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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 2111–2119
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Steady optical spectra and light propagation dynamics in cold atomic samples with homogeneous or inhomogeneous densities

Yan Zhang, Yan Xue, Gang Wang, Cui-Li Cui, Rong Wang, and Jin-Hui Wu  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 2111-2119 (2011)
http://dx.doi.org/10.1364/OE.19.002111


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Abstract

We study both steady and dynamic optical responses of three samples with the same amounts of cold atoms but very different density functions. These samples are driven into the regime of electromagnetically induced transparency by a probe and a coupling in the Lambda configuration. When the coupling is in the traveling-wave pattern, all samples have the same transmission spectra and therefore identical transmitted pulses at the sample exits. In the case of a standing-wave coupling, however, very different reflection and transmission spectra are found for the three samples. Accordingly, reflected pulses at the sample entrances and transmitted pulses at the sample exits are quite sensitive to the spatial inhomogeneity of cold atoms. These interesting phenomena are qualitatively analyzed in terms of constructive and destructive interference between forward and backward probe photons scattered by a standing-wave atomic grating.

© 2011 Optical Society of America

1. Introduction

So far atomic samples dressed by TW or SW laser fields are always assumed to have spatially homogeneous densities for the simplicity in theoretical treatments, which is not true at least for cold atoms confined in magneto-optical traps (MOT). It is well known that cold atomic clouds in MOT usually have the Gaussian [27

27. H.-W. Cho, Y.-C. He, T. Peters, Y.-H. Chen, H.-C. Chen, S.-C. Lin, Y.-C. Lee, and I. A. Yu, “Direct measurement of the atom number in a Bose condensate,” Opt. Express 15, 12114–12122 (2007). [CrossRef] [PubMed]

] or cigar-shaped [28

28. Y.-W. Lin, H.-C. Chou, P. P. Dwivedi, Y.-H. Chen, and I. A. Yu, “Using a pair of rectangular coils in the MOT for the production of cold atom clouds with large optical density,” Opt. Express 16, 3753–3761 (2008). [CrossRef] [PubMed]

] profiles in spatial density. In this paper, we investigate by numerical calculations how the spatial inhomogeneity affects both steady and dynamic optical properties of cold atoms driven (by a weak probe and a strong coupling) into the EIT regime. Steady spectra of probe reflectivity and transmissivity are examined by the transfer-matrix method with the probe susceptibility analytically solved from density matrix equations while the propagation dynamics of an incident pulse is explored via the Fourier-transform method starting from the probe reflectivity and transmissivity.

Our numerical results show that, in the TW driving configuration, the probe transmissivity characterized by an EIT window is irrelevant to the spatial inhomogeneity of cold atoms. Therefore the transmitted pulses are not distinguishable for three typical samples (one homogeneous and two inhomogeneous) with the same amounts of cold atoms but very different density functions. In the SW driving configuration, however, both spectra of probe reflectivity and transmissivity are very sensitive to the spatial inhomogeneity of cold atoms. For instance, coherently induced PBGs denoted by a platform of reflectivity near 100% become narrower and narrower when the inhomogeneity of cold atoms is gradually increased. Consequently, an incident pulse perfectly reflected by the homogeneous sample will be split into a partially reflected one and a partially transmitted one in the two inhomogeneous samples with the same driving parameters. The underlying physics is that constructive and destructive interferences between forward (FW) and backward (BW) probe photons (scattered by an atomic grating generated by the SW coupling) depend critically on the spatial inhomogeneity of cold atoms. These results are expected to be instructive for the design of relevant experiments for light information manipulation in cold atomic samples driven by at least one SW field.

2. Model and Equations

We consider here a simplified three-level Λ system as shown in Fig. 1. The transition from level |1〉 to level |3〉 is probed by a weak light beam of frequency ωp and amplitude Ep while the transition from level |2〉 to level |3〉 is coupled by a strong light beam of frequency ωc and amplitude Ec, which may be either in the TW pattern or in the SW pattern. Under the electric-dipole and rotating-wave approximations, we can write the interaction Hamiltonian of this coherently driven Λ system as
HI=h¯δ|22|h¯Δ|33|h¯[Ωp|31|+Ωc|32|+h.c.],
(1)
where Δ = ωpω31 and δ = ωpωcω21 are, respectively, single-photon and two-photon detunings relevant to the probe and coupling fields. Ωp = Epd13/2c = Ecd23/2) is the probe (coupling) Rabi frequency with d13 (d23) denoting the dipole matrix element on transition |1〉 ↔ |3〉 (|2〉 ↔ |3〉). In particular, when the coupling field is set in the SW pattern, its Rabi frequency takes the form
Ωc(z)=Ωc+e+ikcz+Ωceikcz,
(2)
which varies along the z direction with a spatial periodicity of λc/2 and depends on the space-independent Rabi frequencies Ωc+ and Ωc of the FW and BW coupling beams.

