## Electromagnetically induced transparency and coherent-state preparation in optically thick media

Optics Express, Vol. 4, Issue 2, pp. 107-112 (1999)

http://dx.doi.org/10.1364/OE.4.000107

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### Abstract

The preparation of an optically dense ensemble of three-level systems in dark states of the interaction with coherent radiation is discussed. It is shown that methods involving spontaneous emissions of photons such as Raman optical pumping fail to work beyond a critical density due to multiple scattering and trapping of these photons and the associated decay of the dark state(s). In optically thick media coherent-state preparation is only possible by entirely coherent means such as stimulated Raman adiabatic passage (STIRAP). It is shown that STIRAP is the underlying physical mechanism for electromagnetically induced transparency (EIT).

© Optical Society of America

## 1. Introduction

3. S. E. Harris and Zhen-Fei. Luo, “Preparation Energy for Electromagnetically Induced Transparency,” Phys. Rev. A **52**,R928 (1995). [CrossRef] [PubMed]

4. for a recent review on STIRAP see: K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. **70**, 1003 (1998). [CrossRef]

6. Michael Fleischhauer, “Optical pumping in dense atomic media: Limitations due to reabsorption of spontaneously emitted photons,” http://xxx.lanl.gov/abs/quant-ph/9811017.

4. for a recent review on STIRAP see: K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. **70**, 1003 (1998). [CrossRef]

## 2. Radiation trapping in optically thick three-level systems

*b*⟩ and ∣

*c*⟩ are assumed to be metastable, but we take into account a (e.g. collisional) de-phasing of the ∣

*b*⟩ - ∣

*c*⟩ coherence with rate

*γ*

_{0}.

*b*⟩. We consider the case of large Doppler-broadening of the two optical transitions with the (same) Gaussian distribution of width Δ

_{D}. Two-photon Doppler-broadening of the ∣

*b*⟩ - ∣

*c*⟩ coherence is neglected.

*a*⟩, the decoupled or dark state ∣-⟩, and the orthogonal bright state ∣+⟩:

_{1,2}(

*t*) are the slowly-varying Rabi-frequencies of the laser fields, and Ω

_{1}(

*t*) = Ω(

*t*) cos

*θ*(

*t*) and Ω

_{2}(

*t*) = Ω(

*t*) sin

*θ*(

*t*).

*a*⟩ - ∣+⟩ transition, obeys the equation [7

7. Michael Fleischhauer and Aaron S. Manka, “Propagation of laser pulses and coherent population transfer in dissipative three-level systems: An adiabatic dressed-state picture,” Phys. Rev. A. **54**, 794 (1996). [CrossRef] [PubMed]

*g*is the effective coupling strength proportional to the dipole moment and

*N*is the number density of atoms. When the mixing angle

*θ*(

*t*) is time-dependent, there is an additional coupling between the dark and bright states not shown in Fig. 1b. The effective Rabi-frequency of this (non-adiabatic) coupling is 2

*θ*̇(

*t*). The energy loss of the pulses due to absorption is characterised by the evolution of Ω(

*z*,

*t*). Reshaping of the pulses due to mutual photon exchange via Raman scattering is described by the evolution of

*θ*̇(

*z*,

*t*). We are here interested only in the energy loss and will therefore not consider the propagation of

*θ*̇(

*z*,

*t*). Since furthermore for strong fields and in the adiabatic limit

*θ*̇ is much smaller than Ω we can ignore the non-adiabatic coupling in the present discussion altogether.

*γ*

_{0}in the bare-state basis corresponds to a population exchange between the dark and bright state with rate

*γ*̄

_{0}(

*t*) =

*γ*

_{0}2 sin

^{2}

*θ*(

*t*) cos

^{2}

*θ*(

*t*). (In an asymmetric case, Ω

_{1}≠ Ω

_{2}, it also leads to some cross-coupling rates, which are however of no importance here and are neglected.)

8. Michael Fleischhauer and Susanne F. Yelin, “Radiative atom-atom interactions in optically dense media: Quantum corrections to the Lorentz-Lorenz formula,” http://xxx.lanl.gov/abs/quant-ph/9809087.

6. Michael Fleischhauer, “Optical pumping in dense atomic media: Limitations due to reabsorption of spontaneously emitted photons,” http://xxx.lanl.gov/abs/quant-ph/9811017.

8. Michael Fleischhauer and Susanne F. Yelin, “Radiative atom-atom interactions in optically dense media: Quantum corrections to the Lorentz-Lorenz formula,” http://xxx.lanl.gov/abs/quant-ph/9809087.

*a*⟩ and ∣-⟩ and ∣-⟩ and ∣+⟩ with the rate

*ρ*

_{--}is the dark-state population and

_{1}≈ λ

_{2}is the laser wavelength.

*d*is the smallest effective escape distance of the incoherent photons. If the laser beams are for example cylindrical with a homogeneous transversal intensity profile,

*d*is the radius of the cylinder.

*ρ*

_{--}.We have not taken into account Lorentz-Lorenz local field corrections [9

9. C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A **47**, 1247 (1993). [CrossRef] [PubMed]

*N*λ

^{2}

*d*~ 1. Lorentz-Lorenz local field corrections, on the other hand, require densities of at least one atom per cubic wavelength, i.e.

*N*λ

^{3}~ 1. Thus radiation trapping is usually relevant for much smaller densities then necessary for local field effects.

## 3. EIT with cw-fields

6. Michael Fleischhauer, “Optical pumping in dense atomic media: Limitations due to reabsorption of spontaneously emitted photons,” http://xxx.lanl.gov/abs/quant-ph/9811017.

