## Double-control quantum interferences in a four-level atomic system

Optics Express, Vol. 15, Issue 10, pp. 6484-6493 (2007)

http://dx.doi.org/10.1364/OE.15.006484

Acrobat PDF (143 KB)

### Abstract

A new scheme is suggested to manipulate the probe transitions (and hence the
optical properties of atomic vapors) via double-control destructive and
constructive quantum interferences. The influence of phase coherence between the
two control transitions on the probe transition is also studied. The most
remarkable feature of the present scheme is that the optical properties
(absorption, transparency and dispersion) of an atomic system can be manipulated
using this double-control multi-pathway interferences (multiple routes to
excitation). It is also shown that a four-level system will exhibit a two-level
resonant absorption because the two control levels (driven by the two control
fields) form a dark state (and hence a destructive quantum interference occurs
between the two control transitions). However, the present four-level system
will exhibit electromagnetically induced transparency to the probe field when
the three lower levels (including the probe level and the two control levels)
form a three-level dark state. The present scenario has potential applications
in new devices (*e*.*g*. logic gates and sensitive
optical switches) and new techniques (*e*.*g*.
quantum coherent information storage).

© 2007 Optical Society of America

## 1. Introduction

1. M. Fleischhauer and M. O. Scully, “Quantum sensitivity limits of an
optical magnetometer based on atomic phase
coherence,” Phys. Rev. A **49**, 1973–1986
(1994). [CrossRef] [PubMed]

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

3. S. Y. Zhu and M. O. Scully, “Spectral line elimination and
spontaneous emission cancellation via quantum
interference,” Phys. Rev. Lett. **76**, 388–391
(1996). [CrossRef] [PubMed]

4. J. Q. Shen, “Quantum-vacuum geometric phases in
the noncoplanarly curved fiber system,”
Eur. Phys. J. D **30**, 259–264
(2004). [CrossRef]

5. J. Q. Shen, “Negative refractive index in
gyrotropically magnetoelectric media,”
Phys. Rev. B **73**, 045113(1-5) (2006). [CrossRef]

4. J. Q. Shen, “Quantum-vacuum geometric phases in
the noncoplanarly curved fiber system,”
Eur. Phys. J. D **30**, 259–264
(2004). [CrossRef]

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

3. S. Y. Zhu and M. O. Scully, “Spectral line elimination and
spontaneous emission cancellation via quantum
interference,” Phys. Rev. Lett. **76**, 388–391
(1996). [CrossRef] [PubMed]

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

**50(7)**, 36–42
(1997) and references therein. [CrossRef]

6. J. P. Marangos, “Electromagnetically induced
transparency,” J. Mod. Opt. **45**, 471–503
(1998). [CrossRef]

7. J. L. Cohen and P. R. Berman, “Amplification without inversion:
Understanding probability amplitudes, quantum interference, and Feynman
rules in a strongly driven system,” Phys.
Rev. A **55**, 3900–3917
(1997). [CrossRef]

8. 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) and references therein. [CrossRef]

9. R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, “Spatial consequences of
electro-magnetically induced transparency: Observation of
electromagnetically induced focusing,”
Phys. Rev. Lett. **74**, 670–673
(1995). [CrossRef] [PubMed]

**50(7)**, 36–42
(1997) and references therein. [CrossRef]

8. 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) and references therein. [CrossRef]

10. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal pulse
propagation,” Nature **406**, 277–279
(2000) and references therein. [CrossRef] [PubMed]

8. 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) and references therein. [CrossRef]

11. P. Arve, P. Jänes, and Lars Thylén, “Propagation of two-dimensional
pulses in electromagnetically induced transparency
media,” Phys. Rev. A **69**, 063809(1-8) (2004). [CrossRef]

**50(7)**, 36–42
(1997) and references therein. [CrossRef]

13. Y. Q. Li and M. Xiao, “Transient properties of an
electromagnetically induced transparency in three-level
atoms,” Opt. Lett. **20**, 1489–1491
(1995). [CrossRef] [PubMed]

14. H. Schmidt and A. Imamogğlu, “Giant Kerr nonlinearities obtained
by electromagnetically induced transparency,”
Opt. Lett. **21**, 1936–1938
(1996). [CrossRef] [PubMed]

*i*.

