef] [PubMed]

12.

I. Friedler, G. Kurizki, and D. Petrosyan, “Deterministic quantum logic with photons via optically induced photonic bandgaps,” Phys. Rev. A 71(2), 023803 (2005). [CrossRef]

13.

J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103(13), 133601 (2009). [CrossRef] [PubMed]

14.

P. S. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]

15.

C. L. Cui, J. H. Wu, J. W. Gao, Y. Zhang, and N. Ba, “Double photonic bandgaps dynamically induced in a tripod system of cold atoms,” Opt. Express 18(5), 4538–4546 (2010). [CrossRef] [PubMed]

16.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

17.

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64(5), 053806 (2001). [CrossRef]

18.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003). [CrossRef]

19.

Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30(24), 3371–3373 (2005). [CrossRef]

20.

C. P. Liu and S. Q. Gong, “D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006). [CrossRef]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.1670) Quantum optics : Coherent optical effects
(160.5293) Materials : Photonic bandgap materials

ToC Category:
Quantum Optics

History
Original Manuscript: April 8, 2010
Revised Manuscript: June 4, 2010
Manuscript Accepted: June 16, 2010
Published: July 8, 2010

Citation
Ren-Gang Wan, Jun Kou, Shang-Qi Kuang, Li Jiang, and Jin-Yue Gao, "Controlled light-pulse propagation via dynamically induced double photonic band gaps," Opt. Express 18, 15591-15596 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15591


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References

  1. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36 (1997). [CrossRef]
  2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]
  3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
  4. 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(26), 5229–5232 (1999). [CrossRef]
  5. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84(22), 5094–5097 (2000). [CrossRef] [PubMed]
  6. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82(23), 4611–4614 (1999). [CrossRef]
  7. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006). [CrossRef] [PubMed]
  8. Q. Y. He, J. H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73(5), 053813 (2006). [CrossRef]
  9. J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77(11), 113106 (2008). [CrossRef]
  10. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426(6967), 638–641 (2003). [CrossRef] [PubMed]
  11. A. André, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94(6), 063902 (2005). [CrossRef] [PubMed]
  12. I. Friedler, G. Kurizki, and D. Petrosyan, “Deterministic quantum logic with photons via optically induced photonic bandgaps,” Phys. Rev. A 71(2), 023803 (2005). [CrossRef]
  13. J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103(13), 133601 (2009). [CrossRef] [PubMed]
  14. P. S. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]
  15. C. L. Cui, J. H. Wu, J. W. Gao, Y. Zhang, and N. Ba, “Double photonic bandgaps dynamically induced in a tripod system of cold atoms,” Opt. Express 18(5), 4538–4546 (2010). [CrossRef] [PubMed]
  16. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]
  17. Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64(5), 053806 (2001). [CrossRef]
  18. S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003). [CrossRef]
  19. Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30(24), 3371–3373 (2005). [CrossRef]
  20. C. P. Liu and S. Q. Gong, “D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006). [CrossRef]

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