## Coherent Mechanism of Robust Population Inversion

Optics Express, Vol. 8, Issue 4, pp. 238-245 (2001)

http://dx.doi.org/10.1364/OE.8.000238

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### Abstract

A coherent mechanism of robust population inversion in atomic and molecular systems by a chirped field is presented. It is demonstrated that a field of sufficiently high chirp rate imposes a certain relative phase between a ground and excited state wavefunction of a two-level system. The value of the relative phase angle is thus restricted to be negative and close to 0 or -π for positive and negative chirp, respectively. This explains the unidirectionality of the population transfer from the ground to the excited state. In a molecular system composed of a ground and excited potential energy surface the symmetry between the action of a pulse with a large positive and negative chirp is broken. The same framwork of the coherent mechanism can explain the symmetry breaking and the population inversion due to a positive chirped field.

© Optical Society of America

4. J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular *π* Pulse for Total Inversion of Electronic State Population,” Phys. Rev. Lett. **80**, 1406–1409 (1998). [CrossRef]

5. J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular *π* pulses: Population inversion with positively chirped short pulses,” J. Chem. Phys. **113**, 1898–1909 (2000). [CrossRef]

*N*

_{g}is either the ground state population in the TLS or the expectation of the projection on the ground electronic surface i.e.

*N*

_{g}=〈P̂

_{g}〉=∫|

*ψ*

_{g}(

*r*)|

^{2}

*dr*. In both cases the Heisenberg equation of motion for the change of the ground state population induced by an electromagnetic field becomes [6

6. R. Kosloff, A. D. Hammerich, and D. Tannor, “Excitation without demolition: Radiative excitation of ground-surface vibration by impulsive stimulated Raman scattering with damage control,” Phys. Rev. Lett **69**,2172–2175 (1992). [CrossRef] [PubMed]

*ψ*

_{e}|

*µ̂*|

*ψ*

_{g}〉 is the transition dipole moment, E(t) is the electromagnetic field and

*ϕ*

_{µ}and

*ϕ*

_{E}are phase angles of the transition dipole and the field, respectively.

*ψ*

_{i},

*i*=

*g,e*is a ground and excited state wavefunction, respectively.

*ψ*

_{g}→

*ψ*

_{e}implies

*ϕ*

_{µ}and

*ϕ*

_{E}. The phase of transition dipole 〈

*ψ*

_{e}|

*µ̂*|

*ψ*

_{g}〉 is assembled during the excitation process and is therefore a function of the history of the amplitude and phase of the excitation field. If initially all the population resides on the ground state its initial phase has no relevance since it does not alter the phase of the transition dipole.

4. J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular *π* Pulse for Total Inversion of Electronic State Population,” Phys. Rev. Lett. **80**, 1406–1409 (1998). [CrossRef]

*ω*

_{0}is the transform-limited carrier frequency of the field, Γ is the spectral band-width of the pulse and

*χ*′ is the chirp rate in energy representation given by

*dt/dω*. The chirp rate term causes a phase shift of each spectral component of the field proportional to its ‘distance’ from the carrier frequency. The field in time representation is given by its Fourier transform

*ϕ*

_{E}can be set to zero because a constant phase of the field maps onto the phase of the transition dipole moment. In this case the direction of the population transfer will be determined by the induced instantaneous phase of the transition dipole

*π*<

*ϕ*

_{µ}<0 throughout the process will guarantee a monotonic and robust population transfer.

8. R. Kosloff, *Propagation Methods for Quantum Molecular Dynamics*, Annu. Rev. Phys. Chem.45, 145–178 (1994). [CrossRef]

*π*cycling from the ground state to the excited state and back. The top panel of Fig. 1 shows the total population on the ground state

*N*

_{g}as well as its time derivative

*dN*

_{g}/

*dt*<0. Once all the population is transfered to the excited state, the imaginary part of the transition dipole moment changes sign and redirect the population flow back to the ground state.

*π*/2 and

*π*/2. With increasing chirp rate the trajectories also obtain a real component positive for the positive chirp and

*vice versa*. For the chirped field case more time is spent in the negative imaginary part of the complex plane. With sufficient chirp the whole trajectory is maintained in the negative imaginary quadrants. The perfect symmetry of the trajectories with respect to pulses with positive and negative chirp is obvious for this TLS case.

4. J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular *π* Pulse for Total Inversion of Electronic State Population,” Phys. Rev. Lett. **80**, 1406–1409 (1998). [CrossRef]

5. J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular *π* pulses: Population inversion with positively chirped short pulses,” J. Chem. Phys. **113**, 1898–1909 (2000). [CrossRef]

*|*ψ

_{g}(

*t*)〉 is known to be cos(

*|*

*E*(

*t*)|

*t*)

*|*ψ

_{g}(0)〉 [11

11. G. Ashkenazi, U. Banin, A. Bartana, R. Kosloff, and S. Ruhman, “Quantum Description of the Impulsive Photodissociation Dynamics of **100**, 229–315 (1997). [CrossRef]

*e*

^{iω(t-τ)}, cancels the excited state free evolution term and the integration leads to the following expression

*E*0

*e*-

^{iω(t)}for the sake of simplicity. This solution shows that the alternation of the relative phase angle between the ground and excited state TLS wavefunctions due to the on-resonant Rabi cycling is strictly ∓

*π*/2 in accord with our numerical results, see Fig. 2.

