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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 3 — Feb. 2, 2009
  • pp: 1781–1788
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Detuned slow light in the Doppler broadened multi-level D2 line of Rubidium

Yi Chen, Xiao Gang Wei, and Byoung S. Ham  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1781-1788 (2009)
http://dx.doi.org/10.1364/OE.17.001781


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Abstract

We observed a detuned slow light phenomenon based on electromagnetically induced transparency in 87Rb D2 line composed of multiple excited-hyperfine states within a Doppler-broadened linewidth. The results show that the maximum group delay of a probe occurs at off-detuned two-photon resonance frequency. The observed detuned group delay is analyzed with numerical calculations for a probe pulse interacting with the neighboring excited-states-modified Doppler broadening atoms for a fixed coupling field. The experimental results are in good agreement with the numerical calculations.

© 2009 Optical Society of America

1. Introduction

In the last decade, electromagnetically induced transparency (EIT) [1–3

1. K. -J. Boller, A. Imanoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett . 66, 2593–2596 (1991). [CrossRef] [PubMed]

] has been intensively studied for both fundamental physics and potential applications, where an opaque medium becomes transparent to a resonant probe field when another resonant coupling laser exists. Ultraslow group velocity based on EIT [4

4. S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev . A 46, R29–32 (1992).

] has drawn much attention to quantum optical applications such as quantum memories [5–7

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

], quantum entanglement generations [8

8. M. D. Lukin and P. R. Hemmer, “Quantum entanglement via optical control of atom-atom interactions,” Phys. Rev. Lett . 84, 2818–2821 (2000). [CrossRef] [PubMed]

, 9

9. M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev . A 67, 023811 (2003).

], and quantum routing [10

10. B. S. Ham, “Observations of delayed all-optical routing in a slow-light regime,” Phys. Rev . A 78, 011808(R) (2008).

]. EIT-based ultraslow light has the advantage of active control of the group delay by using a coupling light intensity. So far EIT-based slow light has been observed in Bose Einstein condensate [11

11. 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]

], atomic vapors [12

12. 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 enhance nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett . 82, 5229–5232 (1999). [CrossRef]

], magneto-optically trapped atoms, and solid-state material [6

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

, 13

13. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett . 90, 113903 (2003). [CrossRef] [PubMed]

].

2. Theory

We consider a six-level 87Rb (D2 line) atomic system, where F=1(|1〉) and F=2 (|2〉) of 5S1/2 form two ground levels and F’=0,1,2,3 (|3〉,|4〉,|5〉,|6〉) of 5P3/2 forms excited levels (see Fig. 1). The coupling field with frequency ωc and amplitude Ec couples the levels |4〉 and |2〉, while the probe field with frequency ωp and amplitude Ep couples the levels |4〉 and |1〉. The frequency detuning of the coupling and probe is given by ∆c = ωc - ω 42 and ∆p = ωp - ω 41, respectively. Thus, a typical Λ -type EIT scheme can be satisfied.

In the framework of semi-classical theory, the Hamiltonian for this scheme is given by H = H 0 + H 1, where H 0 and H 1 represent the unperturbed and interaction parts of the Hamiltonian, respectively. The interaction Hamiltonian H 1 can be written as:

H1=ħ2(Ωp31eiωpt31+Ωp41eiωpt41+Ωp51eiωpt51
+Ωc42eiωct42+Ωc52eiωct52+Ωc26eiωct62+H.C.),
(1)

where Ωpi1 = μ i1 Ep /ħ is the Rabi frequency of the probe field for the transition |i〉 ↔ |1〉 (i=3,4,5), and ΚCj2 = μ j2, EC /ħ is the Rabi frequency of the coupling field for the transition |j〉 ↔ |2〉 (j=4,5,6). For the 87Rb D2 line, the transitions 5S1/2, F=2→ 5P3/2, F’=0 and 5S1/2, F=1→ 5P3/2, F’=3 are forbidden.

Fig. 1. Schematic diagram of a six-level atomic system. The corresponding hyperfine levels are for 87Rb D2 line.

