## Steady optical spectra and light propagation dynamics in cold atomic samples with homogeneous or inhomogeneous densities |

Optics Express, Vol. 19, Issue 3, pp. 2111-2119 (2011)

http://dx.doi.org/10.1364/OE.19.002111

Acrobat PDF (845 KB)

### 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

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]

## 2. Model and Equations

*ω*and amplitude

_{p}*E*while the transition from level |2〉 to level |3〉 is coupled by a strong light beam of frequency

_{p}*ω*and amplitude

_{c}*E*, 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 where Δ =

_{c}*ω*−

_{p}*ω*

_{31}and

*δ*=

*ω*−

_{p}*ω*−

_{c}*ω*

_{21}are, respectively, single-photon and two-photon detunings relevant to the probe and coupling fields. Ω

*=*

_{p}*E*

_{p}d_{13}/2

*h̄*(Ω

*=*

_{c}*E*

_{c}d_{23}/2

*h̄*) is the probe (coupling) Rabi frequency with

*d*

_{13}(

*d*

_{23}) 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 which varies along the

*z*direction with a spatial periodicity of

*λ*/2 and depends on the space-independent Rabi frequencies Ω

_{c}_{c+}and Ω

_{c−}of the FW and BW coupling beams.

*Liouville*equations for density matrix elements in the interaction picture

*ρ*

_{11}+

*ρ*

_{22}+

*ρ*

_{33}= 1. In Eqs. (3), Γ

*describes the population decay rate from level |*

_{ij}*i*〉 to level |

*j*〉 whereas

*γ*denotes the dephasing rate of the coherence term

_{ij}*ρ*.

_{ij}*ρ*

_{11}= 1 and

*ρ*

_{22}=

*ρ*

_{33}=

*ρ*

_{23}= 0 so that the steady-state solution of

*ρ*

_{31}can be found from Eqs. (3) to the first order in Ω

*, from which the probe susceptibility may be written as where the atomic density function*

_{p}*N*(

_{a}*z*) has been taken to be space-dependent. With Eq. (4) in hand, we can further attain the complex refractive index which governs absorptive and dispersive properties on the probe transition.

*n*(Δ,

_{p}*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]

*nth*atomic lamina (extending from

*z*=

*nd*−

*d*to

*z*=

*nd*) is described by a 2 × 2 unimodular transfer matrix

*M*(Δ,

_{n}*d*) via where

*E*

_{p+}and

*E*

_{p−}denote, respectively, the FW and BW probe fields. With

*M*(Δ,

_{n}*d*) determined by Eq. (6), the total transfer matrix finally turns out to be

*M*(Δ,

*L*) =

*M*

_{1}(Δ,

*d*) ⋯

*M*(Δ,

_{n}*d*) ⋯

*M*(Δ,

_{N}*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 which is also available for the other case where the coupling field is set in the TW pattern (Ω

_{c−}= 0 or Ω

_{c+}= 0).

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]

*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

*E*(

_{It}*t*) and decompose it into the Fourier components in the frequency domain

*E*(Δ); then to multiply the incident Fourier components

_{If}*E*(Δ) with

_{If}*r*(Δ,

*L*) and

*t*(Δ,

*L*) to attain the reflected and transmitted Fourier components

*E*(Δ) =

_{Rf}*E*(Δ) ·

_{If}*r*(Δ,

*L*) and

*E*(Δ) =

_{Tf}*E*(Δ) ·

_{If}*t*(Δ,

*L*); finally to perform the inverse Fourier transform so that the reflected pulse

*E*(

_{Rt}*t*) at the sample entrance (

*z*= 0) and the transmitted pulse

*E*(

_{Tt}*t*) at the sample exit (

*z*=

*L*) can be reconstructed as 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 where

*t*

_{0}and

*δt*(Δ

_{0}and

*δ*) 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.

_{p}## 3. Results and Discussions

_{c}_{−}= 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

_{c}_{+}and Ω

_{c}_{−}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.

*χ*depends on the spatial variable

_{p}*z*only through the atomic density

*N*(

_{a}*z*) so that the probe transmissivity is proportional to ∫

*N*(

_{a}*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

*χ*depends on

_{p}*z*not only through

*N*(

_{a}*z*) but also through Ω

*(*

_{c}*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.

_{c+}and Ω

_{c−}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.

## 4. Conclusions

## Acknowledgments

## References and links

1. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

2. | M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. |

3. | H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,”Opt. Lett. |

4. | D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A |

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. |

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 |

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 |

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. |

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 |

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 |

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 |

12. | Z. Li, D.-Z. Cao, and K. Wang, “Manipulating synchronous optical signals with a double-Λ atomic ensemble,” Phys. Lett. A |

13. | R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. |

14. | K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature |

15. | A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. |

16. | X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A |

17. | M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. |

18. | 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 |

19. | J.-H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B |

20. | M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature |

21. | K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary Λ atoms,” Phys. Rev. A |

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. |

23. | J. Otterbach, R. G. Unanyan, and M. Fleischhauer, “Confining stationary light: dirac dynamics and klein tunneling,” Phys. Rev. Lett. |

24. | J.-H. Wu, M. Artoni, and G. C. La Rocca, “Decay of stationary light pulses in ultracold atoms,” Phys. Rev. A |

25. | J.-H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. |

26. | J.-W. Gao, J.-H. Wu, N. Ba, C.-L. Cui, and X.-X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A |

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 |

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 |

29. | M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E |

**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

- S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
- M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]
- H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002). [CrossRef] [PubMed]
- X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005). [CrossRef]
- M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003). [CrossRef] [PubMed]
- K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary Λ atoms,” Phys. Rev. A 75, 053802 (2007). [CrossRef]
- 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]
- J. Otterbach, R. G. Unanyan, and M. Fleischhauer, “Confining stationary light: dirac dynamics and klein tunneling,” Phys. Rev. Lett. 102, 063602 (2009). [CrossRef] [PubMed]
- 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]
- J.-H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103, 133601 (2009). [CrossRef] [PubMed]
- J.-W. Gao, J.-H. Wu, N. Ba, C.-L. Cui, and X.-X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010). [CrossRef]
- 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]
- 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]
- M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 046604 (2005). [CrossRef]

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