Fig. 1 (Color online) Schematic diagram of a three-level Λ-type atomic system interacting with a weak probe field ωp and a strong coupling field ωc. The coupling field may be either in the TW pattern or in the SW pattern.

Starting from Eq. (1) and Eq. (2), it is straightforward to attain the following (expanded) Liouville equations for density matrix elements in the interaction picture
ρ11t=Γ31ρ33+iΩp*ρ31iΩpρ13,ρ22t=Γ32ρ33+iΩc*ρ32iΩcρ23,ρ12t=[γ12iδ]ρ12+iΩp*ρ32iΩcρ13,ρ13t=[γ13iΔ]ρ13iΩc*ρ12+iΩp*(ρ33ρ11),ρ23t=[γ23i(Δδ)]ρ23iΩp*ρ21+iΩc*(ρ33ρ22),
(3)
which are constrained by ρij=ρji* and ρ11 + ρ22 + ρ33 = 1. In Eqs. (3), Γij describes the population decay rate from level |i〉 to level |j〉 whereas γij denotes the dephasing rate of the coherence term ρij.

In the limit of a weak probe, we are allowed to assume ρ11 = 1 and ρ22 = ρ33 = ρ23 = 0 so that the steady-state solution of ρ31 can be found from Eqs. (3) to the first order in Ωp, from which the probe susceptibility may be written as
χp(Δ,z)=Na(z)|d13|22ɛ0h¯ρ31(Δ,z)Ωp=Na(z)|d13|22ɛ0h¯i(γ12iδ)(γ12iδ)(γ13iΔ)+|Ωc(z)|2,
(4)
where the atomic density function Na(z) has been taken to be space-dependent. With Eq. (4) in hand, we can further attain the complex refractive index
np(Δ,z)=1+χp(Δ,z),
(5)
which governs absorptive and dispersive properties on the probe transition.

Due to joint modulations of the spatially periodic coupling field and the spatially inhomogeneous atomic density, the complex refractive index np (Δ, z) in Eq. (5) becomes very complicated so that the derivation of relevant transmission and reflection spectra are rather intractable. That is, we have to first partition the atomic sample into a large number of laminas, then derive the individual transfer matrix of each lamina with the complex refractive index, and finally multiply the individual transfer matrices of all laminas in succession to attain the total transfer matrix of the whole sample [29

29. M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005). [CrossRef]

]. To be specific, the propagation of a probe field through the nth atomic lamina (extending from z = ndd to z = nd) is described by a 2 × 2 unimodular transfer matrix Mn (Δ, d) via
[Ep+(nd)Ep(nd)]=Mn(Δ,d)[Ep+(ndd)Ep(ndd)],
(6)
where Ep+ and Ep denote, respectively, the FW and BW probe fields. With Mn (Δ, d) determined by Eq. (6), the total transfer matrix finally turns out to be M(Δ, L) = M1(Δ, d) ⋯ Mn(Δ, d) ⋯ MN (Δ, d) for a sample of length L = Nd. Then we can write the probe reflectivity and transmissivity in terms of matrix elements M(ij) (Δ, L) as
R(Δ,L)=|r(Δ,L)|2=|M(12)(Δ,L)M(22)(Δ,L)|2,T(Δ,L)=|t(Δ,L)|2=|1M(22)(Δ,L)|2,
(7)
which is also available for the other case where the coupling field is set in the TW pattern (Ωc = 0 or Ωc+ = 0).

Now we briefly introduce the Fourier transform method [19

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

] for studying the propagation dynamics of an incident light pulse with r(Δ, L) and t(Δ, L) given in Eqs. (7). In the weak probe limit, the linear response functions r(Δ, L) and t(Δ, L) contain all relevant information on the optical responses of the atomic system under consideration to a monochromatic probe field. The basic procedure for the Fourier transform method is: first to write the incident pulse in the time domain EIt (t) and decompose it into the Fourier components in the frequency domain EIf (Δ); then to multiply the incident Fourier components EIf (Δ) with r(Δ, L) and t(Δ, L) to attain the reflected and transmitted Fourier components ERf (Δ) = EIf (Δ) ·r(Δ, L) and ETf (Δ) = EIf (Δ) ·t(Δ, L); finally to perform the inverse Fourier transform so that the reflected pulse ERt (t) at the sample entrance (z = 0) and the transmitted pulse ETt (t) at the sample exit (z = L) can be reconstructed as
ERt(t)=ERf(Δ)ei(ΔΔ0)td(Δ),ETt(t)=ETf(Δ)ei(ΔΔ0)td(Δ).
(8)
For simplicity without the loss of generality, we will assume in the following that the incident probe pulse has a Gaussian profile in both time and frequency domains as
EIt(t)=E0te(tt0)2/δt2,EIf(t)=E0fe(ΔΔ0)2/δp2,
(9)
where t0 and δt0 and δp) are, respectively, the center and the width of the incident probe pulse in the time (frequency) domain. Eqs. (7) and Eqs. (8) together with Eqs. (9) are the main results that we have derived to examine various effects of the atomic spatial inhomogeneity on both steady and dynamic probe responses.