Under the rotating-wave approximation, the density matrix equation of motion for the interaction Hamiltonian is described by

ρ˙=iħ[HI,ρ]12{Γ,ρ}.
(2)

The susceptibility χ(∆p) = χ′+″ can be obtained by solving this density matrix equation numerically under the steady state condition, where χ′ and χ″ represent dispersion and absorption, respectively. Under the Doppler broadening resulted from random motions of atoms, the total susceptibility for all excited levels becomes:

χDop(ΔP)=χ(ΔP,v)Nvpπev2/vp2dv,
(3)

where N is the atom density, vp = √2kT/m = √2RT/M is the most probable atom velocity, k is the Boltzmann constant, R is the gas constant, and T is the temperature of the atomic system. In Eq. (3) ∆P and ∆C are substituted by ∆P-ω 41 v/c and ∆c - ω 42 v/c.

Because of the steep dispersion spectrum directly resulting from the narrower EIT window according to the Kramers-Kronig relation, the group velocity of the probe pulse can be much smaller than that in vacuum. The group velocity and the group delay are given by:

vg=cn+ωdn,
(4-1)
τg=L(1vg1c),
(4-2)

where L is the length of the medium, c is the speed of light in vacuum, and n is given by n = √1+χ′.

3. Experimental results

The slow light experiments are performed in a Rb vapor cell heated at ~50°C. Two extended cavity diode lasers (ECDL) from Toptica are used for the coupling and the probe. Each laser linewidth is about 1MHz. The coupling and probe beams are combined via a polarizing beam splitter and designed to copropagate through a 75 mm-long Rb cell. Both laser beams are linearly polarized, but perpendicular each other. The power of the coupling and probe beams in a cw case is 19mW and 2.7mW, respectively. A 200ns Gaussian probe laser pulse is generated by an acousto-optic modulator (AOM) driven by an arbitrary function generator at a repetition rate of 100 KHz. After passing through the AOM, we split off a part of the probe laser beam to use it as a reference signal. Both the signal pulse and the reference pulse are detected by using photodiodes. For the group delay measurement, the frequency of the probe laser is scanned slowly across the EIT line center, while the coupling laser frequency is fixed. We record the signal pulse and the reference pulse at the same time. In this way, we can get the group delay as a function of one-phonon detuning of the probe field.

Fig. 2. The variation of delay time with detuning of the probe field when the coupling laser is resonant with (a) the transition |2〉- |4〉; (b) the center line between the level |4〉 and |5〉 from level |2〉; (c) the transition |2〉-|5〉; (d) the center line between the level |4〉 and |6〉 from level |2〉; (e) the center line between the level |5〉 and |6〉 from level |2〉 ; (f) the transition |2〉- |6〉. The blue circles are the experimental data and the red curves are the smoothed data.

We calculate the measured time of the peak value at a basis of reference without the coupling. Figure 2 shows experimental data (blue circles) of the group delay of the probe for different coupling transitions mentioned in Fig.1. For this the probe is scanned across the resonance transition for each fixed coupling frequency tuned to the transition of (a)|2〉-|4〉(5S1/2, F=2-5P3/2, F’=1), (b) the center line between|2〉-|4〉(5S1/2, F=2-5P3/2, F’=1) and |2〉-|5〉 (5S1/2, F=2-5P3/2, F’=2), (c) 2〉-5〉 (5S1/2, F=2-5P3/2, F’=2), (d)the center line between |2〉-|4〉(5S1/2, F=2-5P3/2, F’=1)and |2〉-|6〉(5S1/2, F=2-5P3/2, F’=3), (e) the center line between |2〉-|5〉 (5S1/2, F=2-5P3/2, F’=2)and |2〉-|6〉(5S1/2, F=2-5P3/2, F’=3), and (f) |2〉-|6〉 (5S1/2, F=2-5P3/2, F’=3). In Fig.2, we have observed that the maximum group delay is detuned from the resonance frequency of the probe for each case. The thin red curves in Fig.2 represent the smoothed experimental data obtained by using the moving average method (the moving average calculation span is 15).