*π*/2. Since the relative phase concept does not have any meaning when only one of the levels of a TLS is populated, the turning points of the on-resonant Rabi cycling are singular from the point of view of the relative phase.

*e*

^{-iΔ(t-τ)}appears inside the integral in Eq. 6 due to the difference Δ between the system Bohr frequency (

*ħ*

**Ĥ**

_{e}) and the frequency of the field. This term leads to an extra rotation of the relative phase in one direction and results in faster and less efficient population transfer.

10. J. Vala, O. Dulieu, F. Masnou-Seeuws, P. Pillet, and R. Kosloff, “Coherent control of cold-molecule formation through photoassociation using a chirped-pulsed-laser field,” Phys. Rev. A **63**, 013412 (2001). [CrossRef]

11. G. Ashkenazi, U. Banin, A. Bartana, R. Kosloff, and S. Ruhman, “Quantum Description of the Impulsive Photodissociation Dynamics of **100**, 229–315 (1997). [CrossRef]

## Acknowledgments

## References and links

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

2. | Denise Sawicki and J. H. Eberly “Perfect following in the diabatic limit,” Opt. Express , |

3. | Y. B. Band and O. Magnes, “Chirped adiabatic passage with temporally delayed pulses,” Phys. Rev. A |

4. | J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular |

5. | J. Cao, C. J. Bardeen, and K. R. Wilson, “Molecular |

6. | R. Kosloff, A. D. Hammerich, and D. Tannor, “Excitation without demolition: Radiative excitation of ground-surface vibration by impulsive stimulated Raman scattering with damage control,” Phys. Rev. Lett |

7. | S. A. Rice and M. Zhao, |

8. | R. Kosloff, |

9. | R. Kosloff, |

10. | J. Vala, O. Dulieu, F. Masnou-Seeuws, P. Pillet, and R. Kosloff, “Coherent control of cold-molecule formation through photoassociation using a chirped-pulsed-laser field,” Phys. Rev. A |

11. | G. Ashkenazi, U. Banin, A. Bartana, R. Kosloff, and S. Ruhman, “Quantum Description of the Impulsive Photodissociation Dynamics of |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(030.1670) Coherence and statistical optics : Coherent optical effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 19, 2000

Published: February 12, 2001

**Citation**

Jiri Vala and Ronnie Kosloff, "Coherent mechanism of robust population inversion," Opt. Express **8**, 238-245 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-4-238

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### References

- L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, (Dover Publications, Inc., New York, 1987).
- Denise Sawicki and J. H. Eberly "Perfect following in the diabatic limit," Opt. Express, 4 217-222 (1999), http://www.opticsexpress.org/oearchive/source/9096.htm [CrossRef] [PubMed]
- Y. B. Band and O. Magnes, "Chirped adiabatic passage with temporally delayed pulses," Phys. Rev. A 50, 584-594 (1994). [CrossRef] [PubMed]
- J. Cao, C. J. Bardeen and K. R. Wilson, "Molecular pi Pulse for Total Inversion of Electronic State Population," Phys. Rev. Lett. 80, 1406-1409 (1998). [CrossRef]
- J. Cao, C. J. Bardeen and K. R. Wilson, "Molecular pi pulses: Population inversion with positively chirped short pulses," J. Chem. Phys. 113, 1898-1909 (2000). [CrossRef]
- R. Kosloff, A. D. Hammerich and D. Tannor, "Excitation without demolition: Radiative excitation of ground-surface vibration by impulsive stimulated Raman scattering with damage control," Phys. Rev. Lett. 69, 2172-2175 (1992). [CrossRef] [PubMed]
- S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics, (John Wiley and Sons, New York 2000).
- R. Kosloff, Propagation Methods for Quantum Molecular Dynamics, Annu. Rev. Phys. Chem. 45, 145-178 (1994). [CrossRef]
- R. Kosloff, Quantum Molecular Dynamics on Grids., in R. E. Wyatt and J. Z. Zhang, editor, Dynamics of Molecules and Chemical Reactions, pages 185-230, Marcel Dekker, (1996).
- J. Vala, O. Dulieu, F. Masnou-Seeuws, P. Pillet and R. Kosloff, "Coherent control of cold-molecule formation through photoassociation using a chirped-pulsed-laser field," Phys. Rev. A 63, 013412 (2001). [CrossRef]
- G. Ashkenazi, U. Banin, A. Bartana, R. Kosloff and S. Ruhman, "Quantum Description of the Impulsive Photodissociation Dynamics of I 3 in Solution," Adv. Chem Phys. 100, 229-315 (1997). [CrossRef]

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