*e*., only the two control levels (|2〉 and |2′〉 interacting with the two control fields, respectively) form the dark state. This implies that the total contribution of transitions driven by the two control fields from the two control levels (|2〉 and |2′〉) to the upper level (|3〉) vanishes. Thus the four-level system is equivalent to a two-level system that can exhibit a two-level resonant absorption to the probe field. We study both the constructive and destructive quantum interferences between the two control transitions driven by the control fields, and show that such quantum interferences lead to the transparency and the absorption, respectively, to the probe field. In a conventional three-level EIT system, we have to change the (absolute) intensity of the control field in order to control the optical behaviors of the atomic vapor. However, the optical response of the present four-level atomic vapor can be tunable just by adjusting the relative intensities (the ratio of the intensities) of the two control fields. This means that the double-control scheme would be more convenient and efficient for manipulating the optical properties of the atomic vapors than the conventional three-level EIT scheme did.

## 2. Double-control four-level system and generalized dark state

_{c}, ∆

_{c′}and ∆

_{p}are defined as follows: ∆

_{c}=

*ω*

_{32}-

*ω*

_{c},

*ω*

_{c′}=

*ω*

_{32′}-

*ω*

_{c′}, and ∆

_{p}=

*ω*

_{31}-

*ω*

_{p}, where

*ω*

_{32},

*ω*

_{32′}and

*ω*

_{31}denote the atomic transition frequencies, and

*ω*

_{c},

*ω*

_{c′},

*ω*

_{p}represent the mode frequencies of the control and probe beams, respectively. For the present atomic system, the equation of motion of the probability amplitudes in accordance with the Schrödinger equation is

_{p}= ℘

_{31}𝓔

_{p}/h̄, Ω

_{c}= ℘

_{32}𝓔

_{c}/h̄, and Ω

_{c′}= ℘

_{32′}𝓔

_{c′}/h̄, respectively. Here 𝓔

_{p}, 𝓔

_{c}, and 𝓔

_{c′}stand for the probe and control field envelopes (slowly-varying amplitudes). The decay rates γ

_{2}, γ′

_{2}and Γ

_{3}are defined by γ

_{2}= γ

_{23}+ γ

_{2′,2}- γ

_{2′3}, γ′

_{2}= γ

_{2′3}+ γ

_{2′2}- γ

_{23}and Γ

_{3}= γ

_{23}+ γ

_{2′3}- γ

_{2′2}, where γ

_{ij}’s denote the decay rates (including the contribution of the collisional dephasing and the spontaneous emission decay) of the density matrix elements ρ

_{ij}.

*a*

_{1},

*a*

_{2},

*a*

_{2′}of the atomic levels in the present three-level dark state are restricted by this relation. In the meanwhile, the probability amplitude of level |3〉 is zero (

*i*.

*e*.

*a*

_{3}= 0) according to Eqs. (2) and (4). Here we can define a concept called “driving contribution” that is the product of the Rabi frequency (coupling coefficient) and the probability amplitude of a lower level. For instance, the driving contribution of the probe field is Ω

_{p}

*a*

_{1}. It follows from Eq. (8) that the total driving contribution of the probe and control fields is zero for the dark state (this can be viewed as a quantum destructive interference among the three optical fields). It seems that there is no net interaction between the three lower levels and the three optical fields, and that no population would be excited from the lower levels to the upper level. This leads to the EIT phenomenon.

## 3. Dispersion of atomic electric susceptibility

*i*.

