## Coherent writing and reading of information using frequency-chirped short bichromatic laser pulses

Optics Express, Vol. 4, Issue 2, pp. 113-120 (1999)

http://dx.doi.org/10.1364/OE.4.000113

Acrobat PDF (102 KB)

### Abstract

We propose to use the sensitivity of the population transfer in three-level L-atoms to the relative phases and amplitudes of frequency-chirped short bichromatic laser pulses for coherent, fast and robust storage and processing of phase or intensity optical information. The information is being written into the excited state population which in a second step is transferred in a fast and robust way into a nondecaying storage level. It is shown that an arbitrary superposition of the ground states can be generated by controlling the relative phase between the laser pulses.

© Optical Society of America

## 1. Introduction

*single*desired quantum state is the goal of many important applications of the quantum chemistry, quantum optics, mechanical manipulation and cooling of atoms or molecules by laser radiation. On the other hand, the

*coherent superpositions*of quantum states also play an important role in the above mentioned fields of science with very interesting and useful applications. The population trapping [1

1. E. Arimondo and G. Orriols, “Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping,” Nuovo Cimento Lett. **17**, 333–338 (1976). [CrossRef]

2. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys.Rev.Lett. **61**, 826–829 (1988). [CrossRef] [PubMed]

3. J. Lawall and M. Prentiss, “Demonstration of a novel atomic beam splitter,” Phys.Rev.Lett. **72**, 993–996 (1994). [CrossRef] [PubMed]

4. J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, “Adiabatic population transfer in a three-level system driven by delayed laser pulses,” Phys.Rev. A **40**, 6741–6747 (1989). [CrossRef] [PubMed]

5. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys.Rev.Lett. **67**, p.1855–58 (1991). [CrossRef] [PubMed]

6. M. Weitz, B. C. Young, and S. Chu, “Atomic interferometer based on adiabatic population transfer,” Phys.Rev.Lett. **73**, 2563–2566 (1994). [CrossRef] [PubMed]

8. R. Unanyan, M. Fleischhauer, B. W. Shore, and K. Bergmann, “Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states,” Opt. Commun. **155**, 144–154 (1998). [CrossRef]

8. R. Unanyan, M. Fleischhauer, B. W. Shore, and K. Bergmann, “Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states,” Opt. Commun. **155**, 144–154 (1998). [CrossRef]

*Fast*and

*robus*t writing, storage and reading of optical information is a task to be solved nowadays, especially when

*short laser pulses*are used as the carriers of the optical information. The sensitivity of the population transfer in a multilevel quantum system to the relative phases of exciting laser fields considered above (see also [6–8

6. M. Weitz, B. C. Young, and S. Chu, “Atomic interferometer based on adiabatic population transfer,” Phys.Rev.Lett. **73**, 2563–2566 (1994). [CrossRef] [PubMed]

10. D. Kosachev, B. Matisov, and Yu. Rozhdestvensky, “Coherent population trapping: sensitivity of an atomic system to the relative phase of exciting fields,” Opt.Commun. **85**, 209–212 (1991). [CrossRef]

11. N. V. Vitanov, “Analytic model of a three-state system driven by two laser pulses on two-photon resonance,” J.Phys.B: At. Mol. Opt. Phys. **31**, 709–725 (1998). [CrossRef]

*fast*and

*robust*in the proposed scheme due to the shortness of the applied laser pulses and the robustness of the population transfer produced by the frequency-chirped pulses in the AP regime of interaction.

11. N. V. Vitanov, “Analytic model of a three-state system driven by two laser pulses on two-photon resonance,” J.Phys.B: At. Mol. Opt. Phys. **31**, 709–725 (1998). [CrossRef]

12. C. E. Caroll and F. T. Hioe, “Three-state model driven by two laser beams,” Phys.Rev.A **36**, 724–729 (1987). [CrossRef]

13. G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Theory of the adiabatic passage in two-level quantum systems with superpositional initial states,” J. of Modern Opt. **44**, 1511–1523 (1997). [CrossRef]

## 2. The mathematical formalism

*f*(

*t*) with a three-level Λ-atom having two ground states ∣ 1⟩ and ∣ 3⟩ and excited state ∣ 2⟩, as it is shown in Fig. 1, where also is depicted a nondecaying (metastable) state ∣ 4⟩ which is assumed to be not coupled to the λ -system by BLP. This state will be used only for transfer of the population of the excited state (for information storage, see below) at a second step, by applying a short frequency-chirped laser pulse coupling the states ∣ 2⟩ and ∣ 4⟩ after the interaction with the BLP.