4. Numerical analysis and discussions

To understand this detuned slow light phenomenon, we have made numerical simulations base on the density matrix equations mentioned in the Theory section. In Fig. 3 we numerically calculate the Doppler broadened absorption (Fig.3 (a)) and dispersion (Fig.3 (b)) of the probe for a particular transition with different Rabi frequencies of the coupling field, where the coupling (probe) is tuned to the transition |4〉↔|2〉 (|4〉↔|1〉.Unlike the Doppler free case in an ideal three-level system, where EIT line center locates at two-photon resonance frequency, there exists EIT detuning in the multilevel system of Fig. 1 even with a small coupling Rabi frequency much less than the separation between the nearest neighboring state |3〉 (see the inset of Fig.3(a)). When the Rabi frequency of the coupling increases, the EIT linewidth becomes wider. In particular the EIT position is variable for different Rabi frequencies, whereas in a three-level system, it is not. As the Rabi frequency of coupling field increases, the EIT position becomes more red-shifted due to the extra interactions with the neighboring excited levels and different dipole moment between different transitions.

To explore this phenomenon further, we neglect the level |6〉 in the structure shown in Fig.1 and assume the transition |3〉↔|2〉 is allowed for the coupling field. Furthermore, we assume that the neighboring levels are symmetrically distributed (∆34 = ∆45 = 72MHz). By setting the same decay rates and the same dipole moments of all transition, the system becomes to a symmetrical system. There is no EIT detuning in this system anymore as shown in Fig. 3(c). In 87Rb D2 line, the level 5P3/2, F’=0 is much nearer the level 5P3/2, F’=1 than the level 5P3/2, F’=2 (∆34 =72MHz, ∆45 = 157MHz). In this case, keeping all the decay rates and dipole moments the same as those in Fig. 3(c), we find that the EIT position becomes red shifted (see Fig. 3(d)). On the other hand, if we assume unbalanced dipole moments (μ 52 = 2μ 32 =2μ 42) for the neighboring levels symmetrically distributed (∆34 = ∆45 =72MHz), we also find that the EIT position is red-shifted as shown in Fig. 3(e). If we use another unbalanced dipole moments condition (μ 32 = 2μ 52 = 2μ 42), then the EIT position becomes blue-shifted (not shown).

By using the parameters in 87Rb D2 line, when the frequency of coupling field is resonant with the transition (i)|2〉-|4〉(5S1/2, F=2-5P3/2, F’=1), (ii) the center line between |2〉-|4〉 and |2〉-|5〉(5S1/2, F=2-5P3/2, F’=2), (iii) |2〉-|5〉(5S1/2, F=2-5P3/2, F’=2), (iv)the center line between |2〉-|4〉 and |2〉-|6〉(5S1/2, F=2-5P3/2, F’=3), (v) the center line between |2〉-|5〉 and |2〉-|6〉, and (vi) |2〉-|6〉 (5S1/2, F=2-5P3/2, F’=3), we calculate the Doppler broadened absorption of the probe field as a function of one-photon detuning for corresponding ∆c. As shown in Fig. 3(f), there is always EIT red detuning, because the relative dipole matrix elements are √1/20, 1/2, √7/10 for the transitions |2〉-|4〉 , |2〉-|5〉 and |2〉-|6〉, and the neighboring levels are unsymmetrical distributed (∆34 = 72MHz, ∆45 = 157MHz, ∆56 = 267 MHz).

Fig. 3. The absorption (a) and dispersion spectra (b) for a six-level Doppler-broadened system. Parameters: T = 50° C , Γ21 = 0.3MHz , Γ31 = 6MHz , Γ41 = 5MHz , Γ5l = 3MHz , Γ42 = 1MHz , Γ52 = 3MHz, Γ62 = 6MHz , ΩC42 = √l/20*80MHz , ΩC52 = l/2*80MHz , ΩC62 = √7/110*80MHz , ∆34 = 72MHz , ∆45 = 151MHz , ∆56 = 267MHz,∆c = 0. (c) The absorption spectra for a five-level Doppler-broadened system. ΩC32 = ΩC42 = ΩC52 = 40MHz , ∆34 = ∆45 = 12MHz -(d) The absorption spectra for a five-level Doppler-broadened system., ∆34 = 72,∆45 = 157MHz -(e) The absorption spectra for a five-level Doppler-broadened system., ∆34 = ∆45 = 72MHz , ΩC32 = ΩC42 = ΩC52/2 = 40MHz-(f)The absorption spectra for a six-level Doppler-broadened system. (i) ∆c = 0MHz, (ii) ∆c =157/2MHz, (iii) ∆c =157MHz, (iv) ∆c = 157 + 267/2MHz, (v) ∆c = (1507 + 267)/2MHz , (vi) ∆c = 157 + 267MHz , ΩC42 = √1/20*80MHz ΩC52 =l/2*80MHz, ΩC62 = √7/10*80MHz