*e*., the atomic population at level |1〉 is unity. Under this assumption, Eq. (1) can be reduced to the following form

*β*(∆

_{p}) = 2℘

_{13}ρ

_{31}/(ε

_{0}𝓔

_{p}) with the density matrix element ρ

_{31}=

*a*

^{*}

_{1}

*a*

_{3}≃

*a*

_{3}. Substituting the above results into β(∆

_{p}), one can obtain the explicit expression for the electric polarizability

_{p}) =

*N*β(∆

_{p}), where

*N*denotes the atomic concentration of the EIT vapor. The dispersive behavior of the real and imaginary parts of the electric susceptibility is plotted in Fig. 2, where the typical parameters of the atomic system are chosen as: Γ

_{3}= 1.0 × 10

^{8}s

^{-1}, γ

_{2}= 1.0 × 10

^{5}s

^{-1}, γ′

_{2}= 2.0 × 10

^{5}s

^{-1}, ℘

_{13}= 1.0 × 10

^{-29}C∙m, Ω

_{c}= 1.0 × 10

^{8}s

^{-1}, Ω

_{c′}= 2.0 × 10

^{8}s

^{-1}, ∆

_{c}= 3.0 × 10

^{7}s

^{-1}, ∆

_{c′}= 8.0 × 10

^{7}s

^{-1}, and

*N*= 5.0 × 10

^{20}m

^{-3}. The absorption coefficient α (defined as 2πIm{

*n*

_{r}}/Re{

*n*

_{r}},

*i*.

*e*., the loss in the medium per wavelength) is shown in Fig. 3 as a function of the frequency detuning of the probe beam. Note that in a conventional three-level EIT system, there is only one resonant frequency for the atomic system to exhibit zero absorption (see Fig. 3, where the absorption coefficient of the conventional three-level EIT vapor is plotted under the condition that the control field Ω

_{c′}is absent). For the double-control system, however, there are two resonant frequencies, where the four-level vapor is transparent to the probe beam (zero absorption),

*i*.

*e*., ∆

_{p}→ ∆

_{c}or ∆

_{p}→ ∆

_{c′}. This can be called “double-control electromagnetically induced transparency”.

*i*.

*e*., when ∆

_{p}= ∆

_{c}and ∆

_{p}= ∆

_{c′}). In other words, the two resonant frequencies corresponding to the probe zero absorption are in fact caused by the usual two-level dark states formed by the levels |1〉, |2〉 and |1〉, |′〉, respectively. Thus the so-called three-level dark state composed of all lower levels (|1〉, |2〉, |2′〉) satisfying relation (8) derived using ∆

_{p}= ∆

_{c}= ∆

_{c′}is actually a state of a special case (

*i*.

*e*. completely resonant). However, in general cases (

*e*.

*g*. ∆

_{p}= ∆

_{c}but ∆

_{p}≠ ∆

_{c′}, or ∆

_{p}= ∆

_{c′}but ∆

_{p}= ∆

_{c}), such a three-level dark state would be reduced to the two-level dark state composed of the levels |1〉, |2〉 or |1〉, |2′〉.

## 4. Destructive and constructive quantum interferences

*i*.

*e*.,

*e*

^{iθ}can be obtained by relation (13),

*i*.

*e*.

_{2}→ γ′

_{2}, ∆

_{c}→ ∆

_{c′}, 𝓒 → 1/𝓒, θ → - θ, Ω

^{*}

_{c}Ω

_{c}→ Ω

^{*}

_{c′}Ω

_{c′}. Obviously, the electric susceptibility at the probe frequency depends on the atomic phase-coherence parameter θ. In general, the dephasing rates γ

_{2}, γ′

_{2}are negligibly small (

*e*.

*g*., only one part in 1000 of the spontaneous decay rates in the vapor). Thus, Eq. (18) can be simplified to the form

*e*

^{iθ}is a real number, namely, the phase parameter can only be chosen θ = 0 or π. In order to see how the double-control destructive and constructive quantum interferences influence the optical properties of the atomic vapor, we here consider a simple case with the module 𝓒 = 1. If the phase parameter θ = 0, then from expressions (19) and (20) for the electric polarizability, the present atomic vapor is transparent to the probe field. However, the atomic vapor would be opaque if the phase parameter θ = π. Then expressions (19) and (20) will be reduced to (14) that characterizes the two-level resonant absorption. Thus, the four-level atomic vapor would exhibit the transparency effect once the three levels |1〉, |2〉, and |2′〉 form a

*three-level*dark state, and it would exhibit the resonant absorption once the two levels |2〉 and |2′〉 form a

*two-level*dark state. On the other hand, the present β(∆

_{p}) will be reduced to the atomic electric polarizability of a typical three-level Lambda-configuration EIT system,

*i*.