*A*

_{1,2}(

*t*) =

*f*(

*t*)

*i*Φ

_{1,2}] and time-dependent frequencies

*ω*

_{L1}(

*t*) and

*ω*

_{L2}(

*t*). The pulse

*A*

_{1}(

*t*) couples the states ∣1⟩ and ∣2⟩, and the pulse

*A*

_{2}(

*t*) couples the states ∣3⟩ and ∣2⟩. The carrier frequencies of both pulses of the BLP are assumed to be chirped in time in the same way and the BLP duration is assumed to be much shorter than all relaxation times of the atom. It allows as to deal with the Schrödinger equation for the amplitudes

*a*

_{j},

*j*= 1,2,3 of the states of the • • atom in Fig. 1.

**C**, whose components

*c*

_{i},

*i*= 1,2,3 are proportional to the amplitudes

*a*

_{i},

*i*= 1,2,3 of the atomic states:

_{12}(

*t*) and 2Ω

_{32}(

*t*) with complex amplitudes Ω

_{1}and Ω

_{2}are introduced:

*d*

_{ij}being the dipole moment matrix element for the laser induced transition from state ∣

*j*⟩ to state ∣

*i*⟩, (

*i*,

*j*= 1,2,3).

*ε*

_{21}(

*t*) =

*ω*

_{L1}(

*t*)-

*ω*

_{21},

*ε*

_{23}(

*t*) =

*ω*

_{L2}(

*t*)-

*ω*

_{23}are the detunings from the one-photon resonances with

*ω*

_{21}and

*ω*

_{23}being the resonant transition frequencies between the corresponding states. We assume, that the condition of the Raman resonance has been fulfilled:

*ε*

_{21}(

*t*) =

*ε*

_{23}(

*t*) =

*ε*(

*t*). In what follows, we assume the same linear temporal chirp of the carrier frequencies of the pulses forming BLP:

*ω*

_{Lj}(

*t*) =

*βt*,

*j*= 1,2, where

*β*is the speed of the chirp, see Fig.1.

**C**from the Schrödinger equation for the probability amplitudes

*a*

_{j},

*j*= 1,2,3, [12

12. C. E. Caroll and F. T. Hioe, “Three-state model driven by two laser beams,” Phys.Rev.A **36**, 724–729 (1987). [CrossRef]

*Ĥ*in the rotating wave approximation is:

*g*

^{(+)}and

*g*

^{(-)}amplitudes of the coupled to the excited state by the BLP “bright” and uncoupled “dark” superpositions of the ground states having amplitudes

*a*

_{1}(

*c*

_{1}) and

*a*

_{3}(

*c*

_{3}):

*e*=

*c*

_{2}.

*g*

^{(+)}being coupled to the excited state with amplitude

*e*by the BLP. The “dark” ground state with amplitude

*g*

^{(-)}does not interact with the BLP, as it follows from the third equation in Eq.(2).

*e*

_{in}=

*c*

_{2in}=0. The subscripts

*in*and

*fin*stand for the initial (

*t*- > -∞) and the final (

*t*- > ∞) values.

*a*

_{1fin}and

*a*

_{3fin}of the ground states of our original three-level Λ-atom at the end of the laser pulse using Eq.(3):

_{1}∣ = ∣Ω

_{2}∣) of the laser pulses, we obtain for the final populations

*n*

_{ifin}and the phases

*ϕ*

_{ifin}(i=1,3) of the ground states:

*a*

_{j}= √

*n*

_{j}exp[

*iϕ*

_{j}],

*j*= 1,2,3 and ΔΦ

_{12}= Φ

_{1}-Φ

_{2}is the relative (difference) phase of the laser pulses.

*n*

_{2fin}of the excited state after interaction with the BLP (in the case of ∣Ω

_{1}∣ = ∣Ω

_{2}∣):

*ϕ*

_{13in}and ΔΦ

_{12}, when the initial state of the Λ-atom is a

*superposition*of the ground states ∣1⟩ and ∣3⟩.