The neighboring excited-state-modified Doppler broadened atoms affect on the EIT line center shifted, resulting in the co-called detuned slow light phenomenon as shown in Fig. 2. In Fig. 4 we numerically calculate the group delay of each case observed in Fig. 2, using reasonable parameters according to the experimental condition. The numerical calculations are in good agreement with the experiment results. For all cases, the probe is red-shifted to the slow light, which agrees with the result we observed in the experiment. In Fig.4 (a), for example, the maximum group delay is red-shifted for the resonant transition by ~4MHz. When the coupling field is tuned to the crossover transitions as shown in Figs. 4(b), 4(d), and 4(e), the slow light phenomenon exists (see also Figs. 2(b), 2(d), and 2(e)), and the maximum group delay position is detuned from the crossover line center. Even when the coupling field is resonant with the transition |2〉 - |6〉, which is forbidden to the probe, slow light and group delay detuning also exist. However, the EIT effect results in from other interactions levels |4〉 and |5〉.

Fig. 4. Group delay time of the probe as a function of the detuning of probe field. The coupling field is resonant with (a) the transition |4〉↔|2〉; (b) the center line between the level |4〉 and |5〉 from level |2〉; (c) the transition |5〉↔|2〉; (d) the center line between the level |4〉 and |6〉 from level |2〉;(e) the center line between the level |5〉 and |6〉 from level |2〉; (f) the transition |6〉↔|2〉. Parameters: T = 50°C , L = 1.5cm , ΩC42 = √1/20*80MHz , ΩC52 = 1/2*80MHz , ΩC62 = √7/10*80MHz , Γ21 = 0.3MHz, Γ31 = 6MHz , Γ41 = 5MHz , Γ51=3MHz Γ42 = 1MHz , Γ52 = 3MHz, Γ62 = 6MHz , ∆34 = 72MHz , ∆45 = 157MHz , ∆56 = 267MHz

5. Conclusion

In this paper, we have observed the group delay up to 55 ns in a Doppler-broadened six-level system. The EIT dip shift due to the existence of the neighboring levels has been numerically analyzed. When the coupling field is tuned to the different transition, we have shown the dependence of group delay on the one-photon detuning of the probe. This work gives a deep understanding of EIT and slow light phenomenon in a multi-level system for some potential applications in optical information processing by using slow light.

Acknowledgment

This work was supported by the Creative Research Initiative Program (Center for Photon Information Processing) of MEST via KOSEF, S. Korea. XGW’s permanent address is College of Physics, Jilin University, Changchun 130023, P. R. China.

References and links

1.

K. -J. Boller, A. Imanoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett . 66, 2593–2596 (1991). [CrossRef] [PubMed]

2.

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

3.

Y. Q. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev . A 51, R2703–2706 (1995).

4.

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev . A 46, R29–32 (1992).

5.

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

6.

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

7.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004). [CrossRef] [PubMed]

8.

M. D. Lukin and P. R. Hemmer, “Quantum entanglement via optical control of atom-atom interactions,” Phys. Rev. Lett . 84, 2818–2821 (2000). [CrossRef] [PubMed]

9.

M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev . A 67, 023811 (2003).

10.

B. S. Ham, “Observations of delayed all-optical routing in a slow-light regime,” Phys. Rev . A 78, 011808(R) (2008).

11.

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]

12.

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 enhance nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett . 82, 5229–5232 (1999). [CrossRef]

13.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett . 90, 113903 (2003). [CrossRef] [PubMed]

14.