3. Results and Discussions

With the formulas developed in the last section, we now perform numerical calculations to investigate the probe spectra of reflectivity and transmissivity as well as the propagation dynamics of an incident pulse in one homogeneous sample and two inhomogeneous samples of cold atoms. The three samples under consideration have the same medium length L = 1.5 mm, the same average atomic density N0 = 2.0 × 1010 mm−3, but very different density functions Nai(z), which is constrained by 0LNai(z)dz=N0L (see Fig. 2). In particular, the two inhomogeneous samples have either the sinusoid profile or the Gaussian profile as denoted by the caption of Fig. 2 and the latter is quite typical for cold atomic clouds in MOT [27

27. H.-W. Cho, Y.-C. He, T. Peters, Y.-H. Chen, H.-C. Chen, S.-C. Lin, Y.-C. Lee, and I. A. Yu, “Direct measurement of the atom number in a Bose condensate,” Opt. Express 15, 12114–12122 (2007). [CrossRef] [PubMed]

].

Fig. 2 (Color online) Atomic density functions Nai(z) versus the spatial variable z. Black-solid: Na1(z) = N0; Red-dashed: Na2(z) = N′ sin(πz/L) with N′ = N0π/2; Blue-dotted: Na3(z) = N″ exp[−20(zL/2)2/L2] with N=N020/π. The three samples have the same medium length L = 1.5 mm and the same average atomic density N0 = 2.0 × 1010 mm−3.

First we consider the simple case where a TW coupling field is switched on, which can be examined by setting Ωc = 0 in Eq. (2). We see from Fig. 3(a) that the transmission spectra with a characteristic EIT window are clearly independent of atomic density functions as long as all three samples contain the same amounts of cold atoms. Fig. 3(b) further shows that the three transmitted pulses with a remarkable time delay (relative to the incident pulse) are almost indistinguishable for different atomic density functions constrained by 0LNai(z)dz=N0L. Thus we may conclude that, in the TW driving configuration, it is a good choice to simplify relevant calculations and analyses by ignoring the spatial inhomogeneity of real atomic samples. Note, however, that the pulse distribution in the homogeneous sample is distinct from those in the inhomogeneous samples because the pulse intensity and the group velocity at a given spatial position are inversely proportional to the local atomic density.

Fig. 3 (Color online) Transmissivity versus probe detuning Δp (a) and scaled intensities versus time t (b) with d13 = 1.0375 × 10−29 C·m, λp = 794.983 nm, λc = 794.969 nm, γ21 = 0.2 kHz, γ31 = 5.75 MHz, Δc = 0, Ωc+ = 10 MHz, and Ωc = 0. The black-solid, red-dashed, and blue-dotted curves are respectively obtained with Na1(z), Na2(z), and Na3(z). The thin grey curve in (b) denotes the incident pulse with δt = 20 μs, t0 = 35 μs, and Δ0 = 0.

Then we consider the complicated case where a SW coupling field is switched on, i.e. both Ωc+ and Ωc are nonzero in Eq. (2). We see from Fig. 4(a) that three PBGs, characterized by a platform of reflectivity over 95%, are well developed around the probe resonance with suitably chosen parameters. But these dynamically induced PBGs have quite different widths indicating that they are very sensitive to atomic density functions. Fig. 4(b) further shows that the three transmission spectra depend critically on atomic density functions due to their correlation with the three reflection spectra in Fig. 4(a). In Fig. 4(c) and 4(d), we examine instead the propagation dynamics of a probe pulse with its most carrier frequencies fallen into the widest PBG in Fig. 4(a). It is clear that the incident pulse is perfectly reflected with a short time delay and experiences little energy loss and profile deformation in the homogeneous sample. For the inhomogeneous samples, however, the reflected pulses become depleted and distorted more or less due to the loss of some carrier frequencies. Among the lost carrier frequencies, most make up the transmitted pulses with a long time delay at the sample exits while little are indeed absorbed because we always work in the regime of EIT. It is clear that, in the SW driving configuration, the spatial inhomogeneity of real atomic samples have to be duly addressed, otherwise theoretical predictions may deviate largely from relevant experimental results.