*n*

_{2fin}on the initial phase Φ

_{in}= Δ

*ϕ*

_{13in}+ ΔΦ

_{12}is depicted in Fig. 2.

*n*

_{2fin}= 1, when the initial phase Φ

_{in}= 0 and • there is no excitation at all,

*n*

_{2fin}=

*n*

_{2in}= 0 (dark state) at Φ

_{in}=

*π*in the case of equal initial population of the ground states, when

*n*

_{1in}=

*n*

_{3in}= 1/2.

*n*

_{1in}= 1):

*n*

_{2fin}- 1/2 and

*n*

_{1fin}=

*n*

_{3fin}= 1/4.

_{1}∣ ≠ ∣Ω

_{2}∣) and the atom in the ground state ∣1⟩ initially,

*n*

_{1in}= 1:

## 3. Results of the numerical simulations

*f*(

*t*) = exp[-

*t*

^{2}/2

*t*

_{L}being the duration of the BLP), and the same linear chirp has been assumed for the pulses forming the BLP.

_{in}= Δ

*ϕ*

_{13in}+ ΔΦ

_{12}is clearly seen in Figs.3 for the Λ-atom being initially in the superposition of the ground states: There is effective excitation of the atom when Φ

_{in}= 0 (Fig. 3a) and the excitation of the atom is suppressed when Φ

_{in}=

*π*(Fig. 3b). Note that in the latter case we have a “dark” state in case of equal initial populations of the two ground states (for ∣Ω

_{1}∣ = ∣Ω

_{2}∣).

## 4. Writing/reading of phase and intensity information

*phase*information. As it follows from Eq.(5), the relative phase of the probability amplitudes of the ground states is equal to Δ

*ϕ*

_{13fin}=

*π*+ ΔΦ

_{12}, at the end of the BLP. So, an arbitrary phase difference Δ

*ϕ*

_{13in}(arbitrary value of the Raman, or Zeeman coherence) may be generated by controlling the phase difference ΔΦ

_{12}of the pulses forming the BLP.

*n*

_{2fin}of the excited state is a function of the phase ΔΦ

_{12}. So, the phase information contained in the BLP may be written in the population

*n*

_{2fin}of the excited state if the atoms are prepared initially in the same coherent superposition of the ground states, for e.g., in a “dark” state (with the same initial parameters

*n*

_{1in},

*n*

_{3in}and Δ

*ϕ*

_{13in}). This information however will be distorted due to the spontaneous decay from the excited state during the decay time of this state. One of the way to preserve this information is to transfer the population of the excited state ∣2⟩ (with the information written therein) into the additional nondecaying metastable state ∣4⟩, see Fig. 1, by acting with a subsequent frequency-chirped short laser pulse in the AP regime of interaction. Reading of the phase information stored in this nondecaying state may be produced by acting by the same chirped laser pulse which transforms the phase information into the population of the excited state. The latter may be detected for example, by analysing the spontaneous or stimulated emission from this state.

_{2}∣

^{2}/ ∣Ω

_{1}∣

^{2}. So, optical intensity information (image) contained in the relative intensity of the pulses forming the BLP may be written into the population

*n*

_{2fin}of the excited state. One have to store this information (image) to preserve it from distortion due to the spontaneous decay by transfer the population of the excited state into the nondecaying state (for e.g., into the state ∣4⟩ in Fig. 1). This may be done just like to the case of the considered above phase information storage using a subsequent frequency-chirped short laser pulse. The reading of the stored information may be produced also in the same way as that one of the phase information reading.