*γ*

_{0}as a function of the density parameter

*K*

_{0}=

*N*λ

^{2}

*dγ*/ (Δ

_{D}√8

*π*) . One recognises that beyond a certain critical density, which depends on the de-phasing rate

*γ*

_{0}, a considerable amount of population remains in other states.

*z*/

*z*

_{0}, where

*z*

_{0}corresponds to the absorption length for a single laser field,

*ρ*̄

_{aa}and

*ρ*̄

_{++}are the stationary populations in the excited and bright state respectively.

*R*, which is proportional to the average intensity of the fields as function of the normalised propagation distance. One recognises a linear decrease of the intensity in the initial phase.

## 4. Pulsed EIT and stimulated Raman adiabatic passage

_{1}and Ω

_{2}are applied in counterintuitive order, i.e. if Ω

_{1}is switched on first, the dark state coincides initially with the bare atomic state ∣

*b*⟩. Switching on the second field at a rate small compared to the total Rabi-frequency adiabatically rotates the dark state into the coherent superposition of ∣

*b*⟩ and ∣

*c*⟩ without significantly populating the excited state, i.e. without spontaneous emission. To ensure adiabaticity, the time

*T*of creating the dark state has to be large compared to the Rabi-oscillation period,

*T*≫ Ω

^{-1}and thus can still be much smaller than the time for reaching the stationary state.

_{2}switched on after Ω

_{1}(lower dashed and dashed-dotted curve).

_{2}(

*t*)/Ω

_{1}(

*t*) = const.) evolve into a counterintuitive pair of pulses in the curse of propagation due to Raman scattering [3

3. S. E. Harris and Zhen-Fei. Luo, “Preparation Energy for Electromagnetically Induced Transparency,” Phys. Rev. A **52**,R928 (1995). [CrossRef] [PubMed]

*θ*̇ is built up and a counterintuitive pulse sequence established.

## 5. Summary

## Acknowledgements

## References

1. | for a recent review on EIT see: S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

2. | for a recent review see: E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. |

3. | S. E. Harris and Zhen-Fei. Luo, “Preparation Energy for Electromagnetically Induced Transparency,” Phys. Rev. A |

4. | for a recent review on STIRAP see: K. Bergmann, H. Theuer, and B. W. Shore, “Coherent Population Transfer Among Quantum States of Atoms and Molecules,” Rev. Mod. Phys. |

5. | T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev. A |

6. | Michael Fleischhauer, “Optical pumping in dense atomic media: Limitations due to reabsorption of spontaneously emitted photons,” http://xxx.lanl.gov/abs/quant-ph/9811017. |

7. | Michael Fleischhauer and Aaron S. Manka, “Propagation of laser pulses and coherent population transfer in dissipative three-level systems: An adiabatic dressed-state picture,” Phys. Rev. A. |

8. | Michael Fleischhauer and Susanne F. Yelin, “Radiative atom-atom interactions in optically dense media: Quantum corrections to the Lorentz-Lorenz formula,” http://xxx.lanl.gov/abs/quant-ph/9809087. |

9. | C. M. Bowden and J. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A |

10. | A. Rahman, R. Grobe, and J. H. Eberly, “Two-Photon Beer’s Law for Coherently Prepared Three-Level Media,” |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(020.7010) Atomic and molecular physics : Laser trapping

**ToC Category:**

Focus Issue: Laser controlled dynamics

**History**

Original Manuscript: December 7, 1998

Published: January 18, 1999

**Citation**

M. Fleischhauer, "Electromagnetically induced transparency
and coherent-state preparation in optically thick media," Opt. Express **4**, 107-112 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-2-107

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### References

- for a recent review on EIT see: S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36 (1997) and references therein.
- for a recent review see: E. Arimondo, "Coherent population trapping in laser spectroscopy," Prog. Opt. 35, 259 (1996).
- S. E. Harris and Zhen-Fei. Luo, "Preparation Energy for Electromagnetically Induced Transparency," Phys. Rev. A 52, R928 (1995). [CrossRef] [PubMed]
- for a recent review on STIRAP see: K. Bergmann, H. Theuer, and B. W. Shore, "Coherent Population Transfer Among Quantum States of Atoms and Molecules," Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]
- T. Holstein, "Imprisonment of resonance radiation in gases," Phys. Rev. A 72, 1212 (1947).
- Michael Fleischhauer, "Optical pumping in dense atomic media: Limitations due to reabsorption of spontaneously emitted photons," http://xxx.lanl.gov/abs/quant-ph/9811017.
- Michael Fleischhauer and Aaron S. Manka, "Propagation of laser pulses and coherent population transfer in dissipative three-level systems: An adiabatic dressed-state picture," Phys. Rev. A. 54, 794 (1996). [CrossRef] [PubMed]
- Michael Fleischhauer and Susanne F. Yelin, "Radiative atom-atom interactions in optically dense media: Quantum corrections to the Lorentz-Lorenz formula," http://xxx.lanl.gov/abs/quant-ph/9809087.
- C. M. Bowden and J. Dowling, "Near-dipole-dipole effects in dense media: Generalized Maxwell- Bloch equations," Phys. Rev. A 47, 1247 (1993). [CrossRef] [PubMed]
- A. Rahman, R. Grobe, and J. H. Eberly, "Two-Photon Beers Law for Coherently Prepared Three-Level Media," Coherence and Quantum Optics VII p.449 (Plenum, New York, 1996).

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