Fig. 4 (Color online) Reflectivity and transmissivity versus probe detuning Δp (a, b) and scaled intensities versus time t (c, d). Black-solid, red-dashed, and blue-dotted curves are respectively obtained with Na1(z), Na2(z), and Na3(z). Thin grey curves in (c, d) denote the incident pulse with δt = 10 μs, t0 = 15 μs, and Δ0 = −0.72 MHz. Other parameters are the same as in Fig. 3 except Ωc+ = 30 MHz and Ωc = 20 MHz.

It is straightforward to understand why the inhomogeneity of cold atomic samples is trivial in the TW driving configuration: the probe susceptibility χp depends on the spatial variable z only through the atomic density Na(z) so that the probe transmissivity is proportional to ∫Na(z)dz (i.e. the total amounts of cold atoms) while the probe reflectivity is always negligible. In the SW driving configuration, however, it is not so simple to explain the importance of the inhomogeneity of cold atomic samples. In this case, the probe susceptibility χp depends on z not only through Na(z) but also through Ωc(z), which makes both transmissivity and reflectivity significant around the probe resonance. In particular, dynamically induced PBGs may be observed as a platform of high reflectivity when perfect constructive (destructive) interference occurs between the BW (FW) probe photons scattered by an atomic grating generated by the SW coupling. In a homogeneous sample, different lattices of the atomic grating are exactly the same in optical responses and thus have identical scattering abilities, which is favorable for achieving perfect interference between the probe photons in a large spectral region. When the sample becomes inhomogeneous, however, each lattice of the atomic grating may be very different from others in optical responses, especially in scattering abilities, so that perfect interference can only be achieved between the probe photons in a small spectral region. This is why distinct steady optical spectra and light propagation dynamics have been observed in Fig. 4 for three samples with the same amounts of cold atoms but very different density functions.

One may wonder whether can we compensate the undesirable effect of the spatial inhomogeneity of cold atomic samples to attain wide enough PBGs? Fig. 5 shows that, for the two inhomogeneous samples with fixed medium lengths and density functions, it is viable to widen the dynamically induced PBGs while maintaining their high reflectivities by suitably increasing Rabi frequencies Ωc+ and Ωc of the FW and BW coupling beams. It is also clear that the amended PBGs (red-dashed) in Fig. 5(a) and Fig. 5(b) well match the widest PBG (black-solid) in Fig. 4(a) as far as widths and heights are concerned.

Fig. 5 (Color online) Reflectivity versus probe detuning Δp for Na2(z) in (a) and Na3(z) in (b). The black-solid curves are attained with Ωc+ = 30 MHz and Ωc = 20 MHz. The red-dashed curve in (a) is attained with Ωc+ = 35 MHz and Ωc = 24 MHz while that in (b) is attained with Ωc+ = 41 MHz and Ωc = 30 MHz. Other parameters are the same as in Fig. 3.

4. Conclusions

In summary, we have calculated steady optical spectra and light propagation dynamics in three samples of cold atoms with either homogeneous or inhomogeneous density functions. These samples are assumed to be driven into the EIT regime by a weak probe and a strong coupling in the Λ configuration. When the coupling is a TW field, the transmissivity spectra characterized by typical EIT windows are exactly identical for the three samples with the same amounts of cold atoms but very different spatial inhomogeneity while the reflectivity spectra are not necessary to be taken into account. Consequently, an incident pulse with its most carrier frequencies fallen into the overlapped EIT windows will go through the three samples with the same time delay while suffering little energy loss and profile deformation. The spatial inhomogeneity of cold atomic samples becomes rather important when the coupling is a SW field instead. That is, both transmissivity and reflectivity spectra may change dramatically if the spatial inhomogeneity of a sample is increased or decreased. In particular, the dynamically induced PBG for the homogeneous sample looks much wider than those for the two inhomogeneous samples. As a result, an incident pulse perfectly reflected by the homogeneous sample is seen to split into a partially reflected one and a partially transmitted one when impinging upon the two inhomogeneous samples. Qualitative analyses have been given in terms of constructive (destructive) interference between the BW (FW) probe photons scattered by an atomic grating generated by the SW coupling. These findings should be helpful to amend relevant theoretical results attained without considering the spatial inhomogeneity of real atomic samples coherently driven by one or more SW fields.