## 5. Conclusions

*fast*and

*robust*in the proposed scheme. These processes have time scales equal to those of the laser pulses, which durations may be chosen to be very short. The restriction on the duration of the laser pulses is mainly connected with the conditions for the AP regime of interaction [9,13

13. G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Theory of the adiabatic passage in two-level quantum systems with superpositional initial states,” J. of Modern Opt. **44**, 1511–1523 (1997). [CrossRef]

## Acknowledgements

## References and links

1. | E. Arimondo and G. Orriols, “Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping,” Nuovo Cimento Lett. |

2. | A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys.Rev.Lett. |

3. | J. Lawall and M. Prentiss, “Demonstration of a novel atomic beam splitter,” Phys.Rev.Lett. |

4. | J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, “Adiabatic population transfer in a three-level system driven by delayed laser pulses,” Phys.Rev. A |

5. | M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys.Rev.Lett. |

6. | M. Weitz, B. C. Young, and S. Chu, “Atomic interferometer based on adiabatic population transfer,” Phys.Rev.Lett. |

7. | P. Marte, P. Zoller, and J. L. Hall, “Coherent atomic mirrors and beam splitters by adiabatic passage in multilevel systems,” Phys. Rev.A |

8. | R. Unanyan, M. Fleischhauer, B. W. Shore, and K. Bergmann, “Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states,” Opt. Commun. |

9. | L. Allen and J. H. Eberly, |

10. | D. Kosachev, B. Matisov, and Yu. Rozhdestvensky, “Coherent population trapping: sensitivity of an atomic system to the relative phase of exciting fields,” Opt.Commun. |

11. | N. V. Vitanov, “Analytic model of a three-state system driven by two laser pulses on two-photon resonance,” J.Phys.B: At. Mol. Opt. Phys. |

12. | C. E. Caroll and F. T. Hioe, “Three-state model driven by two laser beams,” Phys.Rev.A |

13. | G. P. Djotyan, J. S. Bakos, G. Demeter, and Zs. Sörlei, “Theory of the adiabatic passage in two-level quantum systems with superpositional initial states,” J. of Modern Opt. |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(070.4560) Fourier optics and signal processing : Data processing by optical means

**ToC Category:**

Focus Issue: Laser controlled dynamics

**History**

Original Manuscript: December 7, 1998

Published: January 18, 1999

**Citation**

G. Djotyan, J. S. Bakos, and Zs. Sorlei, "Coherent writing and reading of information using frequency-chirped short bichromatic laser pulses," Opt. Express **4**, 113-120 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-2-113

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### References

- E. Arimondo and G. Orriols, "Nonabsorbing atomic coherences by coherent two-photon transitions in a three- level optical pumping," Nuovo Cimento Lett. 17, 333-338 (1976). [CrossRef]
- A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping," Phys.Rev.Lett. 61, 826-829 (1988). [CrossRef] [PubMed]
- J. Lawall, and M. Prentiss, "Demonstration of a novel atomic beam splitter," Phys.Rev.Lett. 72, 993-996 (1994). [CrossRef] [PubMed]
- J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, "Adiabatic population transfer in a three-level system driven by delayed laser pulses," Phys. Rev. A 40, 6741-6747 (1989). [CrossRef] [PubMed]
- M. O. Scully, "Enhancement of the index of refraction via quantum coherence," Phys.Rev.Lett. 67, 1855-58 (1991). [CrossRef] [PubMed]
- M. Weitz, B. C. Young, and S. Chu, "Atomic interferometer based on adiabatic population transfer," Phys. Rev. Lett. 73, 2563-2566 (1994). [CrossRef] [PubMed]
- P. Marte, P. Zoller, and J. L. Hall, "Coherent atomic mirrors and beam splitters by adiabatic passage in multilevel systems," Phys. Rev.A 44, R4118-R4121.
- R. Unanyan, M. Fleischhauer, B. W. Shore, K. Bergmann, "Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states," Opt. Commun. 155, 144-154 (1998). [CrossRef]
- L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
- D. Kosachev, B. Matisov and Yu. Rozhdestvensky, "Coherent population trapping: sensitivity of an atomic system to the relative phase of exciting fields," Opt. Commun. 85, 209-212 (1991). [CrossRef]
- N. V. Vitanov, "Analytic model of a three-state system driven by two laser pulses on two-photon resonance," J.Phys. B: At. Mol. Opt. Phys. 31, 709-725 (1998). [CrossRef]
- C. E. Caroll and F. T. Hioe, "Three-state model driven by two laser beams," Phys. Rev. A 36, 724-729 (1987). [CrossRef]
- G. P. Djotyan, J. S. Bakos, G. Demeter and Zs. S”rlei, "Theory of the adiabatic passage in two-level quantum systems with superpositional initial states," J. of Modern Opt. 44, 1511-1523 (1997). [CrossRef]

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