## Normal mode oscillation in the presence of inhomogeneous broadening

Optics Express, Vol. 1, Issue 12, pp. 370-375 (1997)

http://dx.doi.org/10.1364/OE.1.000370

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### Abstract

We investigate effects of inhomogeneous broadening of excitons on normal mode oscillation in semiconductor microcavities using a coupled oscillator model. We show that inhomogeneous broadening can drastically alter the coherent oscillatory energy exchange process even in regimes where normal mode splitting remains nearly unchanged. The depth, frequency, and phase of normal mode oscillations of excitons at a given energy within the inhomogeneous distribution depend strongly on the energy separation between the exciton and the normal mode resonance. In addition, for an inhomogeneous broadened system, pronounced oscillations in the intensity of the optical field or the total induced optical polarization no longer imply a similar oscillatory coherent energy exchange between excitons and cavity photons.

© Optical Society of America

1. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. **69**, 3314 (1992). [CrossRef] [PubMed]

2. T. B. Norris, J. K. Rhee, C. Y. Sung, Y. Arakawa, M. Nishioka, and C. Weisbuch, “Time-resolved vacuum Rabi oscillations in a semiconductor quantum microcavity,” Phys. Rev. B **50**, 14663 (1994). [CrossRef]

_{inh}, HWHM of the inhomogeneous distribution, is considerably smaller than Ω and where inhomogeneous broadening results in no reduction in the normal mode splitting. Calculations based on a coupled oscillator model reveal that the frequency, phase, and depth of normal mode oscillation of localized excitons at a given energy within the inhomogeneous distribution depend strongly on the energy separation between the exciton and the normal mode resonance. Strongest normal mode oscillation occurs for excitons interacting with equal strength with optical fields at the two normal mode resonances, whereas only very weak oscillations are expected for excitons nearly resonant with a normal mode. In an inhomogeneous broadened system, pronounced oscillation in the intensity of the optical field or the total induced optical polarization no longer implies similar oscillations in the exciton population.

8. H. J. Carmichael, “Quantum fluctuations in absorptive bistability without adiabatic elimination,” Phys. Rev. A **33**, 3262 (1986). [CrossRef] [PubMed]

9. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear dispersion theory,” Phys. Rev. Lett. **64**, 2499 (1990). [CrossRef] [PubMed]

*α*is the expectation value of the annihilation field operator for the cavity mode at the position of the QW inside the cavity, Ω is the collective dipole coupling rate determined by spectrally integrated oscillator strength,

*ε*(

*t*) represents the external driving field,

*ω*

_{c}is the resonant frequency of the cavity, and

*f*(

*ω*) is the normalized inhomogeneous distribution of the exciton energy. We have scaled

*β*(

*ω*) such that

*P*(

*ω*)=

*μβ*(

*ω*)/

*N*

^{1/2}represents the optical polarization for an exciton at frequency

*ω*where

*N*is the total number of excitons allowed within the inhomogeneous distribution and

*μ*is the optical transition dipole moment. Note that for simplicity we have used as Eq. (1) a first order Maxwell equation that was originally derived for CW excitations. The first order Maxwell equation can also be used to describe transient processes with a time scale much longer than the cavity round trip time (which is a few fs for typical semiconductor microcavities), as shown in recent theoretical studies [10

10. M. Kira, F. Jahnke, and S. W. Koch, “Ultrashort pulse propagation effects in semiconductor microcavities,” Solid State Commun. **102**, 703 (1997). [CrossRef]

*ω*can be determined in the weak excitation limit from the following equation:

_{0}at ω

_{c}. As shown in Fig. 1a, the magnitude of normal mode splitting is nearly the same for both microcavities in agreement with earlier studies [6

6. S. Pau, G. Bjork, H. Cao, E. Hanamura, and Y. Yamamoto, “Theory of inhomogeneous microcavity polariton splitting,” Solid State Commun. **98**, 781 (1996). [CrossRef]

_{0}(assuming ω

_{0}=ω

_{c}) will interact with equal strength with optical fields at both normal mode resonances, leading to a strong oscillation in the excitonic polarization as well as the exciton population. In comparison, excitons near a given normal mode resonance will couple much more strongly to optical fields at the given normal mode resonance than to optical fields at the other normal mode resonance. Only very weak oscillations are expected for these excitons. The oscillation in the total exciton population is thus much weaker than that of the corresponding homogeneously broadened systems. Oscillation in the total optical polarization, however, remains pronounced because of interference between polarizations arising from excitons at different energies.

11. R. Zimmermann and E. Runge, “Exciton lineshape in semiconductor quantum structures with interface roughness,” J. Lumin. **60&61**, 320 (1994). [CrossRef]

12. D.M. Whittaker, P. Kinsler, T.A. Fisher, M.S. Skolnick, A. Armitage, A.M. Afshar, M.D. Sturge, and J.S. Roberts, “Motional narrowing in semiconductor microcavities,” Phys. Rev. Lett. **77**, 4792 (1996). [CrossRef] [PubMed]

## References and links

1. | C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. |

2. | T. B. Norris, J. K. Rhee, C. Y. Sung, Y. Arakawa, M. Nishioka, and C. Weisbuch, “Time-resolved vacuum Rabi oscillations in a semiconductor quantum microcavity,” Phys. Rev. B |

3. | J. Jacobson, S. Pau, H. Cao, G. Bjork, and Y. Yamamoto, “Observation of exciton-polariton oscillating emission in a single-quantum-well semiconductor microcavity,” Phys. Rev. A |