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev . A 66, 015802 (2002).

15.

G. Huang, C. Hang, and L. Deng, “Propagation of shape-preserving optical pulses in inhomogeneously broadened multi-level systems,” Eur. Phys. J . D 40, 437–444 (2006).

16.

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

17.

L. Deng, M. G. Payne, and E. W. Hagley, “Propagation of light pulse in an ultra-cold atomic vapor: mechanism for the loss of the probe field,” Opt. Commun . 198, 129–133 (2001). [CrossRef]

18.

S. Chakrabarti, A. Pradhan, B. Ray, and P. N. Ghosh, “Velocity selective optical pumping effects and electromagnetically induced transparency for D2 transitions in rubidium,” J. Phys. B: At. Mol. Opt. Phys . 38, 4321–4327 (2005). [CrossRef]

19.

D. Bhattacharyya, B. Ray, and P. N. Ghosh, “Theoretical study of electromagnetically induced transparency in a five-level atom and application to Doppler-broadened and Doppler-free Rb atoms,” J. Phys. B: At. Mol. Opt. Phys . 40, 4061–4075 (2007). [CrossRef]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(270.1670) Quantum optics : Coherent optical effects
(270.5530) Quantum optics : Pulse propagation and temporal solitons

ToC Category:
Slow and Fast Light

History
Original Manuscript: December 15, 2008
Revised Manuscript: January 27, 2009
Manuscript Accepted: January 27, 2009
Published: January 29, 2009

Citation
Yi Chen, Xiao Gang Wei, and Byoung S. Ham, "Detuned slow light in the Doppler broadened multi-level D2 line of Rubidium," Opt. Express 17, 1781-1788 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1781


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References

  1. K. -J. Boller, A. Imanoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991). [CrossRef] [PubMed]
  2. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997). [CrossRef]
  3. Y. Q. Li and M. Xiao, "Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms," Phys. Rev. A 51, R2703-2706 (1995).
  4. S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-32 (1992).
  5. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409, 490-493 (2001). [CrossRef] [PubMed]
  6. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, "Observation of ultraslow light and stored light pulses in a solid," Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]
  7. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004). [CrossRef] [PubMed]
  8. M. D. Lukin and P. R. Hemmer, "Quantum entanglement via optical control of atom-atom interactions," Phys. Rev. Lett. 84, 2818-2821 (2000). [CrossRef] [PubMed]
  9. M. Paternostro, M. S. Kim, and B. S. Ham, "Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime," Phys. Rev. A 67, 023811 (2003).
  10. B. S. Ham, "Observations of delayed all-optical routing in a slow-light regime," Phys. Rev. A 78, 011808(R) (2008).
  11. 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]
  12. 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 enhance nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229-5232 (1999). [CrossRef]
  13. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]
  14. E. Paspalakis and P. L. Knight, "Electromagnetically induced transparency and controlled group velocity in a multilevel system," Phys. Rev. A 66, 015802 (2002).
  15. G. Huang, C. Hang, and L. Deng, "Propagation of shape-preserving optical pulses in inhomogeneously broadened multi-level systems," Eur. Phys. J. D 40, 437-444 (2006).
  16. S. E. Harris, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999). [CrossRef]
  17. L. Deng, M. G. Payne, and E. W. Hagley, "Propagation of light pulse in an ultra-cold atomic vapor: mechanism for the loss of the probe field," Opt. Commun. 198, 129-133 (2001). [CrossRef]
  18. S. Chakrabarti, A. Pradhan, B. Ray, and P. N. Ghosh, "Velocity selective optical pumping effects and electromagnetically induced transparency for D2 transitions in rubidium," J. Phys. B: At. Mol. Opt. Phys. 38, 4321-4327 (2005). [CrossRef]
  19. D. Bhattacharyya, B. Ray, and P. N. Ghosh, "Theoretical study of electromagnetically induced transparency in a five-level atom and application to Doppler-broadened and Doppler-free Rb atoms," J. Phys. B: At. Mol. Opt. Phys. 40, 4061-4075 (2007). [CrossRef]

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