*e*.,

*e*.

*g*., the intensity of the control field Ω

_{c}is much larger than that of the control field Ω

_{c′}(

*i*.

*e*. Ω

^{*}

_{c}Ω

_{c}≫ Ω

^{*}

_{c′}Ω

_{c′}), or the control field Ω

_{c′}drives the atomic system at resonance (∆

_{p}-∆

_{c}→0, ∆

_{p}-∆

_{c′}≠0).

**397**, 594–598
(1999) and references therein. [CrossRef]

10. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal pulse
propagation,” Nature **406**, 277–279
(2000) and references therein. [CrossRef] [PubMed]

15. R.Y. Chiao, “Superluminal (but causal)
propagation of wave packets in transparent media with inverted atomic
populations,” Phys. Rev. A **48**, R34–R37
(1993). [CrossRef] [PubMed]

16. E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical pulse propagation at
negative group velocities due to a nearby gain
line,” Phys. Rev. A **49**, 2938–2947
(1994). [CrossRef] [PubMed]

17. S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, “Logic and functional operations
using a near-field optically coupled quantum-dot
system,” Phys. Rev. B **69**, 115334(1-13)(2004). [CrossRef]

18. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic
switching operation by optical near-field energy
transfer,” Appl. Phys. Lett. **82**, 2957–2959
(2003). [CrossRef]

19. T. Kawazoe, K. Kobayashi, and M. Ohtsu,“A nanophotonic NOT-gate using near-field optically coupled quantum dots,” 2005 Conference on Lasers & Electro-Optics (CLEO), Baltimore, MD, USA, 728–730 (2005). [CrossRef]

*negative group velocity*: It follows from the dispersive behavior of the absorption coefficient in Fig. 3 that the double-control four-level EIT system experiences a dramatic absorption enhancement (a very sharp increase of α between the two resonant frequencies) as compared with a usual three-level EIT system. This means that the property (particularly the real part of the susceptibility between the two EIT transparency windows) of the double-control four-level atomic vapor is more sensitive to the probe frequency. This may lead to a dramatic change of the speed of the light in the four-level medium. In the literature, the ultraslow light and the superluminal propagation (negative group velocity) in the three-level EIT media have received attention from many researchers [8**397**, 594–598 (1999) and references therein. [CrossRef], 15**406**, 277–279 (2000) and references therein. [CrossRef] [PubMed], 1615. R.Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A

**48**, R34–R37 (1993). [CrossRef] [PubMed]]. As the dispersion in both the real and imaginary parts of optical ‘constants’ is stronger than that in a three-level EIT vapor, the ultraslow and superluminal propagations of light also deserve consideration in the case of double-control four-level vapor.16. E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical pulse propagation at negative group velocities due to a nearby gain line,” Phys. Rev. A

**49**, 2938–2947 (1994). [CrossRef] [PubMed]*logic gates*: Recently, ideas of realizing logic gates by using new optoelectronic materials have captured attention of some researchers [17, 1817. S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, “Logic and functional operations using a near-field optically coupled quantum-dot system,” Phys. Rev. B

**69**, 115334(1-13)(2004). [CrossRef], 1918. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett.

**82**, 2957–2959 (2003). [CrossRef]]. We believe that the double-control interference effects can also be used to realize some logic and functional operations (19. T. Kawazoe, K. Kobayashi, and M. Ohtsu,“A nanophotonic NOT-gate using near-field optically coupled quantum dots,” 2005 Conference on Lasers & Electro-Optics (CLEO), Baltimore, MD, USA, 728–730 (2005). [CrossRef]