4. | Hailin Wang, Jagdeep Shah, T. C. Damen, W. Y. Jan, J. E. Cunningham, M. H. Hong, and J. P. Mannaerts, “Coherent oscillations in semiconductor microcavities,” Phys. Rev. B |

5. | D. Bogavarapu, D. McAlister, A. Anderson, M. Munroe, M. G. Raymer, G. Khitrova, and H. M. Gibbs, “Ultrafast photon statistics of normal mode coupling in a semiconductor microcavity,” Quantum Electronics and Laser Science Conference, OSA Technical Digest |

6. | S. Pau, G. Bjork, H. Cao, E. Hanamura, and Y. Yamamoto, “Theory of inhomogeneous microcavity polariton splitting,” Solid State Commun. |

7. | F. Jahnke, M. Ruopp, M. Kira, and S. W. Koch, “Ultrafast pulse propagation and excitonic nonlinearities in semiconductor microcavities,” Adv. in Solid State Phys. |

8. | H. J. Carmichael, “Quantum fluctuations in absorptive bistability without adiabatic elimination,” Phys. Rev. A |

9. | Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear dispersion theory,” Phys. Rev. Lett. |

10. | M. Kira, F. Jahnke, and S. W. Koch, “Ultrashort pulse propagation effects in semiconductor microcavities,” Solid State Commun. |

11. | R. Zimmermann and E. Runge, “Exciton lineshape in semiconductor quantum structures with interface roughness,” J. Lumin. |

12. | D.M. Whittaker, P. Kinsler, T.A. Fisher, M.S. Skolnick, A. Armitage, A.M. Afshar, M.D. Sturge, and J.S. Roberts, “Motional narrowing in semiconductor microcavities,” Phys. Rev. Lett. |

13. | V. Savona, C. Piermarocchi, A. Quattropani, F. Tassone, and P. Schwendimann, “Microscopic theory of motional narrowing of microcavity polaritons in a disordered potential,” Phys. Rev. Lett. |

14. | Different line width for lower and upper cavity-polaritons can also be obtained by using the simple coupled-oscillator model and an asymmetric inhomogeneous lineshape. |

**OCIS Codes**

(270.1670) Quantum optics : Coherent optical effects

(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

**ToC Category:**

Focus Issue: Coherent Phenomena in Solids

**History**

Original Manuscript: September 30, 1997

Published: December 8, 1997

**Citation**

Hailin Wang, Young-tak Chough, S. E. Palmer, and Howard Carmichael, "Normal mode oscillation in the presence of
inhomogeneous broadening," Opt. Express **1**, 370-375 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-12-370

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### References

- C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314 (1992). [CrossRef] [PubMed]
- T. B. Norris, J. K. Rhee, C. Y. Sung, Y. Arakawa, M. Nishioka, C. Weisbuch, "Time-resolved vacuum Rabi oscillations in a semiconductor quantum microcavity," Phys. Rev. B 50, 14663 (1994). [CrossRef]
- J. Jacobson, S. Pau, H. Cao, G. Bjork, Y. Yamamoto, "Observation of exciton-polariton oscillating emission in a single-quantum-well semiconductor microcavity," Phys. Rev. A 51 2542 (1995). [CrossRef] [PubMed]
- Hailin Wang, Jagdeep Shah, T. C. Damen, W. Y. Jan, J. E. Cunningham, M. H. Hong, and J. P. Mannaerts, "Coherent oscillations in semiconductor microcavities," Phys. Rev. B 51, 14713 (1995). [CrossRef]
- D. Bogavarapu, D. McAlister, A. Anderson, M. Munroe, M. G. Raymer, G. Khitrova, and H. M. Gibbs, "Ultrafast photon statistics of normal mode coupling in a semiconductor microcavity," Quantum Electronics and Laser Science Conference, OSA Technical Digest 9, 33 (1996).
- S. Pau, G. Bjork, H. Cao, E. Hanamura, and Y. Yamamoto, "Theory of inhomogeneous microcavity polariton splitting," Solid State Commun. 98, 781 (1996). [CrossRef]
- F. Jahnke, M. Ruopp, M. Kira, and S. W. Koch, "Ultrafast pulse propagation and excitonic nonlinearities in semiconductor microcavities," Adv. in Solid State Phys. 37, accepted for publication (1997).
- H. J. Carmichael , "Quantum fluctuations in absorptive bistability without adiabatic elimination," Phys. Rev. A 33, 3262 (1986). [CrossRef] [PubMed]
- Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Vacuum Rabi splitting as a feature of linear dispersion theory, Phys. Rev. Lett. 64, 2499 (1990). [CrossRef] [PubMed]
- M. Kira, F. Jahnke, S. W. Koch, "Ultrashort pulse propagation effects in semiconductor microcavities," Solid State Commun. 102, 703 (1997). [CrossRef]
- R. Zimmermann and E. Runge, "Exciton lineshape in semiconductor quantum structures with interface roughness," J. Lumin. 60&61, 320 (1994). [CrossRef]
- D.M. Whittaker, P. Kinsler, T.A. Fisher, M.S. Skolnick, A. Armitage, A.M. Afshar, M.D. Sturge, and J.S. Roberts, "Motional narrowing in semiconductor microcavities," Phys. Rev. Lett. 77, 4792 (1996). [CrossRef] [PubMed]
- V. Savona, C. Piermarocchi, A. Quattropani, F. Tassone, and P. Schwendimann, "Microscopic theory of motional narrowing of microcavity polaritons in a disordered potential," Phys. Rev. Lett. 78, 4470 (1997). [CrossRef]
- Different line width for lower and upper cavity-polaritons can also be obtained by using the simple coupled-oscillator model and an asymmetric inhomogeneous lineshape.

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