Acknowledgments

The authors would like to thank the financial supports from NSFC under grants 10874057 and 10904047, from NBRP under grant 2011CB921603, from the Basic Scientific Research Foundation of Jilin University under grant 200905019, and from the Graduate Innovation Foundation of Jilin University under grant 20101051.

References and links

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S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50 (7), 36–42 (1997). [CrossRef]

2.

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

3.

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4.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003). [CrossRef]

5.

S.-J. Li, X.-D. Yang, X.-M. Cao, C.-H. Zhang, C.-D. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008). [CrossRef] [PubMed]

6.

C. Hang and G.-X. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010). [CrossRef] [PubMed]

7.

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]

8.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

9.

C.-L. Cui, J.-K. Jia, J.-W. Gao, Y. Xue, G. Wang, and J.-H. Wu, “Ultraslow and superluminal light propagation in a four-level atomic system,” Phys. Rev. A 76, 033815 (2007). [CrossRef]

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C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulsed,” Nature 409, 490–493 (2001). [CrossRef] [PubMed]

11.

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature 438, 833–836 (2005). [CrossRef] [PubMed]

12.

Z. Li, D.-Z. Cao, and K. Wang, “Manipulating synchronous optical signals with a double-Λ atomic ensemble,” Phys. Lett. A 341, 366–370 (2005). [CrossRef]

13.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007). [CrossRef] [PubMed]

14.

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008). [CrossRef] [PubMed]

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A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002). [CrossRef] [PubMed]

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X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005). [CrossRef]

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

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S.-Q. Kuang, R.-G. Wan, P. Du, Y. Jiang, and J.-Y. Gao, “Transmission and reflection of electromagnetically induced absorption grating in homogeneous atomic media,” Opt. Express 16, 15455–15462 (2008). [CrossRef] [PubMed]

19.

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

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

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K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary Λ atoms,” Phys. Rev. A 75, 053802 (2007). [CrossRef]

22.

Y.-W. Lin, W.-T. Liao, T. Peters, H.-C. Chou, J.-S. Wang, H.-W. Cho, P.-C. Kuan, and I. A. Yu, “Stationary light pulses in cold atomic media and without bragg gratings,” Phys. Rev. Lett. 102, 213601 (2009). [CrossRef] [PubMed]

23.

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J.-H. Wu, M. Artoni, and G. C. La Rocca, “Decay of stationary light pulses in ultracold atoms,” Phys. Rev. A 81, 033822 (2010). [CrossRef]

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27.

H.-W. Cho, Y.-C. He, T. Peters, Y.-H. Chen, H.-C. Chen, S.-C. Lin, Y.-C. Lee, and I. A. Yu, “Direct measurement of the atom number in a Bose condensate,” Opt. Express 15, 12114–12122 (2007). [CrossRef] [PubMed]

28.

Y.-W. Lin, H.-C. Chou, P. P. Dwivedi, Y.-H. Chen, and I. A. Yu, “Using a pair of rectangular coils in the MOT for the production of cold atom clouds with large optical density,” Opt. Express 16, 3753–3761 (2008). [CrossRef] [PubMed]

29.

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005). [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: September 27, 2010
Revised Manuscript: November 29, 2010
Manuscript Accepted: January 8, 2011
Published: January 20, 2011

Citation
Yan Zhang, Yan Xue, Gang Wang, Cui-Li Cui, Rong Wang, and Jin-Hui Wu, "Steady optical spectra and light propagation dynamics in cold atomic samples with homogeneous or inhomogeneous densities," Opt. Express 19, 2111-2119 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2111


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References

  1. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
  2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]
  3. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996). [CrossRef] [PubMed]
  4. D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801 (2003). [CrossRef]
  5. S.-J. Li, X.-D. Yang, X.-M. Cao, C.-H. Zhang, C.-D. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008). [CrossRef] [PubMed]
  6. C. Hang and G.-X. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010). [CrossRef] [PubMed]
  7. 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]
  8. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]
  9. C.-L. Cui, J.-K. Jia, J.-W. Gao, Y. Xue, G. Wang, and J.-H. Wu, “Ultraslow and superluminal light propagation in a four-level atomic system,” Phys. Rev. A 76, 033815 (2007). [CrossRef]
  10. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulsed,” Nature 409, 490–493 (2001). [CrossRef] [PubMed]
  11. T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature 438, 833–836 (2005). [CrossRef] [PubMed]
  12. Z. Li, D.-Z. Cao, and K. Wang, “Manipulating synchronous optical signals with a double-Λ atomic ensemble,” Phys. Lett. A 341, 366–370 (2005). [CrossRef]
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