*e*.*g*. the operations of NOT and AND gates). Here we give an example to show how a NOT gate works based on the double-control interference effects: choose the proper Rabi frequencies of the two control fields that satisfy the relation (*i*.*e*. Ω_{c}*a*_{2}+ Ω_{c′}*a*_{2′}= 0) for the destructive quantum interference between the two control levels (|2〉 and |2′〉). Once the control field Ω_{c′}is switched off (logic operation IN= 0), the present scheme will exhibit a three-level EIT effect (logic operation OUT= 1). But when the control field Ω_{c′}is switched on (logic operation IN= 1), the present double-control scheme will exhibit a two-level resonant absorption to the probe field (logic operation OUT= 0). This is the working mechanism of the double-control NOT gate.*optical switches*: The double-control destructive and constructive interference effects can be applicable to some quantum optical and photonic devices. For example, such a coherent effect can be utilized for designing optical switches. This switches might be a useful technique for future photonic microcircuits on silicon, in which light replaces electrons. At present, the all-optical switch on silicon where one controls light with light on chip has been increasingly developed. We hope the present optical switches based on double-control interference would have potential applications in this field and other related areas,*e*.*g*. integrated optical circuits.

## 5. Concluding remarks

_{p}= ∆

_{c}, ∆

_{p}= ∆

_{c′}), where the four-level system can exhibit the EIT effect. This is a new feature that is different from the conventional three-level EIT system, where there is only one resonant frequency. As the double-control four-level EIT vapor exhibits a large dispersion in both the real and imaginary parts of optical ‘constants’, the optical properties would be more sensitive to the probe frequency as compared with a three-level EIT vapor. The present double-control quantum interference scheme can hence be applicable to many new techniques such as sensitive optical switches, optical magnetometers and wavelength sensors. For example, the optical magnetometers could be used to detect magnetic fields with very high sensitivity because of the strong dispersion caused by the double-control atomic phase coherence, and the double-control EIT-based wavelength sensor can be utilized to measure the probe wavelength. Such a device can be applied to some practical areas like color matching and sorting, where precise measurements of light wavelengths and frequencies are needed.

11. P. Arve, P. Jänes, and Lars Thylén, “Propagation of two-dimensional
pulses in electromagnetically induced transparency
media,” Phys. Rev. A **69**, 063809(1-8) (2004). [CrossRef]

13. Y. Q. Li and M. Xiao, “Transient properties of an
electromagnetically induced transparency in three-level
atoms,” Opt. Lett. **20**, 1489–1491
(1995). [CrossRef] [PubMed]

*quantum coherent information storage*. In order to see how fast the optical behaviors respond to the switching on of the control fields, in the literature, Yao

*et al*. first studied the transient optical properties of the four-level N-configuration system under certain approximation conditions [20]. As there are two resonant frequencies and large dispersion in the double-control EIT, the scheme can also be applied to the technique of coherent information storage. Thus, the transient evolutional behavior also deserves consideration for the present double-control four-level system.

## Acknowledgments

## References and links

1. | M. Fleischhauer and M. O. Scully, “Quantum sensitivity limits of an
optical magnetometer based on atomic phase
coherence,” Phys. Rev. A |

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

3. | S. Y. Zhu and M. O. Scully, “Spectral line elimination and
spontaneous emission cancellation via quantum
interference,” Phys. Rev. Lett. |

4. | J. Q. Shen, “Quantum-vacuum geometric phases in
the noncoplanarly curved fiber system,”
Eur. Phys. J. D |

5. | J. Q. Shen, “Negative refractive index in
gyrotropically magnetoelectric media,”
Phys. Rev. B |

6. | J. P. Marangos, “Electromagnetically induced
transparency,” J. Mod. Opt. |

7. | J. L. Cohen and P. R. Berman, “Amplification without inversion:
Understanding probability amplitudes, quantum interference, and Feynman
rules in a strongly driven system,” Phys.
Rev. A |

8. | 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 |

9. | R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, “Spatial consequences of
electro-magnetically induced transparency: Observation of
electromagnetically induced focusing,”
Phys. Rev. Lett. |

10. | L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal pulse
propagation,” Nature |

11. | P. Arve, P. Jänes, and Lars Thylén, “Propagation of two-dimensional
pulses in electromagnetically induced transparency
media,” Phys. Rev. A |

12. | J. Q. Shen and S. He, “Dimension-sensitive optical
responses of electromagnetically induced transparency vapor in a
waveguide,” Phys. Rev. A |

13. | Y. Q. Li and M. Xiao, “Transient properties of an
electromagnetically induced transparency in three-level
atoms,” Opt. Lett. |

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

15. | R.Y. Chiao, “Superluminal (but causal)
propagation of wave packets in transparent media with inverted atomic
populations,” Phys. Rev. A |

16. | E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical pulse propagation at
negative group velocities due to a nearby gain
line,” Phys. Rev. A |

17. | S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, “Logic and functional operations
using a near-field optically coupled quantum-dot
system,” Phys. Rev. B |

18. | T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic
switching operation by optical near-field energy
transfer,” Appl. Phys. Lett. |

19. | T. Kawazoe, K. Kobayashi, and M. Ohtsu,“A nanophotonic NOT-gate using near-field optically coupled quantum dots,” 2005 Conference on Lasers & Electro-Optics (CLEO), Baltimore, MD, USA, 728–730 (2005). [CrossRef] |

20. | J. Q. Yao, H. B. Wu, and H. Wang, “The transient optical properties in
four-level atomic medium induced by quantum interference
effect,”Acta Sin. Quantum Opt. |

**OCIS Codes**

(120.4530) Instrumentation, measurement, and metrology : Optical constants

(270.0270) Quantum optics : Quantum optics

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: January 18, 2007

Revised Manuscript: March 19, 2007

Manuscript Accepted: March 27, 2007

Published: May 11, 2007

**Citation**

Jian Qi Shen and Pu Zhang, "Double-control quantum interferences in a four-level atomic system," Opt. Express **15**, 6484-6493 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6484

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### References

- M. Fleischhauer andM. O. Scully, "Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence," Phys. Rev. A 49, 1973-1986 (1994). [CrossRef] [PubMed]
- S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997) and references therein. [CrossRef]
- S. Y. Zhu and M. O. Scully, "Spectral line elimination and spontaneous emission cancellation via quantum interference," Phys. Rev. Lett. 76, 388-391 (1996). [CrossRef] [PubMed]
- J. Q. Shen, "Quantum-vacuum geometric phases in the noncoplanarly curved fiber system," Eur. Phys. J. D 30,259-264 (2004). [CrossRef]
- J. Q. Shen, "Negative refractive index in gyrotropically magnetoelectric media," Phys. Rev. B 73, 045113(1-5) (2006). [CrossRef]
- J. P. Marangos, "Electromagnetically induced transparency," J. Mod. Opt. 45,471-503 (1998). [CrossRef]
- J. L. Cohen and P. R. Berman, "Amplification without inversion: Understanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system," Phys. Rev. A 55,3900-3917 (1997). [CrossRef]
- 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) and references therein. [CrossRef]
- R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995). [CrossRef] [PubMed]
- L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal pulse propagation," Nature 406, 277-279 (2000) and references therein. [CrossRef] [PubMed]
- P. Arve, P. J¨anes, and Lars Thyl´en, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 69, 063809(1-8) (2004). [CrossRef]
- J. Q. Shen and S. He, "Dimension-sensitive optical responses of electromagnetically induced transparency vapor in a waveguide," Phys. Rev. A 74, 063831(1-6) (2006).
- Y. Q. Li andM. Xiao, "Transient properties of an electromagnetically induced transparency in three-level atoms," Opt. Lett. 20, 1489-1491 (1995). [CrossRef] [PubMed]
- H. Schmidt and A. Imamo¡glu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21,1936-1938 (1996). [CrossRef] [PubMed]
- R.Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34-R37 (1993). [CrossRef] [PubMed]
- E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994). [CrossRef] [PubMed]
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